I



1IIIi
UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Radio Meteorology

B. R. Bean and E. J. Dutton

Central Radio Propagation Laboratory"^

Natio7ial Bureau of Stafidards

Boulder, Colorado

*The CRPL was transferred to the Enviromental Science Services Administration

while this Monograph was in press.

^^ational Bureau of Standards Monograph 92

Issued March 1, 1966

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D. C. 20402 Price $2.75
National Bureau of $xmi^^

m Z 1 B65

QCl.OO

. a 556

Library of Congress Catalog Card No. 65-60033
Foreword

This volume brings together the work done in radio meteorology

over the past ten years at the National Bureau of Standards' Central

Radio Propagation Laboratory (CRPL). This decade has seen the

development, on an international scale, of great emphasis upon the effects of the lower atmosphere on the propagation of radio waves. The CRPL

group has concentrated upon the refraction of radio waves as well as the

refractive index structure of the lower atmosphere on both synoptic and

climatic scales. These are the areas of radio meteorology that are treated

in this volume, with additional chapters on radio-meteorological parameters and the absorption of radio waves by the various constituents of the

lower atmosphere. An effort has been made to include results obtained

in other laboratories both in the United States and abroad.

A. V. AsTiN, Director.

Ill
Preface

The authors express their gratitude and appreciation to the many

members of the Central Radio Propagation Laboratory staff and others

with whom they have had the pleasure of working. This volume would

not have been possible without their assistance, encouragement, and

stimulation. Particular acknowledgements are due : R. L. Abbott, Mrs.

L. Bradley, Miss B. A. Cahoon, C. B. Emmanuel, L. Fehlhaber,* V. R. Frank, Miss S. Gerkin, Mrs. B. J. Gibson, J. Grosskopf,* J. W. Herbstreit,

J. D. Horn, J. A. Lane,t J. R. Lebsack, R. E. McGavin, Miss F. M.

Meaney, C. M. Miller, K. A. Norton, A. M. Ozanich, Jr., the late Mrs.

G .E. Richmond, L. P. Riggs, P. L. Rice, C. A. Samson, W. B. Sweezy,

G. D. Thayer, Mrs. B. J. Weddle, E. R. Westwater, Mrs. P. C. Whittaker

and W. A. Williams.

B. R. Bean

E. J. Button

*Fernmeldetechnisches, Darmstadt, Germany.

tRadio Research Station, Slough, England.

IV
Contents

Foreword iii Preface iv Chapter 1. The radio refractive index of air 1 1.1. Introduction 1 1.2. Dielectric constant of moist air 2

1.3. Constants in the equation forA^ 4

1.4. Errors in the practical use of the equation for A'' 9

1.5. Presentation of iV data 13

1.6. Conclusions 20

1.7. References 21 Chapter 2. Measuring the radio refractive index 23

2.1. The measurement of the radio refractive index 23

2.2. Indirect measurement of the radio refractive index 23

2.3. Direct measurement of the refractive index 30

2.4. Comparison between the direct and indirect methods of measure- ment 38

2.5. Radiosonde lag constants-. 38

2.5.1. Introduction 38

2.5.2. Theory of sensor time lags 29

2.5.3. Radiosonde profile analysis 41 2.5.4. Conclusion 43

2.6. References 45 Chapter 3. Tropospheric refraction 49

3.1. Introduction 49

3.2. Limitations to radio ray tracing 52

3.3. An approximation for high initial elevation angles 53

3.4. The statistical method 54

3.5. Schulkin's method 54

3.6. Linear or effective earth's radius model 56

3.7. Modified effective earth's radius model 59

3.8. The exponential model 65

3.9. The initial gradient correction method 77

3.10. The departures-from-normal method 77

3.11. A graphical method 80

3.12. Derivations 82

3.13. References 87 Chapter 4. N climatology 89

4.1. Introduction 89

4.2. Radio-refractive-index ch mate near the ground 89

4.2.1. Introduction 89

4.2.2. Presentation of basic data 89

4.2.3. Worldwide values of .Vo 98

4.2.4. Climatic classification by 102

4.2.5. Applications 105 4.2.6. Critical appraisal of results 107 4.2.7. Conclusions 108 V
4.3. On the average atmospheric radio refractive index structure over North America 109

4.3.1. Introduction . 1 09

4.3.2. Meteorological data and reduction techniques 109 4.3.3. Average A'^o structure 111 4.3.4. Continental cross sections 122 4.3.5. Delineation of climatic characteristics 129

4.4. The climatology of ground-based radio ducts 132 4.4.1. Introduction 132 4.4.2. Meteorological conditions associated with radio

refractive index profiles 132 4.4.3. Refractive conditions due to local meteorological

phenomena 134

4.4.4. Background 135 4.4.5. Description of observed ground-based atmospheric

ducts 140

4.5. A study of fading regions within the horizon caused by a surface duct below a transmitter 146

4.5.1. Introduction 146 4.5.2. Regions and extent of fading within the horizon in the presence of superrefraction 147 4.5.3. Theory and results 149

4.5.4. Sample computations 157

4.6. Air mass refractive properties 163

4.6.1. Introduction 163

4.6.2. Refraction of radio rays 164

4.6.3. Conclusions 170

4.7. References 170 Chapter 5. Synoptic radio meteorology 173

5.1. Introduction 173

5.2. Background 173

5.3. Refractive index parameters 185

5.4. A synoptic illustration 195

5.5. Surface analysis in terms of iVo 197

5.6. Constant pressure chart analysis 205

5.7. Vertical distribution of the refractive index using A units 211

5.8. Summary 223

5.9. References 224

Chapter 6. Transhorizon radio-meteorological parameters 229

6.1. Existing radio-meteorological parameters 229

6.1.1. Introduction _ 229

6.1.2. Parameters derived from the profile 233

6.1.3. Parameters involving thermal stability 238

6.1.4. Composite parameter 238

6.1.5. Potential refractive index (or modulus), if 239

6.1.6. Vertical motion of the atmosphere 240

6.1.7. Discussion of the parameters 241

6.1.8. Comparison of some parameters 243

6.1.9. Some exceptions and anomalies 243

6.1.10. Conclusions 246

VI
6.2. An analysis of VHF field strength variations and refractive index

profiles 247

6.2.1. Introduction 247

6.2.2. Radio and meteorological data 247

6.2.3. Classification of radio field strengths by profile types. 248

6.2.4. Prediction of field strength for unstratified conditions. 250

6.2.5. The effect of elevated layers on the Illinois paths 251

6.2.6. Elevated layers at temperature inversions 252

6.2.7. The influence of small layers 255

6.2.7. Conclusions 257

6.3. A new turbulence parameter 258

6.3.1. Introduction 258

6.3.2. The concept of thermal stability 258

6.3.3. The adiabatic lapses N 261

6.3.4. The turbulence parameter, 11 262

6.3.5. Comparison of radio-meteorological parameters 263

6.3.6. Conclusion 266

6.4. References 266

Chapter 7. Attenuation of radio wave.s 269

7.1. Introduction 269

7.2. Background 269

7.3. Attenuation by atmospheric gases 270

7.4. Estimates of the range of total gaseous absorption 280

7.5. Total radio path absorption 283

7.6. Derivation of absorption estimate for other areas 286

7.7. Attenuation in clouds 291

7.8. Attenuation by rain 292

7.9. Rainfall attenuation climatology 297

7.10. Rain attenuation effects on radio systems engineering 298

7.11. Attenuation by hail 302

7.12. Attenuation by fog 303

7 . 13 . Thermal noise emitted by the atmosphere 304

7.14. References 308

Chapter 8. Applications of tropospheric refraction and refractive index models 311

8.1. Concerning the bi-exponential nature of the tropospheric radio

refractive index 311 8.1.1. Introduction and background 311

8.1.2. structure in the I.C.A.O. atmosphere 312

8.1.3. Properties of the dry and wet term scale heights 314

8.1.4. Refraction in the bi-exponential model 321

8.1.5. Extension to other regions 321

8.2. Effect of atmospheric horizontal inhomogeneity upon ray tracing __ 322

8.2.1. Introduction and background 322

8.2.2. Canterbury 323

8.2.3. Cape Kennedy 324

8.2.4. Ray bending 326

8.2.5. Comparisons 328

8.2.6. Extension to other regions 331

8.2.7. Conclusions 333

VII
8.3. Comparison of observed atmospheric radio refraction effects with

values predicted through the use of surface weather observations. _ 333

8.3.1. Introduction 333

8.3.2. Theory 334

8.3.3. The CRPL standard atmospheric radio refractive index profile sample 340

8.3.4. Comparison with independent data 342

8.3.5. Comparison with experimental results 344

8.3.6. Discussion of results 353

8.4. Correction of atmospheric refraction errors in radio height finding. 356

8.4.1. Introduction 356

8.4.2. Refractive index 357

8.4.3. Ray theory 357

8.4.4. Use of the effective earth's radius 360

8.4.5. Meteorological parameters 362

8.4.6. Calculation and correlation of height errors 363

8.4.7. Estimation of the average gradient 363

8.4.8. Regression analysis 364

8.4.9. Height error equations 372

8.4.10. Conclusions 373

8.5. References 373 Chapter 9. Radio-meteorological charts, graphs, tables, and sample computations 375

9.1 . Sample computations of atmospheric refraction 375

9.2. Tables of refraction variables for the exponential reference atmosphere 393

9.3. Refractivity tables 396

9.4. Climatological data of the refractive index for the United States __ 404

9.5. Statistical prediction of elevation angle error 404

9.6. References 423

VIII
Chapter 1. The Radio Refractive

Index of Air

1.1. Introduction

The last few decades have seen a remarkable increase in the practical

utilization of the radio spectrum above 30 Mc/s. This, in turn, has

focussed attention upon the mechanisms by which these radio waves are propagated. Since radio energy at these frequencies is not normally

reflected by the ionosphere, variability in the characteristics of the re- ceived fields is attributed to variations in the lower atmosphere and, in particular, to the radio refractive index.

The radio refractive index is central to all theories of radio propagation

through the lower atmosphere. The atmosphere causes a downward

curvature of horizontally launched radio waves which is normally about

one quarter of that of the earth. Under unusual meteorological conditions, however, the radio energy may be confined to thin layers near

the earth's surface with resultant abnormally high field strengths being

observed beyond the normal radio horizon. At other times a transition

layer between differing air masses will give rise to the reflection of radio

energy. In addition to these gross profile effects, the atmosphere is always more or less turbulent, with the result that radio energy is scattered out of the normal antenna pattern.

It is not the purpose of the present discussion to emphasize the inter- play of various propagation mechanisms, as has been done, for example,

in a classic paper by Saxton [1]^, but rather to emphasize that the refractive index of the troposphere is of central concern in the propagation of radio waves at frequencies above 30 Mc/s. In what follows, then, the

classical equation for the radio refractive index will be considered, a sum- mary of the recent determinations of the constants in this equation given,

the errors in the practical use of the equation will be analyzed, and, finally,

the normal gross features of atmospheric refractive index structure will be described.

1.2. Dielectric Constant of Moist Air

Debye [2] has considered the effect of an impressed electric field upon

the dielectric constant of both non-polar and polar molecules, a polar

^ Figures in brackets indicate the literature references on p. 21.

1
2 RADIO REFRACTIVE INDEX OF AIR

molecule being one with a strong permanent dipole moment. He derives

the same result from both classical and quantum mechanical methods;

i.e., the polarizability is composed of two effects: one due to the distortion

of all of the molecules by the impressed field and the second arising from an orientation effect exerted upon polar molecules. The polarization, P,

of a polar liquid under the influence of a high-frequency radio field is given by

6-iM 47rArr , 1 1 ....

where: e is M is P is N is do is

M is k is T is T is

assuming no interaction between molecules.

tations of the molecules to return to random distribution

after the field is removed,

CO = 27r/ where / is the frequency of the external field. One concludes from Debye's analysis that for external fields with fre- quencies less than 100 Gc/s, cor « 1 with the result that (1.1) is written^

The dispersive effect of the 22.5 Gc/s water-vapor absorption line is not

expected to be important below 30 Gc/s although, as shall be seen, experimental evidence indicates some dispersion due to the 60 Gc/s absorption

2 The effect of the relaxation time upon interpretation of measurements of dielectric constant has been discussed by Saxton [1]; he has also given a description of measure- ment techniques involved in the determination of dielectric constant during the war

years [3].
DIELECTRIC CONSTANT OF MOIST AIR 3

line of oxygen. For non-polar gases (/x = 0), this equation becomes

e - 1 M _ 4:7rNao , . 6 + 2 p ~ 3

^^"^^

which is well approximated by

e - 1 ^-^^irNao (1.4)

for gases at low pressures.

This equation may be rewritten, by assuming the perfect gas law, as ,-l=K[^ (1.5)

where K[ is a constant. The result for polar gases, (1.2), may be written

ao + ^] (1.6) M

which, assuming (1.5), can be rewritten as

6 - 1 = /^^^(a + l) (1.7) T

where K'^, A, and B are constants. For a mixture of gases, Dalton's

law of partial pressures is assumed to hold with the result that we can sum the effects of polar and non-polar gases and hence obtain

(1.8)

For the troposphere, however, we need only consider the effects of CO2, dry air (non-polar gases), and water vapor (a polar gas) such that

. - 1 = if,', + X/i I ( ^ + I ) + K!^ ^ (1.9)

where Pd is the pressure of dry air, e is the partial pressure of water vapor, and Pc the partial pressure of CO2. The equation for refractive index, n, is obtained using the expression n = \/^, where ju, the permeability, may
4 RADIO REFRACTIVE INDEX OF AIR

be assumed to be approximately unity for air. Since n = Vl + (ne - 1) , one may employ the approximation

1 Ate — 1 n - 1^—^,

and obtain the familiar form of practical application^

N = {n-l)10' =K,-^ + K, + Ks~ + K,^ (1.10)

where Ki, K^, K3, and are constants.

1.3. Constants in the Equation for N

A survey of the various determinations of Ki, K2, and was recently

made by Smith and Weintraub [4] to arrive at a set of ''best values" to represent the mean of previous independent determinations. In radio

work one is interested in propagation through the troposphere, therefore,

the composition of air should be taken to include an average amount of carbon dioxide. However, laboratory measurements usually are made

on C02-free air due to variable concentrations of CO2 in the laboratory.

Hence Smith and Weintraub have adjusted the the values of e — 1 originally published for C02-free air by raising them to 0.02 percent to correspond to a 0.03 percent CO2 content. It should be noted that this correction of 0.02 percent is essentially of academic interest, since as

shall be seen, the final equation for N will be considered accurate to 0.5 percent. The value of 0.02 percent was obtained by noting that the

value of dielectric constant for C02-free air, e\ in the expression - 1 =

T

could be utilized for applications in the real atmosphere by expressing

total atmospheric pressure, P<, as Pt = Pd + Pc

3 Henceforth, will be used to denote (n — 1)10^ and will not be used again for Avogadro's number. This slight inconsistency was adopted to maintain notational agreement with both Debye's early work and the later work in radio meteorology.
CONSTANTS IN EQUATION FOR N 5 By assuming a constant CO2 content of 0.03 percent one obtains, from

Dalton's law of partial pressures,

and

= (1 - 3 X 10-') Pt

Pc = 3 X 10-'Pt

with the result that the dielectric constant for the real, dry atmosphere,

e, may be written= - 3 X 10~') Pt K,{S X 10~') Pt

One may then adjust measurements of C02-free air upwards by - 1] 10' = 3 X - 1) percent ^ 0.02 percent,

since KJKi ^ 5/3. Such values are given in table 1.1 on a real, rather

than an ideal, gas basis. The first determination shown, that of Barrell

[5], is an average of the constant term (n for X = 00) of the optical Cauchy

dispersion equations for standard air used in three of the principal

measurement laboratories of the world. Theoretical considerations indicate

that the dielectric constant for dry air will be the same for optical and

radio frequencies. Barrell's value is converted to dielectric constant from the relationship V^M^ with /x, the permeability, taken as unity at optical frequencies. Unless otherwise noted, the standard error is used

here and throughout the remainder of this discussion.

Table 1.1. Dry air refractive index and dielectric constant at 0°C and 1 atm

Reference Frequency of measurement Measured

A'

(e - 1) 10« Year

Barrell [5] Optical'^.-. _ - . 287. 7 ± 0. 0 575. 6 =h 0.

1951 Bimbaum,Kryder,and Lyons[40] 9,000 Mc/s 575. 8 ± 0. 3 1951 Essen and Froome [11] 24,000 Mc/s 288. 2 ± 0.

576. 1 ± 0. 2 1951 Mean value of (« — 1) 10« Mean value of A''^ 288.0. ±0.0

575. 7 ± 0.

a Maryott and Buckley [6] have determined a mean value for the dielectric constant from the simple

average of eight different determinations that, when adjusted to the pressure, temperature, and value of m assumed in table 1.1, yield a value of A'" = 287. 7 ± 0. 15, which is in agreement with Barrell's value.

^ Derived from n = Vm« where n — I = 0. 4 X 10"^ is taken for radio frequencies to account for the per- meability.
6 RADIO REFRACTIVE INDEX OF AIR

The statistical mean of the dielectric constant is then converted to refractive index. The constant Ki is evaluated from

N = K?^, (1.11)

which stems from (1.10) when e = 0. Setting = 288.04, = 1013.25

mbar, T = 273 °K and solving for then^

= 77.607 =1= 0.13 -T— . (1.12) mbar

The constants and Kz have been evaluated from a survey of determinations of the dielectric constant of water vapor in the microwave

region by Birnbaum and Chatterjee [7]. These values, determined from

the data of table 1.2, are - 71.6 ± 8.5 °K/mbar, (1.13)

and

Kz = (3.747 =b 0.031) 10'(°K) '/mbar (1.14)

and were obtained as the weighted means of the various determinations,

the weights being taken inversely proportional to the square of the probable errors. Although Birnbaum and Chatterjee considered only those

data that would allow a determination of K2 and over a wide range

of temperature, they conclude their results are in satisfactory agreement

with those of Stranathan [8] at 1 Mc/s, Phillips [9] at 3000 Mc/s, Grain

[10] at 9000 Mc/s, and Essen and Froome [11] at 24,000 Mc/s.

Table 1.2. Values of the constants K2 and K3 used by Birnbaum and Chatterjee

Observer K2 K} X 10-5 Temperature range

Birnbaum and Chatterjee [7] ... . Groves and Sugden [12] ... 69.43 ± 13.02

77. 75 ± 21. 70

61. 47 ± 21. 70

72. 86 ± 7. 05

3. 774 ± 0. 043

3. 742 ± 0. 097

3. 765 ± 0. 096

3. 736 ± 0. 025

°C

25-103

110-211 111-249

21-189 Hurdis and Smyth [13] Stranathan [8] . ....

* The development here follows that of Smith and Weintraub. Grain's [10] de- termination of the dielectric constant of dry air (e = 1.000572) indicates that Ki may- be somewhat lower than the "best value" of Smith and Weintraub, since his value, Ki = 77.10, is about 40 standard deviations lower than that of Smith and Weintraub.
CONSTANTS IN EQUATION FOR 7 The full equation for the refractive index is now N = 77.6 ^ + 72 I + 3.75 XlO'jr^, (1.15)

after reducing the values to three figures where significant. These are the constants recommended by Smith and Weintraub to yield an overall

accuracy of =b 0.5 percent in A^.^ Equation (1.15) may be simplified by

assuming P< = P = + e and obtaining

N = 77.6 I

- 5.6 I + 3.75 X lO' jr, • (1-16)

For practical work in radio-meteorological studies, (1.16) may be simplified to the two-term expression

AT = 77.6 ^ + 3.73 X 10' , (1.17)

which yields values of N within 0.02 percent of those obtained by (1.16) for the temperature range of —50 to 40 °C; i.e., a maximum error of 0.1 N unit, and a standard error of 0.5 percent of about 4.5 N units. This is accomplished by dividing the second and third terms of (1.16)

by e/T and solving for the composite constant, K^, in relation

3.75 X 10' K, ^ - D.6 = — , (1.18)

which, for T = 273 °K, results in: K, = 3.73 X 10'(°KVmbar). (1.19)

Equation (1.17) is comm.only written as ^ = ^(P + . (1.20)

Although this last equation is in widespread use by radio scientists throughout the world, it is by no means adopted on an international scale.

° The values of K2 and K3 given in (1.13) and (1.14) are those of Birnbaum and

Chatterjee as reported by Smith and Weintraub. A consideration of the ratio of K0/K3 shows the Smith-Weintraub value to be about 0.2 percent greater than that

obtained from the original data, indicating an arithmetic conversion error. This

error is, however, totally absorbed in the rounding to three significant figures in

(1.15).
8 RADIO REFRACTIVE INDEX OF AIR

Another set of values in common use is that of Essen and Froome (11)

which is listed in table 1.3 along with values determined by several other

authors. Subsequent microwave refractometer determinations of these constants by Saito [14] in Japan yielded a value of Ki lower than that of either Smith and Weintraub or of Essen and Froome and nearer to Grain's

determination. Saito's values of K2 and lie between those of Smith

and Weintraub, and Essen and Froome. A recent French microwave

determination of these constants by Battaglie, Boudouris, and Gozzini

[15], has found the Smith and Weintraub values of Ki, K2, and given

by (1.16) fit their experimental data within measurement error. The use

of the Essen and Froome values in radio geodesy appears to be based

upon Wadley's [16] comparison of optical and radio surveys and his conclusion that the best (radio) consistency is obtained when using Essen

and Froome's constants in conjunction with their determination of the

velocity of light.

Table 1.3. Table of constants used by different authors

iV = (n - 1) 106 - — + ^ + —

Reference Date Ki K2

Schelleng, Burrows, and Terrell [20] 1933 79.0 67. 5 1.35 Englund, Crawford, and Mumford [41].. 1935 79.1 68.3 3.81 Waynick [35] 1940 79.0 68.5 3. 72 Smith-Rose and Stlckland [36] 1943 79.0 68.0 3.8 Burrows and Attwood NRDC [37] 1946 79.0 68.0 3.8 Meteorol. Factors in Prop. [40]. . 1946 79.0 68.0 3.8 Grain [10] 1948 77. 10 Craig et al. [32] 1951 79.0 (79) 3.8 Essen and Froome [11] 1951 77.64 64. 68 3.718 Smith and Weintraub [4] - . . 1952 77.6 72.0 3. 75 Essen [38] 1953 75.0 3.68

Saito [14] 1955 77. 26 67.5 3. 77 Battaglia, Boudouris, and Gozzini [15]—. 1957 77.6 72.0 3. 75 Magee and Grain [42].. . . 1958 77.5 65.0 3. 70

The particular constants in the equation for given by Smith and

Weintraub are considered to be good to 0.5 percent in N for frequencies

up to 30,000 Mc/s and normally encountered ranges of pressure, temperature, and humidity. Experimental determinations of the variability of the radio refractivity, (n — 1)10®, with frequency have been carried out by Essen and Froome [11] and are summarized in table 1.4. Remembering that the dispersion of refractive index would be expected

to be greatest at frequencies slightly off the water vapor resonance at 22 Gc/s and the oxygen resonance at 60 Gc/s [17] one concludes that for the normally used frequencies, / < 30 Gc/s, the various gases give a frequency variation of N well within the limits of accuracy given by Smith
ERRORS IX USE OF EQUATION FOR N

Table 1.4. Refractivities of water vapor and dry gases (Dry gases at 0 °C, 760 mm Hg; water vapor at 20 °C, 10 mm Hg)

Gas 9 Gc/s 24 Gc/s 72 Gc/s Water vapor . Oxygen

60. 7 d= 0. 2

288. 10 ± 0. 10

266. 2 ± 0. 2

60. 7 ± 0. 2

288. 15 ± 0. 10

266. 4 ± 0. 2

61. 0 ± 0. 2

287. 66 db 0. 11

263. 9 ± 0. 2

and Weintraub. With the exception of oxygen the same is true up to 72 Gc/s. This conclusion is somewhat strengthened by the results of Hughs and Armstrong [18] at 3000 Mc/s which agree within 1.5 percent

of those of Essen and Froome. It is important to note that the differ- ences in formulas are greater than the frequency variation and, further, as shall be seen in the next section, the errors of determining P, T, and e are sufficiently large to mask even the differences in formulas.

1.4. Errors in the Practical Use of the Equation for N

A high degree of accuracy of temperature, pressure, and water vapor

pressure measurements is necessary for precise determinations of the

refractive index from (1.20). If one assumes that the formula for N is exact, then a relation between small changes in and small changes in temperature, pressure, and vapor pressure may be evaluated from

dN .dN , . dN

ciN = ^dT + ~de + ^dP, (1.21)

assuming that the errors in P, T, and e are unrelated.

The partial derivatives may be evaluated by reference to some standard

atmosphere to yield the approximate expression

AN' = aAT + hAe + cAP. (1.22)

The root-mean-square (rms) error is then

AN = { {aATf + Q)AeY + (cAP) 2) 1/2 (1.22)

Typical values of the constants a, 6, and c, based upon the International

Civil Aviation Organization (ICAO) standard atmosphere and assuming

60 percent relative humidity, are given in table 1.5 for various altitudes.
10 RADIO REFRACTIVE INDEX OF AIR

It was mentioned earlier that the differences in refractive index obtained by the two most commonly used formulas, those of Essen and

Froome and of Smith and Weintraub, are small compared to the error made in observing P, T, and e. For example, by assuming errors of ±2

mbar in P, ±1 °C in T, and dz5 percent in relative humidity (RH) com- mon in radiosonde measurements with sea-level values of P = 1013

mbar, T = 15 °C and 60 percent RH, a standard error of 4.1 units may

be obtained compared to the difference of about 0.5 units between the

values obtained from the two formulas. It is seen from table 1.6 that

(over the normal range of sea-level temperatures, —50 °C to +40 °C)

the differences in the formulas are comparable to those associated with

surface meteorological measurements but are significantly less than the

errors that may arise in radiosonde observations. In addition to these errors, there is an additional source of error in the

uncertainty of the constants in the equation for N. Equation (1.22)

may be written to include this additional source of error as AN' = aAT + hAe + cAP + dAKi + fAK^ + gAK^,

and, again assuming that the errors are uncorrected, the rms error may

be evaluated as before. Table 1.7 lists the percentage error arising from

errors in the various surface meteorological observations as well as those

of the constants. It is seen that the errors in constants constitute no more than 30 percent of the total error. All sources of error combined

will yield errors no larger than 1.5 percent in A^. Thus, although the

Smith and Weintraub equation has a stated accuracy of 0.5 percent due to the constants alone, total errors of nearly twice that figure may occur under conditions of extremely high humidity.

Currently one must use radiosonde data for estimation of A^ gradients,

with the result that the overall accuracy is determined much more by the

errors in the nieteorological sensors than by errors between constants in the equation for A^. Until such time as better measurement methods for T and e are developed, there appears to be little or no need for more

accurate determinations of the constants in (1.10). The different deter- minations of the constants now available are in essential agreement. For

example, the use of the Essen and Froome expression will give values of

A^ that generally lie within the standard error of d=0.5 percent of the

Smith-Weintraub expression. It appears desirable for our purposes to use the Smith-Weintraub constants since they represent the weighted mean

of several independent determinations, noting also that the value of K2

given by Essen and Froome was obtained by extrapolation from optical

measurements rather than direct measurements at radio frequencies.
ERRORS IX USE OF EQUATION FOR N 11 Table 1.5. Values of the constants a, b, and c in the expression AN = aAT + bAe + cAP, for the ICAO Standard Atmosphere and 60 percent relative humidity

Altitude T P € a b c km °C mbar mbar °K-i mbar-^ mbar-^

0 319 15.0 1013 10.2 -1.27 4. 50 0. 27

1 277 8.2 893 6.5 -1.09 4.72 0. 28

3 216 -4.5 701 2.6 -0. 86 5. 17 0. 29

10 92 -50.3 262 0.04 -0. 50 7. 52 0. 30

20 20 -56.5 55 0 -0. 09 7. 96 0. 35

50 0.2 9.5 0.8 0 -0. 0008 4. 67 0. 27

Table 1.6. Comparison of errors in determining ^ from meteorological measurements assuming no error in the equation for N

[p = 1013 mbar and RH = 60 percent]

Source of error -50 °C 0°C +15 °C +40 °C

Difference in formulas: Smith-Weintraub [4] (three term)... Essen and Froome [11] (three term)..352. 61

352. 79

306. 18

306. 07

318. 79

318. 28

419. 55

417. 19 Difference from Smith-Weintraub

value in A^ units +0. 18 -0.11 -0. 51 -2. 34 Surface weather observations (±1 mbar of pressure, ±0. 1 °C of T, and ±1 percent RH), (all errors in rms

A'' units). error ±0. 38 ±0. 43 d=0. 82 ±2. 83 Radiosonde observations (±2 mbar of pressure, ±1 °C

of T, and ±5 percent RH) (all errors in units). rms

error ±1.73 ±2. 02 ±4. 07 ±14. 19

Table 1.7. Percentage contribution of errors in surface meteorological measurements and constants in the equation for N

Percent of total variance due to: Temp (°C) Total rms

error (A^-units) AT Ae AP AivTi -50 16.7 0.0 81.0 2.3 0.0 0.0 0. 387 0 6.2 41.3 35.6 1.0 5.7 10.2 .476

15 1.8 64.1 7.9 0.2 10.0 16.0 .955

40 0.3 69.3 0.5 0.0 12.7 17.2 3.378 rms error in A^-units -50 0. 158 0.005 0. 348 0.059 0. 001 0. 002 0. 387

0 . 119 .306 .284 .048 . 114 . 152 .476

15 . 127 .765 .269 .045 .302 .382 .955 40 .188 2.811 .248 .040 1.202 .399 3.378

This opinion is particularly reinforced by Essen's subsequent determination of K2 and Kz at 9200 Mc/s, where closer agreement was obtained

with the Smith-Weintraub values (note items 9, 10, 11 in table 1.3). The

difference in K2 and accounts for most of the final difference in obtained from the two formulas. Although the great precision of Essen and

Froome' s work is reflected in their standard deviation of experimental
12 RADIO REFRACTIVE INDEX OF AIR

values being about one-tenth that of other workers, one must recognize

that systematic errors undoubtedly contribute to the disagreement in values obtained by different workers. It appears that there exists a real need for a new determination carried out with the greatest possible care and over as large a pressure and temperature range as possible.

The need for further measurements is particularly evident by the

several Essen and Froome determinations. Their values of the dielectric constant have been determined with impressive precision and absolute

accuracy over a wide range of frequencies, but their concomitant determinations of the constants in the equation for N differ from one another

as much as the Smith-Weintraub and the 1951 Essen and Froome values.

It is clear that the use of either set of Essen and Froome values permits

the calculation of either relative or absolute values of with equal precision. However, the question of comparative absolute accuracy with

the results of other workers remains.

As a further example, drawn from radio geodesy, consider the electrical path length

(1.23)

where S indicates the radio path. If we assume that the measurements

are made in a stratum of constant n, then (1.23) becomes

R = n ds = nS = S{1 + N X 10"'), Jo and

S = R/{1 + AT X 10-') ^R{1 - N X 10"'). (1.24)

An error in thus directly produces an error in radio distance determination. In typical radio geodesic field work (see, for example, Wadley,

[16]) would be determined from pressure, dry-bulb temperature, and

wet-bulb temperature readings. The partial pressure of water vapor may

be obtained from Sprung's [19] (psychrometric) formula

e = e's - 0.00067 (r - T') P (1.25)

where e's denotes the saturation vapor pressure at the wet-bulb temperature, r\ For sea-level conditions of P = 1013 mbar, T = 15 °C and

T — T' = 4.1 °C (60 percent RH) one finds, neglecting errors in the

constants, that

AA^' 0.28AP - 4.32AT - 6.96Ar. (1.26)
PRESENTATION OF N DATA 13 Assuming errors of d=0.1 mbar for P and =b0.25 °C for T and (derived from considering Wadley's field data as being accurate to d=5 units

of the next significant figure beyond the tabulated whole degrees Fahrenheit for wet and dry bulb temperatures and hundredths of an inch of mercury for total air pressure) one finds that errors in N may be as large as 2.75 units, which is much greater than the 0.5 unit difference given

by the two formulas for A^. Again it is seen that the difference in the constants used in the two most widely accepted expressions for yields an error of the same size or smaller than that produced by errors of meas- urement of the necessary meteorological parameters.

1.5. Presentation of N Data

There is now, in the literature, an almost bewildering choice of modifications to basic A^ data when presented as a function of height. The

underlying principle is always to remove the systematic decrease of N with

height, h, in an assumed standard atmosphere. This arises from the fact that,

the curvature of a radio ray, C, is given by ^

C = - i^f cos^ (1.27) n dh

where 6 is the local elevation angle of the ray. Since the curvature of a

radio ray is proportional to the gradient of the refractive index, specification of a model for dn/dh specifies the curvature of radio rays in that model

atmosphere. For example, it is customary for radio engineers to think in terms of effective earth radius factors [20]. This convenient fiction,

discussed in chapter 3, makes straight the actual curved path of a radio

ray in the atmosphere by presenting it relative to an imaginary earth

larger in radius by a factor k than the radius of the real earth, a, thus

maintaining the relative curvature between earth and radio ray. This is expressed as

1 . 1 dn . 1 + ~ , cos 0 = — -I- 0 . a n dh ka

curvature of curvature of curvature of curvature of earth radio ray effective earth straight ray

^Equation (1.27) as well as general ray curvature considerations are elegantly derived by Millington [21] for the effective earth's radius model. The same equation

is also derived in chapter 3.
14 RADIO REFRACTIVE INDEX OF AIR

Thus k is defined by

-, , a dn ^

^

^ 1 + - cos d n dh

which, for rays tangential to the earth (^o = 0) and assuming n in unity,

is approximated by

^ (1.29)

1 + a dn/dh

It is customary to set

dn/dh = - (1.30)

and thus obtain k = 4/3. Since l/4a is close to the observed gradients

of n, this represents a '^standard atmosphere" gradient and one may re- move the ''standard" decrease of with height by adding a quantity, 8N

8N = (h/U) 10' (1.31)

to N{h) thus obtaining the '^B unit" or B(h) = [{n - I) + h/4:a] lO' = N{h) + (h/4.a) 10\ (1.32)

This method of presenting refractive index profiles has been shown to be very useful in emphasizing the departure from standard of the atmosphere over southern California [22]. The principal advantage of the

B unit is that any departure from a vertical line, and thus constant n

gradient of dn/dh = — 1 /4a, equals a departure in refractive bending from

that in the effective earth's radius atmosphere with k = 4/3.

A very similar approach is used to aid in the study of extended ranges

of radio waves. The most pronounced case of this phenomena occurs when dn/dh = — (1/a) which, from (1.29), gives /c = oo or an effective earth of infinite radius. This then implies, for radio purposes, the earth is flat and communication is between ''radio-visible" terminals. For this special study the modified index of refraction, ilf, is defined by

Mih) = lin - 1) + h/a] lO' = N(h) + (h/a) 10\ (1.33)

A gradient of dM/dh = 0 implies that A;= oo and also implies a theoretically infinite range of radio signals. M units were used to present the
PRESENTATION OF N DATA 15 meteorological data gathered by the Canterbury Project [23] in their

intensive study of ranges of over-water radar signals. Both B and M

units assume a standard atmosphere with a linear decrease of with

height and thus introduce a correction which increases linearly with

height. If the actual atmosphere had in fact the assumed N distribution

then, for example, B{h) = B(0). Recent studies [24, 25, 26] have shown

than an exponential decrease of with height is a more realistic model of the true atmosphere. • For example, the data given on figure 3.3 were chosen to represent the

extremes of average profile conditions over the United States. The

Miami, Fla., profile is typical of warm, humid, sea-level stations that

tend to have maximum refraction effects while the Portland, Me., profile

is associated with nearly minimum sea-level refraction conditions. Although Ely, Nev., has a much smaller surface N value than either Miami

or Portland, its N profile falls within the limits of the maximum and

minimum sea-level profiles. The distribution for the 4/3 effective earth's radius atmosphere is also shown on figure 3.3. It is quite evident that the 4/3 earth distribution has about the correct

slope in the first kilometer above the earth's surface but decreases much

too rapidly above that height. It is also seen that the observed refractivity distribution is more nearly an exponential function of height than a

linear function as assumed by the effective earth's radius model. The

exponential decrease of A^ with height is sufficiently regular as to permit a

first approximation of average A^ structure from surface conditions alone. Consider that

where ^Z" is a scale height appropriate to the value of A^ at zero height, A^s. Average values of A^^ and H for the United States are approximately 313

and 7 km respectively. As used here, scale height is simply that height

at which Nih) is 1/e of A^^ under the assumption of (1.34).

One would expect the gradient

also to be sufficiently well-behaved as to allow prediction of at least its general features. In fact a high correlation between AN, the simple

difference between A^ at the earth's surface and at 1 km above the earth's

surface, and Ns has been observed in a number of regions. This is

N(h) = NsQxp i-h/H) (1.34)

dN(h)

dh = -(Ns/H) exp i-h/H) (1.35)
16 RADIO REFRACTIVE INDEX OF AIR

particularly true of climatological data such as 5-year averages for given

months. In the United States the expression [26] -AN = 7.32 exp (0.005577iV.), (1.36)

^InA.V . Ns = 0.93,

where r denotes the standard correlation coefficient, has been found, while

in Germany [27] the expression

-AN = 9.30 exp (0.004565iV.), (1.37)

f\nAN • Ns = 0.6,

has been reported. A recent article by Lane [28] indicates that a similar

correlation, rin^" •ns= 0.85, is found over the British Isles. Although

the slopes and intercepts of this relation between AA^ and Ns obviously

vary with climate, one concludes, for the purposes of the present discussion, that the observational evidence corroborates the assumption of an

average exponential decrease of N with height. The general application

of the exponential model must await considerable work upon the part of the radio climatologists.

This has naturally led to the definition of a new unit that adds a correction factor to account for this exponential decrease [29, 30]. This new

unit. A, is given by

A(h) = N{h) + Ar,[l - exp {-h/H)] (1.38)

It is possible to calculate a theoretical value of the scale of N by

assuming a distribution of water vapor. This is, however, quite a complex procedure. Futhermore, the value obtained depends upon the

model of the water vapor distribution, and no definite conclusion can be

justified considering the extreme variation of water vapor concentration

with season, geographic location, and height above the earth's surface. A

convenient and simple alternative is to adopt a value for H from the

average {n — 1) variation with height in the free atmosphere. Several

such values of H were determined by reference to the National Advisory

Committee for Aeronautics (NACA) standard atmosphere and recent

climatological studies of atmospheric refractive index structure [26]. It

is seen in table 1.8 that H varies from 6.56 to 7.63 km in the NACA

standard atmosphere, depending on the value of the relative humidity

assumed. The value oi H = 7.01 km for 80 percent relative humidity is
PRESENTATION OF N DATA 17

in close agreement with H = 6.95 km obtained from ciimatological

studies of (n — 1). Table 1.8. Determination of the effective scale height, H, for the radio refractive index

Source Humidity H

percent NACA standard atmosphere.. . . 100 6. 56

80 7.01 60 7. 63 Ciimatological data As observed 6.95 adopted for this discussion. 7.0*

*The value of H = 7.0 km was arbitrarily adopted for use in the present discussion. Note that this value is smaller than the usual scale height near sea level of

8. 0 km for the density distribution of the air. Although the value H is quite arbi- trary, it is evident from table 1. 8 that //should lie between 6. 5 and 7. 5 km. A comparison of B, M, N, and A units is shown on figure 1.1 for arctic,

temperate, and tropical climates. It is quite evident that both M and B

over-correct the profile to produce increasing values with height while

the A unit tends to yield a constant value at heights in excess of 2 km.

It is noted, however, that important departures of the N{h) profile from

normal within the first few kilometers are emphasized by both the A and

B profiles. In radio meteorology, as in other branches of meteorology, it is often

convenient to express the potential value of the refractive index referred

to some standard pressure level. In practice this is found by adding to the values at any height the product of the N lapse rate and the height.

This has already been done in the case of the B unit, while the A unit

simply adds the total decrease of in an exponential atmosphere from

the surface to the height under consideration. Yet another approach is to replace the values of pressure, temperature, and vapor pressure in the

expression for N with their values at some desired pressure level. This

unit, called the potential refractive modulus, 0, is then defined by

77.6

Po + 4810 (1.39)

where 6 is the potential temperature and eo the potential partial pressure

of water vapor, both referred to the reference pressure Po- This formula

is that of Katz [31] with the Smith-Weintraub constants. The potential temperature is defined as

e = r(Po/p)'''"^ (1.40)
RADIO REFRACTIVE INDEX OF AIR
PRESENTATION OF N DATA 19 and ^0 as

eo = e(Po/p), - (1.41)

where R is the universal gas constant, m the molecular weight of air, and

Cx the specific heat of air at constant pressure. By setting Po = 1000

mbar and noting that R/mCp = 0.286, one obtains

. = ^(i^°r + 3.73X10^^(1^)- (1.42)

Thus, it is seen that </> is obtained by correcting the dry air term of A^, 77.6p/r, by the factor (1000/p)o-7i4 ^nd the water vapor term by the

factor (1000/p)°-^-^. In well-mixed air, 6 and eo are independent of height

and consequently 0 is independent of height. If the temperature lapse

rate is less than the dry adiabatic rate, the potential temperature increases

with height and, consequently, the first term of 0 decreases with height.

If the lapse of the partial pressure of water vapor is less than than given

by (1.41) then eo and the second term of 0 increase with height, conversely,

if e decreases more rapidly with height than the decrease given by (1.41),

eo and the second term of <f) decrease with height. Abundant examples of the vertical distribution of </> under various meteorological conditions are given throughout volume 13 of the Radiation Laboratory Series [32]. The potential refraction modulus has recently been used by Jehn [33] to

illustrate the variability of N about a classic polar-front wave. A conceptually similar approach to that above is to arbitrarily adopt a

reduced-to-sea-level value of refractivity

No = Ns exp (+V7.0) (1.43)

where /i is in kilometers, that effectively removes the station elevation

dependence of Ns [34] and allows the emphasis of air-mass differences. Since the effects of the atmosphere upon the propagation of radio waves

are dependent upon either the absolute value of n or its gradient, we may

use this as a criterion for the choice of appropriate units to represent the

N profile. One may recover both n and its gradient from a knowledge of the height structure alone. The corrections are additive in the case of B, M, and A and multiplicative for No. The values B, M, and A are in convenient form for refraction calculations since it is the gradient above

the surface of the earth that is needed. The value, A^'o, removes the effect of station elevation and thus is convenient for mapping of surface conditions in a fashion similar to sea-level pressure rather than station pressure. One cannot obtain the gradient and absolute value of n from a <f){h) distri- bution, although <j>{h) may be graphically calculated from the familiar
20 RADIO REFRACTIVE INDEX OF AIR

characteristic diagram of the meteorologist; even for that process one must first calculate the potential values of both temperature and water

vapor pressure. The relative merits of the various units are summarized

in table 1.9.

1.6. Conclusions

The above discussion has emphasized the following points:

(a) The differences in constants in the expression for N are small compared with the error inherent in the formula.

(b) The refractive index is effectively nondispersive for frequencies

below, say, 50 Gc/s.

(c) The choice of atmospheric model for A^, and concomitant units,

depends upon the application at hand.

(d) The atmosphere, on the average, yields an exponential distribution

of with height.

(e) The use of an exponential model facilitates the preparation of climatic maps of and makes clear the effect of non-normal structure upon the bending of radio rays.

Table 1.9. Comparison of various units used in radio meteorology

Unit Gradient Referred

to: Advantages Disadvantages

B(/i)=iV(/i) + (/i/4a)10«, dBih)

dh

Surface.-. Vertical distribution

identifies "4/3 earth"

conditions; absolute value and gradient of n

easily recovered.

Over-corrects N{h) above

the first few

kilometers. M(h)=N{h) + {h/a)10^ dM(h)

dh =^^+a/»).o., Surface... Trapping or ducting

layers identified by

idM/dh)<.0, absolute value and gradient of n easily recovered.

Grossly overcorrects N(,h) above the trapping

layer. No{h)=N{h) oxp(/i/7.0), dNoih)

dh dN{h) ( h \ = dh ^''Pl7.o)

, N{h) ( h \

Sea-level. Removes effects of both

height and station ele- vation; facilitates mapping since N{h) is easily recovered.

Gradient on n

not easily re- covered.

A{h)=N{h)

+313[l-exp(-/i/7.0)],

dA{h) dN{h) 313 Surface... Removes effects of height n and gradient easily recovered. Does not re- dhexp(

~ dh ^ 7 -h/7.0), move the ef- fect of station

elevation.

d0(/i) dh

d(i>(h) de

dd dh

dec,

deo dh

1000 mbar Removes the effects of height and station ele- vation, may be de- rived from characteristic meteorological diagram.

Neither n or dn/dh may

be recovered from <)l)(/i); must calculate both from basic p, T, and e data
REFERENCES 21 1.7. References

[1] Saxton, J. A. (Sept. 1951), Propagation of metre radio waves beyond the normal

horizon, Proc. lEE 98, 360-369.

[2] Debye, P. (1957), Book, Polar Molecules, pp. 89-90 (Dover Publ. Co., New

York, N.Y.).

[3] Saxton, J. A. (1947), The anomalous dispersion of water vapour at very high

frequencies, Parts I-IV, Book, Meteorological Factors in Radio Wave Propagation, pp. 278-316 (The Physical Society, London, England).

[4] Smith, E. K., and S. Weintraub (Aug. 1953), The constants in the equation for atmospheric refractive index at radio frequencies, Proc. IRE 41, 1035-1037.

[5] Barren, H. (May 1951), The dispersion of air between 2500 A and 6500 A, J. Opt.

Soc. Am. 41, 295-299.

[6] Maryott, A. A., and F. Buckley (1953), Table of dielectric constants and electric dipole moments of substances in the gaseous state, NBS Circular 537.

[7] Birnbaum, G., and S. K. Chatterjee (Feb. 1952), The dielectric constant of water vapor in the microwave region, J. App. Phys. 32, 220-223.

[8] Stranathan, J. J. (Sept. 1935), Dielectric constant of water vapor, Phys. Rev. 48, 538-544.

[9] Phillips, W. C. (July 1950), The permittivity of air at a wave-length of 10 centi- meters, Proc. IRE 38, 786-790.

[10] Crain, C. M. (Sept. 1948), The dielectric constant of several gases at a wavelength of 3.2 centimeters, Phys. Rev. 74, No. 6, 691-693.

[11] Essen, L., and K. D. Froome (Oct. 1951), The refractive indices and dielectric constants of air and its principal constituents at 24,000 Mc/s, Proc. Phys.

Soc. (London, England) 64, 862-875.

[12] Groves, L. G., and S. Sugden (1935), Dipole moments of vapors II, J. Chem.

Soc. England 30, 971-974.

[13] Hurdis, E. C, and C. P. Smyth (1942), Dipole moment, induction and resonance

in nitroethane and some chloronitroparaffins, J. Am. Chem. Soc. 64, 28-29.

[14] Saito, S. (Aug. 1955), Measurement at 9,000 Mc of the dielectric constant of air containing various quantities of water vapor, Proc. IRE 43, No. 8, 1009.

[15] Battaglia, A., G. Boudouris, et A. Gozzini (May 1957), Sur I'indice de refraction de I'air humide en microondes, Ann. de Telecomm. 12, 181-184.

[16] Wadley, T. L. (1957), The tellurometer system of distance measurement, Empire

Survey Review 14, No. 105, 100-111, and 14, No. 106, 1232-1239.

[17] Born, M., and E. Wolf (1959), Book, Principles of Optics, p. 89 (Pergamon Press, New York, N.Y., and London, England).

[18] Hughs, J. v., and H. L. Armstrong (May 1952), The dielectric constant of dry

air, J. Appl. Phys. 23, 501-504.

[19] Sprung, A. (1888), Bestimm. d. Luftfeuchtigk. mit Hiilfe d. Assmann'schen

Asperationspsychrometers, Das Wetter 5, 105-109.

[20] Shelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave

propagation, Proc. IRE 21, 427-463.

[21] MilHngton, G. (Jan. 1957), The concept of the equivalent radius of the earth in tropospheric propagation, Marconi Rev. 20, No. 126, 79-93.

[22] Smyth, J. B., and L. G. Trolese (Nov. 1947), Propagation of radio waves in the lower atmosphere, Proc. IRE 35, 1198-1202.

[23] Report of Factual Data from the Canterbury Project (1951), Vols. I-III (Dept.

Sci. Indus. Research, Wellington, New Zealand).

[24] Anderson, L. J. (Apr. 1958), Tropospheric bending of radio waves. Trans. Am.

Geophys. Union 39, 208-212.
22 RADIO REFRACTIVE INDEX OF AIR

[25] Bauer, J. R., W. C. Mason, and F. A. Wilson (27 Aug. 1958), Radio refraction in a cool exponential atmosphere. Tech. Rept. No. 186, Lincoln Laboratory, Massachusetts Institute of Technology, Cambridge, Mass.

[26] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric re- fractive index, Proc. IRE 47, No. 5, 740-755.

[27] Bean, B. R., L. Fehlhaber, and J. Grosskopf (Jan. 1962), Die Radiometeorologie

und ihre Bedeutung fiir die Ausbreitung der m-, dm,- und cm-Wellen auf

grosse Entfernungen, 15, 9-16, Nachrichtentechnische Zeitschrift.

[28] Lane, J. A. (1961), The radio refractive index gradient over the British Isles, J. Atmospheric Terrest. Phys. 21, Nos. 2/3, 157-166.

[29] Bean, B. R., and L. P. Riggs (July-Aug. 1959), Synoptic variation of the radio

refractive index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97.

[30] Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the

vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio

Prop.), No. 2, 249-258.

[31] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient

of refractive modulus in homogeneous air, potential modulus. Book, Propagation of Short Radio Waves, ed. D. E. Kerr, pp. 198-199 (McGraw-Hill Book

Co., Inc., New York, N.Y.).

[32] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient

of refractive modulus in homogeneous air, potential modulus, Book, Propagation of Short Radio Waves, ed. D. E. Kerr, p. 189 (McGraw-Hill Book Co.,

Inc., New York, N.Y.).

[33] Jehn, K. H. (June 1960), The use of potential refractive index in synoptic scale radio meteorology, J. Meteorol. 17, 264.

[34] Bean, B. R., and R. M. Gallet (Oct. 1959), Applications of the molecular re- fractivity in radio meteorology, J. Geophys. Res. 64, No. 10, 1439-1444.

[35] Waynick, A. H. (Oct. 1940), Experiments on the propagation of ultra-short-wave

Proc. IRE, 28, 468-475.

[36] Smith-Rose, R. L., and A. C. Stickland (Mar. 1943), A study of propagation

over the ultra-short-wave radio link between Guernsey and England on wavelengths of 5 and 8 meters, J. lEE 90, No. Ill, 12-24.

[37] Burrows, C. R., and S. S. Atwood (1949), Radio wave propagation, Consolidated Summary Technical Report of the Committee on Propagation, NDRC, p. 219 (Academic Press, Inc., New York, N.Y.).

[38] Essen, L. (Mar. 1953), The refractive indices of water vapour, air, oxygen,

nitrogen, hydrogen, deuterium, and helium, Proc. Phys. Soc. B 66, 189-193.

[39] Birnbaum, G., S. J. Kryder, and H. Lyons, (Aug. 1951), Microwave measure- ments of the dielectric properties of gases, J. Appl. Phys. 22, 95-102.

[40] Meteorological Factors in Radio Wave Propagation (1946), (The Physical Society, London, England) Foreword.

[41] Englund, C. R., A. B. Crawford, and W. W. Mumford (July 1935), Further

results of a study of ultra-short-wave transmission phenomena, Bell Sys. Tech. J. 13, 369-387.

[42] Magee, J. B., and C. M. Grain (Jan. 1958), Recording microwave hygrometer, Rev. Sci. Instr. 29, 51-54.
Chapter 2. Measuring the Radio

Refractive Index

2.1. The Measurement of the Radio

Refractive Index

The radio refractive index is defined as the ratio of the speed of propagation of radio energy in a vacuum to the speed in a specified medium.

The radio refractive index may be measured directly if the measuring

instrument is sensitive to the speed of propagation. Refractive index is measured indirectly by measuring temperature, pressure, and humidity

with subsequent conversion to the refractive index as indicated in the

previous chapter. The direct method employs radio frequency refrac- tometers to determine the refractive index; the indirect method, standard

weather observations. The direct method appears preferable because

accuracy is dependent upon a single sensor, rather than three and, of course, because values are obtained directly. However, refractometers

are relatively complex and expensive devices requiring no small degree of

skill to maintain. Hence, as yet, refractometers are not in general use, or even available in sufficient quantity to permit large-scale mapping of refractive index structures. The bulk of the synoptic and climatological

mapping of refractive index is still based on the indirect method of measurement. The laborious task of converting the measured parameters to refractive index has been somewhat alleviated by the development

of special analog computers [1]^ and digital computers.

2.2. Indirect Measurement of the Radio

Refractive Index

The accuracy of the determination of the refractive index from standard

weather observations has been discussed in chapter 1. Figure 2.1 illus- trates the degree of accuracy to be expected in the measurement of the

refractive index as a function of the accuracies of the sensors for sea-level conditions. It is assumed that the errors are additive. Planes of equal

Figures in brackets indicate the literature references on p. 45. 23
24 MEASURING THE RADIO REFRACTIVE INDEX

LlJ

liJ IT<

Figure 2.1. Accuracy of the determination of refractivity as a function of the accuracies

of the meteorological sensors.

accuracy are shown for accuracies of ±0.1, dzO.5, ±1.0, and ±2.0 N.

It can be seen that a measurement error of ±1.0 can be contributed

by each sensor if the sensor errors exceed ±0.8 °C for temperature, ±3.7

mbar for total pressure, and ±0.22 mbar for vapor pressure measurements. It is apparent that extreme accuracy is required in the measure- ment of the water pressure. The temperature can be measured easily to within several tenths of a degree, the total pressure to within several

millibars, but the humidity is considerably less responsive to the same

degree of relative accuracy. If the wet and dry bulb hygrometer technique is used to measure the humidity with subsequent conversion using

either the psychrometric formula of Sprung^ [2] or List^ [3], the degree of accuracy is seen to be a function of temperature. For an error of no

greater that ±1.0 A^, the relative humidity must be accurate to within

0.33 percent at 35 °C, while at 0 °C an accuracy of 3.0 percent is necessary.

2 e = gg — c , c is a constant. 566

3 e = Cs — [ci(l + C2tw)] pi^T), Ci and C2 are constants.
INDIRECT MEASUREMENT 25 Experimenters at the National Physical Laboratories [4] have shown that

when an Assman hygrometer is used with thermometers accurate to within 0.1 °C, the optimum expected accuracy in the determination of the

w^ater vapor pressure is 0.2 mbar. Errors three times as great can be

expected at extreme conditions. Hence, it would appear that measure- ment of water vapor limits the accuracy of the determination of the

refractive index to approximately ±1.0 units. Temperature, pressure, and humidity are standard measurements of the w^orld's weather services. Methods of measuring these parameters

are fairly standard. Usually the temperature is read from mercurial or

alcohol thermometers, the pressure from mercurial barometers, and the

humidity from a conversion of wet- and dry-bulb thermometers. The

degree of accuracy of these measurements is usually a function of the care

exercised by the observer. Thermometers protected by radiation shields are usually accurate to within ±0.1 °C, barometers to within ±1.0 mbar.

Reading errors can easily be in excess of the instrument error, especially

in the determination of humidity where the wet-buJb depression is subject

to many sources of error. Contamination of the wick or water, insuffi- cient wetting, and inadequate aspiration are common sources of error. The wet-bulb determination can be used below freezing, if proper pre- cautions are observed.

Automatic-recording systems have been devised for measurement of temperature, pressure, and humidity. The simplest are the hygrothermograph and the microbarograph, in which the sensors are connected by

mechanical linkages to pens or chart recorders. The pressure is recorded

by means of an aneroid capsule, the temperature by means of a bimetal

strip or a curved Bourdon tube, and the humidity by means of a hair

hygrometer. The accuracy of such devices limits the determination of

.V to within 2 or 3 units. Sensors producing electrical outputs are used to measure meteorological

parameters in a variety of automatic-recording systems, such as stripchart recorders, punched-paper tape, and magnetic-tape recorders. The

more common temperature sensors include resistance thermometers,

thermocouples, and thermistors. The platinum resistance thermometer, an international standard, is capable of measuring temperature in still air to within ±0.05 °C if used in a well-compensated bridge circuit. However, the platinum resistance thermometer is velocity-sensitive in a moving air stream. Thermocouples avoid this problem and yield short- term accuracies of approximately ±0.1 °C with time constants measured

in milliseconds.

The total atmospheric pressure can be measured by a variety of electrical sensors. The simplest device is the pressure potentiometer where

an aneroid capsule is mechanically linked to a potentiometer. One or two millibars can be considered the limit of accuracy in a differential
26 MEASURING THE RADIO REFRACTIVE INDEX

device operating over a range of ±100 mbar with reference to an average

value with time constants of perhaps 5 msec. The capacitive microphone is capable of accuracies of ±0.01 mbar and lag constants of several

milliseconds over a limited range (±2 mbar). The resolution diminishes

to about ±0.1 mbar over the normal range of variations of atmospheric

pressure. The capacitive microphone requires correction for significant

temperature variations. Strain-gage pressure transducers have approximately the same characteristics as capacitive microphones with some

degradation in lag constants. Differential strain gauge transducers operating over a range of 4 to 5 mbar may yield accuracies of several hundredths of a millibar and are relatively free of temperature effects. Relative humidity requires the greatest care in measurement. The

common lithium chloride strip is accurate to within ±5 percent in relative

humidity. In general, the lag constant is of the order 8 to 10 sec for temperatures in excess of 0 °C (see sec. 2.5). Other sensors under development show promise. The barium fluoride strip [5] yields accuracies,

hysteresis effects, and response times far superior to lithium chloride but

has the disadvantage of rapid aging. Phosphorous pentoxide sensors [6] and aluminum oxide sensors [7] appear to be quite promising. The most

accurate method that is available for measuring water vapor pressure is the wet-dry bulb technique using electrical thermometers (±0.02 mbar).

The lag constant will be relatively large, as it is a function of the wetting

properties of the wick, the rate of aspiration, and even the relative

humidity, and is thus recommended only for temperatures above —24 °C. Measurements up to several thousand feet can be made from towers or with tethered balloons or a wiresonde. Only the sensors need be sent

aloft. All auxiliary devices are on the ground, and long cables transmit

the information from the sensors to the recording equipment [8, 9]. The radiosonde is in almost universal use for high-altitude measure- ment of the meteorological parameters influencing the refractivity. There

are many models of the radiosonde, differing from country to country.

The principle of operation for present American radiosondes is illustrated

in figure 2.2. An aneroid capsule is used as the active element of a

baroswitch. The temperature sensor, humidity sensor, and a reference

resistance are alternately switched into the grid circuit of a blocking

oscillator as the baroswitch wiper moves under the action of decreasing

pressure. The blocking oscillator controls the pulse rate of the rf trans- mitter; hence, the pulse rate of the transmission is indicative of the value

of the sensor being sampled. The number of the contact energized is a measure of the pressure; the switching sequence permits identification

of temperature and humidity values. A constant rate of ascent of the

balloon is assumed so that the values of the parameters can be identified with the proper altitude.
INDIRECT MEASUREMENT 27

RF BLOCKING TRANSMITTER OSCILLATOR

RH

ANEROID

CAPSULE

i

BAROSWITCH

REFERENCE

Figure 2,2. Block diagram of the radiosonde.

There is no uniformity of sensors used in the world's radiosondes.

Hence, it is difficult to compare measurements from different countries.

At one international comparison at Payerne, Switzerland, in 1956, 14

natiors participated [10]. The results indicated significant differences. Temperature measurements corresponded to within d=1.5 °C for night

flights but corresponded only to within d=3.5 °C for daytmie flights. Pressure measurements agreed well at low altitudes but indicated a dis- persion of d=1.5 mbar above 9,000 m (29,000 ft), and ±2.5 mbar above

16,000 m (50,000 ft). Humidity comparisons were poor, indicating that

15 percent would be the most optimistic estmiate of the standard deviation from the mean for all flights. On the average, it is estimated that

the standard American temperature sensor indicated a value approximately 1.5 °C below the mean for all sondes used in the test. The pres- sure determination was below the mean by 0.5 mbar, whereas the humidity sensor could not be quantitatively evaluated due to the erratic behavior of all sensors. Xo attempt was made at these comparison trials to determine the

absolute accuracy of any radiosonde. Although the absolute accuracy

of the American sonde has not been determined, satistical evaluation of
28 MEASURING THE RADIO REFRACTIVE INDEX

the uniformity of American radiosondes has been conducted by both the

U. S. Air Force and the U. S. Army Signal Corps. Results indicate a standard deviation for the temperature sensor of 0.8 °C to 6,000 m and

1.0 °C above that altitude. The standard deviation for the pressure determination was 2.2 mbar below 9,000 m and 1.1 mbar above 9,000 m. Under ideal conditions the humidity sensor (lithium chloride) exhibited a standard deviation of 5 percent. This accuracy is possible only if the

element is not subjected to high humidity (95 to 100 percent), or saturation by liquid water, and if the temperature is above 0 °C. The response

of the element is especially poor where both temperature and humidity

are low. The measurement means little if the relative humidity is below

15 percent at a temperature of 20 °C, 20 percent at a temperature of 0 °C, or 30 percent at a temperature of —30 °C. Hence, ''under ideal conditions" at sea level, the standard deviation

in the determination of the refractivity from radiosonde data is approximately 3 units; at 1 km, the standard deviation would be approxi16.7% 33.3% 50% 66.7%

KX)-83.3*

8:

6(

5.3-

57°/h

667-

50%^

333%1

100- 66.-7%

50%

k

^^66.7*33.3%

DECREASING

CHANGE IN RH

100 — 50'Yo-

'83.3*:53.3 % IOO-33.3%

16.7% 33.3% 50% 66.7%

INCREASING

CHANGE IN RH

50-100%

'*^3.3-*83.3%

733.3*100%

20 -40 0 -20 -40 0 -20 -40 0 TEMPERATURE (°C)

20 -40

Figure 2.3. Factors affecting the lag constant of the lithium chloride humidity sensor.

(After Wexler.)
INDIRECT MEASUREMENT 29

TEMPERATURE

SENSOR

1RF

TRANSMITTER ®

PRESSURE

SENSOR

(1)

JEMPERATURE

HUMIDITY

'2^ J SENSOR

®1

BLOCKING OSCILLATOR

[an /AMT-6(xa-l)]

Figure 2.4, Clinger-Siraiton radiosonde.

(Transmission in N units.)

mately 2 units. This then would appear to be the uhimate precision

with which present conventional radiosonde sensors can yield the refractivity.

The lag constants of the radiosonde sensors are also of importance.

Since the sonde is rising at a relatively rapid rate, it passes into regions

of changing refractivity before the sensors are aware of it. The lag

coefficient associated with the radiosonde introduces an error in the esti- mation of the true gradient. Lag constants of sensors have been analyzed by Wexler [11] and by Bean and Button [12]. Some correction can be made to the radiosonde data. Wexler (fig. 2.3) shows that the lag

coefficient of the lithium chloride strip is a function not only of the

temperature, but also of the absolute value of the relative humidity, as

well as of the size and direction of the gradient.

The radiosonde samples temperature and humidity in sequence rather

than simultaneously. Several experimenters have devised means to correct this deficiency. Misme [13] decreased the cycling time in one radio-
30 MEASURING THE RADIO REFRACTIVE INDEX

300 GRAM

RADIOSONDE

BALLOON

Tw 395 MG

FREQ a Tv

RECEIVER FREQUENCY METER

395 MC UNIT

RECORDER

ANALOG COMPUTER

UNIT

T 405 MG

T

1 Y. RECEIVER FREQUENCY

METER

405 MC UNIT FREQ. a T

Figure 2.5. Schematic diagram of the Navy Electronics Laboratory (Thiesen) system.

sonde so that many more samples of each parameter were produced per

unit time, dinger and Straiton [14] developed a radiosonde that combines the parameters so that the output signal is in terms of the refractivie index (see fig. 2.4). Since the wet and dry terms are additive, a parallel

combination of independent conductances can be used. The dry term

sensor has a conductance proportional to temperature and inversely proportional to pressure; the wet term sensor is such that it not only yields a value proportional to the relative humidity, but also adjusts the relative

magnitude of the two terms. Thiesen [15] devised a similar radiosonde

utilizing two separate sondes on the same balloon (see fig. 2.5). The

temperature and humidity information are combined at the ground sta- tion by means of an analog computer. For captive balloon application,

Hirao and Akita [8] and Crozier [9] developed similar systems using wet

and dry bulb thermistors to produce direct output in refractive index.

2.3. Direct Measurement of the Refractive Index

The resonant frequency of a microwave cavity is a function of its dimensions and the refractive index of its contents. Hence, if a cavity is open to the atmosphere, the resonant frequency changes as the refractive

index of the air passing through it changes. If a sealed reference cavity is
DIRECT MEASUREMENT 31 used for comparison, the difference between resonant frequencies becomes

a convenient measure of the refractive index variations in the sampling

cavity. This method of measurement was used in three different instruments: the Grain type, the Birnbaum type, and the Vetter type. The

Grain refractometer utilizes the cavities as frequency determining ele- ments in ultra-stable oscillators. The difference in frequency is then the

unit of measure. The Birnbaum refractometer utilizes the cavities as

passive resonance frequencies of the cavities. The time difference between

resonances is the unit of measure. The Vetter instrument, an improvement on Sargent's [16] modification of the Birnbaum's refractometer,

utilizes servo techniques to achieve a null system, thus eliminating the

necessity for extreme electronic stability.

The resonant cavities are significant components in any of these instru- ments. Of prime importance is the temperature coefficient of the cavity.

Most cavities today are made of invar having a temperature coefficient of approximately one part per million per degree centigrade [17, 18]. This is equivalent to 1 iV unit per degree centigrade. Further temperature compensation of the cavity has produced temperature coefficients of 0.2 and 0.1 unit per degree centigrade [19]. Most recently, improved

invar cavities, when compensated, yielded temperature coefficients of 0.03 N units per degree centigrade [20]. The possibility of improving

cavity performance by use of special ceramics [21, 22], with temperature

coefficients (without compensation) of 0.1 N units per degree centigrade

has been investigated.

The response time of the refractometer is a function of how much of the end plates of the cavity can be opened to the air without appreciable

loss of resonant characteristics. It has been found [23, 24], that as much

as 92 percent of the end plate area could be eliminated without serious resonant degradation. Such end plates do not impede the flow of air, and the response time of the instrument is essentially instantaneous. In general, the accuracy of the refractometer is at least one order of magnitude smaller than can be achieved by indirect measurement. With

proper care, these instruments are capable of discerning changes of the

refractive index that are less than a tenth of an N unit. As a relative

instrument, i.e., used to measure variations about an undetermined mean,

the time constant is such as to easily allow detection of rates of up to 100 c/s. The development of microwave refractometers led to an immediate in- creased interest in the fine structure of refractive index variations and its application to radio wave propagation. A summary of this development

has been given by Herbstreit [25]. Detailed investigations of the inhomogeneities of refractive index structure using spaced cavities yielded signi- ficant results pertinent to the spectrum of turbulence [26, 27]. Refrac- tometers have been widely used in aircraft [18, 28, 29]. The Grain
32 MEASURING THE RADIO REFRACTIVE INDEX

refractometer [29, 30], as illustrated in figure 2.6, utilizes one sealed cavity

as a reference cavity and a ventilated cavity as the sampling element.

Each of these cavities serves as a frequency-determining element of a Pound [31] oscillator. Very good cavity resonance is required. The

Grain refractometer is so designed that the cavity resonators operating

in the 9400-Mc/s range are slightly different in resonant frequency. The

resultant difference frequency is the center frequency from which departures due to changing refractive index are measured. Recent models

utilize a 43 Mc/s center frequency. As the refractive index of the contents of the sampling cavity changes, the difference frequency at the

output of the mixer changes according to

Then, since

AN = An W,

a change of 1 iV unit corresponds to a change in the resonant frequency

SAMPLING

CAVITY

REFERENCE

CAVITY

OSCILLATOR

MIXER

AMP

DISCRIMINATOR RECORDER

Figure 2.6. The Grain refractometer.
DIRECT MEASUREMENT 33

SAWTOOTH

GENERATOR

SAMPLING

CAVITY

PHASE METER

RECORDER

REFERENCE

CAVITY

Figure 2.7. The Birnhaum refradometer.

of the sampling cavity of approximately 9.4 kc/s if the nominal operating

frequency is 9400 Mc/s.

For the measurement of small scale variations of refractive index, a

modified Grain refractometer utilizing a 10.7-Mc/s center frequency with

a 200-kc/s linear range has been used [29]. Scales of 1 to 20 units for

full scale deflections of a 1-mA chart recorder have been used.

The Birnbaum refractometer [32], illustrated in figure 2.7, applies the resonance principle in a somewhat different fashion. Both the reference, or sealed, cavity and the sampling, or open, cavity are passive elements.

The cavities are slightly different in frequency and are of the transmission

type having crystal detectors at the output.

A single klystron, frequency modulated by a sawtooth voltage and

having an output frequency linear with time, excites each cavity in se- quence. During the frequency excursion of the klystron, the respective

cavity resonances are excited, and a pulse formed at each crystal detector. Since the resonant frequencies of the cavities differ, the two output pulses

will be displaced in time. If the modulation on the klystron is periodic,

then the output pulses will be also displaced in phase, the displacement

being a function of the difference in resonant frequencies. Any change in the refractive index of the contents of the sampling cavity will alter the

phase difference between the two pulse trains. The relative phase between the two outputs can then be measured by an electronic phase meter.
34 MEASURING THE RADIO REFRACTIVE INDEX

The phase meter in use is merely a multivibrator having a constant amplitude output. The pulse from each cavity alternately switches the multivibrator ''on" and ''off." The width of the constant amplitude output

pulse from the multivibrator is a measure of the time difference between

pulses. The resulting train of constant amplitude variable width pulses

is then applied to appropriate recording circuits. The Birnbaum refrac- tometer is adaptable to multi-cavity operation. A single klystron may

be used to sweep simultaneously a number of spaced sampling cavities. A single reference cavity permits simultaneous comparison between the

several sampling cavities, and important consideration when the scale and

form of refractive index variations are desired ([27]. The accuracy of the Birnbaum instrument is dependent upon maintaining a linear frequency sweep. This problem can be circumvented in the manner of Sargent [16], who modified the Birnbaum refractometer to operate as a microwave hygrometer. A servomechanism is used to tune

the sampling cavity to the resonant frequency of the reference cavity.

The servomechanism positions a tuning probe in the sampling cavity.

The depth of penetration is the measure of the refractive index of the contents of the sampling cavity. This technique minimizes the dependence on the sweep characteristics of the klystron.

The Vetter refractometer [20], figure 2.8, virtually eliminates the dependence of the refractometer on electronic characteristics and shifts the

limitations to the cavities themselves. In addition, while previous in- struments are primarily relative refractive index indicators, the Vetter

refractometer was developed as an absolute refractive index device by

using klystron stabilization techniques [33, 34, 35, 36]. Figure 2.8 is a simplified diagram of the Vetter refractometer illustrat- ing the basic principle of operation. The reference cavity is excited by a klystron which is modulated by a small sine voltage on the repeller. Any

output at the fundamental modulating frequency at the detector of the

reference cavity is compared in phase to the original modulating signal,

and an error signal is applied to the repeller of the klystron to lock the

center frequency of the klystron to the reference cavity. At coincidence,

the fundamental disappears in the output of the reference cavity. The

same klystron excites the sampling cavity. Any fundamental appearing

at the output after phase comparison to the modulating signal develops

another error signal. This error signal is used to drive a mechanical

servomechanism which tunes the reference cavity to the resonant fre- quency of the sampling cavity with concomitant shifting of the klystron

center frequency. This then is a double loop system; the reference cavity

controls the klystron ; the sampling cavity controls the reference cavity.

Tuning of the reference cavity is accomplished by a motor-driven

probe that penetrates the reference cavity. The design of the probe is
DIRECT MEASUREMENT 35

PHASE SENSITIVE AMPLIFIER

ERROR SIGNAL

SERVO

AMPLIFIER

SERVO

MOTOR

ERROR SIGNAL

READOUT

PHASE SENSITIVE AMPLIFIER

Figure 2.8. Simplified diagram of the Vetter refTactometer.

such as to permit very nearly a linear function of AA^ versus probe penetration over the range of operation. Once the probe is calibrated with

the cavity, the relative calibration is static and practically independent

of the contents of the cavity. Although the reference cavity is sealed,

there is no necessity for a vacuum. Absolute calibration is accomplished

by determining the zero point, or the intercept, where A^ = 0, by evacuating the sampling cavity.

Since the Vetter refractometer is mechanically tuned by a servosystem

having considerable inertia, the high-frequency response is relatively poor. The upper frequency response is 10 c/s. Accuracy and simplicity of operation are its distinct advantages.

Refractometers have been used to measure both the surface value and

the gradient of the refractive index. Although the units so far described were designed originally as ground-based equipment, modifications have
36 MEASURING THE RADIO REFRACTIVE INDEX

been made to convert them to airborne use. However, where vertical

gradients to high altitude are desired, practical aspects of aircraft operation such as the turning radius and the angle and rate of climb or glide are factors influencing how well the vertical gradients are estimated.

Considerable horizontal variation in is often observed. In addition,

the time involved in ascent or descent is relatively long, and the measure- ments are valid only under static conditions for the total time of measurement. Due to expense and weight, conventional refractometers are not economical or practical for use in balloon ascent, whether free or tethered.

This led to the development of several lightweight refractometers to be

used either in wiresonde, radiosonde, or dropsonde applications.

Figure 2.9 is a block diagram of a lightweight (6 lb), expendable refrac- tometer developed by Deam [37] and Deam and Cole [38] for use as a

balloon-borne unit or a dropsonde. The device is a Pound oscillator [31], operating at nominal frequency of 403 Mc/s; the instantaneous frequency

is determined by coaxial cavity. The refractive index is sampled by the

cavity, and is reflected as a capacitance in the tuned circuit of the oscil- lator. Operational tests indicate the electronics are sufficiently stable to pro- duce the desired accuracy although the cavity size and mass appear to introduce a sizable time constant. Present accuracy is estimated to be

better than zL5 N units for a complete profile.

The Hay refractometer [39] was developed as a compromise between

the microwave refractometer and the conventional radiosonde (see fig. 2.10). It lacks the accuracy of the microwave refractometer by an order

of magnitude but, weighing only 7 lb, it incorporates the lightweight

feature of the radiosonde. As a balloon-borne instrument it fills a need

for more accurate, faster probing of the refractive index at the higher

elevations.

This refractometer is a 10-Mc/s oscillator whose frequency is determined by an air-sensing capacitor. A change in the refractive index of the air passing through the capacitor is reflected as a frequency change

in the oscillator. The sensing capacitor is alternately switched with a

reference capacitor of identical design but which is protected from the

free atmosphere. The use of a reference capacitor diminishes the problem

of temperature compensation since both capacitors will be at very nearly

the same temperature. To further improve the temperature characteristics, the 12 plates of each capacitor are made of invar and are sepa- rated by 0.25-in. quartz spacers. The output of the 10-Mc/s oscillator

is doubled in frequency and transmitted to the ground station.
DIRECT MEASUREMENT 37

DETECTOR AHP

PHASE

DETECTOR

CORRECTION

VOLTAGE.

HF

OSCILLATOR

4.5 MC

OSCILLATOR

CRYSTAL MODULATOR

CAVITY

Figure 2.9. The Deani expendable refraciometer (modified Pound oscillator).

REFERENCE

CAPACITOR _ _J

SWITCH OSCILLATOR DOUBLER

AMPUFIER

SENSING

CAPACITOR

Figure 2.10. The Hay refraciometer.
38 MEASURING THE RADIO REFRACTIVE INDEX

2.4. Comparison Between the Direct and

Indirect Methods of Measurement

Absolute accuracy may not be the only consideration in the comparison

between the direct and indirect method of measurement. Although

refractometers may be capable of superior accuracy, the factors of requirements, economics, and availability of competent technical personnel may

outweigh this advantage. Refractometers are relatively expensive, some- what complex, and require competent technical personnel to maintain,

calibrate, and operate them.

In many cases where average values or long-term statistics are adequate,

the use of refractometers may not be indicated. The data on refractive

index structure derived from weather service data has long been used

successfully for the determination of average conditions.

Where extreme accuracy is required, the use of refractometers is indi- cated. Radar and radio navigation are examples where accurate esti- mates of both surface values and gradient are necessary to determine the

refraction through the atmosphere. The necessity for true vertical

gradients would demand the use of a balloon-borne or dropsonde type of refractometer. In many applications the indirect method may be suffi- cient ; however, determination of the fine structure of the refractive index

appears to be presently limited to some type of radio-frequency refrac- tometer.

2.5. Radiosonde Lag Constants

2.5.1. Introduction

The determination of from radiosonde data is subject to all of the

errors inherent in the radiosonde observation. Recently Wagner [40] has analyzed the errors in TV arising from time lag of the sensing elements,

data transmission techniques, and significant level-selection criteria. Of

these sources of error, Wagner concludes that the time lag of the sensing

elements is the most serious source of error. Further, for the southern

California coastal inversions, Wagner concludes that only the time lag of the humidity strip need be considered. A similar conclusion has been

reached by Clarke [41] for practical applications involving ship-borne

radar and over-water air-to-air communications. Although there is a

significant correction associated with the time lag of the U.S. radiosonde's

lithium chloride humidity-sensing element, there is also a time lag in the

temperature element, which, as will be shown, must also be taken into

consideration. The correction for the temperature element yields a twofold correction to due to the actual error in temperature and the ancil- lary correction in vapor pressure resulting from the more correct estimate

of the true saturation vapor pressure. This arises from the fact that
RADIOSONDE LAG CONSTANTS 39 when the lithium chloride element measures relative humidity it must be

used with the saturation vapor pressure. Since the saturation vapor

pressure of water is a function of temperature, an error in temperature

produces an error in the estimated water vapor pressure. The lag constant of the lithium chloride (LiCl) humidity element be- comes significantly larger for temperatures below 0 °C [42]. Wexler [11] has made detailed studies of the lag constants of the LiCl elements at low

temperatures under laboratory conditions. Bunker [43], however, has

found quite different lag constants in the free atmosphere. He attributes

this discrepancy in lag constants to the laboratory-determined values

which were obtained for isothermal conditions; whereas, in rising through

the free atmosphere, the radiosonde normally passes from warm to cooler

air. Bunker has raised a serious question, namely that the temperature

lag of the lithium chloride element is possibly as important as the iso- thermal humidity lags studied by Wexler. It is quite possible that the

interplay of these two lags could produce a total effect either greater or

less than the humidity lag alone. We now have a quandary since there is not currently in the literature a complete analysis of the interplay of the

temperature and humidity lags of the LiCl humidity element. Although

Bunker considers some aspects of this problem, he does not consider the

case of decreasing humidity and increasing temperature, a typical condition giving rise to the superrefraction of radio waves. The differences

between Wexler's and Bunker's estimates of the LiCl lag constants are so great as to make one wonder at the validity of applying any correction for

this effect. Since neither Wexler's nor Bunker's tabulation of lag constants is complete, the choice of lag constant appears to be arbitrary.

Wexler's lag constants will be adopted for the present discussion. Both

the temperature and humidity elements are corrected (as much as is possible) for their respective time lags in this chapter for the purpose of preparing refractive index profiles. In what follows the theory of sensor time lag will be examined. Data

from several climatically diverse locations will then be examined to

illustrate the relative importance of the various lag constant corrections

under conditions of superrefraction of radio waves.

2.5.2. Theory of Sensor Time Lags

Middleton and Spilhaus [44] give

dSi

dt = - \

(di - Be) (2.1)

as the basic differential equation of the time lag of a meteorological

sensor, measuring the variable B, where t is time, X the appropriate lag
40 MEASURING THE RADIO REFRACTIVE INDEX

constant, and the subscripts i and e stand for the indicated and environmental values, respectively. The solution to (2.1) depends upon the manner in which the environmental value of 6 varies. For example, if it

is assumed that de varies linearly with time,

ee = + ^t, (2.2) one obtains for the solution to (2.2)

de - Si = +i3X [1 - exp {-t/\)], (2.3) as compared to ^ [(^X - 1) exp (VX)] Si

' ^X exp m - exp {t/\) ' ^^-^^

under the assumption that de varies exponentially with time,

Be = do exp i-^t). (2.5) For a column of air, one normally knows the initial reading of the

sensing strip, ^o, and for an assumed linear decrease of 6 with height the

coefficient /3 becomes

^ ^ ^ - X[l - exp'(-^/X)] ' ^^'^^

with the result that (2.2) is written = + r- X[f- exp\'-VX)] ^^-^^ Once the value of X is determined, estimates of the true properties of the

air can be found at all heights up to the point where the gradient changes.

One may proceed by a different course by noting in (2.1) that

dSi ^ddi dh ^ dSi , . dt dh ' dt ^ dh' ^

where R is ascension rate of the radiosonde (assumed a constant 300

m/min).

If it is further assumed that di varies linearly between reported values

{an assumption compatible with radiosonde reporting procedure), then

j^^^j^ e,^,-e,,,(2.9) dh hk+i - hk where the kth and the (k + l)st layers are the boundaries considered.
THEORY OF SENSOR TIME LAGS 41 Thus the environmental value of 6 can be determined from

de,Jc+l = ^...+ 1 + R\ ~

l''" , (2.10)

llk+l — flk which involves, in a simple fashion, only the indicated or actually meassured values of the parameter 6. One assumes that di and de are identically the same at /i = 0 and that

successive application of (2.10) will yield a more realistic estimate of the

distribution of d with height. When one applies a correction procedure

of the form of (2.10) to temperature, where the temperature lag constant,

Xr, is always 3 sec [45], one obtains immediately the corrected temperature

profile. The same is true for humidity (provided the temperature is greater than 0 °C) , when X/ is assumed to be always 10 sec [40]. Although

there is some indication [46] that, for room temperatures, the humidity

lag constant may be nearer 5 sec. For temperatures less than 0 °C, however, X/ is a function orf true temperature, true value of relative humidity,

fe, and change of the true value of fe (see fig. 2.3). This means that one must use an iterative solution for/^ since X/ will change as one's estimate

of the true value of fe and A/e changes. Since our knowledge of X/ is essentially empirical, the correction procedure is limited to the temperatures

and values of fe and A/g reported by Wexler [11]. In applying the above equations one generally assumes that the time

lags are always known and that the environmental and indicated values

are identical at the base of each layer [47]. These two conditions are approximately satisfied only for the ground layer since the total lag constant of the humidity strip is not known. It is not clear that any correc- tion for sensor lag may be made above the initial layer since, for subsequent layers, the initial indicated and environmental values are not

identical and, further, lag constants have not been determined for this

case.

2.5.3. Radiosonde Profile Analysis

The utility of the above lag constant corrections is illustrated by

analyzing past radiosonde data for the occurrence of ground-based radio

ducts. A ground-based radio duct is one in which the gradient of is sufficient to refract a radio ray to the same curvature as that of the earth. Thus, for ducting, since ray curvature is given by the gradient of the

refractive index,

4t < — = -157 A units/km (2.11) Ah ~ Vq where ro is the earth's radius. The data analyzed were from the months

of expected maximum duct occurrence at Fairbanks, Alaska (Feb.),
42 MEASURING THE RADIO REFRACTIVE INDEX

Washington, D.C. (May), and Swan Island, W.I. (Aug.). All data were

for the year 1953. As an example of past work, Wagner [40] has assumed that the refractivity lagged its environmental value according to (2.3) with the result that

After an analysis of the meteorological conditions of his area of application, he set \^ = \f = 10 sec; i.e., the time lag of N derived from pressure, temperature, and humidity was identically that of the humidity

sensor for temperature <0 °C. Comparing this method of A^-lag correc- tion with the uncorrected data (first and last columns of table 2.1), one

notes an increase in both intensity and incidence of ducts in all climates.

This is particularly marked for Fairbanks, where a sixfold incidence in- crease is obtained. However, when one makes individual time-lag correc- tions for both temperature and humidity by means of (2.10) a quite

different picture of duct statistics is obtained. For example, the Fairbanks data indicate a twofold increase rather than a sixfold increase. On

the other hand, Washington shows an incidence of nine rather than six ducts with a marked increase in N gradient when one corrects for both

temperature and humidity.

The change of ducting statistics at all three locations obtained by the

two methods of time-lag correction yields paradoxical results. The near

contradiction of the two correction procedures is easily explained and

serves as an illustration of the necessity of correcting both the temperature and humidity elements for general application to ducting statistics. Consider typical temperature and humidity conditions associated with

ground-based ducts within each climate. Such cases are shown in figure

2.11. It is sufficient to note that the temperate ducts arise from typical

radiation inversion conditions of increasing temperature and decreasing

relative humidity with height; the arctic ducts are associated with the

intense arctic radiation inversion with ground temperature near —25 °C

and nearly constant relative humidity with respect to height; the tropical

ducts, however, appear to be due to slight decreases of both temperature

and humidity with height at temperatures near +25 °C. The effect of sensor lag upon these different gradients is always to make the indicated

gradient appear less than the true or environmental gradient. Thus

correcting for sensor lags makes the temperature and humidity gradients more intense. This, in turn, effects the resultant N gradient. One may

write

Ne - Ni = Xa. [1 - exp {-t/\N)l (2.12)

(2.13)
RADIOSONDE PROFILE ANALYSIS 43 Table 2.1. Comparison of ground-based ducting statistics derived from various sensor time lag corrections

Dtation IN 0. profiles

Reported data Corrected for T and /

sensor lag Assuming X.v=X/=10sec and eq (12) No.

ducts Average

gradient No.ducts Average

gradient No.

ducts Averagegradient N units/km N units/km Washington, D.C., (May), temperate

climate ... . 123 1 -163 9 -202 6 -181 Swan Island, (Aug.),

tropical climate . . 62 15 -186 20 -206 23 -238

Fairbanks, Alaska,

(Feb.), arctic climate.. 51 2 -211 4 -212 12 -250

which, for normal conditions of 15 °C, 1013 mbar, and 60 percent RH, is approximated by

^N ^ -1.27 ^T + 4.50 Ae + 0.27 AP. (2.14)

Since, for ducting, a large negative gradient of is required, the effect of temperature sensor lag correction is to make temperate and arctic

temperature increases with height more pronounced with a resultant

larger contribution to a negative gradient. Humidity lag corrections

under all conditions lead to a larger decrease of RH with height than

indicated and concomitant more rapid decrease of with height. Such

an explanation does not seem so evident for tropical conditions, however,

where both temperature and relative humidity decrease with height.

This apparent paradox is explained by the relatively great change in saturation vapor pressure of water associated with a small change of temperature near 25 °C which then produces the required large decrease

of e and N with height. It is seen, then, that the temperature sensor lag

correction produces an added humidity correction. Thus one must correct both sensing elements for the purpose of preparing profiles. Even

in the case of the arctic inversion the interplay of the temperature and

humidity lag corrections is very important, as has been noted by Yerg

[48].Thus, the correction for humidity sensor lag alone tends to overestimate

ducting incidence since the corrected relative humidity decrease, coupled

with the indicated temperature increase, produces a greater decrease in water vapor pressure than is actually present.

2.5.4. Conclusion

It appears from the present study that if studies of refractive index

profile characteristics are to include sensor lag correction, then allowance

should be made for both temperature humidity sensor time lags, regardless of climate. Any systematic application of these conclusions to large
44 MEASURING THE RADIO REFRACTIVE INDEX

bodies of data, however, must await the appearance in the literature of

effective lag coefficients that combine the effects of both temperature and

humidity lags upon the LiCl element.

\

Du

1 1 SWAN ISLAND 8-2C)-53 I50C GMT

-Duct rn

Thick

1

ness -

i I WASHINGTON, D.C. 8-19 -53, 0900 GMT

t Thick

i ness 7

FAIRBANKS, ALASKA

3-2-53, 1500 GMT

0 25 50 75 100 -40 -30 -20 -10 0 10 20 30 40

Relative Humidity Temperature in Degrees Centigrade

Figure 2.11. Temperature and humidity profiles.
REFERENCES 45 2.6. References

[1] Johnson, W. E. (Dec. 1953), An analogue computer for the solution of the radio

refractive index equation, J. Res. NBS 51, No. 6, 335-342.

[2] Sprung, A. (1888), Bestimm. d. Luftfeuchtigk. mit Hiilfe d. Assmann'schen

Asperationspsychrometers, Das Wetter 5, 105-109.

[3] List, R. J. (1958), Smithsonian Meteorological Tables, Washington, D.C. (Published by the Smithsonian Institution).

[4] National Physical Laboratory, Dept. Science Research (1960), The refractive index of air for radio waves and microwaves (Teddington, Middlesex, England).

[5] Jones, F. E., and A. Wexler (July 1960), A barium fluoride film hygrometer

element, J. Geophys. Res. 65, No. 7, 2087-2095.

[6] Macready, P. B., Jr. (1960), Field applications of MRI Model 901 water vapor

meter. Internal Memo Report, Meteorol. Research Inc. of Altadena, Calif.

[7] Stover, C. M. (Apr. 1961), Preliminary report on a new aluminum humidity

element. Internal Tech. Memo, Sandia Corporation, Albuquerque, N. Mex.

[8] Hirao, K., and K. I. Akita (Oct. 1957), A new type refractive index variometer,

J. Radio Res. Lab. 4, No. 18, 423-437.

[9] Crozier, A. L. (Apr. 1958), Captive balloon refractovariometer, Rev. Sci. Instr. 29, No. 4, 276-279.

[10] Cline, D. E. (1957), International radiosonde comparison tests, Tech. Memo

NRM-1907, U.S. Army Signal Eng. Lab. Task NR 3-36-11-402.

[11] Wexler, A. (July 1949), Low temperature performance of radiosonde electric hygrometer elements, J. Res. NBS 43, 49-56.

[12] Bean, B. R., and E. J. Dutton (Nov. 1961), Concerning radiosondes, lag constants, and refractive index profiles, J. Geophys. Res. 66, No. 11, 3717-3722.

[13] Misme, P. (Jan. 1956), Methode de mesure thermodynamique de I'indioe de

refraction de I'air—description de la radiosonde MDI, Ann. Telecommun. 11, No. 1, 81-84.

[14] Clinger, A. H., and A. W. Straiton (May 1960), Adaptation of the radiosonde for direct measurement of radio refractive index. Bull. Am. Meteorol. Soc. 41, No. 5, 250-252.

[15] Thiesen, J. F. (Apr. 1961), Direct measurement of the refractive index by radio- sonde, Bull. Am. Meteorol. Soc. 42, No. 4, 282.

[16] Sargent, J. A. (May 1959), Recording microwave hygrometer, Rev. Sci. Instr. 30, 345-355.

[17] Lement, B. S., C. S. Roberts, and B. L. Averbach (Mar. 1951), Determination of small thermal expansion coefficients by a micrometric dilatometer method,

Rev. Sci. Instr. 22, No. 3, 194-196.

[18] Bussey, H. E., and G. Birnbaum (Sept.-Oct. 1953), Measurement of variation in atmospheric refractive index with an airborne microwave refractometer, J. Res. NBS 51, No. 4, 171-178.

[19] Grain, C. M., and C. E. Williams (Aug. 1957), Method of obtaining pressure and

temperature in sensitive microwave cavity resonators. Rev. Sci. Instr. 28, No. 8, 620-623.

[20] Vetter, M. J., and M. C. Thompson, Jr. (June 1962), An absolute microwave

refractometer, Rev. Sci. Instr. 33, 656-660.

[21] Thompson, M. C, Jr., F. E. Freethey, and D. M. Waters (Oct. 1958), Fabrication

techniques for ceramic X-band cavity resonators. Rev. Sci. Instr. 29, No. 10, 865-868.

[22] Fraser, D. W., and E. G. Holmes (1953), Precision frequency control techniques, Final Report Proj., 229-298 (Georgia Inst. Tech., State Eng. Exp. Sta.).
46 MEASURING THE RADIO REFRACTIVE INDEX

[23] Adey, A. W. (1957), Microwave refractometer cavity design, Can. J. Tech. 34, 519-521.

[24] Thompson, M. C, Jr., F. E. Freethey, and D. M. Waters (July 1959), End plate modification of X-band TE cavity resonators, IRE Trans. Microwave Theory

Tech. MTT-7, No. 3, 388-389.

[25] Herbstreit, J. W. (July 1960), Radio refractometry, NBS Tech. Note 66.

[26] Birnbaum, G. (Apr. 1951), Fluctuations in the refractive index of the atmosphere

at microwave frequencies, Phys. Rev. 82, 110-111.

[27] Birnbaum, G., and H. E. Bussey (Oct. 1955), Amplitude, scale, and spectrum of refractive index inhomogeneities in the first 125 meters of the atmosphere,

Proc. IRE 43, 1412-1418.

[28] Grain, C. M., and A. P. Deam (Aug. 1951), Measurement of Tropospheric Index

of Refraction Profiles with an Airplane-Carried Direct Reading Refractometer,

Electrical Engineering Research Laboratory, University of Texas, Austin, Tex.

[29] Grain, C. M. (Oct. 1955), Survey of airborne microwave refractometer measurements, Proc. IRE 43, No. 10, 1405-1411.

[30] Grain, G. M. (May 1950), Apparatus for recording fluctuations in the refractive index of the atmosphere at 3.2 centimeter wavelength. Rev. Sci. Instr. 21, No. 5, 456-457.

[31] Pound, R. V. (Nov. 1946), Electronic frequency stabilization of microwave

oscillators. Rev. Sci. Instr. 17, No. 11, 490-505.

[32] Birnbaum, G. (Feb. 1950), A recording microwave refractometer. Rev. Sci. Instr. 21, No. 2, 164-176.

[33] Murray, G., and T. B. Watkins (Apr. 1957), Automatic frequency control, U.S. Patent 2,788,445. Filed Feb. 5, 1952, granted Apr. 9, 1957.

[34] Silsbee, R. H. (Jan. 1956), A high sensitivity paramagnetic resonance spectrometer, Cruft Lab. Tech. Report 221, Harvard Univ., Cambridge, Mass.

[35] Jung, P. (Oct. 1960), Transistorized frequency stabilization for reflex klystrons used in magnetic resonance, J. Sci. Instr. 37, 372-374.

[36] Smith, M. J. A. (Oct. 1960), Frequency stabilization of klystrons, J. Sci. Instr. 37, 398-399.

[37] Deam, A. P. (May 1959), An expendable atmospheric radio refractometer, EERL

Report 108, Univ. of Texas, Austin, Tex.

[38] Deam, A. P., and C. F. Cole, Jr. (Jan. 1960), Development of a lightweight

expendable microwave refractometer, EERL Report 5-46, Univ. of Texas,

Austin, Tex.

[39] Hay, D. R., H. C. Martin, and H. E. Turner (June 1961), Lightweight refrac- tometer. Rev. Sci. Instr. 32, No. 11, 693-697.

[40] Wagner, N. K. (July 1960), An analysis of radiosonde effects on measured fre- quency of occurrence of ducting layers, J. Geophys. Res. 65, 2077-2085.

[41] Clarke, L. C. (1960), Theory of atmospheric refraction. Part II of Meteorological

Aspects of Radio-Radar Propagation, pp. 31-82, NWRF31-0660-035 (U.S. Navy Weather Research Facility, Norfolk, Va.).

[42] Dunmore, F. W. (1938), An electric hygrometer and its application to radio meteorography, J. Res. NBS 20, 723-744.

[43] Bunker, A. F. (1953), On the determination of moisture gradients from radiosonde

records. Bull. Am. Meteorol. Soc. 34, 406-409.

[44] Middleton, W. E. K., and A. F. Spilhaus (1953), Book, Meteorological Instru- ments, p. 63 (Univ. of Toronto Press, Toronto, Canada).

[45] Sion, E. E. (Jan. 1955), Time constants of radiosonde thermistors. Bull. Am.

Meteorol. Soc. 36, 16-21.

[46] Wexler, A., S. Garfinkel, F. Jones, S. Hasegawa, and A. Krinsky (Aug. 1955), A fast responding electric hygrometer, J. Res. NBS 55, 71-78.
REFERENCES 47

[47] Wagner, N. K. (May 1961), The effect of time constant of radiosonde sensors on

the measurement of temperature and humidity discontinuities in the atmosphere, Bull. Am. Meteorol. Soc. 42, 317-321.

[48] Yerg, D. G. (May 1950), The importance of water vapor in microwave propagation at temperatures below freezing, Bull. Am. Meteorol. Soc. 31, 175-177.

Chapter 3. Tropospheric Refraction

3.1. Introduction

If a radio ray is propagated in free space, where there is no atmosphere,

the path followed by the ray is a straight line. However, a ray that is propagated through the earth's atmosphere encounters variations in atmospheric refractive index along its trajectory that cause the ray path

to become curved. The geometry of this situation is shown in figure 3.1, which defines the variables of interest. The total angular refraction of the ray path between two points is designated by the Greek letter r, and

is commonly called the ' 'bending" of the ray. The atmospheric radio

refractive index, n, always has values slightly greater than unity near the

earth's surface (e.g., 1.0003), and approaches unity with increasing height.

Thus ray paths usually have a curvature that is concave downward, as shown in figure 3. 1 . For this reason, downward bending is usually defined

as being positive.

If it is assumed that the refractive index is a function only of height

above the surface of a smooth, spherical earth (i.e., it is assumed that the

refractive index structure is horizontally homogeneous), then the path of a radio ray will obey Snell's law for polar coordinates:

the geometry and variables used with this equation are shown in figures

3.1 and 3.19. With this assumption r may be obtained from the following

integral,

which can be derived as shown later in the chapter or as derived by Smart

The elevation angle error, e, is an important quantity to the radar

engineer since it is a measure of the difference between the apparent

Figures in brackets indicate the literature references on p. 87.

n2r2 cos 02 = niri cos Oi, (3.1)

(3.2)

[1].

49
50 TROPOSPHERIC REFRACTION

Figure 3.1. Geometry of the refraction of radio waves.

elevation angle, ^o, to a target, as indicated by radar, and the true elevation angle. Under the same assumptions made previously, e is given as a function of r, n, and B by
INTRODUCTION 51

Figure 3.2. Differential geometry used in the derivation of the effective-earth's-radius- model atmosphere.

The apparent range to a target, Re, as indicated by a radar, is defined as an integrated function of n along the ray path.

Re = I'^ndR = \ (3.4)

Jq Jo smd

However, the maximum range error {Re minus the true range) hkely to be encountered is only about 200 m, hence the evaluation of (3.4) is not

of great importance unless one is dealing with an interferometer or phasemeasuring system.
52 TROPOSPHERIC REFRACTION

The integral for r, (3.2), cannot be evaluated directly without a knowledge of the behavior of n as a function of height. Consequently, the

approach of the many workers in this field has been along two distinct

lines: the use of numerical integration techniques and approximation

methods to evaluate r without full knowledge of n as a function of height,

and the construction of model n-atmospheres in order to evaluate average

atmospheric refraction. The following sections are devoted to a discussion of these methods.

3.2. Limitations to Radio Ray Tracing

The user should keep in mind that the equations given in the preceding

section are subject to the following restrictions of ray tracing:

(1) The refractive index should not change appreciably in a wavelength.

(2) The fractional change in the spacing between neighboring rays

(initially parallel) must be small in a wavelength.

Condition (1) will be violated if there is a discontinuity in the refractive

index (which will not occur in nature), or if the gradient of refractive

index, dn/dr, is very large, in which case condition (2) will also be violated.

Condition (1) should be satisfied if

(dn/dh) per km

N

U.UUiJfcc,

where refractivity, A^, is defined as = (n — 1) X 10® and/kc is the carrier frequency in kilocycles per second [2]. Condition (2) is a basic requirement resulting from Fermat's principle for geometrical optics. An atmospheric condition for which both conditions (1) and (2) are violated is known as ''trapping" of a ray, and it can occur whenever a layer exists with a vertical decrease of greater than 157 iV-units per kilometer. A

layer of this type is called a "duct," and the mode of propagation through

such a layer is similar to that of a waveguide [3]. Taking into account

refractive index gradients, a cutoff frequency may be derived for waveguidelike propagation through a ducting layer [4], In addition to the above limitations, it should be remembered that the

postulate of horizontal homogeneity, made in order to use (3.1), is not

realized under actual atmospheric conditions; some degree of horizontal

inhomogeneity is always present (see chapter 8).
HIGH INTIAL ELEVATION ANGLES 53 3.3. An Approximation for High Initial

Elevation Angles

A method may be derived for determining ray bending from a knowledge only of n at the end points of the ray path, if it is assumed that the

initial elevation angle is large. Equation (3.2) in terms of refractivity,

N, is equal to ^12= - / cot edN 10~\ (3.5)

assuming ?7 ^ 1 in the denominator. Integration by parts yields:

ri,2 = — cot d dN • 10'

A'l ,e. N cot d 10"' - / -^^dd 10-\ (3.6)

Note that the ratio, N/sin^d, becomes smaller with increasing 6 for values

of 6 close to 90°. If point 1 is taken at the surface, then di = Oq and

iVi = Ns. Then for = 10°, No = 0 and $2 = 7r/2, the last term of

(3.6) amounts to only 3.5 percent of the entire equation, and for the same

values of A^2 and 62, but with ^0 = 87 mrad ('^5°), the second term of (3.6)

is still relatively small (~10 percent). Thus it would seem reasonable

to assume that for

^0 > 87 mrad (~5°),

the bending, n ,2, between the surface and any point, r, is given sufficiently

well by

ri,2 = —

or N cot d X 10'

A'r.^r

ri.2 = Ns cot ^0 X 10-6 - Nr cot dr X 10-^ (3.8) The term —Nr cot dr X 10"^ is practically constant and small with re- spect to the first term, for a given value of ^0 and r, in the range ^0 > 87

mrad. Thus ri,2 is seen to be essentially a linear function of Ns in the

range do > 87 mrad. For bending through the entire atmosphere (to a

point where Nr = 0), and for ^0 < 87 mrad, (3.8) reduces to

T = Ns cot ^0 X 10-^ (3.9)
54 TROPOSPHERIC REFRACTION

For initial elevation angles less than about 5°, the errors inherent in this method exceed 10 percent (except near the surface) and rise quite rapidly

with decreasing ^o3.4. The Statistical Method

Another method for determining high-angle bending is the statistical linear regression technique developed by Bean, Gaboon, and Thayer [5]. It has been found that for normal conditions and all heights the righthand

integral of (3.6) is approximately a linear function of Ns {do, r constant)

for ^0 < 17 mrad (^^1°) and that the second term of (3.8) tends to be

constant. Thus (3.6) reduces to a linear equation,

Ti,2 = bNs + a, (3.10)

where b and a are constants (as in tables 9.1 to 9.9) and Ns is the surface

refractivity.

The form of (3.10) is very attractive, since it implies two things:

(1) The value of n ,2 may be predicted with some accuracy as a function

only of Ns (surface height and ^0 constant), a parameter which may be

observed from simple surface measurements of the common meteorological

elements of temperature, pressure, and humidity.

(2) The simple linear form of the equation indicates that, given a

large number of observed ri,2 versus Ns values for many values of h and

^0, the expected (or best estimate) values of h and a can be obtained by

the standard method of statistical linear regression.

This is what was done to obtain values listed in tables 9.1 to 9.9. Tables 9.1 to 9.9 also show the values of the standard error of estimate,

SE, to be expected in predicting the bending, and the correlation coeffi- cients, r, for the data used in predicting the lines. Linear interpolation can be used between the heights given to obtain a particular height that

is not listed in the tables. For more accurate results, plot the values of

r from the tables (for desired Ns) against height, and then plot the values

of the standard error of estimate on the same graph. Then connect these

points with a smooth curve. This will permit one to read the r value and

the SE value directly for a given height.

3.5. Schulkin's Method

Schulkin has presented a relatively simple, numerical integration

method of calculating bending for A^-profiles obtained from ordinary

significant-level radiosonde (or "RAOB") data [6]. The N profile obtained from the RAOB data consists of a series of values of N for different
SCHULKIN'S METHOD 55 heights; one then assigns to N{h) a linear variation with height in between

the tabulated profile points, so that the resulting versus height profile

is that of a series of interconnected linear segments. Under this assumption, (3.2) is integrable over each separate linear iV-segment of the profile

(after dropping the n term in the denominator, which can result in an

error of no more than 0.04 percent in the result), yielding the following

result

Ari.2(rad) ^ - r^''^ cot 6 dn - ^^""^ ~ ""'^ tan ^1 + tan 62

' or Ari,2(mrad) ^ —7 ——7 — . (3.11)

tan Bi + tan 62

For the conditions stated above, this result is accurate to within 0.04

percent or better of the true value of An, 2, an accuracy that is usually

better than necessary. Thus it is possible to simplify (3.11) further by

substituting 6 for tan 6; this introduces an additional error that is less than

1 percent if 6 is under 10° (-^175 mrad). Now (3.11) becomes

(mrad) (mrad)

where 6 may be determined from (3.58).

The bending for the whole profile can now be obtained by summing

up the An. 2 for each pair of profile levels: (mrad)^i;^(f.^f^. (3.13)

(mrad) (mrad)

This is Schulkin's result. The degree of approximation of (3.13) is quite high, and thus most recent "improved" methods of calculating r will reduce to Schulkin's result for the accuracy obtainable from RAOB or other similar data. Thus, provided that the iV-profile is known, (3.13) is the most useful form for computing bending (for all practical purposes)

that should concern the communications or radar engineer. Some other

methods have been published which are actually the same as Schulkin's,

but have some additional desirable features; e.g., the method of Anderson

[7] employs a graphical approach to avoid the extraction of square roots to obtain dk.
56 TROPOSPHERIC REFRACTION

3.6. Linear or Effective Earth's Radius Model

The classical method of accounting for the effects of atmospheric

refraction of radio waves is to assume an effective earth's radius, = ka,

where a is the true radius of the earth and k is the effective earth's radius

factor. This method, advanced by Schelleng, Burrows, and Ferrell [8], assumes an earth suitably larger than the actual earth so that the curvature of the radio ray may be absorbed in the curvature of the effective earth so that the relative curvature of the two remains the same, thus

allowing that radio rays be drawn as straight lines over this earth rather

than curved rays over the true earth. This method of accounting for atmospheric refraction permits a tremendous simplification in the many

practical problems of radio propagation engineering although the height

distribution of refractive index implied by this method is not a very

realistic representation of the average refractive index structure of the

atmosphere. This section will consider the refractive index structure assumed by the effective earth's radius model and how this differs from

the observed refractive index structure of the atmosphere. Further, the

limits of applicability of the effective earth's radius approach will be ex- plored and a physically more realistic model, the exponential, will be

described for those conditions where the effective earth's radius model is most in error.

It is instructive to give a derivation of the expression relating the curvature of radio rays to the gradient of refractive index. In figure 3.2 a wave front moves from AB to A'B' along the ray path. If the phase

velocity along AA'isv and v -\- dv along BB\ then, from considering the

angular velocity,

I = (3.14) p p

-\- dp

or

V p where p is the radius of curvature of the arc AA\ Now, since the phase

velocity, v, is

V = - (3.16) n

where c is the velocity of light in vacuo, one obtains

dv^ _dn

V n
EFFECTIVE EARTH'S RADIUS 57 combining (3.15) and (3.17), the familiar equation,

(3-18)

p n dp

is obtained. If the ray path makes an angle 6 with the surface of constant

refractive index

dr = dp cos d (3.19)

and

1 = _ 1 ^ cos ^. (3.20)

p n dr

If the curvature of the effective earth is defined as

_1_ ^ 1 _ 1 ae a p then

(3.21)

ae = ka =

^ (3.22) 1/a - l/p

and

1

, , a dn

1 + - -77- cos 6 n dh

(3.23)

For the small values of 6 normally used in tropospheric propagation, cos 6 may be set equal to unity. Further, by setting

one obtains the familiar value of ^ = 4/3 for the effective earth's radius

factor. By assuming that the gradient of n is constant, a linear model

of N versus height has been adopted.

For this model, the bending

Ti.2 = - / cot e dn (3.25)

J Til is written

cot d

Tl,2 1: \a

dh (3.26)
58 TROPOSPHERIC REFRACTION

since

and

N = No - ^ 10

4a

dn = dN X 10"' =

dh

4a

(3.27)

(3.28)

Further, for the case hi = ho = 0, and

0 < ^0 < 10°,

where is the initial elevation angle of a ray, (3.26) may be approximated

by

TO,h i:

dh_

4:ad (3.29)

The angle 6 may be determined from

dh =={\ ei-{- 2(N - No) + - (h - ho) • 10^ (3.30)

(3.31)

For the case when = 0, (3.29) becomes = —i— r = J2VQa''' Vh VQ^""

(3.32)

To,h = Vh/a. (3.33)

Now, from the geometrical relationship.

ro,h = ~~ + (^0 — Oh), (3.34)

one finds, for = 0,

do,h = o,(To,h + Sh), (3.35)
MODIFIED EFFECTIVE EARTH'S RADIUS 59 which upon substitution from (3.33) and (3.31) gives

d,,h = V2/i(4/3) a, (3.36)

or, more famiharly,

do,h = V2kah . (3.37)

A very convenient working formula is derived from (3.37) by A: = 4/3, a = 3960 miles and using units of miles for the ground distance to the radio

horizon, dQ,h, and feet for the antenna height, h:

do,h = V2^ miles (3.38)

This is the familiar expression often used in radio propagation engineering for the distance to the radio horizon.

3.7. Modified Effective Earth's Radius Model

The effective earth's radius model, although very useful for engineering

practice, is not a very good representation of actual atmospheric structure. For example, the data on figure 3.3 represent the average of individual radiosonde observations over a 5-yr period at several locations

chosen to represent the extremes of refractive index profile conditions

within the United States. The Miami, Fla., profile is typical of w^arm,

humid sea-level stations that tend to have maximum refraction effects while the Portland, Me., profile is associated with nearly minimum sealevel refraction conditions. Although Ely, Xev., has a much smaller

surface value than either Miami or Portland, it is significant that when

its profile is plotted in terms of altitude above sea level, it falls within

the limits of the maximum and minimum sea level profiles. It is to take

advantage of this simplification, that altitude above sea level rather than

height above ground is frequently used throughout this monograph. The

N distribution for the 4/3 earth atmosphere is also shown on figure 3.3. It is quite evident that the 4/3 earth distribution has about the correct

slope in the first kilometer above the earth's surface but decreases much

too rapidly above that height. It is also seen, by noting that figure 3.3 is plotted on semi-logarithmic paper, that the observed refractivity distri- bution is more nearly an exponential function of height than a linear

function of height as assumed by the 4/3 earth atmosphere. One might

expect the refractivity to decrease exponentially with height since the

first term of the refractivity equation (1.20) involving P/T, comprises at

least 70 percent of the total and is proportional to air density, a well- known exponential function of height.
60 TROPOSPHERIC REFRACTION

N

0 1 4 6 8 10 12 Altitude Above Mean Sea Level in Kilonneters

Figure 3.3. Typical N versus height distributions. One might wonder, in the light of the data of figure 3.3, why the effective earth's radius approach has served so well for so many years. It appears

that this success is due to the 4/3 earth model being in essential agree- ment with the average structure near the earth's surface which largely

controls the refraction of radio rays at the small values of common in tropospheric communications systems.

It would seem that the deficiency of the effective earth's radius ap- proach could be lessened by modifying that theory in the light of the

average structure of the atmosphere. An indication of the average N

structure was obtained by examining a variety of profiles which were

carefully selected from 39 station-years of individual radiosonde observations to represent the range of profile conditions during summer and

winter at 13 climatically diverse locations. The results of this examination are given in table 3.1.
MODIFIED EFFECTIVE EARTH'S RADIUS 61 Table 3.1. Refmctivity statistics as a function of altitude above sea level as derived from individual radiosonde observations

Aiiiiuae {km) iVlaximum A iViiniiTiiim i\ rtange

. .—

.

4 197.

1

209.5 186.5 23.0

5 172.3 184.0 165.0 19.0

6 151.4 161.0 146.0 15. 0 134.0 139.5 129.5 10.0

8 118.4 121.5 113.3 8.2

9 104.8 108.0 100.0 8.0

10 92.4 97.0 86.0 11.0

11 81.2 86.0 70.0 16. 0

12 70.7 76.0 60.5 15.5

14 53.2 60.0 44.5 15. 5 *Range = maximum iV — minimum A''.

It is interesting to note that the range of N values has a minimum at 8 to 9 km above sea level but is systematically greater above and below

that altitude. The average value of 104.8 at 9 km corresponds to a

similar value reported by Stickland [9] as typical of the United Kingdom.

Further the altitude of 8 km corresponds to the altitude reported by

Humphreys [10] where the atmospheric density is nearly constant regardless of season or geographical location. Since the first term in the expression for refractivity is proportional to air density, and the water-vapor

term is negligible at an altitude of 9 km, the refractivity also tends to be

constant at this altitude. It seems quite reasonable, then, to adopt a constant value of N = 105 for 9 km, thus further facilitating the specification of model atmospheres. Further, as also noted in chapter 1, when

the values of table 3.1 are plotted such as on figure 3.3, it is seen that the

data strongly suggest that may be represented by an exponential

function of height of the form:

N(h) = No exp {-bh],

in the altitude range of 1 to 9 km above sea level. The following recommendation is made when dealing with problems

involving ground-to-ground communications systems or other types of low-altitude radio propagation problems where the ray paths involved do

not exceed 1, or at most 2, km above the earth's surface: use the effective earth's radius method to solve the associated refraction problems. The

user should refer to the tables in chapter 9, where effective earth's radius

factors are tabulated along with other refractivity variables. Table 9.27 may be entered with Ns and table 9.28 may be entered with AN{N s subtracted from the N value at 1 km above the surface). In both these

tables linear interpolation will suffice for any practical problem. The

variables listed in these tables are for the exponential model of N{h) that

is covered in the following subsection.
62 TROPOSPHERIC REFRACTION

When the effective earth's radius treatment is used, height is calculated as a function of distance, for a ray with = 0, with the

equation h = d^/2ka, where d is the distance, k is the effective earth's radius factor, and a is the true radius of the earth ('^6373 km). The

errors likely to be incurred when using this equation, assuming as a true atmosphere an exponential N{h) profile as given in the following subsection will not exceed 5 percent for heights of 1 km or less. The preceding background discussion has presented the material neces- sary for the consideration of the suitability of various models of refrac- tivity to describe atmospheric refraction of radio waves. As a guide to what follows, let us ask what a logical sequence of models (or assumptions)

would be to describe the effects of atmospheric refraction.

One such sequence might be:

(1) Assume an invariant model that is near to the actual average

conditions and facilitates the calculation of radio field strengths. This

has been done by the 4/3 earth model.

(2) Assume a variable effective earth's radius factor for the calculation

of radio field strengths in various climatic regions. This approach has

been followed by Norton, Rice, and Vogler [11]. When it has become

apparent that the effective earth's radius approach is inadequate, one might proceed by

:

(3) Correctmg the effective earth's radius model by assuming a more

realistic N structure in the region where that model is most in error. This

''modified effective earth's radius" model would then maintain, for some

applications, the advantages of the original model.

(4) Assume an entirely new model of structure guided by the

average N structure of the atmosphere.

It is assumed that models (3) and (4) would allow for seasonal and

climatic changes of the average refractive index structure of the atmosphere. In the following sections, models (3) and (4) will be developed and

tested by their relative agreement with the ray bendings obtained from

actual long-term average N profiles.

The first model of atmospheric refractivity that will be considered is based upon the effective earth's radius concept in the first kilometer. In

this atmosphere N is assumed to decay linearly with height from the sur- face /is to 1 km above the surface /is + 1. This hnear decay is given by

N(h) = Ns -\- {h - hs) AN,hs < h < hs -\- 1 (3.39)

where

-AA^ = 7.32 exp (0.005577 Ns). (3.40)
MODIFIED EFFECTIVE EARTH'S RADIUS 63 This last relationship comes from the observed relationship between

6 to 8 year averages of daily observations of Ns and AiV, the difference

between and the value of at 1 km above the earth's surface: -AN = N. N{1 km),

It is evident from figure 3.4 that for average conditions, a relationship

exists between AA^ and Ns. The least squares determination_given by

(3.40) was based upon 888 sets of monthly mean values of AA^ and Ns

from 45 United States weather stations representing many diverse cli- mates. This relationship between AA^ and Ns is expected to represent

the best estimate of a majority of individual profiles and certainly will closely agree with average conditions for the United States with one

notable exception, southern California in the summer. These data, al- though shown on figure 3.4, were excluded from the least squares deter- mination due to their singular large range of AA^ compared to their small

range of Ns which resulted in a marked "finger" of points rising from the

main body of the data. This obvious departure of data points (24 points out of a total of 912) plus the well-known unique nature of the southern

California summer climate were taken as sufficient justification for ignoring these points, although, as shall be seen, the ray bendings based upon

100 90

-AN —

888 sets of data

230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400

Figure 3.4. Eight-year average Ne versus 6-year average AN at 0300 and 1600 GMT
64 TROPOSPHERIC REFRACTION

the AN obtained from (3.40) are in rather good agreement with the values

calculated from actual N profiles, even those observed in southern

California.

Equation 3.40 offers a convenient method of specifying various models

of the refractivity structure of the atmosphere, since it allows an estimation of the value of at 1 km in addition to the two values already

known; i.e., Ns and N = 0 at h = oo. It may be further assumed that decreases exponentially from hs I to a constant value of 105 at 9 km above sea level. In this altitude

range is defined by: = A^i exp {-c(h - hs - I)], hs + 1 < h < 9km, (3.41)

where

1 , Nr

and A^i is the value of A^ at 1 km above the surface. Above the altitude of 9 km, where less than 10 percent of the total bending occurs, a single exponential decrease of A^ may be assumed. The

coefficients in the exponential expression

:

A^ = 105 exp {-0.1424 (h - 9)\ , h > 9 km, (3.42) were determined by a least squares analysis of The Rocket Panel data

[12]. This expression is also in agreement with the ARDC Model Atmosphere 1956 [13] and Dubin's [14] conclusion that a standard densitydistribution may be used to determine the refractivity distribution at altitudes in excess of 20,000 ft. The three-part model of the atmosphere expressed by (3.39)-(3.42) has

the advantage of the effective earth's radius approach, particularly for such applications as point-to-point radio relaying over distances up to, say, 100 mi, where the radio energy is generally confined to the first kilometer, plus being in reasonably good agreement with the average A^ structure of the atmosphere. The reader is cautioned, however, that

application of this model to mode-theory calculations would be misleading, since the resultant difi'raction region fields would be enhanced by the

addition of strong reflections from the n-gradient discontinuities at /i^ + 1 and at 9 km. The specific combinations of A^^, hs, and AA^, that define

the CRPL Reference Atmosphere—1958 are given in table 3.2. The station elevations corresponding to given combinations of Ns and

AA" were chosen to correspond with an average decay of A^ with station

elevation. Although the error in neglecting this height dependence has
EXPONENTIAL MODEL 65 Table 3.2. Constants for the CRPL Reference Atmosphere—1958

Ns h. a' k Qe c

ft mi mi per km

0 0 3960. 0000 0 1. 00000 3960. 00 0 200 10, 000 3961. 8939 22. 3318 1. 16599 4619. 53 0. 106211 250 5, 000 3960. 9470 29. 5124 1. 23165 4878. 50 0. 114559

301 1,000 3960. 1894 39. 2320 1. 33327 5280. 00 0. 118710 313 700 3960. 1324 41. 9388 1. 36479 5403. 88 0. 121796 350 0 3960. 0000 51. 5530 1. 48905 5896. 66 0. 130579

400 0 3960. 0000 68. 1295 1. 76684 6996. 67 0. 143848

450 0 3960. 0000 90. 0406 2. 34506 9286. 44 0. 154004

Note: Oe is the effective earth's radius and is equal to a'k, a' = a + hs, where h, is the altitude of the earth's surface above sea level, a = 3960 miles and c = 1/8—hs In A^i/105.

been estimated to be no more than a few percent, it could be important

in such high precision applications as radar tracking of earth satellites. It should be remembered in subsequent applications that a unique feature

of these reference atmospheres is the dependence of Ns on the altitude of surface above sea level. This feature was built in so that the reference

atmospheres would be completely specified by the single parameter Ns.

3.8. The Exponential Model

The next model of the atmosphere to be considered may be specified

by assuming a single exponential distribution of N:

N = Ns exp { -Ce {h - hs)},

where _ , Ns _ , Ns

^""Nilkm) ^^Ns + AN

and these equations are used to determine at all heights. This model

of atmospheric refractivity is a close representation of the average re- fractivity structure within the first 3 km. Further, the single exponential

model has the advantage of being an entire function, and therefore is easily

used in theoretical studies. This model of the atmosphere has been

adopted for use within the National Bureau of Standards with specific

values of the constants in (3.43) and (3.44). These constants are given in table 3.3 and specify the CRPL Exponential Reference Atmosphere

1958.

Figure 3.5 compares the structure of the above two models plus the

4/3 earth model. It can be seen that the assumption agrees with the

reference atmosphere in the first kilometer, which is to be expected since

Ns = 301 is the value required to yield the 4/3 gradient from figure 3.5. Figure 3.5 illustrates the essential agreement of the reference atmosphere

(3.43)

(3.44)
66 TROPOSPHERIC REFRACTION

Table 3.3. Table of the constant Ce for the CRPL exponential radio refractivity atmospheres

N=Nsexp [-Ce{h-hs)]

AN Ns Ce (per km)

0 0 0

22. 3318 200.0 0. 118400

29. 5124 250.0 . 125625

30. 0000 252.9 . 126255

39. 2320 301.0 . 139632

41. 9388 313.0 . 143859

50. 0000 344.5 . 156805

51. 5530 350.0 . 159336

60. 0000 377.2 . 173233

68. 1295 400.0 . 186720

70. 0000 404.9 . 189829

90. 0406 450.0 . 223256

with the Rocket Panel and ARDC data. The exponential reference at- mosphere is also shown on figure 3.5 for Ns = 313, the average value of the United States. The exponential reference atmosphere appears to be

a reasonable single line representation of N throughout the height interval

shown. The differences between the various models becomes more

apparent by examining their agreement with observed N profiles over the

first 10 km as in figure 3.6. The reference and exponential reference atmospheres are given for the

profiles corresponding to near-maximum Ns (Lake Charles, La.) and

near minimum-at-sea-level Ns conditions (Caribou, Me.). The two

reference atmospheres were determined solely from the Ns values of each

profile. Several observations can be made of these data. First, the 4/3

earth model closely represents the slope of the minimal Ns profile over

the first kilometer, but then decreases too rapidly with height. Note,

however, that the 4/3 earth model with its constant decay of 39.2 N units

per kilometer would be a very poor representation of the maximum profile which decreases over 66 N units in the first kilometer. The exponential reference atmosphere is in good agreement with the initial N distribution but tends to give values systematically low above approximately

3 km. At first glance, the exponential reference atmosphere does not appear to be as good a representation of the two observed profiles as the

reference atmosphere, particularly above approximately 5 km. Subsequent analysis of the refraction obtained from the two model atmospheres

will show that this systematic disagreement of the exponential reference

atmosphere in the 5- to 20-km interval is a minor defect of the model

compared to its closer agreement with observed distributions over the

first 1 to 3 kni. This is particularly true for the higher values of Ns such

as that for Lake Charles.

The above models are more in agreement with long-term mean N profiles than is the 4/3 earth model. The application at hand would aid in deciding which of the reference atmospheres would be most useful. To
EXPONENTIAL MODEL 67

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 ALTITUDE IN KILOMETERS ABOVE SEA LEVEL

Figure 3.5. CRPL reference refraciivity atmospheres—1958.

aid in distinguishing between the various models, the following sections

will be concerned with comparing the ray bending between the models.

A comparison of the ray paths in the reference atmospheres in the 4/3

earth model will illustrate the systematic differences between these

models. Such a comparison is given in figure 3.7 for a distance of 200 mi

and a height of 14,000 ft and in figure 3.8 for a distance of 800 mi and a

height of 240,000 ft. The particular graphical presentation used in figures 3.7 and 3.8 shows the 4/3 earth rays as straight lines. It is noted

that the 4/3 earth ray at = 0 is in relatively good agreement with the

values from the reference atmosphere for distances out to 200 mi and

heights up to 14,000 ft, but systematically departs from the reference
TROPOSPHERIC REFRACTION

N

0 12 3 4 5 6 7 8 HEIGHT IN KILOMETERS

Figure 3.6. Comparison of reference atmospheres with observed N profiles.
EXPONENTIAL MODEL 69 atmosphere for greater distances and heights. For a range of 600 mi,

where the ray reaches heights of about 200,000 ft, the 4/3 earth ray is some 9000 ft lower than the Ns = 400 reference atmosphere and 36,000 ft lower than the Ns = 250 reference atmosphere. This height discrepancy

is due to the 4/3 earth model's unrealistically large gradient at great

heights with resultant increased bending.

Figure 3.7. Comparison of rays in the CRPL reference refradivity atmospheres—1958 and the 4/3 earth atmosphere.

Figure 3.8. Comparison of rays in the CRPL reference refractivity atmospheres—1958 and the 4/3 earth atmosphere.
70 TROPOSPHERIC REFRACTION

20 1

1

Expo

1

nential

1

referen

1 1 1

mean profi

nding,

les L

/J earth bend/ng T at 70 km

Mean profiles: 15.12 mrad Mean exp. ref.: 15.23 mrad

4/3 earth : 30.25 mrad

\ /

1 1 1 1 1 , i , ! , 0 2 4 6 8 10 12 14 16 18 20 h-hs IN KILOMETERS

Figure 3.9. Bending versus height.

Further, the bending in the 4/3 earth atmosphere is compared with

that in the exponential reference atmosphere in figure 3.9. The bending

in an ^'average" atmosphere is also given. This average atmosphere is a composite of the 5-year mean profiles for both summer and winter at the 11 U.S. radiosonde stations enumerated in the following paragraph,

and was used as a readily available measure of average conditions. The

important point made by figure 3.9 is that the 4/3 earth model is sys- tematically in disagreement with average bending; at low heights it gives too little bending, while at high altitudes it gives too much bending. The

exponential reference atmosphere does not appear to be systematically

biased, and deviates less than 5 percent from the average atmosphere.

It is significant that the exponential reference and the average atmosphere

are in essential agreement as to the shape of the r-height curve.

It would now be instructive to compare the bendings obtained from

the various models with values obtained from each of the 5-year mean N profiles from different climatic regions. The 5-year mean profiles were obtained for both summer and winter for a variety of climates as represented by the states of Florida, Texas, Maine, Illinois, Nevada,

California, North Dakota, Washington, Nebraska, Wyoming, and by the

District of Columbia.

Comparisons of the bending obtained from the 4/3 earth model and

the bendings obtained from the 5-year mean N profiles with the reference

atmospheres are shown in figures 3.10 and 3.11. These figures were

selected to illustrate the range of agreement between the models and the

expected long-term average bendings. Figure 3.10 gives a comparison
EXPONENTIAL MODEL 71

4 [ \ \ 1 200 300 4 00 500

Ns

Figure 3.10. Comparison of r versus Ng as obtained from CRPL reference atmospheres—1958 and 5-year mean radiosonde data for 0o = 0, h — hg = 5 km.

for a small initial elevation angle, do = 0, and a small height increment,

h — hs = S km, and shows that both reference atmospheres tend to set a

lower limit to the bendings. In this case, the exponential reference atmosphere appears to be in better agreement with the expected long-term mean bendings than does the reference atmosphere. The numbered data

points for Washington, D.C., Omaha, Nebr., and Santa Maria, Calif., are

of special interest. Washington and Omaha have the only long-term
72 TROPOSPHERIC REFRACTION

mean N profiles with initial gradients (i.e., — 112/km and — 106/km,

respectively) that are significantly greater than would be expected from

the AA^ versus Ns relationship. Both of these stations have an unusually

large humidity decrease near the ground. The third point, Santa Maria,

Calif., is of interest since it is in relatively good agreement with the refer- ence atmospheres, even though it represents the southern California summer climate which was excluded from the original AN versus Ns
EXPONENTIAL MODEL 73

relationship. This agreement is attributed to the fact that the reference

atmosphere is a good representation of the N distribution below the

California elevated inversion and to the fact that a majority of the bending is accomplished below the elevated inversion height of about 500 m.

Further, it can be easily shown that the bending integral is increasingly

insensitive to strong discontinuities as the height increases. Figure 3.11 shows a similar comparison for a high initial elevation angle,

^0 = 15° and a large height increment, h — hs = 70 km. This comparison shows that both of the reference atmospheres are in closer agreement

with the long-term mean bendings than are the 4/3 earth bendings. Note

that, whether r is predicted from Ns or AN, the 4/3 earth model gives but

a single value of bending that is outside the limits of the values of r obtained from the long-term mean profiles. In considering the comparisons of figures 3.10 and 3.11, one might ask

if they reflected the form of the basic equation for bending; namely, at low angles is r determined by the N gradient throughout the profile,

and at high angles is r essentially a function of the value of N at both ends

of the profile (i.e., the limits of integration). Thus one might expect

the deviations to be smaller if the comparisons were made on the basis of a function of the gradient such as AA^, particularly for small values of

$0. Such a comparison is given by figures 3.12 and 3.13 for the same

initial elevation angles and height increment as before. It is seen that

the AA^-specified reference atmospheres improve the agreement for the

low-angle case, but decidedly decrease the agreement for the high-angle

case. A numerical evaluation of the root mean square (rms) deviation of the

long-term mean bendings from both the reference atmospheres determined as a function of both AN and A^s was made for a variety of initial elevation angles for the height increments 3 and 70 km. Root mean

square deviations were not calculated for the 4/3 earth model since it was

felt that this model was obviously in marked disagreement with the longterm mean bendings under these conditions. Figure 3.14 summarizes

the rms deviations for the h — hs = 3-km case. It is seen that for

do < 10 mrad (about 0.5°), the AA^-specified reference atmospheres have

the smaller rms deviations. Also, the exponential reference atmospheres,

whether specified by AN or Ns, have smaller rms deviations than the

reference atmosphere.

It is seen for the 70-km case, figure 3.15, that the As-specified reference

atmospheres have a significantly smaller rms deviation than the AAspecified atmospheres for 6o > 5 mrad. Again it is seen that the exponential reference atmosphere generally has the smaller rms deviation for values of do less than 10 mrad. However, the slightly smaller rms deviations associated with the reference atmosphere for > 10 mrad reflect that model's closer agreement with the actual A structure of the atmosphere at high heights.
Figure 3.12. Comparison of t versus AN as obtained from CRPL reference atmospheres—1958 and 5-year mean radiosonde data for do = 0, h. — = 3 km.

r
EXPONENTIAL MODEL 75

Figure 3.13. Comparison of t versus AN as obtained from CRPL reference atmospheres—1958 and 6-year mean radiosonde data for do = 15° h — hg = 70 km.
76 TROPOSPHERIC REFRACTION

100

10

0.1

h-h5 = 3knri

Parameter AN

Mean bendi

reference cng is that for tnnosphere = 330

>Parameter Ng

1

1 Reference Atmosphe —Exponenfial Referenc

re

;e Atmosphere

0.01 O.I I 10

IN MILLIRADIANS

100 1000

Figure 3.14. Comparison of percent RMS deviations of 5-year mean profile bendings

about CRPL reference atmospheres using two parameters, Ns and AN for h — hs = 3 km.

Figure 3.15. Comparison of percent RMS deviations of 5-year mean profile bendings

about CRPL reference atmospheres using two parameters, Ng and AN for h - hs = 70 km.
DEPARTURES-FROM-NORMAL METHOD 77 3.9. The Initial Gradient Correction Method

The importance of the initial gradient in radio propagation, where the

initial elevation angle of a ray path is near zero, has long been recognized.

For example if dn/dh = —1/a (the reciprocal of the earth's radius), then

the equation for r is indeterminate, an expression of the fact that the

ray path remains at a constant height above the earth's surface. This is called ducting, or trapping of the radio ray. The effect of anomalous

initial A^-gradients on ray propagation at elevation angles near zero, and

for gradients less than ducting {\dNJdh \ < 157 N units/km, or dN/dh >

— 157 N units/km) may also be quite large. A method has been developed for correcting the predicted refraction (from the exponential refer- ence atmosphere) to account for anomalous initial gradients, assuming

that the actual value of the initial gradient is known [2]. The result is

Th = Th {Ns, do) + [rioo(A^s*, ^o) - rioo(iVs, ^o)], (3.45)

where th {N s,do) = r at height h, for the exponential reference atmosphere

corresponding to A^^, and A^s* is the Ns for the exponential reference

atmosphere that has the same initial gradient as the observed intiial gradient; t^^q is r at a height of 100m.

This procedure has the effect of correcting the predicted bending by

assuming that the observed initial gradient exists throughout a surface

layer 100 m thick, calculating the bending at the top of the 100-m-thick

layer, then assuming that the atmosphere behaves according to the ex- ponential reference profile corresponding to the observed value of A^s for

all heights above 100 m. This approach has proved quite successful in predicting r for initial elevation angles under 10 mrad, and will, of course, predict trapping when it occurs.

3.10. The Departures-From-Normal Method

A method of calculating bending by the use of the exponential model

of N{h) together with an observed N{h) profile can sometimes be advantageously employed [15]. This method is primarily intended to point out

the difference between actual ray bending and the average bending that is predicted by the exponential N{h) profile and is a powerful method of identifying air mass refraction effects. The exponential model described in section 3.8 can be expected to represent average refractivity profile characteristics at any given location,

but it cannot be expected to depict accurately any single refractivity profile selected at random, even though it may occasionally do so. In order
78 TROPOSPHERIC REFRACTION

to study the differences between individual observed N{h) profiles and the mean profiles predicted by the exponential model, a variable called the

A unit has been developed; it is defined simply as the sum of the observed

at any height, h, and the refractivity drop from the surface to the

height, h, which is predicted by the exponential profile for a given value

of Ns.

Thus

A{Ns,h) = Nih) + Ns {l-exp{-Ceh}). (3.46)

Thus (3.46) adds to N{h) the average decrease of with height, so that

if a particular profile should happen, by coincidence, to be the same as the corresponding exponential profile, the value of A {N s,h) for this profile would be equal to Ns for all heights. The above analysis shows that

the difference between A(Ns,h) from Ns, 8 A(N s,h), is Si measure of the

departure of N{h) from the normal, exponential profile:

8A{Ns,h) = AiNs,h) - Ns = N{h)—Nsexp{-Ceh]. (3.47)

It seems logical that the application of the A unit to bending would indi- cate the departures of bending from normal, in some way, just as it indi- cates departures of refractivity, N, from normal. This is indeed the case, as can be seen in figure 3.16, where for an Ns = 313.0 exponential atmosphere, A (313.0, h) is plotted on one set of graphs for various typical air masses, and the corresponding bending departures from normal are shown

in the second set of graphs corresponding to the same air masses. Obviously, the bending departures between layers are highly analogous to the A unit variation. It can be seen from figure 3.16 that the similarity

exists, although it is less, for higher initial elevation angles. The similarity also decreases with increasing height, owing to the fact that the

bending departures from normal are an integrated effect, and at low initial elevation angles are more sensitive to variations at the lower heights.

This causes an apparent damping of the bending departures from normal

at greater heights. However, the A-unit variation is not similarly in- fluenced; hence, a loss of similarity arises at large heights above the earth's

surface.

If (3.46) is differentiated and substituted into (3.2), the following equation results: '

TO., ^ r^,ih) + i: - T-nffl— I AA(Ar.)

I"'"" X 10-", (3.48)

(.rad) irad) (rad) (rad)
DEPARTURES-FROM-NORMAL METHOD 79 S CP MP MT cT

WINTER WINTER WINTER SUMMER SUMMER

1—I—

r

I /I L 300 320 320 340 310 330 350 370 320 340

53

//

/

I

I

J c:

1 1 \

1

\

1 -

1 -

\) - /

f

1 -2 0 -2 2-4 -2 00

T^i^-Tdnrad)

Figure 3.16. A-unit profiles for typical air masses and refraction deviation from normal.

where

AA{Ns) = AN{h) + A[Ns{l-exp{-Ceh)]] = AN{h) + NsCe e^p(-Ceh)Ah,

and TNs(h) is the value of r tabulated for various atmospheres in tables

9.10 to 9.17, dk and dk+i are in milliradians and must he from the Ns ex- ponential atmosphere used. The value of AA{N s) is obtained from subtraction of the A value at layer level, k, from the value of A at layer, /c + 1. The A value may be obtained by adding any given N{h) value, obtained

from RAOB or other similar data, to a value of A/'s[l — exp{ —ch\] for the same height which may be obtained from figure 3.17. Since TNs{h) has

been calculated only for a few of the exponential atmospheres, these being

the Ns = 200.0, 252.9, 289.0, 313.0, 344.5, 377.2, 404.9, and 450.0 atmospheres, one of these atmospheres must be used in the calculation of bending by the departures method. The selection of the particular atmosphere to be used is based on the value of the gradient of N, dN/dh, between the surface of the earth and the first layer considered. In table 3.2 are shown the ranges of the gradient for the choice of a particular ex- ponential atmosphere.
TROPOSPHERIC REFRACTION

50 100 150 200 250 300 350

Ns[l-exp(ch)] IN N UNITS

Figure 3.17. Graphic representation of Ns[l — exp( —ch)] in N units versus height.

3.11. A Graphical Method

Weisbrod and Anderson [16] present a handy graphical method for computing refraction in the troposphere. Rewriting and enlarging (3.11), one obtains

T(mrad) = X)

N,- N,k+\ S 500 (tan 6*^ + tan Ou+iy

(3.49)

where r will be the total bending through n layers. Terms for the denominator can be determined from figure 3.18. Equation 3.49 is essentially Schulkin's result with only the approximation, tan 6k = Ok, for small angles, omitted.

The procedure for using figure 3.18 follows. Enter on the left margin

at the appropriate Ns — N{h). Proceed horizontally to the proper

height, h, interpolating between curves if necessary. Use the solid height curves when Ns — N{h) is positive and the dashed curves when Ns —

N{h) is negative. Then proceed vertically to the assumed 6o and read

500 tan 6 along the right margin.
GRAPHICAL METHOD 81
82 TROPOSPHERIC REFRACTION

3.12. Derivations

The approximate relation between 6i and 62 is derived here. This rela- tion holds for small increments of height and small ^'s. The relationship was used in making all sample computations in preceding sections. Since for small ^'s

2 2 cos di^l - I

and cos ^2 ^ 1 - | , (3.50)

and knowing

r2 = ri + Ah (fig. 3.19)

then substituting in (3.1) yields

n,(n + Ah)(^ 1 -

i ) - ^ ~2 ) ' ^^-^^^

or

n^ri + UiAh — UiTi — — n^Ah -~ = niVi — niri . (3.52)

Dividing by Vi

. 712 A/l n2^2 Ah $2 ^ 0\ /Q rQ\

712 H -TT - ^2 — — ^ ni - ni -- . (3.53)

ri 2 ri 2 2

Since the term —712 (A/i/^i) (^2/^) is small with respect to the other

terms of (3.53) it may be neglected, and thus:

2 2

, 712 A/i ^2 ^ B\ /o

7i2 + —— - 712 7>- = ni - 7li - (3.54)

ri 2 2

or

2 — '"^ 2 ri

,,2 1^ ^ -7ii - + (m - 712). (3.55)

If one now divides both sides of (3.55) by 712 and assumes

{7ii — 7i2)/7i2 =711 — 712 and ni/7i2 = 1, (3.55) may be arranged to yield
DERIVATIONS 83 Writing (3.56) in terms of N units,

^2(mrad) ^ Jd' + — X lO' - 2(A^i - N2) (3.57)

if 61 is in milliradians.

Generalizing (3.57) for the A;th and the (A: + l)st layers,

^,+i(mrad) ^ J^?.(mrad) +

^'-^ ^

X lO' - 2(N, - N,^,), (3.58)

Also from the geometry shown in figure 3.19, a useful relationship for

Ti,2 can be obtained. Tangent lines drawn at A and B will be respectively perpendicular to ri and r2, since ri and r2 describe spheres of refrac- tive indices rii and 712 concentric with 0. Therefore,

angle AEC = angle AOB = <i>

also, in triangle AEC

angle ACE = 180° - angle CAE - angle AEC

= 180° - ^1 - </). (3.59)

But from triangle DCB

angle ^C^; = angle DCB = 180° - ri.2 - 02. (3.60)

Thus

or 180° - T1.2 - ^2 = 180° - di - cf>,

ri,2 = 0 + (^1 - 62). (3.61)

Now since 0 in radians = d/a, where d is distance along the earth's surface

:

T1.2 = - + (^1 - 62), (3.62)

CL

or the bending of a ray between any two layers is given in terms of the

distance, d, along the earth's surface from the transmitter (or receiver),

the earth's radius, a, and the elevation angles 61 and 62 (in radians) at the

beginning and end of the layer.
84 TROPOSPHERIC REFRACTION

Figure 3.19. Bending geometry on a spherical earth with concentric layers.

If one considers figure 3.20, Snell's law in polar coordinates (3.1) and

the refraction formula (3.2) may be obtained from the more familiar

form of Snell's law.

Assume that the earth is spherical and that the atmosphere is arranged

in spherical layers. In figure 3.20 let C be the center of the earth, 0 the

observer and COZ the direction of his zenith. Let n and n-\-dn be the

indices of refraction in two adjacent thin layers M and M'. Let LP be

the section of a ray in M' which finally reaches the observer at 0. At

F it is refracted along PQ. Similarly, it is refracted at the surfaces between successive layers and the final infinitesimal element of its path is TO.

Draw the radii CP and CQ. Let angle PQF = 6, angle LPS = d + dS, and angle QPF = xp. Then, since the radius CP is perpendicular at P to
DERIVATIONS 85

Z

Figure 3.20. Geometry for the derivation of SnelVs Law in spherical coordinates.

the bounding surface between layers M and M', by Snell's law we have

(n+rfn) sin [90° - {d+dd)] = n sin i/^. (3.63)

Now from the triangle CQP"^, in which CQ = r and CP = r -\- dr, and

2 The assumption involved in this triangle is that the path of the ray in M' is a

straight hne, which, of course, can only be true in an isotropic medium. Hence, it can only be true for an infinitesimal layer in the troposphere. Thus only a differential form of Snell's law, (3.65), in polar coordinates, can be obtained by the use of the geometry of figure 3.20; not the finite form, (3.68), which has the same appearance.
86 TROPOSPHERIC REFRACTION

angle CQP = 90° + 6, we have, from the law of sines,

r sin (90° + ^) = r sin (90° - 6) = {r + dr) sin i/^. (3.64)

Eliminating sin xj/ from (3.63) and (3.64) we then have

in + dn){r + dr) sin [90° - (<9 + dd)] = nr sin (90° - 6), or

(n + dn){r + <ir) cos (^ + dd) = nr cos 6. (3.65)

Multiplication of the (n + c?n) and (r -f rfr) terms, ignoring differential products, yields

(nr + nrfr + rdn) cos (^ + dd) = nr cos ^, or

(nr + nrfr + rdn) [cos ^ cos (c?^) — sin 6 sin (d^)]= nr cos ^. (3.66)

Since cos {dd) = 1, and sin (dd) = dd, another multiplication, again ignoring products of differentials, yields

ndr cos 6 + rdn cos 6 — nr sin 6 dd ^- 0,

or, dividing all terms by nrcos d,

-- + --- tan ddd = 0. (3.67)

r n Now if (3.67) is integrated between any two thin layers of refractive

indices ni and n2, whose radial distances from the earth's center are ri and

r2, and the initial elevation angles of a radio ray entering the layers are di and d2:

f"' dr , h dn f^' ^ ^ ,^ . ^2 , , , , cos d^ ^

/ — + / — — / tan ddd = h ^n — + ^n = 0

Jr, r Jn, n Je. n ni cos ^1

or, taking ahtilogs of both sides,

r2n2 cos d2 _ J rini cos di

'
REFERENCES 87 whence

nifi cos di = UoTo cos 02, (3.68)

which is Snell's law^ for polar coordinates, (3.1). In figure 3.20, it can be seen that

''^' = § = ^^' (3.69)

where <f> is the angle at the earth's center between r and COZ. Substituting (3.69) in (3.67)

dti tan dd(j) + — — tan Odd = 0, n

or

(d<j> - de)tsind = - ^ . (3.70)

Since, by considering (3.61) for infinitesimal angles,

dr = d<t) — do,

or, in (3.70)

dr tan B = — ~ , n

or

dr = -cote — . (3.71) n

Integration of (3.71) yields (3.2).

3.13. References

[1] Smart, W. M. (1931), Book, Spherical Astronomy, Ch. 3 (Cambridge Univ.

Press, London, England).

[2] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric re- fractive index, Proc. IRE 47. No. 5, 740-755.

[3] Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric

refraction and its relation to wave guides and diffraction, Book, Meteorological

Factors in Radio-Wave Propagation, pp. 80-127 (The Physical Society,

London, England).

[4] Freehafer, John E. (1951), Tropospheric refraction, Book, Propagation of Short Radio Waves, pp. 9-22 (McGraw-Hill Book Co., Inc. New York, N.Y.).

[5] Bean, B. R., and B. A. Cahoon (Nov. 1957), The use of surface weather observations to predict the total atmospheric bending of radio waves at small elevation

angles, Proc. IRE, 45, 1545-1546.
88 TROPOSPHERIC REFRACTION

[6] Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere,

Proc. IRE 40, 554-561.

[7] Anderson, L. J. (Apr. 1958), Tropospheric bending of radio waves. Trans. Am.

Geophys. Union 39, 208-212.

[8] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave

propagation, Proc. IRE 21, 427-463.

[9] Stickland, A. C. (1947), Refraction in the lower atmosphere and its apphcation to the propagation of radio waves. Book, Meteorological Factors in Radio Wave

Propagation, pp. 253-267 (The Physical Society, London, England).

[10] Humphreys, W. J. (1940), Book, Physics of the Air, p. 82 (McGraw-Hill Book

Co., Inc., New York, N.Y.).

[11] Norton, K. A., P. L. Rice, and L. E. Vogler (Oct. 1955), Use of angular distance

in estimating transmission loss and fading range for propagation through a

turbulent atmosphere over irregular terrain, Proc. IRE 43, 1488-1526.

[12] The Rocket Panel (1952), Pressures, densities, and temperatures in the upper

atmosphere, Phys. Rev. 88, 1027-1032.

[13] Handbook of Geophysics for Air Force Designers (1957), Geophysics Research

Directorate (Air Force Cambridge Research Center, ARDC, USAF).

[14] Dubin, M. (Sept. 1954), Index of refraction above 20,000 feet, J. Geophys. Res.

59. 339-344.

[15] Bean, B. R., and E. J. Dutton (May-June 1960), On the calculation of departures

of radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3, 259-263.

[16] Weisbrod, S., and L. J. Anderson (Oct. 1959), Simple methods for computing

tropospheric and ionospheric refractive effects on radio waves, Proc. IRE 47, 1770-1777.
Chapter 4. N Climatology

4.1. Introduction

The contents of this chapter include a study of the surface variation of the radio refractive index on a worldwide scale in terms of a reduced-tosea-level form of the index that gives a significantly more accurate description of refractive index variations than the nonreduced form.

The mean vertical structure of the refractive index parameter in the

troposphere over central North America is presented, again in terms of a reduced-to-sea-level form of A^. A climatological treatment of the phenomenon of the atmospheric duct,

or waveguide, and associated fading regions is also presented. The

chapter is concluded with a discussion of refraction of radio waves in various air masses. It is demonstrated that refraction differences within

air masses arise from departures of refractive index structure from normal.

4.2. Radio-Refractive-Index Climate Near the Ground

4.2.1. Introduction

The radio refractive index of air, n, is a function of atmospheric pressure, temperature, and humidity, thus combining in one parameter three

of the normal meteorological elements used to specify climate. In the

following sections we will examine the variability of n during different seasons of the year and in differing climatic regions. The systematic

dependence of n upon station elevation will make it necessary to consider a method of expressing n in terms of an equivalent sea-level value in order to see more clearly the actual climatic differences of the various

parts of the world. After a consideration of the n climate of the world,

the application of this information to such practical problems as the

prediction of radio field strength and the refraction of radio waves will be

discussed.

4.2.2. Presentation of Basic Data

Near the surface of the earth, for VHF and UHF frequencies, n is a number of the order of 1.0003. Since, for air, n never exceeds unity by

89
90 N CLIMATOLOGY

more than a few parts in 10^, it is convenient to consider the climatic

variation of n in terms of N = (n - 1)106, (4.1)

as defined in chapter 1. The notation Ns is used to indicate that (4.1)

has been evaluated from standard surface weather observation.

To obtain long-term average values of N, one should properly average

individual observations over many years. This is difficult to do since, in general, only summaries of weather observations are readily available.

However, long-term average values of temperature, pressure, and

humidity are available and may be converted into an ''average" value

of A^. This ''average" differs from the true average since the inter- correlation of pressure, temperature, and humidity is neglected. This

difference was examined by an analysis of 2 years of weather records of the months of February and August at an arctic location (Fairbanks,

Alaska), a temperate zone location (Washington, D.C.), and a tropical

location (Swan Island, W.L). These data, given in table 4.1, indicate

that the difference between the two methods was never more than 1.5 A" units and that the average difference was less than 1 A^ unit, which is small

compared to commonly observed seasonal and geographic variations of 20 to 100 A^ units. Table 4.1. Two-year average value of Ng versus the value of Ns calculated from average temperature, pressure, and humidity

Ns (P, T, RH) Ns-N> {P, T, RH)

Fairbanks: February 314.0 313.0 1.0 August... . ... . 320.5 320.0 0.5 Washington:

February 305.5 340.5 1.0 August... 356.0 354.5 1.5 Swan Island: February . . 362.0 362.5 0.5 August .. ... 387. 5 388.0 0.5

Average 0.83

On this basis it was decided to use the long-term means given in the

United Nations' monthly publication. Climatic Data for the World. This

publication is particularly advantageous for our present study since it re- ports the fictitious value of the relative humidity needed to obtain the

actual average vapor pressure from the saturated vapor pressure of the

reported mean temperature [1].^

^Figures in brackets indicate the literature references on p. 170.
PRESENTATION OF BASIC DATA 91 Data from 306 weather stations were obtained in order to give reasonable geographical coverage. In general, 5 years of records were obtained

for each station from the period 1949 to 1958, preference being given to the years 1954 through 1958. A noticeable exception, however, was

Russia, for which only 1 year of data (IGY) is reported in Climatic Data

for the World; thus all charts are drawn with dashed contours for Russia.

There are vast expanses of ocean for which there are no meteorological

observing stations. Climatic atlases were utilized in order to present

estimates of world climate in these locales. A reasonable coverage of the sparse data areas of the world was made by estimating temperature

from sea surface isotherms [2] and humidity from charts of seasonal

average depression of the wet bulb temperature [3]. Pressure was esti- mated for these locations from average winter and summer pressure

charts. When these data were converted to [4] and charts prepared, a pro- nounced altitude dependence could be seen, as in figure 4.1. Figure 4.1 and the following charts of N variations across the United States are from

an extensive climatology now being prepared at the Central Radio

Propragation Laboratory. Although the present study is primarily

aimed toward worldwide variations, it is felt that the U.S. data better

illustrate the height dependence of Ns and the subsequent reduction

process employed. It is noted that the coastal areas display high values of Ns, while the inland areas have lower values. There are low values of Ns corresponding to the Appalachian and Adirondack Mountains and a

decrease with increasing elevation of the Great Plains until the lowest

values are observed in the Rocky Mountain region and the high plateau

area of Nevada. A corresponding gradient is observed from the west

coast eastward. Crosshatching encloses areas where the terrain changes

so rapidly that it was felt the data were inadequate to obtain realistic contours of Ns. The altitude dependence of N can be studied in terms of the ''dry" and

'Vet" components of A^. These components are those of the two-term

expression in (1.17). The dry term, D,

is proportional to air density and normally constitutes at least 60 percent

The average variation of density with altitude in the atmosphere may

be expressed in the first approximation as

D =

77.6 P

T

(4.2)

of A^.

P = Po exp { —z/H] (4.3)
92 CLIMATOLOGY

Figure 4.1. Mean Ng, August 0200 local time.

where z is the altitude, po the average sea level density of moist air and

H the average scale height between zero and z. It is useful to introduce

the concept of an effective scale height, H*, for the average variation of refractive index in the atmosphere. Many studies have shown that the

average refractive index variation with height is quite well represented,

to a first approximation, by a formula similar to (4.3) [5, 6, 7]. It is possible to calculate a theoretical value of this effective scale height using a distribution of water vapor. This is, however, quite a complex procedure. Furthermore, the value obtained depends upon the model of the

water vapor distribution, and no definite conclusion can be justified

considering the extreme variation of water vapor concentration with

season, geographic location, and height above the earth's surface. A

convenient and simple alternative is to adopt a value for i/* from the

average (n — 1) variation with height in the free atmosphere. Several

such values of H* were determined by reference to the NACA standard

atmosphere [8] and recent climatological studies of atmospheric refractive

index structure [9]. It is seen from table 1.8, chapter 1, that varies from 6.56 to 7.63

km in the NACA standard atmosphere, depending on the value of rela- tive humidity assumed. The value of H* = 6.95 km obtained from

climatological studies oi (n — 1) variations over the first kilometer above
PRESENTATION OF BASIC DATA 93 the earth's surface from nearly 2 million radiosonde observations from

many diverse climates.

At the time of analysis of the map series presented here, a value of

effective scale height, i/* = 9.46 km, was in use. This form was

Do = Ds exp jj^l

z ) (4.4)

in which the value H* = 9.46 km was determined from the NACA dry

standard atmosphere. A Do chart is shown on figure 4.2, presenting a

gradient that is remarkably free of detail as compared to the Ns chart of

figure 4.1, and is easily drawn for all areas of the country.

An investigation of the elevation dependence of the surface wet term = 3-^3 X 10- e (^^5^

revealed low correlations of log Ws and height, indicating that Ws is not

a marked exponential function of elevation. Contours of Ws for all sections of the country are shown on figure 4.3. The maps that follow are completed in terms of a single reduced form

No ^ (D. + TF.) exp j~[ = exp j^j, (4.6)

where Ns is reduced by the dry term effective scale height, H* = 9.46 km.

Figure 4.4 gives the No contours for the same time as the previous maps

of Do and Ws. The No maps are no more difficult to prepare than the Ws

maps and have effectively removed the station height dependence of Ns. One might wonder at the advisability of arbitrarily reducing the wet term

by the dry term correction. For the coastal areas of the country, where

the exponential height correction factor is nearly unity, this amounts

simply to adding the Do and Ws maps for the mountain areas. Where the

height correction factor is large, the Ws values are small with the result that the gradient of the N isopleths obtained from the Do and Ws maps is essentially maintained on the No maps. As an example, for the series of maps under discussion, the (Do -\- Ws) difference between Reno, Nev.

(1,340 m elevation), and Oakland, Calif. (5.5 m elevation), is 21 units,

while the No difference is 19 N units. The effects of this correction on the worldwide values can be seen from

figure 4.5, where Ns is plotted versus station elevation in kilometers. A

sample line illustrates the decay of Ns with height for No = 348. The

value of A^s for any other value of iVo would be obtained from a line parallel to the A^o = 348 line but having a zero intercept equal to the new value of
94 CLIMATOLOGY

Figure 4.2. Mean Do, August 0200 local time.

Figure 4.3. Mean We, August 0200 local time.

I
PRESENTATION OF BASIC DATA 95

Figure 4.4. Mean No, August 0200 local time.

No. The advantage of adopting A^o is illustrated by the reduction in range from 190 X units for Ns to 115 iV units for iVo, thus diminishing

the number of contours of the resulting maps.

It would appear that by removing the influence of station elevation it would be more efficient to estimate Ns from .Vo charts rather than from

charts of Ns. As a test of this hypothesis, Ns and A^o contour charts were

prepared for both summer and winter from only 42 of the 62 U.S. Weather

Bureau stations for which 8-year means of Ns are available. The remaining 20 stations, distributed at random about the country, were used as a

test sample by estimating their 8-year mean value of Ns from the A^o and

Ns contours. Summertime examples of these charts are given by figures 4.6 and 4.7. Xote that due to the reduced range of A^, the A^o charts are drawn every 5 A^ units as compared to the lO-A'-unit contours of the N

s

charts. The individual deviations of the values obtained from the contour maps with the actual 8-year means are listed in table 4.2. By comparing the root mean square (rms) deviations of 10.7 A" units in winter

and 13.0 A^ units in summer obtained by estimating Ns from the Ns contours with the 2.7-A^-unit rms of estimating Ns from A'o contours, one

concludes that it is at least 4 times more accurate to estimate Ns from A^o contours than from those of Xs. An inspection of the individual deviations in table 4.2 indicates that the A^o contour method is particularly

efficient at elevations in excess of 1,200 m or where the terrain is changing
N CLIMATOLOGY

Figure 4.5. Worldwide values of Ng versus height for August.
Figure 4.7. Test chart of mean No August 0200 local time.
98 N CLIMATOLOGY

rapidly with respect to horizontal distance. As a further practical consequence, one notes the remarkable similarity between the A^o contours

of figures 4.4 and 4.7, even though the latter contours were derived from

only two-thirds of the original data. This indicates that any desired level of accuracy may be maintained with fewer stations (and less expense) by

the use of No. Table 4.2. Deviations of estimated 8-year means of Ng (calculated from contour charts) from actual 8-year means selected at random from 20 Weather Bureau stations

Test station Height

February 1400 August 0200

Actual

8-yr value

of iVs

Deviation* Actual

8-yr value

of Ns

Deviation Ns map

No mapNs map

No map

meters N units N units N units N units N units N units

7 315.6 7.6 0.8 329.6 1.0 1.8 Portland, Greg . . . 8 316.2 4.2 -.5 337. 7 19.7 6.0 San Diego, Calif 11 314.2 -5.8 -2.4 348.1 16.1 3.5 Mobile, Ala 66 326.6 6.6 4.8 376.0 6.0 0.6 Fresno, Calif 86 310.6 9.6 3.4 326.2 5.2 4.2 Boston, Mass 89 308.6 -6.4 -0.5 347.5 -7.5 -0.4 Grand Rapids, Mich 210 304.4 0.4 -5.9 340.5 -1.5 .1 Columbia, Mo 239 300.8 -.2 2.4 348.7 -2.3 -2.5

Minneapolis, Minn 255 301. 1 . 1 0.6 338.5 -0.5 2.7 Cincinnati, Ohio..- 271 302.5 -.5 .7 344.

1

-2.9 -2.8 Des Moines, Iowa 294 300.9 3.9 2.3 343.1 -1.9 -0.1

Pendleton, Oreg. 455 295.9 1.9 0.4 300.9 2.9 -3.1

Billings, Mont- 1,088 269.3 -2.3 . 1 285.6 5.6 1.2 Burns, Oreg 1,262 268. 1 -23.9 -3.2 271.3 -15.7 -4.4

Salt Lake City, Utah 1,288 266.3 1.3 -0.8 279.5 8.5 4.4 Reno, Nev 1,340 259.6 -20.4 -6.8 277.6 -29.4 1.6

Pocatello, Idaho. 1,355 264.7 -2.3 0.4 269.7 -3.3 0.0 Denver, Colo . 1,625 244.9 -8.1 .7 276.6 -1.4 .3 Colorado Springs, Colo 1,882 237.

1

-15.9 -.6 272.4 -6.6 1.1

Flagstaflt, Ariz 2, 131 237.8 -26.2 2.3 261.4 -36.6 -2.2

Root mean square deviation . 10.7 2.7 13.0 2.7

* Deviation equals the actual long-term mean minus the value obtained from map contours.

4.2.3. Worldwide Values of

Mean values of A^o were calculated at each of the 306 selected stations

and charts were prepared for each month of the year. The charts for February and August are given on figures 4.8 and 4.9. It is seen that

the values of A^o for sea-level stations vary from 390 in the maritime

tropical areas to 290 in the deserts and high plateaus. The interiors of continents and mountain chains in middle latitudes are reflected by low

values as compared to those of coastal areas. Further, such pronounced

climatic details as the Indian monsoon and the effects of coastal mountain

ranges blocking prevailing winds and producing rain shadows are indi- cated by these A^o contours. Regional climatological data of A^o for the

United States are given in chapter 9.
i WORLDWIDE VALUES OF A^o 99

Figure 4.8. Worldwide values of mean No /or February.

Figure 4.9. Worldwide values of mean No /or August.
Figure 4.11. Minimum monthly mean value of No.
Figure 4.13. Year-to-year range of monthly mean Ns/or August.
102 N CLIMATOLOGY

The annual variation of A^^ is indicated on figure 4.10 by contours of the difference between the maximum and minimum monthly means observed throughout the year. It is quite remarkable how clearly climatic

differences are evidenced by the yearly range of Ns. The prevailing

transport of moist maritime air inland over the west coasts of North

Am^erica and Europe is indicated by relatively small annual ranges (20 to 30 N units), while, for example, the east coast of the United States with a range of 40 to 50 units or more reflects the invasion of that area from

time to time by such diverse air masses as arctic continental and tropical

maritime. The largest annual ranges of Ns (90 N units) are observed in the Sudan of Africa and in connection with the Indian monsoon. An additional A^o map (fig. 4.11) was prepared from the minimum

monthly mean value of A^^ observed throughout the year to supplement

the range map in order that an estimation might be made of both the

minimum and maximum monthly mean A^^ expected durmg the year. A measure of the variability of the February and August mean values

of Ns is given by monthly range maps (fig. 4.12 and 4.13) determined

from monthly averages from 5 years of data. Ranges are given by the

maximum difference of the five individual monthly mean values. In contouring the two variability maps only those terrestrial regions having

reasonable data coverage are included. Dashed contours are shown for areas of sparse or unreliable data. The general picture of the worldwide

distribution of Ns variability is that of a number of continentally located

cells of moderate range accompanied by somewhat random small-scale variation over ocean areas. Regions of large range, from 40 to as much

as 70 N units, are present, however, in Australia and on islands of the

adjoining oceans, on the African equatorial plateau near the Cameroons,

and in the Great Basin of the southwestern United States. Common to

all these areas of large year-to-year variability, at least during the summer

season, are high mean temperatures ranging from about 25 to 30 °C, the

variability being due to relatively small variations of humidity. It is felt that when a more dense network of stations is available for a longer period

of record, say 10 years, areas of high monthly variability are likely to be more extensive in tropical and desert areas than indicated on our present

maps.

4.2.4. Climatic Classification by

The annual cycle of A^s at each station was examined for the purpose

of deriving similarities of climatic pattern. As one form of climatic

classification, the annual mean value of A^s at each station was plotted versus the annual range at the station. When this was done, several

distinct groupings of data seemed evident. These groupings, described
CLIMATIC CLASSIFICATION BY No 103 in table 4.3, are intended to give a general idea of the geographic and

climatic character of the majority of the stations found within given

values of range and yearly mean of A^sTable 4.3. Characteristics of climatic types

Type Location Annual mean N,Annual

range of Ns

Characteristics

I. Midlatitudecoastal. Near the sea or in lowlands on

lakes and rivers, in latitude

belts between 20° and 50°. N units 300 to 350 N units 30 to 60 Generally subtropical with marine or modified marine climate.

II. Subtropical- Savanna. Lowland stations between 30°N and 25°S, rarely far from the

ocean.

350 to 400 30 to 60 Definite rainy and dry

seasons, typical of Savanna climate.

III. MonsoonSudan. Monsoon—generally between

20° and 40°N, Sudan— across

central Africa from 10° to 20°N

280 to 400 60 to 100 Seasonal extremes of

rainfall and temperature. IV. Semiarid- mountain.

In desert and high steppe regions

as well as mountainous re- gions above 3,000 ft.

240 to 300 0 to 60 Year-round dry climate

V. ContinentalPolar. In middle latitudes and polar

regions. (Mediterranean cli- mates are included because of the low range resulting from

characteristic dry summers.)

300 to 340 0 to 30 Moderate or low annual mean temperatures.

VI. Isothermalequatorial. Tropical stations at low elevations between 20°N and 20°S, almost exclusively along sea- coasts or on islands.340 to 400 0 to 30 Monotonous rainy

climates.

For a given classification of refractive-index climate, diverse meteorological climates and geographical regions may be represented. Note,

for example, that type V of table 4.3 includes stations from Mediterranean

and marine as well as polar climates. Mediterranean stations in this category fail to attain a high range because of the characteristic dryness

of the subtropical high-pressure pattern that is generally found in this area during the summer months. Polar and marine climates in this group maintain a low range due to suppressed humidity effects as a result of low to moderate year-around average temperatures.

Annual trends of Ns for stations typical of each climatic division are shown by figure 4.14. Yet another facet of the climate is the year-to-year variation of the

monthly mean value of Ns. Five consecutive years of monthly means

were prepared for each of the six typical stations whose annual cycles are shown in figure 4.14. Then, for each month, the absolute value of the

difference between consecutive years was obtained. These values were

then averaged for all months and are listed in the second column of table

4.4.
104 N CLIMATOLOGY

400

380

360 340

320 300

280

260

240

400 380

360 340

320 300

280 260

240

TYPE I MID-LATITUDE COASTAL WASHINGTON. D.C.

I I

I I II

I I I I I I I I I I TYPE m

SEMI-ARID MOUNTAIN

DENVER, COLO.

I I I I I I I I I I I

TYPE H

SUBTROPICAL - SAVANNA

MIAMI, FLA.

TYPE m

MONSOON -SUDAN

JODPHUR. INDIA

I I I I I I

TYPE 2 TYPE 3Zr CONTINENTAL -POLAR ISOTHERMAL EQUATORIAL

OSLO, NORWAY CANTON ISL., S. PACIFIC 0.

I

I

I I I

I I I I I I I I I I I

I 1 I

I I I I I I JFMAMJJASONOJ JFMAMJJASONDJ JFMAMJJASONDJ

Figure 4.14. Representative annual cycles of Ns /or the major climatic types.

Another measure of the variation of monthly mean values of Ns is obtained by differencing the maximum and minimum values occurring for a given month during the 5-year period. These differences are also given

in table 4.9 for the months of February, May, August, and November.

Table 4.4. Year-to-year differences of monthly mean Ng

Maximum differences between monthly

Differences between monthly means means over a 5-yr period Climatic* in successive years for the same

type month, averaged for all seasons over

a 5-yr period Feb. May Aug. Nov.

5.7 6.0 16.5 17.0 7.0

II 5.4 8.5 6.5 8.0 11.0

III 8.9 16.5 14.5 20.0 6.5 IV 5.4 10.5 11.5 13.5 6.5 V 4.7 5.5 11.5 10.0 5.0 VI 7.1 9.5 25.5 8. 5 8.5

*Climatic types are the same as those in table 4.3.
APPLICATIONS 105 4,2.5. Applications

The communications engineer usually has available a small amount of measured field-strength data from limited tests of a particular system.

He must then estimate the expected signal level or practical range of that

system, or other systems, for other tmies of the year, other years, and in other areas. The variation of signal level from month to month and

climate to chmate can be explained, in part, by its observed correlation

with Ns. Pickard and Stetson [10, 11] were among the first to note the correlation

of Ns and received field strengths. The correlation of Ns and field strength over a particular path has been studied quantitatively [12, 13] and found to be highest (correlation coefficients of 0.8 to 0.95) when the

variables are averaged over periods of a week to a month. This latter study has shown that the regression coefficient (decibel change in field strength per unit change in N' s) varies diurnally from 0.14 dB in the after- noon hours to 0.24 dB per unit change of Ns in the early morning hours.

This correlation is so sufficiently consistent that Gray [14] and Norton [15] have utilized it in their recent prediction methods of transmission loss in a band from 100 to 50,000 Mc/s. In addition, the coefficient 0.2 dB per

unit change in Xs has been tentatively adopted by CCIR Study Group V

in their revision of the 30- to 300-Mc/s tropospheric-wave propagation

curves to account for the geographic and seasonal variations of field strengths. The estimates of field strength variations attributed to Ns

given in table 4.5 are based upon the CCIR coefficient.

If one assumes, for comparison only, that the worldwide average value

of Ns is 330 and that one is able to estimate the field strength level of a particular communications system at a given distance and for A^^ = 330,

then the above correlations would indicate that the climatic variations of

fields given in table 4.5 might be expected.

Table 4.5. Climatic variation of hypothetical communications system relative to predicted value for Ns = 330, assuming a 0.2-dB variation per unit change in Ns

Climatic type* Expected yearly mean field strength

level relative to Expected annual

range on the above assumption

II-.

III. IV.

v..

VI.

I. -6 to +4

+4 to +14

-10 to +14

-18 to -6

-6 to +2

+2 to +14

dB dB

6 to 12

6 to 12

12 to 20

0 to 12 0 to 6

0 to 6

* Climatic types are the same as those in table 4.3.
106 N CLIMATOLOGY

The data of table 4.5 indicate, for example, that identically equipped

tropospheric communications systems could display as much as a 32-dB

difference in mean signal-strength level due to the climatic difference of say, Denver, Colo., and the tropics. Further, one might expect the

monthly mean field strength of this hypothetical system to vary throughout the year from less than 12 dB in the high plains near Denver to as much as 20 dB in the African Sudan.

Under this same assumption, figures 4.10 and 4.11 allow the communications engineer to estimate the expected maximum and minimum

monthly mean field strength expected throughout the year. The year-to-year variations of the monthly mean Ns listed in table 4.4 indicate that the monthly mean of field strength for a particular month

may differ in successive years by as little as 1.0 dB for climatic category

V in November or as much as 5.1 dB for category VI in May.

Another application of these worldwide charts is to aid in estimating the

refraction of radio waves. The most convenient method of accounting

for the effects of atmospheric refraction is by means of the effective- earth's-radius concept (see chapter 3) of Schelleng, Burrows, and Ferrell

[16]. The effective earth's radius, ae, is determined from

where a is the true radius of the earth, n is the refractive index, and

dn/dh is the initial n gradient with respect to height. A great simplification of propagation calculations is accomplished by assuming that dn/dh

is a constant, thus allowing radio rays to be drawn as straight rays over a

fictitious earth of radius a e rather than curved rays over the true earth of radius a. This simplification allows, for example, the distance to the

radio horizon, d, of a radio ray leaving an antenna of height, h, to be

calculated from d = 2ae h. One notes, however, that the determination of involves dn/dh as well as n and that our A^o charts allow only an estimation of n. This dis- parity may be resolved by utilizing the observation that Ns is highly

correlated with the value of A" at 1 km above the surface. The difference

between Ns and at 1 km is denoted AN. It has been noted [9] that the

correlation coefficient between ^n|AA| and Ns is 0.926 for 888 sets of data from 45 U.S. weather stations representing many diverse climates.

The regression equation

results when both variables are averaged over 6 to 8 years of record.

(4.7)

-AN = 7.32 exp {0.005577 JV,| (4.8)
CRITICAL APPRAISAL OF RESULTS 107 Approximating dn dh in (4.7) by we may determine that the radio

horizon distance of an antenna located 150 m above the earth would vary

from 48 km when Ns = 200 to 59 km when Ns = 400. Yet another application of the Ns charts is to the exponential models of the decrease of refractive index Tvdth height which have been proposed to date [9, 17]. These models are completely specified by Ns and may be used to account

for seasonal and geographic variations of such refraction effects as radar

range and elevation angle errors.

4.2.6. Critical Appraisal of Results

The world maps presented above were based upon data from 306

weather stations. This number of stations appears to be consistent with

the scale of map used. The map scale is so small, however, that only

large climatic differences can be expected to be discerned. For the

climate of any given area one should refer to detailed studies of N such

as those currently in preparation for the Ignited States at the Central

Radio Propagation Laboratory.

The accuracy of the present charts may be assessed from the charts of maximum range, R, of monthly means as given by figures 4.12 and 4.13. The standard deviation of the individual monthly means may be esti- mated from [18] 0.43 R, where the coefficient 0.43 is appropriate for five individual observations. Since, in general, R < 20 units, then

0.43i? < 9.V units, although this standard deviation may be as large as 26

N units for the month of February in Australia and 17 N units in the

southwest of the United States during August, or in the African Sudan

during February.

Further, the standard error of estimating a 5-year mean from five individual monthly values is determined from

0.43 R

) \/n

where n for our case is 5 and thus the error of the 5-year mean would be

0.192 R. Remembering that R < 20 N units and assuming perfect skill in drawing the contours, one would expect the standard error of estimate

to be less than 4 A" units. This standard error can be as large as 12 N

units in Australia where R = 60 N units. The value of N' s at each of the 20 test stations of table 4.2 was estimated

from the No contours with an rms error of 5 A units which is consistent

with the standard error of estimate obtained from the range charts. In

the large areas of sparse data, such as the oceans and Russia, this uncertainty rises to about 10 .V units and thus the contours in these regions

are dashed.
108 N CLIMATOLOGY

At the time the present study was initiated it was felt that Ns should

be reduced to sea level by at least the dry term correction factor as in

(4.6). The absence of published work on models of structure in the

free atmosphere encouraged the decision to rest on prudence and adopt

this dry term correction factor. Since that time several effective ex- ponential models of the free atmosphere have been demonstrated [17, 9]. In future work a smaller value of H* on the order of 7.0, which corresponds

to the N decay in the free atmosphere, will be adopted. The adoption

of any value of H* between 6.5 and 7.5 km would have reduced the range

of A^o values on figure 4.4 by no more than 2 units. Since this reduction

in range is more than an order of magnitude less than the reduction of

(4.6) used to obtain A^o, it appears that the basic advantage of the method

has been realized. A map of No, such as that of figure 4.4, which represents a large continental area, may easily be compared and merged with maps for ocean

areas. This would be more difficult with Ns, since, for example, the

strong gradient over California mainly represents the rapid altitude

variations of the Sierra Nevada Mountains. It has also been demonstrated that A^o is a better indicator of tropospheric storms and air masses than Ns when considered on a synoptic or "daily weather map"

basis [19]. Also (4.6) aids in comparing air properties as a function of altitude at the same position. The variations of A^o will represent the local departures between this quantity and the value in a standard atmosphere and

will show the perturbations in the structure of the atmosphere produced

by fronts, air masses, and other synoptic features. Although any value of H* between 6.5 and 7.5 km will remove the gross altitude dependence of the refractive index, the choice of value of i7* within this range could depend on the application. The synoptic application, which is discussed in detail in chapter 5, would be best served by a scale height near 8.0 km,

whereas the objectives of the climatic chart usage would best be met by

a scale height near 6.5 km. This seeming paradox is easily understood in terms of the physical interpretation of the various scale heights. For

example, i/* = 8.0 km is typical of a perfectly dry atmosphere and its use

results in emphasis of humidity differences between air masses, whereas ^* = 6.5 km corresponds to a saturated atmosphere and minimizes

moisture differences. Thus it would appear that eventually one might

use a value of H* as indicated by the application at hand. As a practical

matter, however, H* = 7.0 km appears to reach a desirable compromise

between the objectives of the two preceding examples.

r 4.2.7. Conclusions

With the above critical appraisal in mind, the salient conclusions of the

present study are

:
DATA AND REDUCTION TECHNIQUES 109

(a) The radio refractive index varies in a systematic fashion with

climate and different climates may be identified by the range and mean

values of the refractive index.

(b) It is 4 or 5 times more accurate to estimate the station value of the

index from charts of the reduced-to-sea-level index than from charts of the station value. This improved accuracy results from using a method

that allows height dependence to be accurately taken into account.

(c) Identically equipped tropospheric communications systems might

be expected to vary as much as 30 dB in monthly mean signal level in different climatic regions, and the annual range of monthly mean field strength could be as high as 20 dB in the Sudan of Africa and as low as 0 to 6 dB in the high plains of the western United States.

4.3. On the Average Atmospheric Radio Refractive

Index Structure Over North America

4.3.1. Introduction

As has been already noted, the radio refractive index of the atmosphere

combines three of the meteorological elements normally used to specify the

state of the atmosphere on either a synoptic or a climatological basis. This fact has led to its being used as a synoptic tool [19, 20, 21, 22, 23] and

as a measure of climatic characteristics [7, 24, 25, 26, 27]. The present treatment is concerned with the degree to which the average

N structure in the vertical direction reflects the gross differences in climate over the North American continent. Diurnal and seasonal range graphs

of N at the earth's surface also shed light upon climatic characteristics.

4.3.2. Meteorological Data and Reduction Techniques

The basic data used in this study are the significant level data of the

radiosonde observations from the 18 weather stations shown on figure 4.15

for the 5-year period 1952-57. These observations were converted by means of (1.20) to radio refractivity, A^. The significant level data were collected for the values of at the

earth's surface and within height increments centered upon 0.25, 0.5, 0.75,

1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 km. Each value of was referenced to the

center of its height increment by use of the average atmosphere,

N{h) = Noexp {-h/7.0\, (4.9) which has been shown to be a reasonable model for the decrease of the

refractive index with height for the United States. By use of this ex- ponential model, the gross height dependence of N within each height
110 N CLIMATOLOGY

180° 150° 120° 90° 60° 30° WEST 0° EAST 30° 60°

Figure 4.15. Radiosonde stations used in this study.

increment was effectively removed and there remains a more reliable estimate of the mean N at the center of the height increment. The final value of No, for example, at 1 km over Washington, D.C., would be 300

with a standard deviation of 15.7 units for 152 observations. The

error in the mean is then determined to be 1.3 from

s (N) = , (4.10)

Vk

with k = 152. An examination of similar data for all stations and levels used shows that s{N) is generally less than 1.0 unit. In addition, when considering atmospheric cross sections, the mean

values of N were referred to sea-level by means of (4.9) thus further emphasizing chmatic differences [28].
AVERAGE A^o STRUCTURE 111 4.3.3. Average Structure

The first series of charts presented in this analysis are those showing

the time variation of A^o both vertically above each station as well as at the standard ground-observing level. A slight dichotomy in data sample

exists since the vertical data are monthly averages of radiosonde observations taken twice a day, corresponding in time to the hours 0300 and 1500

GMT, while the surface data are 8-year means for the even hours of the

day local time [29]. This climatic variation at each station is represented

by a two-part chart prepared for each station, the first part depicting

seasonal changes in the mean value of No throughout the first 3.5 km

above the station and the second showing the seasonal and diurnal changes

of A^o at station elevation. These two presentations give a more complete

climatic picture of a location than do the usual unidimensional annual

cycle graphs.

Tatoosh Island (TTI), figure 4.16, off the coast of Washington State,

illustrates typical features of a marine west coast climate. The seasonal

profile changes indicate moderate gradients the year around with a small summer maximum in the A^o gradients in keeping with the cool temperature regime and small diurnal temperature range of this maritimedominated climate. This consistency of weather conditions is further

emphasized by the almost complete lack of diurnal pattern in the surface

data throughout the year and small seasonal change in iVo for any hour

of the day.

The seasonal profile chart of Oakland (OAK), figure 4.17, indicates less maritime effect than at Tatoosh Island and shows clearly the influence of the summer subsidence inversion on the northeastern edge of the Pacific high-pressure area by both the low values of No and increased gradients

at about 1.0 km. The surface data show the moderate seasonal and

diural cycles for all months of the year expected in a Mediterranean

climate.

The effects of the Pacific High are more pronounced further south along

the California coast. The vertical profile data for San Diego (SAN),

figure 4.18, show clearly that during the winter months the strongest

gradients of No are near the surface as one would expect in a maritime

dominated climate while, by contrast, in the summer the well-known

Southern California elevated inversion produces strong gradients aloft. The subsiding air above the inversion is reflected by a characteristic in- crease of No with height. The maritime effect is also evident at the

surface in the low diurnal range of about 10 A" units. It is significant

that the presence of the elevated layer produces a smaller diurnal range

in the summer than in the winter.

The semitropical nature of the humid periphery of the Gulf of Mexico

is reflected by the data for Brownsville, Tex. (BRO), figure 4.19, which

show a strong seasonal cycle and pronounced diurnal range in the summer
112 N CLIMATOLOGY

Figure 4.16. Diurnal, seasonal, and vertical variation of No for Tatoosh Island, Wash. (TTI).

due in large measure to the very high summer temperatures in this locale with resultant high water-vapor capacity. It is also quite striking that

the general level of No is 25 at 30 N units above that of the west coast. The high A^o values of the Gulf regions are also found at Miami (MIA),

figure 4.20, although the moderating effect of continual onshore winds

produces less pronounced vertical gradients and smaller diurnal ranges

than Brownsville, particularly during the summer months.
AVERAGE A^o STRUCTURE 113

I

Figure 4.17. Diurnal, seasonal, and vertical variation ofNofor Oakland, Calif. (OAK).

The interplay of polar and maritime air masses along the middle east coast of the United States is reflected in the strong seasonal range of 50

units at Washington, D.C. (DCA), figure 4.21. The summertime diurnal

range of 15 N units reflects the moderating influence of maritime air so common along the central east coast in the summer. The latitudinally controlled lower mean temperatures of the northeast

coastal regions are reflected in the generally lower values of No plus
114 N CLIMATOLOGY

Figure 4.18. Diurnal, seasonal, and vertical variation of No/or San Diego, Calif. (SAN).

smaller seasonal, diurnal, and vertical ranges as illustrated by Portland,

Me. (PWM), figure 4.22. The curious low in late autumn appears to be

the result of advection of air from the continental interior and is perhaps

indicative of the Indian summer of New England. The long New England winter appears in the surface data as a large area of nearly constant

No.The above data reflect the south-to-north change from humid subtropical to marine-modified continental climates typical of the lee coasts
AVERAGE A^o STRUCTURE 115

Figure 4.19. Diurnal, seasonal, and vertical variation 0/ No/or Brownsville, Tex. (BRO).

of continents. By contrast, the island-like station of Cape Hatteras

(HAT), figure 4.23, reflects both the characteristic strong seasonal range

of a lee coast station that is dominated by dry, cold continental air in the

winter and warm, moist maritime air in the summer and the small diurnal

ranges for all months of the year of a maritime modified climate.

^ The data for Denver, Colo. (DEN), figure 4.24, strongly reflect the

influence of the climatic controls of altitude and continentality. The level of iVo is intermediate to that of Oakland and Washington. The vertical
116 N CLIMATOLOGY

Figure 4.20. Diurnal, seasonal, and vertical variation of No /or Miami, Fla. (MIA).

profile chart for this station indicates the mild influx of tropical air in the summer at the surface and the convective mixing of this air to great heights

by the strong thunderstorm activity common to the area. The high A^o values between 2 and 3 km are perhaps due to superior air subsiding on

the lee slopes of the Rocky Mountains. Relatively intense diurnal ranges

are apparent in both summer and winter, as the day to night variations of

this high plains climate are controlled to a large extent by radiational heating and cooling through the thin atmospheric blanket at high elevations.
AVERAGE No STRUCTURE 117

Figure 4.21. Diurnal, seasonal, and vertical variation 0/ No for Washington, D.C. {DCA).

The strong warm-season influx of tropical maritime air from the Gulf

of Mexico up the Mississippi Valley is spectacularly in evidence at Columbia, Mo. (CBI), figure 4.25. The most significant feature of these

charts is the weak year-round diurnal range of 10 units or less accompanied by a strong seasonal range of nearly 50 units. In summary, it is seen that the west-coast stations display a rather

uniform low average value of No accompanied by small seasonal and

diurnal ranges. This is due to the continual onshore advection of cool
118 N CLIMATOLOGY

Figure 4.22. Diurnal, seasonal, and vertical variation of No/or Portland, Me. (PWM).

maritime air that keeps the mean temperatures both low and uniform.

By comparison, the continental stations show relatively large diurnal and

seasonal ranges controlled in large measure by the radiative heating and

cooling, summer to winter and day to night. The east-coast stations

display a general increase of A^o from north to south arising from the
AVERAGE No STRUCTURE 119

general increase of mean temperature with resultant increase in water- vapor capacity. These same stations have a marked seasonal range due

to the interplay of continental and maritime effects. The relatively

small summertime diurnal ranges along the east coast reflect the strong

influence of maritime air upon this region during that season.
120 N CLIMATOLOGY

Figure 4.24. Diurnal, seasonal, and vertical variation of No /or Denver, Colo. (DEN).
AVERAGE A^o STRUCTURE 121

Figure 4.25. Diurnal, seasonal, and vertical variation of No /or Columbia, Mo. (CBI).
122 N CLIMATOLOGY

4.3.4. Continental Gross Sections

The vertical profile data obtained for the stations of figure 4.15 provide

a means to construct A^o cross sections traversing varied climatic zones and geographic regions.

The first cross section of this series, figure 4.26, is taken along the

Pacific coast of central North America from Canada to Mexico. Intrusion of polar air at the northern end of the wintertime cross section is responsible for the relatively flat gradient over Tatoosh. The 320

A^o-isopleth covering most of the coast southward from Tatoosh shows the

uniform modifying mfluence of the ocean which, at this time of year, is considerably warmer than the continent. Over southern California the

low at 500 m results from the drying-out effects of the Pacific high inversion. On the summer cross section moderate iVo-values typify the coast. The minimum at Oakland stems perhaps from the upwelling of relatively

cool ocean waters off the California coast. Striking evidence of the

Pacific inversion is apparent from the low at 1 km between Oakland and

San Diego that results from significant decreases in the vapor pressure term contribution to within the dry subsiding air aloft.

DISTANCE (KmxIO') Figure 4.26. North-south No cross section along the western coast of the United States.
CONTINENTAL CROSS SECTIONS 123

The cross section for the eastern seaboard of the United States, figure

4.27, presents a considerably different picture of refractive index chmate

than the Pacific coast. On the winter cross section the area between

Portland and Washington falls within a low produced by frequent occur- rence of polar air masses during this season of the year. Southward to- ward Cocoa, Fla. (COF) there is a significant increase that corresponds

to the considerable latitudinal temperature gradient that exists on the

east coast of the continent during the winter months. On the summertime cross section the temperature gradient difference between the two

coasts is even more pronounced and is reflected in the strong east-coast No

difference of about 40 N units as compared to the Tatoosh-San Diego

difference of about 10 units. The vertical gradients are also seen to increase in a systematic fashion from north to south, reflecting the in- creased vapor pressure gradients in the warmer southern regions.

The extensive continental cross section from Isachsen (IC) to Balboa

(BLB), figure 4.28, discloses several interesting features. On the wintertime chart the relatively high Nq values between Isachsen and Churchill

(YQ) indicate the presence of very cold, dense, dry air. To the south over Bismarck, N.Dak. (BIS), and the Great Plains, relatively warm, dry air of
124 CLIMATOLOGY

Figure 4.28. North-south No cross section from Isachsen, North West Territories (IC). through Churchill (YQ) to Balboa, Canal Zone (BLB).
COXTIXEXTAL CROSS SECTIONS 125

IFEBRUARYI | AUGUST!

0 12 3 4 0 12 3 4 DISTANCE (KmxIO' Figure 4.29. West-east Xo cross section for the northern United States.

lower density creates a refractive index low. Southward from the

Great Plains a considerable .Yo gradient is encountered as the Gulf source

region for tropical maritime air is approached. Latitudinal temperature

gradients produce a continued increase in Xq to the southern extreme of the chart at Balboa in the tropics. The .Yo gradients of the summer cross section are largely thermally controlled, rangmg from cool temperatures

and low A^o in the polar regions to a maximum between Brownsville and

Balboa near the warm-season heat equator.

Three zonal cross sections have been prepared extending across the

northern, central, and southern portions of the United States. The

February chart for the northern cross section, Tatoosh to Portland, figure

4.29, exhibits lows at both coasts, with slightly higher values of No in the

cold interior of the continent. The sunnmertime chart clearly shows the

intrusion of tropical maritime air from the Gulf of Mexico pushing northward over the Great Plains. The central U.S. cross section, figure 4.30,

shows, on the wintertime chart, slightly lower values of iYo on the west

coast at Oakland than at Hatteras. East of the Rockies, the chinook

winds of the prevailing westerlies assist in the formation of a relatively

warm, dry air mass of characteristically low .Yq. On the summer map,
126 N CLIMATOLOGY

the influence of the Pacific high is in evidence over Oakland, while in the

interior of the continent the influence of tropical maritime air is noticeable

up the Mississippi Valley. The zonal cross section across the southern

United States, figure 4.31, shows a transition from the Mediterranean

climate of the Pacific coast to the humid subtropical type from Brownsville eastward. Here again the east-coast No values, from a higher

temperature regime, are larger than those in the west. The summertime chart is similar to the winter one with the V-gradient intensified all along the route. Figure 4.32 is for the longest cross section, extending approximately

22,000 km from Canton Island (CAN) to Wiesbaden (WSB) in northern

Europe. The winter chart shows a gradual decrease in A^" from Canton

Island to Brownsville and Cocoa, with very uniform structure over the

Atlantic to Ship and onwards to Wiesbaden. The summer chart

shows a double maximum, one at Canton and the second over the warm

North American continent between Brownsville and Cocoa. The gradual

decrease in mean temperature northward across the Atlantic Ocean is evidenced by the rather uniform decrease of No between Cocoa and

Wiesbaden.
CONTINENTAL CROSS SECTIONS 127

DISTANCE (Km) Figure 4.31. West-east No cross section for the southern United States.
128 CLIMATOLOGY

Figure 4.32. West-east No cross section from Canton Island, South Pacific (CTN)

through Wiesbaden, Germany (WSB).

r
CLIMATIC CHARACTERICS 129 4.3.5. Delineation of Climatic Characteristics

Since climate is really a synthesis of weather elements taken over a

period of time, it is apparent that no fixed climatic boundaries exist in nature. These boundaries shift from year to year with changing weather

and with the addition of new data into the climatic averages. Climatic

borders, then, represent transition zones between so-called ''core climates"

of one type or another. A ''core climate" presumably maintains its climatic personality consistently over a period of time. The coniferous

forest regions of the far north of Canada, for example, maintain a continuously frigid climate under the arctic inversion during the heart of the

winter season. Since a simple system is required for effective classifica- tion, the congeries of minute climatic areas arising from extensive climatic

division are, for practical purposes, coordinated into areas having broad

similarity of climatic character. Thus we shall only attempt here to look

for general patterns in our limited data sample.

The A^o data for the station elevation of figures 4.16 to 4.25 contain

information about the radio refractive index climate of the respective

stations that may be converted to a pair of indices, one seasonal, one

diurnal, useful in classifying climate. The diurnal index is the ratio obtained by dividing the difference between the highest and lowest hourly means for August by a similar difference for February. This ratio is then

plotted versus a ratio of the maximum difference for the 12 mean A^o values at 0200 divided by the maximum differences for the 1400 means. The results of this analysis can be seen on figure 4.33, where the largest

pair of ratios are observed for Denver and Bismarck, a consequence of the strong continentality effects at these inland stations. Denver, it is to be noted, displays a large seasonal ratio whereas Bismarck represents

the extreme in diurnal ratios. Brownsville, Washington, and Columbia

show sizable seasonal ratios coupled with slightly greater than normal

diurnal ratios. Summer-to-winter climate changes are considerable at these three locations.

The remainder of the stations display the near-unity va)ue of both

ratios that would be expected of maritime-dominated climates.

A comprehensive, though simple, index of climatic variation of the

vertical distribution of is particularly difficult to envisage. One such

index is the ratio, R{AN), of the summer to winter value of AN versus the mean value of AA^" (the absolute difference in A^ at the earth's surface and

at 1 km above the earth's surface). The gradient, AA^", has received wide

engineering application and is currently being mapped on a worldwide

basis by the International Consultative Committee for Radio [30]. When

this ratio is plotted versus the yearly mean value of AA^^, as on figure 4.34, one again obtains about the same climatic demarcation as above. For

example, the maritime-dominated climates generally have values of R{AN) of 1.2 or less.
130 CLIMATOLOGY

2.0^

l.8_

(s) DEN CO

l.b- <Si <=>

-ea: 1.4. ®

U-l BRO CO

1.2. ®DCA

I.0_

,8

SAN®"^MIA PWM®"^^^'

COF^ ®

OAK

®BIS

() —1

1 1 I 1 1 1,0 2,0 3,0 4,0 MEAN DIURNAL RATIO

Figure 4.33. Comparison of mean seasonal to mean diurnal ratio of No at the earth's surface.

The more southerly and humid stations have relatively high mean AA^

values in excess of 55 units/km. The single European station, Wiesbaden, shows both low average value of AA^ arising from the low average

temperatures and small seasonal variation associated with the continuous

onshore advection of cool, moist air from the Atlantic Ocean. The values

R{AN) < 1.0 for the Canadian stations of Churchill and Isachsen reflect the steep gradients associated with the intense stratification during the

long winter night. The high, dry climate of Denver produces low mean values of AN but a high seasonal range. The interplay of continental and maritime effects along the east coast produces relatively high

values of R{AN) while the presence of the Pacific high inversion layer

below 1 km accounts for similar values along the west coast. The ex- tremely high value of R{AN) of 1.8 for Tripoli (TPI) arises principally

from the intense summertime N gradients that exist near the surface in
CLIMATIC CHARACTERISTICS 131

CO

CO

-a: U_J oO

2.0

1.8

1,6.

1.4.

I.2_

1.0.

DCA

;CBI

®

TPI

®HAT

DEN PWM.

^BIS ®

TTI ®WSB

OAK®

MIA: ®YQ

®IC

SAN

CTN OOF

30 40 50 60 70 YEARLY MEAN AN

Figure 4.34. Yearly mean and range ratio of AN.

this locale. In the lowest layers of the atmosphere warm, moist air from

the Mediterranean Sea is advected inland underneath superheated air from the Sahara Desert. Along the zone of contact between these air masses of widely differing moisture content, spectacular drops in vapor

pressure and, consequently, in the wet term component of N occur. During the winter, moderate temperatures and prevailing offshore flow

combine to produce near-normal gradients of .Y. Consequently the

summer-to-winter AN ratio turns out to be quite high for this borderland

station. The analysis above represents a rough and crude first attempt to understand the climatic variations of .V. It is expected that such classifica- tions will improve with the addition of data from more diverse locations

and, in particular, as our knowledge of the role of water vapor in climatic

variability increases.
132 N CLIMATOLOGY

4.4. The Climatology of Ground-Based Radio Ducts

4.4.1. Introduction

The occurrence of atmospheric ducts places Hmitations upon ray tracing

of VHF-UHF radio waves. Ducting is defined as occurring when a radio

ray originating at the earth's surface is sufficiently refracted so that it either is bent back towards the earth's surface or travels in a path parallel to the earth's surface. Although the rigorous treatment of ducting in- volves consideration of the full wave equation solution [31] rather than a

simple ray treatment, the present study will be based upon a geometrical

optics definition of the limiting case in which ray tracing techniques may

be used. This simple criterion is then applied to several years of radiosonde observations from stations typical of arctic, temperate, and tropical

climates to derive estimates of the variation of the occurrence of radio

ducts with climatic conditions.

4.4.2. Meteorological Conditions Associated With Radio

Refractive Index Profiles

The path followed by a radio ray in the atmosphere is dependent upon

the gradient of the refractive index along that path. Of the vertical and

horizontal gradient components that compose the path gradient, the

horizontal gradient is normally negligibly small. Thus, the atmosphere

is considered horizontally homogeneous and only the vertical gradient of the refractive index is utilized. The numerical value of the vertical

gradient of the index of refraction depends on the vertical distribution

of atmospheric temperature, humidity, and pressure. Normally, temperature and humidity decrease with height in the atmosphere, since turbulence prevents any great changes in structure. However, there are periods of time in which the air becomes fairly calm,

whereupon temperature inversions and humidity lapses can be built up

and maintained. Temperature inversions have a twofold importance in that (a) they can be widespread in area and persist over a relatively long

period of time, and (b) they exercise a stabilizing influence on air motion

such that turbulence is suppressed and strong humidity gradients may

develop. Layers in which there is intense superrefraction to the point of duct formation may be formed as a result of these gradients and trapping

of radio waves may follow. The inversions may start at ground level or

at some greater height. The thickness of the layer can show great

variability. Three processes that form temperature inversions are: Advection: Advection is the horizontal flow of air having different heat

properties. Such a process is of importance in microwave propagation

since it may lead to a different rate of exchange of heat and moisture between the air and the underlying ground or ocean surface, thus affecting
RADIO REFRACTIVE INDEX PROFILES 133 the physical structure of the lowest layers of the atmosphere. This process results in air with different refractive index characteristics being

brought into the area. The most common and important case of advection is that of dry air above a warm land surface flowing out over a cold

sea. This type of advection frequently appears in the English Channel

during the smnmer when the weather has been fine for several days.

Advective duct formation depends on two quantities: (a) the excess of the unmodified air temperature above that of the water surface, and (b)

the humidity deficit (the difl'erence between the water vapor pressures of the modified and unmodified air). If these quantities are large, especially

the humidity deficit, an intensive duct may form.

Important advective processes can also occur over land, but the conditions required for duct formation occur less frequently. However, a duct

may be formed when dry. warm air flows over cold, wet ground with

resultant temperature and humidity structure previously discussed.

Radiation: Differences in daytime and nighttime radiation are the

causes of diurnal variation in refractive conditions. A subrefractive layer

may be present during the day, especially at the time of maximum surface

heating. Clear skies and hght surface winds at night result in considerable coohng of the earth, thus causmg the formation of temperature in- versions. The surface heat loss produced by nocturnal radiation is a prime factor in the formation of temperature inversions. Atmospheric

stratification formed by such a combination of meteorological parameters

may produce a trappmg layer. A temperature mversion is seldom strong

enough to produce a duct in the middle and low latitudes, but it is of major importance in the formation of ducts in the northern latitudes. Low stratus clouds or extreme amounts of moisture (as in the tropics)

tend to prevent loss by radiation which lessens the possibility of duct

formation.

If the ground temperature in a nocturnal inversion falls below the

dewpoint temperature, the water vapor m the lowest layers of the air condenses and the heat of condensation is released directly to the air. Under conditions of radiative fog formation, the humidity lapse tends to counteract any temperature inversion present and may cause substandard

refraction if the humidity "inversion'' is sufficiently strong. However,

the temperature inversion may be strong enough to keep the layer

standard or superrefractive.

Subsidence: Subsidence is the slow settling of air from a high-pressure

system. The air is heated by adiabatic compression as it descends and

spreads out in a layer well above the earth's surface. This process pro- duces stable layers and inversions of temperature with an accompanying

decrease in relative humidity. Since the air has come from a high level

in the atmosphere, it is dry and may overlay a cooler, moist air mass.
134 N CLIMATOLOGY

This type of inversion may cause the formation of an elevated superrefractive layer where the air temperature usually decreases immediately

above the ground, rises through the inversion layer and decreases above

the layer. This is a common occurrence which may be observed at any

time. Subsidence has a tendency to destroy subrefractive layers and to intensify superrefractive layers. Although the effects of subsidence are generally observed at high levels, they are occasionally observed at lower

levels, especially in the subtropics. Since subsidence frequently occurs

in the lee of mountains and in the southeastern regions of northern

hemisphere highs, elevated ducts may also be observed.

Conditions inimical to ducting are those which mduce mixing of the

lower atmosphere. Small scale atmospheric motions (turbulence) and

consequent mixing and mass exchange result from differential surface

heating and surface roughness effects. Both of these processes work to destroy stratification. They ultimately result in uniform vertical distri- bution of moisture through considerable depths of the lower atmosphere

and the establishment of neutral temperature lapse rates. Accordingly,

where the process of mechanical and convectively induced mixing are at work, the probability of the occurrence of ducting is vanishingly small. Thus, few, if any, ducts are observed over snow-free, low albedo land

areas from midmorning to late afternoon when the skies are clear, or in areas of moderate to great surface roughness when the surface winds are more than a few meters per second, irrespective of cloud conditions or the

time of the day.

4.4.3. Refractive Conditions Due to Local

Meteorological Phenomena

Land and sea breezes may produce ducts along the coastal regions since

the winds are of thermal origin, resulting from temperature differences

between land and sea surfaces. During the day, when the land gets warmer than the sea, the air above the land rises and is replaced by air from the sea, thus creating a circulation from the sea to the land, called a

sea breeze. During the night, the land becomes colder than the sea and a

circulation, called a land breeze, is set up in the opposite direction. This

type of circulation is generally shallow and does not extend higher than

a few hundred meters above the land or sea surface. A land or sea breeze may modify the refractive conditions in different

ways depending upon the distribution of moisture in the lower layers.

Since these breezes are of a local nature and generally extend only a few

miles, only coastal locations are usually affected. The very nature of sea and land breezes results in a marked refractive index pattern. With a
BACKGROUND 135

sea breeze a duct may be formed over the water due to subsidence. The

land breeze is accompanied by subsiding air over the land with resultant

duct formation.

The formation of fog results in a decrease of superrefractive or ducting

possibilities. When a fog forms by nocturnal radiation, the w^ater content

of the air remains practically the same; however, part of the water

changes from the gaseous to the liquid phase thus reducing the vapor

pressure. The resulting humidity lapse rate tends to counteract the

temperature inversion and cause above standard refraction. However,

the temperature inversion may be strong enough to keep the layer standard or superrefractive. This process may also occur wdth advection fogs.

The nighttime temperature profile is a result of the interaction between

nocturnal radiation, turbulence, and heat conduction. The associated

refractive index profiles are such that a radar duct begins to form about

the time of sunset, developing quickly during the early evening, more

slowly after midnight, and dissipating rapidly after sunrise. This is mainly an inland effect resulting from large diurnal temperature variations observed in the interiors of large continents. However, a shallow

body of water may have an appreciable diurnal temperature variation, as compared to the open ocean so that superrefraction may occur over such

a location from time to time.

It is generally recognized that radiosonde observations (RAOB's) do

not have a sufficiently high degree of accuracy to be completely acceptable

for use in observing changes in the degree of stratification of the very

lowest layers of the atmosphere; however, until more accurate methods

such as meteorological towers and refractometer measurements are more commonly used, the RAOB will continue to be used as a basis for fore- casting the occurrence of superrefractive conditions.

4.4.4. Background

The property of the atmosphere basic to radio ray tracing is the radio

refractive index of the atmosphere, n, which for VHF-UHF frequencies

at standard conditions near the surface, is a number of the order of 1.0003. Although the refractive index is used in ray tracing theory, it is more

convenient w^hen evaluating refraction effects from common meteorological observations to use the refractivity N, which, for the frequency

range 0 to 30,000 Mc/s, is given by (1.20).

When evaluating the meteorological conditions that give rise to refrac- tive phenomena, it is frequently instructive to examine separately the

behavior of the dry component, D, and wet component, W, of N.
136 N CLIMATOLOGY

The gradient of the refractivity, AA^, with respect to height may then

be expressed:

AA^ = AD + AW. (4.11)

Average values of AA^, AD, and AT^ are given in table 4.6 for two increments between the earth's surface and 1 km above sea level for Fairbanks,

Alaska; Washington, D.C.; and Swan Island, W.I.

Table 4.6. Gradient of N, D, and W in N units per kilometer

Station Height

increment February August -^N -AB -AW -AN -AD -AW

Fairbanks, Alaska surface—0. 5 km 37 41 -4 31 27 4

0. 5 km-1. 0 km 35 35 0 36 24 12 Washington, D.C surface— 0. 5 km 41 34 7 60 28 32 0.5 km-1. 0 km 30 26 4 46 24 22 Swan Island, W.I surface—0. 5 km 39 24 15 47 26 21 0.5 km-1. 0 km 58 24 34 66 24 42

Several general observations may be made of the data of table 4.6: The gradient of the dry term is relatively less variable than that of the

wet term when considered as a function of season or height ; the increase

of AN from winter to summer at a particular location or from arctic to

tropical climate at a given time is most strongly reflected in AIF rather

than in AD. The marked increase of gradient with height for Swan Island

reflects the drop of refractivity across the interface of the trade wind

inversion where dry subsiding air overlies the moist oceanic surface layer.

A fundamental equation used in radio ray tracing is Snell's law, which,

for polar coordinates, is given in chapter 3 as

nr cos 6 = rioro cos ^o, (4.12)

where n is the radio refractive index of the atmosphere, r is the radial

distance from the center of the earth to the point under consideration, and

6 is the elevation angle made by the ray at the point under consideration

with the tangent to the circle of radius r passing through that point. The

radius to any point, r, is equal to (a + h), where a is the radius of the

earth and h is the height of the point above sea level. The zero subscript

refers to the value of n, r, or d at the earth's surface. For the present study, geometrical optics techniques, similar to those

considered by Bremmer [32], are used to indicate when refraction is
BACKGROUND 137

sufficiently great to direct a ray either back to earth or in a circular path

at a constant height above the earth; i.e.

noVo cos do ^ ^ nr ~

'

This condition then allows one to obtain the value of Oq which divides the

rays into two groups: those that penetrate the duct and those that are trapped within the duct. This particular value of ^o, called the angle of penetration and designated dp, is obtained by noting those instances where

^

> 1 (4.14) nr ~

and solving for the value of such that (4.13) is equal to unity.

It is instructive to consider the order of magnitude of refractive index

gradient needed for trapping for several commonly observed refractive

index profiles. If we rewrite Snell's law,

ntVt cos dt = rbAVA cos dd, (4.15)

where the subscripts t and A refer to the values of the variables at the

transmitter height and the top of the trapping layer, then trapping

occurs when

> 1, (4.16) rtAVA ~

and the angle of penetration at the transmitter, dp, is given by setting

cos dp = 1. (4.17)

riA Ta The maximum permissible n gradient for a given value of dp is then given

by:

|n^»i_^ (4.18)

Ar Va — rt where Ua must satisfy (4.17); i.e.: = cos dp. (4.19) ^A
138 N CLIMATOLOGY

By designating va = ft + Ha, (4.18) becomes

An nt r, rt . 1 = ~" iT 1 nr cos dp • Ar HaL rt -h Ha

By rewriting (4.20)

(4.20)

An

Ar

n^

hA

1 - cos 6t 1 + rt

(4.21)

and expanding (1 + hA/rt)~^ and cos dp, one obtains the expression:

An

Ar = — n.

by neglecting terms ^p/4! and QiA/rtY, and terms beyond these. For

the case of Bp = 0, (4.22) reduces to: An _ n 1 Ar a a

157 N units/km. (4.23)

It is seen from (4.22) that the n gradient necessary to trap a radio ray

at a given value of dp is practically independent of transmitting antenna

height above the earth. For example, a = 0 ray will be trapped by

an n gradient of —157.0 N units/km at sea level, where n< = 1.0003,

while the necessary n gradient at 3 km above sea level will be —156.9

units/km for an n< ^ 1.0002. This indicates, for all practical applications, that the necessary n gradient for trapping is independent of

altitude. Further, by considering the temperature and humidity gra- dients encountered in the troposphere, one is led to the conclusion that

ducting gradients would not be expected to occur at altitudes greater

than 3 km. In fact. Cowan's [33] investigation indicates that trapping

gradients are nearly always confined to the first kilometer above the

surface. A consideration of (4.22) indicates that the magnitude of the negative

gradient necessary for ducting is 1/a for dp = 0 but increases in proportion to dp'^/2h.SiS dp increases. The gradients necessary for atmospheric

ducts as a function of do are given for several different profiles in figure

4.35 An analysis of radiosonde data indicates that gradients in excess of 0.5 A units per meter are seldom exceeded within atmospheric layers.

It is interesting to note how rapidly the necessary gradients increase to
BACKGROUND 139

the approximate upper limit of radiosonde-observed gradients: a groundbased layer 100 m thick attains this gradient at 8.3 mrad while the

maximum observed gradient is intercepted by the 30-m layer curve at

^0 = 4.5 mrad. A third example was calculated for an elevated layer 0.5 km above the ground and 100 m thick by assuming normal refraction

between the ground and the base of the layer and solving for the necessary

ducting gradient within the layer. The large values of n gradient neces- sary for this case explain why elevated ducts were not observed. Although the preceding examples were calculated for a ground transmitter,

the combinations of Bp, An/Ar, and Ar are very nearly the same as would

be obtained for any other transmitter height within the first 3 km above

the surface.
140 N CLIMATOLOGY

4.4.5. Description of Observed Ground-Based

Atmospheric Ducts

Approximately three years of radiosonde data typical of an arctic climate (Fairbanks, Alaska), a temperate climate (Washington, B.C.),

and a tropical maritime climate (Swan Island, W.L) were examined by means of a digital computer for the occurrence of ducts during the months

of February, May, August, and November. The percentage occurrence

of ducts is shown on figure 4,36. The maximum occurrences of 13.8 percent for August at Swan Island and 9.2 percent for Fairbanks in February are significantly greater than the values observed at other times

of the year. The Washington data display a summertime maximum of 4.6 percent. These data indicate that the temperate zone maximum

incidence is about one-half the wintertime maximum incidence in the

arctic, and about one-third of the summertime tropical maximum.

The range of observed values of dp is shown in figure 4.37. The mean

value calculated for each month as well as the maximum and minimum

values of dp observed for the limiting cases are given for each month and

location. The mean value of the angle of penetration under these conditions is between 2 and 3 mrad and appears to be independent of climate.

The maximum value of dp observed during ducting is 5.8 mrad.

The refractivity gradients observed during ducting are given on figure

4.38. The maximum gradient of 420 N units per kilometer was observed

during February at Fairbanks, Alaska. The mean values of N gradient

appear to follow a slight climatic trend from a high value of 230 N units

per kilometer at Fairbanks to a value of 190 A" units per kilometer at Swan Island. Another property of radio ducts is their thickness, which is given in

figure 4.39. Again there is observed a slight climatic trend as the median

thickness increases from 66 m at Fairbanks to 106 m at Swan Island. These values of thickness correspond to the gradients given in figure 4.38. One can then obtain, by linear extrapolation, the thickness at which the

gradient is equal to — 1/a; i.e., the height corresponding to the gradient

just sufficient to trap the ray at = 0. These values, shown on figure

4.40, display an increase in the median thickness of about 25 percent for Swan Island, 100 percent for Washington, and 200 percent for Fairbanks,

which results in a reversal of the climatic trend of the observed thickness

between Fairbanks and Swan Island. This increase in height emphasized

the preceding conclusion. Fairbanks is characterized by shallow layers

with relatively intense gradients.

These maximum duct widths may be used to estimate the maximum

radio wavelengths trapped. Kerr [34] gives the maximum wavelength,

X, trapped by given thickness, d:

(4.24)
GROUND-BASED ATMOSPHERIC DUCTS 141

16

15

14

too 13

Z5

(~)yD 1

2

CDCOo

1

1

CD

TDc

10 o

9

x:~>

:> 8

t/^ co 7 >

6

oo o

H

o

C 4

CDo

CD 3

a. 2

Numbers o number of

n curvt

profile

?5 ^JAf" / 5 ona/y

*otol sed. .218

226

SWL7/7 Isia 166

\

v496

/

151 /

14 472

175 Fa

i

1 irbanks, Alaska —

,

1 1 ^

247

447

247^ 247

J FMAMJJASON

Months of the Year

Figure 4.36. Frequency of occurrence of ground-hased ducts.

by assuming a linear decrease of n within the duct. The coefficient c

is a constant and 7 is a function of the n gradient excess over the minimum

value of An/Ah = 1/a. Expressing \max in centimeters, and d in meters,

then

0.1 10 (4.25)
142 N CLIMATOLOGY

and

c = 2.514 X 102. The maximum wavelengths trapped during ducting conditions were

estimated by (4.24) for the maximum duct thicknesses of figure 4.40. These values, given in table 4.7, were determined for the month with the

maximum occurrence of ducts, thus allowing an estimate of the radio

frequencies likely to be affected by ducting conditions. Note, for ex- ample, that the data of table 4.8 indicate that 1000 Mc/s rays (X = 30 cm)

will be trapped by 50 percent of the ducts regardless of location.
GROUND-BASED ATMOSPHERIC DUCTS 143

420

340

140

IRBANK

1 S, ALASk(A

\ Numbers on curves are the

\ fatal number of observed ducts.

1

18 \ r

^Mean

\(Vlin

1

1 WASHINGTON, D.C

23 \ /

^

Max

9 Mean

Min

FEB MAY AUG NOV FEB MAY AUG NOV

SWAN SLAND

M ax

3 3( Jan r 2 < n

FEB MAY AUG NOV

Months of the Year Figure 4,38. Refradivity gradients of ground-based ducts.

Table 4.7. Estimated maximum wavelength trapped at 6o = 0

station

Percentage of ducts that trap wavelengths in centimeters equal to or shorter than tabulated value

95 90 75 50 25 10 5

Fairbanks, Alaska (February). 20.5 22.6 25.5 34.0 43.5 61.5 69,0 Washington, D.C. (August)... 7 10 27 50 112 164 200 Swan Island (August) 12 23 41.5 60 82.3 92 132

Table 4.8. Ranges of surface ducting gradients

Duct gradients A'' units/km A'' units/km Nunits/km Nunits/km

at Washington, D.C. February May August November Max -200 -372 -395 -195 Mean -185 -230 -240 -170 Min -175 -162 -158 -159
144 N CLIMATOLOGY

The reader is cautioned that an atmospheric duct does not have the

sharp boundaries of a metaUic waveguide. Thus the maximum wavelengths obtained by (4.24) do not represent cutoff frequencies but, as Kerr is so careful to emphasize, merely suggest lower limits under the

assumptions of this rudimentary theory.

The results given above were derived from a consideration of radiosonde data. Although the radiosonde is not an extremely sensitive in- strument, it is readily available and is the only source of climatic information involving the temperature and humidity structure of the atmosphere.

It is beheved that the radiosonde data will at least yield the chmatic trend

of radio ducts as well as their probable temperature and humidity distri- butions. Further, it is evident that the choice of stations will definitely

affect the percentage of ducts observed. For example, it is almost certain

that a greater percentage of ducts would be observed over water in the

subtropics than over Swan Island.
GROUND-BASED ATMOSPHERIC DUCTS 145 With these reservations in mind, the present study has shown:

(a) Ducts occur no more than 15 percent of the time.

(b) The annual cycle of the incidence of ducts is reversed for the arctic and tropical stations studied. The arctic station has a wintertime maximum and the tropic station a summertime maximum. The temperate

station has a summertime maximum incidence of less than 5 percent.

(c) The maximum initial elevation angle is limited to 5.8 mrad with

a mean value of about 3 mrad.

(d) The steepest gradient of N observed is —420 units per kilometer.

(e) The maximum thickness of observed ducts is such as to trap radio waves of 1000 AIc/s and above at all locations for at least 5 percent of the

observed ducts.

(f) Ducts in the arctic appear to be associated with temperature in- versions at ground temperatures of —25 °C or less; temperate zone, with

the common radiation inversion and accompanying humidity lapse;

tropics, with a moderate temperature and humidity lapse for temperatures

of about 30 °C.
146 N CLIMATOLOGY

4.5. A Study of Fading Regions Within the Horizon

Caused by a Surface Duct Below a Transmitter

4.5.1. Introduction

Among the factors influencing the choice of an antenna site for a micro- wave receiver operating on a line-of-sight path is the desirabihty of locating in a region relatively free from space-wave fadeouts. Although

line-of-sight propagation is normally characterized by steady, high, dependable signals, deep, prolonged space-wave fadeouts are observed from

time to time. Since serious disruptions occur during such fadeouts, a

systematic discussion of fading phenomena is of considerable interest to the radio circuit engineer.

There are [35] results in detail of a year's study of signal fadeouts

occurring at certain points within the horizon caused by ground-based

superrefractive layers. The path studied in this case [35] has its trans- mitter at Cheyenne Mountain near Colorado Springs, Colo., at an elevation of 8800 ft. Receivers are at Kendrick, Colo., 49.3 mi from the

transmitter, and at an elevation of 5260 ft; Karval, Colo., 70.2 mi from

the transmitter, and at an elevation of 5260 ft; and Haswell, Colo., 96.6

mi from the transmitter, and at an elevation of 4315 ft. A study of the

profile of this path reveals that Kendrick is well within the radio horizon,

Karval is near, but still witnin the radio horizon, and Haswell is beyond

the radio horizon.

The Cheyenne Mountain study found that throughout the period of observation, fadeouts (of 5 dB or more) occurred regularly at Karval on a

frequency of 1046 Mc/s, in conjunction with superrefractive A^-profile

conditions, and coincided with enhanced field strengths at Haswell; A

particular instance of this, the night of June 21-22, 1952, is shown in figures 4.53 and 4.54. Figure 4.53 shows the field strengths recorded

simultaneously in a 12-hr period at Haswell, Karval, and Kendrick. Here

it is to be noted that with the sudden intense rise in field strength at Haswell, a progressively deepening fadeout appears in the Karval data

relative to the monthly median, while only insignificant changes are noted

at Kendrick. Meanwhile, figure 4.54 indicates the shift in the refrac- tivity profile at Haswell (presumably the same for Kendrick and Karval)

towards conditions of superrefraction throughout the evening of 21-22

June, 1952. The great number of fadeouts at Karval as compared to Kendrick

strongly suggested a dependence of fading on distance from the trans- mitter within the horizon. The following sections describe the conditions

under which fading, variations in fading, and locations that favor the occurrence of fading within the radio horizon (in the presence of a groundbased duct) occur.
FADING REGIONS WITHIN THE HORIZON 147 Fading, or fadeout, of a radio wave may be defined as a drop in power

or field strength below a specified level of intensity. For a given site, criteria may be set up by the communications engineer to establish the

magnitude of drop that defines the onset of fading conditions. This

magnitude usually ranges from about 5 to 30 dB below the specified level

of intensity. The specified level is derived from some ideal propagation

condition.

Fading is also a time-dependent occurrence. It may be classified as

(a) prolonged fading, which is fading of sufficient interval to cause con- tinued communication disruptions, and (b) short-term Rayleigh fading,

which is only instantaneously observed b^^ a receiver. One of the main causes of deep fading of field strength within the

horizon as compared with the free space field strength value is the N

structure of the atmosphere. Theoretically, the atmosphere can be considered horizontally homogeneous and in spherical stratifications concentric with the earth, and can be considered to decrease exponentially

with increasing height above the earth's surface. However, in reality, this picture of the atmosphere is rarely, if ever, realized because of the

synoptic meteorological conditions that are perpetually present. Stratifications caused by the synpotic meteorological pattern give rise to field strength fading within the horizon by defocusing the lobe pattern of the

transmitter along a given path. WTienever the rate of change of refractivity from the surface value with height (called the gradient of N with

height) is less than —157 units per kilometer, a ''ducting" condition is said to exist at the surface, and, as shown in figure 4.35, certain rays will tend to be ''trapped" or guided within the surface duct. It is this atmospheric condition of surface ducting which will be further explored herein

with respect to fadmg within the radio horizon.

It should be realized that not all within-the-horizon fading has been

attributed to refractivity gradient discontinuities in the lower atmosphere, ^lisme [36] shows the influence of frontal effects and frontal

passage on signal fading within the horizon, the frontal passage even

occurring at a time when one would most expect a fadeout caused by a

surface duct.

4.5.2. Regions and Extent of Fading Within the Horizon

in the Presence of Superrefraction

Serious disruptions in reception from a transmitter above a duct to a receiver within a duct can occur at particular points within the horizon.

It is these disruptions, and these locations, which are of interest to the

communications engineer establishing a given transmitting-receiving

path. Whenever these disruptions occur between a transmitter and

receiver within the horizon, a corresponding increase in the field strength

characterized by steady, high, dependable signals, is usually expected
148 N CLIMATOLOGY

beyond the horizon. The high signal strength is in keeping with the

properties of a surface duct. Nevertheless, deep, prolonged fadeouts can occur in regions beyond the horizon as well as within the horizon.

Price [37] has theoretically determined regions of deep radio fading

associated with surface ducts, which he has termed ' 'shadow zones." In

the case of a transmitter above a surface duct, a representation of what

occurs is shown in figure 4.43. It shows the location of a shadow zone above the radio horizon line in the normal interference region. In the

interference region, a fadeout of signal strength due to the presence of superrefraction must be compared with the value of the field when only

the interference pattern is present. Papers such as those by Norton [38] and Kirby, Herbstreit, and Norton [39] give methods for calculation of the normal interference field. Ikegami [40] gives a more general method

for the calculation of received power in the presence of ground based ducts. Ikegami's procedure is based on a simple geometrical optics ray-tracing

technique of determining the power relative to free space transmission

that is received at different locations from the transmitter. A more

refined, yet more complex, procedure is the field-strength calculations

along the lines followed by Doherty [41], in which a strict mode-theory

treatment of the problem reveals that geometrical optics is not sufficient in the presence of refraction anomalies. Doherty expands techniques

originally considered by Airy [42] to determine relative field strengths in the neighborhood of caustics (apparent ray intersections) that result vicinal to refraction anomalies such as ducts. In the procedure that follows, a model for the determination of the

location and extent of shadow zones, rather than the determinations of the

actual received field strengths and powers at any particular point, will be

given for conditions typical of the temperate climate to aid the radio cir- cuit engineer in avoiding these troublesome interference areas. Washington, D.C., is taken to represent an average temperate zone

climate. Fairly extensive work has been done in determination of various

ducting conditions at this station as well as with the average exponential

refractivity above this station. Therefore, Washington, D.C., will be

used as a model for all following calculations.

The gradient of with height determines whether or not a surface

duct will exist. Therefore, a surface ducting atmosphere corresponding

to conditions at Washington, D.C., consisting of a ducting gradient up to any desired height, and the Ns = 313.0 exponential atmosphere [9] above

this height, have been chosen for this study. The ducting gradients

chosen are maximum, mean, and minimum values for February, May,

August, and November (representing the middle of winter, spring, summer, and fall, respectively) taken from figure 4.38 and given in table 4.8. Also in this particular model the duct is assumed to be of uniform height

throughout the region considered in the calculations.
THEORY AND RESULTS 149 4.5.3. Theory and Results

In figure 4.41, although it is known that ray theory does not explain

what happens to the radio ray at the point A when the ray is tangent to the top of the duct, because of the apparent ray intersection, it is sufficient to define the first shadow zone as that zone between the two possible paths

of the ray tangent to the duct at a point A (this is the ray emitted from the

transmitter with an initial angle of with the horizontal). This is possi- ble because it can be readily seen that no radiation will be able to enter the

shadow zone, and thus a receiver located in the shadow zone will theoretically receive no signal from the transmitter. The portion of the ''split"

ray that enters the duct at A will be reflected at point B to point C, where

it "splits" again. The distance, d from point A to C projected along the

earth's surface will be designated the ''length" of the first shadow zone.

If the surface duct is homogeneous throughout, the ray will be symmetrical about point B and, therefore,

di = d2 = d/2 = dhi (4.26)

where di and c?2 are the "half-lengths," dht, of the shadow zone.

Figure 4.41. Shadow zone occurrence of a curved earth with the transmitter above

the duct.
150 N CLIMATOLOGY

Chapter 3 shows that the bending, ri,2, of the radio wave between any

two points in the atmosphere is given by

T1.2 = (di- 62), (4.27)

where di,2 is the distance between the two points 1 and 2 (see fig. 4.42)

projected along the earth's surface, a is the radius of the earth, and 61 and 02 are the angles in radians the ray makes with the horizontal at points

1 and 2 respectively. As described in chapter 3, Schulkin [43] obtained

Ti,2 in radians as = - ^"^

for 0 < 9„ < 10°, (4.28)

where Ani = ni — 1, An2 = 722 — 1, = (^1 + ^2)/2. In terms of refractivity, (4.28) becomes

(Ni - N2) X 10-' AN X 10-'

T1.2 = = 7 . (4.29)

Considering now the case of the half-length, dht, (4.27) becomes

r,,2 = — + e„ (4.30) a

since ^2 = 0 at the top of the duct, and di = 6 p. Now substituting (4.29) in (4.30),

2AiV X 10- ^ d,, ^ ^^ 3^)

Up a

since here dm = 6p/2. Rearranging (4.31)

Bp (rad)

Since

r2AiV X 10"

, n r AW (A oo\

( = cl\ —a /^^A\— + (^^^) I • (4-32)

Bp = cos-^ ^^ J2[AN X 10-^ - . (4.33)

(rad)

^oa \ L a J
THEORY AND RESULTS 151

RADIO RAYx

Figure 4.42. Geometry of radio ray refraction.

where Ua is the index of refraction at the top of the duct, Ta is the distance

from the center of the earth to the top of the duct, and no is the index of refraction at the earth's surface, it is seen that c?a^ is uniquely determined

from the surface gradient of with height inside the duct, provided the

height of the duct is known.

If one desires to know the distance, dh, for a particular height, h, from

the reflection point of the ray in the duct (point B in fig. 4.41), where dh

is such that

(4.34)
152 CLIMATOLOGY

1 ST SHADOW ZONE

rrrA DIRECT AND REFLECTED RAY

INTERFERENCE REGION

ZONE BEYOND THE HORIZON

GROUND BASED DUCT

Figure 4.43. Regions of fading from different effects of the radio rays.

then (4.37) becomes= (4.35)

since 02 is no longer just equal to zero, but is such that

0 < ^2 < Op, (4.36)

and AN' will no longer be the same as the AN for the duct width. The

value of 02 may be determined from the formula:

-40/ + ^ - 2AN' . (4.37)

In figures 4.44 to 4.46 the shadow zone half-lengths are given as a function of duct height for the various seasons of the year. As shown in

figure 4.39, the 50 percent level, or median of observed ground-based duct

thickness at Washington, D.C., is about 100 m. Figure 4.40 shows that

the percent level of observed maximum ground-based duct thickness is about 200 for Washington, D.C. In table 4.9, the values of dht during the

four seasons are shown for the 100- and 200-m surface duct heights.

As previously mentioned, the atmosphere above the assumed duct was

taken to coincide with the Ns = 313.0 exponential atmosphere [9] because

this value most nearly corresponds to the average annual Ns value at
THEORY AND RESULTS 153

DUCT HEIGHT ,h, IN m

Figure 4.44. Shadow zone half-lengths for maximum ducting conditions at Washington, D.C.

. NpisN ! I U-n I 1 i l-l-n I n M I M M I M LM. 1 U- I n'^n-y-n I i 1 11! U. I 11 I i I 0 20 40 60 80 100 120 140 160 180 200

DUCT HEIGHT , h , IN m

Figure 4.45. Shadow zone half-lengths for mean ducting conditions at Washington, D.C.
154 N CLIMATOLOGY

DUCT HEIGHT , h, IN m

Figure 4.46. Shadow zone half-lengths for minimum ducting conditions at Washington, D.C.

Table 4.9. Half-lengths at Washington, D.C. for various duct heights and surface gradient conditions

February.

May.

August.

November.

Cond.

Max. Mean

Min.. Max. Mean

Min.. Max. Mean

Min.. Max. Mean

Min..

Duct height

100 m

68. 21

84. 50

106. 35

30. 53

52. 37

199. 42

29. 02

49. 12

440. 73

72. 56

123. 93

313. 61

200 m

119. 50

148. 98 43.17

74.05

281. 95 41.04

69. 46

622. 53

102. 60

175. 24

443. 23

Washington, D.C. Figures 4.47 and 4.48 show the distance measured

along the earth's surface that a ray with an initial elevation angle of zero

will travel in the Ns = 313.0 exponential atmosphere. The value of zero for an initial elevation angle is used because the ray can be thought

of as leaving the position A, tangent to the duct in figure 4.41 and

''arriving" at the transmitter, even though the reverse process is actually

what is occurring. Thus, from figures 4.47 and 4.48, knowing the height

of the transmitter, ht, and the height of the ground-based duct. Ha, the
THEORY AND RESULTS 155

HEIGHT , h , IN m

Figure 4.47. Distance versus height above duct representing average Washington,

D.C., and Fairbanks, Alaska, atmosphere.

distance, do, that the radio ray travels from the transmitter to the top of the duct is given by

do = d(ht - hA)', (4.38)

i.e., the distance obtained by using the difference of ht and Ha as the value

of height on the abscissa in figures 4.47 and 4.48 gives the distance do. The distance, d ho, to the standard radio horizon from the transmitter is dho = V2ht (4.39)

if h t, the transmitter height, is in feet and dho is in miles. Using this fact and figures 4.47 and 4.48, table 4.10, which shows the ratio of do to dho,

Table 4.10. Ratio of onset point of fading zones from transmitter to distance of radio horizon from the transmitter, do/dho, for various transmitter heights at Washington, D.C.

Duct height, meters Height of transmitter

10 20 50 100 200

200 0. 795 0. 699 0. 512 0. 300 0 300 .837 .759 .605 .432 0. 187 500 .878 .818 699 .565 .375 800 .907 .860 .765 .659 .509

1000 .918 .876 .792 .697 .563

1500 .935 .900 .831 . 754 .644 2000 .944 .914 .854 .787 .692

3000 .953 .928 .880 .825 .747 5000 .959 .940 .902 .859 .799

7000 .959 .943 .911 .875 .825
156 N CLIMATOLOGY

HEIGHT, h, IN METERS

Figure 4.48. Distance versus height above duct representing average Washington, D.C., and Fairbanks, Alaska, atmosphere.

is derived for various transmitter heights and various duct heights. This

table is useful in determining the portion of the distance (measured from

the transmitter) within the horizon where fading due to the presence of superrefraction has no effect on transmission.
SAMPLE COMPUTATIONS 157

4.5.4. Sample Computations

It can be seen from figure 4.52 that

hi < loss region < /i^ + h^, (4.40)

where hi can be determined from figures like 4.49 to 4.51, which depend

upon the station location and the season of the year; hA is the height of the duct; and /12 is determined from figures 4.47 and 4.48. The following

example will illustrate the determination of the shadow zone. As an illustrative problem, assume a 500-m transmitting antenna in the

spring of the year at 40°X latitude, and that one desires to place the re- ceiving antenna at (a) 40 km and (b) 90 km from the transmitter. How

high should the receiving antenna be to be free from fading loss caused

by a ground-based superrefractive layer?

Since both of the distances required are within the standard horizon

distance, dho, which, for a 500-m transmitter is

dho = 92.181 km,

this is a case of within-the-horizon propagation.

Since the 40°N latitude position is within the temperate zone, use will be made of the Washington, D.C., model ducting atmosphere. As pre- viously mentioned, 100 m is the median mean duct thickness at Washington, D.C., and 200 m is the median maximum duct thickness observed at Washington. These are the two duct thicknesses to use in this calculation. Also, since it is the spring of the year, figures 4.49 to 4.51 must be

used to determine hi. For the 40-km distance, case (a), the subscript of (4.38) becomes

ht - hA = 500 - 100 = 400 m,

which is then entered on the abscissa of figure 4.47, where,

c?o = c?4oo = 83 km.

For a 200-m-thick duct, - hA = 300 m.
158 N CLIMATOLOGY

0 20 40 60 80 100 120 140 160 180 200 HEIGHT ,h , IN m

Figure 4.49. Height versus distance at Washington, D.C., for maximum midspring

(May) ducting conditions.

20 40 60 80 100 120 140 160 180 200

HEIGHT , h , IN m

Figure 4.50. Height versus distance at Washington, D.C., for mean midspring

(May) ducting conditions.
i SAMPLE COMPUTATIONS 159

20 40 60 80 100 120 140 HEIGHT , h , IN m

160 180 200

Figure 4.51. Height versus distance at Washington, D.C., for minimum midspring

(May) ducting conditions.

Therefore, in this case,

do = dzQo = 72 km.

In either case, as can be seen from figure 4.41, do is greater than 40 km.

Therefore, any height of the receiving antenna will be satisfactory at this point, since the first shadow zone has not yet been reached.

For the 90-km distance, case (b), it can be seen that this distance is beyond the onset point of the first shadow zone, and figures 4.49 to 4.51 must be used here. Figures 4.49 to 4.51 are used differently depending upon which side of the reflection point B of figure 4.52 one desires to locate an antenna. If on the lefthand side of point B, one must subtract the desired distance

from the distance from B to the top of the duct traversed by the ray and

use this new distance to determine height, h, in figures 4.49 to 4.51. This

procedure must be followed every time the ray heads downward. If on

the righthand side of point B of figure 4.52 (or any other reflection point), one simply uses the distance from the reflection point directly. One must

keep in mind that any previous half-lengths must be subtracted from the

desired distance from the transmitter, as well as subtracting do, in order

to obtain the distance used in figures 4.49 to 4.51.
160 N CLIMATOLOGY

Figure 4.52. Geometry of the shadow zone loss region.

For the maximum midspring ducting conditions (fig. 4.49) and the

100-m-thick duct, recalling do = 83 km, the antenna is to be located

90 - 83 = 7 km

beyond the onset point of the shadow zones. For the 200-m duct, the

antenna is located

90 - 72 = 18 km

beyond point A of figure 4.52. From figure 4.44, the half-length traveled

by a ray in a 100-m duct under maximum midspring ducting conditions

is 30.5 km. Thus, as described above, the distance used in figure 4.49

to find h is

30.5 - 7.0 = 23.5 km

because at 25 km the ray is still sloping downward and hence is on the

lefthand side of point B of figure 4.52. From figure 4.44, dht = 43 km

for a 200-m duct; therefore, the distance used in figure 4.49 is 43 - 18 = 25 km.

Similarly, utilizing figures 4.45 and 4.46 for mean ducting and minimum

ducting conditions for a 100-m and 200-m duct:

Figure 4.50 distance to be used for the 100-m duct = 45 km,

Figure 4.50 distance to be used for the 200-m duct = 56 km,

Figure 4.51 distance to be used for the 100-m duct = 191 km.

Figure 4.51 distance to be used for the 200-m duct = 262 km,
! SAMPLE COMPUTATIONS 161

Figure 4.53. 1,046-Mc/s fields for the night of June 21-22, 1952, at Kendrick, Karval,

and Haswell, Colo. and hence

hi for a 100-m duct under maximum spring ducting conditions = 92.3 m,

hi for a 200-m duct under maximum spring ducting conditions = 166.0 m,

hi for a 100-m duct under mean spring ducting conditions = 96.0 m,

hi for a 200-m duct under mean spring ducting conditions = 187.0 m,

/ii for a 100-m duct under minimum spring ducting conditions = 99.0 m,

and

hi for a 200-m duct under minimum spring ducting conditions = 199.0 m.

Figures 4.47 and 4.48 will be used to determine /12 from the total distance, d. The distance to be used is obtained by subtracting the dis- tance, do, from the desired distance. For the 100-m duct, do = 83 km and = 90 - 83 = 7 km.

Likewise for the 200-m duct, do = 72 km and

d = 90 - 72 = 18 km.

Thus it is found that for the model assumed

:

(a) /i2 for the 100-m duct height case will always be 3 m,

(b) /i2 for the 200-m duct height case will always be 18 m.

Since from (4.40)

hi < loss region < /i^ + h2.
162 CLIMATOLOGY

Therefore, in answer to part (b) of the problem, for maximum ducting

conditions with a 100-m-thick duct:

92.3 m < loss region < 3 + 100 m, or

92.3 m < loss region < 103 m.
INTRODUCTION

With a 200-m-thick duct

163

166 m < loss region < 18 + 200 m, or

166 m < loss region < 218 m.

For mean ducting conditions with a 100-m-thick duct:

96 m < loss region < 103 m. With a 200-m-thick duct:

187 m < loss region < 218 m.

For minimum ducting conditions with a 100-m-thick duct:

99 m < loss region < 103 m,

with a 200-m-thick duct:

199.3 m < loss region < 218 m.

4.6. Air Mass Refractive Properties

4.6.1. Introduction

The foregoing material on refractive index climatology concerned the

geographical variation of mean (or A^o) over the surface of the earth

and the three-dimensional distribution of mean N over North America.

This section develops another aspect of the climatological variation of the

refractive index: the mean profile of N in various air masses. The

meteorologist defines air masses as bodies of air in the troposphere having

approximately homogeneous character at the surface. An air mass is described in terms of its origin, as, for example, polar continental or tropical

maritime.

Profiles of and values of bending of radio rays are given in terms of departures from normal for various air masses. Recent studies have led to the evaluation of radar elevation angle

errors in different climates and air masses [43, 44]. In general, these

studies did not emphasize the relation between air mass refractive index

structure and the refraction of radio waves within the air mass. For

example, mean angular bendings for radio rays passing completely through
164 CLIMATOLOGY

Ol I \ \ I I I I I I 0 20 40 60 80 100 120 140 160 180 200

ANGLE OF ARRIVAL OR DEPARTURE, ^o,(mrad)

Figure 4.55. Comparison of total angular bending of radio rays within air masses.

After Schulkin [1952].

the earth's atmosphere are given for extremes of air mass types and a range of initial elevation angles, ^o, on figure 4.55 [43]. It is seen that

maritime tropical air produces 30 percent more bending than continental

polar air at initial angles or arrival or departures of about 10° (175 mrad)

and that this difference increases to about 70 percent at zero initial elevation angle. Further, the magnitude of the total bending increases

rapidly with decreasing values of ^o- Figure 4.55 does not make it clear that differences in radio ray refrac- tion arise from differences in the refractive index profiles of the two air masses. It is the purpose of this section to show that various air masses have characteristic refractive index profiles and that the radio ray refrac- tion within each air mass is mirrored by the difference of the actual re- fractive index structure from a standard atmosphere of exponential form.

r 4.6.2. Refraction of Radio Rays

The angular bending of a radio ray, ri,2 (see fig. 4.56) between two

points in the atmosphere of refractive indices Ui and n2 is given by the
REFRACTION OF RADIO RAYS 165

Figure 4.56. Geometry of radio ray refraction.

geometrical optics formula [45, 43],

cot e dn (4.41)

under the assumption that dn/n = dn and 6 is the local elevation angle.

It is also assumed that n is spherically stratified and concentric with the

earth. The value of cot B is determined from Snell's law, and ri,2 is completely determined from a knowledge of the variation of n with height.

An empirical formula to describe the long-term average variation of n
166 N CLIMATOLOGY

with height isN{h) = [n{h) - 1] 10^ = 313 exp {-^6.951 (4.42)

where h is the height above the earth's surface in kilometers and 313 is the

long-term average value of (n — 1) 10^ at the earth's surface for the United

States. In practice it is convenient to utilize this average N{h) function

to refer the observed N{h) data to the common level oi h = 0 by means of

Chapter 1 shows that A(h,3l3) is analogous in concept to potential temperature but utilizes the normal N gradient rather than the adiabatic

gradient of the potential refractive modulus of Lukes [46] and Katz [20], The notation A{h, 313) is used to indicate that A is determined from the

refractive index at the height h and the 313 exponential atmosphere. It has been found to be advantageous to use several atmospheres of exponential form in applications involving different climatic regions [47]. The

particular form of (4.43) emphasizes departures of structure from

normal as shown by recent studies of synoptic variations of N{h) about

frontal systems [21] and, in addition, permits direct calculation of radio

ray bending for any observed refractive index structure. When A (/i 313)

is introduced into (4.41) for ray bending, one obtains (see chapter 3, sec. 3.10).

which indicates that the bending can be regarded as the resultant of the

bending in the normal atmosphere, r(/i, 313), and a perturbation component representing departures of refractive index structure from that of the

313 exponential atmosphere. The value of T{h, 313) can be obtained from

refraction tables [9], and the perturbation term can be evaluated by

graphical methods to yield an overall accuracy of a few percent in esti- mating ri,2 [47]. Some A{h, 313) profiles were prepared from typical temperature, pressure, and humidity distributions within a range of air masses as published

in the literature [48, 49, 50] and climatic summaries of upper air data [31]. Two of these profiles, one for maritime tropical air and the other repre- senting continental tropical air, are plotted on figures 4.57 and 4.58 to represent the extremes of A{h, 313) profiles. These profiles clearly show

that the two air masses have quite different refractive index structures. The difference is most pronounced near the ground. At heights of 20

km, however, A{h,313) rapidly approaches the asymptotic value of 313,

A(h, 313) = Nih) 313 [1 - exp {-V6.951]. (4.43)

'A. (4.44)
REFRACTION OF RADIO RAYS 167

40

36

32

SAN JUAN, W.I. (July Average)

310 320 330 340 350 360 370 380 390

A(h, 313), N UNITS

I \ \ \ L_ \ \ \ I -8 •6 -5 -4 -3 -2

T313-T (mrad)

Figure 4.57. Comparison of A(h,313) profiles with departures from normal hendiiig of radio waves for maritime tropical air. After Byers [1944].
168 N CLIMATOLOGY

40

36

32 -

28

1 EL PASO, TEXAS

(June Average)

310 320 330 340 350 360 370 380

A(h,3l3),N UNITS

I \ \ \ __L \ \ I -2 -1012 3 4 5

T3,3-T(mrad)

Figure 4.58. Comparison of A(h,313) profiles with departures from normal bending

of radio waves for continental tropical air. After Ratner (1945).
REFRACTION OF RADIO RAYS 169 regardless of air mass type. The radio ray bending is also plotted on

these figures relative to the value expected in the N{h) = 313 exp

{—h/6.9b} atmosphere. Near the ground, bending departures are seen

to be mirrored by the A unit profiles. That is, a negative gradient of N or A produces a positive increase of bending. Above a few kilometers,

however, the bending departures approach a fixed, usually nonzero, value.

It is apparent that the asymptotic value of T313 — r at large heights is determined by the bending in the first few kilometers, where 60 to 75

percent of the low-angle bending normally occurs. The marked differ- ences in air mass n structure and bending are confined, therefore, to the

first few kilometers above the earth, as is illustrated by figure 3.16. The

distribution of the meteorological elements within each air mass is re- flected in the A{h, 313) profiles. For example, the steep humidity

gradient characteristic of maritime tropical air is reflected by the rapid

decrease of A{h, 313) with height. Comparatively, the high ground-level

temperatures and rapid decrease of temperature with height in continental

tropical air are reflected by the increase of A{h, 313) with height within

that air mass. These, and -profile characteristics of other air masses,

are listed in table 4.11 and are evident from the form of the equation for the refractive index at radio frequencies [see chapter 1, (1.20)].

Table 4.11. Refractive characteristics of air masses.

Air Mass Ref- erence Meteorological characteristics Refractive characteristics

Superior (S) S/mT, [48] typical of Gulf coast (Lake Charles, La.). Formed from subsidence of highlevel air with resulting dry adiabatic temperature lapse rate and low humidity. Often found overlying other, more humid air masses that show a re- sultant characteristic drying with

height in the lower levels.

Rapid decrease of N and A

with height in the lowermost

layer produces a superrefractive* layer. The overlying

superior air is nearly normal in bending characteristics.

Continental Polar (cP) [49] source region. Characterized by low temperature and humidity with a pronounced temperature inversion at the surface created by progressive nocturnal cool- ing during the arctic night.

Ground-based superrefractive layer arising from the temperature inversion.

Maritome Polar (mP), [48] typical low-level ground modification

(Seattle, Wash.).

Cool air, nearly saturated to a height

of several kilometers. Example shows

typical drying in lower levels when

this air mass moves over land. Overlying mP has increasing humidity and

decreasing temperature with height.

Surface layer produces near- normal bending. Increasing N in the overlying mP air pro- duces an elevated subrefractive layer.

Continental Tropical [51] (cT) source region

(El Paso, Tex.).

Characterized by superheated lower

layers, rapid decrease of temperature with height, and very low humidity.

Strong temperature lapse produces a subrefractive layer reaching to several kilometers. Maritime Tropical [48] (mT) source region

(Pensacola, Fla.).

Relatively warm air, high water- vapor content in lowest layer, which

decreases rapidly with height. Moderate changes in temperature and hu- midity structure produce large re- fractive gradient changes.

Strong superrefraction to great heights arising chiefly from rapid decrease of hu- midity with height.

*Normal refraction is taken to mean t(/i, 313), superrefraction to mean r(/i)>r(/i, 313), and subrefraction T{h)<T{h, 313).
170 CLIMATOLOGY

The average decrease of pressure, temperature, and humidity with

height produces a normal decrease of A^. If, however, the temperature

increases with height, as in a temperature inversion, decreases more

rapidly and radio rays are superrefracted, dependent, of course, upon the

vapor pressure. Conversely, an unusually rapid decrease of temperature,

or an mcrease of humidity, with height produces a subnormal decrease of with height, or subrefraction of radio waves. In any event it is evident

from these figures that ^-profile effects on ray bending are most pro- nounced at ^0 = 0, are significantly less pronounced at = 52 mrad

('~3°), and continue to diminish with increasing until, at = 7r/2, there is no bending at all and consequently all departures are zero.

4.6.3. Conclusions

The work of Schulkin and others has shown that characteristic total bending differences in radio ray refraction exist between various air masses. The present study extends Schulkin's conclusion by identifying

abnormal bending of radio rays with departures of refractive index struc- ture from average in the lowermost layers of the air masses. Consideration of departures of both ray bending and refractive index structure from

their value in a standard exponential atmosphere results in a suitable

method of cataloging air masses in terms of either refractive index struc- ture or bending characteristics.

4.7. References

[1] Resolution No. 71 (1948), Conf. of Directors, Internat. Meteorol. Organization

(Lausanne, Switzerland).

[2] World Atlas of Sea Surface Temperature Charts (1944), Hydrographic Office Publ. 225, chap. 4, No. 2.

[3] Atlas of climatic charts of the oceans (1938), U.S. Weather Bureau Publ. 1247, Washington, D.C.

[4] Johnson, W. E. (Nov.-Dec. 1953), An analogue computer for the solution of the

radio refractive index equation, J. Res. NBS 51, No. 6, 335-342.

[5] Gerson, N. C. (1948), Variations in the index of refraction of the atmosphere,

Geofis. Pura Appl. 13, 3-4.

[6] Bean, B. R. (Apr. 1953), The geographical and height distribution of the gradient

of refractive index, Proc. IRE 41, No. 4, 549-550.

[7] Misme, P. (Nov.-Dec. 1958), Essai de radiochmatologie d'altitude dans le nord

de la France, Anna. Telecommun. 13, No. 11-12, 303-310.

[8] Smithsonian Meteorological Tables (1951), Table 63, Sixth Revised Ed., Book

(Washington, D.C.).

[9] Bean, B. R., and G. D. Thayer (1959), CRPL Exponential Reference Atmosphere,

NBS Mono. 4. [10] Pickard, G. W., and H. T. Stetson (1959), Comparison of tropospheric reception,

J. Atmos. Terrest. Phys. 1, 32-36.

[11] Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception

at 44.1 Mc with 92.1 Mc over the 167-mile path of Alpine, New Jersey to Needham, Mass., Proc. IRE 38, No. 12, 1450.
REFERENCES 171

[12] Bean, B. R. (1956), Some meteorological effects on scattered radio waves, IRE

Trans. Commun. Syst. CS4, 32-38.

[13] Onoe, M., M. Hirai, and S. Niwa (Apr. 1958), Results of experiment of long- distance overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79.

[14] Gray, R. E. (Jan., Feb., Mar. 1959), The refractive index of the atmosphere as a

factor in tropospheric propagation far beyond the horizon, IRE Natl. Conv.

Record, Pt. 1, 3-11 (1957), Elec. Commun. 36, No. 1, 60.

[15] Norton, K. A. (Mar. 1956), Point-to-point radio relaying via the scatter mode of tropospheric propagation, IRE Trans. Commun. Syst. CS-4, 39-49.

[16] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave

propagation, Proc. IRE 21, 427-463.

[17] Anderson, L. J. (1958), Tropospheric bending of radio waves. Trans. Am. Geophys. Union 39, 208-212.

[18] Snedecor, G. W. (1946), Statistical Methods, Book, 4th Ed., pp. 97-98 (Iowa

State College Press, Ames, Iowa).

[19] Bean, B. R., and L. P. Riggs (July-Aug. 1959), Synoptic variation of the radio

refractive index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97.

[20] Katz, I. (1951), Gradient of refractive modulus in homogeneous air, potential modulus. Propagation of Short Radio Waves by D. E. Kerr, Book, pp. 198-199 (McGraw-Hill Book Co., Inc., New York, N.Y.).

[21] Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the

vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio

Prop.), No. 2, 249-258.

[22] Jehn, K. H. (June 1960), The use of potential refractive index in synoptic-scale radio meteorology, J. Meteorol. 17, 264.

[23] Bean, B. R., and J. D. Horn (Nov.-Dec. 1959), The radio refractive index near

the ground, J. Res. NBS 63D (Radio Prop.), No. 3, 259-273.

[24] duCastel and P. Misme (Nov. 1957), Elements de radio climatologie, L'Onde

Electrique 37, 1049-1052.

[25] Hay, D. R. (Dec. 1958), Air mass refractivity in central Canada, J. Phys. 36, 1678-1683.

[26] Tao, K., and K. Hirao (Mar. 1960), Vertical distribution of radio refractive index

in the medium height of atmosphere, J. Radio Res. Labs. 7, No. 30, 85-93.

[27] Fehlhaber, L., and J. F. Grosskopf (1959), Beitrage zur Radioklimatologie

Westdeutschlands, technische Bericht 5546, FTZ.

[28] Bean, B. R., and R. M. Gallet (Oct. 1959), Applications of the molecular re- fractivity in radio meteorology, J. Geophys. Res. 64, No. 10, 1439-1444.

[29] Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1960), CHmatic charts and data

of the radio refractive index for the United States and the world, NBS Mono. 22.

[30] Report 147 (1959), IX Plenary Assembly, International Radio Consultative Committee, pp. 299-337 (Los Angeles, Calif.).

[31] Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric

refraction and its relation to wave guides and diffraction. Book, Meteorological Factors in Radio Wave Propagation, pp. 80-127 (The Physical Society and

Roy. Meteorol. Soc, London, England).

[32] Bremmer, H. (1949), Terrestrial Radio Waves, Book, pp. 131-138 (Elsevier Publ. Co., New York, N.Y.).

[33] Cowan, L. W. (1953), A radio climatological survey of the U.S., Proc. Conf. Radio

Meteorol. (Univ. of Texas, Austin, Tex.).

[34] Kerr, D. E. (1951), Propagation of Short Radio Waves, Book, pp. 9-22 (McGrawHill Book Co., Inc., New York, N.Y.).

[35] Bean, B. R. (May 1954), Prolonged space wave fadeouts at 1046 Mc observed in Cheyenne Mountain propagation program, Proc. IRE 42, 848-853.
172 N CLIMATOLOGY

[36] Misme, P. (Apr. 1956), Methode de mesure thermodynamique de I'indice de

refraction de I'air-description de la radiosonde MDI, Ann. Telecommun. 11, No. 4, 81-84.

[37] Price, W. L. (July 1948), Radio shadow effects produced in the atmosphere by

inversions, Proc. Phys. Soc. 61, 59-73.

[38] Norton, K. A. (Dec. 1941), The calculation of ground-wave field intensity over a

finitely conducting spherical earth, Proc. IRE 29, 623-639.

[39] Kirby, R. S., J. W. Herbstreit, and K. A. Norton (May 1952), Service range for air-to-ground and air-to-air communications at frequencies above 50 Mc,

Proc. IRE 40, No. 5, 525-536.

[40] Ikegami, F. (Aug. 1959), Influence of an atmospheric duct on microwave fading,

IRE Trans. Ant. Prop. AP-7, 252-257.

[41] Doherty, L. K. (1952), Geometrical optics and the field at a caustic with applications to radio wave propagation between aircraft. Res. Rept. EE138 (Cornell

Univ., Ithaca, N.Y.).

[42] Airy, G. B. (1938), On the intensity of light in the neighbourhood of a caustic, Cambridge Phil. Trans. 6, 379-402.

[43] Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere,

Proc. IRE 40, No. 5, 554-561.

[44] Fannin, B. M., and K. H. Jehn (May 1957), A study of radar elevation angle

error due to atmospheric refraction, IRE Trans. Ant. Prop. AP-5, 71-77.

[45] Smart, W. M. (1931), Spherical Astronomy, ch. 3, Book (Cambridge Univ. Press, London, England).

[46] Lukes, G. D. (1944), Radio meteorological forecasting by means of the thermo- dynamics of the modified refractive index. Third Conf. on Prop., NRDC,

pp. 107-113 (Committee on Propagation, Washington, D.C.).

[47] Bean, B. R., and E. J. Button (May-June 1960), On the calculation of departures

of radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3, 259-263.

[48] Byers, H. R. (1944), General Meteorology, Book, pp. 255-277 (McGraw-Hill

Book Co., Inc., New York, N.Y.).

[49] Willett, H. C. (1944), Descriptive Meteorology, Book, pp. 190-220 (Academic

Press, Inc., New York, N.Y.).

[50] Trewartha, G. T. (1943), An Introduction to Weather and CHmate, Book, pp. 206-215 (McGraw-Hill Book Co., Inc., New York, N.Y.).

[51] Ratner, B. (1945), Upper air average values of temperature, pressure, and relative humidity over the United States and Alaska (U.S. Weather Bureau).
Chapter 5. Synoptic Radio Meteorology

5.1. Introduction

This chapter treats the variation of refractive index structure in the

troposphere with synoptic tropospheric disturbances. Within the scope

of the synoptic field are timewise and spacewise variations in the atmosphere from microscale fluctuations to broad-scale systems of weather map

dimensions.

The microscale fluctuations of the refractive index are those that one would expect to observe at a particular point along a radio path that

reflect detailed terrain and weather conditions in the immediate vicinity

of, say, the transmitter or receiver site. Mesoscale variations, by way of contrast, are those which cover tens

of kilometers and thus encompass a substantial portion of a radio path.

Examples of this type of variation are land-sea breeze effects and con- vection cells of thunderstorm activity.

Large-scale weather systems, affecting vast areas, perhaps even on a

continental scale, fall under the classification of macroscale variation.

Examples of this type of activity in the atmosphere are sweeping air mass

changes and frontal systems traversing thousands of kilometers on the

earth's surface. A detailed analysis of such a system is given as an illus- tration later in this chapter.

5.2. Background

The problem of determining the vertical and horizontal distribution of the radio refractive index has engaged the attention of radio meteorologists on an international scale for the better part of two decades [1, 2, 3, 4, 5,

6].'By analysis of current synoptic conditions from standard weather

charts, one may ascertain the air mass type appearing over a given region

and, likewise, may predict with reasonable accuracy the air mass type

expected over a particular locale in, say, 24 hours. Then, from the air mass profile characteristics table of section 4.4, one is able to estimate

what the departures of refractive index (and radio ray bending) from

normal will be over a certain region. The bending predictions permit an

1 Figures in brackets indicate the literature references on p. 224. 173
174 SYNOPTIC RADIO METEOROLOGY

estimate of refraction errors and the introduction of appropriate correc- tions for radio range and elevation angle errors for radio navigational

equipment.

VHF-UHF radio field strengths beyond the normal radio horizon will also differ from air mass to air mass. It has been known for many years

that the seasonal cycle of VHF radio field strengths received far beyond

the normal radio horizon were correlated with the refractive index [7, 8,

9, 10] and that significant changes in field strength level are observed from

air mass to air mass [11, 12, 13.] Speaking about signal level on a 60

Mc/s beyond the horizon radio path near Boston, Mass., Hull [11] states that during the winter low signal levels prevail during the presence over

the path of fresh polar air. Periods of high signal occur when a cold, dry

polar air mass is overrun by warm, moist air of tropical maritime origin.

Hull's analysis represents early recognition of refraction and reflection phenomena on a synoptic scale. Later work on seasonal changes of fields and represents, in a way, a summary of synoptic conditions over a

period of time.

Gerson [2] was one of the first to consider the variation of the radio

refractive index, n, in terms of seasonal and air mass changes. Gerson

divided n into two parts, one density-sensitive and the other moisturesensitive. This division is equivalent to the wet- and dry-term separation

of A/' = (n — 1) 10^ in chapter 4, section 1. Gerson was able to measure

seasonal thermal changes by variation in the dry term and seasonal

moisture changes by observed variation in the wet term. Gerson pre- pared graphs showing a sinusoidal variation of the dry term with a warm

season trough and a cool season crest, indicating density changes in inverse

proportion to the temperature. The wet-term component of n, on the

other hand, was observed to attain its maximum during the warm season when the dry term was at its minimum. In arctic and antarctic locales,

the surface variation of the wet term was found to be quite small while in temperate and tropical climates there was a sizable annual variation of the moisture component. As a pertinent aside, Yerg [14] showed that even during the long, cold, arctic night, vertical variations in moisture

made significant contributions to the A^ profile. Apparent ducting

gradients, obtained by neglecting the wet term at low temperatures may,

in actuality, be only slightly more refractive than standard.

Continuing his investigation, Gerson next turned to the analysis of refractive index changes within various air masses. Using air mass data

available in the meteorological literature of the day, he charted mean

refractive index profiles for different air mass types. The largest initial values and also the largest vertical gradients of n occurred with tropical

maritime air. The air mass with the weakest gradients and, therefore,

the poorest refraction properties was found to be the polar continental

type. Common to all of Gerson's air mass refractive index graphs is an
BACKGROUND 175 approximately exponential decrease of n with respect to height. Schelleng, Burrows, and Ferrell [15] attempted to remove this systematic n

decrease with height by utihzing a correction factor of linear form,

although, as has been seen in chapter 3, this leads to serious overestimation of refraction effects in the more modern problem of satellite tele- communications.

The work of Hay [6] confirmed and extended the observations Gerson

had made concerning air mass profiles. Measuring air mass characteristics at Alaniwaki, Quebec, Hay concluded that each large-scale air mass

type in central Canada has a distinctive refractive index profile. Hay

determined the height distribution of X for four basic air masses by fitting a second-degree polynominal to each of the four sets of air mass data,

indicating that an average .V profile cannot be approximated effectively

by a linear curve except over , small height increments. Hay obtained

further discrimination by constructing a ''dry-term" curve for each air mass. The dry-term curves display a smaller standard deviation than

the total .V curves for all air masses except the continental arctic which

is, of course, itself very dry. The largest variations in total are due

to fluctuations in the wet term. The saturation vapor pressure is approximately an exponential function of the temperature, so that during the

warm season normal temperature changes cause the saturation vapor

pressure and, therefore, the wet term is vary sharply.

Contained also in Hay's paper [6] is a discussion of the use of profiles to estimate corrections for radio ray refraction by use of a table of mean effective earth's radius factors for each of four air masses within

95 percent probability limits. Misme [5, 16] has had an interest in synoptic radio meteorology in connection with telecommunications networks in France and North Africa. Interests of this French radio engineer include vertical gradients of the

refractive index and atmospheric reflection of radio waves. A reasonable question at this point is, to what extent are model ex- ponential atmospheres applicable to various climatic zones around the

earth? While it is readily apparent that individual profiles vary widely

from any sort of exponential norm, there is an increasing backlog of experimental evidence [17] that shows that long-term averages of synoptic

refractive index variations do tend toward an exponential form with

respect to height. Indeed, the 5-year mean N profiles of figures 5.1 and

5.2, representing arctic and tropical climates respectively, bear out this contention by showing close agreement with models developed from midlatitudinal data. Indeed, these examples verify the results obtained by

Gerson [2] for various air masses. Randall [4] investigated the relationship of surface meteorological data

to surface A^, Ns, and field strength in the FAI frequency band. The

results described were draAvn from a very limited data sample that covered
176 SYNOPTIC RADIO METEOROLOGY

ATMOSPHERE, =332.5

0 2 4 6 8 10 12 14 h(km)

Figure 5.1. Mean N distribution for Isachson, N.W.T., Canada.

less than a month during the summer of 1947. Within the framework of

this speciahzed study, however, Randall found that polar continental air masses were associated with low field strengths and low Ns while tropical

maritime air masses were associated with high field strengths and high

Ns, as shown on figure 5.3. Randall advanced the hypothesis that the

observed field strength changes were due to the existence of characteristic

N profiles typical of each air mass type. Randall was curious also as to the behavior of N and radio fields during the passage of fronts and squall

lines. Figure 5.4 shows the results of this investigation and indicates that
BACKGROUND 177 definite field strength changes may occur during frontal and squall line passage. Caution should again follow in the interpretation of these

results since they represent but a single example of frontal and squall line passage. Gray [18] made measurements of correlation of N and the gradient on N with path losses at UHF frequencies for observing points representative

of various climatic areas around the world and concluded that changes in transhorizon telecommunications are strongly dependent on atmospheric

changes.

0 2 4 6 8 10 12 14 h (km)

Figure 5.2. Mean N distribution for Canton Island, Pacific Ocean.
178 SYNOPTIC RADIO METEOROLOGY

PLOTTING KEY

+ Average wind speed equol

to or more than IQnaph

cP Polar contlnentol air moss mP Polor maritime oir moss mT Tropical monfime air mass mT

+

cPamP

+

^mP cPamP

c

ml

f mT

PamT

*mT " mT

+ cP

f mP

f

cP

cPamT

--+ cP

cParnT mT mT

a mT fmT Ut

i

h + f cP cP cP

hcP

cPa mT

cPamr

320 340 350 360 370 380 39C

Hourly Mean Surface Refractivify Figure 5.3. Field strength versus refractivity.

Scatter diagram of select hourly median field strength vs. hourly mean refractivity July 17 to Aug. 8, 1947.

In another article Gray [19] considered radio propagation and related meteorological conditions over the Caribbean Sea. Utilizing the effective earth's radius factor as a representative index, Gray presented an empirical curve of annual median scatter loss versus effective propagation

distance designed to fit both Caribbean and temperate regions. Effective

distance as defined by Gray is the angular distance in radians multiplied

by the radius of the earth modified for normal refraction. Gray reports

that the refractive gradient in the first 100 m is in general the determining

factor in median scatter loss for transhorizon telecommunications, as one would expect from earlier refraction studies [20]. Other studies on refraction problems during the 1950's have led to systematic computation of refraction effects and to significant applications such as the evaluation of radar elevation angle errors in differing air masses and climates [21, 23]. Schulkin [21] advanced a practical and very

fundamental method for numerical calculation of atmospheric refraction

(radio ray bending) from radiosonde data. Figure 4.55 [after Schulkin]

gives mean angular bendings for radio rays passing completely through

the earth's atmosphere for two extremes of air mass type. Fannin and

Jehn concluded that a particular refractive index profile depends on air
BACKGPjjrXD 179 mass type and the climatic controls of season and latitude, as is substantiated by mean A' profiles for 34 weather observing sites located in or near

four distinct air mass source regions about the world. These data show

that a definite difference does exist in the profiles of various air masses. Refraction effects were found to be largest ia tropical maritime air, inter- mediate in polar continental and polar maritime, and least in tropical

continental. Fannin and Jehn also published graphs showing day-to-day

variations ia profiles representing the effects of air mass changes over a given observing station and graphs of diurnal profile variations.

In other studies along these lines, Bean. Horn, and E:^g- ^23', demonstrated that radio ray refraction within the lower layers of an air mass is mirrored by the difference between the observ^ed refractive index structure and that of a standard atmosphere. Figure 5.5 shows a graph of bending,

r, plotted together with a n. ;i:- : ::::: 'iv- ::.d-x profxle for an example

of sommertune tropical mar:::mc a:r. I: appard.: that near the grotmd

bending departures reflect refractive index profile departures from standard. Figure 3.16 presents a series of graphs of departures of refractive

3J

II

I!

1.7

f

s t

12 Ccd pijsses — Xiy 9-1 S-

.7 Sq<-oii !.:^^ cnssses

pcT) I320 >jti i9

:3C 63-:

C.9

C.7

_ 5 -3

'^-i posses Ricftmond _4 2i30Jul>i9 -2

390

FlGTriZ

Hourly //'f J" S^'"o:e ^e'^ac'rvify

Field strength and N changes during frontal passage.

:. : ~.z.r passage of a add fnmt sjstem over the WasMngton-Eiclimond path and graph

i: ::f:in.g the fluctuation, oi honrly median field strength with time.
180 SYNOPTIC RADIO METEOROLOGY

index and bending from normal for each air mass and emphasizes the

close relation between the two. This affords the synoptic radio meteorologist a set of standard reference profiles for the study of a given air mass

or the confluence of contrasting air masses at a frontal zone. Arvola [24] discussed the changes in refractive index profiles caused by

migratory weather systems. Examining a series of synoptic situations in the midwestern portion of the United States during November 1951 that

gave rise to greater-than-normal refraction, Arvola found that ridges and

accompanying subsidence effects generally gave rise to strong gradients

and enhanced signal strength over a 200-km link broadcasting at 71.75

Mc/s. Refractive gradients were stronger in the warmer air masses and

at times when moist air was present below the inversion created by the

subsidence mechanism. Strong gradients which appeared behind a squall

line later weakened with the approach of a cold front. After the passage

of this front stratification in the cold air again increased the gradient.

Subsequent investigations of polar continental air across central North

America make use of reduced-to-sea-level forms of the radio refractive

index as synpotic parameters. The reduced forms are sensitive indicators of synoptic changes and afford a clearer picture of storm structure than that obtained using analyses in terms of unreduced N or B units

(defined in chapter 1). Later portions of this chapter consider in detail

these two synoptic parameters.

Jehn [25], at the University of Texas, used a form of potential refractive

modulus, K, developed by Lukes [26] and Katz [27] to account for the

height dependence feature of the refractive index. Articles by Jehn

[28, 29] on synoptic climatology use composite analysis techniques to study the synoptic properties of the Texas-Gulf cyclone and the central United States type of cold outbreak. In another application of the potential refractive modulus, Flavell and

Lane [30] have utilized Katz's K unit (see chapter 6) to study tropospheric wave propagation. Field strength measurements on VHF-UHF trans- horizon radio links over the British Isles are analyzed in terms of cross sections in terms of K and regional charts of AK, the difference between

K at the surface and Kgsombar. These charts show features similar to those obtained by use of No or A (see chapter 1). The authors cite a series of measurements on a 500-km path at 877

Mc/s on which the received signal was ordinarily below the noise level. The singular occasions on which the received signal could be measured at the normal times of radiosonde ascent all exhibited a symmetric variation

of AK over the transmission path. These results lend credence to the

hypothesis that synoptic disturbances play an important role in trans- horizon telecommunications.

Moler and Arvola [31] advanced the hypothesis that the vertical

gradient of the refractive index is affected by broad-scale vertical motion
BACKGROUND 181

SAN JUAN, W. L

(July Average)

310 320 330 340 350 360 370 380 390

A(h,3l3), N UNITS

8 -7 -6 -5 •3 -2 0

T3,3-T (mrad)

Figure 5.5. Departures of t and N from normal for maritime tropical air.
182 SYNOPTIC RADIO METEOROLOGY

in the troposphere and suggest that moisture and temperature stratifica- tion is modified principally by changes in vertical velocity. This latter work was extended [32] by a study of mesoscale centers of horizontal air mass convergence and divergence in the troposphere. Horizontal con- vergence takes place in a low-pressure area, where the winds around the

low have a predominant component toward a local vortex at the center. Upward vertical motion results from the pile-up of air in the vortex region. In a high-pressure area, winds have a predominant component away from

the center of a high and subsiding air descending from higher levels takes

the place of air transported outward from the center of the high. Local

convergence, then, implies upward vertical motion in the lower levels of the troposphere, while local divergence implies downward vertical motion

in the lower levels. It follows that local convergence centers (smallscale low-pressure cells) produce updrafts in the atmosphere that result in considerable mixing and the destruction of atmospheric layers. Centers

of divergence (small-scale high-pressure cells) create strong temperature

and humidity inversions by the motion of subsiding air. Such inversions

produce large vertical refractive index gradients that partially reflect microwaves traveling through this meteorological environment [33]. In a well-mixed atmosphere, on the other hand, the primary propagation

mechanism is believed to be scattering by turbulent fluctuations of the

refractive index [34, 35]. Sea-level measurements showed iVs to be essentially invariant during

the experiment, yet signal levels ranged over more than 60 dB, a power

factor on the order of 10^. Since scattering theory would account for a

rise in signal level of only about 13 dB, Moler and Holden [32] conclude

that refractive layering and thermal stability over the oceans are principally functions of vertical wind velocities. These conclusions bear out

earlier ones of Saxton [36] and later ones of Flavell and Lane [30] stating

that high signal levels from a distant transmitter may well be the con- sequence of refractive layering and subsequent reflection of radio waves.

In the same article Saxton also considered both the scattering of radio

energy by turbulent eddies and the effects of superrefraction of radio waves produced by departures from normal of the height variation of the

tropospheric refractive index.

A knowledge of the vertical motion of the atmosphere becomes at this point central to the problem of refractive layering. A brief resume of the

Moler-Holden method for estimating the relative magnitude and direction

of the vertical component of the wind velocity follows. Moler and Holden postulate a model atmosphere bound by the follow- ing conditions:
BACKGROUND 183

(1) A barotropic level (level of nondivergence) exists at some pressure

level greater than 500 mbar.

(2) The horizontal wind velocity divergence changes sign at the level of nondivergence (LND).

(3) The vertical velocity vector in the troposphere is proportional to that at the LND.

(4) The vertical velocity vector (a) vanishes at the surface of the earth

and (b) approaches zero at the outer reaches of the atmosphere.

Commencing with the equation of continuity,

|f+F.Vp + pV.7 + ^ (pW) = 0, (5.1)

where p is the density of air, V is the horizontal del operator, V Is the

horizontal wind velocity vector, W is the vertical wind speed, and z is the

vertical coordinate; and employing horizontal velocity divergence in the

natural coordinate system of the form

F = ^ + .f, (5.2)

ds dn

Moler and Holden derive for the magnitude of the vertical vector at the

LND, upon employing the vorticity equation,

(I) (11)

P LJlnd f at Jlhd { as

(III)

where f is the vertical component of the absolute vorticity. Moler and

Holden reason that of the labeled integrals (I) decreases in proportion

to the height above the LND and is « (II) at 300 mbar. Since W « V,

(III) makes only a small contribution. The expression for (TF)lnd be- comes

P J LND r ds \dx dy/
184 SYNOPTIC RADIO METEOROLOGY

Figure 5.6. Isobaric and streamline maps.

which simplifies to the approximate relation

(W) = relative magnitude oi^ — T^i (5.5) ^ dS \dX dy/200 mbar

for purposes of calculation.
REFRACTIVE INDEX PARAMETERS 185 —50

Figure 5.7. X-hand signal level versus time.

Santa Barbara X-band signal level for 26 Mar. 1958.

Moler and Holden then continue with a description of the large signal

enhancement and deep fading that occur as the propagation mechanism

varies between partial reflection and scattering on a transhorizon radio

path along the California coastline. Reflection occurs when strong re- fracting layers are present within a kilometer of the surface, typifying

meteorological conditions generally associated with the subsidence inversion frequently found along the California coast. Figure 5.6 shows the sea level pressure chart and a series of streamline

analyses by Moler and Holden depicting mesoscale centers of convergence

and divergence for a day in ^larch along the southern California coast. Figure 5.7 shows the X-band signal level received at Point Loma, San

Diego (SD) from Santa Barbara (SB) during the same day. As the

centers of convergence along the radio path weaken, refractive layers are formed and the signal level rises sharply during the middle of the day.

Later the signal level lowers again with the regeneration of convergence

centers and the destruction of stratified layers during the afternoon hours.

5.3. Refractive Index Parameters

In later sections, the analysis of a synoptic disturbance in the troposphere will be described in detail. Certain reduced forms of the refrac- tive index that will be useful in the ensuing discussion will be developed

here. These forms, already discussed in chapter 1, are revisited here for the purpose of comparison in synoptic example.
186 SYNOPTIC RADIO METEOROLOGY

Figure 4.1 shows contours of the mean value of N at the surface, Ns, determined from eight years of data for August, 0200 local time. Miniature circles indicate the 62 observing stations used to analyze this chart.

It is evident that coastal areas display high values of A^s as compared with

inland locations. Low values of Ns are apparent along the Appalachian

mountain chain and in the great mountain systems and inter-mountain

plateaus of the western United States. There is a marked similarity between the Ns contours on figure 4.1 and the elevation contours of figure

5.8. As a sensitive indicator of changes in atmospheric density, Ns dis- plays a strong elevation dependence. To remove this effect the reducedto-sea-level expression. No, was introduced in chapter 4 as:

No = Ns exp (5.6)

where z is height in kilometers and i/* = 7.0 km is the scale height.

Scale height is the height at which the mean value of N has decreased to a

fraction 1/e of its initial value. A scale height of 7.0 km is in close agree- ment with //* = 7.01 km for the NACA standard atmosphere with 80

percent relative humidity and H* = 6.95 km obtained from climatic

studies utilizing over two million observations of the variation in over

the first kilometer above the surface of the earth [20]. The A'o contours

Figure 5.8. Ground elevation above sea level.
REFRACTIVE INDEX PARAMETERS 187

LEGEND

COLD FRONT ^ ISOTHERM

R.H. ISOPLETH CLOUD PATTERN _

PRECIP PATTERN

DISTANCE, km

Figure 5.9. Idealized diagram of a fast-moving cold front.

of figure 4.4 utilize the same data as figure 4.1. The use of A^o produces a

simpler map with a smaller range of variation. Additionally, Ns may

easily be estimated from the smooth and slowly varying contours of A^o providing only that station elevation is known. It was shown in chapter

4 that A^s may be more accurately estimated from charts of No than

from charts of Ns itself by a factor of 4 or 5 to 1 [37]. The attempt to find a workable method to compensate for the decrease

of N with height has brought about the development of various model

atmospheres discussed in chapter 3. The paragraphs that follow will outline briefly steps in this development that are relevant to synoptic

studies. Vertical refractive index cross sections are standard working charts for synoptic studies. Such charts constructed from observed values of

suffer from a serious shortcoming in that the natural decrease of A^ with

respect to height effectively masks contrasts between air masses in the

lower troposphere. An idealized synoptic example depictmg the confluence of contrastmg air masses is presented on figures 5.9 and 5.10. When these idealized systems are analyzed in terms of A" as on figures 5.11 and 5.12 the most prominent feature is the laminar structure of the A^

field.
188 SYNOPTIC RADIO METEOROLOGY

LEGEND

WARM FRONT —^— ISOTHERM

R H. ISOPLETH CLOUD PATTERN • PRECIP PATTERN \\\\\\\\\\ AIR FLOW 9-

DISTANCE, km

Figure 5.10. Idealized diagram of a warm front.

DISTANCE IN KILOMETERS Figure 5.11. Idealized cold front in N units.
REFRACTIVE INDEX PARAMETERS 189

WARM FRONT

N ISOPLETH

DISTANCE, km

Figure 5.12. Idealized warm front in N units.

Early attempts to compensate for the decrease of N with height used

the constant gradient of the effective earth's radius theory, l/4a, where a

is the radius of the earth. As an illustration, the strong elevated layer

found during the summer in southern California was studied in terms of a form of (1.32) for B, given by

B = N{z) + (39.2)0, (5.7)

where N{z) is the value of at height z in kilometers [38]. Since N tends

to be an exponential function of height rather than the linear function

assumed by the effective earth's radius theory, the B unit approach overcorrects when z is greater than about 1 km.

This point is illustrated by figures 5.13 and 5.14 where the data of

figures 5.11 and 5.12 are plotted in terms of B units. Note that the overcorrection produces a cross section in which increases with height from

a value of 310 at the surface to 360 at 5 km. A function of exponential

form was designed to account for the systematic decay of density with

height that characterizes the terrestrial atmosphere, as given by

A = N{z) +313 [l - jexp (- 7^) (]

(5.8)
190 SYNOPTIC RADIO METEOROLOGY

-500 -400 -300 -200 -100 0 100 DISTANCE IN KILOMETERS

200 300 Figure 5.13. Idealized cold front in B units.

The quantity, A, enables one to discern departures of N structure from

the model atmosphere

N = 313 exp ]- (5.9)

Further, the radio-ray bending,

ri,2 = - L cot e dN • 10"', (5.10)

where B is the local elevation angle of the radio ray to spherically stratified surfaces of constant N, may be approximated by

71.2 = -

Ia'^^^^^-^-^

dA(z, 313) + t(z, 313). (5.11)

The term t{z, 313) is the bending in the average atmosphere given by (5.9), while the integral term represents the departures in bending produced by
REFRACTIVE INDEX PARAMETERS 191

300 400 500 600

DISTANCE IN KILOMETERS

900

Figure 5.14. Idealized warm front in B units.

various synoptic and air mass effects. The values of bending in the

average atmosphere are tabulated [39] and approximate methods of calculating the integral term in (5.11) to within a few percent have been given

[40].The next logical step is to plot the frontal cross sections, previously

analyzed in terms of N and B, in A units. This has been done on figures 5.15 and 5.16. The range of refractivity values on the new charts is re- duced from more than 60 to about 25 units and a pattern emerges that

displays sharp contrasts for air mass differences associated with the

frontal zone. Note, for the warm front case (fig. 5.16), that the A values

increase with height until they reach a maximum associated with the upgliding warm moist air overriding the frontal surface. The region of precipitation in advance of the front is shown as an area of high surface

N. In the cold-front case (fig. 5.15), the classic push of warm air aloft by the encroaching cold air is evidenced by the ''dome" of high A values

just before the front. Stratification in the cold air due to inversion

effects, although impossible to detect in the N charts, is clearly seen by

the use of A unit charts.
192 SYNOPTIC RADIO METEOROLOGY

LEGEND

COLD FRONT

A ISOPLETH A=Ni-NoCI-exp(chO WHERE No= 313

c = -.l43859

h= ht. in km.

DISTANCE IN KILOMETERS Figure 5.15. Idealized cold front in A units.

A units were used in subsequent cross-section analyses in order to throw frontal discontinuities and air mass differences into sharp relief. The Potential Refractivity Chart of figure 5.17 facilitates the rapid conversion of iV to A. This simplification eliminates the necessity of using

exponential tables for each individual calculation of A and thus lends considerable ease to the preparation of charts of the new parameter.

The reader probably has already observed that the A^'o and A correc- tions do substantially the same thing. Their primary distinction is that ^ is a nonlinear ''add-on" correction while A^o is a multiplicative one. The disparity between A^o and A is tabulated in table 5.1. These figures are obtained by taking the zero values of (for example,

300), subtracting the add-on correction for, say, 3 km (300 — 109 = 181),

and reducing this number, A^ = 181, to zero elevation by the A^o reduction.

No = N exp (3/7). The 313 exponential atmosphere is adopted for a

single reference atmosphere. The large discrepancies of table 5.1 may be

avoided for practical applications by choosing a model near to the mean

value of A^ of the site under study.

The A unit has the additional advantage that, while it is a convenient

method for height reduction, the ray bending is also readily recoverable
REFRACTIVE INDEX PARAMETERS 193

lOr LEGEND

WARM FRONT A=N+NoCI-exp(-l)j WHERE No=3l3 H=70 Km

330-

325

300 400 500 DISTANCE , km

600 700 800 900

Figure 5.16. Idealized warm front in A units.

from it, requiring only a knowledge of the altitude of the observed refrac- tive index measurement and the local elevation angle of the radio ray. The potential refractive modulus of I. Katz [27],

4> = Po +

Co (5.12)

where 6 is the potential temperature and eo the potential vapor pressure,

is also in current use. The constants 5 and c of (5.12) are given in the

development of modification to N data in chapter 1 (1.39). The potential

refractive modulus has been employed by Jehn [25] to study polar waves

over North America. Refractivity, A^, cannot accurately be recovered

from <f) for bending calculations unless additional information is available;

namely, observed temperature and vapor pressure. The concept of the

potential refractive modulus arose out of the earlier refractive modulus,

M (see ch. 1), which may be defined by

M = n - 1 + X 10^ Niz) + - X 10 (5.13)

where z = height above the earth's surface and a = earth's radius.
194 SYNOPTIC RADIO METEOROLOGY

170 190 210 230 250 270 290 310 330 350 370 390 410 430 N=(n-l)IO^

Figure 5.17. Potential refractivity chart.

Table 5.1. Differences between No and A at various elevations Value ofN at z=0

Elevation 250 300 313 350 400 (km)

1 -10 -2 0 5 13

2 -21 -5 0 7 29

3 -49 -22 0 5 31

4 -48 -29 0 30 68
A SYNOPTIC ILLUSTRATION 195 The M unit came into being out of an approach similar to that which

led to the development of the B unit. The condition dn/dz = — 1/a (a

radio duct) implies an effective earth of infinite radius (effective earth's radius factor, k = oo, see ch. 1). The M unit is designed so that dM/dz = 0 when k = oo. M units are employed from time to time in radio

meteorological analysis. The Canterbury Project [41], for example, used

M unit analyses in the study of ranges of over-water radar signals.

5.4. A Synoptic Illustration

The specialized field of synoptic radio meteorology attempts a description of the variations in atmospheric refraction that arise from large

scale weather changes such as the passage of a polar front or the movement of an air mass over a particular geographic region. The term air mass is used to describe a portion of the troposphere that has at the surface

generally homogeneous properties. Although no air mass is in fact homogeneous, the advantages of the air mass concept as a convenient

fiction are evident in the cataloging of meteorological observations for Climatic or synoptic purposes. The region of interaction between the cold air of the poles and the

warm air of the tropics is referred to as the polar front and is generally

located between 30 and 60°N. From time to time a section of the polar

front is displaced northward by a flow of warm tropical air while an adja- cent section is simultaneously displaced southward by a flow of polar

air. The interaction of the flow of polar and tropical air results in the

formation of a 'Vave" that moves along the polar front, often for thousands of kilometers. An example of a fully developed polar front wave is shown on figure 5.18(a), in the same manner that it would appear on a

daily weather map. Across the Great Plains and eastern seaboard of the

United States the polar front wave normally moves along the line AB in

figure 5.18(a). An idealized space cross section along the line AB is shown in figure 5.18(b). The warm tropical air that flows into the warm

sector of the wave overrides the cool air before the wave to form the

transition zone denoted as a warm front. The cold front represents the

transition between the generally humid air of the warm sector that has

been forced upwards and the advancing cold polar air. Squall lines are drawn to represent belts of vigorous vertical convection, intense thunder

showers, and sharp wind shifts that frequently precede fast-moving cold

fronts. The fronts shown on a daily weather map represent the ground

intersection of the transition zones between various air masses.^

2 The reader who wishes a critical appraisal of recent meteorological thinking on

fronts, air masses, squall Hnes, etc., is referred to Dynamic Meteorology and Weather

Forecasting, by Godske, Bergeron, Bjerknes, and Bundgaard, American Meteorological Society and Carnegie Institute of Washington, D.C., 1957.
196 SYNOPTIC RADIO METEOROLOGY

(b) Figure 5.18. The Polar Front wave.
SURFACE ANALYSIS IN TERMS OF A^o 197 As a representative example of the application of these newly developed

units to a synoptic situation, a large-scale outbreak of continental polar

air which took place over the United States during February 1952 is analyzed in terms of No and A [42, 43]. For this synoptic illustration the reduced expression, A^o, is used in preparing constant, level charts of the storm at the same levels and times

as those used in the daily weather map series of the U.S. Weather Bureau.

The A unit, on the other hand, is employed to construct vertical cross- sections through the frontal system to give a three-dimensional picture of the synoptic changes taking place.

5.5. Surface Analysis in Terms of No

A pronounced cold front developed and moved rapidly across the

United States during the period 18 to 21 February 1952. Prior to February 18, a polar maritime air mass had been moving slowly

eastward across the Great Basin and Rocky Mountain regions. This

system included a slow-moving cold front extending from northern Utah

southwards into Arizona and a quasi-stationary front extending northeastwards into Wyoming. With the outbreak of polar continental air east of the Rocky Mountains, the maritime front became more active

and, as it moved ahead of the fast-moving polar continental front sweeping across the Great Plains, was reported as a squall line by the time it crossed the Mississippi River early on the morning of February 20. During the latter stages of the storm system, the polar maritime cold

front-squall line was located in the developing warm sector of the polar

front wave. The entire ensemble of cold front, polar front wave, and

squall line then moved rapidly to the east coast by the morning of the

21st, thus completing the sequence. Charts of A^o were prepared from Weather Bureau surface observations

taken at 12-hour intervals from 0130 EST, February 18 until 0130 EST,

February 21, 1952, or, in other words, the period of time that it took the

polar front wave to develop and move across the country. The synoptic

sequence is seen on figures 5.19 through 5.25., where contours of A^o are derived for various stages of the storm and compared to the superimposed

Weather Bureau frontal analysis. The same procedure of comparing

derived contours with the existing frontal pattern was followed throughout

the present example. Observations from 62 weather stations were used

in preparing the surface weather maps. Figure 5.19 indicates that the

cold front extending from Utah southward displays weak A^o changes

across the frontal interface. In the early stages of the sequence (figs. 5.19 and 5.20) this lack of air mass contrast is evidenced in another way

by the slight change of the position of the A^o = 290 contour encircling west Texas and New Mexico as the frontal system moves through that

area.
Figure 5.20. No chart for storm system 1330E 18 Feb. 1952.
SURFACE ANALYSIS IN TERMS OF A^o 199

Figure 5.22. No chart for storm system 1330E 19 Feb. 1952.
200 SYNOPTIC RADIO METEOROLOGY

Figure 5.24. No chart for storm system 1330E 20 Feb. 1952.
SURFACE ANALYSIS IN TERMS OF No 201

Figure 5.25. No chart for storm systems 0130E 21 Feb. 1952.

By comparison, the cold front sweeping down across the Great Plains

(figs. 5.20 to 5.22) has a rather marked No gradient across the front, due

in large measure to the northward flow of warm, moist air that forms a

definite warm sector by 1330 EST on the 19th. It is perhaps significant

that the A^o contours indicate that the various frontal systems are transi- tion zones rather than the sharply defined discontinuities of textbook

examples, a point that has been enlarged upon by Palmer [44]. Figures 5.22 to 5.25 trace the trajectory of this vigorous push of cold

air across the Gulf Coastal Plain and the southeastern states. The most

spectacular gradients on this map series are in the eastern half of Texas

where the marked contrast of cold, dry polar air of low A^o and warm

moist Gulf air of very high A^o occurs. A prominent feature of meteorological significance on figures 5.20 to 5.23 is the northward advection of tropical maritime air in the warm sector ahead of the cold front. The

advection is evidenced by the northward bulge of high A^o over the

Mississippi Valley on both charts. Figure 5.25 shows the synoptic situa- tion as the front moves off into the Atlantic Ocean and refractivity

gradients across the continent gradually weaken.

The variation of A^o due to the passage of the frontal system can be

seen on figures 5.26 to 5.30 where the 24-hour changes of A^o have been

contoured. The 24-hour change, designated AVo, is obtained by subtracting from the current value the value of Nq observed 24 hours ago.
202 SYNOPTIC RADIO METEOROLOGY

Figure 5.27. 24-hour ANo chart, 1330E 19 Feb. 1962.
SURFACE ANALYSIS IN TERMS OF A^o 203

Figure 5.29. 24-hour ANo chart, 1330E 20 Feb. 1962.
204 SYNOPTIC RADIO METEOROLOGY

Figure 5.30. 24-hour ANo chart, 0130E 21 Feb. 1952.

The change is determined on a 24-hour basis in order to remove effects of the diurnal cycle of Nq. The ATVo charts show a general rise of A^'o in the

warm sector and a drop in Nq behind the front amounting, in the warm

sector, to 35 to 40 units by 1330 EST on February 20 (fig. 5.29) accompanied by a 40 to 50 A^" drop behind the front. The relative sensitivity of A^o to humidity changes is emphasized by

the AA^o charts. The N drop behind the cold front occurs in a region of increasing pressure and decreasing temperature—a combination that

increases the dry term and depresses the wet term. The decrease in the

wet term from rapidly dropping dewpoint more than compensates for the

increased dry term. As an example, in the 24-hour period ending 0130

EST on the 19th, the station pressure at Oklahoma City increased 13 mbars.

The dry term increased 12 N units while the wet term dropped 42 A^ units,

giving a net change of minus 30 A^ units. This A^o rise in the warm sector and the drop behind the cold front is consistent throughout the development of the polar front wave and appears to be what one would expect

for this type of weather system. The present system had about a 35 A^ unit rise and a 40 A^ unit drop. This general pattern might be expected

to occur in all fast-moving cold fronts with varying intensity, depending

upon the individual synoptic pattern. In any case, it appears that the

Nq pattern is a sufficiently stable and conservative property of the atmosphere so that it should be possible to develop forecasting rules for A^o, but not, of course, without the analysis of many more A^ patterns.
CONSTANT PRESSURE CHART ANALYSIS 205 5.6. Constant Pressure Chart Analysis

The same frontal system as above was analyzed for selected constant

pressure levels. The 850 mbar charts (about 5,000 ft above mean sea

level) and the 700 mbar charts (about 10,000 ft above mean sea level) were prepared for the times of radiosonde ascent (10 A.M. and 10 P.M.

EST) throughout the synoptic sequence from the radiosonde reports of 43 U.S. sounding stations. It is not necessary to reduce the 850 mbar

or 700 mbar level data, since it is already referenced to the indicated constant pressure level. Contours for the charts aloft are shown on

figures 5.31 to 5.36 while their respective 24-hour changes, A^85o and iVyoo, are given on figures 5.37 to 5.40. The A^85o charts show that the northerly flow of warm humid air within

the warm sector that was so prominent on the A^o maps is also clearly in evidence at the 850-mbar level. Further, a change pattern similar to that on the iVo maps is also observed at the 850-mbar level, particularly on figure 5.34. That is, a rise in Nsbo values in the warm sector and a

decrease behind the cold front is apparent. Surprisingly enough, by the

time the frontal system is well-developed, at 1000 EST on the 20th, the

A^85o values are nearly as large as those on the surface. The A^7oo charts are more difficult to interpret than those of A^o or

A'850. It appears that at this altitude the wet term is usually negligible

and N will normally vary inversely as temperature since the pressure is, of course, constant at the 700-mbar level. By 1000 EST on the 19th (fig. 5.34) an intrusion of low N values is observed in the 700-mbar warm sector due to the advection of warm, low-density air northwards. The chart

for 24 hours later (fig. 5.42) displays two prominent highs in which

Ntoo = 225. One of these highs lies between the squall line and the cold

front and the other just south of the apex of the 700-mbar wave. Inter- estingly enough, these two highs are due to quite different causes. The

high centered over Atlanta appears to have arisen from the unusually high

transport of moisture to the 10,000-ft level, since the 700-mbar wet term

at Atlanta increases from 4.5 to 25 A^ units in the 24-hour period ending

with 1000 EST on February 20. The second high, centered over Omaha,

appears to be due to an intense dome of cold air, as indicated by the drop

of the 700-mbar temperature from -7.3 °C to -21.4 °C in 12 hours

preceding map time. When temperatures are below 0 °C, the wet term

contribution to N is quite small and density changes become significant

in producing changes in A^. Falling temperatures produce higher density

air and, consequently, a region of high A^ values around Omaha as depicted on the A^7oo chart of figure 5.31 and AA700 chart of figure 5.40 which

shows this change more clearly.

1
206 SYNOPTIC RADIO METEOROLOGY

Figure 5.32. N700 chart, lOOOE 18 Feb. 1952.
CONSTANT PRESSURE CHART ANALYSIS 207

Figure 5.33. Ngso chart, lOOOE 19 Feb. 1952.

Figure 5.34. N700 chart, lOOOE 19 Feb. 1962.
Figure 5.36. N700 chart, lOOOE 20 Feb. 1952.
CONSTANT PRESSURE CHART ANALYSIS 209

Figure 5.37. 24-hour ANgso chart, lOOOE 19 Feb. 1952.

Figure 5.38. 24-hour AN700 chart, lOOOE 19 Feb. 1962.
Figure 5.40. U-hour AN700 chart, lOOOE 20 Feb. 1952.
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 211 5.7. Vertical Distribution of the Refractive

Index Using A Units

The synoptic study of the vertical distribution of the radio refractive

index extends the foregoing constant-level analyses by considering the

problem of whether the air mass properties associated with this typical

wintertime outbreak of polar air are reflected in the vertical refractive

index structure. Charts showing the structure of the storm have been

prepared using radiosonde measurements from stations located along a

line normal to the frontal zone between Glasgow, Mont., and Lake

Charles, La. (fig. 5.41). Plots of N versus height along this cross section

line were obtained at 12-hour intervals during a 4-day period and con- verted to A units. Figure 5.42 is an example cross section along the Glasgow-Lake Charles

line analyzed in terms of unmodified A'" as in the idealized cases of figures 5.11 and 5.12. Compare this figure with the A unit analysis of figure 5.48 for the highlighting of air mass differences refractive-index-wise.

Examples of the distribution of components, temperature and humidity,

around the front are charted on figures 5.43 and 5.44. Various stages

in the advance of this intense storm system across the continent are represented by figures 5.45 to 5.50 in terms of modified N {A units). At

the outset of the period of observation (fig. 5.45), the polar front was

located over the northern Great Plains, between Rapid City, S. Dak.

Figure 5.41. Station identification chart.
212 SYNOPTIC RADIO METEOROLOGY
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 213 and North Platte, Xebr. At this time the entire cross section is characterized by weak to moderate gradients of refractivity. On figure 5.46 12 hours later, this front had moved some 300 km southward. The

contrast of the southward push of polar air and the northerly advection

of tropical maritime air from the Gulf of Mexico into the developing warm

sector of the polar front wave is evidenced by the relatively large gradients

in the neighborhood of Dodge City. The core of tropical maritime air has

evidently not progressed far enough northward to displace the warm but

dry air that had been over the Great Plains prior to the outbreak, with

the result that a region of low A values is found between the front and the

tropical maritime air. On figure 5.47 the effects of the Pacific front are apparent in the buildup of a secondary region of high A some 400 km

ahead of the polar front, located at this time over Dodge City. Twelve

hours later (fig. 5.48) the core of tropical maritime air has become more

extensive and now reaches to a height of 3 km. The second (Pacific) front is picked up now on the cross section and the area of low A values is confined between the two fronts. By 0300 UT on the 20th, figure 5.49,

the polar front is approaching Lake Charles and the Pacific front is re- ported on the daily weather map as a squall line. Finally, by the morning

of February 20 (fig. 5.50), both fronts have passed to the south of Lake

Charles, and the polar air just behind the front is characterized by rela- tively low A values. The use of space cross sections does not always yield measurements

at the most desirable points along a frontal zone. Another method of arriving at the probable refractive index structure about the frontal

interface is to plot radiosonde observations for a single station arranged

according to observation times as on figures 5.51 to 5.57. The time cross section for Glasgow, Mont. (fig. 5.51), which is in the

cold air behind the polar front for the entire period of observation of the

storm, displays gradients of A values that are generally weak. An exception occurs on the evening of the 19th (20/0300 UT) apparently as a

result of subsidence effects. Rapid City, S. Dak. (fig. 5.52), similar to Glasgow, shows generally weak gradients throughout in the cold air behind the front. North Platte, Nebr. (fig. 5.53), exhibits moderate

gradients with increasing A values in the post-frontal, higher-density polar

air. This station is too far north to record much in the way of moisture

effects at this time of year. Dodge City, Kans. (fig. 5.54), represents

dry low ahead of the polar front and increasing A values in the cold air just behind the front. Oklahoma City, Okla. (fig. 5.55), represents a

classic synoptic situation in which advection of tropical maritime air from the Gulf of Mexico produces a strong high ahead of the front and a low within the cool, dry, polar continental air. Again in the region

around Little Rock, Ark. (fig. 5.56), there is warm, moist air of extremely

high refractivity ahead of the front being replaced by polar continental
Figure 5.45. Space cross section in A units, 0300Z, 18 Feb. 1952.
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 215

DISTANCE IN KILOMETERS GGW RAP LBF DDC OKC LIT Figure 5.47. Space cross section in A units, 0300Z, 19 Feb. 1962.
216 SYNOPTIC RADIO METEOROLOGY

Figure 5.49. Space cross section in A units, 0300Z, 20 Feb. 1962.
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 217

I I I I I I I I I I 500 DISTANCE IN KILOMETERS

I GLASGOW DODGE CITY LAKE CHARLES Figure 5.50. Space cross section in A units, 1500Z, 20 Feb. 1952.

I80300Z I8I500Z I90300Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z DAY AND HOUR

Figure 5.51. Time cross section, Glasgow, Mont., in A units.
218 SYNOPTIC RADIO METEOROLOGY

I80300Z I8I500Z I90300Z I9I500Z 200300Z 201500Z 2I0300Z 2II500Z DAY AND HOUR

Figure 5.52. Time cross section, Rapid City, S. Dak., in A units.

I80300Z I8I500Z I90300Z I9I500Z 200300Z 20I500Z 2I0300Z DAY AND HOUR

Figure 5.53. Time cross section, North Platte, Nebr., in A units.

2II500Z
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 219

I- 2.0

I80300Z iSISOOZ I90300Z I9I500Z 200300Z 20I500Z 2I0300Z DAY AND HOUR

Figure 5.54. Time cross section, Dodge City, Kans., in A units.

2I1500Z

3,000

2,500

2.000

l,OOC

I80300Z I8I500Z I90300Z I9I500Z 200300Z DAY AND HOUR

20I500Z 2I0300Z 2II500Z

Figure 5.55. Time cross section, Oklahoma City, Okla., in A units.
220 SYNOPTIC RADIO METEOROLOGY

I80300Z I8I500Z I90300Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z DAY AND HOUR

Figure 5.56. Time cross section, Little Rock, Ark., in A units.

3,000

2,500

2,000

1,500

1,000

I80300Z I8I500Z I90300Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z DAY AND HOUR

Figure 5.57. Time cross section, Lake Charles, La., in A units.
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 221

-2700 -2500 DISTANCE FROM FRONT IN KILOMETERS

Figure 5.58. Epoch chart, Glasgow, Mont., in A units.

air of characteristically low A value. The time cross section for Lake

Charles, La. (fig. 5.57), is complex but represents again the same general

features: high A values ahead of the front and low ones behind.

The exponential correction to the refractive index height distribution

used in this storm series allows air mass properties to be clearly seen. By

use of such an exponential correction, one may construct an idealized

refractive index field about a frontal transition zone that shows the temperature and humidity contrasts of the different air masses. Further,

when this technique is applied to the analysis of a synoptic tropospheric

disturbance, it does indeed highlight air mass differences. The time cross section presentation is referred to as an epoch chart

when the observations are presented as plus or minus time deviations with

respect to the frontal passage. Thus, as a frontal system advances and

passes over a station, one obtains yet another perspective of the space

cross section. Such a presentation is given on figures 5.58 to 5.64. Figure 5.59 represents a typical continental station located in the polar
222 SYNOPTIC RADIO METEOROLOGY

continental air mass throughout the occurrence of the storm. The essential feature here, as in figure 5.52, is the absence of detail of A structure due to the presence of a uniform air mass over this station. Compare

this figure with the epoch chart for Oklahoma City (fig. 5.62), where the

structure of the idealized model is clearly reflected by the prefrontal A

unit high, strong gradient across the frontal zone, and the A unit low

behind the front. This rather fortuitous agreement is felt to be due to the strategic location of Oklahoma City with respect to the motion of contrasting air masses about the polar front. That is, this epoch chart

represents a point of confluence of virtually unmodified polar continental

and tropical maritime air.
SUMMARY 223

0.5 —

-1600-1500 -1000 -500 0 300 DISTANCE FROM FRONT IN KILOMETERS

Figure 5.60. Epoch chart, North Platte, Nebr., in A units.

5.8. Summary

A survey such as this is designed to indicate the direction of radio

meteorological research. Among the brighter prospects is the work of Moler et al., concerning the interrelation of refractive index structure and

mesoscale weather changes. Other synoptic features likely to have refractive index significance are the migratory high and low pressure cells of the

middle latitudes, since certain surface patterns and vertical profiles occur

frequently with a particular type of synoptic system.

A knowledge of the meso-macroscale behavior of in a synoptic sense enables the propagation engineer to anticipate the occurrence of superrefraction or ducting conditions. Thus, meteorological conditions that

give rise to such phenomena as prolonged space-wave fadeouts and inter- ference effects from elevated layers in the troposphere can to some extent be planned for in advance.
224 SYNOPTIC RADIO METEOROLOGY

DISTANCE FROM FRONT IN KILOMETERS

Figure 5.61. Epoch chart, Dodge City, Kans., in A units.

5.9. References

[1] Sheppard, P. A. (1947), The structure and refractive index of the atmosphere,

Book, Meteorological Factors in Radio Wave Propagation, (Phys. Soc. and

Roy. Meteorol. Soc, London, England).

[2] Gerson, N. C. (1948), Variations in the index of refraction of the atmosphere,

Geofis. Pura Appl. 13, 88.

[3] Perlat, A. (1948), Meteorology and radioelectricity, L'Onde Elec. 28, 44.

[4] Randall, D. L. (1954), A study of the meteorological effects on radio propagation

at 96.3 Mc between Richmond, Va., and Washington, D.C., Bull. Am. Meteorol.

Soc. 35, 56-59.

[5] Misme, P. (1957), Influence des discontinuites frontales sur le propagation des ondes decimetriques et centimetriques, Ann. Telecommun. 12, 189-194.

[6] Hay, D. R. (1958), Air-mass refractivity in central Canada, Can. J. Phys, 36, 1678-1683.

[7] Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception,

J. Atmos. Terrest. Phys. 1, 32.

[8] Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception

at 44.1 Mc with 92.1 Mc over the 167-mile path of Alpine, N.J., to Needham,

Mass., Proc. IRE 38, 1450.

[9] Bean, B. R. (1956), Some meteorological effects on scattered radio waves, IRE

Trans. Commun. Syst. CS-4, No. 1, 32.
REFERENCES 225

-1000 -500 0 500 800 DISTANCE FROM FRONT , km

Figure 5.62. Epoch chart, Oklahoma City, Okla., in A units.

[10] Onoe, M., M. Hirai, and S. Niwa (1958), Results of experiment of long-distance overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79.

[11] Hull, R. A. (1935), Air-mass conditions and the bending of ultra-high-frequency waves, QST 19, 13-18.

[12] Hull, R. A. (1937), Air-wave bending of ultra-high-frequency waves, QST 21, 16-18.

[13] Englund, C. R., A. B. Crawford, and W. W. Mumford (1938), Ultra-short-wave

radio transmission through the non-homogeneous troposphere, Bull. Am.

Meteorol. Soc. 19, 356-360.

[14] Yerg, D. G. (1950), The importance of water vapor in microwave propagation at temperatures below freezing, Bull. Am. Meteorol. Soc. 31, 175-177.

[15] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (1933), Ultra-short-wave

propagation, Proc. IRE 21, 427-463.

[16] Misme, P. (1960), L'influence du gradient equivalent et de la stabilite atmos- phe rique dans les liaisons transhorizon au Sahara et au Congo, Ann. Telecom- mun. 16, 110.

[17] Misme, P., B. R. Bean, and G. D. Thayer, (1960), Comments on "Models of the atmospheric radio refractive index," Proc. IRE 48, 1498-1501.

[18] Gray, R. E. (1957), The refractive index of the atmosphere as a factor in tropospheric propagation far beyond the horizon, IRE Nat. Convention Record,

Pt. 1, 3.
226 SYNOPTIC RADIO METEOROLOGY

500 0 500 DISTANCE FROM FRONT IN KILOMETERS

1700

Figure 5.63. Epoch chart, Little Rock, Ark., in A units.

[19] Gray, R. E. (1961), Tropospheric scatter propagation and meteorological conditions in the Caribbean, IRE Trans. Ant. Prop. AP-9, No. 5, 492-496.

[20] Bean, B. R., and G. D. Thayer (1959), On models of the atmospheric radio

refractive index, Proc. IRE 47, No. 5, 740-755.

[21] Schulkin, M. (1952), Average radio-ray refraction in the lower atmosphere, Proc. IRE 40, No. 5, 554-561.

[22] Fannin, B. M., and K. H. Jehn (1957), A study of radio elevation angle error due

to atmospheric refraction, IRE Trans. Ant. Prop. AP-2, No. 1, 71-77.

[23] Bean, B. R., J. D. Horn, and L. P. Riggs (1960), Refraction of radio waves at low angles within various air masses, J. Geophys. Res. 65, 1183.

[24] Arvola, W. A. (1957), Refractive index profiles and associated synoptic patterns,

Bull. Am. Meteorol. Soc. 38, No. 4, 212-220.

[25] Jehn, K. H. (1960), The use of potential refractive index in synoptic-scale radio meteorology, J. Meteorol. 17, 264.

[26] Lukes, G. D. (1944), Radio meteorological forecasting by means of the thermo- dynamics of the modified refractive index. Third Conf. Prop., NDRC, pp. 107-113 (Committee on Propagation, Washington, D.C.).

[27] Katz, I. (1951), Gradient of refractive modulus in homogeneous air, potential modulus. Book, Propagation of Short Radio Waves, pp. 198-199 (McGrawHill Book Co., Inc., New York, N.Y.).

[28] Jehn, K. H. (1960), Microwave refractive index distributions associated with

the Texas-Gulf cyclone. Bull. Am. Meteorol. Soc. 41, 304-312.
REFERENCES 227

DISTANCE FROM FRONT IN KILOMETERS

Figure 5.64. Epoch chart, Lake Charles, La., in A units.

[29] Jehn, K. H. (1961), Microwave refractive-index distributions associated with the

central United States cold outbreak, Bull. Am. Meteorol. Soc. 42, 77-84.

[30] Flavell, R, G., and J. A. Lane (1962), The application of potential refractive index in tropospheric wave propagation, J. Atmospheric Terrest. Phys. 24, 47-56.

[31] Moler, W. F., and W. A. Arvola (1956), Vertical motion in the atmosphere and its effects on VHF radio signal strength. Trans. Am. Geophys. Union 37.

[32] Moler, W. F., and D. B. Holden (1960), Tropospheric scatter propagation and

atmospheric circulations, J. Res. NBS 64D (Radio Prop.), No. 1, 81-93.

[33] Gossard, E. E., and L. J. Anderson (1956), The effect of super-refractive layers on

50-5,000 Mc nonoptical fields, IRE Trans. Ant. Prop. AP-4, 175-178.

[34] Megaw, E. C. S. (1950), Scattering of electromagnetic waves by atmospheric

turbulence. Nature 166, 1100-1104.

[35] Booker, H. G., and W. E. Gordon (1950), A theory of radio scattering in the

troposphere, Proc. IRE 38, 401-412.

[36] Saxton, J. A. (1951), Propagation of metre radio waves beyond the normal

horizon, Proc. lEE 98, 360-369.

[37] Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1960), Climatic charts and data

of the radio refractive index for the United States and the world, NBS Mono. 22.

[38] Smyth, J. B., and L. G. Trolese (1947), Propagation of radio waves in the troposphere, Proc. IRE 35, 1198.
228 SYNOPTIC RADIO METEOROLOGY

[39] Bean, B. R., and G. D. Thayer (1959). CRPL exponential reference atmosphere,

NBS Mono. 4. [40] Bean, B. R., and E. J. Button (1960), On the calculation of the departures of radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3, 259-263.

[41] Canterbury Project (1951), Vols. I-III (Department of Scientific and Industrial Research, Wellington, New Zealand).

[42] Bean, B. R., and L. P. Riggs (1959), Synoptic variations of the radio refractive index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97.

[43] Bean, B. R., L. P. Riggs, and J. D. Horn (1959), Synoptic study of the vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio Prop.), No. 2, 249-258.

[44] Palmer, C. E. (1957), Some kinematic aspects of frontal zones, J. Meteorol. 14, No. 5, 403-409.
Chapter 6. Transhorizon RadioMeteorological Parameters

6.1. Existing Radio-Meteorological Parameters

6.1.1. Introduction

A method of predicting the statistical distribution of field strength on

transhorizon paths is an important requirement m tropospheric wave

propagation. Consequently, considerable attention has been given in recent years (see figs. 6.1 to 6.6) to studies of the correlation between the

measured signal level (e.g., the monthly median value) and some quantity

derived from surface or upper air meteorological data. Figures 6.1 to 6.6 show relationships between field strength and various meteorological

parameters. It can be seen that there are some quite marked similarities (high correlation coefficients) between the two. It thus appears that if a reliable "radio-meteorological parameter" could be developed, then

generally available meteorological data would replace expensive radio

measurements in deriving the required distribution.

Progress has already been made in this difficult problem, [1, 2]^ In

these investigations, special attention has been given to two parameters:

(a) the surface value of refractivity, Ns, and (b) the difference, AA^,

between Ns and at a height of 1 km, A^i. Other groups have studied

different parameters [3, 4, 5, 6, 7, 8], either as possible alternatives to Ns and AA^ in the prediction process or as quantities which clarify the

effect of meteorological features, such as anticyclonic subsidence, on

signal strength. It is evident from the literature that some difference of emphasis exists regarding the relative merits of the parameters proposed

to date, and particularly on the value of studies of Ns. This chapter provides a critical survey of the present position in this

field of radio meteorology, and indicates a new approach which incorporates some aspects of all existing treatments. Section 6.1 contains a study of previous work and attempts to put the various views in proper

perspective; section 6.2 discusses some selected radio data from VHF

paths and its classification in terms of refractive index profiles, while

section 6.3 introduces a parameter combining the concepts of refraction

and atmospheric stability, and compares its properties with those of existing parameters.

^ Figures in brackets indicate the literature references on p. 266. 229
230 TRANSHORIZON PARAMETERS

1947 J948 Figure 6.1. Weekly means of measured field intensities of W2WMN and W2XEA at Needham, Mass., compared with corresponding atmospheric surface refraction at Boston, Mass.

(After Pickard and Stetson, 1950).

I 1 1 1 1 1 1 1 1 1 \ fX —I 1——I 1 1 1 1 1 1 1 1 1 1 DO^'NAMOLASO 30 40 SO 60 V BO Mese ^

Figure 6.2. Correlation of the received monthly mean field strength at Campu Sa Spina

and the gradient AN {between 0 and 1000 m.) of the monthly median index of refraction

at Elmas.

(The values of field strength are those obtained after omission of superrefraction). (After Bonavoglia, 1958)

.
INTRODUCTION 231

58

56

54

52 5048

46

44

4240

3852

50 4846

444240

38

36

f

1 1 1 )900 C.S.1

1 —

;

/ \

1

1 \ / \

\ / -1

1 V

1 7 \ —h /\N

I—\

'—

\ \

V \ \

—

\—

= : 7 \

r \

1

-)

\

\ V

/ —

f—V- '/ \ —

T-r

i— -n

/ \ /\ 1 LbH1 (a) —

2

1 1 1 !I00 C.S.1 -0 r. - 1 v17 — -AN— /

i > V-

-\- \ -t

i\ / , \

\\ -\

-A A \- > \- r-V

-t I

\

f \ \ \\ \

1

/ T /

1 — -^

(b)

166

168

170

172

174

176

178

180

182

184

186

172

174

176

178

180

182

184

186

188

oQ

c

ONDJFMAMJJ ASONDJFMAMJJASOND 1950 1951 1952 Figure 6.3. Comparison of the monthly median basic transmission loss and refractivity gradient for KIXL-FM, Dallas, Tex., recorded at Austin, Tex.

The development of prediction techniques is especially difficult in the

case of

(a) any path with terminals just within or just beyond the normal

horizon, and

(b) signal enhancements which occur for small percentages of the time.

Section 6.2 therefore includes some discussion of propagation characteristics on VHF paths for which considerable radio and meteorological

data are already available. The results obtained are particularly relevant

to an understanding of the large differences observed between median
232 TRANSHORIZON PARAMETERS

signal levels on VHF paths of comparable length, frequency and angular

distance.

The primary purpose of a radio-meteorological parameter is to provide

the best estimate of the statistical distribution of field strength (in terms of

hourly, daily, weekly, or monthly median values as required on a specific

path). The reliability of the parameter must be judged solely in terms of

this requirement, and care must be exercised in assessing the value of any

given parameter in terms of data obtained over limited intervals of time

or from restricted geographical areas. Discussions later in the chapter

will consider to what extent it is possible to develop a parameter which^

in addition to being statistically reliable, is also characteristic of the

physical structure of the atmosphere.

The present practice in applying radio-meteorological parameters consists in determining an average signal level for a given distance and then

adjusting this average level for climatic and seasonal differences by refer- ence to the changes in some function of the refractive index of the atmosphere. We may express this procedure mathematically, for a linear

regression model, thus:

E = b fin) + a (6.1) where E is the field strength, and b is the regression coefficient expressing
INTRODUCTION 233

•1.0

•0.6

0

0.2

0.4

0.6

0.8

1.0

1 1

' o

1 1 1 1 1 o

° k'^

1 °

° ° \.a-^<^^o o °

°

o X

1 \

° < o ESTIMATED AVERAGE

" ° 1 r , CORRELATION COEFFICIENT

^° —

\x

1 ° X o

o

ESTIMATED I - MEASUREMENT

ERROR 1

1

i

o U.S. DATA

« GERMAN DATA

1 1 1 1 1

s(Lb), dB

Figure 6.5. Correlation coefficients of seasonal cycles of Ng and Lb: night

{2000, 2200, 2400) versus the standard deviation of Lb.

the sensitivity of £J to a unit change in/(n). The intercept, a, is a function of path length, antenna heights, and terrain characteristics, and can be derived from existing prediction procedures [9]. Comments on some

results obtained are given in the remainder of this section.

6.1.2. Parameters Derived From the N-Profile

It has long been recognized that the variations in field strength on

transhorizon paths are intimately connected with changes in the vertical

gradient of refractive index over the path. Although our knowledge of the detailed fluctuations in refractive index in the troposphere is still in- adequate for many requirements, it is nevertheless possible to relate,
234 TRANSHORIZON PARAMETERS

WJAS FM - PATH 12 TRANSMITTER: PITTSBURGH, PA. RECEIVER: STATE COLLEGE, PA. DISTANCE: 188.5 KILOMETERS

9: 30.44 MILLIRADIANS

295 300 305 310 315 320 325 330 335 340 350 No

165 L^ = 0.42Iaq,->-I9I.56 x

\ \ \ r L^=0.475a No,2*I93.35

rL^.^^N = 0.858

35 40 45 50 55 30 35 40 45 50

Figure 6.6. Monthly median basic transmission loss versus Ns or ANh for a path from

Pittsburgh, Pa., to State College, Pa.
PARAMETERS DERIVED FROM THE A^-PROFILE 235

statistically, changes in signal level and functions of a parameter derived

from routine surface and upper-air measurements of pressure, temperature, and humidity.

Following the work of Pickard and Stetson [10], Ns and AN have been

the subject of detailed studies by several workers [1, 2, 11, 12, 13, 14]. Only the more important conclusions are summarized here. Values of the

correlation coefficient, r, relating monthly median values of either Ns or AA^ and field strength, derived from a number of paths in diverse climatic

conditions [2], range from 0.4 to 0.95 with a median value of about 0.7. An analysis has also been made of the results obtained by using (a) values

of AA^ obtained from the surface readings and at heights other than 1 km,

and (b) values of AN between different levels on the profile up to a height

of 3 km. A comparison of measured field strength at frequencies near

100 Mc/s on 20 paths, 130 to 446 km long, located in various part of the

United States, yields the following result: the use of Ns gives as good a

correlation as any of the AA^ values, due to the high correlation between

the surface value and these differences. The values of r relating monthly

median values of Ns and AA^ (the decrease in the first kilometer) have

been obtained for the United States [15], France, [16], Germany, [17] and

the British Isles [18]. They range from 0.60 to 0.93. Moreover, the data

are consistent with the assumption of a reference atmosphere in which A^ decreases exponentially with height [15]. These results lead at once to a consideration of the value of A^s in predicting the geographical variation

of monthly median field strength. This question is considered in more

detail later in this section.

It has been shown that, where only past radio or meteorological results are available, one obtains at least as good a prediction of the diurnal and

seasonal variations of field strength from long-term meteorological data as from relatively short-term (say 1 year's) radio data. The annual cycle

may be represented by a single regression coefficient of 0.18 dB/A^-unit

for either night or day ; however, the regression coefficients for the diurnal

cycles lie between 0.2 and 1.1 dB/V-unit and vary with distance and

season, being greatest for paths between 175 and 200 km long and for the

winter months. The possibility of predicting the variation of hourly

median values of field strength within any given month by combining the

seasonal and diurnal correlations has also been discussed [14]. It is recog- nized that the development of a prediction procedure based on this ap- proach must account for the complex sensitivity of field strength, E, to changes in A^s; this requirement in turn leads to considerations of season,

climatic region, distance and frequency. The results for the distance

dependence of the E-N s regression coefficient are, of course, intimately

connected with the propagation mechanism. In particular, in the case

of VHF paths about 200 km long there will usually be components in the

received field due to diffraction around the earth's surface and scattering
236 TRANSHORIZON PARAMETERS

from randomly dispersed eddies; on occasions there will also be a semicoherent field arising from partial reflection at elevated stable layers.

Some aspects of this complex situation are discussed in section 6.2. To sum up other important conclusions reached by the authors mentioned above, the correlation between E and Ns (1) increases with increasing variation of E or Ns, (2) is greater for seasonal cycles of night-time values of the variables

than for the midday values, and

(3) is greater for summer diurnal cycles than for winter ones. Conclusions (1) and (2) are particularly important if we try to assess the

utility of A^s in prediction work in terms of signal data from areas in which

the variation of E is small (e.g.. Western Europe), or in terms of data for afternoon periods only.

Another parameter, closely related to AN, is the ''equivalent gradient,"

Qe, proposed by Misrne [19]. This is defined as that linear decrease of n

with height which produces the same amount of bending as the actual

inhomogeneous atmosphere over a given transmission path. The problem is illustrated in figure 6.7, where the dotted line represents the actual

ray path between T and P above an earth of radius a. It is required to determine the curvature, p, of the circular, full-line path which is tangential to the real path at P and which corresponds to a constant value of dn/dh in a fictitious atmosphere. Boithias and Misme [20] have described a graphical method for calculating Qe ( = l/p), for example, when P is located in the center of the common volume of the antenna beams.

Monthly median values of are obtained from the corresponding monthly

median values of dn/dh, and Misme [19] has given tables of Qe for various

path lengths for different months.

It is evident that Qe, like AN, expresses the amount of refraction pro- duced by the atmosphere and one might expect a high correlation between

the two parameters. This is known to be the case in some results quoted

by Misme [19] and which are illustrated in figure 6.8. Here the monthly

median values of AA^ and Qe (for a 300 km path) are compared, together

with the variation in Ns, for Leopoldville in the Congo area. The corre- lation between Qe and AA^ is high, but the maximum values in Ns in December and January are accompanied by a local minimum in the

values of Qe and AA^. These results, and similar ones from Dakar (W.

Africa), have been quoted in support of arguments that A^s is of limited

value in predicting seasonal and geographical variations in field strength

[19]. It is essential, Misme claims, to investigate the nature of the

Ns-AN correlation in separate climatic areas; the correlation seems poor

for some equatorial climates, probably because of the presence of semipermanent elevated layers. In these conditions, Misme feels that an

exponential reference atmosphere and a correlation between Ns and AA^

are not to be expected.
PARAMETERS DERIVED FROM THE iV-PROFILE 237

MONTH (JULY-JUNE)

Figure 6.8. Comparison of monthly medians of ge, AN, and Ng.

(Leopoldville, Congo)
238 TRANSHORIZON PARAMETERS

While admitting the importance of an explanation of the results dis- cussed above, it is difficult at present to accept the argument that Qe

should be used instead of AN. Indeed, it has been shown [19] that for a 471-Mc/s link, 160 km long, located in the Sahara region, there is a

difference of 20 dB between the signal levels exceeded for 99 percent of the time during daylight hours in January and June, whereas the January

to June variation in is less than 2A^/km. This result indicates the in- fluence of atmospheric stability on signal strength. We defer, therefore,

further consideration of Qe until later in the chapter merely noting at this stage that its derivation requires more detailed calculation (i.e., ray

tracing) than in the case of AA^, without any appreciable benefit.

6.1.3. Parameters Involving Thermal Stability

The role of stable elevated layers in tropospheric propagation has been

discussed by a number of scientists; for example, by Saxton [21], Friis,

Crawford, and Hogg [22], and in work done by the French [23] in an

important series of papers. These contributions have stimulated interest

in the development of radio-meteorological parameters which depend, in part at least, on thermal stability, and this section considers some of the

characteristics of these parameters.

6.1.4. Composite Parameter

Misme [19] has discussed the Sahara results mentioned above in rela- tion to the theoretical work of Voge [23]. In considering the effect of elevated layers, Voge introduces a parameter, defined by:

area of one or several layers at a given height

rj = ; ; ;—; the horizontal area, at the same height, visible from

receiver and transmitter

The value rj is an increasing function of the atmospheric stability measured

between two levels in the atmosphere. Stability may be defined as the

work, AW, required to raise a unit mass of air from one level to another.

We have, therefore:

rj tends to 0 as AW tends to 0.

T) tends to 1 when AW is large.

The values of AW (in joules per gram) for Aoulef (Sahara) are given

by Misme for the January and June months in which measurements were made over the 160 km path at 471 Mc/s. The relevant data are summarized in table 6.1.
COMPOSITE PARAMETER 239 Table 6.1. Influence of thermal stability on signal level

(d = 160 km: f = 471 Mc/s: North Sahara Region)

Jan. June

ge (-N/km) 34.0 32.5 AW for 0.7 to 2 km 0. 17 0. 0024 Relative signal level, 99 percent value, (dB) 20.0 0.0

The dominant propagation mechanisms in January and June are thought to be ''diffuse reflection" and ''scattering," respectively, and on

this basis Misme uses the equations given by Voge to calculate the ex- pected difference in the January and June signal levels. With various

assumptions regarding the properties of stable and turbulent layers, a value of 19 (=b5) dB is obtained for the ratio of predicted field strengths in January and June, in good agreement with the measured value exceeded

for 99 percent of the time. This analysis is facilitated by the fact that the

two selected months are characteristic of well-defined climatic situations

in the Sahara region. Nevertheless, the results indicate that a parameter

which combines the concepts of the equivalent gradient and thermal

stability may be of general application. Misme [19] has therefore sug- gested a parameter M of the form

:

where a, h, and n are constants, Qe is the equivalent gradient, and AW is the thermal stability defined above for a 1 km height interval. With

a = 0.5 dB/A^/km, M provides an estimate of the variation in field strength, E, in decibels, caused by changes in equivalent gradient and

stability. It remains to define E in a "standard" atmosphere by selecting a mean value of stability and a suitable value for the term h. This composite parameter is thought by Misme to be more representative of the real atmosphere (especially in tropical areas) than Ns and AA^,

and it is certainly of great potential value. However, only fragmentary

radio data are available for purposes of comparison in the references

quoted, and here again the precise value of the parameter in prediction

work can only be determined by a more comprehensive study.

6.1.5. Potential Refractive Index (or Modulus), K

This parameter is defined as the value of TV = (n— 1)10^ which an air mass would have if brought adiabatically to a standard pressure, say

lOOOmbar, assuming a constant humidity mixing ratio. Examples of the use of K in radio meteorology have been given by Katz [24], and by Jehn

M = a (ge - ^0 -\- h [AWY) (6.2)
240 TRANSHORIZON PARAMETERS

[25]. It has also been applied by Flavell and Lane [6] in studies of the

effect of anticyclonic subsidence on tropospheric propagation. Values of K can be derived very rapidly from upper-air data and, since plots of K

against height or pressure do not exhibit the large systematic decrease of N with height in the conventional Nih) profile, the structure and motion

of meteorological features are clarified considerably. (Below the condensation level, a lapse rate of — 20A^/km = dK/dh = 0.) In this respect,

K is superficially similar to the A unit [26] derived from the exponential

reference atmosphere. However, no quantitative results are yet available with this parameter and, furthermore, the method of deriving K

assumes a dry adiabatic lapse rate and a constant humidity mixing ratio. The presence of a condensation level in the actual atmosphere is therefore

neglected. In addition, accurate values of N can only be derived directly

from the K profile in certain restricted conditions. Pending further

studies in this direction therefore, K and dK/dh remain more suitable for qualitative synoptic studies than for quantitative predictions of field strength variations. However, it is of interest to note that a close connection exists between K and the composite parameter M discussed above.

Misme [27] has shown that the change of K in a given height interval,

AK/Ah, is of the form:

AK/Ah = ki AN/Ah - k2AW/Ah (6.3)

where ki and /c2 are constants, and AW is a measure of thermal stability as defined earlier.

6.1.6. Vertical Motion of the Atmosphere

The influence of stability has also been discussed by Moler and Arvola

[7], and Moler and Holden [8]. These authors assume that the average

lapse rate, dn/dh and the magnitude of local irregularities on the profile, are primarily determined by changes in vertical velocity. Local centers

of convergence (low pressure cells) produce updrafts which result in considerable mixing and the dissipation of any stable layer structure. Horizontal divergence from local high-pressure centers create temperature

inversions and associated layer-type discontinuities in the n-profile.

These latter features are most pronounced in conditions of anticyclonic

subsidence.

The direction and relative magnitude of the vertical component of wind velocity can be estimated by techniques outlined by Moler and

Holden, and a correlation between hourly median field strength and calcu- lated values of vertical velocity has been noted by these authors. These

fundamental studies represent an important attempt to explain signal

variations in terms of atmospheric motion, and a survey of available ex- perimental evidence supports the basic assumptions in this approach. It
VERTICAL MOTION OF ATMOSPHERE 241

is particularly interesting to consider whether local centers of convergence

and divergence, ill-defined on Daily Weather Reports, can be distin- guished by observations with microbarographs and whether the results are correlated with signal characteristics. Some results of such an investi- gation, forming part of a more general study of VHF propagation in the

United Kingdom, are shown in figure 6.9. The signal records obtained

at a frequency of 186 Mc/s on a 140 km path are shown for 12 September

1959 together with the pressure variations recorded at transmitting and

receiving sites. The steady high signal at 0900 hours begins to fall as the

pressure at the transmitter begins to decrease, and at about 1030 hours

the median level and fading rate change abruptly as the pressure at the

midpoint falls. Between 1200 and 1700 hours the pressure over the whole

path is falling, and the signal characteristics remain essentially uniform.

However, a significant increase in median level and a reduction in fading

rate are evident at 1700 hours as the pressure begins to increase at the

transmitter. By about 2300 hours the pressure at the center of the path

has reached a steady value and the signal level is again high with negligi- ble fading. The total barometric variation between 0900 and 2200 hours

is only 2 mbar.

There is obviously a fruitful field of radio-meteorological study sug- gested by the work outlined above, but we have to conclude that the

results obtained, while extremely valuable in clarifying a qualitative

relationship between signal strength and vertical motion (or stability), are not immediately applicable in the problem of predicting field strength

variation.

6.1.7. Discussion of the Parameters

The above review emphasizes the need for a critical inspection of representative data, radio and meteorological, with the objective of comparing the merits of the several parameters and explaining, if possible, some of their relative merits and limitations. Some relevant points have

been mentioned already, and these issues are developed m more detail in the following discussion.

It is important to bear in mind that in the particular problem of predicting field strength changes we are concerned with a statistical relationship

between two quantities, the median signal level and a radiometeorological parameter. It may be advantageous, of course, to investigate

the merits of special parameters developed for particular climatic

areas; for example, the value of the gradient, dn/dh, in a semipermanent

elevated layer in the trade winds area. Nevertheless, such a parameter

must also be statistically ''reliable" if it is to be applied in prediction

analysis.
242 TRANSHORIZON PARAMETERS

12 SEPTEMBER 1959

60 I- 40

20O «-

1019

e 1018 o

E

LxJ OC3

CO

cn

UJ

Gl

1017

1016

10150600

— — »^ \

\

t\ \

V

\

V \ / /

/ /

/

/

CLEAR SKY

i

1

1

V

CLOUD

1

1

1

/ ^ — ^

CLEAR SKY

1

1 1 1200 TIME (GMT)

1800 12 SEPT 1959

2400

Figure 6.9. Signal records and pressure variations at transmitter, T, and receiver, on a 140 km path; f = 186 Mc/s.
COMPARISON OF SOME PARAMETERS 243

6.1.8. Comparison of Some Parameters

It is probably fair to say that, at present, the greatest interest is con- centrated on the relative merits of Ns, AN, and ge, for these quantities

have received more detailed attention than any others. (It might be

argued that, since the initial gradient of refractive index with height has a

strong influence on refraction, this quantity should be highly correlated

with signal level. However, the errors of measuring the initial gradient

by radiosonde techniques effectively mask the correlation with the signal.) In a comparison of Ns, AN, and Qe, the monthly median values of trans- mission loss, L, were determined for the hours of radiosonde ascents for 20 radio paths in various parts of the United States. The equivalent gradient, Qe, was then calculated using standard raytracing techniques [15] assuming:

(a) the actual antenna heights,

(b) a smooth earth with the sea-level radius,

(c) horizontal stratification of N with a vertical distribution the same

as that of the monthly mean.

Standard statistical methods were then employed to determine the correlation between monthly median values of (a) E and Qe, (b) E and Ns, and (c) E and AA^ (surface to 1 km height). The average values of these

correlation coefficients for the 20 paths are given in table 6.2. The paths

studied in this comparison were the same as those listed by Bean and

Cahoon [2] in their analysis oi Ns — AN correlation. The data used were

representative of climatic conditions ranging from those of New England

and the Great Lakes area to Texas and the Pacific coast. This investi- gation, at least, would seem to justify existing prediction procedures

based on Ng and AN. It is valuable, however, to consider some of the

results for specific paths in more detail, since they illustrate some features

of interest directly relevant to the arguments concerning the value of Ns and AA^. Table 6.2. Average correlations, r(z), of refraction variables with field strength

(20 paths, 130 to 446 km long, /=92 to 106 Mc/s)

Variable 9e AN

r(z) 0. 59 0. 70 0. 71

6.1.9. Some Exceptions and Anomalies

In so complex a matter as field-strength prediction, it would be unreasonable to expect any radio-meteorological parameter to be of worldwide application to a uniform degree of reliability, and some conditions

have already been quoted in which Ns may not be a reliable parameter.

The significance of these examples is really the issue on which views
244 TRANSHORIZON PARAMETERS

diverge most at the present time. Unfortunately, adequate radio data

are not available for several areas of interest (e.g., equatorial Africa) and

in these cases we can only propose tentative explanations based on a

critical examination of existing results, for similar but not identical conditions. An examination of available data, in published and unpublished reports,

has shown two particular examples which deserve further study; these

are (1) climatic conditions in which stable elevated layers are persistent

during certain seasons of the year, and (2) equable climates in which the

annual range of Ns, AN, and field strength is relatively low. Attention

has already been drawn to an example in the first category; namelj'', the

path from San Diego to Santa Ana, Calif. [2]. The well-defined coastal

inversion in this area occurs at a height of about 0.7 km, and the asso- ciated stable layer has a strong influence on radio field strength. Here

the correlation between field strength and N gradient for heights up to

0.7 km above the surface is small and negative (i.e., opposite to the general

trend). The correlations with Ns, however, and with N differences to heights above the base of the inversion, are about 0.8. This result suggests an explanation of some of the results, already mentioned, for Dakar

and Leopoldville. Consider the profile of figure 6.10 typical of ascents made through daytime inversions in equatorial and Mediterranean areas. (For comparison purposes, an exponential reference atmosphere is also shown.)

At Dakar, for example, an elevated layer is observed in 40 to 50 percent

of the daytime soundings during August, frequently with the base of the

inversion above 1 km height. In these conditions, at least, the high N

s

values measured in August are not accompanied by high values of AA^" (0 to 1 km), for the lapse rate, dN/dh, below the inversion is generally less steep than would be expected for the given value of N It would be

valuable, therefore, to examine in more detail the distribution of the

height of the layer, particularly in the summer months. If the data given

by Misme contain an appreciable fraction of such profiles, then the poor

Ns — AN correlation he discusses is not surprising. The correlation between Ns and signal level, however, could still be significant, as in the

San Diego-Santa Ana link [2]. Further radio data are obviously required

for areas in which elevated inversions are persistent. In many temperate regions, a somewhat different situation exists. In western Europe, for example, the annual range of monthly median values

of A' s is 10 to 20 N units; similarly, the variation in the monthly median

values of field strength is also small and frequently lies within the esti- mated measurement error. Figure 6.11 shows some results which illus- trate these features. Monthly median values of relative field strength

are shown for the months January-November, 1959, for a 300-km path at a frequency of 174 Mc/s; the terminals being located at Lille (northern
SOME EXCEPTIONS AND ANOMALIES 245

1.5

1.0 X

CD

UJX

0.5

1 r

' \

1 \ I \

1 IN \\ \

VERSION

^YER

EXPON

REFERENCE ATM

V 1 \ *^TYPI entialX \ osphereX \

CAL PROFILE

\\^

\

\ \^ \

250 300 350 400

Figure, 6.10. Typical profile through an inversion

areas.) (equatorial and Mediterranean

J \ \ L M A H J J A S 0 MONTH OF YEAR, 1959

Figure 6.11. Variation of monthly median values of Ns and field strength for a 300 km

path from Lille, France, to Reading, England.
246 TRANSHORIZON PARAMETERS

France) and Reading (England). The variations in A^^ obtained from the

Crawley radiosonde station (close to the midpoint of the path) are also shown. Here Ns has a range of zb 10 iV units, and the signal level a range

of no more than ±4.5 dB. In fact, from March to September the total variation in observed monthly median field strength is only =bl.5 dB.

The measurements of field strength in these experiments were estimated

to be subject to possible errors of about d=3 dB. Consequently, one would not be justified in assessing the value of A^s as a radio-meteorological

parameter on the basis of the correlation coefficient (about 0.25) calculated

from the two curves in figure 6.11. During the period studied in this work (0900-2300 UT daily, January-November 1959) the highest signals were observed during anticyclonic winter weather, with elevated inversion

layers at a height of about 0.6 km. Extended stable layers at this height were rarely seen on the sonde ascents during the summer months. This

result may partly explain the fact that the seasonal variation in Ng

(highest values in the summer months) is not accompanied on the Lille- Reading path by a corresponding variation in monthly median field strength.

6.1.10. Conclusions

A review of available data shows that no radio-meteorological parameter has yet been proved to be superior to A^s or AA^ for general application

in the prediction of field strength distributions. The value of these

parameters has been established by studies of many paths in diverse

climatic conditions. However, it should be noted that much of the

radio data has been obtained at frequencies near 100 Mc/s, and there is a clear need for further analysis of the several parameters in conjunction

with field strength measurements at higher frequencies.

Pending more detailed results, it does not seem likely that the equivalent gradient, Qe, affords any significant advantage over AA^, especially

in view of the many calculations required in its derivation. Some data

from selected areas (Congo, Sahara, west Africa) suggest that Ns and AA^

may have limited value in these regions; however, the present lack of adequate radio data precludes any definite conclusions. These examples,

and allied work on vertical motion and thermal stability, emphasize the

importance of a parameter (such as that suggested by Misme) which takes

account of the influence of elevated layers. This approach would probably prove fruitful not only in equatorial areas but also in temperate

regions where the annual range of Ns and AA^ is small. Analysis of some results obtained at a frequency of 100 Mc/s has shown

that it has proved feasible to provide at least as good a prediction of the

annual cycle of field strength variations from long-term meteorological

data as from relatively short-term radio data.
INTRODUCTION 247 6.2. An Analysis of VHF Field Strength Variations

and Refractive Index Profiles

6.2.1. Introduction

It is evident from the discussion in section 6.1 that the further development of radio-meteorological parameters would be assisted by a better

understanding of the propagation mechanism on non-optical paths. In

particular, the influence of thermal stability on signal level, fading rate and wavelength dependence is an important topic requiring further study.

The effect of varying meteorological conditions on signal characteristics

is especially marked in the case of paths of ''intermediate" length, with

terminals just beyond the radio horizon. As noted in section 6.1 on such a path the residual scattered field in the absence of stable layers or surface

ducts will often be comparable with the weak diffracted field. Further- more, at frequencies up to, say 300 Mc/s (X > Im) relatively strong

fields will frequently be observed in conditions favorable to the production

of temperature inversions in the first 2 km or so above the earth's surface.

It is the purpose of this section to discuss some aspects of radiometeorology relevant to this situation, especially the field strength distribution

observed on a 200 km path at frequencies between 72 and 180 Mc/s.

The signal characteristics are analyzed in terms of a classification of refractive index profiles, with the objective of clarifying the relative importance of different propagation mechanisms and their influence on the

measured field strength distribution.

Table 6.3 lists characteristic profile types, the assumed mechanism

associated with each type, and typical meteorological conditions. Selected references are given for each category, and special mention may be

made here of recent work [28] in France which, to a large extent, unifies and extends earlier analyses based on the separate concepts of ''reflection" and "scattering."

6.2.2. Radio and Meteorological Data

The analysis to follow is limited to paths between Chicago and Urbana,

111. (flg. 6.12), since several years of radio data were available for four

separate wavelengths between 1.67 and 4.18 m. Moreover, two radiosonde stations are located on or near the path, a unique situation in radiometeorological investigations. Details of the radio paths are given in table 6.4 (in which d is the total angle between the horizon rays from trans- mitter and receiver on a 4/3-earth proflle).

The meteorological data were obtained from the simultaneous radiosonde observations made by the Weather Bureau at Joliet and at Chanute Air Force Base, near Rantoul, 111. The results used were those

from the significant levels reported whenever the temperature or humidity

departed by d=10 percent from predetermined values.
248 TRANSHORIZON PARAMETERS

Table 6.3. Refractive index 'profile classification, probable propagation mechanisms

and meteorological conditions

Profile Assumed propagation mechanism

Ref- erence Meteorological

conditions

Unstratified (U), monotonic decrease with height, gradient nowhere ex- ceeds twice normal for that height.

Scattering plus diffraction [29] [30]

[31]

Well-mixed atmosphere due to thermal convection and, or wind shear

Elevated Layer (EL), monotonic de- crease with height with one or more

distinct layers with gradients at least twice normal for that height.

Scattering plus diffraction plus reflection.

[32]

[21] [28]

Layer formed by subsi- dence inversion or lift- ing of radiation inversion. Super-refractive (SR), same as EL but

the layer is ground-based. Extended radio horizon pro- ducing enhanced dif- fracted and scattered components.

[31] Radiation inversion formed during the night or rapid evaporation from soil after

rain. Ducting (D), same as SR but the gra- dient exceeds the earth's curvature,

1/a.

Extension of radio horizon

to include the receiver.

[33] Same as SR.

Table 6.4. Chicago-to-Urbana radio path characteristics

Station Distance 9 X / Period of record km mrad cm Mc/s

WBKB-TV 203.

16.3 4. 18 71. 75 5/51- 5/53 WNBQ-TV 202.9 16.7 3. 67 81. 75 10/50-10/51 WMBI-FM 202.7 16. 1 3.15 95. 50 7/50- 6/52 WENR-TV 202.9 16.4 1.67 179. 75 7/51- 6/53

6,2.3. Classification of Radio Field Strengths

by Profile Types

The RAOB significant level data were converted to refractive index by

use of (1.20). The gradient of was then determined between the re- ported significant levels of each profile and examined as to whether the

gradients fell into the category of linear, subrefractive, or superrefractive

depending upon the criteria set down in table 6.5, wherein superrefractive

is approximately twice normal and subrefractive has a positive gradient.

Simultaneous observations of similar profile types at Rantoul and Joliet were necessary for entry as a distinct profile occurrence. If a superrefractive layer occurred above the crossover heights of the radio horizon

tangent rays from both transmitter and receiver then it was classified as an elevated layer provided the reported layer heights were within 1 km

of one another at both radiosonde stations. Table 6.5. N gradient classification of profile types in ^-units/km

, p h Subrefractive Unstratified Superrefractive mb km

1000-850 0 -1. 46 -{dn/dh)<0 20<-(rfrt/d/i)<60 100<- (dn/dh) 850-700 1. 46-3. 01 -{dn/dh)<0 20<-(d?i/d/i)<50 80< -idn/dh)

700-600 3. 01-4. 20 -{dn/dh)<0 15<-(dn/dft)<40 70<- (dn/dh) 600-500 4. 20-5. 57 -{dn/dh)<0 10<-(dn/d/i)<30 50<- idn/dh)

500-400 5. 57-7. 18 -{dn/dh)<0 10<-(dn/d/i)<25 iO<- idn/dh)
RADIO FIELD STRENGTHS BY PROFILE TYPES 249 Elevated layers below the crossover height were classified as groundbased superrefractive layers. Elevated layers below the crossover height

at one weather station and above that height at the other were classified as tilted elevated layers.

After these characteristic profiles were isolated, the median field strengths for the 3-hourly periods centered upon the radiosonde observation time were arranged into cumulative probability distributions for each profile type. The results are shown in figure 6.13. (There were

relatively few examples of subrefractive profiles and no distributions for

this category are given.)

Figure 6.12. Location of radio path and radiosonde stations used in this study.
250 TRANSHORIZON PARAMETERS

Generally, the unstratified samples have the lowest overall field strengths throughout the entire distribution range. The presence of any- layer (elevated or ground-based) tends to increase the field strength by

10-25 dB at any percentage level of the distribution. (The exception to

this observation, WNBQ-TV, is probably explained by the fact that the

observations were limited to six winter months, rather than the 2-year

period of the other stations.) Tilted elevated layers appear to produce

the greatest enchancement of signal strength, probably as a result of focusing effects due to the layer tilt.

6.2.4. Prediction of Field Strength for

Unstratified Conditions

The field strengths recorded during the times when the radiosonde

ascents at both Joliet and Rantoul indicated nonstratified conditions were compared with the values predicted by Norton, Rice, and others [31] for the case of diffracted plus scattered radio waves. This particular prediction process is adjustable for the average refractive conditions over the

path in that it adjusts the effective earth's radius factor to the initial gradient of for the calculation of diffracted field strengths. One also needs the angular separation of the radio horizon rays at their intersection near midpath. The average initial gradient of was obtained for each

instance of unstratified profile by simply averaging the initial gradients

from Rantoul and Joliet, while the angular separation was obtained by

determining the amount of radio ray refraction expected over each particular path in atmospheres of exponential decrease with height that closely

match the observed N conditions.

Figure 6.14 illustrated a comparison of the predicted and observed field strengths. For WNBQ and WENR there is approximate agreement between the two sets of data. However, the predicted values for WBKB

and WMBI are approximately 10 to 12 dB higher than the observed

values.

This tendency to predict field in excess of the measured values suggests

that the empirical data, on which the predictions are based may include

meteorological conditions with some degree of stratification in the first 2 km or so, even though most of the empirical data refer to afternoon

hours in winter. It will be shown that elevated layers of moderate size (say a few kilometers in horizontal dimensions), that may exist undetected

by the radiosonde, could produce field strengths on the Illinois paths comparable with the median values for ''unstratified" conditions shown in figure 6.13. Furthermore, the limits placed on the profile gradients

specifying unstratified conditions in table 6.5 are such that some layer

type profiles may be included in the unstratified category. Consequently,

it is important to study in more detail the properties of the elevated layer,
ELEVATED LAYERS ON THE ILLINOIS PATH 251

not only as a feature occasionally producing high field strengths, but also as a mechanism which, in less intense form, partly determines the strength

of the weaker fields observed for large percentages of the time.

6.2.5. The Elfect of Elevated Layers on the Illinois Path

The influence of elevated layers of VHF transmission beyond the

horizon has been studied by several workers [21, 22, 28, 34, 35, 36, 37]. However, few investigations have contained any detailed comparisons of theory and experimental results. The following analysis presents such

a comparison, using simple models of the elevated layer for the four

Illinois paths.
252 TRANSHORIZON PARAMETERS

6.2.6. Elevated Layers at Temperature Inversions

Recent radar and refractometer investigations of tropospheric structure have shown that elevated layers in the refractive index distribution are frequently observed in the stable air of temperature inversions [38]. A typical value of layer thickness is 100 m, with horizontal dimensions

of tens of kilometers. On occasions, extended layers no more than 10 m

in thickness have been detected by refractometer soundings. In the

present discussion we attempt to evaluate the reflection coefficient of these elevated layers. We may express the modulus of the reflection

coefficient |p|, for a wave incident at a glancing angle a on a layer of thickness h, and horizontal extent, x, in the form:

\p\ = ^J{a,h,x) (6.4)

where f{a,h,x) is the ratio of the reflection coefficient of the model to that of the infinitely sharp case (i.e., the Fresnel discontinuity value,

An/2a^). This function has been evaluated for several layer profiles [28],
ELEVATED LAYERS AT TEMPERATURE INVERSIONS 253 and preliminary calculations based on this work were made to determine

the most suitable model in the present application. It was evident from

these calculations that a simple linear profile would yield the best agree- ment with the measured data, and this model was therefore adopted in the subsequent analysis.

Consider the layer profile shown in figure 6.15, i.e., a linear decrease of An over a height interval h, with transition regions of height d. This

model and others have been discussed by several authors, but the most

detailed treatment is that of Brekhovskikh [39]. His analysis shows that

for this linear model

:

|p| = An ' X/Sirh sin^ a ^ An ' X/87r/ial (6.5)

This equation is valid if

:

(a) An • X « Trha^ and (b) Aad « X.

In the present problem, with values of X of 1.7 to 4.2 m, An '-^ 10~^, a '~ 0.02 condition (a) is satisfied for layer thicknesses greater than about

20 m. In addition, condition (b) is fulfilled for the stated conditions if the

thickness of the transition region is less than a few meters. These conditions do not seem inconsistent with available refractometer data on

elevated layers, but a rigorous justification of the model is impossible at the present time. In any case, there is almost certainly no unique profile

representative of all elevated layers. We assume here, therefore, the

linear profile of figure 6.15 merely as a simple analytical model. It may

be noted here that the value of | p | given by (6.5) agrees with that quoted

by du Castel [40] but is half the value obtained in an earlier analysis [28]. Equation 6.5 was used to calculate reflection coefficient of the layers on each occasion on which these were observed in the sonde ascents. The

results, expressed in terms of a reflection loss are compared with the

measured values of field strength in figure 6.16. The general agreement

is satisfactory for the assumed model. As might be expected, there is a considerable scatter in the data and two considerations are important in assessing the significance of these results. These concern sonde response

and layer structure. The work of Wagner [41] on the response of radiosondes shows that, for an elevated inversion layer with An = 3 X 10~^,

h = 100 m, a sonde with a 10 sec time constant in the sensing elements,

and rising at 5 m/sec will give an indicated value for An of approximately

half the true value. The above procedure, using sonde data, therefore

underestimates the value of \p\ for an idealized infinite layer.
254 TRANSHORIZON PARAMETERS

On the other hand, the analysis assumes a smooth layer extending

horizontally at least over a distance x equal to the first Fresnel zone. We have:

X = V2a\/a (6.6)

where 2a = path length = 203 km for the Illinois paths. Hence x is of the order of a few tens of kilometers. In addition, we have assumed that

the layer is horizontal and smooth over a distance x, thus neglecting convergence. If we adopt the Rayleigh criterion, the height of the surface
INFLUENCE OF SMALL LAYERS 255

irregularities on the layer, Ah say, must not exceed zt\/Sa for the layer

to be considered smooth; i.e.. Ah < ±7 m (X = 1.67 m)

Ah < ±17 m (X = 4.18 m).

These values apply for a = 0.03 rad, corresponding to a layer height of 2.5 km; for lower layers Ah Avill be greater due to the decrease in a. These conditions are not likely to be satisfied in all the examples studied

and the analysis therefore overestimates the value of | p | in this respect.

(Some discussion of this point has been given by Bauer and Meyer [42].) The limitations of sonde soundings, and the effects of layer tilt and surface

irregularities therefore pro\'ide a partial explanation of the scatter of points in figure 6.16. Further detailed measurements of layer structure

are obviously desirable.

6.2.7. The Influence of Small Layers

The above discussion has dealt with the particular case of extended

elevated layers such as are often associated with temperature inversions.

However, it seems quite possible that these layers are merely the more

extreme examples of anisotropic irregularities which are thought to be

prevalent in the troposphere. There is already some preliminary evidence

supporting this concept in the results of refractometer and radar soundings

and recent theoretical work [22, 28, 40] has developed this approach in

detail. The relationship of this work to earlier analyses in terms of a

"scattering" model is discussed in the references quoted and need not concern us here. For our purpose it is sufficient to utilize the essential

features of the argument as the basis for a simple calculation.

It seems reasonable to assume that even in an atmosphere which sonde

data would lead us to classify as ''well-mixed" there are often layer-type

irregularities. Detailed evidence on the spatial form and stability of this type of layer or "feuillet" is so far lacking, but an inspection of some

refractometer results suggests that horizontal dimensions of a few kilo- meters represent a realistic assumption. Such a layer might exist as a

separate entity for, say, several minutes (as compared with a period of several hours for the extended layer in a stable inversion).

For the following analysis, let us consider two layers of horizontal

dimensions, x, of 2 and 10 km respectively with the following characteristics: An = 10-5

h = 100 m

a = 0.01-0.03 rad (i.e., layer height of 0.4 to 2.5 km on the Illinois paths).
256 TRANSHORIZON PARAMETERS

-Lr REFLECTION LOSS IN dB ^ -Lr REFLECTION LOSS IN dB

-Lr REFLECTION LOSS IN dB -Lr REFLECTION LOSS IN dB Figure 6.16. Observed hourly median field strength versus reflection loss assuming the linear model of figure 6.15.

For these conditions, the layers correspond to those of ''intermediate"

size in the analysis of Friis, Crawford, and Hogg [22]. They are defined

by the inequality:

y/2a\la > X > VSaX (6.7)

where 2a is the path length. In this case, the power received, Fr^ from

an antenna of effective aperture, Ar^ with a transmitter radiating a power Ft from a antenna of effective aperture v4r is given by:

FrIFt = (AtAr XW)/(2X3a3). (6.8)
INFLUENCE OF SMALL LAYERS 257 We can use this equation to calculate the corresponding field strength,

for the Illinois paths, in terms of lA'/m for 1 kW radiated from a half- wave dipole. We have the following relations:

where E is the field strength in volts/meter if Pr is in watts. From

(6.8), (6.9), and (6.10), we can calculate E for the two layers specified

above. The results obtained are shown in figure 6.17 for various layer

heights and the following models of reflection coefficient:

Model (b) is the Fresnel discontinuity equation which gives the limiting

value of |p| towards which all models tend as the layer thickness decreases. The curves in figure 6.17 show that the calculated field strength

depends considerably on the assumed n-profile. If |p| = An X/Sira^h,

values of field strength comparable with the long-term median value may

be produced by layers of about 10 km in lateral dimensions in the height

range 0.5 to 1 km. If |p| = An/2a'^, similar field strength may be pro- duced by layers in this height range if the lateral dimension is of the order

of 2 km. The effect of the layer decreases with increasing height, but

even with layer heights of 3 km, the field strength is still 1 /xV/m or

greater at both wavelengths for a 10 km layer with | p | = An/2a^. However, it should be pointed out that the assumed value of An = 10"^ is probably somewhat large for layers as high as 3 km. The results also show that model (b) (i.e., p = An/2a'^) gives field strength values which

are higher at X = 1.67 m (/ = 179.75 Mc/s) than at X = 4.18 m (/ = 71.75

Mc/s).

Any departure of refractive index structure from a smooth monotonic

decrease with height produces an increase in field strength on a 200-km

path in the frequency band 70 to 180 Mc/s. In the particular case

studied, elevated tilted layers result in signal enhancement of 10 to 25 dB

over the values for unstratified conditions, at all percentage levels. (The

importance of the tilted layer is possibly a consequence of the asymmetry

of the path, the transmitting antenna being 200 m above ground and the

receiving antenna 30 m.)

The predicted field strengths, for conditions classified as unstratified

in terms of sonde data, are in approximate agreement with observed

results, although the scatter of the points (plus the tendency to predict

A (X/2 dipole) = 0.127 X^ Pr (X/2 dipole) = E^X^/SOOt^

(6.9) (6.10)

(a) IpI = An • \/STva%

(b) IpI = An/2a\

6.2.8. Conclusions
258 TRANSHORIZON PARAMETERS

values in excess of the measured ones) point to the influence of anisotropic

layers or eddies of varying size and degrees of stability. This interpretation is consistent with numerical calculations based on the properties of ''intermediate" size layers, suggested in the analysis of Friis, Crawford,

and Hogg [22]. Calculations of the field strength produced by extended stable layers,

using sonde data and a model profile with a linear lapse of n with height, are in reasonable agreement with the experimental results. However,

there is probably no unique profile characteristic of elevated layers.

6.3. A New Turbulence Parameter

6.3.1. Introduction

We shall here attempt to unify past work utilizing the surface value

of the refractive index and the concept of atmospheric stability. To do

so we shall first review the concepts of atmospheric stability from the

viewpoint of temperature structure. In the process we shall show that

past efforts have neglected the important role of the conditions at the

earth's surface; then we shall extend this work to derive an analogous

expression for the radio refractive index.

6.3.2. The Concept of Thermal Stability

The stratification of the earth's atmosphere by and large reflects the

control of the earth's gravitational field. This average structure is, however, constantly disturbed by the intrusion of thermal plumes of heated air rising from the differentially heated surface of the earth, as well as by the mechanical mixing produced by the passage of air currents.

It is customary to assume that parcels of air are forced upwards through

the normal stratified atmosphere without mixing with the environmental

air mass. Since the parcel is forced to rise, be it by convection or by

mechanical mixing, it is also assumed that it expands and cools without

exchange of heat, i.e., adiabatically. In such a process it can be shown

that the pressure, P, is given by

(6.11)

where T is the temperature (°K) ; g, the acceleration of gravity ; R, the

gas constant for air; and, where a, the constant lapse of temperature with

height is (6.12)
CONCEPT OF THERMAL STABILITY 259

-40

0.5 1.0 1.5 2.0 2.5 3.0 LAYER HEIGHT (km)

Figure 6.17. Field strength produced by layers of horizontal dimensions 2 km and 10 km on the Chicago-Urbana paths for two model profiles.
260 TRANSHORIZON PARAMETERS

The zero subscripts in (6.11) refer to conditions at the earth's surface. Equations (6.11) and (6.12) are taken to hold as long as saturation does

not occur. It is further assumed that the mixing ratio e^eo-^ (6.13)

is constant throughout the process. In (6.13) e is the partial pressure of water vapor.

A parcel of air which is caused to rise will cool to the dew point if forced

aloft far enough. Further rising will lead to condensation. Conversely, a parcel of air which sinks down will have work done upon it adiabatically

and will become warmer than it was in its original elevated position. A

parcel of air following a condensation curve has a value of a, of — 6°C/km

denoted at a*. In addition, as water vapor condenses out of the parcel,

the vapor pressure will decrease. We here assume that the vapor pressure

will follow that of the saturation vapor pressure curve, Csz, as given by

Csz = est exp [-(3a*iz-t)], z > (6.14)

where i is the height of the lifting condensation level and /3 = 0.064 (°C)~^

The value of I has been found to be given [44] with sufficient accuracy for practical applications by

^ ^ 0.125 (r-rrf)o (km) (6.15)

where the zero subscript indicates that the difference between the temperature and the dewpoint, T^, need be evaluated only at the earth's

surface. Consider now a parcel rising through some environmental distribution

of temperature. The dry adiabatic lapse of temperature with height is greater than that observed on the average in the atmosphere (6 °C/km)

and thus the parcel becomes cooler, and more dense, than the environmental air and will sink to its initial conditions with the removal of the

lifting force. If, on the contrary, anywhere in its trajectory it becomes

warmer, and thus less dense, it will become unstable and rise of its own

buoyancy through the environmental air. Past radio-meteorological

studies have taken the area between the adiabatic curves and the environmental temperature distribution on a pressure (^ height) versus temperature chart as a measure of atmospheric stability. On the general argu- ment that the effects of atmospheric stability, or turbulence, influence

radio waves only through the radio refractive index, we shall extend the

above concepts to the radio refractive index.
CONCEPT OF THERMAL STABILITY 261

6.3.3. The Adiabatic Lapses of N

Introducing the dry adiabatic variations of P, T, and e into (1.20),

chapter 1. one obtains the dry adiabatic decrease of X, Xd'.

which is vahd for z < i. The distribution of A'^ Avith z may be ev aluated

by taking the logarithms of both sides of (6.16)

ln.V.= lnA-« + (|^-2) In (l -

and expanding the logarithms on the righthand side of (6.17) with the

omission of second-order terms where appropriate for our applications

of 0 < 3 km:

In Xd ^ In .Vo - TdZ (6.18) or A'^^^Yoexp (-r^<;) (6.19)

where

y g_ 2a 77.6 Ppg ,

^lien evaluated for the standard conditions of To = 288 °K (15 °C) and

Pq = 1013.2 mbar, T^ becomes

r, = 0.08 (km)-\ (6.21)

It is sometimes more convenient to use the drv adiabatic scale height,

Hd-. Hd = ^ = 12.5 km. (6.22)

1 d Above the local condensation level, the expression for .V becomes that of the wet adiabatic, .V^::

+ exp (/3a*^)| (6.23)
262 TRANSHORIZON PARAMETERS

where, for convenience, n 77 A ^ w 77.6 (4810) est Dt = 77.6 ~ and W( = ^r^^

Applying the same approximations as in the derivation of (6.18), one

obtains

In = InN, - T^z (6.24)

where

2c W, - — + + ir^«* (6-25) Tt TtNt '

For the standard conditions assumed above and eo corresponding to 60 percent relative humidity, ^ 7.5 km. Again, the form of the wet

adiabatic lapse of is given by

A^,, = A^, exp {-Ty,z), z> i. (6.26)

We shall now apply these results to the derivation of a refractive index

turbulence parameter.

6.3.4. The Turbulence Parameter, IT Analogously to the concept of thermal stability, we define IT as the area between the environmental N{z) curve and the appropriate adiabatic

decrease of A^. That is

CZ km / \

n = [Nobserved " Nadiabatic j

dz (6.27)

where the integration is arbitrarily taken from the surface to 3 km. This

then extends the integration well above the crossover height of the radio

horizon rays from transmitter and receiver normally encountered in tropospheric propagation. The integral for IT may be written

n = / Ndz - No exp (-Tdz) dz

Jo Jo - j

^ exp [-T^{z-i)] dz. (6.28)
TURBULENCE PARAMETER, II 263 Noting that

iV, = A^o exp (-r^^),

(6.28) then becomes, upon integration,

n =

Jo Id

A^o exp (-TdO {exp [-r^(3-0]-l}- (6.29)

n is thus dependent upon both profile characteristics and initial conditions. In fact, all the terms on the righthand side of (6.29) save

are determined from the conditions at the earth's surface. One might expect, then, that 11 would constitute a useful radio-meteorological parameter since it includes both the time-proven parameter A^o, the concept of atmospheric stability, and the integrated A^-profile characteristics. We

shall now apply this new parameter to the radio data presented in detail

in section 6.2. The values of Hd and utihzed are those for standard

sea-level conditions; i.e., 12.5 km and 7.5 km, respectively.

6.3.5. Comparison of Radio-Meteorological Parameters

The various radio-meteorological parameters discussed in section 6.1 plus n as derived from the Rantoul and Joliet radiosondes were tested

for their relative utility by comparison with the radio data of WBKB-TV

and WEXR-TV (see table 6.4). Correlation coefficients between the

radio data (3-hourly medians) of each path and the various radiometeorological parameters were determined for each of the profile categories given in table 6.5 as well as for all available data. It was not

possible to fit all of the data into the various categories since these were

deliberately made very strict in order to explore the utility of each

parameter within each of the propagation mechanisms assumed for the

profile categories (table 6.3). Thus it is observed that the sum of the

individual categories does not equal all available data. The results of these correlations are given in table 6.6. Before discussing these results

it is well to note that the crossover height of the tangent rays from each

end of the path were calculated by detailed ray tracing [15] for each radiosonde ascent. The values as determined from the Rantoul and Joliet soundings were averaged. This average crossover height was then used

to determine the contribution to the values of n below the crossover

Ndz
264 TRANSHORIZON PARAMETERS

height (W), above the crossover height (W), and Qe- As hsted in table

6.6, n represents the value of (6.27) for the height increment zero to 3 km.

Before proceeding, the method of calculating Qe will be described. The

concept of the equivalent gradient is closely related to that of the effective earth's radius. The curvature of the effective earth is given by:

1 1 1 -- = - + - ^ -T- cos d

Te a n ah

where fg is the effective earth's radius; a is the radius of the earth; and

(1/n) (dn/dh) cos d is the curvature of the radio ray. The value of is determined from the geometrical relationship

where c?,., the ground distance to the crossover height, h^, has already been

determined by the ray-tracing procedures discussed in chapter 3. Thus

one obtains

/dn\ _2hc _1

\dh/ dc^ a

for ^ = 0 and setting (cos 6)/n ^ I. In order to obtain the equivalent

gradient as a positive quantity in the more convenient refractivity units,

the following expression is used:

'-ii' !;)»

It is evident from table 6.6 that the results obtained with n are comparable with those using n', due supposedly to the crossover height being on the order of 100 m for these paths. The value 11'' appears to contribute

little to the final value of 11 and is poorly correlated with the radio data.

An interesting exception to this observation is the subrefractive case where n'' yields a larger correlation coefficient than either n or n'. The

overall conclusion that one reaches from the data of table 6.6 is that the

most promising of the parameters considered is 11, closely followed by

AA^ and Ns. In fact, the difference between the results obtained with

n and AiV gives one pause to consider if the added complexity of deter- mining n is worth the relatively small gain in correlation with the radio

data. The results obtained with Qe and go (the N gradient of the initial layer as reported by the radiosonde) are disappointingly small and are of no practical significance when compared with those of n, AN, and even
COMPARISON OF RADIO-METEOROLOGICAL PARAMETERS 265 Table 6.6. Summary of correlations

Cate- gory oampic

size

Radio-meteorological variable n n

tt" n ye AN

WENR.TV

68 —0. 3794 —0 3795 —0 1859 0. 1889 —0 1884 0. 1414 —0 3465

2 21 -^2064 -. 2683 ." 2003 !0035 !l346 ." 0531 -. 2653

3 101 -. 1748 -. 1754 .0087 .0291 -. 0303 -. 0940 -. 4553

4 6 -.3632 -. 3503 -.4315 .4990 -.4149 .0905 .0873

5 196 -. 5665 -. 5732 -. 0312 .0555 -. 1772 .3870 -. 5640

6 518 -. 4624 -. 4602 -. 0169 .0722 -. 1762 .2889 -. 4-544 WBKB-TV

(71.75 Mc/s) 1 67 -.2952 -. 2976 -. 2231 -.0926 .0791 .0107 -. 1687

2 27 -. 2071 -. 1843 -. 1378 -. 1037 .0939 .1678 -. 1608

3 113 -. 2691 -. 2709 .0763 . 1652 -. 0810 .0703 -. 3486

4 6 -.2445 -. 2202 -. 6849 .3084 -. 3686 .2573 .3485

5 213 -. 5519 -. 5562 -. 0576 .0624 -. 1152 .4168 -. 4635

6 564 -. 4384 -. 4398 .0087 .0430 -. 1027 .2128 -. 3505

*Category

1 = Unstratified.

2 = Ground-based duct.

3 = Elevated layer.

4 = Subrefractive layer. 5=1 through 4 combined. 6=All data for period of record.

The above data also allow one to evaluate a parameter of the form

Hi = age -\- h W,

which is analogous to Misme's recently suggested M parameter [19]. The

parameter Hi given above is physically more realistic for radio purposes

since it incorporates a measure of the refractive index turbulence rather

than thermal stability. The correlations obtained between field strength

and Hi are WENR-TV: 0.46

WBKB-TV: 0.44

which represents, for the significant figures carried, no improvement over

the use of 11 alone. This last analysis suggests that it might be desirable

to consider other combinations such as H2 = a^AN + 62 n. The correlations for this case are: WENR-TV: 0.49

WBKB-TV: 0.44

which represents a slight improvement over n alone.
266 TRANSHORIZON PARAMETERS

One might well wonder why the correlations of ge and field strengths

yield a higher correlation relative to that with AN or A^s on a monthly

median basis than on a daily basis. This is apparently due to the nonlinear nature of the averaging process of Qe which is dependent upon the

nonlinear weighting of the N{h) profile inherent in the refraction process. On the other hand, the average AiV or Ns obtained from the average N{h)

curve is the same as that obtained from averaging AN or Ns- Correlations on n and field strength on a monthly median basis are now under

study.

6.3.6. Conclusions

In conclusion, we reach the opinion that the results of previous experi- ence with the dependence of radio refraction upon Ns and the concept

of stability are incorporated into E. The resultant correlation with

radio data on a 3-hourly median basis is perhaps the most encouraging

obtained to date, but still, appears to be a marginal improvement over

the time-honored, and much simpler, parameter AA^. This conclusion

could well change with broader experience on other paths and radio

frequencies.

6.4. References

[1] Bean, B. R., and F. M. Meaney (Oct. 1955), Some applications of the monthly

median refractivity gradient in tropospheric propagation, Proc. IRE 43, No.

10, 1419-1431.

[2] Bean, B. R., and B. A. Cahoon (Jan.-Feb. 1961), Correlation of monthly median

transmission loss and refractive index profile characteristics, J. Res. NBS 65D

(Radio Prop.), No. 1, 67-74.

[3] Misme, P. (Mar.-Apr. 1960), Le gradient equivalent mesure direct et calcul theorique, Ann. Telecommun. 15, No. 3-4, 93.

[4] Misme, P. (Nov.-Dec. 1960), Quelques aspects de la radio-climatologie, Ann. Telecommun. 15, No. 11-12, 266.

[5] Misme, P. (Jan.-Feb. 1961), Essais de radio climatologie dans le bassin du

Congo, Ann. Telecommun. 16, No. 1-2, 29.

[6] Flavell, R. G., and J. A. Lane (Jan. 1963), The application of potential refractive index in tropospheric wave propagation, J. Atmos. Terrest. Phys. 24, 47-50.

[7] Moler, W. F., and W. A. Arvola (Aug. 1956), Vertical motion in the atmosphere

and its effect on VHF radio signal strength. Trans. Am. Geophys. Union 37, 399-409.

[8] Moler, W. F., and D. B. Holden (Jan.-Feb. 1960), Tropospheric scatter propagation and atmospheric circulations, J. Res. NBS 64D (Radio Prop.), No. 1, 81-93.

[9] Rice, P. L., A. Longley, and K. A. Norton (1959), Prediction of the cumulative

distribution with time of ground wave and tropospheric wave transmission

loss. Part I, NBS Tech. Note 15.

[10] Pickard, G. W., and H. T. Stetson (Jan. 1950), Comparison of tropospheric

reception, J. Atmos. Terrest. Phys. 1, 32-36.

[11] Bonavoglia, L. (Dec. 1958), Correlazione fra fenomeni meteorologici e propagazione altre I'orizzonte sul Mediterraneo, Alta Frequenza 27, 815.
REFERENCES 267

[12] Gray, R. E. (Sept. 1961), Tropospheric scatter propagation and meteorological

conditions in the Caribbean, IRE Trans. Ant. Prop. AP-9, 492-496.

[13] Onoe, M., M. Hirai, and S. Niwa (Apr. 1958), Results of experiment of long- distance overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79.

[14] Bean, B. R., L. Fehlhaber, and J. Grosskopf (Jan. 1962), Die Radiometeorologie

und ihre Bedeutung fiir die Ausbreitung der m-, dm-, and cm-Wellen auf

grossen Enternungen, Nachrichtentechnische Zeit. 15, 9-16.

[15] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric

refractive index, Proc. IRE 47, No. 5, 740-755.

[16] Misme, P. (Nov.-Dec. 1958), Essai de radioclimatologie d'altitude dans le nord

de la France, Ann. Telecommun. 13, No. 11-12, 303-310.

[17] Bean, B. R., (Mar. 1962), The radio refractive index of air, Proc. IRE 50, No. 3, 260-273.

[18] Lane, J. A. (Feb.-Mar. 1961), The radio refractive index gradient over the

British Isles, J. Atmos. Terrest. Phys. 21, No. 2-3, 157-166.

[19] Misme, P. (May-June 1961), L'influence du gradient equivalent et de la stabilite atmospherique dans les liaisons transhorizon au Sahara et au Congo, Ann. Telecommun. 16, No. 5-6, 110.

[20] Boithias, L., and P. Misme (May-June 1962), Le gradient equivalent; nouvelle determination et calculgraphique, Ann. Telecommun. 17, No. 5-6, 133.

[21] Saxton, J. A. (Sept. 1951), Propagation of metre radio waves beyond the normal

horizon, Proc. lEE 98, 360-369.

[22] Friis, H. T., A. B. Crawford, and D. C. Hogg (May 1957), A reflection theory for propagation beyond the horizon. Bell Syst. Tech. J. 36, 627.

[23] duCastel, F., P. Misme, A. Spizzichino, and J. Voge (1958-1960), Resultats experimentaux en propagation tropospherique transhorizon (a series of 10

papers), Ann. Telecommun. 13, 14, and 15.

[24] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Meteorology of the refraction problem. Book, Propagation of Short Radio Waves by

D. E. Kerr, pp. 198-199 (McGraw-Hill Book Co., Inc., New York, N.Y.).

[25] Jehn, K. H. (June 1960), The use of potential refractive index in synoptic scale radio meteorology, J. Meteorol. 17, 264.

[26] Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the

vertical distribution of the radio refractive index, J, Res. NBS 63D (Radio

Prop.), No. 2, 249-258.

[27] Misme, P. (1962), Private communication.

[28] duCastel, F., P. Misme, and J. Voge (1960), Sur le role des phenomenes de

reflexion dans la propagation louitaine des ondes ultracourtes. Electromagnetic

Wave Propagation, Book, p. 671 (Academic Press, London and New York).

[29] Booker, H. G., and W. E. Gordon (Apr. 1950), A theory of radio scattering in the troposphere, Proc. IRE 38, No. 4, 401-412.

[30] Villars, F., and V. F. Weisskopf (Oct. 1955), On the scattering of radio waves by

turbulent fluctuations in the atmosphere, Proc. IRE 43, No. 10, 1232-1239.

[31] Norton, K. A., P. L. Rice, H. B. Janes, and A. P. Barsis (Oct. 1955), The rate of fading in propagation through a turbulent atmosphere, Proc. IRE 43, No. 10, 1341-1353.

[32] Smyth, J. B., and L. G. Trolese (Nov. 1947), Propagation of radio waves in the lower atmosphere, Proc. IRE 35, No. 11, 1198-1202.

[33] Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric

refraction and its relation to wave guides and diffraction. Book, Meteorological Factors in Radio Wave Propagation, pp. 80-127 (The Physical Society,

London, England).
268 TRANSHORIZON PARAMETERS

[34] Gossard, E. E., and L. J. Anderson (Apr. 1956), The effect of super refractive layer on 50-5000 Mc nonoptical fields, IRE Trans. Ant. Prop. AP-4, No. 2, 175-178.

[35] Starkey, B. J., W. R. Turner, S. R. Badcoe, and G. F. Kitchen (Jan. 1958), The

effects of atmospheric discontinuity layers up to the tropopause height on

beyond-the-horizon propagation phenomena, Proc. lEE, Pt. B, 105, Suppl.

8, 97.

[36] Abild, V. B., H. Wensien, E. Arnold, and W. Schilkorski (1952), tJber die Ausbreitung ultrakurzer Wellen jenseits des Horizontes unter besonderer Berucksichtigung der meteorologischen Einwiskungen, Tech. Hausmitteilungen des Nordwestdeutschen Rundfunks, p. 85.

[37] Northover, F. H. (Feb. 1952), The anomalous propagation of radio waves in the 1-10 metre band, J. Atmospheric Terrest. Phys. 2, 106-129.

[38] Lane, J. A., and R. W. Meadows (Jan. 1963), Simultaneous radar and refrac- tometer soundings of the troposphere. Nature 197, 35.

[39] Brekhovskikh, L. M. (1960), Waves in layered media, Book (Academic Press, New York and London, England).

[40] duCastel, F. (1961), Propagation tropospherique et faisceaux hertziens trans- horizon, Book, p. 90 (Editions Chiron, Paris).

[41] Wagner, N. K. (1960), An analysis of radiosonde effects on measured frequency

of occurrence of ducting layers, J. Geophys. Res. 65, 2077-2085.

[42] Bauer, J. R., and J. H. Meyer (Aug. 1958), Microvariations of water vapor in the lower troposphere with applications to long-range radio communications,

Trans. Am. Geophys. Union 39, 624.

[43] Saxton, J. A. (July-Aug. 1961), Quelques reflexions sur la propagation des ondes

radioelectrique a travers la troposphere, L'Onde Elect. 40, 505.

[44] Hewson, E. W., and R. W. Longley (1944), Meteorology, Theoretical and Applied, Book, p. 352 (John Wiley & Sons, New York).
Chapter 7. Attenuation of Radio Waves

7.1. Introduction

The advent of tropospheric forward scatter techniques has made possi- ble communications over longer distances with higher frequencies than

has been heretofore thought practicable. The limitations imposed by

gaseous absorption, and by scattering by raindrops, upon the power re- quirements of a communications system for this application become more

important with increasing distance and frequency. It has been common

in the past to evaluate propagation path attenuation due to absorption by

multiplying the ground separation of the terminals by the value of the

absorption calculated for surface meteorological conditions [1]^ or avoid

the problem by restricting the communications system to frequencies

that are essentially free of absorption [2]. This is in contrast to another

approach [3] which actually used the absorption along the ray path.

The following sections of this chapter will be devoted to a descriptive

treatment of absorption of radio waves by raindrops and gaseous oxygen

and water vapor in the atmosphere.

Unless otherwise specified, the following conditions will be assumed in this chapter: (1) All attenuations will be expressed in terms of decibel

loss per unit length of the propagation path (dB/km). The attenuations

due to different causes are simply added to give the total attenuation in decibels. (2) In this treatment average conditions of temperature, droplet size, and droplet distribution are assumed for the radio path in order

to approximate conditions met in practice.

7.2. Background

The attenuation experienced by radio waves is the result of two effects:

(1) absorption and (2) scattering. At wavelengths greater than a few

centimeters, absorption by atmospheric gases is generally thought to be

negligibly small except where very long distances are concerned. However, cloud and rain attenuation have to be considered at wavelengths

less than 10 cm, and are particularly pronounced in the vicinity of 1 and

3 cm.

^ Figures in brackets indicate the literature references on p. 308. 269
270 ATTENUATION OF RADIO WAVES

It is helpful to recall that when an incident electromagnetic wave passes over an object whose dielectric properties differ from those of the sur- rounding medium, some of the energy from the wave is (a) absorbed by

the object and heats the absorbing material (this is called true absorption), and (b) some of the energy is scattered, the scattering being generally smaller and more isotropic in direction the smaller the scatterer is with respect to the wavelength of the incident energy. In the case of point-to-point radio communications we are interested

in the total attenuation of the scattering energy caused by losses resulting

from both the true absorption and the scattering.

7.3. Attenuation by Atmospheric Gases

The major atmospheric gases that need to be considered as absorbers

in the frequency range of 100 to 50,000 Mc/s are water vapor and oxygen.

For these frequencies the gaseous absorption arises principally in the

1.35 cm line (22,235 Mc/s) of water vapor and the series of lines centered

around 0.5 cm (60,000 Mc/s) of oxygen [4]. The variations of these

absorptions with pressure, frequency, temperature, and humidity are described by the Van Vleck [4, 5] theory of absorption. The frequency

dependence of these absorptions is shown in figure 7.1 [4]. In connection with figure 7.1, the water vapor absorption values have

been adjusted to correspond to the mean absolute humidity, p, (grams of water vapor per cubic meter) for Washington, D.C., 7.75 g/m^. The

reason for this adjustment is that water vapor absorption is directly pro- portional to the absolute humidity [6] and thus variations in signal in- tensity due to water vapor absorption may be specified directly in terms

of the variations in the absolute humidity of the atmosphere.

It can be seen from figure 7.1 that the water vapor absorption exceeds

the oxygen absorption in the frequency range 13,000 to 32,000 Mc/s,

indicating that in this frequency range, the total absorption will be the

most sensitive to changes in the water vapor content of the air, while

outside this frequency range the absorption will be more sensitive to changes in oxygen density. Only around the resonant frequency corres- ponding to X = 1.35 cm is the water vapor absorption greater than the

oxygen absorption. The absorption equations and the conditions under

which they are applicable have been discussed by Van Vleck [4], and the

best values to use for this section of the report have been taken from

Bean and Abbott [3]. The Van Vleck theory describes these absorptions from 100 Mc/s to 50,000 Mc/s in the following manner. The oxygen absorption at
ATTENUATION BY ATMOSPHERIC GASES 271

0.0002

0.0001

100 200 2,000 5,000 10.000 20,000 50,000

Frequency in Mc/s

Figure 7.1. Atmospheric absorption by the 1.36 cm line of water vapor and the 0.5 cm

line of oxygen.
272 ATTENUATION OF RADIO WAVES

T = 293 °K and standard atmospheric pressure in decibels per kilometer,

7i, is given by the expression:

0.34

7i =

1 . A2 +

+ A.;+ (7.1)

where X is the wavelength for which the absorption is to be determined

and where ^vi and are line-width factors with dimensions of cm~^

This formula is based on the approximations of collision broadening theory.

This theory postulates that, although the electromagnetic energy is freely exchanged between the incident field and the molecules, some of the electromagnetic energy is converted into thermal energy during

molecular collisions and thus a part of the incident electromagnetic energy

is absorbed. The term in (7.1 )involving Avi gives the nonresonant

absorption arising from the zero frequency line of oxygen molecules while

the terms involving Ai^2 describe the effects of the several natural resonant

absorptions of the oxygen molecule which are in the vicinity of 0.5 cm

wavelength. The (2 =b 1/X) (cm~^) terms are the portion of the shape

factors that describe the decay of the absorption at frequencies away from

the resonant frequency (the number 2 is the reciprocal of the centroid resonant wavelength 0.5 cm).

The water vapor absorption at 293 °K arising from the 1.35 cm line, 72, is given by:

72

P

3.5 X 10

(1 - -L)^

AX 1.35/

+

\X ^ 1.35/

(7.2)

where p is the absolute humidily and Aj^s is the line width factor of the

1.35 cm water vapor absorption line. The additional absorption arising

from absorption bands above the 1.35 cm line, 73, is described by:

73

P 0.05 Aj/4 (7.3)
ATTENUATION BY ATMOSPHERIC GASES 273 where is the effective hne width of the absorption bands above the

1.35-cm hne. The nonresonant term has been increased by a factor of 4 over the original Van Vleck formula in order to better satisfy experimental results [7]. Although Van Vleck gives estimates of the various line widths, more

recent experimental determinations were used whenever possible. The

line-width values used in this chapter are summarized in table 7.1. Table 7.1. Line width factors used to determine atmospheric absorption

Line Temperature Value Sources width

293 °K 0. 018 em-i atm-i Birnbaum & Maryott [8] Aj'2 300 °K . 049 cm~i atm~' Artman & Gordon [9] AV3 318 °K . 087 cm-i atm-i Becker & Autler [7] A;.4 318 °K . 087 cm-i atm-i Becker & Autler [7]

The preceding expressions for gaseous absorption are given as they

appear in the literature and do not reflect the pressure and temperature

sensitivity of either the numerical intensity factor or the line widths.

This sensitivity must be considered for the present application since it is necessary to consider the manner in which the absorption varies with

temperature and pressure variations throughout the atmosphere. The

dependence of intensity factors upon atmospheric pressure and temperature variations was considered to be that given by the Van Vleck theory.

The magnitude and temperature dependence of the line widths is a question not completely resolved. Both theory and experiment indicate

the line width to vary as x <0. Different measurements on the same line of oxygen have given values of x ranging from 0.71 to 0.90

with differences in the magnitude of Av of about 2 percent [10, 11]. Experiments have also clearly indicated that the line width changes from

line to line, with maximum fluctuations of about 15 percent. In the

frequency region considered in this chapter (10 to 45 Gc/s) the centroid

frequency approximation for oxygen is valid and a mean line width can be

used with good accuracy, but in the region of the resonant frequencies of oxygen, the line-to-line hne width variations must be taken into account. The expressions used to calculate the absorptions are given in table 7.2. The reference temperatures given are those at which the appropriate

experimental determinations were made, and the pressures are to be

expressed in millibars. A detailed discussion of the theoretical aspects of the pressure and temperature dependence is given by Artman [12]. Experimental measurements on the absorption of microwaves by the

atmosphere (performed after our original work) show different values of the loss than those obtained by theoretical prediction methods. There

is reasonably good agreement between the predicted and measured loss for oxygen, but the measured loss of water vapor is considerably greater
274 ATTENUATION OF RADIO WAVES

Table 7.2. Values used in the calculation of atmospheric absorption

Absorption Intensity factor Line width dB/km

0. 34 / P \ /293\ 2

X2 \1013. 25/ \ T / \1013. 25/ \ T /

and .J "

\1013. 25/)r)"' \ T /

72*

P

0. 0318 /293\ 5/2 / 644\

X2 \ T / ^^^\ T

)

/ P \ /318\ 1/2 Avzl- )( ) (l+.0G46p)

\1013. 25/ \ T /

73*

p

0. 05 /293\

X2 \ T/

/ P \ /318\ 1/2 A»'4( )( ) (l+.0046p)

\1013. 25/ \ T /

*p is water vapor density in g/m^

than that of the predicted amount, particularly above 50,000 Mc/s [13]. These observed discrepancies have little effect upon the present study,

which is confined to frequencies less than 50 Gc/s. The results of the

present study, for the frequency range 100 to 50,000 Mc/s, agree with

those reached by Tolbert and Straiton [14] in their field experiments at Cheyenne Mountain and Pikes Peak, Colo., at altitudes of 14,000 ft. The above approach represents that presented by Bean and Abbott

[3]. The following treatment was given by Gunn and East [15] and

based on Van Vleck's two papers [4, 5]. This latter presentation is only

valid when single line absorption with no appreciable overlap from

adjacent lines is considered.

By taking into account the temperature and pressure dependence of the line widths, it is seen that for a given quantity of water vapor, the

attenuation is proportional to

P-' and exp (- ^) ,

at the resonance line, toP and exp ^)

at the sides of the curve, and to P and T~^''^ well away from resonance. In applying the above considerations to absorption approximations it also must be remembered that for a given relative humidity, the density

will vary considerably with temperature. Table 7.3 shows attenuation

by water vapor at various temperatures and wavelengths.
ATTENUATION BY ATMOSPHERIC GASES 275 The behavior of water vapor attenuation near the resonant line is very

remarkable, as can be seen by inspecting (7.2). Since Av^ is small compared to 1/X, it may be neglected in the denominator of (7.2) for nonresonant wavelengths. The attenuation per unit density is thus directly

proportional to Av^ and hence to the total pressure for these frequencies.

But at the resonant frequency, the dominant term in the expression is proportional to 1/Avz, and thus inversely proportional to the pressure. In the atmosphere, the water vapor density is proportional to the total pressure. Therefore, the attenuation is independent of pressure at the resonant frequency and now depends only on the fraction of water vapor

present. For practical purposes, this means that attenuation can occur

at high altitudes with the same effectiveness as in the lower, denser layers

if the mixing ratio is the same. On the other hand, oxygen absorption occurs because of a large number

of lines around 60 Gc/s. In the region from 3 to 45 Gc/s the attenuation

is proportional to and to [15]. As the temperature decreases the

attenuation increases gradually. At —40 °C oxygen attenuation is about

78 percent higher than at 20 °C due to increased density at low temperatures. Table 7.4 shows the pressure and temperature corrections for oxygen attenuation at wavelengths between 0.7 and 10 cm.

Figure 7.2 shows the attenuation at a pressure of 1 atmosphere and

20 °C as a function of wavelength [15]. The solid lines represent values

of attenuation measured by Becker and Autler [7]. The dashed line shows values calculated from Van Vleck's theory. The water vapor

absorption curve, c, corresponds to a water content of 1 g/ml

Since absorption is so sensitive to the absolute humidity level, it is helpful to have information on the chmatic variation of absolute humidity

throughout the 1 to 99 percent range of values normally used in radio

engineering. Estimates of the values of absolute humidity at the surface

expected 50 percent of the time for the United States for February and

August are given in figures 7.3 and 7.4 respectively [16]. It is evident

that for either month the coastal regions display greater values of absolute

humidity than do the inland regions. Note that for any location the

August values are consistently greater than the February values. Figures

7.5 through 7.8 show the values of absolute humidity expected to be

exceeded 1 and 99 percent of the time throughout the United States in both summer and winter. In addition to oxygen and water vapor, there are a number of other

atmospheric gases which have absorption lines in the microwave region

from 10 to 50 Gc/s. These gases normally constitute a negligible portion

of the general composition of the atmosphere, but could conceivably

contribute to attenuation. Table 7.12 shows the resonant frequencies,

maximum absorption coefficients at 300 °K (attenuation coefficient if the

fraction of molecules present were equal to unity), expected concentration
276 ATTENUATION OF RADIO WAVES

Table 7.3. Water vapor attenuation (one way) in dB/km

[After Gunn-Eastj.

P, pressure in atmospheres; W, water vapor content in g/m~^ T

(°C) X(cm) 10 5.7 3.2 1.8 1.24 0.9

20

0 -20 -40

0. 07X10-3PW

0. 08X10-3PW

0. 09X10-3PW

0. 10X10-3PW

0. 24X10-3PW

0. 27X10-3PW

0. 30X10-3PW

0. 34X10-3PW

0. 7X10-3PW

0.8X10-3PW

0. 9X10-3PW

1. OXIO-SPW

4.3X10-3PW*

4. 8X10-3PW*

5. 0X10-3PW*

5. 4X10-3PW*

22. 0X10-3PW*

23. 3X10-3PW*

24.6X10-3PW*

26. 1X10-3PW*

9. 5X10-»PW

10.4X10-3PW

11. 4X10-3PW

12. 6X10-3PW

*The pressure dependencies shown are only approximate. Near the 1. 35 cm water vapor absorption line (between 1. 0 cm and 2. 0 cm) no simple power dependency of P and W is accurate.

Table 7.4. Pressure and temperature correction for oxygen attenuation for wavelengthi between 0.7 and 10 cm

[After Gunn-East]

r(°C)

20

0 -20

-40

Factor (P is pressure in atmospheres)

1. 00 P2

1. 19 P2

1. 45 P2

1. 78 P2
ATTENUATION BY ATMOSPHERIC GASES 277

Figure 7.4. Estimate of the value of absolute humidity expected SO percent of the time

for August.
278 ATTENUATION OF RADIO WAVES

Figure 7.6. Values of absolute humidity expected to be exceeded 1 percent of the time

for August.
ATTENUATION BY ATMOSPHERIC GASES 279

Figure 7.8. Values of absolute humidity expected to be exceeded 99 percent of the time

for August.
280 ATTENUATION OF RADIO WAVES

in the atmosphere, and expected absorption coefficients due to these

trace constituents. The data on molecular absorption coefficients was

taken from Ghosh and Edwards [33], that on concentrations from the

Glossary of Meteorology [29]. It is readily seen that the attenuation due

to these sources is negligible compared to the high absorption due to oxygen and water vapor.

7.4, Estimates of the Range of Total

Gaseous Absorption

The range in gaseous absorption can be seen by considering the data

for the months of February and August at Bismarck, N. Dak., and

Washington, D.C., two stations with very different climates. The values

of total gaseous absorption (defined as the sum of 71, 72, and 73, where

7i = oxygen absorption in decibels per kilometer, 72 = water vapor

absorption arising from the 1.35 cm line and 73 = additional absorption

arising from absorption lines whose frequencies are considerably higher

than that corresponding to the 1.35 line) at each station and elevation up

to 75,000 ft are shown in figures 7.9 and 7.10 for each of the four station

months for the frequency range of 100 to 50,000 Mc/s. Above 75,000 ft the absorption values for all four station months are identical and are given for each frequency in figure 7.11. The absolute humidity was

calculated using the upper air monthly average values of temperature,

pressure, and humidity as reported by Ratner [17]. Readings for the

relative humidity are not generally given in this report for altitudes

greater than about 15 km due to the inability of the radiosonde to measure the small amount of humidity present at these altitudes. It is believed that the climates represented by these station months encompass

the range of those of the majority of the continental United States radio

propagation paths.

An interesting property of the annual range of absorption as a function

of the frequency may be seen in figures 7.9 and 7.10. For the first 5,000

ft above the surface, it is noted that in the frequency range of 10 to 32.5 Gc/s, the summer values are greater than the winter values due to in- creased humidity of the summer months. Outside of this frequency

range, however, the winter values of absorption are greater due to the

increased oxygen density. In the frequency range 6 to 45 Gc/s, atmospheric absorption, 7^, at a frequency p, arises primarily from oxygen absorption, jdy, and water

vapor absorption, 7^;^; i.e..

7>' y dv ~\~ ywv (7.4)
RANGE OF TOTAL GASEOUS ABSORPTION 281
282 ATTENUATION OF RADIO WAVES

I 1 i I WASHINGTON, D.C.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Height Above the Surface in Thousands of Feet Figure 7.10. Total gaseous atmospheric absorption from the surface to 76,000 feet: Washington, D.C.
TOTAL RADIO PATH ABSORPTION 283 Zhevankin and Troitskii [31] have indicated that dv and y^v can be repre- sented as exponential functions of height, Z, above the earth's surface,

idv = 7dvo exp

where dv^ and 7u,vo are the values of ^ dv and ^wv, respectively, at the

earth's surface, and Udv and Uy^^ are called the ''scale heights" of dv and

7«,^. This model is known as the "bi-exponential" model of absorption,

and and 7,rv are often called the ''dry" and "wet" terms of 7^. The

scale height for the dry term in the frequency range 6 to 45 Gc/s can be

written as [32]

Ed. (in km) ^ (7.6)

where To is the surface temperature in °K, a is the temperature lapse rate with height in °K/km, and 6, c, are constants determined from thermodynamic considerations. Because of the hump in the lly,v curves as op- posed to the flat Udv curve in figure 7.28, such a handy expression as (7.6) for R is not possible in the 6 to 45 Gc/s frequency range (fig. 7.28 was

determined from actual radiosonde data at Verkhoyansk, U.S.S.R.).

7.5. Total Radio Path Absorption

The total path absorption is determined by calculating the various

absorption coefficients as functions of the heights along the ray path and

then numerically integrating the values along the entire path using standard ray tracing techniques outlined in chapter 3. The values of total path

integration over a 100-km path thus obtained are presented in figure 7.12

for Bismarck, N. Dak., and Washington, D.C. The difference between

the two climates is evident principally at the higher frequencies, where

the Washington absorptions are consistently above the Bismarck values. This is apparently due to a combination of generally greater humidities

and greater refractive effects. These two effects are related. The in- creased humidity at Washington enhances the water vapor absorption

and increases the refraction causing the radio ray to travel consistently

through lower levels of the atmosphere with consequent increase in total path absorption.
284 ATTENUATION OF RADIO WAVES

Height Above the Surface in Thousands of Feet Figure 7.11. Common values of total gaseous atmospheric absorption for elevations greater than 76,000 feet.
TOTAL RADIO PATH ABSORPTION 285

I I I I

I 1—I I I I 1—I

I M I

X = BISMARCK, FEB. BISMARCK, AUG.

= WASHINGTON, D.C., FEB. A = WASHINGTON, D.C., AUG.

J I I I I I I J \

I I I I I I J \ \ I I I I I 100 20 0 50 0 7 0 0 10 0 0 20 0 0 5 0 0 0 7 0 0 0 1 0 0 0 0 20 0 0 0 FREQUENCY (Mc/s)

50000 70000 100000

Figure 7.12. Total path absorption over a 1000 km propagation path with the climate

of Bismarck, N. Dak.
286 ATTENUATION OF RADIO WAVES

7.6. Derivation of Absorption Estimate for Other Areas

The values of total path absorption given above are for two specific

locations. For this particular study to be of practical use, a means

should be provided for arriving at estimates of geographic and annual

variations of total path absorption for various surface distances and

frequencies. The method chosen utilizes the correlation between total path absorption and the surface value of the absolute humidity, expressed

in grams of water vapor per cubic meter, which appears explicity in Van Vleck's water vapor absorption formulas [4,5]. The basis for expecting a correlation to exist between the absolute humidity and the total path absorption is that the absorption at those frequencies for which

water vapor absorption is dominant (approximately 10 to 32 Gc/s) varies

directly as the absolute humidity while, for those frequencies at which

oxygen absorption is dominant, it varies inversely as the absolute humidity due to the inverse relationship of oxygen density and water vapor

density. That is, during the warm seasons of the year the total atmospheric pressure tends towards its yearly minimum (as does the oxygen

absorption), while the absolute humidity tends towards its yearly maximum (as does the water vapor absorption). Conversely, during the

colder seasons of the year, the pressure tends towards its maximum, value

while the absolute humidity tends to\/ards its minimum value [3]. As an example of the correlation method, the surface absorption was

calculated at a water vapor-dominated frequency (22.2 Gc/s) and

oxygen-dominated frequency (50 Gc/s) for each month throughout the

year for both Washington and Bismarck. These values are plotted on

figure 7.13. The term surface absorption is used for the values of absorption calculated from standard ground level weather observations.

The water-dominated 22.2 Gc/s data fall on a smooth curve despite the pressure and temperature differences of the two stations. The oxygendominated 50 Gc/s data, however, display an interesting separation of points for each station, although the distribution of points at the two

locations display similar slopes. The 50 Gc/s absorption is more sensi- tive to the atmospheric density dift^erence between the stations. If the

pressure differences were taken into account, the Bismarck data would

increase about 12 percent and the two curves would be distributed along a common line with the same slope as the original two curves. This

figure, then, indicates that the absorption is correlated with the absolute

humidity. The above illustration is for surface values rather than for integrated propagation path values. Variations in the upper air meteorology that are not reflected in the surface values will tend to diminish

the correlation. (For comparison of percentage absorption over a 300-km

path, the first few hundred feet contribute about 5 percent at 100 Mc/s

increasing to 42 percent at 10 Gc/s and remaining constant to 50 Gc/s.)

Keeping these reservations in mind, one may utilize the method of least
ABSORPTION ESTIMATE FOR OTHER AREAS 287

0.07

0.05

0.03

0.02

0.01

0.007

0.005

0.003

0.002

0.001

o Washington O, at 50,000 Mc/s

• Washington H^O at 22,200 fVlc/s 9 Bismarl< O2 at 50,000 D/lc/s ® Bismar/< H^O at 22,200 iWc/s

0.1 0.2 0.3 0.4 0.5 OB 0.7 0.8 0.9 1.2 1.3 Log of Absolute Humidity (gm/m ) Figure 7.13. Surface value of 22.2 and 60 Gc/s absorption versus absolute humidity at Washington, D.C., and Bismarck, N.Dak.
288 ATTENUATION OF RADIO WAVES

Table 7.5. Values of m and b in the regression line y = mx + b, where y is the logarithm of the total path attenuation, y, b is the value of y when the absolute humidity

is 1 g/m^, and x is the logarithm of the absolute humidity.

[After Bean and Abbott]

Distance 100 km 300 km 1000 km

Freq. (Gc/s) m b m b m b

100 -0. 07263 -3. 69822 -0. 06324 -2. 60042 -0. 13298 -1. 83711 300 -. 08212 -1.72139 -. 06363 -0. 73037 -. 15276 -0. 17792 -. 10203 -0. 49427 -. 07078 +. 31846 -. 16932 + .75687 6000 +. 06996 -. 24077 +. 06872 +. 54167 -. 04731 + .96517

10, 000 +. 26022 -. 14038 +. 23447 +. 64637 +. 10831 +1. 06172

15,000 +. 48097 +. 20891 +. 44214 +. 98393 +. 31506 +1. 36994

18, 000 +. 62034 +. 65324 +. 58044 +1. 42477 +.46112 +1. 77604

21, 000 +. 75045 +1. 41086 +. 70693 +2. 25807 +. 60558 +2. 56360

22, 406 +. 77630 +1. 67001 +. 72874 +2. 57270 +. 62885 +2. 87766

23, 076 +. 77153 +1. 69105 +. 72508 +2. 57734 +. 62963 +2. 87575

24, 000 +. 74979 +1. 62838 +. 70659 +2. 46633 +. 60458 +2. 77175

30, 000 +. 54155 +1. 34408 +. 50126 +2. 11161 +. 37634 +2. 48455

33, 000 +. 46305 +1. 51053 +. 42416 +2. 28497 +. 29646 +2. 67460

42, 000 +. 25936 +2. 46361 +. 23331 +3. 25085 +. 10712 +3. 66659

45, 000 +. 18460 +2. 91681 +. 16668 +3. 70472 +. 04360 +4. 12531

FREQUENCY IMc/S) Figure 7.14. Values of b and m in the regression line y = mx + b, where y is the logarithm of the total path absorption, x is the logarithm of the absolute humidity, b is the value of y when the absolute humidity is 1 g/m^.
ABSORPTION ESTIMATE FOR OTHER AREAS 289 squares to obtain the total path absorption as a linear function of the

absolute humidity. The parameter is a statistical regression line at each

frequency for path propagation distances of 100, 300, and 1,000 km as given in table 7.5, and also plotted on figure 7.14, allowing the reader to estimate the regression line for frequencies other than those of the

present study [3]. Statistics for the variation of absolute humidity have already been

given, and contours of the mean values of absolute humidity for the world

are given on figures 7.15 and 7.16 for the months of February and August

respectively. It is noted that the absolute humidity for either month is greater in the coastal regions than in the continental interiors and that

the windward sides of continents have larger values than do the leeward

sides. By reference to figures 7.15 and 7.16 [16] for the United States, and

figure 7.14, estimates of radio power loss due to absorption may be

obtained. As an example, figure 7.17 gives the values expected to be

exceeded 1 percent of the time over a 300 km propagation path at 10,000

Mc/s for the United States during the month of August.

Figure 7.15. Average absolute humidity (g/m^), February.
290 ATTENUATION OF RADIO WAVES

Figure 7.17. August values of 300-mi propagation path absorption loss {in dB) to he exceeded 1 percent of the time for a frequency of 10,000 Mc/s.
ATTENUATION IN CLOUDS 291 7.7. Attenuation in Clouds

Cloud droplets are regarded here as those water or ice particles having

radii smaller than lOO^t or 0.01 cm. For wavelengths of incident radiation

well in excess of 0.5 cm, the attenuation becomes independent of the

drop size distribution. The generally accepted equations for attenuation

by clouds usually show the moisture component of the equations in the

form of the liquid water content (g/m^). Observations indicate that the

liquid water concentration in clouds generally ranges from 1 to 2.5 g/m^

[18], although Weickmann and aufm Kampe [19] have reported isolated

instances of cumulus congestus clouds with a reading of 4.0 g/m^ in the

upper levels. In ice clouds, it will rarely exceed 0.5 and is often less than

0.1 g/m^. The attentuation of cloud drops may be WTitten as: K = K,M,

where K = attenuation in dB/km,

Ki = attenuation coefficient in dB/km/g/m^, and

M = liquid-water content in g/m^

Values of Ki by ice and water clouds are given for various wavelengths

and temperatures by Gunn and East in table 7.6.

Table 7.6. One-way attenuation coefficient, Ki, in clouds in dB/km/gm/m^

Temperature

(°C) Wavelength (cm)

0.9 1.24 1.8 3.2

20 0.647 0. 311 0. 128 0. 0483 Water cloud 10 .681 .406 .179 .0630 0 .99 .532 .267 .0858 -8 1.25 .684 .34 . 112 (extrapolated) (extrapolated)

0 8. 74X10-3 6. 35X10-3 4. 36X10-3 2. 46X10-3

Ice cloud —10 2. 93X10-3 2. 11X10-3 1. 46X10-3 8. 19X10-3 -20 2.0 XlO-3 1.45X10-3 1.0 xio-3 5. 63X10-4

Several important facts are demonstrated by table 7.6. The decrease

in attenuation with increasing wwelength is clearly shown. The values

change by about an order of magnitude for a change of X from 1 to 3 cm. The data presented here also show that attenuation increases with decreasing temperature. Ice clouds give attenuations about tw^o orders of magnitude smaller than water clouds of the same water content. The

attenuation of microw^aves by ice clouds can be neglected for all practical

purposes [20]. The comprehensive works of Gunn and East [15] and

Battan [20] on attenuation offer excellent sources of detailed information on this subject.
292 ATTENUATION OF RADIO WAVES

7.8. Attenuation by Rain

Ryde and Ryde [21] calculated the effects of rain on microwave propagation and showed that absorption and scattering effects of raindrops

become more pronounced at the higher microwave frequencies where the

wavelength and the raindrop diameters are more nearly comparable. In

the 10 cm band and at lower wavelengths the effects are appreciable, but

at wavelengths in excess of 10 cm the effects are greatly decreased. It is also known that suspended water droplets and rain have an absorption

rate in excess of that of the combined oxygen and water vapor absorption [3]. In practice it has been convenient to express rain attenuation as a function of the precipitation rate, R, which depends on both the liquid

water content and the fall velocity of the drops, the latter in turn depending on the size of the drops.

Ryde studied the attenuation of microwaves by rain and showed that

this attenuation in dB/km can be approximated by:

Kr = K RirTdr (7.7)

where Kr = total attenuation in dB

K = constant

R{r) = rainfall rate

r = length of propagation path in km, and

a = constant. Laws and Parsons [22] observed the distribution of drop sizes for various rates of fall on a horizontal surface. The higher the rainfall rate, the larger the drops, and also the greater the range in size of the drops.

However, in order to derive the size distributions occurring while the

drops are falling in the air, each rainfall rate must be divided by the

particular velocity of fall appropriate to the corresponding drop diameter.

Figure 7.18 shows the resulting distribution in air, expressed as the rela- tive mass of. drops of given diameter, for three chosen representative

values of precipitation rate, p, namely, 2.5 mm/hr, 25 mm/hr, and 100

mm/hr [23]. The excess path loss per mile, according to Ryde, for the

three carrier frequency bands of 4, 6, and 11 Gc/s is shown on figure 7.19. Figure 7.20 [24] is a scatter diagram showing transmission loss versus rainfall rate. For comparison, Ryde's equation is plotted on figure 7.20. The greatest uncertainty in predictions of attenuation due to rainfall,

using theoretical formulae as a basis for calculation, is the extremely

limited knowledge of drop size distribution in rains of varying rates of

fall observed under differing climatic and current weather conditions.
ATTENUATION BY RAIN 293

<

\ p = ^1 . 5 mm/

h

r

\ P = 2.3 mm/ hr

/ V

/ /\

J / \ \ P = 100 mm/lir

1 \ \

0-a c-s O-^ C-5 O-G

D. cm

FiGUEE 7.18. Relative total rna.ss of liquid water in the air contributed by rain drops of diameter D for various precipitation rates. (Derived from Laws and Parsons' distributions for a horizontal surface by dividing by the appropriate terminal velocities.)
294 ATTENUATION OF RADIO WAVES

40

RAINFALL RATE IN INCHES PER HOUR

Figure 7.19. Rain attenuation versus rainfall rate.

(Theoretical, after Ryde and Ryde).

There is little evidence that a rain with a known rate of fall has a unique

drop-size distribution though studies on this problem seem to indicate

that a most probable drop size distribution can be attached to a rain of given rate of fall [22]. Table 7.8 shows the percentage of total volume

of rainfall occupied by raindrops of different diameters (cm) and varying

rainfall rates (mm/hr). Tables 7.8 and 7.9 are offered as an aid to estimating, through a qualitative approach, the attenuation of radio waves by raindrops [25]. Table 7.8 gives the decibel attenuation per kilometer in rains of different rates of fall and radio wavelengths between 0.3 and 10 cm. In table 7.9,
ATTENUATION BY RAIN 295

similar to table 7.8 an additional set of results is contained for rains of measured drop size distribution.

Since the total attenuation cross section [26] depends on the temperature

(because of its effects on the dielectric properties of water), it is important

to evaluate the attenuation of rains whose drops are at different temperatures from those in the preceding tables. Table 7.10 contains the neces- sary data relative to the change of attenuation with temperature and is to be used with table 7.8. For example, in table 7.8, with a precipitation

rate of p = 0.25 mm/hr, temperature of 18 °C, X = 1.25 cm, the attenuation is 0.0215 dB/km. Using the correction factors obtained from table

7.10, for the same general conditions of precipitation and wavelength,

for a temperature reading of 0 °C, the attenuation reads 0.02043 dB/km;

for a value of 30 °C an attenuation of 0.019350 dB/km is noted; and for a temperature of 40 °C the attenuation is 0.01742 dB/km.

Figure 7.20. Transmission loss in decibels per mile versus rainfall in inches per hour.

(After Hathaway and Evans, 1959).
296 ATTENUATION OF RADIO WAVES

Table 7.7. Dro-p size distribution (Burrows-Attwood)

Precipitation rate, p, in mm/hr

Drop diameter, D, in 0. 25 1.25 2.5 12.5 25 50 100 150 centimeters Percentage of a given volume containing drops of diameter, D

0.05 28.0 10.9 7.3 2.6 1.7 1.2 1. 0 1.0

. 10 50.

37. 1 27.8 11.5 7.6 5.4 4.6 4.

. 15 18.2 31.3 32.8 24.5 18.4 12.5 8.8 7.6 .20 3.0 13.5 19.0 25.4 23.9 19.9 13.9 11.7 .25 0.7 4.9 7.9 17.3 19.9 20.9 17.1 13.9 .30 1.5 3.3 10. 1 12.8 15.6 18.4 17.7

.35 0.6 1.

4.3 8.2 10.9 15.0 16.1 .40 .2 0.6 2.3 3.5 6.7 9.0 11.9 .45 .2 1.2 2.

3.3 5.8 7.7 .50 0.6 1.

1.8 3.0 3.6

.55 .2 0.5 1.1.7 2.2

.60 .2 0.5 1.0 1.2 .65 .2 0.7 1.0

. 70 0.3

Table 7.8. Attenuation in decibels per kilometer for different rates of rain precipitation

Temperature 18 °C

(Burrows-Attwood)

Precipi- tation Wavelength, X in cm

rate, p, in mm/hr X=0.3 X=0.4 X=0. 5 X=0.6 X = 1.0 X = 1.25 X=3. 0 X=3.2 X = 10

0. 25 0. 305 0.230 0. 160 0. 106 0.037 0. 0215 0. 00224 0. 0019 0. 0000997

1.25 1. 15 0. 929 0. 720 0. 549 0. 228 0. 136 0. 0161 0. 0117 0. 000416

2.5 1.98 1. 66 1.34 1.08 0. 492 0. 298 0. 0388 0. 0317 0. 000785

12.5 6. 72 6. 04 5. 36 4. 72 2. 73 1.77 0. 285 0. 238 0. 00364 25.0 11.3 10.4 9. 49 8. 59 5. 47 3. 72 0. 656 0. 555 0. 00728

50. 19.2 17.9 16.6 15.3 10.7 7. 67 1.46 1.26 0. 0149

100. 33.3 31. 1 29.0 27.0 20.0 15.3 3. 24 2. 80 0. 0311

150. 46.0 43.7 40.5 37.9 28.8 22.8 4. 97 4. 39 0. 0481

Table 7.9. Attenuation in rains of known drop size distribution and rate of fall

(decibels per kilometer)

( Burrows-Attwood)

Precipitation Wavelength X in cm

rate, p, in mm/hr

1.25 3 5 8 10 15

2. 46

' 1. 93 10-1 4.92 10-2 4. 24 10-3 1.23 10-3 7. 34 10-1 2. 80 10-1 4.0 3. 18 10-1 8. 63 10-2 7.11 10-3 2.04 10-3 1.19 10-3 4. 69 10-4 6.0 6. 15 10-1 1.92 lO-i 1.25 10-2 3. 02 10-3 1.67 10-3 5.84 10-4 15.2 2. 12 6. 13 10-1 5. 91 10-2 1. 17 10-2 5. 68 10-3 1.69 10-3 18.7 2. 37 8. 01 10-1 5. 13 10-2 1.10 10-2 6. 46 10-3 1.85 10-3 22.6 2. 40 7. 28 10-1 5. 29 10-2 1.21 10-2 6. 96 10-3 2. 27 10-3 34.3 4. 51 1.28 1. 12 10-1 2. 32 10-2 1. 17 10-2 3.64 10-3

43.6.17 1.64 1.65 10-1 3. 33 10-2 1.62 10-2 4. 96 10-3
RAINFALL ATTENUATION CLIMATOLOGY 297 Table 7.10 Correction factor {multiplicative) for rainfall attenuation

(Burrows-Attwood)

Precipitation

rate, p, in X cm 0 °c 10 °c 18 °C 30 °C 40 °C mm/hr

0. 25 0. 5 0. 85 0. 95 1. 0 1.02 0. 99

1. 25 .95' 1. 00 1. 0 0.90 .81 3.2 1. 21 1. 10 1. 0 .79 .55

10. 0 2. 01 1.40 1. 0 .70 .59 2.5 0.5 0. 87 0. 95 1.0 1.03 1. 01

1. 25 . 85 . 99 1. 0 0. 92 0. 80 3.2 . 82 1. 01 1. 0 .82 .64

10. 0 2. 02 1. 40 1. 0 .70 .59 12.5 0.5 0.90 0. 96 1.0 1.02 1.00 1.25 .83 .96 1.0 0. 93 0. 81 3.2 .64 .88 1.0 .90 .70 10.0 2. 03 1.40 1.0 .70 .59 50.0 0.5 0. 94 0. 98 1.0 1.01 1.00

1. 25 .84 .95 1.0 0. 95 0. 83 3.2 .62 .87 1.0 .99 .81 10.0 2.01 1.40 1.0 .70 .58

150. 0.5 0. 96 0. 98 1.0 1.01 1.00 1.25 .86 .96 1.0 0. 97 0. 87 3.2 .66 .88 1.0 1.03 .89 10.0 2.00 1.40 1.0 0. 70 .58

7.9. Rainfall Attenuation Climatology

The above paragraphs have been concerned with a descriptive presentation of the theoretical and technical background of the problem of power loss due to attenuation by rain and atmospheric gases. In an

attempt to circumvent the difficulties of the above methods of attenuation prediction it was considered important to try a climatological ap- proach to this problem. However, the results of such a study were

disappointing due to the fact that the problems of a systematic climatological estimation of rainfall attenuation are many and varied. Answers

are needed for such questions as; how often do various rainfall rates and

drop sizes occur in geographical areas and over how large an area do

these rate and drop size statistics apply? Furthermore, to what height

in the atmosphere do these data apply [1]? Unfortunately, the present state of meteorological knowledge concerning these problems in such that no conclusions can be drawn on a systematic and climatological basis. In view of these facts, it appears

prudent at this time to provide only engineering estimates of the combined gaseous and rain absorption. In this regard, Bussey [1] has shown that the absorption due to rainfall exceeds that of gaseous constituents about 5 percent of the time for frequencies around 6,000 Mc/s. The 5 percent figure was obtained by a study of the rainfall rate distribution for various locations in the United

States. Figure 7.21 shows the combined rain and gaseous absorption

to be exceeded 1 per cent of the time.
298 ATTENUATION OF RADIO WAVES

7.10. Rain Attenuation Effects on Radio

Systems Engineering

Attenuation due to rainfall is obviously a dominating factor in determining the reliability of a communications system, especially at frequencies in excess of 30 Gc/s. Rain varies greatly in frequency of occurrence from one region to another, so it is important to have an effective method

of predicting the performance of a radio system in any region in order

that the communications engineer will be able to gain the widest possible

application and degree of reliability, consistent with cost, in any system

of his design.

This section will be concerned, in the main, with the results of the

Bell System's [24] field experiment in the Mobile, Ala., area, which was

designed to establish a relationship between excess path attenuation and

instantaneous rate of rainfall and to seek out any relationship between

the profile of rate of rainfall along a radio path and rainfall measured

at a point.

The main problem concerns the ability to predict outage time due to

rainfall (time the system noise exceeds the system objective) at 11 Gc/s

in all areas of the country. This is obviously a difficult problem since,

due to reasons of cost, it is not feasible to measure rainfall attenuation

in all parts of the country. Therefore, it is desirable to be able to use what rainfall data are available and to couple the data, through what are thought to be reasonable assumptions, to the relationships between rainfall and attenuations.
RAINFALL ATTENUATION CLIMATOLOGY 299 In approaching the problem of predicting the outage time due to rainfall, it has been assumed that the annual distribution of one-hour point

rainfall rates is indicative of the instantaneous values over 30-mi radio

paths [1] and that the frequency of severe rainfall of the type measured

in the Mobile area will be reduced in other parts of the country in proportion to the distribution of annual point rates of 1 in. or more rainfall per hour.

Figure 7.22 indicates the expected outage time due to rainfall for various path lengths in different rain areas of the United States. The

curves A through H correspond to the areas contoured in figure 7.23

which illustrates contours of constant path length for fixed outage times

for different areas of the United States. The longer paths have been

somewhat weighted to take into account the less severe rainfall covering

larger geographical areas than the intense storms typical of the Gulf

region. In a complete 11-Gc/s Bell System point-to-point relay system, the

rain outages of the individual hops must be added to obtain the performance of the system. It is desirable that the individual hops meet the same objective, but this is not always possible. Sometimes one or more

6 7 8 9 10 15 20 25 30 35 40 45 50 60 PATH LENGTH IN MILES Figure 7.22. Expected outage time in hours per year versus path length in miles for various areas of the United States.

(After Hathaway and Evans, 1959). The curves A through H correspond to the areas contoured in figure 7.23.
300 ATTENUATION OF RADIO WAVES

Figure 7.23. Contours of constant path length for fixed outage time.

(After Hathaway and Evans, 1959).

hops of a system are electrically long; they will have insufficient fading

margin (the number of dB the receiver input level can be reduced before

the noise exceeds the system objective) and hence contribute more than

their share of the outage time. So, this excess must be made up by

imposing tighter requirements on the remaining hops. To meet the

overall objective of the Bell System, it is necessary to know the contributions of the long hops—those having a fading margin less than 40 dB.

Figure 7.24 shows the excess path loss due to rain, versus hours per year

for the Mobile area study. Since the shape of this curve is nearly

identical to Bussey's curve of cumulative distribution for point rates in Washington, D.C. (if we assume the shape of this curve to be representative for other areas of the country), then the additional outage time for path lengths given by figure 7.23 can be estimated for hops having a fading margin less than 40 dB. The data shown on figure 7.24 have been

rationalized and are shown in figure 7.25 as an estimate of additional

outage time.

Sometimes it is practical to shorten a proposed path to bring the fading

margin up to 40 dB. An approximation of the necessary reduction path

length can be made if uniform rainfall rate is assumed over the path.
RAINFALL ATTENUATION CLIMATOLOGY 301

45

0 1 I I I I

\

I I I I I I I I 6 7 8 9 10 20 30 40 50 60 70 80 100 200

HOURS THAT THE ORDINATE VALUE WAS EXCEEDED

Figure 7.24. Excess path loss due to rainfall versus hours per year at Mobile, Ala.

(After Hathaway and Evans, 1959).

151 \

I I I I I I I

\

I 20 30 40 50 60 70 80 100 200 300 400

Figure 7.25. Additional outage time expected for 11 Gc/s systems having a fading

margin less than 40 dB.
302 ATTENUATION OF RADIO WAVES

Under this condition the attenuation due to rainfall should be directly

proportional to the path length. Thus the path length in figure 7.24 can

be shortened to correct for insufficient fading margin.

Bell System results indicate that for their microwave relay links in the

extreme southeastern region of the United States rainfall will limit 11 Gc/s radio systems having a 40 dB fading margin to path lengths of approximately 10 to 15 mi, depending on the number of hops, if normal

reliability objectives are to be met. Path lengths of 20 to 30 mi should

be acceptable in the central area and paths as long as 35 mi should be

acceptable in the northwestern part of the country. However, in their

existing point-to-point radio relay systems, the paths average about 23

mi due to other considerations than those of propagation. It appears

that the 11 Gc/s systems will not be penalized unduly except in the

southeastern part of the United States. An illustration of the correlation between the rainfall and path loss on March 15, 1956, of the Mobile study is presented in figure 7.26 in support

of the above conclusions.

40

, CALCULATED

(RAINFALL DATA) ^ rv

( 1 / 1 MEASURED

' 1 K

' 1 ' 1 M/ iV"\ 1 r / » ¥ \/

1

1

1

1—7 —IT \> A /

^ lU

A

\ \ V\

f j

30

u- 20

1130 1145 1200 TIME OF DAY

1215 1230 1245

Figure 7.26. Correlation between rainfall and path loss, March 15, 1956.

(After Hathaway and Evans, 1959).

7.11. Attenuation by Hail

Ryde concluded that the attenuation caused by hail is one-hundredth

that caused by rain, that ice crystal clouds cause no sensible attenuations,

and that snow produces very small attenuation even at the excessive rate

of fall of 5 in. an hour. However, the scattering by spheres surrounded

by a concentric film of different dielectric constant does not give the same

effect that Ryde's results for dry particles would indicate [24, 27]. For

example, when one-tenth of the radius of an ice sphere of radius 0.2 cm

melts, the scattering of 10 cm radiation is approximately 90 percent of the value which would be scattered by an all-water drop.
ATTENUATION BY FOG 303 At wavelengths of 1 and 3 cm with y - 0.126 (7 = 2a/X; a = radius

of drop) Kerker, Langleben, and Gunn [27] found that particles attained

total-attenuation cross sections corresponding to all-melted particles when

less than 10 percent of the ice particles were melted. When the melted mass reached about 10 to 20 percent, the attenuation was about twice that

of a completely melted particle. These calculations show that the attenuation in the melting of ice immediately under the 0 °C [28] isotherm can

be substantially larger than in the snow region just above, and under some

circumstances, greater than in the rain below the melting level. Further

melting cannot lead to much further enhancement, apparently, and may

lead to a lessening of the reflectivity of the particle by bringing it to sphericity or by breaking up of the particle. This effect, combined with

the fact that hail has greater terminal velocities than rain, gives rise to the so-called ''bright band" near the 0° isotherm.

The characteristic feature of a fog is the reduction in visibility. Visibility is defined as the greatest distance in a given direction at which it is just possible to see and identify with the unaided eye (a) in the daytime, a prominent dark object against the sky at the horizon, and (b) at

night, a known, preferably unfocused moderately intense light source [29]. Although the visibility depends upon both drop size and number of drops and not entirely upon the liquid-water content, yet, in practice,

the visibility is an approximation of the liquid-water content, and therefore, may be used to estimate radio-wave attenuation [28]. On the basis

of Ryde's work, Saxton and Hopkins [30] give the figures in table 7.11

for the attenuation in a fog or clouds at 0 °C temperature. The attenuation varies with the temperature because the dielectric constant of water

varies with temperature; therefore, at 15 and 25 °C the figures in table

7.11 should be multiplied by 0.6 and 0.4 respectively. It is immediately

noted that cloud or fog attenuation is an order of magnitude greater at

3.2 cm than at 10 cm. Nearly another order of magnitude increase occurs between 3.2 cm and 1.25 cm.

7.12. Attenuation by Fog

Table 7.11. Attenuation caused by clouds or fog [30] Temperature =0 °C

Attenuation (dB/km)

Visibility X = l. 25 cm X=3. 2 cm X = 10cm m

30

90 300

1.25 0.25 0.0450.20

0.04

0. 007

0. 02

0. 004

0. 001
304 ATTENUATION OF RADIO WAVES

7.13. Thermal Noise Emitted by the Atmosphere

General laws of thermodynamics relate the absorption characteristics

of a medium to those of emission. Good absorbers of radiation are also good emitters, and vice versa. Thus, in the microwave region, the atmosphere is also a good emitter, as well as a strong absorber, of radiation.

We may, therefore, describe quantitatively both emission and absorption by the same parameter; namely, the absorption coefficient. The emission characteristics of any real body at a fixed frequency may

be compared to those of a blackbody at the same temperature. In the

microwave region, the noise intensity emitted by a blackbody is given by

the Rayleigh -Jeans law:

where \l/{v) = emitted blackbody flux density per unit frequency

V = frequency

T = absolute temperature, °K

c = the velocity of light, and

k = Boltzmann constant (1.38054 X IQ-^^ erg/°K)

The emission per unit length along an actual ray path may now be ex- pressed as

where 7(1') = attenuation per unit length. Remembering that the fraction

of energy absorbed in a path length, ds, is given by the optical depth,

dr = y{v)ds, we may obtain the differential equation for transmission of radiation through the atmosphere:

(7.8)

(7.9)

dl{v) = -I{v) + ^P{v) (7.10)

dr

where I{v) is the flux density per unit frequency.

The solution to this radiative transfer equation is

(7.11)
THERMAL XOISE EMITTED BY ATMOSPHERE 305 where the summation extends over all discrete noise sources which may

be present, I m{v) is the unattenuated flux density transmitted from the

mth discrete source located at position rm, s is the point of reception of energy, and the other symbols have their previous meaning. It should

be recognized that the above integrals extend over a ray path determined

by the refractive properties of the medium and cannot be evaluated

unless these refractive properties are known.

In analogy to the temperature dependence of the noise energy as by

the Rayleigh-Jeans law, we may, in the microwave region, relate the

intensity of radiation received from a particular direction, I{v), to an

equivalent temperature, Tn{v), by the following relation

/W=?^ (7.12)

or, from (7.8) TM = ^Tn,miy) exp

This equivalent temperature is called the thermal noise temperature.

It is apparent that the thermal noise temperature of the atmosphere,

as measured by an antenna, will depend explicitly upon the antenna angle

and the frequency, and implicitly upon the atmospheric conditions along

the ray path giving rise to attenuation and emission of energy. It seems

plausible, therefore, that one could exploit this dependence of thermal

noise on atmospheric conditions as a probe of atmospheric structure. Thermal noise is equally important in communications receiving, since

it represents the lowest possible noise level that can be attained by an antenna immersed in the atmosphere. This minimum noise level will, of course, vary, depending on atmospheric conditions, the frequency, and

the antenna orientation. For example, in the microwave region, the

antenna noise temperature at vertical orientation may be as low as 1 °K,

and in a horizontal position, where more of the lower layers are ''seen" the noise temperature may be of the same order as the actual ternpera- ture of the lower atmosphere; i.e., around 280 °K. Figure 7.27 shows

sky temperature as frequency for various antenna angles for mean atmospheric conditions at Bismarck, X. Dak., during February 1940-43.
306 ATTENUATION OF RADIO WAVES

1 r

0 14 18 22 26 30 34 38 42 46 FREQUENCY (Gc/S)

Figure 7.27. Thermal noise versus frequency for mean profile conditions

at Bismarck, N.Dak.
THERMAL NOISE EMITTED BY ATMOSPHERE 307 Table 7.12

Gas ymax

Percent by

volume

at ground

7 at ground

SO2

Mc/s

12, 258. 17 {dB/km

1.9X10-1 -

(0 to l)X10-6

(dB/km)

(0-1.9)X10-7

12, 854. 54 8. 7X10-1 (0-8. 7) X10-7

23, 433. 42 1.2X10-1 (0-1. 2)X10-7

24, 304. 96 2.3 (0-2.3)X10-«

25, 398. 22 2.1 (0-2. l)X10-6

29, 320. 36 3.3 (0-3.3)X10-6

44, 098. 62 5.2 (0-5.2)X10-6

52, 030. 60 9.5X10-1 (0-9. 5) X10-7

N2O

24, 274. 78 2.5

0.5X10-«

1. 25X10-6

22, 274. 60 2. 5 1. 25X10-6

25, 121. 55 2.5 1. 25X10-6

25, 123. 25 2. 5 1. 25X10-6 NO2 26, 289. 6 2.9 (0 to 2) X 10-8 (0 to 5. 8) X10-8

O3

10, 247. 3 9. 5X10-2 Summer

(0 to.07)X10-«

"Winter

(0 to .02)X10-«

(0 to 6. 3) X10-9 11,075.9 9.1X10-2 (0 to 6. 3) X10-9

42, 832. 7 4.3X10-1 (0 to 2. 8) X10-8

8

E^ 6

1 1 1 1 1 1

AUGUST,

FEBRUARY, ' —

\ FEBRUARY, H,., V_ AUGUST, H,., -

1

1

1 1 1 1 0 10 20 50 40 50 60

FREQUENCY IN Gc/S

Figure 7.28. Variations with frequency of the scale heights of the bi-exponential

absorption model at Verkhoyansk, U.S.S.R.
308 ATTENUATION OF RADIO WAVES

7.14. References

[1] Bussey, H. E. (Aug. 1950), Microwave attenuation statistics estimated from

rainfall and water vapor, Proc. IRE 38, No. 7, 781.

[2] Davidson, D., and A. Pote (Dec. 1955), Designing over-horizon communication

links. Electronics, 28, 126.

[3] Bean, B. R., and R. Abbott (May-Aug. 1957), Oxygen and water vapor absorption of radio waves in the atmosphere, Geofis. Pura Appl. 37, 127-144.

[4] Van Vleck, J. H. (Apr. 1947), Absorption of microwaves by oxygen, Phys. Rev.

71, 413-424.

[5] Van Vleck, J. H. (Apr. 1947), The absorption of microwaves by uncondensed

water vapor, Phys. Rev. 71, 425-433.

[6] Van Vleck, J. H. (1951), Theory of absorption by uncondensed gases, Book,

Propagation of Short Radio Waves, pp. 646-664 (McGraw-Hill Book Co., Inc., New York, N.Y.).

[7] Becker, G. B., and S. H. Autler (Sept. 1, 15, 1946), Water vapor absorption of electromagnetic radiation in the centimeter wavelength range, Phys. Rev. 70, Nos. 5 and 6, 300-307.

[8] Birnbaum, G., and A. A. Maryott (Sept. 15, 1955), Microwave absorption in compressed oxygen, Phys. Rev. 99, 1886.

[9] Artman, J. O., and J. P. Gordon (Dec. 1954), Absorption of microwaves by oxygen

in the milhmeter wavelength region, Phys. Rev. 96, No. 5, 1237-1245.

[10] Tinkham, M., and M. W. P. Strandberg (July 15, 1955), Line breadths in the microwave magnetic resonance spectrum of oxygen, Phys. Rev. 99, No. 2, 537-539.

[11] Hill, R. M., and W. Gordy (Mar. 1954), Zeeman effect and line breadth studies

of the microwave lines of oxygen, Phys. Rev. 93, 1019.

[12] Artman, J. O. (1953), Absorption of microwaves by oxygen in the millimeter wavelength region, Columbia Radiation Lab. Rept. (Columbia Univ. Press, New York, N.Y.).

[13] Straiton, A. W., and C. W. Tolbert (May 1960), Anomahes in the absorption of radio waves by atmospheric gases, Proc. IRE 48, No. 5, 898-903.

[14] Tolbert, C. W., and A. W. Straiton (Apr. 1957), Experimental measurement of the absorption of millimeter radio waves over extended ranges, IRE Trans. Ant. Prop. AP-5, No. 2, 239-241.

[15] Gunn, K. L. S., and T. W. R. East (Oct.-Dec. 1954), The microwave properties of precipitation particles. Quart. J. Roy. Meteorol. Soc. 80, 522-545.

[16] Bean, B. R., and B. A. Cahoon (Sept. 1957), A note on the climate variation of absolute humidity. Bull. Am. Meteorol. Soc. 28, No. 7, 395-398.

[17] Ratner, B. (1945), Upper air average values of temperature, pressure, and relative humidity over the United States and Alaska (U.S. Weather Bureau).

[18] Donaldson, Ralph J., Jr. (June 1955), The measurement of cloud liquid-water content by radar, J. Meteorol. 12, No. 3, 238-244.

[19] Weickmann, H. K., and H. J. aufm Kampe (June 1953), Physical properties of cumulus clouds, J. Meteorol. 10 204-221.

[20] Battan, L. J. (1959), Radar Meteorology, Book, p. 43 (Univ. of Chicago Press, Chicago, 111.). [21] Ryde, J. and D. Ryde (1945), Attenuation of centimeter waves by rain, hail, fog, and clouds (General Electric, Wembly, England).

[22] Laws, J. O., and D. A. Parsons (Apr. 1943), The relationship of raindrop size to

intensity, Trans. Am. Geophys. Union, 24th Annual Meeting, 452-460.

[23] Ryde, J. W. (1946), The attenuation and radar echoes produced at centimetre wave-lengths by various meteorological phenomena, Meteorological Factors in Radio Wave Propagation, pp. 169-188 (The Physical Society, London, England).
REFERENCES 309

[24] Hathaway, S. D., and H. W. Evans (Jan. 1959),, Radio attenuation at 11 kMc and some implications affecting relay systems engineering, Bell Syst. Tech. J. 38, No. 1. [25] Burrows, C. R., and S. S. Atwood (1949), Radio wave propagation, Consolidated Summary Technical Report of the Committee on Propagation, NDRC, p. 219 (Academic Press, Inc., Xew York, N.Y.).

[26] Mie, G. (1908), Beitrage zur Optik triiber Medien, speziell KoUoidaler Metallasungen, Ann. Physik, 25, 377.

[27] Kerker, M., M. P. Langleben, and R. L. S. Guim (Dec. 1951), Scattering of microwaves by a melting spherical ice particle, J. Meteorol. 8, 424.

[28] Best, A. C. (1957), Physics in Meteorology (Pittman and Sons, London, England).

[29] Glossary of Meteorology (1959), Am. Meteorol. Soc. 3, 613.

[30] Saxton, J. A., and H. G. Hopkins (Jan. -Feb. 1951), Some adverse influences of meteorological factors on marine navigational radar, Proc. IRE 98, Pt. Ill, 26.

[31] Zhevankin, S. A., and V. S. Troitskii (1959), Absorption of centimetre waves in the atmosphere, Radioteknika i Elektronika 4, No. 1, 21-27.

[32] Button, E. J., and B. R. Bean (June 1965), The biexponential nature of tropospheric gaseous absorption of radio waves. Radio Sci. J. Res. NBS 69D, No. 6, 885-892.

[33] Ghosh, S. N., and H. D. Edwards (Mar. 1956), Rotational frequencies and

absorption coefficients of atmospheric gases, Air Force Surveys in Geophysics

(Air Force Cambridge Research Center, ARDC, USAF).

Chapter 8. Applications of Tropospheric

Refraction and Refractive Index Models

8.1. Concerning the Bi-Exponential Nature of

the Tropospheric Radio Refractive Index

8.1.1. Introduction and Background

The recent explosive growth of space science and telecommunications

has spurred the development of new models of the tropospheric radio

refractive index to account for the systematic refraction of radio waves

and the calculation of theoretical radio field strengths at satellite heights.

The simple exponential model has been found to represent, to a first approximation, the average refractive index structure within the first few

kilometers above the ground for the United States [1],^ France [2], and

Japan [3]. All of the above investigations have reported varying degrees

of departure of the atmosphere from this model and Misme [4] has en- deavored to delineate the regions of the world where the exponential

model is most applicable, although subsequent analysis of several types

of data has shown this model to be more generally applicable than at first sight and not unreasonable for use even in arctic and tropic locations [5]. If one considers that is composed of a dry term,

D = (8.1)

and a wet term,

3.73 X lO'e W = ^ ^ (8.2)

then one may consider the height variation of each term separately. We

shall examine the possible advantages of a model of the form

N{z) = Do exp {- J^} + ^0 exp {-^} (8-3)

to describe the average decrease of with height, where Do and Wo are the values of the dry and wet components at the earth's surface and Hd

Figures in brackets indicate the literature references on p. 373. 311
312 REFRACTION AND REFRACTIVE INDEX MODELS

and Hy, are the scale heights of D and W, respectively. This particular

form has been found useful by Katz [6], in his derivation of the potential

refractive modulus, and by Zhevankin and Troitskii [7] in their treatment

of atmospheric absorption. It would be well for the reader to recall that scale height, as used in this study, is merely the height at which the

value of the atmospheric property has decreased to 1/e of its surface value. Typical values of Do, TTo, and iVo are listed for arctic, temperate, and

tropical locations in table 8.1. It is seen that the contribution of W to the total value of N is nearly negligible in the arctic but becomes greater as one passes from temperate to tropical climates. There is, of course, generally an inverse correlation between the magnitude of D and W since, at sea level, where P ^ 1,000 mbar, the low arctic temperature increases

the D term and, combined with low atmospheric water vapor capacity,

decreases the wet term. Conversely, the higher temperatures of the

temperate and tropical climates depress the D term and provide a greater

water vapor capacity with the result that W may have a sizable contribution to the total N.

Table 8.1. Typical average values of the dry and wet components of N

Station and climate -Do Wo No

Isachsen (78°50' N), arctic 332.0 0.8 332.8 Washington, D.C. (38°50' N), temperate Canton Island (2°46' S), tropic

266. 1 58.5 324.6

259.4 ni. 9 371.3

8.1.2. N Structure in the I.G.A.O. Atmosphere

One may examine structure in a standard atmosphere as a guide to

its general distribution in the free atmosphere. On this basis the I.C.A.O. standard atmosphere [8] (fig. 8. 1) was examined. The conditions specified for this atmosphere are an approximately exponential pressure decrease

with respect to height and a linear temperature decrease from ground level to the tropopause. It is evident, then, that in this atmosphere D decreases in an exponential fashion with height.

When these data are converted into refractive index and plotted on

semilogarithmic paper, as on figure 8.2. both D and W are seen to display an approximately exponential distribution from the surface to the tropopause. This conclusion is based upon the observation that the distribution is nearly linear as one would expect if one inverted the function

y = A exp ( — h/c) (8.4)
N STRUCTURE IN I.C.A.O. ATMOSPHERE 313

P(mbar) T(°K) ^^_,_,g_P

Figure 8.1. The U.S. extension to the I.C.A.O. Standard Atmosphere.

into

Iny = - h/c + In A, (8.5)

which is the equation of a straight Une on semilogarithmic paper. The

exponential distribution of W with height in this atmosphere follows

naturally from the definition of constant relative humidity, since the

saturation vapor pressure, Ss, is itself, to a first approximation, an ex- ponential function of temperature. It is evident that the value of W can

significantly affect the surface values of N but has no appreciable effect upon the value of at the tropopause.

An examination of long term means from observations in the actual

atmosphere shows that this same general bi-exponential trend is observed

in practice, for temperate climates at least. Examples are given on

figures 8.3 and 8.4 for Bismarck, N.Dak., and Brownsville, Tex. Bismarck is typical of the high, dry great plains region of North

America which is frequently subjected to strong intrusions of arctic air, while Brownsville represents the humid periphery of the Gulf of Mexico.

Even in these very dissimilar climates one finds a strong tendency towards

a bi-exponential distribution of A^, particularly when W is large.
314 REFRACTION AND REFRACTIVE INDEX MODELS

8.1.3. Properties of the Dry and Wet Term Scale Heights

The average values of D and W versus height were determined for 22 U.S. radiosonde stations located about the country. The data used were the published values of mean pressures, temperatures, and humidities

for the United States [9] which may be converted into mean values of the

refractive index with negligible error [10]. The method of least squares was then used to determine the scale heights of the wet term, Hw, and of the dry term, H^. Examination of these scale heights did not reveal any

simple method of predicting their geographic and seasonal behavior, other

than simply to map them. Such maps were prepared for the United States
DRY AND WET TERM SCALE HEIGHTS 315

for both winter (February) and summer (August) . The immediate conclusion that one reaches from these maps, figures 8.5 and 8.8, is that Hy,

has a year round, country-wide average value of perhaps 2.5 km while

Hd has an average value of about 9 km.

Since the scale height is the height at which the value of an atmospheric property has decreased to \/e of its surface value it reflects the

degree of stratification of the property. For example, cold arctic air is very stratified with very little vertical motion, with the result that its density scale height would be expected to be low. By contrast, tropical

maritime air that has moved over land is characteristically unstable with

convective activity thoroughly mixing the original moist surface air

N

1,000

700

500

300

200

100

70

50

30

20

107

5

3

2

1 ELEVATION =:505m _

- ^AUGUST

-\

\

^D(h)

\

\ ^AUGUST V

\

- FEBRUARY^\

1 1 \ \ 1

1 2 4 6 8 10 12 ALTITUDE ABOVE M.S.L. IN km

14

Figure 8.3. N distribution for Bismarck, N.Dak.

Note that altitude rather than geopotential height is used here to faciUtate the eventual calculation of radio- ray bending through actual atmospheric layers.
316 REFRACTION AND REFRACTIVE INDEX MODELS

1,000

700

500

300

200

100

70

50

N 30

20

107

5

3

2

"0 2 4 6 8 10 12 14 ALTITUDE ABOVE M.S. L. IN km

Figure 8.4. N distribution for Brownsville, Tex.

throughout the entire troposphere, with the result that the density scale height is relatively larger than in the case of arctic air. The dry term

scale height on figures 8.5 and 8.7 show a distinct tendency to be larger

during the warm summer months when the atmosphere is well mixed to great heights. Consequently D decreases slowly with height.

A slight geographic pattern is observed in the Ha maps: the coastal

regions display somewhat higher values than the inland regions. The

north-south direction of the isopleths along the west coast on the February

Hd map definitely reflects the uniform onshore advection of low-density

maritime air. By contrast, the east coast shows an east-west isopleth

pattern, the high values in Florida reflecting the well-mixed nature of subtropical air, and the lower values in New England indicating the presence

n \ \

\ r ELEVATION = 7m

- ^FEBRUARY
DRY AND WET TERM SCALE HEIGHTS 317

Figure 8.6. Wet term scale height, H^, in kilometers for February.
Figure 8.8. Wet term scale height, Hw, in kilometers for August.
DRY AND WET TERM SCALE HEIGHTS 319

of the more dense and stratified continental air that customarily flows

offshore during the winter months. The same pattern is repeated on the

summertime map along the west coast but is less pronounced on the east coast due to the combination of more uniform heating and also onshore

advection of maritime air produced by the circulation pattern of the

Bermuda high-pressure area. The high value oi Hd = 10.5 km observed

in the southwest during the summer appears to be due to the intense heating with resultant convective mixing to great heights so common in that

desert area. A somewhat opposite pattern is evidenced by the Hy, maps.

For example, the coastal areas generally have the lowest values and thus

reflect the characteristic strong humidity stratification of maritime air. The smaller humidity gradients of the inland regions produce somewhat

larger scale heights for that area. The summer Hy, map is quite sur- prising in that very little variation is shown, perhaps indicating uniform

vertical convection of the available moisture at all locations throughout

the country. The strong convection indicated in the southwest on the summer Hd map is again reflected by the high value of = 3.0 km for that same area.

It is quite evident from figure 8.9 that within the troposphere, h < 10

km, the bi-exponential model has a lower rms error for the common,

near-zero angles of departure used in tropospheric propagation of radio waves. Both models yield about a 12 percent error in determining

T for ^0 = 0 and h = 11 km. At = 100 mrad, however, the percentage

error decreases to 4 percent for the bi-exponential model and 7 percent for the exponential reference atmosphere. The rather marked errors of the

single exponential model at 10 km simply reflect that that model is deliberately fitted to the average structure over the first few kilometers

with the result that this model systematically departs from the average

atmospheres in the vicinity of the tropopause. This is particularly

apparent at the higher values of where the integral tends to become a

function of the limits of integration. That is, using the theorem of the mean for integrals.

under the assumption that cot 6 may be replaced by cot do over the interval

of integration. For < 20 mrad this assumption introduces less than

a 10 percent error for the interval 0 < /i < 10 km. It is apparent from

(8.9) that the error in predicting r for large do is then simply a matter of how closely the model approaches the true value of n in the atmosphere.

At ^0 near zero, however, the integral for r is very heavily weighted to- wards the effect of n gradients near the earth's surface [1]. Since the

values of Oq commonly used in tropospheric propagation are as near as

(8.9)
320 REFRACTION AND REFRACTIVE INDEX MODELS

practically possible to zero, and both models show comparable rms errors

for ^0 = 0, one concludes that there is no clear advantage to the bi- exponential model for this application. This conclusion is furthered by

the fact that charts of Hd, H^, Do, and Wo are not available, while to use the single exponential model one need only to refer to existing regional or worldwide maps [10, 12].

Figure 8.9. Root-mean-square errors of predicting bending for both the bi-exponential

refractive index model and the C.R.P.L. (single) exponential atmosphere.

Height is used here to indicate the actual thickness of atmosphere traversed by the radio ray
REFRACTION IN BI-EXPONENTIAL MODEL 321 8.1.4. Refraction in the Bi-Exponential Model

The test of a model of atmospheric refractive index is the degree to which it represents the average structure of the atmosphere. A

further critical test is the degree to which the refraction, or bending, of a

radio ray is represented by this atmosphere. The bending is given as the angular change of a radio ray as it passes from ni to 722 in a spherically

stratified atmosphere and is adequately represented by

rni Ti.2 = - / cot 6 dn (8.6)

where 6 is the local elevation angle and is determined at any point from

SnelFs law, nr cos 6 = noVo cos ^0, (8.7)

where r is the radial distance from the center of the earth and zero subscript denotes the initial conditions. It is customary to evaluate (8.6)

numerically [11] since the integral is intractable for all but the most

simple models of n versus r. The bending was obtained for the mean

profiles for one-half of the 22 U.S. weather stations mentioned earlier. The values of r predicted by the bi-exponential model for these same

stations were obtained by preparing U.S. maps of Hd and Hy, from the

other half of our data, selecting values oi Ha and for the test stations,

calculating the bending and obtaining rms differences between these

values and those obtained from the mean profiles. These rms differ- ences are shown on figure 8.9. Also, for comparison, the rms errors obtained from the CRPL exponential atmosphere are shown. This

atmosphere, based upon a single exponential curve passing from the

surface value, Ns, to the value at 1 km, Ni, is founded upon the expression = Ns - 7.32 exp {0.005577 A^.l (8.8)

which has been found to be applicable in the United States [1]. An obvious advantage of the bi-exponential model is that the scale heights do reflect the physical properties of the atmosphere in a much

clearer way than does the single exponential model.

8.1.5. Extension to Other Regions

The present study is based upon data from the continental United

States and one wonders if the same approach might be of utility in other

regions. As a brief check, the D and W term scale heights were determined for conditions typical of the long arctic night (Isachsen, Northwest
322 REFRACTION AND REFRACTIVE INDEX MODELS

Territory, Canada, for February) and for humid tropical areas (Canlon

Island, South Pacific, February) from 5-yr means of N versus height

and are listed in table 8.2. The extreme meteorological differences of these two locations are quite

evident. The Hd = 9.4 km and H^, = 2.0 km at Canton Island indicate a warm atmosphere with a strong humidity gradient, while the value

of Hd = 6.3 km at Isachsen indicates very stratified air with high surface

density and a strong density decrease with height. The value of Hy, =

6.5 km at Isachsen reflects a very low humidity gradient; in fact, at no

point in the troposphere does W exceed 3 units for this location and

season.

Table 8.2. and Hu, /or arctic and tropical locations

Station Hd

km km

Canton Island 9.4 2.0 Isachsen, N.W.T., Canada.. 6.3 6.5

The value of Hy, = 2.6 km reported for the characteristic altitude of water vapor for the middle belt of the U.S.S.R. [7] also appears to be in agreement with the conclusions of the present study.

8.2. Effect of Atmospheric Horizontal

Inhomogeneity Upon Ray Tracing

8.2.1. Introduction and Background

It is common in ray tracing studies to assume that the refractive index

of the atmosphere is spherically stratified with respect to the surface of the earth. Thus, the effect of refractive index changes in the horizontal

direction is normally not considered, although Wong [13] has considered

the effect of mathematically smooth horizontal changes in airborne propagation problems.

Neglecting the effect of horizontal gradients seems quite reasonable in the tropospheric because of the relatively slow horizontal change of refractive index in contrast to the rapid decrease with height. In fact,

examination of climatic data indicates that one must compare sea-level stations located 500 km from each other on the earth's surface in order to observe a difference in refractive index values which would be comparable

to that obtained by taking any one of these locations and comparing its surface value with the refractive index 1 km above the location. Although

the assumption of small horizontal changes of the refractive index appears
CANTERBURY 323

to be true in the average or climatic sense, there are many special cases, such as frontal zones and land-sea breeze effects, where one would expect

the refractive index to change abruptly within the 80-odd kilometers of horizontal distance traversed by a tangential ray passing through the

first kilometer in height.

It is these latter variations that are investigated in this section. Two

cases of marked horizontal change of refractive index conditions were

studied, one which occurred over the Canterbury Plain in New Zealand,

and the other at Cape Kennedy, FJa. Ah hough these particular sites were chosen for several reasons, such as land-to-sea paths and subtropical

location (where marked changes in refraction conditions are common), the

major consideration was that detailed aircraft and ground meteorological

observations were available for prolonged periods.

These detailed measurements allow a quantitative evaluation of the

error apt to be incurred by assuming that the refractive index is horizontally stratified. The procedure used was to determine the refractive

index structure vertically over the transmitter and assume that this same

structure vertically described the atmosphere everywhere. Rays were

then traced through this horizontally laminated atmosphere. These ray

paths were then compared with those obtained by the step-by-step ray

tracing through the detailed convolutions of refractive index structure in the two cases under study. In the sections that follow we will discuss the two cases chosen for study, the methods of calculation used to evaluate refraction effects, and

the degree of confidence to which standard prediction methods may be

used under conditions of horizontal inhomogeneity.

8.2.2. Canterbury

The Canterbury data were compiled by a radio-meteorological team

working from September 1946, through November 1947, on the South

Island of New Zealand under the leadership of R. S. Unwin [14]. This

report proved invaluable in this investigation, as it was very carefully

prepared, giving minute details of the experiment on a day-to-day basis. Anson aircraft and a trawler were used for meteorological measurements

over the sea, and three mobile sounding trucks for observations on land. The trucks and the trawler carried wired sonde equipment, whereby

elements for measuring temperature and humidity up to a height of from

150 m to 600 m (depending on wind conditions) were elevated by means

of balloons or kites. Standard meteorological instruments provided a continuous record of wind, surface pressure, temperature, and humidity

at stations at the coast and 14 km and 38 km inland. The headquarters

of the project were at Ashburton Aerodrome, and the observations ex- tended out to sea on a line perpendicular to the coastline of Canterbury
324 REFRACTION AND REFRACTIVE INDEX MODELS

Plain. Aircraft were equipped with a wet-bulb and dry-bulb psychrometer, mounted on the portside above the wing. Readings were taken

three or four times on each horizontal flight leg of 2 or 3 min duration.

Special lag and airspeed corrections were applied, resulting in accuracy

of zbO.l °C. It was found that, under the variety of conditions in which

observations were made, the aircraft flights were more or less parallel to the surface isobars; hence, the sea-level pressure as recorded at the beach

site was considered to hold over the whole track covered by the aircraft. The relationship used for calculating the pressure, P, in millibars at a height h in feet was

P{h) = Po - h/30

where Po is the surface pressure. This approximation (determined by

averaging the effect of the temperature and humidity distributions on

pressure in a column of air) resulted in a maximum error in the refrac- tivity of 0.5 percent at 900 m. Radiosonde ascents at Hokitika on the

west coast of South Island and Paraparaumuo and Auckland on North

Island were used to supplement the aircraft measurements, particularly

in the altitude levels above 1 km.

The observations, diagrams, and meteorological records were studied,

and a profile of unusually heterogeneous nature was chosen. The

synoptic situation for the morning of November 5, 1947, was selected, as

it revealed a surface-ducting gradient near the coast with an elevated layer

about 100 km off shore. A cross section of the area from Ashburton to a

point 200 km offshore was plotted with all available data, and isopleths of modified refractive index, M, were drawn to intervals of 2.5 units

M = N + (Keh)W, (8.10)

where Ke = (15.70)(10-8)/m, and is as defined in chapter 1, (1.20). A

simplified version of the lower portion of this cross section with the corres- ponding M curves is accompanied by a sketch of the general location of the experiment in figure 8.10. Some smoothing was necessary, particularly near the sea surface and in those areas where aircraft slant ascents and descents caused lag errors in altimeter readings and in temperature

and humidity elements. Isopleths over land were plotted above surface

rather than above sea level, with an additional adjustment in the scale ratios of height and distance in an attempt to simplify the reading of values from the diagram.

8.2.3. Cape Kennedy

The second area studied was the Cape Kennedy to Nassau path for the

period of April 24 to May 8, 1957. This material was supplied by the

Wave Propagation Branch of Naval Research Laboratories and the
CANTERBURY 325

METERS

900i

ISOPLETHS OF MODIFIED REFRACTIVE INDEX

400. 390 380 -^

370- —

360

-370 380- 380„„ - 355-^^°

40 20 0 20 —LAND SEA^

FEET

r!000 -2500

-2000

-1500 -1000 800

600

400 200

METERS

300i —LAND SEA—

40 60 80 100 120 140 RANGE FROM COAST (KILOMETERS)

20 0 10 20 40

a;

NOTE:

340 340 340 340 340 340

HORIZONTAL SCALE

VERTICAL SCALE CHANGE ABOVE 300 KM

160 200 "^^^ ^1000

-[-800 -MOO

340 340 340

(ADAPTED FROM CANTERBURY PROJECT)

-NEW ZEALAND- \\_ \ 1

\iasman"1

AUCKLAND-5x^^ j I j —

L6l««» south <^^«p> PACIFIC OCEAN

/

36° S T" (SeT^ ^PARAPARAUMUO 1 40° S

\ \ 1 HOKITIK/^l^^^' ASHBURTON / / / W^4k(CANTERBURY PLAIN) / / / ^'^^^^.[fSLOGICAL / X DENOTES RAOB SITES

1

1 FLIGHT AKhA ^ q^^r [jqq p^^j

44° S

158' E 166" E 174' E 178' W 170° W

Figure 8.10. Isopleths and curves of refractive index for November 5, 1947, Canterbury, with a map locating sources of meteorological data.

(Courtesy of the Canterbury project).
326 REFRACTION AND REFRACTIVE INDEX MODELS

University of Florida. The particular case chosen for study was the

meteorological profile of May 7, 1957 (2000 L.T.) due to its heterogeneous

nature, showing a well-defined elevated layer at about 1,500 m. Fourteen

refractometer soundings from aircraft measurements taken at various

locations along the 487-km path (fig. 8.11) and six refractive index

profiles (deduced from radiosonde ascents from Cape Kennedy, Grand

Bahama Island, and Eleuthera Island) were read in order to plot a cross section of the atmosphere which would represent as closely as possible

the actual refractive conditions at that time. Unfortunately, the data

near the surface (up to 300 m) were quite sparse compared to those

recorded in the Canterbury Project, and calibration and lag errors had

not been noted as carefully in this preliminary report; therefore, some

interpolation and considerable smoothing of refractive index values were

necessary when drawing isopleths.

8.2.4. Ray Bending

The classic expression for the angular change, r, or the bending of a ray passing from a point where the refractive index is ni to a second point

where the refractive index is 712 is given in chapter 3 as

J ni dn

r,^^ = - I —cote, (8.11)

where 6 is the local elevation angle. Equation (8.11) was evaluated by

use of

Ari.2 = " X 10-' cot 0, (8.12)

^1,2 where

e =

di + d 2 The value of 6 at each point was determined from Snell's law:

rtiVi cos di = n2r2 cos 62 = constant, (8.13)

where r is the radial distance from the center of the earth and is given

by a -\- h, where a represents the radius of the earth and h the altitude

of the point under consideration. For simplicity one may rewrite

(8.13) as

(1 + A^i X 10-6) (a + hi) cos di = {1 + N2X 10-6) (a + h2) cos 02. (8.14)
RAY BENDING 327

REFRACTOMETER FLIGHT PATH '=^^»-=^PE KENNEDY u.r.j // , /\ ELEUTHERA ISLAND ^^4^"-^^^ fiRANn RAHAMyf«^BAH^^ _ SAN SALVADOR^X^

FLORIDA \

n X DENOTES RAOB SITES

Figure 8.11. Isopleths of refractive index and map of refractometer flight path for May 7, 1957, Cape Kennedy to Nassau.
328 REFRACTION AND REFRACTIVE INDEX MODELS

Then, when d is small, one may expand (8.14) [as in chapter 3, (3.50) to

(3.58)] and obtain the convenient expression:

where all values of 6 are in milliradians.

After obtaining r by use of (8.12) and (8.15), one may determine the

distance, d, along the earth's surface that the ray has traveled from:

di,2 = a [ti.2 + (^2 - di)]. (8.16)

Thus by successive application of the above formulas, one may trace the progress of the radio wave as it traverses its curved path through

the atmosphere. Normally the use of these equations is quite straightforward. When considering horizontal changes in n, however, one must

satisfy these equations by iterative methods. In the present application,

since n had to be determined by graphical methods, it was felt to be

sufficient to assume a constant distance increment of 250 to 500 m, solve

for appropriate height increment from

Ah = Ad tan di a} 1 + 7 , (8.17)

graphically determine for the point di + Ad, hi + Ah, and then determine $2 and ri,2. This latter type of ray tracing was done for various rays of initial elevation angles between 261.8 mrad (15°) and 10 mrad (^^0.5°). The

calculations were not carried to smaller elevation angles, since this type

of ray tracing is not valid within surface ducts for initial elevation angles

below the angle of penetration [16].

8.2.5. Comparisons

Although both of the calculated ray paths consisted of an oversea

itinerary with coastal transmission sites, they are quite different in other

aspects. Canterbury Plain is located southeast of the 10,000-ft chain

of the Southern New Zealand Alps at a latitude of 44°S (the equatorward

edge of the westerly belt of winds in November). Cape Kennedy is located on a sea-level peninsula at 28°N (the poleward edge of the northeast trade circulation in May). While the Canterbury profile showed

superrefractive tendencies, the Cape Kennedy profile illustrated subrefraction at the surface counterbalanced by an elevated trade wind

inversion layer, indicating that the total bending values of Canterbury
COMPARISONS 329 would be higher than normal, while the Gape Kennedy example would

have values near or lower than normal.

These differences are illustrated by figures 8.12 and 8.13 where the

bending, r, in milliradians is plotted versus altitude in kilometers. The

effect of horizontal changes is most pronounced for rays with initial eleva- tion angles of 10 mrad. On these figures the term "vertical" ray is used

to designate the ray path through the horizontally homogeneous n struc- ture determined from the refractive index vertically over the station. The term "horizontal" ray designates the ray path through the complex

actual n structure. It is quite evident that a consistent difference in bending of about 1 mrad exists between the "vertical" and "horizontal" rays at Canterbury above 1 km for = 10 mrad. This would be expected

since the vertical M profile (fig. 8.10) at the beach (our hypothetical

transmitter site) is nearly normal in gradient while as little as 10 km to sea a duct exists, thus indicating a near maximum difference between the

"horizontal" and the "vertical" rays at any initial elevation angle small

enough to be affected by the duct. This is in contrast, however, with

the case of Cape Kennedy where, except for the region of the elevated

duct centered at about 1,500 m, the "vertical" and "horizontal" rays are

in quite close agreement. These two examples illustrate that horizontal

variations must be near the surface to be most effective. The importance

of the altitude of the variation is due to the fact that refraction effects are very heavily weighted toward the initial layers [16]. Also shown on figures 8.12 and 8.13 are the values of the bending which

would be predicted from the Central Radio Propagation Laboratory

corrected exponential reference atmosphere [1]. The values shown are obtained from the value of at the transmitter site as corrected by the

vertical gradient over the first 100 m. It is noted that, for = 10 mrad

at Canterbury, the value of bending predicted by the model is in essential

agreement with the "vertical ray" bending but under estimates the

"horizontal ray" (which has the largest variation of n with horizontal

distance) by about 1.25 mrad. For Cape Kennedy at = 10 mrad,

the model atmosphere overestimates the bending by about 1.25 mrad for altitudes in excess of 2 km. It should be emphasized that, although the

model exponential atmosphere appears to represent the average of the

two specific cases studied, the departure from this average arises from

quite different causes in each case. The differences in the Canterbury

case arise from the marked effect of horizontal variation of n as is indi- cated by the agreement of the vertical ray bending with the model atmosphere. The disagreement in the Cape Kennedy case is due to the

presence of a very shallow surface layer of nearly normal gradient topped

by a strong subrefractive layer; therefore, it represents a shortcoming of the model rather than an effect of horizontal changes of n.
330 REFRACTION AND REFRACTIVE INDEX MODELS

BENDING IN MILLIRADIANS

Figure 8.12. Canterbury, 0 to 10 km, altitude versus ray bending.

BENDING IN MILLIRADIANS

Figure 8.13. Cape Kennedy, 0 to 10 km, altitude versus ray bending.
EXTENSION TO OTHER REGIONS 331 The preceding analysis of bending throws the refractive differences in each case into sharp rehef. The effect of refraction, of course, is to vary

the ray path. Figures 8.14 and 8.15 show the ray paths corresponding

to the bendings of figures 8.12 and 8.13. Note that for Canterbury at

^0 = mrad, the effect of the horizontal variation of n is to produce a

difference in estimation of about 1 km in height or 20 km in ground dis- tance at 300 km from what would be obtained from considering the vertical n profile as a representation of the entire path. The effect of the

subrefractive layer at Cape Kennedy is not so large, but it does cause an

overestimation of the ground distance by about 5 km and an underestimation of the height by less than one quarter of a kilometer at a ground

distance of 300 km by assuming that the vertical profile may be used

throughout the entire ray path.

8.2.6. Extension to Other Regions

It should be pointed out that the ducting case at Canterbury represents an extreme refraction condition and is not necessarily typical of conditions

observed in other regions or, indeed, at Canterbury. The Canterbury

project was purposely restricted to a study of ducting conditions with the

result that less than 20 percent of the total observations for the fifteen
332 REFRACTION AND REFRACTIVE INDEX MODELS

months are reported. Therefore, because one of the more extreme cases

is represented by the November 5 example, one might conclude that much

less than 20 percent of the observations would show the same degree of horizontal n change as the profile studied.

If one further hypothesizes that the greatest horizontal n change would

be associated with ducting conditions, then the percentage incidence of ducts as evaluated from radiosonde observations, listed for various

stations in table 8.3 would indicate that the effects of horizontal changes

of n sufficient to cause variations in the ray path as large as those of the

present study would be observed less than 15 percent of the time, regardless of geographic location.

The probable importance of subrefractive layers upon the prediction

of refraction effects has emerged as a secondary result of the present study.

Although subrefraction is normally neglected, it is potentially a very important refractive factor for distances of, say, less than 40 km. Even

though the percentage occurrence of subrefractive layers can be as large as 6 percent (see table 8.4), this effect is frequently offset by the concurrent occurrence of an adjacent superrefractive layer, as is illustrated by the Cape Kennedy example.
CONCLUSIONS 333 Table 8.3. Percentage occurrence of surface ducts during the years 1952 to 1956

Percent incidence

station —

February May August November

Fairbanks, Alaska . 9.4 0.4 0.4 6.2 Columbia, Mo 0.7 2.5 8.4 1.3 Washington, D.C 0.7 4.8 4.3 1.4 Canton Island.. ... 10.0 9.2 12.4 11.5 Miami, Fla 0.7 3.5 8.5 2.7

Table 8.4. Percentage occurrence of surface suhrefractive layers during the years 1952 to 1956

Percent incidence

Station February May August November

Fairbanks, Alaska 0.0 0.0 1.2 0.4 Columbia, Mo 0.0 1.6 0.6 4.0 Washington, D.C 0.9 2.2 5.8 2.7 Canton Island 0.0 0.0 0.0 0.3 Miami, Fla 0.7 0.3 0.9 0.7

8.2.7. Conclusions

The conclusions of the present study could be considerably modified

by the analysis of many more examples, although it is evident that horizontal variation of n near the earth's surface produces the most marked

deviations from the ray paths obtained by assuming horizontal stratifica- tion of n. The effect of horizontal changes occurring more than a kilometer above the surface appear from our present examples, to have little effect. Further, the effects of horizontal changes appear to be most

pronounced in the presence of surface ducts and at small elevation angles.

The tentative conclusion is reached that the effect of horizontal n change

is normally small, since ducting will occur less than 15 percent of the time.

8.3. Comparison of Observed Atmospheric Radio Refraction Effects With Values Predicted Through

the Use of Surface Weather Observations

8.3.1. Introduction

The atmospheric radio refraction effects considered in this section are

of two general types: errors in measuring distance by means of timing

the transit of radio signals between two points, known as radio range

errors, and errors in estimating the elevation angle of a target by means

of measuring the angle of arrival of radio signals from the target, known
334 REFRACTION AND REFRACTIVE INDEX MODELS

as elevation angle errors. Many methods have been proposed to take

into account these refraction effects for the purpose of improving measure- ments by removing systematic bias. One of these involves the use of the surface value of the radio refractivity, Ns, Si quantity which can be

measured directly with a microwave refractometer, or calculated from

the ordinary meteorological variables of temperature, pressure, and

humidity, to predict values of either range error or elevation angle error;

this method has been shown theoretically to be useful, with the accuracy

increasing with increasing initial elevation angle [17, 18, 19]. It is the

purpose of the present note to compare recent experimental determinations of atmospheric refractive effects with values estimated theoretically

from surface meteorological conditions.

8.3.2, Theory

The operation of a radio tracking system depends on the measurement,

in some manner, of radio signals received from the target. The radio

signals are transmitted in the form of radio waves which travel from the

target to the tracking system. The form of these radio waves is distorted

by the presence of the earth's atmosphere. Since solutions of the wave

equation are extremely difficult to obtain for the case of general atmospheric propagation over a spherical earth, it is common practice to evaluate refraction effects by means of ray tracing, a process which is based on the use of Snell's law.

One of the two types of refraction errors considered in this appendix

is the elevation angle error, e, which is the difference between the apparent

direction to a target, as indicated by the angle of arrival of a normal to the radio wave front, and the true direction. This error is primarily a

function of the refraction, or bending, of the radio ray. For targets

beyond the atmosphere, the two quantities are asymptotically equal (with

increasing range). , The values of e and r at any point on the ray path

obey the following inequality:

r/2 ^ e ^ r.

Recalling that (chapter 3) the bending of a radio ray may be expressed

by an equation of the form

T = a + bNs (8.18)

where a and b would be functions of the initial elevation angle of the ray,

do, and the height (or range) along the ray path at which the bending is to be calculated. Such an assumption can be checked by examining the

behavior of values of r, ray traced for a number of observed height profiles of radio refractive index, plotted against the corresponding values of A^^.
THEORY 335 Such a plot is shown in figure 8.16 for a small initial elevation angle,

50 mrad (about 3°), and a ^^target" height beyond the atmosphere, 70 km.

The family of profiles used in ray tracing this sample of bending values

is referred to as the CRPL Standard Sample.^ It can be seen from inspection of figure 8.16 that the assumption of hnearity expressed in (8.18) is justified for this case. A similar conclusion can be reached from examination of data for other cases, including low target heights and elevation

angles down to zero degrees, although for these extremes the degree of correlation between e and Ns is not as marked as that shown in figure 8. 16. The other refraction variable treated in this section is the radio range

error, ARe, which is here defined as being that error incurred in measuring

the distance between two points by means of timing the transit of radio

signals between the points, and assuming that the velocity of propagation

is equal to that of free space. For the case of a radio ray, this error is composed of two parts : the difference between the curved length of the ray

path, called the geometric range, Rg, and the true slant range, Ro; and

the discrepancy caused by the lowered velocity of propagation in a refrac- tive medium. The geometric range is given by

and the apparent, or radio, range by

Thus the total radio range error, ARe = Re — Ro, is given by

or

ARe = 10"' J 0 CSC d dh esc 6 dh — Ro.

(8.19)

2Meaning explained in section 8.3.3.
336 REFRACTION AND REFRACTIVE INDEX MODELS

Table 8.5. Typical and extreme values of range errors for targets beyond the atmosphere

Typical N3==320 Extreme Ns==400 &Xi inu.ID p6rc6nt

do ^Rg ARe ARg ARn ARe ARg/ARe

meters

0 10 100 110 60 165 225 27 20 mrad 2.5 62.5 65 4.5 73 77.5 6 50 mrad 0.7 38.1 38.8 1.0 43 44 2.3

100 mrad 0. 14 22. 26 22.4 0.2 24.8 25 0.8

200 mrad 0. 02 11.9 11.9 0. 03 13.0 13.0 0. 23 500 mrad 0. 001 5. 01 5. 01 0. 002 5. 50 5. 50 0.04 The first term on the right-hand side of (8.19) is the 'Velocity" or

'^refractivity" error, ARn; the last two terms represent the geometric

range error, ARg, which is the difference in length between the straight

path, Ro, and the curved ray path, Rg. Table 8.5 gives some typical and

extreme values of range errors ray traced for observed N profiles.

7.0

I

1 ^ ^

1

^ ^ ^ ^

1

^ . ^ r

N3

Figure 8.16. Total refraction at 0o equals SO mrad, h equals 70 km, for the CRPL

standard sample.
THEORY 337 From table 8.5 it can be seen that the geometric range error, A/^^, does

not represent a significant portion of the total range error except at very

small initial elevation angles, between zero and about 3°. This being so, the behavior of the total range error will be primarily a function of the

first integral in (8.19) for elevation angles greater than about 3°. The

integral expression

^Re = 10~' /

' N cscd dh, Jo may be rewritten as

ARe ^ esc ^0 X 10" Ndh

1 - 2 sin' y-^) + cot ^0 sin {d - do)

or

fht ARe = CSC ^0 / N dh

Jo + E f^'

iv[cot^osin(^ - ^o) -2sin' (--^)]' dh, (8.20)

for sin ^ < 2 sin ^o, 0 < ^ < 7r/2. This expression is analogous to that

derived for the case of ray bending, (3.6) of chapter 3, and, similarly, the

integral series on the right-hand side of (8.20) contributed only 3 percent

or less to the value of ARe for larger than about 10°. From (8.20) one would thus suspect that the radio range error might be well estimated as

a linear function of the integral of N with respect to height. In treating

this integral, it is informative to note that any given N{h) profile may be

''broken up" into three primary components:

N{h) = N\Ns, h) + N"{h + hs) + m{h)

where A^' is that part of the profile which can best be expressed as a

function of A^^ and height, A^'' is a standard distribution of refractivity

with respect to altitude above mean sea level {h -\- hs) which is independent of Ns, especially above the tropopause, and bN represents a random

component of the profile which cannot in general be accounted for a priori.
338 REFRACTION AND REFRACTIVE INDEX MODELS

The component is generally effective over the first few kilometers,

while above 6 or 7 km altitude, the N" component forms the bulk of the

profile [1]. Thus the integral of the N profile with respect to height may

be written as:

fht fht fht+h, fh,

/ Ndh= N' (Ns, h)dh+ / N" (h + hs)dh + / 8N(h) dh, Jo Jo Jh^ Jo

or, fht

/ Ndh = Fi (Ns, ht) + F2 (hs, ht) + 8F (ht) Jo

where 8F is the random contribution to the integral. For any particular

ht then

Jo Ndh = F, (Ns) + F2 (hs) + 8F

0

or N dh = F2 (K = 0) + Fi (Ns) - Fz (hs) + 8F (8.21)

where

'hs N"(h + hs) dh, and F2(hs = 0) is a constant.

It was found empirically, from integrated N(h) profiles, that

fht N dh^a + biNs - h^hs in S.E. . (8.22)

The analogy between (8.21) and (8.22) is plain (the standard error of estimate of (8.22), SE, represents the standard deviation, 8F, of (8.21)).

The results of such an empirical study are shown in figure 8.17 for the

CRPL Standard N Profile Sample, for ht beyond the atmosphere.

For any particular application of (8.22) at a single location, the term

62/is will be absorbed into the constant a, since hs does not vary. However the introduction of this term is necessary to explain the station
THEORY 339

Figure 8.17. Integrated refractive index -profiles for the CRPL Standard Sample.

elevation dependence of integrated N'ih) profiles when taken from a sample containing stations at widely differing elevations, such as the

Standard Sample.

It is thus apparent that radio range errors, at least at the higher elevation angles, are primarily a linear function of Ns. That this is also true

at comparatively low angles is shown in figure 8.18, for = 50 mrad

(about 3°) for the same profile sample. The reader should especially note

the similarity of the distributions of the points about the regression lines between figures 8.17 and 8.18, showing that the range errors at about 3° are still primarily a function of the integral of N with respect to height, or the range error at 90°.

It has thus been demonstrated that, theoretically, it should be possible

to estimate both the angle of refraction of radio rays and errors in radio

range measurements from measurement of the refractive index at the

surface of the earth. This should be true for targets in or beyond the

atmosphere, at elevation angles down to, and possibly lower than, 3°. In addition, if the behavior of refractive index profiles is similar in differ- ent parts of the world, it should be possible to specify ' 'universal" values
of the coefficients in (8.18) and (8.22) and to predict these values in ad- vance by analysis of a large heterogeneous sample of refractive index

profiles. In the succeeding subsections it will be shown how this has

been done, and a comparison will be made between the results so derived

and the results of some measurements over actual radio paths.

8.3.3. The GRPL Standard Atmospheric Radio

Refractive Index Profile Sample

In the preceding section it was shown that, theoretically, it should be

possible to estimate either radar elevation angle errors or radio range

errors at any particular location by means of a system of linear equations

in Ns, where the coefficients are functions of the target position. The

target position can be specified by either the apparent elevation angle and

target height, or the apparent range and target height (or as a third possibility, the apparent range and elevation angle), each having advantages

in different situations [19]. The equations recommended are

e = ai (^0, ht) + hi (do, ht) Ns ± S.E.i ((9o, ht), (8.23)
REFRACTIVE INDEX PROFILE SAMPLE 341 and

^Re = a. (Re, ht) + 62 {Re, k t) Ns =b S.E.2 (Re, k t) , (8.24)

where e is the elevation angle error, ARe is the radio range error, is the

apparent elevation angle, ht is the target height. Re is the apparent radio

range, and S.E. is the standard error of estimate about the regression

line of € or ARe on Ns. Values of the coefficients may be obtained by

performing linear regressions of e or ARe, as ray traced for an appropriate

sample of radio refractive index profiles, upon Ns for a large number of target positions. As a byproduct of these calculations, one also obtains,

for each target position, a value of the residual error (the standard error of estimate) to be expected for the particular type of profile sample used. In order to obtain a general set of equations to be useful under arbitrary conditions of location, climate, and weather, a large sample of

profiles has been assembled which is believed to be representative of both mean climatic and geographic trends and the larger synoptic variations which may be encountered. This was done by choosing 13 radiosonde stations representative of the major geographic and climatic types

of the world, and then choosing from each station six profiles of particular types, two of which are typical of the extremes of monthly mean

conditions for that location, and the other four of which are typical of some of the variations which are found at that location [18]. The result

is a sample of 77 profiles,^ which has been found over a period of years

to be a sound cross section of general refractive conditions and has thus

been named the CRPL Standard Atmospheric Radio Refractive Index

Profile Sample, hereafter referred to as the CRPL Standard Sample.

Although the locations chosen for this sample are heavily weighted towards

the United States, it has been found that the general behavior of the

refractive index structure as inferred from the standard sample is typical

of conditions experienced in most parts of the world [5]. The remainder of this section will be devoted to some comparisons of observed radio refraction data with the predictions supplied by the

CRPL Standard Sample, as derived from the linear regressions mentioned above.

Since the refraction measurements reported here consist of samples

taken at particular locations over comparatively short periods of time,

they should provide a test for the general set of coefficients derived from

the Standard Sample; not only is the general theoretical approach tested

against measured values, but the measurements coming from places of more or less homogeneous nature, they provide a check as to whether oi not coefficients derived for a large heterogeneous sample of data are applicable also to individual places and times; i.e., they should reveal

^ One of the types could not be found for one of the stations used.
342 REFRACTION AND REFRACTIVE INDEX MODELS

how much of the observed correlation of the heterogeneous sample is derived from correlation between ^'classes" of data (in the statistical sense). For a more thorough treatment of the CRPL Standard Sample

and the associated regression coefficients for range error and elevation

angle error, the reader is referred to Bean and Thayer [19]. For the Standard Sample, the standard error of estimate is equal to the standard prediction error within ±1 percent over the range of Ns from

200 to 470, and will be used interchangeably with the latter.

8.3.4. Comparison With Independent Data

Before turning to an examination of the experimental refraction data

and the degree of success realized in applying the theoretical prediction

model to those data, it seems appropriate to examine the accuracy of the

prediction model when applied to some independent theoretical (i.e., raytraced) data. For this purpose, four check stations were selected which

were not only independent in the sense of not having been included in the original 13-station Standard Sample, but were from locations widely

differing from the region of selection of the original sample. It was decided to select one station representative of an arctic type climate, one

temperate, one tropical, and one from a ^'problem" climate area. Amundsen-Scott station at the South Pole (lat. 90°S) was chosen as the

arctic type; this station was expected to present the most rigorous test of the prediction model (as based on the Standard Sample) that could be

obtained anywhere in the world. In the first place the extreme arctic- continental climate, with almost no water-vapor contribution to the re- fractive index and the nearly incessant temperature inversion, is more

alien to the Standard Sample than any other type ; in the second place the

station elevation is 2800 m, which is 900 m in excess of the highest station

(Ely, Nev., 1908 m) included in the Standard Sample. These two effects were expected to augment each other as regards refraction. Dakar, Senegal, on the western coast of Africa, was selected as a "problem" climate station; an inverse relationship exists there between

Ns and AN (the A^-gradient over the first kilometer above the surface).

A Congo basis station, Bangui, in what was French Equatorial Africa, was selected as the tropical location, and Moscow, U.S.S.R., was selected as the temperate location. In order to combine brevity with comprehensiveness, ray tracings were done of the total refraction (bending at 70 km target height) at two

elevation angles, 20 mrad and 100 mrad, for six profiles from each location. The six profiles were selected as representing roughly the range of Ns in winter (February), summer (August), and spring-fall (May and

November), two profiles being selected from February, two from August,

and one from each of May and November. The 20-mrad elevation angle
COMPARISON WITH INDEPENDENT DATA 343

MOSCOW, U S S R LAT=55°49' N. LONG. = 37° 37' E. ELEV. = l86m

•STANDARD SAMPLE

-2,87». 0.0358 Ns OBSERVED FIT T=-4.l9.0 0390Ns

S.E = »0 IZmrod

STANDARD SAMPLE r=-OI40»000963Ns

(PREDICTION ERR0R=t0 0l mrod)'

1

\ 1 1 1 \ BANGUI, CENTRAL AFRICAN REPUBLIC

LAT. = 4° 23| N LONG = 18° 34' E. ELEV.=385 m

^STANDARD SAMPLE

T = -2 87*00358 Nj •STANDARD PREDICTION ERROR LIMITS (67% CONFIDENCE) OBSERVED FIT

= -6 I7*0.0457N

S E = i0 34mrai

STANDARD SAMPLE

r=-0 I40*000963N5

(PREDICTION £RR0R = «0.0l mrad)

290 300 310 320 330 340 350 320 330 340 350 360 370

Figure 8.19. Comparison of predicted refraction at h equals 70 km for do's of 20 mrad

and 100 mrad from regressions of the CRPL standard sample and ray-traced values from four independent locations.
344 REFRACTION AND REFRACTIVE INDEX MODELS

was selected as representing roughly the lower limit of elevation angles

for which the bending is expected to be strongly correlated with Ns (say

r > 0.9), while at 100 mrad (about 6°) the correlation is expected to be

extremely high (say r > 0.99) and the refraction should be reasonably

free of random profile effects. The results of the ray tracings and the comparison with predicted

values are shown in figure 8.19. As expected, the results from the South

Pole seem to depart significantly from the predicted values, at least for the 20-mrad elevation angle. At the 100-mrad elevation angle some of the calculated points lie more than one standard deviation from the pre- dicted line (the theoretical prediction error is too small to show on the

graph clearly) ; however, in all four cases the differences are less than 50

/xrad, a figure which as shall be seen may represent the limit of accuracy

obtainable from the atmosphere in actual practice. At angles over 100

mrad the errors would be smaller; in fact they should tend to decrease in inverse proportion to the square of the initial elevation angle, as indeed

they do between 20 and 100 mrad.

A conclusion which may be drawn from the above results is that any

regions where the prediction model based on the Standard Sample would

not be expected to provide the theoretical accuracy are probably regions

of climatic extremes, and at least for the case of angular errors the effects

will be negligible for elevation angles of a few degrees or more. As an

interesting aside it can be noted that apparently the Antarctic may be a

desirable area for tracking systems location, at least with respect to atmospheric refraction effects, since (most likely because of the lack of substantial water vapor and the relatively homogeneous conditions) the

prediction error for = 20 mrad in figure 8.19 is only about one-fifth as

large as for temperate climates, indicating a possibly more stable atmosphere (even 90 percent confidence limits for the SE in figure 8.19 yield a value less than half of the theoretical temperate value of ±0.286 mrad).

8.3.5. Comparison With Experimental Results

Before comparing the theoretical and experimental results, it is appropriate at this point to examine what one would expect to observe on the

basis of propagation theory. In the case of angular errors it is expected

that propagation through the real, turbulent atmosphere will produce

random variations in the shape of the incoming wavefront, so that meas- urements made with systems in which the receiving antenna is alined with

the incoming signal will have random variations introduced in addition to the ordinary refraction effects. Since these variations will probably not be a function of elevation angle to any great extent, this implies that the

residual variance in predicting the elevation angle errors will probably

always be greater than predicted from theoretical (static) considerations,

and that there will probably be some minimum value of this variance for
COMPARISON WITH EXPERIMENTAL RESULTS 345 very large elevation angles. Thus, in some cases, the residual errors will probably not decrease steadily with increasing elevation angle, but will tend to flatten out at some point and assume a more or less constant value

above that point. These effects will be complicated in comparing one set of data with another by such things as differences in the location or time

of day or season in which data are taken, and instrumental effects such as aperture averaging.

The case of range errors is more straightforward. The effects of turbulent atmospheric inhomogeneities are expected to average out over

regions of abnormally high or low density, when considering the transit

time of particular points on the wave front. Hence the effect on the

residual range errors is expected to be small, and the observed values are expected to compare rather well with the predicted (theoretical )values.

Turning first to the comparison of observed and predicted elevation

angle errors, figure 8.20 shows some data on the mean refraction of 1.85- cm radio waves received from the sun, a target at essentially infinite range

so that the elevation angle error is identical with the total angular bending

of the radio ray, r. The data shown in figure 8.20 were obtained by

tracking the sun with a precise radio sextant developed by the Collins

Radio Company, and were collected in August through December of 1959

at Cedar Rapids, Iowa [20]. These data represent essentially instantaneous measurements. The mean of all observations at each elevation

angle is plotted for elevation angles ranging from 2 to 65°, and the mean

value of Ns associated with each point is about 332; the curve for the mean bending of the CRPL Standard Sample corresponds to the mean

value of A/" s of 334.6 for that sample and hence the data should be comparable. The standard deviation 'Vings" refer to the standard deviation

of the individual "instantaneous" data, not to the standard error of estimate of the mean value. The close agreement observed for elevation

angles between 2 and 35° constitutes not only a confirmation of the usefulness of the Standard Sample, but also a verification of the accuracy

of ray-tracing theory in estimating radio wave refraction in the actual,

and thus heterogeneous, atmosphere. The standard deviation of the

Collins data (shown on the lower part of fig. 8.20) is generally lower than

for the standard sample, but this is to be expected in view of the larger

range of climatic variation contained in the CRPL Standard Profile Sample. The apparent discrepancies in the measurements made at elevation angles over 40° are apparently due to some slight inaccuracies in the

calibration procedure used on the radio sextant during the period of data

acquisition. In fact, the data shown in figure 8.20 are almost precisely

^ The data for the highest elevation angles in figure 8.20 were necessarily collected during the early part of the period when the sun was higher in the sky. In a private

communication, Anway states that the mean Ns applicable to the data at 60° to 65° was 358 rather than 332; this difference would account for about one-third of the

discrepancies noted, reducing the residual bias to a maximum of about 40 ursid.
346 REFRACTION AND REFRACTIVE INDEX MODELS

ELEVATION ANGLE, DEGREES

2° 4° 8° 14° 26° 40° 55° 65°

0.01 0.02 0.05 01 02 0.5 I 2 5 10 TAN 9o

Figure 8.20. Comparison of measured total atmospheric refraction of 1.85 cm radio waves at Cedar Rapids, Iowa, with values predicted from Ns.

what one would expect to observe if all of the measured values of refrac- tion were increased by a systematic calibration error of about 50 M^ad

over their correct values. The standard deviations in figure 8.20 tend

to flatten out at high elevation angles, an effect which is to be expected

theoretically as pointed out previously. At any rate, the largest differ- ence between the observed data and the predicted curve in figure 8.20

at elevation angles over 30°, is only about 50 jurad or 10 sec of arc (the

angular diameter of the planet Mars at its average distance from the

earth is 10 sec of arc, an angle not discernible to the naked eye). Although this discrepancy might be significant militarily, it is only about

0.5 percent of the diameter of the target sun and is probably near the

limit of accuracy of the equipment used.
COMPARISON WITH EXPERIMENTAL RESULTS 347 Figure 8.21 shows the results of the specific measurements reported by

Anway for the radio sextant for all cases at an elevation angle of 8° =b 0.09°; each point represents an ''instantaneous" reading. The solid hne

represents the linear regression of the measured refraction data on the

values of A^^; the dashed line shows the predicted linear relationship

derived from least squares fits to the CRPL Standard Sample ray traced

refraction data. The mean bias between the two lines is about 40 ^rad,

interestingly close to, and in the same direction as, the apparent cali- bration error noted in the mean refraction data at high elevation angles.

The standard error of estimate is considerably higher than predicted ; however, the rms uncertainty of =t0.052°, or d=0.91 mrad, in the apparent

elevation angle would be sufficient by itself to increase the standard error

of estimate to about zbO.017 mrad, which is 4 times larger than the pre- dicted value. It is not known how much of the total standard error of dz0.12 mrad is due to measurement errors as opposed to unforeseen

fluctuations in actual atmospheric refraction.

2.5

1.5

T = -0.216 + 0.00754 Ns (mrad)

r = 0.87, S.E. = 0.12 mrad 5 j

5 = MEASURED REFRACTION OF 1.85 cm RADIO WAVES

AT 8° ± 0.052° ELEVATION ANGLE , WITH UNCERTAINTY WINGS DUE TO INEXACT ELEVATION ANGLE ; [^1 . ^„ = 0.0163

L dop

^LINEAR REGRESSION OF C.R.P.L. STANDARD N- PROFILE SAMPLE •, PREDICTED VALUES'- T = -0.058 + 0.00698 Ns (mrad)

r = 0.999, S.E. = 0.004 mrad i

\ \ \ \ 290 300 310 320 330 340

Nc

350 360 370 380 390

Figure 8.21. 1.86 cm radio refraction at an elevation angle of 8 degrees, at Cedar Rapids

Iowa.

(After Anway, 1961).

i
348 REFRACTION AND REFRACTIVE INDEX MODELS

Figure 8.22. Elevation angle fluctuations from phase differences taken across a 24-ft

vertical baseline, at Cape Kennedy, Fla.

Figure 8.22 shows some results of measurements taken at Cape

Kennedy, Fla., on November 1-3, 1959, at a very low elevation angle,

about 0.7 mrad or 0.04°. These are

' ^instantaneous" measurements,

taken at half-hourly intervals, of the phase difference fluctuations between

the signals from a beacon as they arrived at the upper and lower terminals

of a vertical 24-ft baseline, thus being very closely equivalent to a meas- urement of the fluctuations in the angle of arrival of the wave front at the

centerpoint of the baseline (the altitude difference between this point and

the target beacon is referred to as the "mean" target height). Since only

the fluctuations and not the total phase differences were measured, only

the slope and scatter of the elevation angle errors as a function of the

observed Ns data can be compared with the predicted values from the

CRPL Standard Sample. The zero point on the graph is set by the pre- dicted mean value for the sample. The correlation coefficient is, as ex- pected, only 0.57. In this case the scatter of the observed data is well

inside the limits of the standard error of estimate of the regression for the

standard sample, even at this very small elevation angle where horizontal

changes in the profile can exert a large effect on elevation angle errors.
COMPARISON WITH EXPERIMENTAL RESULTS 349 Figure 8.23 shows the results of a comparison between predictions of elevation angle errors estimated from the CRPL Standard Sample and

some measurements made with a 6-cm radar at Tularosa Basin, N. Mex.

[21]. Each point represents the miean of five ''instantaneous" readings

made at 1-min intervals over a period of 4 min. The standard deviation

of each five-reading group averaged 0.16 mrad, and the maximum range

in any one group was 0.58 mrad. The radio energy was propagated over

a 45-mi path at a mean apparent elevation angle of 18 mrad; the target was a beacon located on a mountain peak 5610.5 ft higher than the desert

floor where the radar was located. The data in figure 8.18 show that even

for this rather extreme case, where the degree of correlation between Ns and € is expected to be only 0.4, agreement is obtained between:

(a) the predicted and observed mean refraction,

(b) the observed and predicted slopes of the e versus Ns relation, and

(c) the observed and predicted residual errors of predicting e from Ns alone.

-TARGET HEIGHT=l.7i km PREDICTED FROM

OF ESTIMATE

230 240 250 260 270 280 290 300 310 320 330 Ns

Figure 8.23. Measured refraction of C-band radar at Tidarosa Basin, N.Mex.
350 REFRACTION AND REFRACTIVE INDEX MODELS

340 350 360 370 380 Ng-PUUNENE AIRPORT U.S.W.B. Figure 8.24. Range error fluctuations observed on Maui path.

The small discrepancy between the intercepts (i.e., between the mean

refraction) of the observed and predicted e versus Ns lines may be perhaps

attributed to, for example, antenna lobe pattern distortion caused by

differential refraction, or defocusing [22]. The remaining data which are examined were of necessity taken in such a manner as to have a rather high degree of autocorrelation (trends).

Such data are not as suitable for confirming the accuracy of a regression

prediction process as are independent data. A discussion of this is in- cluded at the end of this section. Turning to examination of radio range errors, figure 8.24 shows the

results of some measurements of apparent radio range fluctuations over a 25-km path on the island of Maui, Hawaii, on November 9-11, 1956 [23]. These measurements were made at 1-hr intervals, and are essentially

''instantaneous" values. The target beacon was situated on the summit

of Mount Haleakala at an elevation of 10,025 ft, while the ''ground"

station was near Puunene Airport at an elevation of 104 ft, thus yielding a target height of 3.046 km, in a region of critical target heights for pre- diction of radio range errors in tropical climates [19]. The measured

range fluctuations (absolute errors not measured) are plotted against

values of Ns taken at about the same time (mostly 15 to 20 min later) by
COMPARISON WITH EXPERIMENTAL RESULTS 351 U.S. Weather Bureau personnel at the Puunene Airport weather station. The agreement between observation and prediction is fairly good, especially when one considers that only 32 of the 86 points lie outside of the

predicted standard error of estimate Umits, while chance would indicate

that 29 points would exceed these limits. Also, it should be kept in mind

that in this case, as for all except the Collins data, the target beacon is located on the surface of the earth, whereas the predictions from the

CRPL Standard Profile Sample are derived for targets in the free atmosphere; there is undoubtedly some bias introduced in this way.

As a part of a continuing investigation into the atmospheric limitations

imposed on electronic distance measuring equipment, some measure- ments have been made recently by the Tropospheric Physics Section,

NBS, of both range errors and range difference errors (across a phasedifferencing baseline) over a propagation path near Boulder, Colo.

Figures 8.25 and 8.26 are based on some of the preliminary results of these measurements [24]. Figure 8.25 shows the results of measurements

of the fluctuations in apparent range, made at half-hour intervals on May 9-11, 1961, over a 15.5-km path between a transmitting beacon

'

1 I.I 1 1 1^ 1

LINEAR REGRESSION ^ r=0.86 SLOPE = 0.865 ppm/N a/^ ^-predicted slope, Q879 ppm/N -

°8 r 0

° THE STANDARD ERROR OF ESTIMATE

OF EITHER LINEAR FIT IS 5.4ppm, THE PREDICTED VALUE = 5.8ppm. —

o /

o A> o

° ° S° "i^g o

° o° 0 ,3^^^°° <»o o

/y \ 1 1 1 1

1 1 1

o

tH 290

Figure 8.25. Range error fluctuations observed over the Boulder Creek-Green Mountain

path, Colo.

3-day run, half-hourly readings, range = 15.5 km, target height 688 m, approx elevation angle 26° May

9-12, 1961
352 REFRACTION AND REFRACTIVE INDEX MODELS

N,-AT TWO ANTENNAS Figure 8.26. Range difference fluctuations observed over a 460-m in-line baseline, Boulder Creek-Green Mountain path, Colo.

4.15-hr run, 460 meter in-line baseline, target height = 688 m apparent range 15.5 km. May 9-11, 1961

on Green Mountain at an elevation of 2242 m and a receiving antenna

located near Boulder Creek at an elevation of 1554 m, the true target

height thus being 688 m. The apparent range fluctuations, expressed in parts per million of the

15.5-km path length (with an arbitrary zero since the total range was not measured), are plotted as a function of the surface value of the refractive

index taken at a point quite close to the lower terminal. Quite good

agreement is seen between the simple linear regression of the observed

ARe values on Ns and the predicted linear relationship obtained from

the CRPL Standard Sample. Note that both lines have statistically

equal standard errors of estimate with respect to the observed data.

Figure 8.26 shows the results of the range difference measurements

made over a 460-m baseline essentially in line with the transmission path,

where the second antenna was farther from the target beacon than the

primary antenna. Here the range difference fluctuations (again with an

arbitrary zero) have been plotted as a function of the mean value of Ns measured at each end of the baseline. The zero point on the graph is set by the predicted mean of the sample. In this case there seems to be some

discrepancy between the regression of the data and the predicted slope;
DISCUSSION OF RESULTS 353 however, note that the standard errors of estimate for the two hnes are, to two significant figures, equal, indicating that the difference in the

slopes is probably statistically insignificant.

There are some data points in figure 8.26 having a rather large deviation from the regression lines. Statistical theory (using the ''Student"

^-distribution for 84 deg of freedom) shows that, if the data points are drawn from a normally distributed population, there should be only one

point having a deviation of more than ±9 mm from the observed regres- sion line. There are in fact five points in figure 8.26, four above and

one below the line. If these five points are thrown out, on the grounds

that they weight too heavily the extremes of the distribution of data points

(this is especially true when using least squares regression), and the regres- sion is then redone using the remaining 81 data points, the resulting value

of the slope is —0.385 mm/A^-unit wdth r = 0.77 compared to the pre- dicted slope of —0.381 mm/A^-unit, a rather close agreement.

8.3.6. Discussion of Results

As a summary of the results of the experimental versus theoretical

comparisons given in the preceding section, a statistical analysis has been run on the significance of the differences between the slopes of the observed and predicted regression lines. In order to make the tests more

stringent, it was assumed that the slopes derived from the Standard Sample should be taken to be the slopes of the population regression lines (^3), thus yielding an estimate of the significance of the departure of the observed slope from the assumed population value. A value of t was first calculated for each case using the relation [25]

SE ' ^^-^^^

where h is the observed slope, |8o the assumed population, or theoretical,

slope, X refers to the independent variable in each regression, Ns, SE is the standard error of estimate, and tj-2 is the value of t for j — 2 deg of freedom. From confidence limits for 13 at the 100(1— a) percent

level can be calculated from [25]

^ _ tj.2,aSE < ^ < ^ +

tj~2,.SE

26)

Vxixi - xY \/^{xi - xy.

The probability that the observed value h would have fallen outside of these limits by chance is a. Many statisticians consider a value of tj-2
354 REFRACTION AND REFRACTIVE INDEX MODELS

falling below the lOOa = 5 percent level to be not significant, between the

5 percent and 1 percent levels to be of questionable significance, and over

the 1 percent level to be significant [25]. An observed slope h falling

o I ^i-2.0.0lSE

Po >

would thus be taken to represent a significant departure from the value

/3o, and would thus imply the possibilities

(a) j8o does not represent jS, or

(b) h represents the regression of data from a population different than

that used in determining /3o, or

(c) both.

Before making the significance tests, however, the value of j, the number of independent observations going into the determination of 6, must

be known. In general, data of the type presented here are more or less highly autocorrelated, and hence not all independent. The data pre- sented here, with the possible exception of the Collins data and the

Tularosa Basin data for which the calculations could not be performed,

have autocorrelation coefficients Vk, for lag k {k = 1, 2, 3 units of time between successive measurements) that can be approximately described by

Tk ^ M*,

and for this type of data the effective number of pieces of independent

data, j, is given by [26]

(8.27)

For the data treated here weighted mean values of r' were calculated

from

1 + 4 + 9 + • • • (8.28)

where k was the largest lag for which the autocorrelation coefficient was

calculated, usually 4 or 5. No special justification is offered for the use

of (8.28) other than the obvious fact that is to be approximated by

the kth power of r', and hence a function of k would seem to be the most

logical weighting function to use; the use of k^ as a weighting function

seemed to give the best overall fit to the series of ta; encountered from

these data.
DISCUSSION OF RESULTS 355 Table 8.6 shows the resuhs of the significance tests on the slopes of the

various experimental and theoretical (predicted) regression lines. The

number of pieces of data is shown in the first column, the observed slope

b and theoretical slope in the second and third columns, the autocorrelation coefficient for lag of one time unit in the fourth column, and the

weighted mean r' as defined in (8.28) in the fifth column. In column 6

the effective number of independent pieces of data, j, is shown, while in column 7 the value of tj_2 is shown for the difference between b and ^. The next column shows the value of tj_2, 0.5, the value for the 50 percent

significance level for j — 2 degrees of freedom.

Only one of the t values turns out to be significant at the 50 percent

level, which means that there was a better than even chance that such

differences would have occurred by chance in the other cases. In the case

of the Collins data at ^0 = 8°, the value of t = 1.01 would not be significant at the 25 percent level; the value t^Q = 1.01 corresponds to o: = 0.34, or a 34 percent chance that the observed deviation |5 — /3| is of a random

nature, and thus not significant.

Table 8.6. Experimental versus theoretical slopes

e VS A^, n b Tl r' 3 ti-2 tiO. 50, J-2)

Is [6-/3] sig- nificant at the a =50

percent level?

Collins data 8° — 48 0. 00754 0. 00698 (48) 1.01 0. 68 Yes

Tularosa Basin. 161 0. 0103 0. 0102 (161) 0. 031 0. 676 No

Cape Canaveral.. 86 0. 01356 0. 00648 .870 .860 6.5 0. 708 0. 73 No

ABe VS N, Maui data 86 0.02833 0. 01610 .974 .950 2.2 3.56 7.6 No

Boulder Creek- Green Mt. ... 155 0. 865 0. 879 .944 .950 4.0 0. 32 0. 82 No

A(Ai?) VS A"'. Boulder Creek86 -0. 344 -0. 381 .957 .946 2.4 0. 60 2.0 No.

From the point of view of a statistician, the results of these tests are such that no significance can be attached to any of the apparent discrepancies between theory and observation, and given reason to believe that

the values of /3 are theoretically sound, one could say that the results are

significantly positive in nature. The significance of the differences between the predicted and observed slopes of the regression lines for ^0 =

20 mrad for the independent data check of subsection 8.3.4 were tested

using the same method as the preceding tests, except that the six observations in each case were assumed to be independent. The results are summarized in table 8.7 and confirm the general use of the Standard

Sample for ^0 > 20 mrad.
356 REFRACTION AND REFRACTIVE INDEX MODELS

From the experimental data which are available at the present time

it may be concluded that

(a) Radio range and elevation angle errors can be predicted from the

surface value of the radio refractive index, and the accuracy obtained will be generally commensurate with the estimates of residual errors made

from theoretical ray-tracing considerations.

(b) The functional dependence of either angular refraction or range

errors on the surface value of the refractive index as derived from the

CRPL standard N profile sample may be applied to arbitrary locations or ch mates without noticeable decrease in accuracy over that obtained

with a sample from the location under consideration,

(c) The effects of horizontal inhomogeneities of the refractive index,

which certainly must have been prevalent over the transmission paths for which experimental data have been presented, do not appear to introduce

any bias or additional residual variance into the values of observed re- fraction variables over those predicted from surface observations.

Table 8.7. Comparison of slopes for independent check

Predicted slope at 0o=2O mrad: 0. 358 mrad/Ar^s

Station Observed

slope Difference

6-/3o

t 100a Significance

Amundsen-Scott ... 0. 0520 +0. 0162 37.9 10.2 <0. 1 percent very high Dakar .0316 -. 0042 56.5 0. 72 52 percent none

Bangui- , ... . .0457 +. 0099 47.2 1.37 25 percent very low Moscow .0390 +.0032 44.3 1.18 31 percent very low V 7:{Ns-Ns)^

S. E.

8.4. Correction of Atmospheric Refraction

Errors in Radio Height Finding

8.4.1. Introduction

As a radio ray passes through the atmosphere, the length and direction

of its path varies with the radio refractive index. Uncorrected radar

output determines the position of a target by a straight line path at constant velocity. The difference between the straight path and the actual

path results in an error which becomes increasingly significant as the

distance to the target increases. The height error (the component of the

position error normal to the surface of the earth) constitutes over 95 per- cent of the total error. Until recently, the range of height finding equipment was sufficiently limited so that the refraction errors could be either

neglected, or approximated by the effective (four-thirds) earth's radius

correction [27].
RAY THEORY 357 Bauer, Mason, and Wilson [28] obtained an equation for accurately

estimating radar target heights in a specific exponential atmosphere.

Beckmann [29] presented a probability estimate of the height errors with- out meteorological measurements.

The purpose of the study is to investigate the correlation between

available meteorological parameters and height errors for targets of interest in terminal air traffic control and to develop height error correc- tion procedures using these parameters.

8.4.2. Refractive Index

The radio refractive index, n, of a propagation medium is the ratio of the free space velocity of light, c, to the velocity in the medium, v, (i.e., n = c/v). Since the propagation velocity of the atmosphere is only

slightly less than the free space velocity, it is often convenient to use the

scaled up difference between the refractive index and unity. This quantity is the refractivity.

The refractivity is obtained from (1.20). Normally, the equation for N is dominated by the first term so that the refractivity can be approximated by an exponential function of height as shown in section 3.8.

8.4.3. Ray Theory

If the refractive index is assumed to satisfy (for a spherically strati- fied atmosphere)

f > - P

(«-^«)

then, for frequencies greater than 100 kc/s, the path of a radio ray is determined by Snell's law for polar coordinates (3.1) of chapter 3 (see

fig. 8.27) and the bending angle, r, is determined from (3.2). The dis- tance, d, along the surface of the earth is obtained from (3.62).

The length of the path is called the geometric range and is obtained by

R = CSC ^ dr, (8.30)

and the apparent or radio range is found by

Re = n CSC d dr = R -{- NX 10"' esc 6 dr. (8.31)
REFRACTION AND REFRACTIVE INDEX MODELS

Figure 8.27. Geometry of radio ray refraction.
RAY THEORY 359

Figure 8.28. Effective earth's radius geometry.
360 REFRACTION AND REFRACTIVE INDEX MODELS

Because the difference between Re and the true slant range, Rq, is small

compared to the height error, the slant range and radio range are assumed

to be identical to the geometric range, R. The apparent height of the target, in figure 8.28, is obtained by solving

(ro + haY = + i?^ + 2ro R sin (8.32)

for ha. The following form is useful for numerical calculations:

K = ^(^ + 2rosinM _

ro + Vro^ R{R -\- ro sin ^0) The height error for a target at height h is found by

eh = ha - h (8.34)

which will always be positive if n decreases with height.

If the refractive index is known as a function of height the foregoing

procedure is useful for determining the height error when the true height

and the arrival angle of the ray are hypothesized. Unfortunately, it is not applicable for obtaining the height error from the apparent position

of the target.

8.4.4. Use of the Effective Earth's Radius

The inaccuracy of the ''four-thirds earth" correction stems mainly

from the assumption that all radio rays have the same constant curvature. The accuracy would be greatly enhanced if an ''average" effective radius

could be determined for each ray path.

The following expression, with the effective earth's radius denoted by

Ve in figure 8.28,

(re + hy = + _p 2re R sin do, (8.35) can be combined with (8.32) and (8.34) to obtain

I ha€h el (R^ — hl \l 1 1 \ . .

The difference between the curvature of the actual earth and the "average"

effective earth for the ray path represents the "average" curvature of the

ray. Thus, if the ray curvature can be determined as a function of the
USE OF EFFECTIVE EARTH'S RADIUS 361 target position and the refractive index structure, (8.36) will provide a simple formula for approximating the height error,

since for target heights {h < 70,000 ft) at the maximum range {R 150

mi) to be considered

I ^ - 7^ I < 20 ft, Te 2re

or less than one percent of the total height error.

If the curvature of a ray, 1/p, at any point on the path is called K, then,

from (8.29) and (3.20)

^_ no cos do dn /c qq\ ^ " ~ n^l + h/fo) dK ^^"^^^ From (8.32) ignoring terms of the order one obtains

eos^.„^(l + ^)(«l^) (8.39)

so that (8.38) becomes

ii + ^Y'^

n, {R' - hlY " V To/ dn_ (8.40) K ^ —

n2 R / ^ \ dh

The refractive index usually decreases with height so that the quantity

./2 no . _ . n^ 1 -\-h ~

To

varies only sHghtly with height, and the curvature at a point on the ray

path can be approximated by
362 REFRACTION AND REFRACTIVE INDEX MODELS

Therefore, (8.37) becomes

{r' - hiy /2

where g represents the ''average" gradient on the ray path (i.e., g = dn/dh

at an intermediate point on the path).

Since g depends upon the meteorological conditions along the path,

the basic problem is to determine g for a given target from the conditions

at and/or near the surface.

8.4.5. Meteorological Parameters

Measurement of the refractivity at the radar site will provide an esti- mate of the gradient if a model of the refractive index structure is assumed.

In the exponential model, for example,

n{h) = 1 + A^, exp (-ch) X 10"^

where Ns is the surface refractivity and c is a constant, then

^ = -cNs exp (-ch) X 10-'. an For a target at a height, ht, the average gradient along the ray path is

g = Y^ll - exp i-cht)] X 10"', (8.43)

fit but since ht is not known, g must be approximated as a function of the

apparent height.

Additional meteorological measurements at a sufficient height above

the surface to obtain values significantly different from the surface values can be used to determine the initial gradient,

where A^i/ is the refractivity at the height, H, of the above surface measurements. The initial gradient provides a boundary condition for esti- mating g asa. function of the apparent height. For convenience the initial gradient of refractivity. Go = go X 10^, with H in kilometers, was used

as a prediction parameter.
ESTIMATION OF AVERAGE GRADIENT 363 For the purposes of this study, the average (per kilometer) gradient of the first kilometer of the atmosphere is the only prediction parameter

used which will require upper air measurements. The average 1 km

gradient,

where A^i is refractivity at 1 km above the surface, was selected because

climatological summaries, as in chapter 4, can be used to estimate the

height error when meteorological measurements are unobtainable.

8.4.6. Calculation and Correlation of Height Errors

Bean, Cahoon, and Thayer [18] selected refractive index profiles, determined from radiosonde observations at 13 climatically distinct locations,

which represent a wide variety of mutually exclusive profile types. The

ray paths at arrival angles varying from 0° to near 90° were determined

for each profile by numerical evaluation of (8.29) through (8.33) using

methods similar to those described by Bean and Thayer [1]. The height

errors were calculated with (8.33) and (8.34) at selected height intervals

of 70,000 ft for each ray path. Newton's method of interpolation was

used to determine height errors for fixed ground distances to 150 mi. The

limits of height and distance were chosen to extend beyond the current needs in terminal air traffic control, but are sufficiently restricted to allow some of the previous assumptions.

The prediction parameters, Ns, Go, and AN, were obtained from each

of the refractive index profiles. Linear and multiple regression analysis were employed to obtain least squares estimates of the height error at each height and distance for each prediction parameter and for various

combinations of the parameters.

8.4.7. Estimation of the Average Gradient

Based on the correlations the following forms, suggested by (8.43), were

selected for approximating g:

AN = Ni- N (8.45)

(8.46)

N^

(8.47) or

(8.48)

depending upon the availability of Go and AA^.
364 REFRACTION AND REFRACTIVE INDEX MODELS

To obtain a direct estimate of the height error, (8.46) through (8.48) were combined with (8.42) and the functions /^y {i > j = I, 2, 3) were

determined as least squares polynomials.

8.4.8. Regression Analysis

The volume of data processed is of sufficient magnitude that it is impractical to include it all in this report. Therefore, certain information

obtained from the regression analysis was selected as being the most

significant.

The mean height error provides the best general estimate obtainable

if meteorological data are not available. The standard deviation (about

the mean) of the height errors determines the reliability of this estimate,

since 68 percent of the observed height errors are within ±1 standard

deviation of the mean height error if the observations are normally dis- tributed. In figure 8.29, the mean height error was plotted for each

target position, then contour lines were drawn to display the mean height

error as a function of true height and distance. By similar construction

the standard deviation of the height error as a function of target position

is displayed in figure 8.30. The standard error of estimate establishes the same confidence limits

for prediction with a regression equation as the standard deviation does

for the mean. Thus, comparison of the standard error to the standard

deviation indicates the improvement in accuracy of prediction with

0 10 20 30 40 50 60 JO 80 90 100 NO 120 1 30 140 150 160 DISTANCE (MILES) Figure 8.29. Mean height errors in feet
REGRESSION ANALYSIS 365

0 1 0 2 0 30 4 0 5 0 6 0 70 8 0 9 0 iOO 110 120 130 140 150 160

Figure 8.30. Standard deviation of height errors.

meteorological parameters. The standard error as a function of target

position for

eh = hiNs + a (8.49)

is shown in figure 8.31; for

eh = hiNs + h2Go + a (8.50)

in figure 8.32; and for

eh = biNs + + hzAN + a (8.51)

in figure 8.33. The regression equations, (8.49) through (8.51), were used because the

respective figures demonstrate how each additional parameter enhances

the accuracy of the estimate. The parameters Go and AA^ applied indi- vidually, that is,

eh = bGo + a and

eh = hAN + a, were of significant value only for targets at low heights {h < 10,000 ft). Examination of the figures shows that prediction oi eh with Ns provides

significant improvement over the mean for target heights above 15,000 ft. The addition of Go improves the estimate for heights below 15,000 ft and the addition of AA^ provides a slight overall improvement.
366 REFRACTION AND REFRACTIVE INDEX MODELS

In figures 8.29 through 8.33 the contours do not extend below 15,000 ft for distances greater than 120 mi and 10,000 ft for distances greater than

80 mi. Correlations were not calculated for these target positions, be- cause for certain refractive index profiles they are beyond the radio

horizon and for certain other profiles the arrival angle is too low for the

ray to penetrate a trapping layer. If conditions exist, without violating

the assumptions of sections 8.4.2. through 8.4.4. such that a target at 5000 ft height and 150 mi distance would be visible to radar, the resulting

height error would be about 10,000 ft.

0 10 20 30 40 50 60 70 80 90 100 liO 120 130 140 150 160

Figure 8.32. Standard error of et versus Ns and Go.
REGRESSION ANALYSIS 367

DISTANCE (MILES) Figure 8.33. Standard error of €h versus Ng, Go, and AN.

As an aid to further studies the coefficients for (8.49) through (8.51) are listed in tables 8.9 to 8.17. Table 8.8. The coefficients snj for {8.52) through (8.67)

3 0 1 2 3 4 5 6 7

1 -19. 596 . 014096 0. 77906X10-^ 0. 67545X10-6

2 -17. 849 . 011202 . 13665X10-3 . 58925X10-2 -0. 64975X10-2 0. 12340X10-3

3 -15.319 . 006388 . 18549X10-3 . 39074X10-7 -. 55818X10-2 . 12671X10-3 -0. 023980 -0. 22547X10-4

Table 8.9. Constant term, a, in the regression equation {8.52)

Height, kft mi

5 10 15 20 25 30 35 40 50 60 70

5 -3 -3 -2 -1 1 1 2 2 2

10 -14 -13 -10 -8 -6 -4 -3 -2 1 1 15 -31 -29 -24 -19 -15 -12 -9 -4 -2 -1

20 -55 -53 -44 -35 -28 -23 -18 -14 -9 -5 -3

25 -86 -83 -69 -56 -45 -37 -29 -24 -15 -10 -7

30 -123 -119 -100 -81 -66 -54 -44 -35 -23 -15 -11

35.- -166 -162 -136 -111 -91 -74 -60 -49 -32 -22 -16

40 -215 -211 -178 -146 -120 -98 -80 -65 -43 -30 -22

45 -269 -267 -226 -186 -153 -125 -102 -83 -55 -39 -29

50 -327 -329 -280 -231 -190 -156 -127 -104 -70 -49 -36

60 -454 -470 -404 -335 -277 -228 -187 -153 -103 -73 -54

70 -586 -633 -552 -461 -381 -315 -259 -212 -144 -102 -76

80 -709 -812 -722 -607 -505 -418 -345 -284 -193 -137 -103

90 -1000 -913 -776 -648 -539 -446 -367 -252 -179 -134

100 -1186 -1122 -965 -812 -678 -563 -465 -320 -228 -171

110 -1352 -1342 -1176 -997 -836 -697 -578 -399 -286 -214

120 -1473 -1565 -1404 -1203 -1014 -850 -707 -491 -352 -264

130 -1774 -1644 -1429 -1215 -1022 -852 -597 -429 -322

140 -1946 -1889 -1672 -1434 -1214 -1019 -718 -618 -389

150 -2048 -2123 -1928 -1674 -1428 -1203 -855 -620 -465
368 REFRACTION AND REFRACTIVE INDEX MODELS

Table 8.10. Constant term, a, in the regression equation {8.53)

Height, kft

DistillCG mi

5 10 15 20 25 30 35 40 50 60 70

5 _3 _3 _2 \ 1 2 2 2 10 — 12 — 12 — 10 -6 -4 -3 2 \ 15 -28 -29 -24 -19 -15 -12 -9 -7 -4 -2 -1

20 -50 -51 -43 -35 -28 -22 -18 -14 -9 -5 -3

25 77 —80 —67 55 -45 -36 -29 23 25 7 30 — 110 — 115 —97 —80 -65 -53 -43 —35 —23 25 \\ 35 — 148 — 156 — 133 — 110 —90 —73 —60 43 —32 22 — 16 40 -190 -204 -174 -144 -118 -97 -79 -64 -43 -30 -22

45 —235 —257 -221 —183 -151 -124 -101 -82 —55 —38 —28

50 -284 -316 -273 -227 -187 -154 -126 -103 -69 -48 -36

60 -383 -449 -394 -329 -273 -225 -185 -151 -102 -72 -54

70 -473 -600 -536 -452 -376 -311 -257 -210 -143 -101 -76

80 -537 -762 -700 -595 -497 -413 -341 -281 -191 -136 -102

90 -925 -880 -797 -637 -531 -441 -363 -249 -178 -133

100 -1076 -1075 -940 -797 -667 -556 -460 -317 -226 -170

110 -1189 -1275 -1141 -976 -822 -687 -571 -395 -283 -212

120 -1236 -1468 -1355 -1174 -996 -837 -697 -486 -349 -262

130 -1637 -1576 -1390 -1189 -1005 -840 -590 -425 -319

140 -1748 -1794 -1619 -1400 -1192 -1003 -708 -512 -385

150 -1765 -1989 -1855 -1630 -1399 -1183 -843 -612 -460

Table 8.11. Constant terrn, a, in the regression equation (8.54)

Height, kft mi

5 10 15 20 25 30 35 40 50 60 70

5 -2 -1 -1 1 1 2 2 2 10 -2 -8 -8 -6 -5 -3 -2 -2 1 15 -4 -18 -18 -15 -12 -10 -8 -6 -3 -2 -1

20 -7 -33 -33 -28 -23 -19 -15 -12 -7 -5 -3

25 -9 -51 -51 -44 -37 -31 -25 -20 —13 -9 -6

30 -12 -73 -74 -64 -54 -45 -37 -30 -20 -13 -10

35 -13 -98 -100 -87 -74 -62 -51 -42 -28 -19 -14

40 -13 -125 -130 -114 -97 -81 -67 -55 -37 -26 -19

45 -10 -156 -164 -144 -123 -104 -86 -71 -48 -33 -25

50 -5 -187 -201 -178 -152 -129 -107 -88 -60 -42 -31

60 21 -254 -285 -256 -221 -187 -156 -129 -88 -62 -47

70 70 -318 -378 -346 -302 -257 -216 -179 -123 -87 -66

80 152 -369 -478 -448 -395 -338 -285 -237 -164 -117 -88

90 -396 -578 -560 -499 -431 -366 -306 -213 -152 -115

100 -383 -669 -677 -615 -536 -457 -384 -269 -194 -146

110 -310 -741 -795 -739 -652 -560 -473 -334 -241 -182

120 -163 -776 -906 -867 -776 -673 -573 -408 -296 -224

130 -754 -998 -996 -909 -797 -682 -491 -359 -272

140 —649 -1055 -1117 -1044 -929 -804 -485 -430 -327

150 -436 -1057 -1217 -1179 -1068 -933 -690 -510 -389
REGRESSION ANALYSIS 369

Table 8.12. Coefficient of Ns, bi, in the regression equation {8.52)

Height, kft mi

5 10 15 20 25 30 35 40 50 60 70

5 - 0. 031S 0 0313 0 0296 0 0281 0. 0267 0. 0257 0. 0248 0 0243 0. 0236 0. 0231 0. 0225

10 . 1153 1059 0944 0848 .0770 .0705 .0652 0608 .0543 .0498 .0462

15 . 2545 2303 2025 1795 . 1608 . 1454 . 1326 1218 . 1056 .0943 .0858

20 .4495 4046 3541 3123 .2784 .2504 .2271 2074 . 1776 .1568 .1413

25 . 7005 6291 5494 4835 .4300 .3858 .3489 3177 .2704 .2373 .2127 30 1.0078 9040 7889 6934 . 6159 . 5519 .4983 4531 .3841 .3359 .3003

35 1. 3714 1 2295 1 0728 9424 .8366 .7489 .6756 6136 .5190 .4529 .4042

40 1. 7917 1 6060 1 4018 1 2311 1. 0923 .9774 .8812 7998 .6754 .5885 .5244

45 2. 2688 2 0336 1 7762 1 5599 1. 3837 1. 2377 1. 1154 1 0119 .8536 .7429 .6613

50 2. 8030 2 5129 2 1968 1 9295 1. 7113 1. 5305 1. 3788 1 2503 1. 0539 .9163 .8150 60 4. 0426 3 6268 3 1787 2 7938 2. 4780 2. 2155 1. 9952 1 8085 1. 5223 1. 3216 1. 1738 70 5. 5119 4 9494 4 3513 3 8300 3. 3981 3. 0380 2. 7355 2 4788 2. 0845 1. 8072 1. 6032 80 7. 2127 6 4794 5 7236 5 0453 4. 4788 4. 0046 3. 6053 3 2666 2. 7442 2. 3769 2. 1060 90 8 2133 2966 6 4464 5. 7273 5. 1225 4.6119 4 1779 3. 5078 3. 0342 2. 6854

100 10 1439 9 0736 8 0412 7. 1529 6. 4005 5. 7634 5 2207 4. 3804 3. 7846 3. 3456

110 12 2529 11. 0533 9 8363 8. 7649 7. 8482 7. 0684 6. 4028 5. 3687 4. 6329 4. 0909

120 14. 5240 13 2284 11 8360 10. 5679 9. 4723 8. 5370 7337 6. 4808 5. 5863 4. 9257

130 15. 5836 14 0399 12. 5772 11. 2895 10. 1777 9^ 2192 7. 7254 6. 6512 5. 8573

140 18. 0866 16 4440 14. 7931 13. 3002 12. 0051 10 8816 9. 1125 7. 8354 6. 8904

150 20 7053 19 0333 17. 2171 15. 5220 14. 0251 12. 7148 10. 6504 9. 1488 8. 0339

Table 8.13. Coefficient of Ng, bi, in the regression equation (8.53)

Height, kft mi

5 10 15 20 25 30 35 40 50 60 70

5 ... 0. 0285 0. 0303 0 0292 0. 0279 0. 0267 0. 0257 0. 0249 0. 0244 0. 0237 0. 0232 0. 0227

10 . 1014 .1015 0922 .0835 .0762 .0701 .0649 .0607 .0543 .0498 .0463

15 .2226 . 2202 1972 .1763 .1588 . 1441 .1317 . 1212 . 1053 .0942 .0857

20 .3915 .3862 3444 .3065 .2746 .2479 .2253 .2061 .1768 . 1563 . 1410 25 .6072 .5997 5430 .4742 .4239 .3817 .3459 .3155 .2690 .2365 .2122

30- .8686 .8603 7661 .6797 .6069 .5457 .4939 .4497 .3821 .3346 .2994

35 1. 1741 1. 1680 1 0411 .9234 .8240 .7403 .6694 .6089 .5162 .4511 .4028

40 1. 5217 1. 5225 1 3591 1. 2056 1. 0755 .9658 .8729 .7935 .6716 .5860 .5226

45 1.9089 1. 9235 1 7204 1. 5268 1.3619 1. 2227 1. 1046 1. 0037 .8486 .7396 .6589

50 2. 3325 2. 3701 2 1254 1. 8873 1. 6836 1. 5114 1. 3651 1. 2400 1.0475 .9121 .8119 60 3. 2717 3. 3976 3 0671 2. 7287 2. 4355 2. 1864 1. 9743 1. 7928 1. 5126 1. 3152 1. 1691 70 4. 2984 4. 5945 4 1841 3. 7339 3. 3361 2. 9957 2. 7052 2. 4560 2. 0705 1. 7981 1. 5965 80 5. 3577 5. 9423 5 4790 4. 9073 4. 3907 3. 9449 3. 5629 3. 2348 2. 7249 2. 3643 2. 0968

90 7. 4110 6 9435 6. 2511 5. 6047 5. 0400 4. 5537 4. 1346 3. 4816 3. 0171 2. 6730

lOO 8. 9558 8 5676 7. 7696 6. 9854 6. 2887 5. 6849 5. 1625 4. 3456 3. 7620 3. 3294

110 10. 5049 10 3305 9. 4588 8. 5356 7. 6975 6. 9635 6. 3258 5. 3231 4. 6035 4. 0699

120 11. 9716 12 1954 11.3115 10. 2579 9. 2713 8. 3974 7. 6323 6. 4215 5. 5484 4. 8989

130 14 1047 13. 3104 12. 1560 11. 0200 9. 9945 9. 0868 7. 6489 6. 6027 5. 8234

140 15 9660 15. 4244 14. 2189 12. 9405 11. 7614 10. 7087 9. 0139 7. 7737 6. 8477

150 17 6672 17. 6007 16. 4316 15. 0401 13. 7056 12. 4899 10. 5245 9. 0705 7. 9804

I
370 REFRACTION AND REFRACTIVE INDEX MODELS

Table 8.14. Coefficient of Ns, bi, in the regression equation (8.54)

Height, kft mi

5 10 15 20 25 30 35 40 50 60 70

5 0. 0091 0. 0231 0. 0259 —0. 0265 0. 0265 0. 0262 0. 0260 0. 0258 0. 0255 0. 0252 0. 0248

10 . 0217 . 0689 . 0744 . 0720 . 0687 . 0652 . 0619 . 0589 . 0540 . 0504 . 0474

15. . 0418 . 1447 . 1551 . 1478 . 1391 . 1303 . 1218 . 1141 . 1016 . 0923 . 0850

20 . . . 0682 . 2497 . 2677 . 2539 . 2378 . 2214 . 2058 . 1915 . 1683 . 1511 . 1378

25 — - . 0991 . 3830 . 4119 . 3902 . 3646 . 3386 . 3138 . 2912 . 2542 . 2268 . 2058 30 1325 5430 5873 5567 5198 4821 4462 4133 3595 3196 2891 35 '. 1660 ^7281 17933 ^7532 !7033 ^6520 '6030 '. 5579 ^4843 ^4295 !3878

40 . 1971 .9358 1. 0291 .9795 .9151 .8483 .7843 .7253 .6287 .5567 .5020

45 .2232 1. 1641 1. 2939 1. 2354 1. 1553 1. 0712 .9903 .9155 .7929 .7014 .6318 50 .2418 1. 4090 1. 5863 1. 5206 1. 4239 1. 3209 1. 2213 1. 1290 .9772 .8637 .7775 60 .2501 1. 9361 2. 2485 2. 1771 2. 0459 1. 9009 1. 7588 1. 6263 1. 4070 1. 2423 1. 1170

70 .2231 2. 4816 2. 9993 2. 9439 2. 7804 2. 5892 2. 3986 2. 2192 1. 9202 1. 6944 1. 5223 80 .2005 3. 0016 3. 8207 3. 8126 3. 6248 3. 3863 3. 1423 2. 9101 2. 5191 2. 2224 1. 9953

90 3. 4452 4. 6780 4. 7689 4. 5741 4. 2912 3. 9913 3. 7012 3. 2075 2. 8288 2. 5384

100 3. 7639 5. 5316 5. 7994 5. 6251 5. 3042 4. 9474 4. 5952 3. 9877 3. 5170 3. 1545

110 3. 9201 6. 3291 6. 8710 6. 7588 6. 4182 6. 0094 5. 5936 4. 8629 4. 2900 3. 8464

120 3. 9312 7. 0067 7. 9475 7. 9627 6. 6279 7. 1738 6. 6971 5. 8369 5. 1516 4. 6173

130 7. 4937 8. 9769 9. 2092 8. 9193 8. 4423 7. 9028 6. 9123 6. 1053 5. 4713

140 7. 7293 9. 8930 10. 4569 10. 2723 9. 7952 9.2175 8. 0909 7. 1551 6. 4118

150 7. 7099 10. 6160 11. 6542 11.6633 11. 2325 10. 6218 9. 3770 8. 3049 7. 4432

Table 8.15. Coefficient of Go, b2, in the regression equation {8.53)

Height, kft mi

5 10 15 20 25 30 35 40 50 60 70

5 -0 0065 -0. 0019 -0. 0008 -0. 0003 -0. 0000 0 0001 0 0002 0. 0002 0.0003 0. 0003 0. 0004

10 0268 -. 0084 -. 0043 -. 0024 0015 0009 0005 0003 0000 0001 .0002

15 0617 -. 0196 -. 0102 -. 0060 0039 0026 0018 0012 0006 0003 -. 0001 20 1123 -. 0356 -. 0187 -.0112 0073 0050 0035 0026 0015 0008 -. 0005

25 1807 -.0571 -. 0299 -. 0179 0118 0081 0057 0043 0025 0016 -. 0011 30 2695 -.0846 -. 0441 -. 0264 0175 0120 0086 0064 0039 0025 -. 0018 35 3820 -.1191 -. 0615 -.0368 0244 0168 0120 0091 0055 0036 -. 0026 40 5228 -. 1616 -.0827 -. 0493 0326 0224 0161 0122 0075 0048 -. 0035

45 6968 -.2132 -. 1080 -. 0642 0423 0291 0209 0158 0097 0063 -. 0047

50 9019 -. 2765 -. 1382 -. 0817 0537 0369 0265 0200 0123 0081 -. 0059

60 -1 4924 -. 4437 -. 2160 -. 1260 0822 0564 0404 0305 0187 0123 -. 0090

70 -2 3492 -. 6870 -. 3238 -. 1860 1201 0820 0586 0441 0270 0177 -. 0130

80 -3 5911 -1. 0399 -. 4737 -. 2672 1704 1156 0822 0616 0374 0245 -. 0179

90 -1.5533 -. 6835 -. 3782 2375 1596 1128 0839 0507 0330 -.0239

100 -2. 3002 -. 9795 -.5258 3243 2163 1520 1125 0674 0437 -.0314

110 -3. 3840 -1.3993 -. 7309 4438 2917 2030 1491 0884 0568 -. 0406

120 -4. 9414 -1.9997 -1.0155 6002 3891 2703 1962 1148 0734 -. 0518

130 -2. 8631 -1.4124 8152 5219 3547 2562 1482 0939 -. 0658

140 -4. 1054 -1. 9740 -1 1116 6963 4717 3347 1909 1194 -. 0827

150 -5. 8817 -2. 7735 -1.5206 9329 6185 4353 2437 1515 -. 1037
REGRESSION ANALYSIS 371

Table 8.16. Coefficient 0/ Go, bo, in the regression equation (8.54)

Height, kft mi

5 10 15 20 25

1 30 35 40 50 60 70

0 -0. 0032 -0. 0060 -0. 0002 -0. 0001 -0.0000 0.0000: 0. 0000 0.0000 -0.0000 -0. 0000 0. 0000

10 -. 0133 -. 002« -.0013 -.0005 0002 0001 -.0000' 0000 0000 .0000 . 0000

15 -. 0309 -.0067 -.0031 -.0012 0005 0002: -.0001 0000 0000 .0000 .0000 20 — 0573 —. 0124 —. 0056 —. 0022 0011 00041 —. 0002 0001 0000 . 0000 0000

25 I -.0942 -. 0202 -, 0091 -.' 0036 0017, 0008; -!0003 0001 0000 !oooi ^0000 30 -. 1442 -. 0306 -. 0136 -. 0055 0027 0012 -.0005 0002 0001 .0001 -0000

35 -. 2105 -.0442 -. 0194 -.0079 0038: 00171 -. 0007! 0004 0001 .0001 -.0000

40 -.2974 -. 0618 -. 0265 -. 0109 0053, 0024; -.0010] 0006 0002 .0001 -.0000

45 -.4101 -. 0840 -.0354 -. 0146 0072^ 0034' -. 0015: 0008 0002 . 0002 -.0001

50 -. 5553 -. 1130 -.0464 -. 0193 0095, 0045 -. 00201

1 0012 0004 . 0002 -.0001

60 -.9784 -. 1951 -. 0767 -.0322 0159^ 0078 -.00371 0022 0007 .0001 -. 0002

70 -1. 6559 -. 3276 -. 1222 -. 0516 0256 i 0129 -.00651 0038 0014 -. 0000 -.0003

80 -2. 7138 -. 5396 -.1916 -.0809 0401

1

0206 -. 0107, 0063 0024 -. 0004 -.0006

90 -. 8786 -. 2981 -. 1261 0622i 03231 -.0171: 0102 0040 -. 0010 -.0010

100 -1.4169 -.4631 -. 1907 0929 0488' -. 0266: 0160 0065 -.0020 -.0017

110 -2. 2638 -. 7186 -. 2906 1416 0741 -. 0406 0246 0101 -. 0035 -.0026

120 -3. 5736 -1. 1171 -.4432 2097 1095; -. 0622 0371 0153 -.0059 -.0039

130 -1. 7384 -. 6752 3139, 16461 -.0906 0548 0229 -.0093 -. 0059

140 -2. 7042 -1. 0330 47I61 2424! -. 1372 0810 0339 -.0142 -.0085

150 -4. 1878 -1. 5853 7079' 3584' -. 1978 1175 0485 -. 0212 -. 0123

Table 8.17. Coefficient of AN, bs, in the regression equation (8.64)

Height, kft 1

Dist 1

mi 5 10 15 20 25 30 35 40 50 60 70

5 -0 0806 -0. 0301 -0 0136 -0. 0060 -0. 0011 0. 0021 0.0043 0. 0057 0. 0073 0. 0083 0. (X)87 10 3321 -. 1360 0739 -.0481 -. 0313 -. 0201 -. 0127 -.0075 -.0012 -. 0024 -.0044

15 7533 -. 3146 1754 -. 1189 -.0819 -. 0575 -.0411 -.0296 -.0154 -.0077 -.0029

20 -1' 3470 -. 5687 3196 -. 2191 -. 1535 -. 1104 -. 0813 -.0608 -. 0355 -. 0218 -. 0132

25 -2 1168 -. 9026 5084 -. 3498 -.2468 -. 1793 -. 1336 -. 1015 -. 0616 -. 0401 -. 0265 30 -3 0664 -1.3217 7448 -. 5125 -.3628 -.2648 -. 1986 -. 1519 -. 0940 -.0628 -. 0430

35 -4. 1993 -1.8327 -1 0320 -. 7091 -.5028 -. 3679 -. 2768 -. 2126 -. 1330 -.0901 -.0627

40 -5 5178 -2. 4440 -1. 3744 -. 9818 -. 6680 -. 4894 -. 3690 -.2842 -. 1788 -. 1222 -. 0860

46 —7. 0221 -3. 1634 -1. 7769 -1.2136 -.8604 -. 6308 -. 4761 -.3672 -.2320 -. 1592 -. 1127 50 -8 7091 -4. 0036 -2. 2458 -1. 5276 -1.0819 -. 7935 -. 5992 -. 4625 -.2929 -. 2016 -. 1434 60 -12 5871 -6. 0883 -3 4103 -2. 2978 -1.6230 -1. 1893 -. 8979 -. 6937 -.4402 -. 3038 -.2171

70 -16. 9766 -8. 8019 -4. 9354 -3. 2912 -2. 3149 -1. 6931 -1.2773 -. 9865 -.6263 -.4320 -. 3092 80 -21. 4838 -12. 2504 -6. 9080 -4. 5605 -3. 1905 -2. 3271 -1. 7521 -1.3526 -. 8573 -. 5909 -. 4227

90 -16. 5206 -9. 4378 -6. 1745 -4. 2931 -3. 1194 -2. 3426 -1.8054 -1. 1417 -. 7843 -.5608

100 -21. 6282 -12. 6475 -8. 2077 -5. 6667 -4. 1012 -3. 0720 -2. 3636 -1. 4910 -1. 0207 -. 7284

110 -27. 4311 -16. 6690 -10. 7799 -7. 4021 -5. 3291 -3. 9747 -3. 0501 -1. 9168 -1. 3061 -. 9311 120 -33. 4941 -21.6151 -14. 0135 -9. 5614 -6. 8460 -5. 0972 -3. 8960 -2. 4355 -1.6531 -1. 1731 130 -27. 5400 -18. 0521 -12. 2761 -8. 7510 -6. 4662 -4. 9325 -3. 0686 -2. 0720 -1.4666

140 -34. 3121 -23. 0424 -15. 6717 -11. 1154 -8. 1908 -6. 2123 -3. 8447 -2. 5769 -1.8159

150 -41. 4796 -29. 0964 -19. 9016 -14. 0671 -10. 3024 -7. 7822 -4. 7800 -3. 1896 -2. 2378
372 REFRACTION AND REFRACTIVE INDEX MODELS

8.4.9. Height Error Equations

The equations for approximating €h were determined as D D^'^

eh, = aio -— + gi + k, (8.52)

= ^20 + g2 + k, (8.53)

and

D , D

3/2

eh, = azo Y2 + 9s -\- k (8.54)

where

D = R"^ - 0.03587 /la'.

Values for gi from (8.46) through (8.48) are found from

fii{ha) = a^i + ai'.ha + , (^ = 1, 2, 3) , (8.55)

/z2(/ia) = + a,5/ia , (^ = 2, 3) , (8.56)

and

/33(/ia) = a^6 + asy/ia (8.57)

for R in miles and ha in thousands of feet. The term in D/ha^ was introduced to account for, in part, a large negative constant term which tended

to produce negative height errors for ranges less than 30 mi. Further- more, the inclusion of this term increased the accuracy of the estimate of by about 2 percent. An additional term in hj for (8.55) increased

in the accuracy by about 1 percent but introduced a fictitious minimum

near 60,000 ft, while a term in ha^ for (8.56) and (8.57) increased the

accuracy of (8.53 and (8.54) by less than 0.1 percent. The relative improvement of (8.53) over (8.52) is about one percent. The coefficients

aij are listed in table 8.8. The constant term, k, which would vanish

if the equations were exact, is about —70 for a least squares approximation.
REFERENCES 373 8.4.10. Conclusions

Height-error correction can be significantly improved by accounting

for the surface refractivity at the radar site. The use of the initial gradient, in addition to the surface refractivity, yields a significant improvement only for targers beyond about 60 mi and below 15,000 ft. In this

case. Go is important not only to improve the accuracy but to determine if the assumption in section 2.2 has been violated, namely, if Go < —W/vo. The still further improvement obtained with the use of AN would not, in general, justify the trouble and expense of measuring

this parameter.

If the distance to the target exceeds about 50 mi, the normal decrease

with height of the gradient should be accounted for in a height error

correction.

8.5. References

[1] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric refractive index, Proc. IRE 47, No. 5, 740-755.

[2] Misme, P. (Nov.-Dec. 1958), Essai de radioclimatologie d'altitude dans le nord

de la France, Ann. Telecommun. 13, Nos. 11-12, 303-310.

[3] Tao, K., and K. Hirao (1960), Vertical distribution of radio refractive index in the medium height of the atmosphere, J. Radio Res. Lab. 7, No. 30, 85-93.

[4] Misme, P. (Nov.-Dec. 1960), Quelques aspects de la radio-climatologie, Ann. Telecommun. 15, 266.

[5] Bean, B. R., and G. D. Thayer (Aug. 1960), Rebuttal to P. Misme's comments on "Models of the Atmospheric Radio Refractive Index," Proc. IRE 48, No. 8, 1499-1501.

[6] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient

of refractive modulus in homogeneous air, potential modulus. Book, Propagation of Short Radio Waves, ed. D. E. Kerr, pp. 198-199 (McGraw-Hill Book

Co., Inc., New York, N.Y.).

[7] Zhevankin, S. A., and V. S. Troitskii (1959), Absorption of centimeter waves in the atmosphere, Radioteknika i Electronika 4, No. 1, 21-27.

[8] Minzner, R. A., W. S. Ripley, and T. P. Condron (1958, U.S. extension to the ICAO standard atmosphere, Tables and Data to 300 Standard geopotential

kilometers, U.S. Government Printing Office, Washington, D.C. 20402.

[9] Ratner, B. (1945), Upper air average values of temperature, pressure, and relative humidity over the United States and Alaska (U.S. Weather Bureau).

[10] Bean, B. R., and J. D. Horn (Nov.-Dec. 1959), The radio refractive index near

the ground, J. Res. NBS 63D (Radio Prop.), No. 3, 259-273.

[11] Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere,

Proc. IRE 40, No. 5, 554-561.

[12] Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1960), Climatic Charts and Data of the Radio Refractive Index for the United States and the World,

NBS Mono. 22.

[IS] Wong, M. S. (Sept. 1958), Refraction anomalies in airborne propagation, Proc. IRE 46, No. 9, 1628-1639.

[14] Report of Factual Data from the Canterbury Project (1951), Vols. I-III, (Dept.

of Scientific and Industrial Research, Wellington, New Zealand).
374 REFRACTION AND REFRACTIVE INDEX MODELS

[15] Freehafer, J. E. (1951), Tropospheric refraction, Book, Propagation of Short Radio Waves, ed. D. E. Kerr, pp. 9-22 (McGraw-Hill Book Co., Inc., New

York, N.Y.).

[16] Bean, B. R. (July-Aug. 1959), Climatology of ground-based radio ducts, J. Res. NBS 63D (Radio Prop.), No. 1, 29.

[17] Bean, B. R., and B. A. Cahoon (Nov. 1957), The use of surface weather observations to predict the total atmospheric bending of radio waves at small elevation

angles, Proc. IRE 45, No. 11, 1545-1546.

[18] Bean, B. R., B. A. Cahoon, and G. D. Thayer (1960), Tables for the statistical prediction of radio ray bending and elevation angle errors using surface values

of the refractive index, NBS Tech. Note 44.

[19] Bean, B. R., and G. D. Thayer (May-June 1963), Comparison of observed atmospheric radio refraction effects with values predicted through the use of surface weather observations, J. Res. NBS 67D (Radio Prop.), No. 3, 273.

[20] Anway, A. C. (1961), Empirical determination of total atmospheric refraction at centimeter wavelengths by radiometric means, Collins Res. Rept. CRR-2425

(Collins Radio Company, Cedar Rapids, Iowa).

[21] Anderson, W. L., N. J. Beyers, and R. J. Rainey (Aug. 1960), Comparison of experimental and compeutd refraction, IRE Trans. Ant. Prop. AP-8, 456.

[22] Wilkerson, R. E. (July-Aug. 1962), Divergence of radio rays by the troposphere,

J. Res. NBS 66D (Radio Prop.), No. 4, 479.

[23] Norton, K. A., J. W. Herbstreit, H. B. Janes, K. O. Hornberg, C. F. Peterson, A. F. Barghausen, W. E. Johnson, P. I. Wells, M. C. Thompson, Jr., M. J. Vetter, and A. W. Kirkpatrick (1961), An experimental study of phase variations in line-of-sight microwave transmissions, NBS Mono. 33.

[24] Thompson, M. C, Jr. (1962), Private communication.

[25] Bennett, C. A., and N. L. FrankUn (1954), Book, Statistical Analysis in Chemistry

and the Chemical Industry (John Wiley & Sons, Inc., New York, N.Y.).

[26] Brooks, C. E. P., and N. Carruthers (1953), Handbook of Statistical Methods in Meteorology (Her Majesty's Stationery Office, London).

[27] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave

propagation, Proc. IRE 21, No. 3, 427-463.

[28] Bauer, J. R., W. C. Mason, and F. A. Wilson (1958), Radio Refraction in a Cool

Exponential Atmosphere, Tech. Rept. 186, Lincoln Laboratory, Massachusetts

Institute of Technology, Cambridge, Mass.

[29] Beckmann, P. (1958), Height errors in radar measurements due to propagation

causes, Acta Technika, 3, No. 6, 471-488
Chapter 9. Radio-Meteorological Charts,

Graphs, Tables, and Sample Computations

9.1. Sample Computations of Atmospheric

Refraction

The following problem will serve to illustrate the application of the

various methods of calculating bending of a radio ray as described in chapter 3. A particular daily set of RAOB readings from Truk in the

Caroline Islands yields the following data:

Height above the N value

surface (km) (A^ units)

0.000 400.0 = Ns

0.340 365.0

0.950 333.5

3.060 237.0

4.340 196.5

5.090 173.0

5.300 172.0

5.940 155.0

6.250 152.0

7.180 134.0

7.617 125.5

9.660 98.0

10.870 85.0

What is the total bending up to the 3.270-km level at initial elevation

angles of 0°, 10 mrad, 52.4 mrad (3°), and 261.8 mrad (15°) by (a) Schulkin's approach, (b) the exponential model, (c) the initial gradient method,

(d) the departures from normal method, (e) the use of regression lines,

and (f) the graphical method of Weisbrod and Anderson? Since the

gradient between the ground and the first layer is AN 365.0 - 400.0 .^ooAr :n M = 0340 = -^^^'^ ^

and this is a decrease of per kilometer that is less than the —157 N

units/km required for ducting, no surface duct is present. However,

375
376 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

should a surface duct have been present, it would have been necessary to calculate the angle of penetration,

to find the smallest initial elevation angle that yields a non-trapped ray. Any initial elevation angle less than dp cannot be used in bending calculations. Schulkin's approach (a) of (3.13) yields the results shown in table 9.19

for 0 mrad, table 9.20 for 10 mrad, table 9.21 for 52.4 mrad, and table

9.22 for 261.8 mrad, where ^^+1 is determined from (3.58) using

Tk = CL -\- hk, and a is the radius of the earth,

It should be remembered that Ok = 0, 10, 52.4, or 261.8 mrad only for the

first-level calculation, and that thereafter is equal to the 6k+i computed

for the preceding layer; e.g., for the second layer of table 9.19

(^0 = Omrad), Ok = 6. 15 mrad, which is the calculated for the first layer.

The exponential model solution (b) may be found by using tables

9.10 through 9.17. Interpolation will usually be necessary for Ns, do, and

height; this interpolation may be done linearly. In practice, one of these

three variables will often be close enough to a tabulated value that inter- polation will not be necessary, thus reducing from 7 to 3 the number of interpolations necessary. Since in the problem iorN s = 404.9, h = 10.0 km

and ^0 = 10 mrad

and thus- by linear interpolation for h = 10.870 km, = 10 mrad,

dp = V2lNs - Nh - 156.9 {Ah) (in km)]

a = 6370 km.

To, 10. 0(10 mrad) = 15.084 mrad

and at /i = 20.00 km. Bo = 10 mrad.

To, 20. 00(10 mrad) = 15.946 mrad,

To, 10. 870, do mra(,d) = 15.084 + (15.946 - 15.084) = 15.159 mrad.

10.870 - 10.00

20.00 - 10.00

1
COMPUTATIONS OF ATMOSPHERIC REFRACTION 377 Similarly for iVs = 377.2 in the exponential tables,

To, 10. 870 (10 mrad) = 13.120 Hirad.

Again using linear interpolation, but now between the Ns = 377.2 and

Ns = 344.5 atm, the desired value of r at 3.270 km for Ns = 360.0 and

^0 = 10 mrad is obtained.

Thus

ro,io.870(io mrad) = 13.120 + (15.159 - 13.120) 9 I 377 2 = 14.798 mrad.

For the ^0 = 0, 52.4, and 261.8 mrad cases, by similar calculations, using

linear interpolation:

To, 10. 870, (0 mrad) = 21.386 mrad,

To, 10. 870, (52.4 mrad) = 5.816 mrad,

To, 10. 870, (261.8 mrad) = 1.270 mrad.

The initial gradient correction method (c) may be used if one deter- mines the Ns* which corresponds to the observed initial gradient and then

applies (3.45). The initial N gradient is —102.9 N units/km, which, as can be seen from table 3.17, corresponds to the Ns = 450.0 exponential

atmosphere. Therefore, using the exponential tables of Bean and

Thayer [1]^ and (3.45) to determine the bending for the = 0 mrad case, one finds by linear interpolation

Tio,ooo(o) = Tio,ooo (400.0, 0 mrad) + [rioo (450.0, 0 mrad) - noo (400.0, 0 mrad)] = 21.309 mrad + [5.908 - 3.657] mrad = 23.560 mrad.

The noo (400.0, 0 mrad) is determined by linear interpolation between the

404.9 and 377.2 atm. At h = 20.0 km as given in the tables,

T2o,ooo (0 mrad) = T2o,ooo (400.0, 0 mrad) + [tioo (450.0, 0 mrad)

T 100 (400.0, 0 mrad)] = 22.191 + [5.908 - 3.657] mrad = 24.442 mrad.

^Figures in brackets indicate the literature references on p. 423.
378 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Hence by linear interpolation,

rio.870 (0 .rad) = 23.560 + [24.442 - 23.560]

2^^000 - 10 000 = 23.637 mrad.

The bendings for Oq = 10, 52.4, and 261.8 mrad are as given below:

^0 = 10; no, 870 (10) = 15.053 mrad,

do = 52.4; Tio.sTo (52.4) = 5.864 mrad,

^0 = 261.8; no, 870 (261.8) = 1.280 mrad.

To use the departures-from-normal method (d) of determining bending,

it is first necessary to know the atmosphere which must be used for the

calculation. In the problem, - I = 102.9 N units/km,

Ctrl

I initial which is within the range of the Ns = 450.0 exponential atmosphere, as can be seen from table 9.18. Thus one will use table 9.17 to determine

the ^'s and the r's in the Ns = 450.0 exponential atmosphere, or one can

use the exponential atmosphere tables [1]. For an A^s = 450.0 atmosphere to a height of 10.870 km:

TNsio mrad) = 30.776 mrad,

TNsdQ mrad) = 19.414 mrad,

TArs(52.4 mrad) = 7.024 mrad,

TArs(26i.8 mrad) = 1.506 mrad.

Equation (3.58) should be used for the 6 interpolation in preference to

linear interpolation, although, if no tables or other facilities are present

at the engineering site for easy acquisition of square roots, linear inter- polation will suffice. Proceeding in table 9.17 with (3.58) for the first layer a,t h = 0.340 km and the ^o = 0 mrad case:

= 6.388 mrad. The remaining d's for the various layers are shown in table 9.23. To determine the value of A at the bottom and top of the
COMPUTATIONS OF ATMOSPHERIC REFRACTION 379

layer, one makes use of (3.46) or figure 3.17. First, however, one must

determine the value of c in (3.46) to be used. Usually interpolation will be necessary in table 9.18, but in the Ns = 450.0 case it is not possible,

and thus the straight Ns = 450.0 exponential atmosphere values are used.

From (3.46)

A(Ns, h) = N(h) + iV, [1 - exp (-ch)],

and for the layer running from h = 0 to h = 0.340 km, figure 3.17 yields

377.2 (1 - exp (cO)] = 0.0 and

450.0 [1 - exp (-C X 0.340)] = 32.8,

and, therefore,

A (450.0, 0) = 400.0 + 0 = 400.0

and

A (450.0., 0.340) = 365.0 + 32.8 = 397.8,

whence

AA = A (450.0, 0.340) - A (450.0, 0) = 397.8 - 400.0 = -2.2 units. Therefore, the departure term of (3.23),

becomes

2

r 7Vfc+i

0 + 6.388

[~l365.0

-2.2 =

J4OO.O +0.689 mrad.

The remaining calculations are tabulated in table 9.23 for the ^0 = 0 mrad

case, in table 9.24 for the ^0 = 10 mrad case, in table 9.25 for the ^0 = 52.4

mrad case, and in table 9.26 for the ^0 = 261.8 mrad case. The sum of the departures for the 0 mrad case is

AA (Ns) Lr = -5.335 mrad.

6k + QkJrl - Determination of the bending is required in part (e) of the problem by

using regression lines. By (3.10), using table 9.7 and 9.8, it is found for the ^0 = 0 mrad case that at 10.0 km (from table 9.7)

TO, 10.0 = (0.1149) (400.0) - 18.5627 ± 7.5227 = 27.3973 ± 7.5227 mrad
380 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

and at 20.0 km (from table 9.8)

To.20.0 = (0.1165) (400.0) - 17.9573 zt 7.5131 = 28.6427 ± 7.5131 mrad.

Thus, by linear interpolation

ri.2 = To.10.87 = 27.3973 + (28.6427 - 27.3973) 20 00 I Iq oo ± 7-5227 _ - . 10.87 - 10.00, =F (7.5227 - 7.5131)

20.0O - 10.00

ri.2 = 27.5056 ± 7.5218 mrad.

Similarly, for the remaining ^'s,

Ti.2(io mrad) = 13.9548 =b 0.9701 mrad,

Tl,2(53.4 mrad) = 5.2186 =t 0.0817 mrad,

Ti, 2(261. 8 mrad) = 1.2695 ± 0.0158 mrad.

Determination of the bending by means of the graphical method (f) of Weisbrod and Anderson yields, from figure 3.18 for 500 tan 0, for the

first layer.

At 00 At ^0 At ^0 At 00 h(m) = 0 mrad = 10 mrad = 52.4 mrad = 261.8 mrad

0.000 0.0 5.0 26.2 134.0

0.340 3.0 5.8 27.0 134.0

which yields for the bending in the first layer.

At ^0 At 00 At ^0 At ^0 = 0 mrad = 10 mrad = 52.4 mrad = 261.8 mrad.

~lT.^7~ 3^24 (K66 o7l3

Similarly, the bending for the entire profile may be obtained, and shown

to be

At 00 At ^0 At 00 At 00 = 0 mrad = 10 mrad = 52.4 mrad = 261.8 mrad.

24.42 14.00 5732 TTt
COMPUTATIONS OF ATMOSPHERIC REFRACTION 381 The answers to the several parts of the problem are summarized in the next table of this chapter (immediately preceding the main body of

tables). Bending values for the assumed profile, from a method which

exponentially interpolated layers between given layers and then integrated

between resulting layer^, assuming only a linear decrease of refractivity

between interpolated layers, are included for the sake of comparison.

The computations were performed on a digital computer.

The reason that the answers to part (e) vary so radically from the

remaining answers for the = 0 mrad case and not so much for the

^0 = 261.8 mrad case is the fact that the accuracy of the regression line method increases with increasing initial elevation angle, ^o. It must be

remembered that the statistical regression technique, like the exponential

model, is an adequate solution to the bending problem for all ^o's larger

than about 10 mrad, and all heights above 1 km.

The reason that the answers in part (f) and part (a) agree more closely

than with any other of the answers is because (3.49) is, as mentioned

before, Schulkin's result with only the approximation, tan dk = Bk for small angles, omitted. For this individual profile, the bending obtained

from an exponential atmosphere does not give particularly accurate bendings; however, for 22 five-year mean refractivity profiles, figure 3.9 shows

that exponential bending predicts accurately within 1 percent of the

average bending for these five-year means. Figure 3.19 shows the rms

error in predicting bending at various heights as a per cent of mean bending (not including superrefraction)

In summary, it is recommended that the communications engineer

either use the statistical regression technique or the exponential tables

[1] without interpolation (i.e., pick the values of height, N and that

are closest to the given parameters) for a quick and facile bending result,

keeping in mind the restrictions on these methods. However, as mentioned before, use of Schulkin's method is recommended if accuracy is the primary incentive.

Summary of refraction results for the sample commutation

Prob- lem

part Method used Bending

in mrad at 00=0 mrad

Bending

in mrad at

00 = 10 mrad

Bending

in mrad at 00=52. 4 mrad

Bending

in mrad at 00=261.8 mrad

a. Schulkin's method 24. 248 14. 008 5. 341 1.196

b. Exponential model ... . . 21. 386 14. 798 5. 816 1.270

c. Initial gradient correction method. 23. 637 15. 053 5. 864 1.280

d. Departures from normal method 25. 441 14. 858 5. 350 1. 143

e. Statistical regression method-- .- 27. 506 13. 955 5. 2186 1. 2695 ±7. 522 ±0. 9701 ±0. 0817 ±0. 0158

f. Graphical method... 24. 42 14. 00 5. 32 1. 168 Comparison (exponential layer

interpolation) bending 24- 171 14. 104 5. 343 1. 168

i
382 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Tables 9.1 through 9.9 are tables of coefficients, a and b, standard errors of estimate, SE, and correlation coefficients, r, for use in the statistical method

of refraction prediction.

Table 9.1. Variables in the statistical method of refraction prediction for h — hs =0.1 km

do r b a SE

0. 0 0. 2665 0. 0479 -8. 7011 6. 7277

1.0 .2785 .0257 -4. 1217 3. 4363

2.0 .2881 . 0162 -2. 2732 2. 0960

5. 0 .3048 .0073 -.9181 0. 8792

) 0. 0 . 1915 .0053 -. 6085 1. 0551

20. 0 .2070 .0025 -. 2639 0. 4555

52. 4 .2100 .0009 -. 0973 . 1688

100. 0 .2105 .0005 -. 0507 . 0879

200. 0 .2105 .0002 -. 0250 .0435

400. 0 .2107 .0001 -. 0120 .0208

900. 0 .2108 . 00004 -. 0040 .0070

Table 9.2. Variables in the statistical method of refraction -prediction for h — hs = 0.2 km

do r b a SE

0. 0 0. 2849 0. 05801 -10. 4261 7. 5726

1.0 .2979 .0348 -5. 6431 4. 3330

2.0 .3104 .0239 -3. 5707 2. 8357

5. 0 .3415 .0117 -1.5287 1.2512

10. 0 .2306 .0073 -0. 7436 1. 1990

20. 0 .2550 .0035 -. 3184 0. 5122

52. 4 .2604 .0013 -. 1162 . 1890

100. 0 .2610 .0007 -. 0603 .0983

200. 0 .2613 .0003 -. 0299 .0486

400. 0 .2613 .0002 -. 0143 .0233

900. 0 .2604 . 00005 -. 0047 .0078

Table 9.3. Variables in the statistical method of refraction prediction for h — hs = 0.5 km

do r b a SE

0. 0 0. 3615 0. 0769 -14. 6443 7. 6170

1.0 .3997 .0510 -9. 0567 4. 4954

2. 0 .4369 .0384 -6. 5408 3. 0395

5. 0 .5205 .0228 -3. 6605 1. 4376

r 10. 0 .3933 .0140 -1.9055 1. 2733

20. 0 .4563 .0071 -0. 8926 0. 5365

52. 4 .4731 .0027 -.3308 . 1966

100. 0 .4753 .0014 -. 1721 . 1022

200. 0 .4760 .0007 -. 0851 . 0505

400. 0 .4761 .0003 -. 0408 .0242

900. 0 .4764 .0001 -.0137 .0081
COMPUTATIONS OF ATMOSPHERIC REFRACTION 383

Table 9.4. Variables in the statistical method of refraction prediction for h — hs = 1.0 km

do r b a SE

0.0 0. 3936 0. 0840 -15. 1802 7. 6151

1.0 .4620 .0607 -10. 3739 4. 5217

2. 0 .5238 .04918 -8. 2066 3. 1040

5.0 .6348 .0337 -5. 4816 1. 5931 10.0 .5718 .0224 -3. 2378 1. 2574

20. 0 .6598 .0124 -1.6959 0. 5531 52.4 .6823 .0049 -0. 6495 .2071 100.0 .6851 .0026 -.3388 . 1080

200. 0 .6859 .0013 -. 1676 .0534 400.0 .6860 .0006 -. 0803 . 0256

900. 0 .6864 .0002 -. 0270 .0086

Table 9.5. Variables in the statistical method of refraction prediction for h - hs = 2.0 km

r b a SE

0. 0 0. 4524 0. 0985 -17. 7584 7. 5391 1.0 .5490 .0752 -12.9451 4. 4420

2.0 .6316 .0636 -10. 7566 3. 0277 5.0-- .7707 .0475 -7. 8969 1.5234

10.0 .7634 .0345 -5. 3712 1. 1421

20. 0 .8515 . 02111 -3. 1571 0. 5086

52. 4 .8668 .0089 -1. 2770 .2003

100.0 .8679 .0047 -0. 6705 . 1057

200. 0 .8681 .0023 -0. 3323 .0524

400. 0 .8682 .0011 -0. 1593 .0252

900. 0 .8684 .0004 - .0535 .0084

Table 9.6. Variables in the statistical method of refraction prediction for h — hs = 5.0 km

do r b a SE

0. 0 0. 4962 0. 1115 -19. 1704 7. 5676

1.0 .6101 .0881 -14. 3543 4. 4401

2.0 .7030 .0764 -12. 1589 3. 0001

5. 0 .8504 .0601 -9. 2514 1. 4422

10.0 .8674 .0464 -6. 6445 1.0420

20. 0 .9484 .0308 -4. 0706 0. 4028

52. 4 .9674 .0139 -1.6236 . 1426 100.0 .9695 .0075 -0. 8348 .0739

200. 0 .9701 .0037 -. 4098 .0365

400. 0 .9702 .0018 -. 1960 .0175

900. 0 .9703 .0006 -. 0658 .0059
384 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Table 9.7. Variables in the statistical method of refraction prediction for h - hs = 10.0 km

r b a SE

0. 0 0. 5099 0. 1149 -18.5627 7. 5227

1.0 .6290 .0915 -13. 7469 4. 3895

2. 0 .7250 .0799 -11.5514 2. 9443

5. 0 .8734 .0635 -8. 6434 1. 3733

10.0 .8950 .0498 -6. 0729 0. 9713

20. 0 .9723 .0338 -3. 5012 .3179

52.4 .9907 .0157 -1. 1441 .0844

100. 0 .9927 .0085 -0. 5084 .0406

200. 0 .9931 .0043 -.2310 .0197 400.0 .9932 .0020 -. 1078 .0094

900. 0 .9932 .0007 -. 0359 .0032

Table 9.8. Variables in the statistical method of refraction prediction for h - hs = 20.0 km

do r b a SE

0.0 0. 5155 0. 1165 -17. 9573 7. 5131 1.0 .6367 .0931 -13. 1413 4. 3763

2. 0 .7336 .0814 -10.9463 2. 9281

5. 0 .8815 .0651 -8. 0397 1. 3521 10.0 .9028 .0514 -5. 4747 0. 9573

20. 0 .9785 .0353 -2. 9228 .2909

52.4 .9968 .0169 -0. 6738 .0535

100.0 .9984 .0093 -. 1802 .0203

200. 0 .9986 .0047 -. 0467 .0096

400. 0 .9986 .0023 -.0161 .0046

900. 0 .9986 .0008 -.0048 .0016

Table 9.9. Variables in the statistical method of refraction prediction for h - hs = 70.0 km

Bo r b a SE

0. 0 0. 5174 0. 1170 -17. 9071 7. 5113

1.0 .6391 .0936 -13. 0912 4. 3738

2. 0 .7361 .0820 -10. 8960 2. 9251

r 5. 0 .8837 .0656 -7. 9895 1. 3481

10. 0 .9051 .0519 -5. 4209 0. 9539

20. 0 .9797 .0358 -2. 8696 .2862

52.4 .9979 .0173 -0. 6246 .0445

100. 0 .9997 .0096 -. 1402 .0095

200. 0 1.0000 .0048 -.0212 .0013

400. 0 1.0000 .0024 -.0027 .0002

900. 0 1.0000 .0008 -.0002 .0001
COMPUTATIONS OF ATMOSPHERIC REFRACTION 385

00 00 lO 05 <N ^ 05 Tt< 05

iiiiii iiiiii

iiiiii Iiiiii

iiiisi iiiiii

CS fO M ^'

iiiiii iiiii

^ es CO 'o o "2

SisSig iiiiii

222=32 SSSSSS

iilil iiiiii

liii iiilis

2S3SSSS

iiiiii iiiiii

Mmk iiiiis

II JjJJJJ JJ;sss8

i

I 1

I

II

iiiiii siiiii

iiiiii gggiii mm mm

iiiiii iiiiii

(N cs cc «

Hiiiii

iiiiii mm

111311 iiisis

mmm mmm

222S2S

iiiiii iiiiii

^' (N M "5

iiiiii iiiiii

^' (N CV5 id j;H

JJJJ S8gSS8
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

iiiiii Iiiiii

iiiiii issiig

iiiisi mm

iiiiii iiiiii

C<i CO iO CD

iiiiii iiiiii

tN TfJ CO 00 Oi Oi

iiiiii iiiiii

iiiiii iiiiii

d d Oi O r-H

^2 888888

!

0.

0017

.0034 .0084 .0167 .0331

.

0810

.

1563

.2915 .5966 .8844 1.

0899

1.1519

261.

804

261.808 261.

820

261.841 261.

883

262,

Oil

262.

228

262.

677

264.

116

266.

712

272.

175

298.

662

(M'

0.

0086

.1071 .0427 .0851

.

1687

.

4102

.7844 1.

4372 2.

8133 3.9667 4.6644

-

4.

8332

52.381 52.

403

52.

467

52.

574

52.

788

53.

431

54.

509

56.

676

63.

147

73.

459

91.

688

154.

494

0.

0150

.0299 .0745

.

1480

.2921

.

7025 1.

3202 2.3491 4.3356 5.

8325 6.

6578 6.8439

2

(. iiiiii iiiiii

Sssisi Siisig

i

iiiiii Iiiiii

<I> iiiigi Sfiigis

1

iiiiii giilii

ISi^ii 2ig§i^

Height (km) 0.

01 .02 .05 .10 .20 .50 1.00

2.005.00

10.

00

20.

00

70.

00
COMPUTATIONS OF ATMOSPHERIC REFRACTION 387

a

mmm mmm

mmm 11111

iiisii iiiiii

iiSSil sisisl

mmm 111111

SS22 = S CiSSEiSg

iiiiii iilili

Iiiiii Iiiiii

sddddd dddddd il

iiiiis iiiiii

iiSiis sSillS

giissi mam

iiiiii iiiiii

^' CO- CO td >C CO

iiiiii iiiiii

iilili iiiiii

d • ^^dcocc^-

2222^2

iilili iiiiii

------

iiiiii iilii

d^-^-co^-d dc.-d2cdd

^^•cc^-dd «o«d^^-

Sg§2^S SSSSSS
388 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

iiiiii iiiiii

«>

261.

802

261.

805

251.814 261.

830

261.860 261.

956

262.

124

262.

493

263.

794

266.

327

271.816 298.

365

t

iiiiii iiiiii

<» iigsii

8

o

.0509 .1268 .2518 .4966

1.1915 2.

2284 3.9184 6.9465 8.8659 9.

6438 9.7439

1

111111 ligiii

i

0.

5853

1.0018 1.

9035 2.

9585 4.

4578 7.3331

10.

3092

13.

8820

18.

6069

21.

0224

21.

8972

22.

0033

1

1.

2071 1.

7061 2.6917 3.7918 5.3234 8.2262

11.

2147

14.

7946

19.

5234

21.

9397

22.8146 22.

9208

Height (km) o

.05 .10 .20 .50. 1

00

2.005.00

10.

00

20.

00

70.

00

^'

0.

0037

.0075 .0186 .0371 .0733

.

1773

.

3357

.6040 1.

1262 1.

4919 1.

6487 1.

6670

«) 261.

802

261.

804

261.

810

261.

821

261.

843

262.

049

262.

365

263.

591

266.115 271.

639

298.

218

1

0.

0191

.0382 .0952 .1893 .3740 Qni7 1.

6987 3.

0219 5.

4548 6.

9761 7.

5252 7.

5768

o

0.

0334

.0667

.

1661

.3299 .6504 1.

5581 2.

9040 5.

0619 8.7161

10.

7493

11.4081 11.4653

<=> 1—

0.

1000

.1993 .4924 .9662 1.

8628 4.

2047 7.

2492

11.

3482 iiii

22^8

i

2222d2

0.

8154 1.4279 2.

7880

4.3997 6.

6952

11.

0468

16.

4280

20.

4350

26.

4286

29.

0324

29.

7768

29.

8376

1

1.8854 2.

6642 4.2000 5.

9079 8.

2725

12.

6813

17.

0868

22.

1065

28.

1060

30.

7108

31.

4553

31.

5161

Height (km)

0.01 .02 .05 .10 .20 .501.00 2.

00

a.

uo

10.

00

20. 0070.

00
COMPUTATIONS OF ATMOSPHERIC REFRACTION 389 Table 9.18. Initial N gradients, ANe, in the CRPL

Exponential Reference Atmosphere*

Range of ANe {N units/km) N{h)

ANe <, 27. 55

27. 55 < ANe <. 35. 33

35. 33 < A.Ve <. 47. 13

42. 13 < ANe <. 49. 52

49. 52 < ANe < 59. 68

59. 68 < ANe ^ 71. 10

71. 10 < ANe ^ 88. 65

88. 65 < ANe

200 exp(-0. 1184ft) 252.9exp(-0. 1262ft) 289 exp( -0.1257ft) 313 exp(-0. 1428ft) 344. 5 exp(-0. 1568ft) 377.2 exp(-0. 1732ft) 404.9exp(-0. 1898ft) 450 exp(-0.2232ft) *Note height, ft, is in kilometers.
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

TkMl (mrad)

11.547 3.

331 5.

073 1

444 0.

726

.

029

.473 .080 .451 .242 .597 .255

2

=

24.

248

i

(M (mrad)

3.

031 9.

458

19.

024

28.

050

32.

363

34.

284

35.

911

37.

677

39.

885

42.

199

46.

087

50.

996

i

1

(mrad)

2 36. 75

165.

26

634.

64

955.

33

1143.65 1207.

53

1374.

30

1465.

54

1721.

25

1841.

30

2426.

98

2780.

31

2X?

1

1

106.

75

191.51 662.

38

401.

69

235.

32

65. 88

200. 77

97. 24

291.

71

137.

05

640.

68

379.

33

1 1

f

1

(km)

0.

680

1.220 4.

220 2.560

1.500 .420

1.280 .620

1.860 .874

4.086 2.

420

(km)

6370.

6370.

340

6370.

950

6373.

060

6374.

340

6375.

090

6375.

300

6375.

940

6376.

250

6377.

180

6377.

617

6379.

660

6380.

870

ooiooiooo oooiooo

(km) 0

0.

340 0.

950 3.

060 4.

340

5.090 5.

300 5.940

6.250 7.

180 7.

617

9.660 10.

870

cc csi r-; d

I

sisiiii iilii

I

iisii

liiiii iissis

I msm mm

lit

I

ooo ooot^oo 05 o •^in<X)'-"Ot^

00 O CO 05 IM .-H to « 00

' c6 uri lodt^i^oid
COMPUTATIONS OF ATMOSPHERIC REFRACTION 391

«g S S2 =^ 5S ^SJJSSS? ::: 1

^

++

(mrad)

52.

575

53.

352

5().

048

59.

4S<J

01.001 02.

()20 63.

532

04.

542

65.

866

67.

282

69.

825

73.

130

J

1

(mrad)

2

2782.

51

2911.02 3380.

40

3701.

09

3889.

41

3953.

29

4120.

06

4211.30 4467.

01

4587.

0()

5172.

74

552(i. 07

1

f

1

-ci + mm

1

1 1

1

"

f

1

(km)

0.

680

1.220 4.

220 2.560

1.500 .420 1.

280

.

620

1.860 .874 4.

086 2.420

!

(km)

6370

6370.

340

6370.

950

6373.

0()()

6374.

340

6375.

090

6375.

300

6375.

940

6376.

250

6377.

180

6377.

617

6379.

660

6380.

870

1

oiiiiil iiSsis

1

iiiiii i i

' ' i

(mrad)

.134

.

120

.368

.

154

.089 .004 .064 .011 .068 .032 .103 .049 2

=

1.

196

1

(M

1

(mrad)

261.

870

262.

115

263.

009

263.

618

263.

975

264.

096

264.412 264.

584

265.

067

265.

293

266.

395

267.

057

1

(mrad)

2

68575.

99

68704.

50

69173.

88

69494.

57

69682.

89

69746.

77

9913.

546

70004.

78

70260.

49

70380.

54

70966.

22

71319.

55

1

f

1

1

1 1

i

"

f

1

iiiiiSi

1

f

(km)

6370

6370.

340

6370.

950

6373.

060

6374.

340

6375.

090

6375.

300

6375.

940

6376.

250

6377.

180

6377.

617

6379.

660

6380.

870

(km) 0

.340 .950 3.

060 4.340

5.090 5.

300 5.

940 6.250

7.180 7.

617 9.

660

10.

870

N
392 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Table 9.23. Departures 7nethod sample computation for do = 10 mrad

mh) AiN,h)

dk+dk+i

Departure term

rNsih)

400.0

365.0 333.5

237.0

196.5 173.0

172.0

155.0

152.

134.

125.

98.

85.

400.0

397.8 419.5

459.7

475. 7 478.5

484.1

485.

490.

493.

493.

495.

495.

-2.2

+21.7 +40.0

+16.0 +2.8

+5.6

+1.5 +4.9

+2.9

-. 1 +2.6 -.7

0

6. 388

11. 230

22. 683

28. 195

31. 383

32. 191

34. 654

35. 811

39. 123

40. 581

47. 236

50. 645

0. 3131

. 1135 .0590

.0393 .0336

.0315 .0299

.0284 .0267

.0251 .0228

.0204

+0. 689 -2. 463 -2. 359 -0. 629 -.094

-.176 -.045

-.139

-.077

+. 003 -.059

+. 014 -5. 335

). 776 25. 441

Table 9.24. Departures method sample computation for do = 10 mrad

mh) A{N,h) 6 2 dk+0k+i

Departure term

400.0 400.0 10. 000 365.0 397.8 -2.2 1L873 0. 0914 +0. 201 333.5 419.5 +21.7 15. 038 .0743 -1.613

237.0 459.7 +40.0 24. 789 .0502 -2. 008

196.5 475.7 +16.0 29. 947 .0365 -.585

173.0 478.5 +2.8 32. 938 .0318 -.089

172.0 484.

1

+5.6 33. 712 .0300 -.168

155.0 485.6 +1.5 36. 069 .0287 -.043

152.0 490.5 +4.9 37. 181 .0273 -.134

134.0 493.4 +2.9 40. 383 .0258 -.075

125.5 493.3 -. 1 41. 801 .0243 +.002

98.0 495.9 +2.6 48. 282 .0222 -.058

85.0 495.2 -.7 51. 630 .0200 +. 014 S = -4. 556

TNsih)

19. 414 14. 858

Table 9.25. Departures method sample computation for do = 52.^ mrad (3°)

N(h) AiN„h)

9k+0k+\

Departure term

TNs{h)

0

0. 340

0. 950

3. 060

4. 340

5. 090

5. 300

5.940

6. 250

7. 180 7.617

9. 660

10. 870

400.0

365.0 333.6

237.0

196.5 173.0

172.0

155.0

152.0

134.0

125.5 98.0

85.0

400.0

397.8

419.5 459.7 475.7

478.5

484.1 485.6

490.5

493.4

493.3

495.9

495.2

-2.2

+21.7 +40.0

+16.0 +2.8

+5.6

+1.5

+4.9

+2.9

-. 1 +2.6

52.4

52. 750

53. 550

57. 061

59. 567

61. 045

61. 477

62. 792

63. 433

65. 368

66. 277

70. 512

72. 920

0. 0190

.0188

.0181

.0171 .0166

.0163

.0161 .0158

.0155

.0152

.0146 .0139

+0. 042 -.408

-.724 -.274

-.046

-.091 -.024

-.078

-.045 +.002

-.038 +.010

-1.674

5. 350
EXPONENTIAL REFERENCE ATMOSPHERE 393 Table 9.26. Departures method sample computation for do = 261.8 mrad (15°)

h Nih) AiNs,h) AA e 2 ek+ek+\

Departure term

Tl ,2

0 400.0

365.0

333.5

237.0

196.5 173.0

172.0

155.0

152.0

134.0

125.5 98.0

85.0

400.0

397.8

419; 5 459.7 475.7

478.5

484.1 485.6

490.5

493.4

493.3

495.9

495.2

-2.2

+21.7 +40.0

+16.0 +2.8

+5.6

+1.5

+4.9 +2.9

-. 1 +2.6 -.7

261.8

261. 880

262. 035

262. 758

263. 308

263. 633

263. 733

264. 034

264. 184

264. 647

264. 872

265. 934

266. 592

0. 0038 .0038

.0038

.0038

.0038

.0038 .0038

.0038

.0038

.0038 .0038

.0038

+0. 008 -.083

-. 152 -.061

-.011

-.021 -.006

-.019

-.011

+. 000 -.010 +.003 2 = -0. 363

0. 340

0. 950

3. 060

4. 340

5. 090-

5. 300

5. 940

6. 250

7. 180—

7. 617

9. 660

10. 870...- 1.506 1. 143

9.2. Tables of Refraction Variables for the

Exponential Reference Atmosphere

The following table of estimated maximum errors should serve as a

guide to the accuracy of the tables. Errors in elevation angle, 6:

^0 < 4 mrad =b 0.00005 mrad ) nearly

4 mrad < < 100 mrad d= 0.000005 mrad ? independent

^0 < 100 mrad db 0.00004 mrad ) of Ns.

Errors in r, e (in milliradians)

Ns = 450 404.8 377.2

^0 = 0 ± 0.001 0.00065 0.0005

do = r 0.0003 0.00015 0.0001

^0 = 3° 0.00004 0.000025 0.00002

344.5 313 252.9 200

0.0004 0.0003 0.0002 0.0002

0.00008 0.00006 0.00005 0.00005

0.000017 0.000015 0.000013 0.000012

Errors in Ro, R, Re, or Ah (in meters):

A^. = 450

^0 = 0 ± 5.0

^0 = 1° 0.4

404.8 377.2

2.7 1.8

0.3 0.25

344.5 313

1.2 0.8

0.2 0.17

252.9 200

0.65 0.6

0.15 0.14

Assume that the error in AR or ARe is it 0.5 percent or dzO.l m, whichever

is larger.
394 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

In tables 9.27 to 9.29 the following equations were used in determining

the various profile variables:

-AA^ = 7.32 exp (0.005577 A^.),

k = ' 1 - — ic,N,) X

10-"

Us

Us ^ I + Ns X lo-^

where ro is taken as 6373.024987 km for all values of Ns in these tables. Table 9.29 was prepared using an iterative method for solution of transcendental equations involved. In all these tables the accuracy may

be taken as dzl in the last digit listed.

Table 9.27. Parameters for exponential refraction with height for various values of Ns

Ns -AN Ce -dNo k

200 22. 3317700 0. 118399435 23. 6798870 1. 17769275 210 23. 6125966 . 119280212 25. 0488444 1. 18991401 220 24. 9668845 . 120458179 26. 5007993 1. 20315637 230 26. 3988468 . 121916361 28. 0407631 1. 21752719 240 27. 9129385 . 123642065 29. 6740955 1. 23314913

250 29. 5138701 . 125626129 31. 4065323 1. 25016295 260 31. 2066224 . 127862319 33. 2442030 1. 26873080

270 32. 9964614 . 130346887 35. 1936594 1. 28904048

280 34. 8889558 . 133078254 37. 2619112 1. 31131073 290 36. 8899932 . 136056720 39. 4564487 1. 33579768 300 39. 0057990 . 139284287 41. 7852861 1. 36280330

310 41. 2429556 . 142764507 44. 2569972 1. 39268608

320 43. 6084233 . 146502381 46. 8807620 1. 42587494 330 46. 1095611 . 150504269 49. 6664087 1. 46288731 340 48. 7541501 . 154777865 52. 6244741 1. 50435338 350 51. 5504184 . 159332141 55. 7662495 1. 55104840 360 54. 5070651 . 164177379 59. 1038565 1. 60393724 370 57. 6332884 . 169325150 62. 6503054 1. 66423593

380 60. 9388149 . 174788368 66. 4195799 1. 73349938

390 64. 4339281 . 180581312 70. 4267116 1. 81374807 400 68. 1295015 . 186719722 74. 6878887 1. 90765687

410 72. 0370324 . 193220834 79. 2205420 2. 01884302

420 76. 1686780 . 200103517 84. 0434770 2. 15232187 430 80. 5372922 . 207388355 89. 1769927 2. 31525447 440 85. 1564647 . 215097782 94. 6430240 2. 51823286

450 90. 0405683 . 223256247 100. 4653113 2. 77761532
EXPONENTIAL REFERENCE ATMOSPHERE 395 Table 9.28. Parametersfor exponential refraction with height for various values of —AN

-AA" Ns -diVo k

20 180 00A077 0. 117626108 21. 1993155 lOOi

/

22 ly/. 316142 . 118216356 23. 3259953 I '7A^'7A^ 0 I I '±0 i 24 . . 212 Q1 7QR7 . 119594076 25. 4637276 lyoDDoUo

26 227 270255 . 121491305 27. 6113599 Z1Ot:o0D0 28 240. 558398 . 123746115 29. 7681671 1 23406110

30 252. QOQQAO y/yoo^ . 126255291 31. 9336703

32 501627 . 128950180 34. 1075325

34 275. *370nflO o/zuyy . 131783550 36. 2895127 oUU/'loZo 36 285 621054 . 134721962 38. 4794288 ozi/ooys

38 295 315731 . 137741207 40. 6771452 1 34981825

40 304 513148 . 140823306 42. 8825481 o

t

oyuyo^

42 tSlo ZDllOO . 143955014 45. 0955611 ]^ ±c\'^^ 0A^ a 44 321 dUZooo . 147125889 47. 3161107 loiooooy

46 329 573439 . 150328075 49. 5441405

48 337 204713 . 153555418 51. 7796106 1. 49227226

50 344 524418 . 156803056 54. 0224815 2 / you

52 ooi 557000 . 160067149 56. 2727266 2 54 358 . 163344614 58. 5303179 2 56 364 845143 . 166633002 60. 7952415 2 DoZl'lUOo 58 371 137293 . 169930326 63.0674811 1 67162830

60 01 Ri nc . 173234984 65. 3470266 2 62 coo uyoooi . 176545680 67. 6338699 2 7';7nf;7^9 64 388 7QQQ7Q /ooo/o . 179861358 69. 9280046 1. 80335830

66 394 305974 . 183181171 72. 2294300 1 85232456

68 399 658845 . 186504431 74. 5381454 1. 90419225

70 .. OODOiSO . 189830583 76. 8541525 1. 95922635

72 Qn77Qfi yu/ /yo . 193159183 79. 1774555 2. 01772514

74 414 SzUOOU . 196489873 81. 5080567 2. 08002556

76 419 602477 . 199822385 83. 8459677 2. 14650999

78 424 . 203156494 86. 1911914 2. 21761358 80 428 799768 . 206492043 88. 5437400 2. 29383429

82 433 227348 . 209828917 90. 9036251 2. 37574437

84 437 548229 . 213167031 93. 2708570 2. 46400458

86 441 767432 . 216506335 95. 6454475 2. 55938222

88 445 889634 . 219846812 98. 0274147 2. 66277367

90 449 919193 . 223188453 100. 4167688 2. 77523207

92 453 860184 . 226531281 102. 8135290 2. 89800399

94 457 716416 . 229875327 105. 2177108 3. 03257531 96 461 491458 . 233220637 107. 6293319 3. 18073184 98 465 188659 . 236567271 110. 0484114 3. 34463902

100 468 811163 . 239915290 112. 4749663 3. 52694820
396 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Table 9.29. Parameters for exponential refraction with height for various values of k

k -AN Ce -dNo

1.0 0.0 0.0 0.0 0. 0

1. 2 217. 689023 24. 6471681 . 120160519 26. 1576259 1.3. 275. 037959 33. 9367000 . 131692114 36. 2203304

1. 4 312. 297111 41. 7747176 . 143600133 44. 8459068

1. 5 339. 003316 48. 4839018 . 154339490 52. 3215987

1.6 359. 298283 54. 2941700 . 163827653 58. 8629945

1. 7 375. 341242 59. 3759008 . 172203063 64. 6349115

1.8 388. 391792 63. 8586055 . 179626805 69. 7655765

1.9 399. 243407 67. 8426334 . 186242834 74. 3562234

2.0 408. 424907 71. 4070090 . 192172034 78. 4878451

2. 1 416. 304322 74. 6148487 . 197514185 82. 2260091

2 2 423. 146728 77. 5171828 . 202361472 85. 6243634

2.3 429. 148472 80. 1557288 . 206751820 88. 7272277

2.4 434. 458411 82. 5649192 . 210771674 91. 5715264 2.5 439. 191718 84. 7734613 . 214458304 94. 1883108

2.6 443. 438906 86. 8054237 . 217851443 96. 6038056

2.7 447. 272272 88. 6811886 . 220984823 98. 8403838

2.8 450. 750273 90.4181120 . 223887193 100. 9172131

523. 299600 135. 5109159 . 299693586 156. 829534 4/3-. 289. 036274 36. 6922523 . 135758874 39.2392391

9.3. Refractivity Tables

Tables 9.30 through 9.35 have been prepared on a digital computer to

assist the reader in making an easy determination of the refractivity, N,

value from pressure, temperature, and relative humidity data. Standard

pressure values of 700, 850, and 1000 mbar have been used for the tables. The tables are at every 10 percent relative humidity and for every degree

centigrade ranging for —45 to +45 °C. Interpolation for the tables is accomplished in the following manner.

First, round the temperature to the nearest degree, then interpolate

linearly for the values of pressure, P, and relative humidity, RH, used.

No distortion is introduced by this method as the wet term, W (as described in chapter 4), is a linear function of RH, and the dry term, D, is a linear function of P. Example:

and knowing

Given P = 980 mbar

RH = 47 percent

T = 22 °C

D = D{P, T)

N = N{P, T, RH)

W = W{T, RH),

then

1) (850, 22) = TV (850, 22, 0) = 223.6 N units.

2) (980, 22) = iV(980, 22, 0) = 223.6 (980/850) = 257.8 units.
REFRACTIVITY TABLES 397 Now interpolating for humidity,

A^(850, 22, 50) = 280.3 units. Since

A^(850, 22, 50) - Z)(850, 22) = TF(22, 50),

thus

TF(22, 50) = 280.3 - 223.6 = 56.7 N units, and

W{22, 47) = 56.7 (47/50) = 53.3 N units. Adding

TF(22, 47) to Z)(980, 22) yields

A^(980, 22, 47) = Z)(980, 22) + TF(22,47) = 257.8 + 53.3 = 311.1 N units.

Substitution directly into (1.20) and use of the Smithsonian Meteorological Tables [2] for vapor pressure yields the identical result to four

significant figures.

A^(980, 22, 47) = 311.1 A^ units. Ambient recording thermometers generally read to a tenth of a degree

centigrade. Rounding the temperature to the nearest whole degree

introduces an extraneous error into the interpolation process. At 45 °C,

1000 mbar, and at a relative humidity of 100 percent, this error will have

a maximum of 7.7 A^ units. However, these conditions are an extreme

rarity climatologically speaking, and at normal temperature ranges an

error of more than 2 A^ units is unhkely.
398 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Table 9.30. Refractivity for a pressure of 700 mhar

Relative humidity

ature

0 10 20 30 40 50 60 70 80 90 100

° C

—45 238. 2 238. 3 238. 4 238. 5 238. 6 238. 6 238. 7 238. 8 238. 9 239. 0 239. 0 —44 237. 2 237. 3 237. 4 237. 5 237. 6 237. 6 237. 7 237. 8 237 9 238. 0 238.

1

—43 236. 2 236. 3 236. 4 236. 5 236. 6 236. 7 236. 8 236. 9 237. 0 237.

1

237.

1

—42 235. 2 235. 3 235. 4 235. 5 235. 6 235. 7 235. 8 235. 9 236. 0 236.

1

236. 2 -41 234.

1

234.3 234.4 234. 5 234.6 234.7 234.8 235.0 235.

1

235.2 235.3 —40 233.

1

233. 3 233. 4 233. 5 233. 7 233. 8 233. 9 234. 0 234. 2 234. 3 234. 4 —39 232.

1

232. 3 232. 4 232. 6 232. 7 232. 9 233. 0 233.

1

233. 3 233. 4 233. 4 —38 231.

1

231. 3 231. 5 231. 6 231. 8 231. 9 232. 1 232. 2 232. 4 232. 6 232. 7 —37 230. 2 230. 3 230. 5 230. 7 230. 9 231. 0 231. 2 231. 4 231. 5 231. 7 231. 9 -36 229. 2 229.4 229.6 229.8 230.0 230. 1 230.3 230.5 230.7 230.9 231. 1 —35 228. 2 228. 4 228. 6 228. 9 229. 1 229. 3 229. 5 229. 7 229. 9 230.

1

230. 3 —34 227. 3 227. 5 227. 7 228. 0 228. 2 228. 4 228. 6 228. 9 229.

1

229. 3 229 5 —33 226. 3 226. 6 226. 8 227. 1 227. 3 227. 6 227. 8 228.

1

228. 3 228. 6 228. 8 —32 225. 4 225. 7 225. 9 226. 2 226. 5 226. 7 227. 0 227. 3 227. 6 227. 8 228.

1

-31 224. 5 224.8 225. 1 225.3 225.6 225.9 226.2 226. 5 226.8 227.

1

227.4 —30 223. 5 223. 9 224. 2 224. 5 224. 8 225.

1

225. 5 225. 8 226.

1

226. 4 226. 8 —29 222. 6 223. 0 223. 3 223. 7 224. 0 224. 4 224. 7 225.

1

225. 4 225. 8 226.

1

—28 221. 7 222. 1 222. 5 222. 9 223. 2 223. 6 224. 0 224. 4 224. 8 225. 1 225. 5 -27 220. 8 221. 2 221. 6 222. 1 222. 5 222. 9 223. 3 223. 7 224.

1

224. 5 225. 0 -26 219. 9 220. 4 220. 8 221. 3 221. 7 222. 2 222.6 223.

1

223. 5 224. 0 224.4 —25 219. 0 219. 5 220. 0 220. 5 221. 0 221. 5 222. 0 222. 5 222. 9 223. 4 223. 9 —24 218. 2 218. 7 219. 2 219. 7 220. 3 220. 8 221. 3 221. 9 222. 4 222. 9 223. 5 —23 217. 3 217. 9 218. 4 219. 0 219. 6 220. 2 220. 7 221. 3 221. 9 222. 5 223. 0 —22 216. 4 217. 0 217. 7 218. 3 218. 9 219. 5 220. 2 220. 8 221. 4 222. 0 222. 7 —21 215. 6 216. 2 216. 9 217. 6 218. 3 218.9 219.6 220. 3 221.0 221.6 222. 3 -20 214. 7 215. 4 216. 2 216. 9 217. 6 218.4 219. 1 219.8 220. 6 221. 3 222. 0 — 19 213. 9 214. 6 215. 4 216. 2 217. 0 217. 8 218. 6 219. 4 220. 2 221. 0 221. 8 -18 213. 0 213. 9 214. 7 215.6 216. 4 217.3 218. 1 219. 0 219. 9 220. 7 221. 6 — 17 212. 2 213.

1

214. 0 215. 0 215. 9 216.8 217. 7 218. 6 219. 6 220. 5 221. 4 — 16 211. 4 212. 4 213. 4 214. 3 215. 3 216. 3 217. 3 218. 3 219.3 220. 3 221. 3 —15 . 210. 5 211.6 212. 7 213. 8 214. 8 215. 9 217. 0 218. 0 219.

1

220. 2 221. 3 —14 209.

7

210. 9 212. 0 213. 2 214. 3 215. 5 216. 7 217.8 219. 0 220.

1

221. 3 —13 208. 9 210. 2 211.4 212. 7 213. 9 215.

1

216.4 217. 6 218. 9 220.

1

221. 4 —12 208.

1

209. 5 210. 8 212.

1

213. 5 214. 8 216.

1

217. 5 218.8 220. 2 221. 5 — 11 207.3 208.8 210. 2 211. 6 213.

1

214. 5 216. 0 217. 4 218.8 220. 3 221. 7 JQ 206 5 208 1 209 6 211 2 212 7 214 3 215 8 217 4 218 9 220 4 222. 0 -9 205'

8

207^4 209! 1 2m 7 212^4 214! 1 215^7 217! 4 219^0 220^7 222^3 -8 205.0 206.8 208.5 210.3 212. 1 213.9 215.7 217.4 219.2 221.0 222.8 -7 204.2 206.

1

208.0 29.9 211.8 213.8 215.7 217.6 219.5 221.4 223.3 -6 203.4 205.5 207.5 29.6 211.6 213.7 215.7 217.8 219.8 221.9 223.9 -5 202.7 204.9 207. 1 209.3 211.4 213.6 215.8 218.0 220.2 222. 4 224.6 -4 201.9 204.3 206.6 209.0 211.3 213.7 216.0 218.3 220.7 223.0 225.4 -3 201.2 203.7 206.2 208.7 211.2 213.7 216.2 218.7 221.2 223.8 226.3 -2 200.4 203.1 205.8 208.5 211.2 213.8 216.5 219.2 221.9 224.6 227.3 -1 199.7 202.6 205.4 208.3 211.2 214.0 216.9 219.8 222.6 225.5 228.4

0 199.0 202.0 205.

1

208.2 211.2 214.3 217.3 220.4 223.4 226.5 229.6
REFRACTIVITY TABLES

Table 9.31. Refractivity for a pressure of 700 mhar.

399

Relative humidity

ature

Q 10 20 30 40 50 60 70 80 90 inn

° C

0 199.0 202.0 205.1 208. 2 211.2 214.3 217.3 220.4 223.4 226.5 229.0

198 2 201 5 204. 8 208. 0 211. 3 214. 6 217. 8 221.

224. 4 227 6 230 9 2 197 5 201. 0 204. 5 208. 0 211. 5 214. 9 218. 4 221. 9 225. 4 228 9 232 3 3 196. 8 200. 5 204. 2 207. 9 211. 7 215. 4 219.

222. 8 226. 5 230. 2 233 9 196 1 200 1 204 0 208 0 211 9 215 9 219 8 223 8 227 7 231 7 jioo. 0

5 195'

199^6 203^8 208^0 212^2 21615 220*7 224^9 229! 1 233*. 3 237.5

g 194 7 199 2 203 7 208.

212 6 217 1 221. 6 226.230. 5 235 0 239 5

194. 0 198. 8 203. 5 208. 3 213.217. 8 222. 6 227. 4 232.236. 9 241 7 g 193 3 198 4 203 4 208 5 213 6 218 7 223 7 228 8 233 9 238 9 244 0 Q 192 6 198 0 203 4 208 8 214 2 219 6 224 9 230 3 235 7 241 1 246 5 10 19L9 197^7 203^4 209^1 214.' 8 220^5 226^3 232*0 237> 243^4 249! 1

2 J 191. 3 197. 3 203. 4 209. 5 215. 6 221. 6 227. 7 233. 8 239. 8 245 9 252 0 12 190. 6 197. 0 203. 5 209. 9 216. 4 222. 8 229. 2 235. 7 242.

248. 6 255 0

13 189. 9 196. 8 203. 6 210. 4 217. 3 224. 1 230. 9 237. 7 244. 6 251. 4 258 2

14 189 3 196 5 203. 7 211. 0 218. 2 225. 5 232. 7 239. 9 247. 2 254. 4 261 7

15 m6 m3 204^0 211! 6 219^3 227^0 234^6 242*3 250!0 257^6 265.' 3

16 188. 0 196 1 204. 2 212. 3 220 4 228. 6 236. 7 244. 8 252. 9 261.

269 2

17 187 3 195 9 204 5 213.221. 7 230. 3 238. 9 247. 5 256.

264. 7 273 3

18 186. 7 195. 8 204. 9 213. 9 223. 0 232 1 241. 2 250. 3 259. 4 268. 5 277 6

19 186. 0 195. 6 205. 3 214 9 224 5 234 1 243. 7 253. 3 262. 9 272. 6 282 2 20 185*4 195*6 205*7 215^9 226^0 236^2 246^4 256^5 266^7 276! 9 287^0

20 184 8 195 5 206 2 217 0 227 7 238 4 249. 2 259. 9 270. 6 281. 4 292 1 22 184 1 195 5 206 8 218 1 229 5 240 8 252. 2 263. 5 274. 8 286. 2 297 5 23 183 5 195 5 207 4 219. 4 231 4 243 3 255. 3 267. 3 279. 2 291. 2 303 2 24 182 9 195. 5 208 1 220. 8 233. 4 246 0 258. 6 271. 3 283. 9 296 5 309 1 25 18Z3 195^6 208^9 222^2 235^5 248^8 262^2 275! 5 288^8 302^1 315.' 4 26 181 7 195 7 209 7 223 8 237 8 251 8 265 9 279. 9 293. 9 308 0 322 0

27 181 1 195 9 210 6 225 4 240 2 255 0 269 8 284. 6 299. 3 314 1 328 9 28 180. 5 196 0 211. 6 227 2 242 7 258 3 273. 9 289. 5 305. 0 320. 6 336 2

29 179 9 196. 3 212. 7 229 0 245. 4 261 8 278 2 294 6 311 0 327. 4 343 8 30 179^3 196*5 213^8 23L0 248^3 265*5 282^8 300^0 317^3 334^5 351.' 8

31 178 7 196 8 215 0 233 1 251 3 269 4 287 6 305 7 323. 8 342 0 360 1 32 178 1 197 2 216 3 235 3 254 4 273 5 292. 6 311. 7 330. 7 349. 8 368 9 33 177. 5 197 6 217 6 237 7 257 7 277 8 297. 8 317. 9 337. 9 258. 0 378 1 34 176. 9 198. 0 219 1 240. 1 261 2 282 3 303. 4 324 4 345 5 366. 6 387 6 35 176^4 198^5 220*. 6 242^7 264^9 287* 0 309^ 1 331.* 3 353^4 375.* 5 397^6

36 175.8 199.0 222.3 245.5 268.7 291.9 315.2 338.4 361.6 384.9 408.1 37 175.2 199.6 224.0 248.4 272.7 297.1 321.5 345.9 370.2 394.6 419.0

38 174.7 200.2 225.8 251.4 277.0 302.5 328.1 353.7 379.2 404.8 430.4 39 174.

200.9 227.7 254.5 281.4 308.2 335.0 361.8 388.6 415.4 442.3 40 173.5 201.7 229.8 257.9 286.0 314.

342.2 370.3 398.4 426.5 454.6

41 173.0 202.4 231.9 261.4 290.8 320.3 349.7 379.2 408.6 438.1 467.5

42 172.4 203.3 234.

265.0 295.9 326.7 357.6 388.4 419.3 450.

481.0

43 171.9 204.2 236.5 268.8 301. 1 333.4 365.7 398.0 430.3 462.6 494.9

44 171.4 205.2 239.0 272.8 306.6 340.4 374.2 408.

441.9 475.7 509.5

45 170.8 206.2 241.6 277.0 312.3 347.7 383.1 418.5 453.9 489.2 524.6
400 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Table 9.32. Refractivity for a pressure of 850 mhar

Relative humidity

ature

0

1 n on 40 50 60 70 80 90 100

° C

—45 289. 3 289. 4 289. 5 289. 6 289. 6 289. 7 289. 8 289. 9 289. 9 290. 0 290.

1

—44 288. 0 288.

1

288. 2 288. 3 288. 4 288. 5 288. 6 288. 7 288. 7 288. 8 288. 9 —43 286. 8 286. 9 287. 0 287.

1

287. 2 287. 3 287. 4 287. 5 287. 6 287. 7 287. 8 —42 285. 5 285. 6 285. 8 285. 9 286. 0 286.

1

286. 2 286. 3 286. 4 286. 5 286. 6 -41 284.3 284.4 284.5 284.7 284.8 284.9 285.0 285.1 285.3 285.4 285.5 —40 283.

1

283. 2 283. 4 283. 5 283. 6 283. 7 283. 9 284. 0 284.

1

284. 3 284. 4 —39 281. 9 282. 0 282. 2 282. 3 282. 5 282. 6 282. 7 282. 9 293. 0 283. 2 283. 3 —38 280. 7 280. 8 281. 0 281. 2 281. 3 281. 5 281. 6 281. 8 281. 9 282.

1

282. 3 —37 279. 5 279. 7 279. 8 280. 0 280. 2 280. 4 280. 5 280. 7 280. 9 281. 0 281. 2 -36 278.3 278.5 278.7 278.9 279.1 279.3 279.4 279.6 279.8 280.0 280.2 —35 277. 1 277. 3 277. 6 277. 8 278. 0 278. 2 278. 4 278. 6 278. 8 279. 0 279. 2 —34 276. 0 276. 2 276. 4 276. 7 276. 9 277.

1

277. 3 277. 6 277. 8 278. 0 278. 2 —33 274. 8 275. 1 275. 3 275. 6 275. 8 276. 1 276. 3 276. 6 276. 8 277.

1

277. 3 —32 273. 7 274. 0 274. 2 274. 5 274. 8 275. 0 275. 3 275. 6 275. 9 276.

1

276. 4 -31 272.6 272.9 273.2 273.4 273.7 274.0 274.3 274.6 274.9 275. 2

j 275.5 —30 271. 4 271. 8 272. 1 272. 4 272. 7 273. 0 273. 4 273. 7 274. 0 274. 3 274. 7 —29 270. 3 270. 7 271. 0 271. 4 271. 7 272. 1 272. 4 272. 8 273.

1

273. 5 273. 8 —28 269. 2 269. 6 270. 0 270. 4 270. 8 271.

1

271. 5 271. 9 272. 3 272. 7 273. 0 —27 268.

1

268. 5 269. 0 269. 4 269. 8 270. 2 270. 6 271. 0 271. 4 271. 9 272. 3 -26 267.0 267.5 267.9 268.4 268.8 269.3 269.8 270.2 270.7 271.1 271.6 —25 266. 0 266. 5 266. 9 267. 4 267. 9 268. 4 268. 9 269. 4 269. 9 270. 4 270. 9 —24 264. 9 265. 4 266. 0 266. 5 267. 0 267. 6 268.

1

268. 6 269. 2 269. 7 270. 2 —23 263. 8 264. 4 265. 0 265. 6 266.

1

266. 7 267. 3 267. 9 268. 4 269. 0 269. 6 —22 262. 8 263. 4 264. 0 264. 7 265. 3 265. 9 266. 5 267. 2 267. 8 268. 4 269. 0 -21 261.7 262.4 263.

1

263.8 264.4 265.

1

265.8 266. 5 267.2 267.8 268.5 —20 260. 7 261. 4 262. 2 262. 9 263. 6 264. 4 265.

1

265. 8 266. 6 267. 3 268. 0 — 19 259. 7 260. 5 261. 3 262.

1

262. 8 263. 6 264. 4 265. 2 266. 0 266. 8 267. 6 — 18 258. 7 259. 5 260. 4 261. 2 262.

1

262. 9 236. 8 264. 6 265. 5 266. 4 267. 2 — 17 257. 7 258. 6 259. 5 260. 4 261. 3 262. 3 263. 2 264.

1

265. 0 266. 0 266. 9 -16 256.7 257.6 258.6 259.6 260.6 261.6 262.6 263.6 264.6 265.6 266.6 — 15 255. 7 256. 7 257. 8 258. 9 259. 9 261. 0 262.

1

263. 2 264. 2 265. 3 266. 4 — 14 254. 7 255. 8 257. 0 258.

1

259. 3 260. 4 261. 6 262. 8 263. 9 265.

1

266. 2 — 13 253. 7 254. 9 256. 2 257. 4 258. 7 259. 9 261. 2 262. 4 263. 6 264. 9 266.

1

—12 252. 7 254.

1

255. 4 256. 7 258.

1

259. 4 260. 7 262.

1

263. 4 264. 8 266.

1

— 11 251. 8 253. 2 254. 6 256.

1

257. 5 258. 9 260. 4 261. 8 263.

3

264. 7 266.

1

— 10 250. 8 252. 3 253 9 255. 4 257. 0 258. 5 260.

1

261. 6 263. 2 264. 7 266. 2 -9 249^8 25l'5 253^2 254'

8

256^5 258! 1 159'

8

261 '5 263! 1 264^8 266.4 -8 248.9 250.7 252.5 254.2 256.0 257.8 259.6 261.4 263.1 264.9 266.7 -7 248.0 249.9 251.8 253. 7 255.6 257.5 259.4 261.3 263.2 265.1 267.

1

-6 247.0 249.1 251.

1

253.2 255.2 257.3 259.3 261.4 263.4 265.4 267.5 -5 246.1 248.3 250.5 252.7 254.9 257.1 259.3 261.5 263.6 265.8 268.0 -4 245.2 247.5 249.9 252.2 254.6 256.9 259.3 261.6 264.0 266.3 268.6 -3 244.3 246.8 249.3 251.8 254.3 256.8 259.3 261.9 264.4 266.9 269.4 -2 243.4 246.1 248.8 251.4 254.1 256.8 259.5 262.2 264.8 267.5 270.2 -1 242.5 245.4 248.2 251.1 254.0 256.8 259.7 262.6 265.4 268.3 271.1 0 241.6 244.7 247.7 250.8 253.8 256.9 260.0 263.0 266.

1

269.

1

272.2
REFRACTIVITY TABLES

Table 9.33. Refractivity for a pressure of 850 mbar

401

Relative humidity

ature

0

1 fi lU 20 oU 40 50 fin 70 80 on lOU

° C

0 241.6 244.7 247.7 250. 8 253. 8 256. 9 260.0 OfiQ ZOO. u 266.1 269.

1

272.2

240 7 244 0 247.3 250 5 253. 8 257. 1 260 3 263. 6 266.8 270 1 o 239 9 243 3 246. 8 250 3 253. 8 257. 3 260 7 264. 2 267.7 271 2 274 7 239 0 242 7 246.4 250 1 253. 8 257.5 261. 3 265. 0 268. 7 272 4 276 1 4 ^oo. 1 242 1 246.0 250 0 253. 9 257. 9 261 9 265. 8 269. 8 273 7 277. 7

5 237.3 24L5 245. 7 249^9 254. 1 258.3 262! 5

Ofifi zoo. 7 271. 0 275! 2 279.4

ft0 zoo. 4 240 9 245.4 249 9 254. 3 258. 8 363 3Ofi7 ZD/. a0 272. 3 276 8

Ofil 0 Z8i. Z 235 6 240 3 245.1 249 9 254.6 259.4 264 2 268. 9 273. 7 278 5

Ofl"i 0 ZOO. z

Q0 234 7 239 8 244. 9 249 9 255.0 260.

1

265 1 270 2 275. 3 280 3 285 4

g 233 9 239 3 244.7 250 1 255.4 260. 8 266. 2 271. D 277. 0 282 4 287 8 10 233^1 238^8 244. 5 250^2 256.0 261.7 267! 4 273 1 278. 8 284! 5 280!3

1

1

11 OQO Q 238 3 244.4 250 5 256.5 262. 6 268 7 074 Z/4 a0 280. 8 286 9 OQQ n zyo. u

1 0 231 4 237 9 244.3 250 8 257.2 263. 6 270 1 276. 5 283 0 289 4 zyo. y iO 230 6 237 5 244. 3 251 1 258. 0 264. 8 271 6 278 4 285. 3 292 1 298 9 14 229 8 237 1 244. 3 251 5 258. 8 266. 0 283 3 280 5 287. 7 295 0 302 2 15 229! 0 236'. 7 244.4 252^0 259. 7 267. 4 275! 0 282 7 290 4 298! 1 305! 7 ID ooa 0 236 4 244.5 252 6 260. 7 268. 8 277 0 z85 1 293 2 301 3 Qno r; ouy. 0

1 71 / 007 A 236 0 244. 6 253 2 261. 8 270. 4 279 0 287 6 296 2 304 8 010. i

1 Q18 226 7 235 8 244. 9 253 9 263. 0 272. 1 281 2 290 3 299 4 308 5 317 6

1Qly 225 9 235 5 245.1 254 7 264 274. 0 283 6 293 2 302 8 212 4 322 0

20 225! 1 235^3 245. 4 25516 265! 8 275. 9 286! 1 296 3 306 4 316! 6 326! 7 Zl OOA 4 Zz4. * zoo. 1 245. 8 256 6 267. 3 278 0 288 8 OQQ zyy 5 310. 2 321 0

QQ1 7 001. / OOQ R 0Q4 Q zo^. y 246. 3 257 9 268. 9 280 3 291 6 302 9 314. 3 325 6 QQ7 n 00/. U

OQ 000 Q 234 8 246. 8 258 7 270. 7 272 7 294 6 306 6 318. 6 330 5 242 5 04 222 1 234 7 247. 3 260 0 272. 6 285. 2 297 8 310 4 323 1 335 7 348 3 25 221'

3

234! 7 248. 0 26L3 274. 6 287. 9 301! 2 314 5 327. 8 341! 1 354! 5 OA oon ft OQA fi Z01. D 248. 7 262 7 276. 7 280 9 0U4. 0 0I8 8 332 9 346 9

Qfin Q ODU. y 07 01 Q Q ziy. y 234 7 249. 4 264 2 279. 0 293 8 308 6 323 4 338 1 352 9

Qfi7 7 OD/. / Ofi 219 1 234 7 250. 3 265 8 281. 4 297. 0 312 6 328 1 343 7 359 3

Q7/1 fi 0/^. 8

Ofi 218 4 234 8 251. 2 267 6 284. 0 300. 4 316 8 333 2 349. 6 365 9 382 3 30 217^7 234^9 252. 2 269^4 286. 7 303 9 321! 2 338 4 355. 7 372! 9 390! 2 ol 01 7 n Zl /. u zoo. 1 253. 3 271 4 289. 6 307. 7 ozo. 0 o44 0 362. 1 oou. 0

QQfi A oy8. 4

<10 oZ 01

R

ZID. 0 zoo. 0 254. 4 z/o. 0 292. 6 311. 7 Q^n 7 oou. / 249. 8 368. 9 laa n 000. u

/in7 1 4U/. 1 OO 01 <; fi zio. 0 235 6 255. 7 275 7 295. 8 315. 8 335 9 355. 9 376. 0 296 0 416 1 34 214 9 235 9 257. 0 278 1 299. 1 320 2 341 2 362. 3 383. 4 404 5 425 5 35 214*2 236! 3 258. 4 280^5 302. 7 324. 8 346! 9 369. 0 391. 2 413! 3 435! 4 36 213.5 236.7 259. 9 283.2 305. 4 329. 6 352.8 376. 1 399.3 422.5 445.8

37 212.8 237.2 261. 5 285.9 310. 3 334. 7 359.0 383. 4 407. 8 432.2 456. 5 38 212.

1

237.7 263. 2 288.8 •314. 4 340. 0 365.5 391. 1 416.7 422.2 467.8 39 211.4 238.2 265. 0 291.9 318. 7 345. 5 372.2 399. 1 425. 9 452.7 479.6

40 210.7 238.8 267. 0 295.

1

323. 2 351. 3 379.4 407. 5 435. 6 463.7 491.8

41 210.1 239.5 269. 0 298.4 327. 9 357. 3 386.8 416. 2 445.7 475.1 504.6 42 209.4 240.2 271. 1 302.0 332. 8 363 7 394.5 425. 4 456.2 487.

1

517.9

43 208.7 241.0 283. 3 305.6 338. 0 370. 3 402.6 434. 9 467.2 499.5 531.8

44 208.1 241.9 275. 7 309.5 343. 3 377. 1 411.0 444. 8 478. 6 512.4 546.2 45 207.4 242.8 278. 2 313.6 348. 9 384. 3 419.7 455.

1

490.5 525.8 561.2
402 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Table 9.34. Refractivity for a 'pressure of 1,000 mbar

Relative humidity

ature

0 10 20 30 40 50 60 70 80 90 100

° C

-45 340.4 340.4 340. 5 340.6 340. 7 340.8 340.8 340.9 341.0 241. 1 341.

-44 338.9 339.0 339.0 339.

339. 2 339.2 339.4 339.5 339.6 339.7 339. 7 -43 337.4 337. 5 337. 6 337. 7 337.8 337. 9 338. 0 338.

338.2 338.3 338.4 -42 335.9 336.0 336. 1 336.3 336.4 336. 5 336.6 336.7 336.8 336.9 337.0 —41 334. 5 334. 6 334. 7 334. 8 335. 0 335.

335. 2 335. 3 335. 4 335. 5 335. 7 -40 333.0 333.2 333.3 333.4 333.6 333.7 333.8 334. 0 334. 1 334.2 334.3 -39 331. 6 331.8 331. 9 332.

332. 2 332.3 332. 5 332.6 332.8 332.9 333.

-38 330. 2 330.4 330. 5 330.7 330.8 331.0 331.2 331.3 331. 5 331.6 331.8 -37 328.8 329.0 329.2 329.3 329. 5 329.7 329.8 330.0 330.2 330.4 330.5 —36 327. 4 327. 6 327. 8 328. 0 328. 2 328. 4 328. 6 328. 7 328. 9 329.

329. 3 -35 326. 1 326.3 326. 5 326. 7 326.9 327.

327.3 327. 5 327. 7 327.9 328. 1 -34 324. 7 324.9 325. 1 325.4 325. 6 3-25.8 326.0 326.3 326. 5 326. 7 326.9 -33 323.3 323.6 323.8 234.

324.3 324.6 324.8 235. 1 325.3 325. 6 325.8 -32 322.0 322.3 322. 6 322.8 323.

323.3 323.6 323.9 324. 2 324.4 324. 7 —31 320. 7 321. 0 321. 3 321. 5 321. 8 322.

322. 4 322. 7 323. 0 323. 3 323. 6 -30 319.3 319. 7 320.0 320. 3 320. 6 320. 9 321. 3 321. 6 321. 9 322.2 322. 6 -29 318.0 318.4 318.7 319.

319.4 319.8 320.320. 5 320.8 321.2 321.5 -28 316. 7 317. 1 317.5 317.9 318.3 318.6 319. 0 319.4 319.8 320. 2 320.5 -27 315. 4 315. 9 316.3 316.7 317.

317.5 317. 9 318.4 318.8 319. 2 319.6 —26 314. 2 314. 6 315. 1 315. 5 316. 0 316. 4 316. 9 317. 3 317. 8 318. 2 318. 7 -25 312.9 313.4 313.9 314.4 314.9 315.4 315.8 316.3 316.8 316.3 317.8 -24 311.6 312. 2 312. 7 313.2 313.8 314.3 314.8 315.4 315. 9 316.4 317. 0 -23 310.4 311.0 311.6 312.

312. 7 313.3 313.9 314.4 315.0 415.6 316.2 -22 309. 2 309.8 310.4 311.0 311.7 312.9 313.5 314.2 314.2 314.8 315.4 —21 307. 9 308. 6 309. 3 310. 0 310. 6 311. 3 312. 0 312. 7 313. 3 314. 0 314. 7 -20 306.7 307.6 308.2 308.9 309.6 310. 4 311.

311.8 312.6 313.3 314.0 -19 305.5 306.3 307. 1 307.9 308.7 309.5 310.3 311.0 311.8 312.6 313.4 -18 304.3 305.2 306.0 306.9 307.7 308.6 309.4 310.3 311.

312.0 312.9 -17 303.

304.0 305.0 305.9 206.8 307.7 308.7 309.6 310.5 311.4 312.3 — 16 301. 9 302. 9 303. 9 304. 9 305. 9 306. 9 307. 9 308. 9 309. 9 310. 9 311. 9 -15 300.8 301.8 312.9 403.0 305.

306.307.2 308.3 309.4 310.4 311.5 -14 299.6 300.8 301.9 303.1 304.2 305.4 306.5 307.7 308.9 310.0 311.2 -13 298.5 299.7 300.9 302.2 303.4 304.7 305.9 307.2 308.4 309.7 310.9 -12 297.3 298.7 300.0 301.3 302.7 304.0 305.3 306.7 308.0 309.4 310.7 -11 296.2 297.6 299. 1 300.5 301.9 303.4 304.8 306.2 307.7 309.

310.6 -10 295.

296. 6 298.299. 7 301. 2 302. 8 304. 3 305. 9 307. 4 309. 0 310. 5 -9 203.9 295.6 297.3 298.9 300.6 302.2 303.9 305.5 308.2 308.9 310.5 -8 292.8 294.6 296.4 298.2 299.9 301.7 303.5 305.3 307.

308.8 310.6 -7 291.7 293.6 296.5 297.5 299.4 301.3 303.2 305.

307.0 308.9 310.8 -6 290.6 292.7 294.7 296.8 298.8 300.9 302.9 305.0 307.0 309.0 311.1 -5 289.6 291.7 293.9 296. 1 298.3 300.5 302.7 304.9 307. 1 309.3 311.5 -4 288.5 290.8 293.2 295.5 297.9 300.2 302.5 304.9 307.2 309.6 311.9 -3 287.4 289.9 292.4 294.9 297.4 299.9 302.5 305.0 307.5 310.0 312.5 -2 296.3 289.0 291.7 294.4 297. 1 299.8 302.4 305. 1 307.8 310.5 313.2 -1 285.3 288.2 291.0 293.9 296.8 299.6 302.5 305.3 308.2 311. 1 313.9

0 284.2 287.3 290.4 293.4 296.5 299.5 302.6 305.7 308.7 311.8 314.8
REFRACTIVITY TABLES

Table 9.35. Refractivity for a pressure of 1,000 mbar

403

Relative humidity

ature

0 10 20 30 40 50 60 70 80 90 100

° C

0 284.2 287.3 290.4 293 4 296. 5 299.5 302.6 305.7 308. 7 311.8 314.8

1 283.2 286.5 289.7 293 Q 296 299.6 302.8 306.

1

309 312.6 315.9

2 282.2 285.7 289. 1 292 6

269' 1 299. 6 303.

1

306.6 3io! 0 313.5 317.0

3 281.2 284. 9 288. 6 292 3 296. 0 299.7 303.4 307.1 310. 9 314.6 318.3 280 1 284.1 288. 1 292. 0 296. 0 299. 9 303 9 307 8 311. 8 315 7 319 7 5 279.

1

283. 3 287.6 291. 8 296. 0 300.2 304.4 308.6 213. 8 317.0 321.2

6 278. 1 282.6 287.

1

291 g 296 300.5 305.0 309.5 314 Q 318.5 323.0

7 277.

1

281.9 286. 7 29L 4 296^ 2 301.0 305. 7 310.5 315' 3 320.0 324.8 8 276.2 281.2 286. 3 291. 4 296. 7 301. 5 306.6 311.6 316. 7 321.8 326.8

g 275 2 280. 6 285. 9 291. 3 296. 7 302.

1

307 5 213 9 318. 3 323 6 329 0 10 274.2 279. 9 285.6 291. 4 297. 1 302. 8 308.5 314.2 320. 0 325.7 331.4

11 273.2 279. 3 285.4 291 0 297 0 303 g 309.7 315.7 321 g 327.9 334.0

12 272.3 278.7 285.2 29L 6 298'. 0 304 310.9 317.4 323^ 8 330.3 336.7

13 271.3 278.2 285.0 291. 8 298. 7 305! 5 312.3 319. 1 326. 0 332.8 339.6

14 270 4 277. 6 284. 9 292. 1 299. 3 306. 6 313 8 321 1 328. 3 335 5 342 8

15 269.4 277. 1 274. 7 292. 5 300. 1 307. 8 315.5 323.

1

330. 8 338.5 346.1

16 268.5 276. 6 284.8 292 301 Q 309 1 317.2 325.4 333 5 341.6 349.7

17 267.6 276. 2 284. 8 293 4

302' 0 310. 6 319.2 327.8 336^ 4 344.9 353.5

18 266.7 275. 8 284. 9 293 9 303 0 312. 1 321.2 330.3 339. 4 348.5 357.6

19 265. 8 275. 4 285.0 294 6 304. 2 313. 8 323. 4 333.

1

342. 7 352. 3 361. 9 20 264.8 275. 0 285. 2 295 3 305 5 315. 8 325.8 336.0 346. 1 356.3 366.5

21 363.9 274. 7 285. 4 296 2 306 9 317. 6 328.4 339. 1 349. 8 360. 6 371.3

22 263.1 274. 4 285. 7 297 1 308. 4 319 7 331.

1

342.4 353. 7 365.1 376.4

23 262.2 274. 1 286. 1 298 1 310 0 322. 0 334.0 345.9 357. 9 369.8 381.8

24 261. 3 273. 9 286. 5 299 1 311. 8 324. 4 337. 0 349. 6 362. 3 374. 9 387. 5 25 260.4 273. 7 287 0 300 3 313 6 327 0 340.3 353.6 366. 9 380.2 393.5

26 259.5 273. 6 287. 6 301 g 315 y 329. 7 343.7 357.8 371. 8 385.8 399.8

27 258.7 273. 5 288 2 303 0 317 8 332 6 347.4 362.2 276. 9 391.7 306.5

28 257.8 273. 4 288. 9 304 5 320 1 335. 8 351.2 366.8 382. 4 397.9 413.5

29 257. 0 273. 3 289. 7 306 1 322. 5 338. 9 355 3 371. 7 388. 1 404. 5 420. 9 30 256. 1 273. 4 290. 6 307 9 325 1 342 4 359.6 276.9 384. 1 411.4 428.6

31 255.3 273. 4 291. 6 309 7 327 8 346. 0 364. 1 382.3 400. 4 418.6 436.7

32 254.4 273. 5 292. 6 311 7 330. 7 349. 8 368.9 388.0 407. 1 426.

1

445.2

33 253.6 273. 6 293 7 313 8 333 8 363. 9 373.9 394.0 414. 0 434.

1

454.1

34 252.8 273. 8 294. 9 316. 0 337. 0 358. 1 379.2 444.3 421.3 442.4 463.5

35 251. 9 274. 1 296. 2 318 3 340. 5 362. 6 384. 7 406. 8 429.0 451. 1 473. 2 36 251.

1

274. 4 297. 6 320 8 344. 1 367. 3 390.5 413.7 437.0 460.2 384.4

37 250.3 274. 7 299. 1 323 5 347. 8 372.2 396.6 421.0 445.3 469.7 494.

1

38 249.5 275. 1 300. 7 326 2 351. 8 377.4 403.0 428.5 454.

1

479.7 505.2 39 248.7 275. 5 302. 3 329 2 256. 0 382.8 409.6 436.4 463.2 490.

1

516.9

40 247.9 276. 0 304. 1 332 2 360. 4 388.5 416. e; 444. 7 472. 8 500.9 529.0

41 247. 1 276. 6 306. 0 335. 5 364. 9 394.4 423.9 453.3 482. 8 512.2 541.7 42 246.3 277. 2 308. 1 338. 9 369. 8 400.6 431.5 462. 3 493.2 524.0 544.9

43 245.6 277. 9 310. 2 342. 5 374. 8 407. 1 439.4 471.7 504.0 536.3 568.6 44 244.8 278. 6 312.4 246. 2 380. 0 413. 9 447.7 481.5 515.3 549. 1 582.9

45 244.0 279. 4 314. 8 350. 2 385.5 420.9 456.3 491.7 527. 1 562.5 597.8
404 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

9.4. Climatological Data of the Refractive Index

for the United States

Charts of 8-year mean A^o values for 0200 and 1400 local time are given

for the four seasons of the year on figures 9.1 to 9.8. Selected stations illustrate the range of seasonal and diurnal variations

of A^s in differing climatic regions across the country. These are plotted on figures 9.9 to 9.14. Note the large annual and diurnal ranges of means

for humid subtropical Washington, D.C. (fig. 9.9), compared with the

modest variations exhibited by maritime-dominated Tatoosh Island

(fig. 9.14). The results of these analyses are consistent with those obtained from climatic studies of the refractive index structure over North

America, described in section 4 of chapter 4. That is, large seasonal and

diurnal ranges are exhibited by continental-climate stations (Colorado

Springs) and small ranges by maritime-climate stations (Tatoosh Island).

Cumulative distribution curves for Ns, useful in radio ray bending predictions, are given in figures 9.15 to 9.20 for the example stations above.

Figures 9.21 and 9.22, standard deviation of Ns, provide a measure of the accuracy of the A^o charts of figures 9.1 to 9.8. As has been noted

in chapter 4, a number of climatic features are apparent on the standard

deviation maps. The climatic stability of various regions of the country,

for example, is reflected in these charts. Small standard deviations

characterize the maritime climate of the west coast. By comparison, the

strong air mass changes of wintertime synoptic patterns sweeping across the southeastern United States are indicated by large standard deviations in that region (fig. 9.21).

9.5. Statistical Prediction of Elevation Angle Error

The elevation angle error, e, as defined in chapter 3, can, like bending,

be predicted from surface refractivity, Ns, by an equation of the form

e = mNs + t

In tables 9.36 to 9.44 for a given height h, the coefficients m and t have

been determ.ined by a statistical regression performed on the Standard

CRPL Sample, each having a unique N

This process is analogous to the statistical bending prediction method

described in chapter 3.
ELEVATION ANGLE ERROR 405

Figure 9.2. Mean No, February I4OO.
406 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Figure 9.4. Mean No, May I4OO.
ELEVATION ANGLE ERROR 407

Figure 9.6. Mean No, August I4OO.
408 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Figure 9.8. Mean No, November I4OO.
ELEVATION ANGLE ERROR 409

Figure 9.9. Annual and diurnal cycles 0/ Ns for Washington, D.C.
410 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

JAN MAR MAY JULY SEPT NOV JAN JAM NAD NAY JULY SEPT NOV JAN Month of the Year Month of the Year

0400 0800 1200 IGOO 2000 2K0 0400 0600 1200 1600 2000 2400 Hour of the Day Hour of the Day Figure 9.10. Annual and diurnal cycles of Ng/or San Antonio, Tex.
ELEVATION ANGLE ERROR 411

Figure 9.11. Annual and diurnal cycles of Ns /or Bismarck, N.Dak.
412 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

JMI IMR WY MY SEPT NW M( JMN MM) WY JULY SEFT NOV JAN Month of the Year Month of the Yeor

0000 0400 0800 eOO MOO 2000 2400 0000 0400 0800 1200 OO 2000 2400 Hour of the Day Hour of the Day Figure 9.12. Annual and diurnal cycles of Ns /or Colorado Springs, Colo.
ELEVATION ANGLE ERROR 413

Hour of the Day Hour of the ,Day Figure 9.13. Annual and diurnal cycles of Ns /or Salt Lake City, Utah.
414 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

02C 0 U col 7 me

/

Nc

MAR m UY SEPT Month of the Year

I4C 0 LoCQ.Tme

V

I

- _

m JULY SEPT Month of the Year

Febr lary

Max. A

-0-

Min.

0«00 1200 H Hour of the Day

August

0800 1200 II Hour of the Day Figure 9.14. Annual and diurnal cycles of ^afor Tatoosh Island, Wash.
ELEVATION ANGLE ERROR 415

August

= 0200 n=248-

= 08O0 n=248

= 1 400

lonr 248 248- \\

!

-

—

s

1

)l 10 30 50 70 90

1 98 99.5 99

5 20 40 6 0 80 9 5 99 9 9.9 0.1 I 5 20 40 60

P OF N, > ORDINATE VALUE

95 99 99.9

Figure 9.15. Cumulative probability distribution of Ns, Washington, D.C.
416 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
ELEVATION ANGLE ERROR 417

Ft bruc

i

1 0200 n=226- 0800 n=223

=l4O0 n= =9fVV1 n= 226

226 —

330 — I 1 0200 n= 0600 n= 248" 248

= 1400 n=248 —

>

tove n « 1 1 III

= 0200 n=240' =0800 n=240

= 1400 n=240 =9fWl n^^iin

•v.

01 0.5| 2 10 30 50 70 90 98 99.5 99

5 2 0 4 0 60 8 0 95 99 99.9 0.1 I 5 20 40 60 80 95 99 99.9 P OF N, > ORDINATE VALUE Figure 9.17. Cumulative prohahility distribution of Ns, Bismarck, N.Dak.
418 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Feb ru or y 1 1 n II 1 200 n=226- -0800 n=225 -2OC)0 n = 226

\

M ly 1 1 02C8C

4Ca

1 )0 n = )0 n =

)0 n = )0 n =

248-

248

248 - 248

\

s

0 -2 \

Ns \ \

N

(11 0.5 I 2 5 10 20 30 40 50 60 70 80 90 P of > Ordinate Value

5 10 20 30 40 50 60 TO 80 90 95 98 99 935 999 P of > Ordinate Value

Augus t 20C 8001 n=248-

n = 248 n=248-

n = 248

c Os,

ti, 0(X) \\

\,\ \s \

N\

N ^

Nc ve mber C 2C

8C

4Ca

i 1 )0 n = 240- 0 )0 n = 240 -2 X) n = 240

\\

2 5 10 20 30 40 50 60 70 80 90 95 98 99 995 99.9 P of N,: > Ordinate Value

5 10 20 30 40 50 60 T08C 90 95 98 99 995 993 P of > Ordinate Value Figure 9.18. Cumulative probability distribution of Ng, Colorado Springs, Colo.

Fe brua

1 i II 3200 n= 3800 n= 226

226 226"

226

(10

w.

OC

) n =

n =

300

M >y Mill 0200 n = 248~

0800 n= 248

OC

x;

X) n = 2 48

A u t

1 I II

(3200 n=248"

3800 n=248 400 n = 248"

2000 n = 248 H

N

S S

01 0.5 2 10 30 50 70 90 98 99.5 99

tove n Jer

1 III

0200 n = 240~

0800 n=24C -2

4< OC)0 n=i

n=2 4040

01 0.5 2 10 30 50 70 90 98 99.5 99

5 2 0 40 60 8 0 9 5 99 99.9 0.1 I 5 20 40 60 80 95 99 99.9

P OF N. > ORDINATE VALUE Figure 9.19. Cumulative probability distribution of Ns, Salt Lake City, Utah.
ELEVATION ANGLE ERROR 419

Figure 9.21. Standard deviation of Ns, February 0200.
420 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Figure 9.22. Standard deviation of Ns, August 0200.

Tables 9.36 to 9.44

These tables show values of m and i in the regression equation

e = mNs + ^, where:

€ is the elevation angle error in milHradians to the height h, Ns is the value of the radio refractivity at the earth's surface,

m the change in elevation angle error per unit change in Ns, i is the zero intercept,

r is the correlation coefficient,

SE is the standard error of prediction using the regression line, ^0 is the initial elevation angle in milliradians.

Table 9.36. Variables in the statistical method of elevation angle error for h = 0.1 km

do r m I SE

0. 0 0. 2637 0. 0243 -4. 3954 3. 4429

, 1.0 .2760 .0131 -2. 1055 1. 7712

• 2. 0 .2855 .0083 -1.2185 1. 0837

5. 0 .3017 .0037 -. 4741 0. 4561

10.0 . 1609 .0027 -. 2834 .6501

20. 0 . 1777 .0012 -.1175 .2706

52.4 . 1805 .0005 -. 0426 . 0998

100.0 . 1824 .0002 -. 0227 .0518

200. 0 . 1815 .0001 -. 0110 . 0256

400. 0 . 1844 .0001 -. 0055 . 0123
TABLES 9.36 TO 9.44 421

Table 9.37. Variables in the statistical method of elevation angle error for h = 0.2 km

do r m I SE

0. 0 0. 2739 0. 0310 -5. 5364 4. 2249

1.0 .2838 .0191 -3. 0776 2. 4984

2. 0 .2927 .0132 -1.9685 1. 6702

5.0 .3146 .0065 -. 8506 .7628

10. 0 .1919 .0044 -. 4623 .8906

20. 0 .2125 .0021 -. 1941 .3757 52.4 .2168 .0008 -. 0705 . 1385

100. 0 .2175 .0004 -. 0367 .0721

200. 0 . 2170 .0002 -.0179 .0356

400. 0 .2170 .0001 -.0085 .0170

Table 9.38. Variables in the statistical method of elevation angle error for h = 0.5 km

do r m I SE

0.0 0. 3255 0. 0430 -8. 0534 4. 8023

1.0 .3457 .0289 -5. 0198 3. 0131 2.0 .3661 .0216 -3. 5807 2. 1131 5.0 .4197 .0125 -1.9306 1. 0427

10.0 .2887 .0083 -1.0741 1.0693

20. 0 .3304 .0041 -.4899 .4568

52. 4 .3418 .0016 -. 1803 .1679

100. 0 .3430 .0008 -.0934 .0872 200.0... .3439 .0004 -. 0463 .0431

400. 0 .3408 .0002 -. 0218 .0208

Table 9.39. Variables in the statistical method of elevation angle error for h = 1.0 km

r m I SE

0.0 0. 3371 0. 0472 -8. 2579 5. 1154 1.0 .3772 .0339 -5. 5583 3. 2281 2.0 .4180 .0270 -4. 2916 2. 2775 5.0 .5150 .0179 -2. 7591 1. 1590 10.0 .3983 .0123 -1.6158 1. 1044 20.0--.... .4831 .0066 -.8154 .4703

52.4 .5129 .0026 -.3070 . 1714

100. 0 .5167 .0014 -. 1595 .0890

200. 0 .5177 .0007 -. 0787 .0440 400.0. .5180 .0003 -. 0379 .0211

Table 9.40. Variables in the statistical method of elevation angle error for h = 2.0 km

do r m I SE

0.0 0. 3795 0. 0567 -9. 8908 5. 3659

1.0 .4413 .0427 -7. 0696 3.3704 2.0 .5020 .0355 -5. 7509 2. 3735

5.0 .6351 .0258 -4. 1251 1. 2198

10.0 .5642 .0188 -2. 7184 1. 0773

20. 0 .7007 .0113 -1.5650 .4499

52.4 . 7531 .0047 -. 6256 . 1615

100. 0 .7608 .0025 -.3279 .0835

200. 0 .7633 .0012 -. 1627 .0412

400. 0 .7643 .0006 -.0784 .0198
422 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS

Table 9.41. Variables in the statistical method of elevation angle error for h = 5.0 km

do r m I SE

0. 0 0. 4324 0. 0704 -12. 0857 5. 6994

1.0 .5186 .0554 -9. 0682 3. 5425

2. 0 .5983 .0477 -7. 6679 2. 4766

5. 0 .7543 .0372 -5. 9085 1. 2570

10. 0 .7424 .0290 -4. 3100 1. 0255

20. 0 .8789 .0194 -2. 7570 .4118

52. 4 .9248 .0089 -1. 1721 . 1431 100.0 .9316 .0048 -. 6154 .0733

200. 0 .9336 .0024 -. 3043 .0360

400. 0 .9339 .0012 -. 1460 .0173

Table 9.42. Variables in the statistical method of elevation angle error for h = 10.0 km

do r TO I SE

0.0 0. 4617 0. 0797 -13. 2953 5. 9448

1.0 .5602 .0637 -10.0839 3. 6560

2.0 .6475 .0555 -8. 5935 2. 5360

5. 0 .8057 .0443 -6. 6912 1. 2652

10. 0 .8125 .0353 -4. 9338 .9913

20. 0 .9301 .0245 -3. 1637 .3779

52.4 .9679 .0118 -1.3002 . 1195

100. 0 .9742 .0065 -. 6607 .0584

200. 0 .9763 .0032 -. 3208 .0281

400. 0 .9766 .0016 -. 1533 .0135

Table 9.43. Variables in the statistical method of elevation angle error for h = W.O km

do r TO I SE

0. 0 0. 4805 0. 0874 -14. 0318 6. 1895

1.0 .5866 .0703 -10. 5942 3. 7658 2.0 .6780 .0616 -8. 9972 2. 5905 5.0 .8351 .0495 -6. 9273 1. 2665

10. 0 .8497 .0397 -5. 0189 .9637

20. 0 .9538 .0278 -3. 0650 .3423

52.4 .9861 .0136 -1.0682 .0899

100. 0 .9917 .0076 -. 4672 .0384 200.0 .9934 .0038 -. 2067 .0172

400. 0 .9939 .0018 -. 0947 .0080

Table 9.44. Variables in the statistical method of elevation angle error for h = 70.0 km

do r TO I SE

0. 0 0. 4999 0. 0985 -15. 2202 6. 6214

1.0 .6137 .0794 -11.3649 3. 9661

' 2.0 .0783 .0698 -9. 5814 2. 6984

5. 0 .8617 .0563 -7. 2363 1. 2867

10. 0 .8811 .0451 -5. 0975 .9476

20. 0 .9701 .0317 -2. 9121 .3102

52.4 .9948 .0158 -. 8032 .0630

100. 0 .9983 .0089 -. 2595 .0202

200. 0 .9992 .0045 -. 0822 .0070

400. 0 .9995 .0022 -. 0319 . 0028
REFERENCES 423 9.6. References

[1] Bean, B. R., and G. D. Thayer (1959), On models of the atmospheric radio refrac- tive index, Proc. IRE 47, No. 5, 740-755.

[2] Smithsonian Meteorological Tables (1951), Table 63, Sixth Revised Ed. (Washington, D.C.).

Subject Index

A

A-unit (potential refractivity), 16-20, 78-79, 166-170, 189-194, 197, 211, 213-227, 379

Absolute humidity, 270, 272, 275, 277-280, 286-290

Absorption coefficient {see also Van Vleck's absorption formulas), 272

Adiabatic refractivity decrease, 258-262

dry, 261

wet, 262

Adiabatic temperature lapse rate, 133, 258

Advection, 132, 201, 205, 213, 319

definition, 132

relative to duct formation, 133, 135

Air masses, 79, 102, 114, 119, 123, 163-164, 167-169, 173-176, 179-181, 191, 195-

197 201 213 221-222

refractive characteristics of, 163, 169-170, 174-175, 211 Angle of penetration (critical angle), 137, 140, 142, 376 Apparent height, 360 Apparent range, 51, 335, 357-360 ARDC Standard Atmosphere, 64

Attenuation of radio waves, 269-309 by fog, 303 by gases, 270-290 by hail, 302-303 by rain (see also Climatology), 292-297

in clouds, 291 B

B-unit, 14-15, 17-20, 189-191, 195 Barium fluoride strip, 26 Bending of a radio ray (see also Refraction)

definition, 49 methods of computing and predicting, 53-81, 375-393

Bi-exponential model

for absorption, 283, 307

for refractivity, 311-322 Birnbaum refractometer, 31, 33-34

Bright band, 303 C

Caustic, 148

Climatic types, relation to A^g, 102-104

Climatology, 89-172

of ground based radio ducts, 42-43, 132-145

of refractivity, N, near the ground (No), 89-109, 111-128, 404-420

of rainfall attenuation, 297-302

Clouds, attenuation in, 291 Cold front (see Frontal zone)

Composite parameter, 238

Continental cross sections of No, 122-128

Continental arctic air mass (see also Continental polar air mass), 102, 316

Continental polar air mass, 79, 123, 163-164, 169, 174-176, 179-180, 196-197,

221-222, 319

Continental tropical air mass, 79, 167-169, 175, 179 Convergence

centers of, 182, 240-241

influencing atmospheric refractivity, 182-185, 240 Core climate, 129 425
426 SUBJECT INDEX

Correlation of field strength with meteorological parameters, 105, 174, 229, 233-266

Crain refractometer, 31-33

Cross-over height, 248-249, 263-264 CRPL Reference Atmosphere (see Exponential atmosphere model for refractivity)

CRPL Standard A^-profile Sample, 335, 338-353, 355-356 Curvature of radio ray, 13, 42, 56, 264, 360-361 Deam refractometer, 36-37

Delta-A^ (A N) (see also Refractivity, vertical gradient), 15-16, 61, 63-66, 73-76,

106, 129-131, 139, 229-230, 342, 363, 365, 371, 373, 389, 394-396

as a field strength predictor, 234-238, 240, 243-244, 246, 264-266

usage for 24-hour horizontal changes, 201-204, 209-211

Departures-from-normal method of computing refraction, 77-79, 167-168, 181, 190-

191, 375, 378-379, 381, 392-393

Dielectric constant of air (see also Refractive index), 2-8, 12

Dispersion, 5, 8 Diurnal index (diurnal ratio), 129-130

Diurnal range

of field strength, 105, 235-236

of No, 111-121, 129-130, 204

Divergence

centers of, 182, 240-241

influencing atmospheric refractivity, 182-185, 240 Drop size distribution, 292, 294-297

Dropsonde, 36 Dry term

of atmospheric absorption, 283

of the Smith-Weintraub equation, 30, 91, 93-94, 136, 174-175, 311-322

scale heights of (see Scale height)

surface values of, 93-94, 311-312, 320

Ducting, 42-45, 52, 89, 132-163, 241-246, 248, 324, 328, 331-333, 376 E

Effective earth, 13-15, 51, 56-62, 64-65, 106, 189, 195, 250, 264, 359-362

curvature, 13, 56-57, 62, 106, 195, 361 method of calculating refraction, 56-59

radius, 13-15, 51, 56-57, 59-62, 64-65, 106, 189, 195, 250, 264, 359-362

Elevated layer, 241, 244-258, 324-329, 331

reflection coefficient of, 252-257

Elevation angle (see also Refraction)

approximation for, 82-83

definition, 49-50

errors, 49-50, 178, 333-335, 340-342, 344-345, 348-350, 356, 404, 420-422

prediction of, 174, 334, 340-342, 344-345, 348-350, 356, 404, 420-422 Epoch chart, 221-227 Equation of continuity, 183 Equivalent gradient, 236-239, 243, 246, 264-266

as a field strength predictor, 236-239, 243, 246, 265 method of calculation, 264

Exponential atmosphere models

for absorption, 283

for refractivity, 15-20, 56, 61, 63-80, 92-93, 106, 110, 149, 153-156, 164, 166-

167, 174, 176-177, 186, 189-190, 192-194, 244-245, 261-263, 311-322,

329-332, 357, 362, 375-378, 385-389, 393-396

in radiosonde correction process, 40-41 method of computing refraction, 375-377, 385-388 Eta (v) parameter, 238-239

F

Fading (see also Field Strength), 146-163, 223

caused by ducting, 146-163

definitions, 147 due to rainfall attenuation, 298-302

Field strength

prediction from radio meteorological parameters, 89, 105-106, 109, 174, 177-180,

229-236, 238-251, 253-258, 263-266
SUBJECT INDEX 427 Fog attenuation, 303

Four-thirds earth {see also Effective earth), 14-15, 20, 51, 57, 59-60, 62, 66-75,

247, 360

Fresnel discontinuity value, 252, 257

Frontal zone, influence on refractivity, 176-177, 187-193, 195-224, 323 G

Geometric range, 335, 357

Gradient of refractive index (see Refractive index)

Gradient of refractivity (see Refractivity)

regression expressions for, 363-364, 372

Graphical method of computing refraction, 80-81, 375, 380-381 Ground-based radio ducts (see also Ducting), 42-44, 132-163, 324, 328, 333, 375 frequency of occurrence, 140-141

thickness of, 140, 143-145 Ground-based superrefractive layers, 250 H

ZT-parameters, 265

Hail attenuation, 302-303

Half-length, 150-154, 160 Hay refractometer, 36-37

Height errors, 356-357, 360-373

definition, 356, 360

regression equations for, 364-372 High angle approximation for computing refraction, 53-54

Horizontal homogeniety, assumption of, 49, 52, 163, 322

Horizontal inhomogenieties, 52, 322-333, 356

Horizontal ray, 329

I ICAO Standard Atmosphere, 9, 11, 312-314

Initial gradient of refractivity (see Delta-A^ and Refractivity)

Intensity factor, 273-274

Inversions (see Ducting and Elevated layers)

K

iiC-unit, 180, 182, 239-240 L

Lag times, 29, 38-45

in electrical thermometers, 26

of relative humidity sensor in radiosondes, 39, 41-45

of temperature sensor in radiosondes, 39, 41-45 Land breeze, relative to duct formation, 134-135, 323

Level of nondivergence, 183

Lifting condensation level, 260

Line widths of atmospheric gaseous absorption (see also Attenuation), 272-275 Lithium chloride humidity strip, 26, 28-29, 39, 45 M

iVf-parameter (see Thermal stability) Macroscale fluctuation of refractive index, 173, 223 Maritime polar air mass, 79, 114, 169, 179, 197 Maritime tropical air mass, 79, 102, 119, 125-126, 164, 167, 169, 174, 177, 179-181,

201, 213, 221-222, 319 Maximum wavelengths trapped by a duct, 141-143 Mesocale fluctuation of refractive index, 173, 185, 223

Microscale fluctuation of refractive index, 173 Mixing ratio, 260 Models

of atmospheric absorption structure, 283

of atmospheres, 9, 11, 16-17, 64, 92-93, 180, 186, 312-314

of refractivity structure, 13-20, 40-41, 51, 56-80, 92-93, 106, 110, 149, 153-156,

164, 166-170, 174, 176-177, 180, 186, 189-195, 197, 213-227, 244-245, 250, 261-264, 311-322, 329, 331-332, 357, 360-362, 375-378, 381, 385-389, 393-396
428 SUBJECT INDEX

Modified effective earth's radius method of computing refraction, 59-65

Modified index of refraction (M-unit), 14-15, 17-20, 193, 195, 324 N

A^-unit (see Refractivity)

NACA Standard Atmosphere, 16-17, 92-93, 186 Nocturnal duct formation (see also Ground-based radio ducts), 133, 135 Nonpolar gases, dielectric constant of, 2-3O

Optical depth, 304 Outage time, 298-301 Oxygen

absorption properties of (see Attenuation by gases)

refractivity at STP, 9 P

Permeability of air, 3-4

Pi-factors (turbulence parameters), 262-266

Polar air mass (see Continental polar air mass)

Polar front (see also Frontal zone), 195-197, 213

Polar wave, 193, 195-197, 213

Polar gases, dielectric constant of, 2-3

Potential refractive modulus (<^-unit), 17, 19-20, 166, 193 Pound oscillator (see Deam refractometer)

Precipitation rate, 292

Profile (gradient) classification (see also Air masses and Refractivity), 248-250

Profiles of refractive index (see Refractive index, vertical profile of) Profiles of refractivity (see Refractivity, vertical profile of) Psychrometric equation, 12, 24 R

Radar

applications involving (see also Rain attenuation), 49, 51, 56, 59, 61, 77, 340, 357 anomalous propagation in (see Ducting)

Radiation, relative to duct formation, 133 Radiative transfer equation, 304 Radio

duct (see Ducting)

holes (see Ducting)

horizon equation for, 59

in ducting, 146-148, 152, 155, 157

refractive index (see Refractive index)

Radiosonde (RAOB), 10-11, 26-30, 36, 38-43, 53-55, 59, 61, 70-72, 74-75, 79, 110-

111, 135, 138-140, 144, 179, 181, 194, 205, 211, 213, 243, 246, 247-249, 251, 253,

255, 258, 264, 280, 283, 326, 363, 375

sensors in, 26-27, 29-30, 39-45

Rain attenuation, 292-297 Range

errors (radio), 12, 174, 335-342, 345, 350-353, 356

definitions, 335-336

prediction of, 341, 345, 350-353

total, 335-337

of a radio ray, 51, 335-336, 357-361 Ray tracing (see also Refraction), 52, 132, 135-136, 148, 322-323

limitations of, 52, 132

Rayleigh-Jeans'law, 304-305

Reflection coefficient (of an elevated layer), 185, 252-257

models of, 252-257

Refraction (angular) of a radio ray, 14-15, 19-20, 49-87, 163-170, 178-181, 190, 192, 236, 266, 311, 319-323, 326-337, 339, 341-350, 357, 375-396

definition, 49

derivation in terms of refractive index, 87
SUBJECT INDEX 429

Refractive index, atmospheric radio {see also Refractivity)

applications of, 12-14, 31-34, 36, 49-87, 136-138, 150-152, 165-166, 174, 223, 231-266, 311, 319-323, 326-329, 335-336, 357, 359-362, 394

definition, 23

dispersion, 5, 8 expressions for, 3-7

general discussion of, 1-8, 16-17, 19-20, 49, 89-90, 92, 103, 109-110, 125, 129,

133, 135, 163-166, 169-170, 173-175, 178, 185-195, 211, 222, 311, 314, 332-

333, 342-343, 356, 404

vertical gradient of {see also Delta-iV), 13-14, 19, 36, 38, 52, 56-57, 64, 77, 106-107,

132, 138-140, 142, 180, 182, 189, 195, 236, 240-241, 248, 264, 319, 357, 361-364

vertical profile of, 14-15, 40, 42-45, 52, 59, 62, 135, 173-176, 179-180, 182, 187, 211, 222, 229, 240, 247-252, 254, 257-258, 331-332, 334, 339-342, 363, 366

Refractivitv {see also Refractive index)

applications of, 12-13, 19-20, 49-87, 105-107, 136-170, 178-181, 190, 229-266,

319-373, 375-396

as a function of height {see Models)

definition, 4, 90, 357

diurnal range {see Diurnal range)

expressions for, 4-7

constants in, 4-11

errors in, 9-12 measurement of, 23-45

accuracv and errors in, 24-32, 34-36, 38-45

near the ground (surface values), 15-20, 36, 42-43, 53-54, 58, 61, 63-80, 89-163,

176-182, 186-187, 192, 194, 197-205, 223, 229-230, 233-237, 239, 243-246,

258, 261-266, 311-312, 319-321, 333-356, 361-363, 365-367, 369-370, 373, 375-388, 392-396, 404-422

of atmospheric constituents carbon dioxide, 5 dry air {see also Dry term), 9 oxygen, 9 water vapor {see also Wet term), 9 potential {see A-unit)

scale heights of {see Scale height)

synoptic variation of {see Frontal zone)

vertical gradient of {see also Delta-.Y), 9-12, 15, 20, 33, 36, 42-43, 61, 63-66,

73-77, 79, 106, 114, 123, 129-131, 136, 143, 145, 147-148, 150, 152, 166,

168, 229-230, 234-241, 243-244, 246, 248, 250, 264-266, 342, 362-363,

365-367, 370-371, 373, 375, 378, 389, 394-396

vertical profile of {see also CRPL Standard N-profile Sample), 15, 17-19, 42-45,

54-55, 57, 59-68, 70-71, 73, 75, 77-79, 111, 115, 117, 122, 129, 135, 138-

139, 146, 162-164, 167-170, 174-177, 194, 211-227, 235, 244-245, 263, 266,

321, 329, 336-339, 341, 344, 375, 381

Refractometer, 8, 23, 31-38, 334 Resonant frequencies (absorption lines) of oxygen, 270-275, 280

of water vapor, 270-275, 280 Rocket Panel data, 64 S

Satellite telecommunications, 65, 175 Saturation vapor pressure, exponential assumption for, 260

Scale height

of absorption, 283

of refractivitv, 15-17, 92-93, 108, 186, 261-263, 311-312, 314-322

of drv term, 93, 261, 263, 311-312, 314-322

of wet term, 93, 262-263, 311-312, 314-322

Scattering bv atmospheric constituents {see also Attenuation) 185, 247-248, 255, 269-270, 292

Schulkin's method of computing refraction, 54-55, 80, 179, 375-376, 390-391 Sea breeze, relative to duct formation, 134-135, 323

Seasonal index (seasonal ratio), 129-130
430 SUBJECT INDEX

Seasonal range

of field strength, 174

of No (see also Climatology), 111-121, 129-130 Shadow zones, 148-150, 152-155, 157, 159-160

Slant range, 335, 360

Snell's law in polar co-ordinates, 49, 84-87, 136-137, 165, 326

derivation, 84-87 Snow attenuation, 302 Space cross sections of refractive index, 211-217, 221 Sprung's formula, 12

Stability

atmospheric, 238-241, 258, 260, 262-263, 266

thermal (see Thermal stability)

Standard atmosphere, 9, 11, 16-17, 64, 92-93, 180, 186, 312-314

Statistical method of computing refraction, 54, 335, 342-350, 379-380, 382-384

Stratopause, appoximate location, 313

Stratosphere

approximate extent, 313

meteorological characteristics of, 313

Subrefraction, 133, 248-249, 329, 332-333

Subsidence, relative to duct formation, 133-135, 180, 182, 185 Superior air mass, 79, 169

Superrefraction (see also Ducting), 133-135, 146-148, 156, 169, 182, 248, 250, 328-329, 332-333

Surface duct (see Ground-based radio ducts)

Synoptic illustration of N change (see Frontal zone)

T

Temperature dependence

of gaseous absorption (see Van Vleck's absorption formulas)

of refractivity, 1-8 Thermal noise temperature, 304-307

definition, 305 Thermal stability, 239-241, 262, 265

Tilted elevated layer, 249-251 Time cross sections of refractive index, 213, 217-221 Tracking system, radio, 334

Transhorizon propagation mechanisms, 248

Transition zone (see Frontal zone)

Trapping of a radio ray (see Ducting)

Tropopause, approximate location, 313 Troposphere

approximate extent, 313

meteorological characteristics of, 313 Turbulence

atmospheric, 260 parameter (see Pi-factors) U

Unstratified layer, 248, 250-251 V

Van Vleck's absorption formulas, 270, 272-275

Velocity of radio propagation, 56

Vertical gradient of N (see Refractivity)

profile of N (see Refractivity)

ray, 329

Vetter refractometer, 31, 34-35

W

Warm front (see Frontal zone) Water vapor

absorption properties of (see Attenuation by gases) an average refractivity value of, 9 density (see absolute humidity)

dependence of refractivity, 1-8
SUBJECT INDEX 431 Weather influence on refractivity (see Frontal zone)

Wet term

of atmospheric absorption, 283

of the Smith-Weintraub equation, 30, 91, 93-94, 136, 174-175, 311-322

scale heights of (see Scale height)

surface values of, 93-94, 311-312, 320 Wind velocity

influence on atmospheric stability, 240-241

vertical component of, 182-184

Wiresonde, 326

Author Index

A

Abbott, R. 270, 274, 288, 308

Abild, V. B. 268 Adey, A. W. 46

Airy, G. B. 148, 172

Akita, K. I. 30, 45

Anderson, L. J. 22, 55, 80, 88, 171, 227, 268

Anderson, W. L. 374 Anway, A. C. 347, 374 Armstrong, H. L. 21 Arnold, E. 268 Artman, J. O. 273, 308

Arvola, W. A. 180, 182, 226, 227, 240, 266 Atwood, S. S. 8, 22, 296-297, 309 aufm Kampe, H. J. 291, 308

Autler, S. H. 273, 275, 308 Averbach, B. L. 46 B

Badcoe, S. R. 268 Barghausen, A. F. 374

Barrel!, H. 5, 21

Barsis, A. P. 267

Battaglia, A. 8, 21 Battan, L. J. 291, 308

Bauer, J. R. 22, 255, 268, 357, 374

Bean, B. R. 22, 29, 46, 54, 87-88, 170-

172, 180, 225-228, 243, 266-268,

270, 274, 288, 308-309, 342, 363, 373-374, 377, 423

Becker, G. B. 273, 275, 308 Beckmann, P. 357, 374

Bennett, C. A. 374

Best, A. C. 309

Beyers, N. J. 374 Birnbaum, G. 5-7, 21-22, 31, 33-34, 46-

47, 273, 308

Boithias, L. 236, 267

Bonavoglia, L. 230, 266

Booker, H. G. 87, 171, 227, 267

Born, M. 21 Boudouris, G. 8, 21 Brekhovskikh, L. M. 254, 268 Bremmer, H. 136, 171 Brooks, C. E. P. 374

Buckley, F. 5, 21 Bunker, A. F. 39, 47 Burrows, C. R. 8, 22, 56, 88, 106, 171,

175, 225, 296-297, 309, 374

Bussey, H. E. 46, 297, 302, 308

Byers, H. R. 167, 172

C

Cahoon, B. A. 54, 88, 243, 266, 308, 363, 374

Carruthers, N. 374

Chatterjee, S. K. 6-7, 21

Clarke, L. C. 39, 47

Cline, D. E. 46

Clinger, A. H. 30, 46

Cole, C. F., Jr. 36, 47 Condron, T. D. 373 Cowan, L. W. 171

Craig, R. A. 8, 22, 267, 373

Grain, C. M. 6, 8, 21-22, 31-33, 46-47

Crawford, A. B. 8, 22, 225, 238, 256, 258, 267

Crozier, A. L. 30, 46D

Davidson, D. 308 Debye, P. 2, 4, 21 Deam, A. P. 36-37, 46-47

Doherty, L. H. 148, 172 Donaldson, R. J., Jr. 308 Dubin, M. 64, 88 du Castel, F. 171, 253, 267-268 Dunmore, F. W. 47 Dutton, E. J. 29, 46, 88, 172, 228, 309 E

East, T. W. R. 274, 276, 291, 308 Edwards, H. D. 280, 309 Englund, C. R. 8, 22, 225

Essen, L. 5-6, 8, 10-12, 21-22

Evans, H. W. 295, 299-302, 309 F

Fannin, B. M. 172, 179, 226

Fehlhaber, L. 22, 171, 268

Ferrell, E. B. 8, 21, 56, 88, 106, 171, 225, 374

Flavell, R. G. 180, 182, 227, 240, 267

Franklin, N. L. 374

Eraser, D. W. 46

Freehafer, J. E. 87, 374

Freethey, F. E. 46

Friis, H. T. 238, 256, 258, 267 Froome, K. D. 5-6, 8, 10-12, 21 G

Gallet, R. M. 22, 171

Garfinkel, S. 47

Gerson, N. C. 170, 174-176, 224 Ghosh, S. N. 280, 309 Gordon, J. P. 273, 308 Gordon, W. E. 227, 267 433
434 AUTHOR INDEX

Gordy, W. 308

Gossard, E. E. 227, 268

Gozzini, A. 8, 21 Gray, R. E. 105, 171, 178, 226, 267

Grosskopf, J. F. 22, 171, 268

Groves, L. G. 6 Gunn, K. L. S. 274, 276, 291, 303, 308-309

H

Hasegawa, S. 47 Hathaway, S. D. 295, 299-302, 309 Hay, D. R. 36-37, 47, 171, 175, 224

Herbstreit, J. W. 31, 46, 148, 172, 374 Hewson, E. W. 268

Hill, R. M. 308

Hirai, M. 170, 225, 267

Hirao, K. 30, 45, 171, 373 Hogg, D. C. 238, 256, 258, 267

Holden, D. B. 182-183, 185, 227, 240, 266 Holmes, E. G. 46 Hopkins, H. G. 303, 309 Horn, J. D. 22, 171, 180, 226, 228, 267, 373 Hornberg, K. O. 374 Hughs, J. V. 21

Hull, R. A. 174, 225 Humphreys, W. J. 61, 88

Hurdis, E. C. 6, 21

I Ikegami, F. 148, 172

J

Janes, H. B. 267, 374

Jehn, K. H. 19, 22, 171-172, 179-180,

193, 226-227, 239-240, 267

Johnson, W. E. 45, 170, 374

Jones, F. E. 45, 47

Jung, P. 47 K

Katz, I. 17, 22, 166, 171, 180, 193, 226,

239, 267, 312, 373

Kerker, M. 303, 309

Kerr, D. E. 141, 144, 171 Kirby, R. S. 148, 172

Kirkpatrick, A. W. 374

Kitchen, G. F. 268

Krinsky, A. 47 Kryder, S. J. 5, 22 L

Lane, J. A. 16, 22, 182, 227, 240, 266-267

Langleben, M. P. 303. 309 Laws, J. O. 292-293, 308 Lement, B. S. 46

List, R. J. 24, 45

Longley, A. 266

Longley, R. W. 268

Lukes, G. D. 166, 172, 180, 226

Lyons, H. 5, 22

M

Macready, P. B., Jr. 45 Magee, J. B. 8, 42

Martin, H. C. 47 Maryott, A. A. 5, 21, 273, 308 Mason, W. C. 22, 357, 374 Meadows, R. W. 268 Meaney, F. M. 266 Megaw, E. C. S. 227 Meyer, J. H. 255, 268

Middleton, W. E. K. 40, 47

Mie, G. 309

Millington, G. 13, 21 Minzner, R. A. 373 Misme, P. 30, 46, 147, 170-172, 175, 224-

225, 236, 238-240, 246, 265-267, 311, 373

Moler, W. F. 182-183, 185, 223, 227, 240, 266 Montgomery, R. B. 22, 267, 373 Mumford, W. W. 8, 22, 225 Murray, G. 47 N

Niwa, S. 170, 225, 267

Northover, F. H. 268

Norton, K. A. 62, 88, 105, 148, 171-172,

250, 266-267, 374O

Onoe, M. 170, 225, 267

Ozanich, A. M., Jr. 171, 227, 373 P

Palmer, C. E. 201, 228

Parsons, D. A. 292-293, 308

Perlat, A. 224

Peterson, C. F. 374

Phillips, W. C. 6, 21 Pickard, G. W. 105, 170, 224, 230, 232,

235, 266

Pote, A. 308 Pound, R. V. 32, 36-37, 46

Price, W. L. 148, 172R

Rainey, R. J. 374

Randall, D. L. 176-177, 224

Ratner, B. 168, 172, 280, 308, 373

Rice, P. L. 88, 92, 250, 266-267

Riggs, L. P. 22, 171, 180, 226, 228, 267

Ripley, W. S. 373

Roberts, C. S. 46

Rubenstein, P. J. 22, 267, 373 Ryde, D. 292, 294, 308 Ryde, J. W. 292, 294, 302-303, 308 S

Saito, S. 8, 21

Sargent, J. A. 31, 34, 46

Saxton, J. A. 1-2, 21, 182, 227, 238, 267-

268, 303, 309

Schelleng, J. C. 8, 21, 56, 88, 106, 171,

175, 225, 374

Schilkorski, W. 268
AUTHOR INDEX 435

Schulkin, M. 88, 150, 164, 170, 172, 179, 227 373 Sheppard, P. A. 224

Silsbee, R. H. 47

Sion, E. 47 Smart, W. M. 49, 87, 172 Smith, E. K. 4, 6-8, 10-12, 17, 21 Smith, M. J. A. 47 Smith-Rose, R. L. 8, 22 Smyth, C. D. 21 Smyth, J. B. 6, 21, 227, 267

Snedecor, G. W. 171

Spilhaus, A. F. 40, 47

Spizzichino, A. 267

Sprung, A. 12, 21, 24, 45

Starkey, B. J. 268

Stetson, H. T. 105, 170, 224, 230, 232,

235, 266

Stickland, A. C. 8, 22, 61, 88

Stover, C. M. 45

Straiton, A. W. 30, 46, 274, 308

Stranathan, J. J. 6, 21 Strandberg, M. W. P. 308

Sugden, S. 6 T

Tao, K. 171, 373

Thayer, G. D. 22, 54, 87, 170, 225-226,

228, 267, 342, 363, 373-374, 377, 423

Thiesen, J. F. 30, 46 Thompson, M. C, Jr. 46, 374 Tinkham, M. 308

Tolbert, C. W. 274, 308 Trewartha, G. T. 172

Troitskii, V. S. 309, 312, 373

Trolese, L. G. 21, 227, 267

Turner, H. E. 47

Turner, W. R. 268

U

Unwin, R. S. 323 V

Van Vleck, J. H. 270, 273-275, 308

Vetter, M. J. 31, 34-36, 46, 374

Villars, F. 267

Voge, J. 238-239, 267

Vogler, L. E. 62, 88W

Wadley, T. L. 8, 12-13, 21 Wagner, N. K. 38-39, 42, 47, 253, 268 Walkinshaw, W. 87, 171, 267

Waters, D. M. 46 Watkins, T. B. 47 Waynick, A. H. 8, 22 Weickmann, H. K. 291, 308 Weintraub, S. 4, 6-8, 10-12, 17, 21 Weisbrod, S. 80, 88

Weisskopf, V. F. 267

Wells, P. I. 374

Wensien, H. 268

Wexler, A. 28-29, 39, 41, 45-47

Wilkerson, R. E. 374

Willett, H. C. 172

Williams, C. E. 46

Wilson, F. A. 22, 357, 374

Wolf, E. 21 Wong, M. S. 322, 373Y

Yerg, D. G. 45, 47, 174, 225 Z

Zhevankin, S. A. 309, 312, 373

* U.S. GOVERNMENT PRINTING OFFICE : 1966 0 —775-028

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