IJationa?^ Sr:rNBS MONOGRAPH 4

CRPL Exponential Reference Atmosphere

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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UNITED STATES DEPARTMENT OF COMMERCE • Frederick H. Mueller, Becreiary NATIONAL BUREAU OF STANDARDS • A. V. Astin, DiTccior

CRPL Exponential Reference Atmosphere

B. R. Bean and G. D. Thayer

National Bureau of Standards Monograph 4

" Issued October 29, 1959

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington 25, D.C. - Price 45 cents.
Contents

Page

1. Introduction 1

2. Background 1

3. Basic theory 3

4. Numerical ray tracing 5

5. Accuracy of the tables 7

6. Tables of the CRPL exponential reference atmospheres 8

7. References 9

8. Appendix A. Tables of profile variables for the CRPL exponential reference atmosphere 10

9. Appendix B. Tables of refraction variables for the exponential reference atmosphere 11 n
CRPL Exponential Reference Atmosphere

B. R. Bean and G. D. Thayer

The background and development of an exponential model of atmospheric radio

refractivity, the CRPL Exponential Reference Atmosphere, is outlined. A set of ray

tracings for the model is presented in the form of tables of refraction variables for the complete range of observed values of surface refractive index. A detailed analysis of the accuracy of the ray tracing and tabulation methods is made for these tables. The variables

are presented as numbers between 1 and 10 multiplied by the appropriate power of 10, thus maintaining a maximum number of significant figures. The tables may be used for the

solution of practical refraction problems involving elevation angle errors, range errors, and

similar quantities.

1. Introduction

Some exponential models of atmospheric radio

refractivity have been introduced recently to replace the standard 4/3 earth treatment for applications involving long-range transmission

paths, forward scatter predictions, or radar

tracking at high altitudes [1, 2, 3, 4, 5].^ One of these models, developed at the Central Radio

Propagation Laboratory and called the "CRPL

Exponential Reference Atmosphere," has been

adopted for use by the National Bureau oi Standards in predictions of refraction phenomena.

In this monograph, the background of the

exponential reference atmosphere will be outlined

and a set of radio ray tracings presented. These

ray tracings may be used for the solution oi many

problems involving refraction by means of the

exponential reference atmosphere.

2. Background

In dealing with refraction problems it is convenient to use the radio refractivity defined by:

rather than the radio refractive index n. The

basic predictor of refraction effects used in the

exponential reference atmosphere is the surface

value of refractivity at the transmitting point,

Ng. The reasons for this become clear upon

examination of the normal, or average, N structure of the atmosphere. It is found that the

gradient of A'^ which, with the usual assumption

of horizontal homogeneity, is always with respect

to height, is, on the average, a function of the

surface value, Ns. This is illustrated in figure 1 where values of the difference between Ng

and the value oi N at 1 km. above the stirface, are plotted against A^^. The data represent 888

six-yr monthly means from 45 U.S. weather

stations representing many climatically and geographically diverse locations. As shown, AA^

is approximately an exponential function of Ng,

the relationship being expressed by

:

-a]V=7.32 [exp {0.005577iV,}], (1) which is a least squares fit of In ( — AA^) to Ns.

' Figures in brackets indicate the literature references at the end of this monograph.

It has also been found that the average decay

of A'^ with height is very nearly exponential.

This can be seen from inspection of figure 2 where averages and extremes of A^ structure are shown for medium high elevations on a semilogarithmic plot.

The exponential reference atmosphere was

defined simply as that family of A^ profiles having

a simple exponential decay with height and

passing through the values Ns at the surface and

A^s + AA^ at a height of 1 km above the surface

where AN is found from eq (1). N{h) is then

given by

:

N=Nsexj>{-Ce{h-hs)}, (2)

where hs is the surface elevation, h is the altitude

above mean sea level, and Cg is the decay constant

given by

:

Table 1 shows values of Ns, AN, c„ and hs for those profiles which were used in making up

the tables of refraction variables given in appendix B (a more com.plete table of this sort may be found in appendix A). The values of hs

1
-AN

100 9080

70

60

50

40

30

20

15

10

'. • V ,

"• ' * -AM -

•••;,;0.='. 888 sets of data

230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400

Figure 1. Six-year mean AN as a function of 8-year mean N, at 0300 and 1500 G.m.t. for ^5 U.S. weather stations.

were chosen to reflect the average elevation dependence of Ns.

It is found that several important advantages

accrue from this particular choice of model,

among which are:

(1) The profiles involved prove to be a fairly

good representation of average A/^-structure over

i, Auc

ortlar

•ust, 1

d. Me

5:00

,Feb

GMT.

'uary /5:0OGMT

1

y, Feiiruary

1

, /5:00 Gh

1/IT

rApprox

dividiimotelal ra

data

iiosor

axirriL %A

limits deprc

im

eon

^D/linii

for

files

num

\ \

'Ih pre files

the first 3 km. above the surface, a critical range

for refraction effects at low angles to the horizon.

(2) The resulting relationship between the

A^-gradient near the surface and is found to be

very close to the observed average relationship

(for 5-yr mean profiles), being within ±3 percent over the normally encountered range of Ns

(from 300 to 365 for sea level, temperate zone

stations). This agreement is a requirement

for the successful prediction of average refraction

effects at elevation angles close to the horizontal.

(3) The resulting profiles and their associated

gradients are continuous functions of height and

are in reasonably good agreement with the actual

Table 1. Exponential reference atmosphere profiles contained in appendix B

0 2 4 6 8 10 12

Altitude Above Mean Sea Level in Kilometers

Figure 2. Means and extremes of radio-refractivity height

structure, and a comparison with extreme 4/3 earth

profiles.

N, k. AN

Effective

earth's radius

factor,' k N ft N

200.0 10,000 -22. 3318 0. 118400 1. 17779

252. 9294 5,000 -30. UOOO . 126255 1. 25551 313.0 700 -41. 9388 . 143859 1. 40229

344. 5244 0 -50.0000 . 156805 1. 52479

377. 2161 0 -60. 0000 . 173233 1. 71320

404. 8565 0 -70.oonu . 189829 1. 95921 450.0 0 -90. 0406 . 223256 2. 77761 289.0 0 -36. 684830 0. 13574V71 1. 3332410

*k=-

1

6373.0150 km (3,960 miles).

(note that ~<0).

an r„=a+h, where a is taken as

2
observed refractivity at all heights at frequencies

sufficiently high so that the refractivity is not

liafluenced by the ionization in the ionosphere.

' An illustration of the effectiveness of the expoinential reference atmosphere in predicting refraction effects is given by figure 3, where the values of

: angular ray-bending are plotted against height for

a ray starting at zero elevation angle. The agree-

• ment between the average bending for a large number of profiles and the average of the values predicted by the exponential reference atmosphere is seen to be quite good. Note however that the

bending versus height predicted by the 4/3 earth

model is markedly in disagreement with the

observed for most heights.

The preceding background material is intended

only as an outline of the development of the exponential reference atmosphere. The remainder of

this monograph will be devoted to an analysis of ray tracing theory and its application to the exponential reference atmosphere with particular

emphasis on the preparation of the tables contained

in appendix B.

2

So 0

Expc

1

HGntiol

1

tefsrcn

1

rff

1 1 1 1

^Ave meo'oge be

n profi

nding,

les f

'3 earth bending T at 70 km

es; 15.12 m,r

-

;f,: 15.23 m,r.

. 30.25 m.r.

1 i 1 1 , 1 ,

1

Mean profi Mean exp. r 4/3 earth

1 , 0 2 4 6 8 10 I? 14 16 18 20 ti-hs IN KILOMETERS Figure 3. Average radio-ray bending for 22 5-yr. mean

radio-refractivity "profiles as a function of height, compared with the average of the values predicted by the exponential reference atmosphere for each profile and the values predicted by the 4/3 earth treatment.

3. Basic Theory

The geometry of the radio-ray-refraction problem is illustrated in figure 4, and the variables used

in the following development are defined in the same figure. The path of the radio ray is the

curved line R, which will also be used to represent

the length of the ray path from the souree at Tq to the point at r. If the transmitter is a radar set, the observed elevation angle to the target at r would be do, the true elevation angle would be 0; thus the elevation error angle, e, is given by:

(4) The angle t is a basic nzeasure of refraction

effects, and will be called the bending since it represents the angle through which the ray has

been bent from its original direction due to refraction. It can be seen from inspection of figure 4

that: e-9o=(t>—T, (5) from purely geometric considerations.

If it is assumed that the refractive index is spherically stratified with respect to the earth,

then Snell's law (for the ray path) may be ex- pressed as:

nr cos 5=no?'o cos ^0= constant, (6) or

(l+A^X10-8)(a+A) cos 6 = (l +N,XlO-')(a+hsycosdo, (7) where the radial distance from the center of the

earth, r, equals a -\- h. The zero or s subsciipts

in (7) refer to the initial values at the earth's

surface. Equation (7) may be used to calculate

the local elevation angle, 6, at any point along the

ray and thus affords a complete description of

the ray path. If:

7i2r2 cos 62=ni cos di, and

7i2=?J'i +An,

r2=ri+Ar,

and

where Aw, Ar, and A0 are infinitesimals, then

niTi cos ei = (7i,i+A?i)(ri+Ar)cos (^i+A^),

and in the limits as Ar approaches zero this becomes:

%ri cos di = {ni^dn){ri+dr){coB di—dd sin ^i). (8)

If (8) is expanded and products of differentials are omitted, then one obtains the differential equation:

« 7^ , dr

tan 6d6= —, n r

1^ dn , „ , dr

de=— cot e-\—cot 0. n r

Noting that dr cot dlr—rd4)lr=d<j>, and, from

geometry, d6=d<t>—dT, one obtains the classic ex- pression for the bending of a radio ray [6, 7]: dT=—— cot 6, n

or

J nl I''

3
Figure 4. The geometry of the radio-ray refraction problem.

Note that (9) defines downward bending as a

positive quantity.

If the value of r has been obtained for some

particular point (r, n) then the angle e may be

found by the following scheme:

nr cos d=noro cos do from (6), and:

r cos {d-\-T—e) —ro cos (do—e).

(This is the equation of a straight line along the

slant range, Rq.) Combining these two equations,

employing sum and difference trigonometric

formulae, and solving for e will yield:

e=Arc tan

cos T— sin T tan d~n/ns n

n. tan sin t— cos r tan 6 (10)

A similar expression for the angle 8 may be derived:

5=Arc tan -< —cos T —sm T tan dn

sin r— cos T tan ^o+~ tan 6

^- (11)

The distance along the earth's surface from Vq to r

is found by:

d=ro4>=ro{e-do^T). (12)

The true distance from to r, called the slant

range, i?o, is found from the law of sines or cosines: \

B,=-^^AW^=^,^K^<^- (13) \

The range along the ray path, called the geometric range, is found by integrating over the

differential path lengths:

Jo J To

dr

sin d' (14)

while the effective or electrical (actually radio)

range. Re, is found by:

Re= rndR= V~ (15)

Jo JraSmd

The radio range error, ^Re, and the geometric

path length difference, Ai?, are defined as:

and:

ARe^Re— Ro,

AR=R-Ro.

(16)

(17)

It should be remembered that the above

equations are subject to the two usual restrictions

of ray tracing:

(1) The refractive index does 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 not be violated by the

exponential reference atmospheres since there are no discontinuities in dn/dr.^ Condition (2) should be satisfied if: AN

0.002 A,,

which would restrict the ray tracings for the

exponential reference atmosphere to frequencies over about 110 kc (for 7V,=450, for A^,=200 the

lower limit would be about 55 kc).

2 This assumes also that I 3-^ < — . and 60 > 0.

4
4. Numerical Ray Tracing

Because of the great amount of calculation

necessary to obtain a set of ray tracings for the

reference atmosphere it was advantageous to resort to digital computing machinery. A complex

ray-tracing program had been written for the

IBM 650, utilizing the floating point mode (8

digits plus exponent), which was designed to handle arbitrary refractive index profiles having

up to 100 layers and it was decided to analyze the

exponential reference atmosphere with this program. This section will be devoted to an analysis

of the ray tracing techniques employed in the

machine program.

The utilization of digital computer techniques

for this type of ray tracing necessitates certain

changes and alterations in the equations of

refraction outlined in the previous section.

To begin with, the basic accuracy of the ray

tracing depends upon the accuracy with which the

local elevation angles can be calculated using

SneU's law. In order to determine an angle in the

neighborhood of 1 milliradian (mr) to an accuracy

of 8 significant figures from the cosine of that

angle, it is necessary to know the value of the

cosine to 14 figures (cos x=0.999999xxxxxxxx, i.e.: the 6 nines are nonsignificant) or to 1 part in 10~^*. Rather than employ the relatively slow doubleprecision techniques required for the- machine

evaluation of eq (6) an alternative formulation of

SneU's law was obtained by writing:

1— cos 6=1 cos nr

and taking advantage of the fact that 2 sin^ 5/2=1—

cos 6, the following equation was derived:

9=2 Arc sin

ro (N-N)

X10-' cos do]}" (18)

This equation is exact, and if each of the

variables employed is correct to 8 figures, then 6 will have an error no greater than 5 parts in the

eighth figure. Since in performing ray tracings at

initial elevation angles from 0 to 900 mr the value

of theta will never be as large as 1 radian, the

maximum error in calculating d by this method

should be 5X10~* radians, or in general, not more

than about 10~^ percent of the elevation angle.

With an accurate method of calculating the local elevation angles at various points along the ray

path, the primary remaining problem is to compute accurately the total bending of the ray to each selected point. For this purpose a profile of versus height is entered into the machine.

The local elevation angle for each of the profile

points may be calculated by means of eq (18), and

if it is assumed that

dN

dh =constant (19)

for each layer defined by succeeding pairs of

points of the N profile, it can be shown that:

where:

^[1.2] ATti,2] = ^'-L "^^ XlO-^cotgti.a] (20)

U,2] ^1+^2

and the only approximation is made by letting:

cot e--

r'2

Y n2 dn cot 6=- I dn cot d. Jl '^[l,2lJl

Since Wi—n2 never exceeds about 80X10~® between layer heights in the profiles, the maximum

error entailed by this approximation cannot be

very large (it is actually less than 10~*% of At in the worst case). It is evident from this method that t is essentially the weighted summation of the profile N

gradients, where the cot di are the weighting factors. It is important to note that, since for small

6 cot ^ = ^> the weighting factors are very large

for the first few layers if do=0. It is thus ex- pected that refraction effects at very-low elevation angles would be extremely sensitive to the

initial gradient of N.

The integral for t was thus computed as the

sum of At for each layer:

i=l (21)

where the only error of any magnitude is that

made by assuming dn/dh= const&nt for each

layer. In order to minimize this error it is neces- sary to restrict the thicknesses of each layer of

the profile such that: Ari< constant, where the

constant is some arbitrary angular amount. For

the profiles presented in this paper the layer

thicknesses were chosen such that At was always

less than 2 mr for the first increment, and less than 3 mr for all succeeding increments.

The approximate size of At^ was found from the

relation ATi= ANJOa where 6^ is the approximate

value of Oi. These were calculated for 6o=0, in which case 0a = Va/V^''o/2 where k is the effective

earth's radius factor for the surface gradient and

h is the mean height of the layer for which At, is to be calculated.

Thus:

/=—^V^W2 <C constant.

5
It is thus possible to represent the given profile

adequately with about 25 or less layers of constant

A^-gradient. It is also possible to calculate bending using the approximation: n= constant for each layer. However it has been found that in order to insure sufficient accuracy about 1,000

layers must be used to approximate the profile

with this method [8]. The calculation of the distance is straight

forward:

dt=roie,-eo+T,) (22)

where i refers to the value of each variable at the

top of the ith layer.

Calculations of the elevation error angle are for

convenience made to depend on calculations of the' angle 5: '

e + r-8, (23f"

since 5 is never larger than 1/2 t and is thus more

easily calculated than e. (Over the first few in- crements of height the ray path is approximately

a circular arc, and thus e=d= 1/2 t, however as the'; range increases without limit the elevation angles'

formed by the ray path and the slant range "path"i

both approach 7r/2 and thus 5—>0. This, of course, implies that lim e=lim e=T since T=e+5.) Equation (11) for 5 is modified to:

CiN-N)

5=Arc tan-< X10"^+[l — cos t] — sin r tan do

sm T- cos T tan do-\—- tan d n

(24:),

for accuracy in calculation.

The slant range is calculated with the law of

cosines formulation (13), reduced to:

7?o=V('"-^o)'+2[l-cos<^.] (25)

again for accuracy of calculation.

The geometric path length along the ray is calculated by the summation of the incremental

path length through each layer (which will be

denoted as As to avoid confusion with the range

error, AR):

i = l (26)

which in tiu-n is calculated by a law of cosines

formulation similar to eq (25):

ASt=V(^i-ri-i)'+2[l-cos (<^.,-0,_,)]. (27)

Since the ray path through each layer is very

nearly a circular arc, the maximum error in computing ASi from (27) (assuming At is exact) is:

error (As,) =

2 sin

At,: xioo%

At? = -24^X100% (to7sig. fig.),

and since, as stated earlier, At; is never as large

as 0.003 radians, error (A,,) <0.0000375%, and

hence: error (R) <0.00039% = 2.5 m maximum

since for the "worst case" i.e.: A''s=450, (the errors discussed in this and following sections are generally a maximum for the profile with

the largest A^^ gradient, in the case of the exponential reference atmosphere, theA^,=450 profile),

00=0, total T =31.5 mr, there would be about 10

increments of 3 mr each (if calculated with the* minimum number of layers), and i?o— 650 km

when T=31.4 mr, essentially the total amount

of bending. Actually there are some 30 layers

calculated when t=31.4 mr, and the calculated

maximum error in R is only 0.0000514 percent

of 650 km or about 0.334 ms. It is felt that

calculation of ranges to an accuracy of ±0.5 ms

or less is sufficient for any valid use to which the

exponential reference atmosphere might be put.

It should also be borne in mind that the accuracy

to which the distance along earth's surface, the

slant range, the geometric path length, and the

radio range, can be calculated is also dependent

on the accuracy to which the angle 0 has been

calculated; this point will be discussed further

in the section dealing with the accuracy of the

tables. The radio range, (or "electrical path length",]

"effective path length", etc.), is calculated simply

as:^

Re- j:as'u

i=l (28)

where: ^

ASi• = j ndR=

J i-l n,As, = ^^^?^A.r (29)

Assuming that As, has been calculated accurately,

the error in eq (29) for As'i depends on the extent

to which:

R

ndR=

Since nr^i^i never exceeds 80X10"^,and the error ! (defined as the difference between the above

terms) will never be as large as 2 percent of

the maximum error in eq (29) should be

6
,.,less than 1.6X10~* percent. The worst case

(again A^s=450, 6o=0) yields a calculated maxi-

'mvim error of 2.8X10~^ percent (at i?e = 800 km)

ior about 0.224 m. Note that the weighted

average of n is always smaller than the arithmetic

'.mean of the end-point values, n< and Ui-i; this

.. indicates that the errors due to this approximation

-wUl be of opposite sign to those due to using

^|eq (26) and (27) to calculate R. Since these

; two errors will thus oppose each other when

calculating Re, this quantity will tend to be more

accurate than R, the calculated error for the

.-V,=450, 0o=-O case being only —0.110 m (11

5. Accuracy

In the preceding section on numerical raytracing the errors inherent in the equations and

approximations employed were discussed at some

length. It is important now to deal with those

errors peculiar to the particular mode of computation used in developing the tables, i.e. : those

errors which could be avoided by emplojang a

; more accurate method of calculation using the

: same equations and approximations. The fol-

: lowing material is a rather detailed analysis of

; errors; for a concise presentation of the maximum

; errors likely to occur for each variable the reader

1 is referred to the table at the beginning of ap-

: pendix B. The basic source of error of this type is derived

from the accuracy in the calculations of the N

values used in the refractivity profiles. The

profiles for the exponential reference atmosphere

were calculated to an accuracy of ±5X10"^^ in n (or ±5X10"^ in A'' units). Thus the quantity

A^j—A^o-oi, the difference between surface and

Ai^ at 10 m above the surface can be calculated to an accuracy of only about 5 significant figures

(this is the worst case), resulting in the following

maximum, errors : Error (6) < ± 0.000047 m.r 0o<4

mr; error (e)<± 0.000005 m.r do>4 mr; and

error in At <0.0001 mr (from this cause only).

These are random rounding errors and will not

accumulate over a large number of height increments. For do larger than 10 mr the accuracy

wiU be unaffected by the mentioned errors in A^, with the result that the largest error in 6 (for do of 400 and 900 m.r) would be 3 or 4 units in

cm error in about 800 km, or 1.37X10"= percent).

The errors discussed in the above paragraphs

are, of course, much larger percentage-wise when

applied to the values of the range errors: AR=R-Ro

ARe =Re—Ro

in the case previously mentioned as representing

the largest errors (7V,=450, ^o=0, i?o>800 km)

the percentages are: Error (A/?) = —0.67 percent

maximum; error (Ai?e) = —0.06 percent m.aximu.m)

of the Tables

the eighth place: Error (e)<0. 000004 mr 10

mr<e„<100 mr; error (6')<0.00004 mr &<,>100 mr. Hence the maximum error expected to be

encountered in 6 at any point is less than 5X10~*

radians.

The accuracy with which r can be calculated

depends primarily on the accuracy with which

Ni—Ni^i is known, so that r can be relied upon

to 5 or 6 figures, depending on whether or not it is over 10 mr. Except for the first increment at ^0=0, an error in the calculated value of At for some particular height increment due to a rounding

error in Ni—Ni^i will in general be at least %

offset by an error of opposite sign over the fol- lowing height increment. An error in the first increment at ^o=0 will not usually be made up

later due to the excessively large weighting factor

given the error by cot 0 in that case. Upon in- vestigating the average error in t at anv height

caused by the above mentioned errors, it is found

that: partial error (tXO.0003 m.r. This is an additional error in t due to the differ- ence between the true exponential decay of A^ and

the linear approximation of N versus height as- sumed by the ray-tracing procedure. Errors of

this type wiU be a maximum for the first increment

of T for ^0=0, because of the rapid change of cot 6 near 6—0. An equation has been derived for the

difference in the incremental bending for the first layer at 6o=0 between the linear and exponential

profiles (having the same value of A^'s and A'' at r— ro=AAi), which is the error in Ati due to the

linear approximation:

error (At,) = (AtO[(i-^-^)-AA(^-^+-^- . . .)}

error due solely to difference in A^ gradient assuming sam.e cot 6

+ ryj-^ x3 i-^ (AAO^/^ radians, (30)

y(i-[Ar,-Ar,]X10-«) '-•^

519249 O -59 -2 7
error due solely to difference in cot 6 where the

subscript 1 refers to values of the variables at the

first height increment, r—ro=Ahi. In the case of

the A^s=450 profile the partial error in Atj from

the first part of expression (30), am.ounts to only

0.000760 m.r, while the remainder of the error in Ati from the second part of eq (30) amounts to only 0.00000725 m.r, or only 1 percent of the first part. These errors drop rapidly with decreasing

initial gradient of A^, so that for the Ns=377 .22 profile the error in Ari is only 0.000287 mr, or only about K as much as for the A^s =450 profile.

Errors in the value of the elevation error angle,

€, will be of the same order of magnitude as those

in T at any point, the method of computation being more precise than the values of t used in the

calculation.

Errors in the distance, d, and the ranges, Rq, R,

Re will be determined by the accuracy to which the

angular distance can be calculated,

which in turn depends on the accuracy to which

6 and t are known. Since 6 will have a maximum

error of about 4X10"' mr and r a maximum error

of about 8X10"* mr, the distance and ranges are calculated at least to an accuracy of about ±5 m. However, if it is assumed that the profile is exact

as represented by linear layers, the accuracy of the distance and ranges increases, with a maximum

error of about ±0.6 m. Actually the ranges for the Ns=377 .22 profile are accurate to about ±1.8

m., for the Ns=200 profile the maximum error is of

the order ±0.6 m (about the same as if the profile were actually composed of linear layers).

The preceding estimates of maximum, errors are based on the 6o=0 ray; for any ray at do^O the

errors will in general be no larger than the limits

imposed by computational accuracy (i.e. : errors not caused by the difference between exponential

and linear N layers). Note that for very large

the bending of the ray path is so slight that no

significant differences can be detected between the

arc lengths along the ray path, i.e. : the geometric

range, and the slant range. In these cases the

reported Aff values are almost entirely one to four

parts of the eighth significant figure of R, interspersed with values given as exactly zero. These

are due to random rounding errors, and are often

negative In general, to be considered as significant (or real) a value of AR should be at least as large as two parts in the seventh figm-e of R. An

exception to this rule is for do=900 mr; in this case the angle 4> can only be calculated to two

significant figures at the low heights, since 6— do

is about 10~^ radians and d=6o is about 1 radian,

and as a result of internal inconsistencies in the

triangles formed by the and r„ combinations, erroneous AR values as large as 0.002 percent of

Rq are calculated. However, in none of the raytracings is there an entirely erroneous value of

AR or ARg reported which is larger than a few

centimeters (almost all such values are in the m.illi-j meter class, and differences between Rq and R as

s.mall as 10'^ km., or 0.001 mm are found in some

of the ray-tracings for 9o=200 and 400 mr). A

good rule to follow in using the values of AR and;

ARe given in these ray-tracings woidd be to con-j

sider that any of the values may be subject to an

error of at least ±10 cm, and (in the case of thej

larger values of AR) possibly as much as ±50 cm. The values shown as "DELTA H" in the tables are the corrections necessary to correct the height , given by a simple 4/3 earth atmosphere for the

distance given (assumed exact) to the height given

(also assumed exact). As a result, the values of

j DELTA H have an error approximately equal to: .

error (delta ^) = ± (absolute en or in d) since the

d

distances given are not really exact. The values

shown as "DELTA THETA" are the corrections

necessary to correct the elevation angle given by

a simple 4/3 earth atmosphere at the given distance

(assumed exact) to the value calculated for the

exponential profile. This quantity is found by:

4

where r and <j> are as calculated for the exponential

reference atmosphere at the point where Ad is desired. The maximum error in this quantity

therefore depends mostly on the errors in r, hence:

error (A^XIO"^ radian.

6. Tables of the CRPL Exponential Reference Atmospheres

The tables themselves are simply a presentation

of ray tracing variables. In these tables all angles are given in milliradians and all distances

in kilometers. For each different profile {Ns=

200 through 450) ray tracings are given for 6^ from 0 to 900 mr, in approximately a geometric

progression (i.e.: 0, 0.5, 1, 2, 4, 8, 15, 30, 65,100,

200, 400, and 900 mr), at fixed heights along the

ray path. From left to right: first is given the

SURFACE A^, or A",, to 4 figures, not rounded.

Exact values of A^s used are given in table 1. The next column is the value of 6^, or "INIT(ial)

THETA", in milliradians. The "HEIGHT"

is given in kilometers. The preceding quantities

are given with fixed decimal points, the remaining

quantities are listed as floating point numbers in the following fashion: ±X.XXXX±£' where the

first sign refers to the sign of the number,

X.XXX which is to be multiplied by 10 to the i power ±£' depending on the second sign. !

8
Examples:

0.0000 =

1.2345 =

1.2345 3 =

-1.2345 3 =

1.2345-3 =

-1.2345-3 =

zero 1.2345 (X10°)

1.2345X10^ - 1.2345 XIO^

1.2345X10-^

-1.2345X10-^

The following quantities are listed as floating

point numbers in this manner: "DELTA H" =

correction to 4/3 height at same distance to equal

"HEIGHT", in kilometers; "THETA" = 0 or

local elevation angle, in mUliradians; "DELTA

THETA" = correction to 4/3 earth 6 to equal

THETA at same distance, in milliradians

;

"DISTANCE"=ro<^ or distance along earth's

surface, in kilometers; "TAU" = t, angular bending, in mUlh-adians; "ERROR ANGLE" = e, elevation error angle, in milliradians; "SLANT

RANGE"=i?o. straightline distance from ray

origin to point at h, d, in kilometers; "DELTA

R" =Ai?, difference between distance from h=Q

to given height along ray path and slant range, in

kilometers; "DELTA R—E"^ARe, difference between radiopath length and slant range, in kilometers.

The following ray-tracing variables have not

been printed, but can be easUy calculated from

those given, as follows:

d_ "DISTANCE"

ro "INITIAL RADIUS"'

where ro=6373.0150+^, (kilometers)

<^.= r-e="TAU"-"ERROR ANGLE",

R=Ro+AR="SLANT RANGE"

+ "DELTA R",

Re= Ro+ARe="SLANT RANGE"

+ "DELTA R-E".

Values of elevation of the ray starting-point, h„

are given in table 1. The last profile, for A7^s= 289.0, is a special

profile which was prepared for the purpose of making direct comparisons between 4/3 earth

refraction and an exponential profile having the same gradient at the surface. It is intended

for applications where it is desired to replace

the 4/3 earth treatment with a single standard

atmosphere having the advantage of differing

very little from the 4/3 earth near the surface.

The authors wish to acknowledge the technical

assistance of Miss B. A. Gaboon, and Mrs. B. J. Weddle, and particularly of K. A. Norton whose

help in working out some of the mathematical

methods employed was invaluable.

7. References

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

[2] B. R. Bean, Atmospheric bending of radio waves, paper presented at International Conference on

Electromagnetic Wave Propagation, Liege, Belgium,

October, 1958 (to be published by the Academy

Press, spring 1959).

[3] L. J. Anderson, Tropospheric bending of radio waves.

Trans. Am. Geophys. Union 39, No. 2, 208 (February

1958).

[4] J. R. Bauer, W. C. Mason, F. A. Wilson, Radio refrac- tion in a cool exponential atmosphere, Lincoln

Laboratories, Technical Report No. 186 (August

27, 1958).

[5] G. Millington, Propagation at great heights in the atmosphere. The Marconi Rev. 21, No. 131, 143, 4th

Quarter (1958).

[6] M. Schulkin, Average radio-ray refraction in the lower

atmosphere, Proc. IRE 40, 554 (May 1952).

[7] W. M. Smart, Spherical astronomy. Chapter III (Cambridge University Press, London, 1931).

[8] G. H. Millman, Tropospheric effects on radar target measurements. Proceedings of the Seventh Weather

Radar Conference, held at the Deauville Hotel,

Miami Beach, Florida (November 17-20, 1958).

9
8. Appendix A. Tables of Profile Variables for the CRPL Exponential Reference

Atmosphere

The following equations were used in determining the various profile variables:

-AiV-7.32 exp {0.005577 A^,}

-dNo=CeNs

1

Table A-2

k=

l+-"(c,iV,)X10-«

n.

ro is taken as 6373.014987 km for all values of

Ns in these tables. The table giving Ns, —AN, Cg, and —dNo as functions of k was prepared using an iterative

method for solution of the transcendental equations involved. In all these tables the accuracy

may be taken as ± 1 in the last digit listed.

Table A-1.

N. -AN c. k

200 22. 3317700 0. 118399435 23. 6798870 1. 17769275

21U 23. 6125966 . 119280212 25. 0488444 1. 18991401

220 24. 9668845 . 1204,58179 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

—AN N, Ce —dNo k

20 180. 226277 0. 117626108 21. 1993155 1. 15617524 22 197. 316142 . 118216356 23. 3259953 1. 17457412

24 212. 917967 . 119594076 25. 4637276 1. 19366808 26 227. 270255 . 121491305 27. 6113599 1. 21348565 28 240. 558398 . 123746115 29. 7681671 1. 23406110

30 252. 929362 . 126255291 31. 9336703 1. 25543336

32 264. 501627 . 128950180 34. 1075325 1. 27764560

34 275. 372099 . 131783550 36. 2895127 1. 30074523

36 285. 621054 . 134721962 38. 4794288 1. 32478398

38 295. 315731 . 137741207 10. 6771452 1. 34981825

40 304. 513148 . 140823306 42. 8825481 1. 37590934

42 313. 261483 . 143955014 45. 0955611 1. 40312414

44 321. 602888 . 147125889 47. 3161107 1. 43153539

46 329. 573439 . 150328075 49. 5441405 1. 46122250

48 337. 204713 . 153555418 51. 7796106 1. 49227226

50 344. 524418 . 156803056 54. 0224815 1. 52477960

52 351. 557000 . 160067149 56. 2727266 1. 55884863

54 358. 324138 . 163344614 58. 5303179 1. 59459364

56 364. 845143 . 166633002 60. 7952415 1. 63214058

58 371. 137293 . 169930326 63.0674811 1. 67162830

60 377. 216108 . 173234984 65. 3470266 1. 71321044

62 383. 095581 . 176545680 67. 6338699 1. 75705732

64 388. 788373 . 179861358 69. 9280046 1. 80335830

66 394. 305974 . 183181171 72. 2294300 1. 8.5232456 68 399. 658845 . 186504431 74. 5381454 1. 90419225

70 404. 856538 . 189830583 76. 8541525 1. 95922635

72 409. 907798 . 193159183 79. 1774555 2. 01772514

74 414. 820650 . 196489873 81. 5080567 2. 08002556

76 419. 602477 . 199822385 83. 8459677 2. 14650999

78 424. 260086 . 203156494 86. 1S11914 2. 21761358

80 428. 799768 . 206492043 88. 5437400 2. 29383429

. zuyszoyi

(

yu. yuoDzoi z. 0/0/44.5/ 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 . 233220G37 107. 6293319 3. 18073184 98 465. 188659 . 236567271 110. 0484114 3. 34463902

100 468. 811163 . 239915290 112. 4749663 3. 52694820

Table A-3

k N. -AN Ct -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 L4 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 . 202351472 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

10
9. Appendix B. 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:

i9o<4 mr

4 mr<eo<100 mr

6*0 > 100 mr

Errors in r, e (in milliradians):

±0. 00005 mr 1 ± . 000005 mr

± . 00004 mr )

^nearly independent of Ns450 404. 8 377. 2 344. 5 313 252. 9 200

$0 = 0 ±0. 001 0. 00065 0. 0005 0. 0004 0. 0003 0. 0002 0. 0002

eo=i° . 0003 . 00015 . 0001 . 00008 . 00006 . 00005 . 00005

eo=z° . 00004 . 000025 . 00002 . 000017 . 000015 . 000013 . 000012

Errors in Rq, R, Re, or (in meters):

450 404. 8 377. 2 344. 5 313 252. 9 200 eo=o ±5. 0 2. 7 1. 8 1. 2 0. 8 0. 65 0. 6 eo>i° 0. 4 0. 3 0. 25 0. 2 . 17 . 15 . 14

Assmne that error in AR or AR^ is ±0.5 percent or ±0.1 m, whichever is larger.

11
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-3--3--3--3-*-3--3--3--3--3--3--3--3--3--3--3--3--3--3--3--3--3-<f<JU. S. GOVERNMENT PRINTING OFFICE : 1959 O - 519249



THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories, in Washington,

D.C., and Boulder, Colo., is suggested in the following hsting of the divisions and sections engaged

in technical work. In general, each section carries out specialized research, development, and engineering in the field indicated by its title. A brief description of the activities, and of the resultant pubUcations, appears on the inside front cover. WASHINGTON, D.C.

Electricity and Electronics. Resistance and Reactance. Electron Devices. Electrical Instru- ments. Magnetic Measurements. Dielectrics. Engineering Electronics. Electronic Instru- mentation. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Engineering Metrology.

Heat. Temperature Physics. Thermodynamics. Cryogenic Physics. Rheology. Molecular

Kinetics. Free Radicals Research.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State

Physics. Electron Physics. Atomic Physics. Neutron Physics. Radiation Theory. Radioactivity. X-ray. High Energy Radiation. Nucleonic Instrumentation. Radiological Equipment.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Molecular Structure and Properties of Gases. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Fluid Mechanics. Engineering Mechanics. Mass

and Scale. Capacity, Density, and Fluid Meters. Combustion Controls.

Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Plastics. Dental Research.

Metallurgy. Thermal MetaUtu-gy. Chemical Metallm-gy. Mechanical Metallurgy. Corrosion.

Metal Physics.

Mineral Products. Engineering Ceramics. Glass. Refractories. Enameled Metals. Constitution and Microstructure.

Building Technology. Structural Engineering. Fire Protection. Air Conditioning. Heating

and Refrigeration. Floor, Roof, and Wall Coverings. Codes and Safety Standards. Heat

Transfer. Concreting Materials.

Applied Mathematics. Nimierical Analysis. Computation. Statistical Engineering. Mathematical Physics.

Data Processing Systems. SEAC Engineering Group. Components and Techniques. Digital

Circuitry. Digital Systems. Analog Systems. AppUcations Engineering.

• Office of Basic Instrumentation. • Office of Weights and Measures.

BOULDER, COLORADO

Cryogenic Engineering. Cryogenic Equipment. Cryogenic Processes. Properties of Materials.

Gas Liquefaction.

Radio Propagation Physics. Upper Atmosphere Research. Ionosphere Research. Regular

Prediction Services. Sun-Earth Relationships. VHF Research. Radio Warning Services.

Airglow and Aurora. Radio Astronomy and Arctic Propagation.

Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements. Trophospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Standards. High-Frequency Electrical Standards. Radio Broadcast Service. Radio and

Microwave Materials. Electronic Calibration Center. Microwave Physics. Microwave Circuit

Standards.

Radio Communication and Systems. Low Frequency and Very Low Frequency Research..

High Frequency and Very High Frequency Research. Modulation Systems. Antenna Research.

Navigation Systems. Systems Analysis Field Operations.