PB161598

^ecltnical vtete 92o. 97

^BouUer laboratories U^ ^ ^ ; \ ,

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TECHNIQUES FOR COMPUTING

REFRACTION OF RADIO WAVES

IN THE TROPOSPHERE

BY

E.J.DUTTON AND G.D.THAYER

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

*£ecknical 'Tlote 97

October 17, 1961

(Revised July 15, 1964)

TECHNIQUES FOR COMPUTING REFRACTION OF

RADIO WAVES IN THE TROPOSPHERE

BY

E. J. DUTTON AND G. D. THAYER

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1X1

TABLE OF CONTENTS

PAGE

ABSTRACT v

1. INTRODUCTION 1

2. LIMITATIONS TO RADIO RAY -TRACING * 3

3. AN APPROXIMATION FOR HIGH INITIAL ELEVATION

ANGLES 4

4. THE STATISTICAL METHOD 5

5 . SCHULKIN'S METHOD 6

6. THE FOUR-THIRDS EARTH MODEL 8

7. THE EXPONENTIAL MODEL 10

8. THE INITIAL GRADIENT CORRECTION METHOD 12

9. THE DEPARTURES -FROM-NORMAL METHOD 13

10. A GRAPHICAL METHOD 16

11. SAMPLE CALCULATIONS 16

REFERENCES 26

APPENDIX I , 27

APPENDIX II. Tables of Refraction Variables for the

Exponential Reference Atmosphere

Tables of Coefficients, Standard Errors of

Estimate and Correlation Coefficients for use in

the Statistical Method 36

Refraction Variables in Various N(h) Atmospheres

41

Initial N Gradients, AN , in the C.R.P. L.

Exponential Atmosphere 49

Schulkin's Method Sample Computations 50

Departures Method Sample Computations 54

TECHNIQUES FOR COMPUTING REFRACTION OF RADIO

WAVES IN THE TROPOSPHERE

E. J. Dutton and G. D. Thayer

Eight methods of computing atmospheric refraction of radio rays

are discussed with appropriate theoretical background. These methods

are:

(1

(2

(3

(4

(5

(6

(7

(8

The high-angle, or astronomical, refraction case

The statistical method

The low-angle, or terrestrial, refraction case (Schulkin's method)

The four-thirds earth model

The exponential model

The initial gradient correction method

The departures -from-normal method

A graphical method (Weisbrod's and Anderson's method).

Sample computations are included for each of the above methods

TECHNIQUES FOR COMPUTING REFRACTION OF RADIO

WAVES IN THE TROPOSPHERE

E. J. Dutton and G. D. Thayer

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 1. Figure 1 defines the variables of interest. The

total angular refraction of the ray-path between two points is designated

by the Greek letter t, 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 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 co-ordinates:

n r cos 6 = n r cos 9 ; (1)

the geometry and variables used with this equation are shown in figure 2.

With this assumption t may be obtained from the following integral:,

n2

, 2

1,2

=

" j *<**T'

V e

i
r =a + h

r r=a + h

FIGURE 1. Geometry of the Refraction of Radio Waves
-2-

which can be derived as shown in Appendix I, and also by Smart [ 1931] .

The elevation angle error, e , is an important quantity to the radar

engineer since it is a measure of the difference between the apparent

elevation angle, 9 , to a target, as indicated by radar, and the true

elevation angle. Under the same assumption made previously c is given

as a function of t, n, and by

_ cos t - sin t (tan 9) - — ^

€ = Arctan J s V . (3) — tan 9 - sin t - cos t tan 9

n o

s

The apparent range to a target, as indicated by a radar, is defined

as an integrated function of n along the ray path,

R * h

R = CndR = f

B*i

(4) e J J sin 9

However, the maximum range error (R minus the true range) likely to

be encountered is only about 200 meters, hence the evaluation of (4) is

not of great importance unless one is dealing with an interferometer or

phase-measuring system.

The preceding material should suffice to show the importance of

radio ray bending in radar systems evaluation and allied types of radio

propagation work. Unfortunately, the integral for t (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 t without full knowledge of n as a function of height, and the construction of model n- atmospheres in order to evaluate average atmospheric refraction.
-3-

The following sections are devoted to a discussion of these methods.

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)perkm <0 N kc

where refractivity, N, is defined as N = (n-1) x 10 and f is the carrier

K.C frequency in kilocycles [Bean and Thayer, 1959]. 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 of refractive index exists with a vertical decrease of N greater

than 157 N-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 [Booker and Walkinshaw, 1946] . Taking into account refractive index gradients, a cutoff frequency may be derived for waveguidelike propagation through a ducting layer [Kerr, 1951] .

In addition to the above limitations, it should be remembered that

the postulate of horizontal homogeneity, made in order to use equation

(1), is not realized under actual atmospheric conditions; some degree
-4-

of horizontal inhomogeneity is always present.

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 (2) in terms of refractivity, N, is equal to

N,, 0,

1,2 J cot 9dN • 10

-6 (5)

N1' 9

1 assuming n — 1 in the denominator. Integration by parts yields:

N2' 9

2

1,2 "J

cot 0dN • 10

N

r

e

i

N2' 6

2

N cote • 10

92' N

2

-I

N

~2~

sin 9

de- 10

(6)

Note that the fraction, N/ sin 9, becomes smaller with increasing 9 for

values of 6 close to 90 . If point 1 is taken at the surface, then 9. = 9

1 o and NIs= N . Then for 9 = 10°, N_, = and 9^ = tt/ 2, the last term of o 2 2

(6) amounts to only 3. 5 percent of the entire equation, and for the same

values of N_ and 9. but with 9 = 87 mr (~ 5 ) the second term of (6) is L L O

still relatively small (~ 10 percent). Thus it would seem reasonable to

assume that for

9 > 87 mr (-5°),

o the bending, t , between the surface and any point, r, is given sufficiently

well by

1,2

N cot 9 x 10

N , 9

r r

N , 9

s o

(7)
-5-

or t , = N cot x 10" -N cot x 10" . (8)

1 , 2 s o r r

The term -N cot 0x10 is practically constant and small with resr r pect to the first term, for a given value of and r, in the range ^ 87 mr. Thus t, _ is seen to be essentially a linear function of o 1,2

N in the range ^87 mr. For bending through the entire atmosphere,

(to a point where N = 0), and for > 87 mr, (8) reduces to • r o

t = N cot x 10" . (9)

s o

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 .

4. THE STATISTICAL METHOD

Another method for determining high-angle bending is the statistical

linear regression technique developed by Bean, Thayer, and Cahoon

[ 1959] • It has been found that for normal conditions and all heights the

right-hand integral of (6) is approximately a linear function of N (0 ,

r constant) for > 17 mr (~ 1 ) and that the second term of (8) tends to o be constant. Thus (6) reduces to a linear equation,

t _ = bN + a, (10) 1,2 S

where b and a are constants (as in tables I - IX) and N is the surface

s refractivity.

The form of (10) is very attractive, since it implies two things:

1) t may be predicted with some accuracy as a function only

1 , c of N (surface height and constant), a parameter which may be

observed from simple surface measurements of the common
-6-

meteorological elements of temperature, pressure, and humidity.

2) The simple linear form of the equation indicates that, given a large number of observed t versus N values for many values

of h and 6 , the expected (or best estimate) values of b and a can

be obtained by the standard method of statistical linear regression.

This is what was done to obtain values listed in tables I - IX.

Tables I - IX also show the values of the standard error of estimate, SE, to be expected in predicting the bending, and the correlation

coefficients, 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 t from the tables (for desired N ) 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 t value and the SE value directly for a given height.

5. SCHULKIN'S METHOD

Schulkin has presented a relatively simple, numerical integration

method of calculating bending for N-profiles obtained from ordinary significant-level radiosonde (or "RAOB ") data [ Schulkin, 1952]. The

N-profile obtained from the RAOB data consists of a series of values of

N for different heights; one then assigns to N(h) a linear variation with

height in between the tabulated profile points, so that the resulting N

versus height profile is that of a series of interconnected linear segments

Under this assumption, (2) is integrable over each separate linear

N-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:
At (rad) = - \ cot dn =

1 > " J

-7-

2(n

l - n

2

)

n

1 ,e

l

tan + tan

or

2(N - N ) x 10" 3

AT

l,2 (mr) ^

Tan 0, + tan 6. " (U)

For the conditions stated above, this result is accurate to within

0. 04 percent or better of the true value of At , an accuracy that is

usually better than necessary. Thus it is possible to simplify (11) further

by substituting for tan 0; this introduces an additional error that is less

than 1 per cent if is under 10 (~175 mr). Now (11) becomes

2(N

X - N

2

) AT

i,z (mr) =

e +0 ' Vnde

2

inmr - (12)

1 £

(mr) (mr)

where may be determined from (33), Appendix I.

The bending for the whole profile can now be obtained by summing

up the At for each pair of profile levels:

1 » "

k=° (mk

r) <!&)

This is Schulkin's result. The degree of approximation of (13) is

quite high, and thus most recent "improved" methods of calculating

t will reduce to Schulkin's result for the accuracy obtainable from RAOB

or other similar data. Thus, provided that the N-profile is known, (13)

is the most us eful 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,
-8-

but have some additional desirable features; e. g. , the method of Anderson

[ 1958] employs a graphical approach to avoid the extraction of square

roots to obtain 6. . k

6. THE FOUR-THIRDS EARTH MODEL

Perhaps the earliest attempt to utilize a model of atmospheric

refractive index for the solution of problems in microwave radio propagation dates back to 1933, when Schelling, Burrows, and Ferrell [ 1933]

published their discovery that radio propagation through an atmosphere

with a constant refractive index gradient of -%a, where "a" is the radius

of the earth, was equivalent, for purposes of calculation, to radio propagation over an airless world of radius 4a/ 3. This was a great simplification, since it meant that for the calculation of radio field strengths,

etc. , the atmospheric refractive index could be ignored provided that

4a/ 3 was entered in the calculations instead of "a" wherever it appeared.

This method quickly became known as the "four-thirds -earth, " and has

formed the backbone, until very recently, of radio refraction calculations

since its introduction.

The 4/3 earth method, as originally proposed, suffers from two

serious shortcomings, only one of which may be overcome by use of this

kind of a model. They are as follows:

a) The gradient of refractive index near the earth's surface that

is implied by the ratio 4/ 3 ( —4 ON/ km) is valid only for certain

areas and at certain times, e.g., temperate areas in winter; the

gradient implied is less than average for temperate climates in

summer, always much below average for tropical climates, and

greater than average for arctic climates.
-9-

b) The gradient of refractive index implied by the 4/ 3 earth model

is nearly constant, decreasing with height at uniform rate, and

thus the values of refractive index implied quickly reach unrealistically low values; free space value (N = 0) is attained at about

eight -kilometer -height.

% The first of these drawbacks may be avoided by a simple modification of the original 4/ 3 earth theory. All that is required is to pick a value of the "effective earth's radius factor", e.g. , 4/3, which is consistent with the meteorological data that are available for the area under

consideration. Hence, a location that has a normal gradient of refractivity

near the surface, of -100 N-units/ kilometer, would have an associated

effective earth's radius factor of ll/4, and the effective earth's radius

for this location would be lla/4, or about 17,500 km.

The shortcoming of the 4/ 3 earth model listed under "b) u above is

an objection to the effective earth's radius theory in general, and hence

cannot be avoided by a change in the size of the effective earth's radius

factor (except by making the factor a function of height).

With the above considerations 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 one, or at most two,

kilometers above the earth's surface, the effective earth's radius method

should be used to solve the associated refraction problems. The user

should refer to the tables in Appendix II, where effective earth's radius

factors are tabulated along with other refractivity variables. Table A-l

may be entered with N and table A-2 may be entered with AN(N ) subtractec

s s from the N value at one kilometer above the surface. In both these tables
-10-

Probable Errors When using Effective Earth's Radius Model for a Ray

with 9 = 0; using h = d / 2 ka.

True Height of

Ray for an

Exponential Profile:

1.000 km

2.000 km

Calculated Height and Percent Errors

For Normal Conditions

dN/dh = -50/ km

0.987 km, 1. 3% error

1.950 km, 2.5% error

For Superrefraction:

dN/dh = -100/ km

~ 0.95 km, 5% error

~1.8 km, 10%, error

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.

When the effective earth's radius treatment is used, height is

calculated as a function of distance, for a ray with =0, with the

2, equation h = d / 2 ka, 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 table

above will serve as a guide to 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.

7. THE EXPONENTIAL MODEL

An exponentially decreasing refractive index in the troposphere

has been recognized for some time [Bauer, Mason, and Wilson, 1958;

Anderson, 1958] . Recently Bean and Thayer [ 1959] introduced an exponential model for N(h) based on an analysis of observed profiles from many

climatic areas (mostly in the U.S. ). With this model, the value of N as a function of height is given by the equation

N(h) = N exp{-c h} , s e

(14)
-11-

where c is a function only of N , and is thus a constant for any given

e s profile, and h is the height above the surface. The quantity c can be

related to N and AN by

s

C

e

= ln

{ N-fW }

' (15)

so that a relationship between N and AN would fix each exponential

s profile of form (14) as a function of the single variable, N . A brief

description of the development of such a relationship is given in the

following paragraphs.

If it is assumed that N(h) is indeed an exponential function of

height, then the gradient of N(h) would also be an exponential function of

height. The most extensive amount of data with which to evaluate the

coefficients in the exponential is that of AN (the value of N at one kilometer

minus the surface value, N ) which has received wide application in radio

s propagation problems . Thus one would expect

f£-=kiexp{-k

2

h} (16)

to take the form

AN - k exp{-k

2

},

for the special case of Ah = h = 1 kilometer. Examination of available

AN data reveals that k is dependent upon N , i.e., the higher the surface

" s value of N the greater the expected drop in N over one kilometer. Further

examination indicates that

k = k N , 2 3 s and the resultant equation,

AN = k exp{-k N } , (17)

1 3 s
FIGURE 2. Bending Geometry on a Spherical Earth with

Concentric Layers
-12-

may be solved by least squares. The least squares determination is

facilitated by converting (17) to the form

In I AN I = -k N + Ink , (18)

3 s 1

tnat is, expressing the natural logarithm of AN as a linear function of

N . The values of k, and k„ are established from 888 sets of 8-year

s 1 3

means of AN and N from 45 U.S. weather stations. The results of this

s, study are shown graphically in figure 3, and the least squares exponential fit of AN and N is given by

AN = -7. 32 exp{0.005577N } . (19)

With this equation the C.R.P. L. Exponential Reference Atmosphere

[Bean and Thayer, 1959a] is determined; the profiles are completely

defined by equations (14), (15), and (19).

Ray tracings have been computed for this model covering more

than the normal range of N , and the results found in Tables X through

XVII may be used to predict r for any normal combination of N , , s o and height [Bean and Thayer, 1959a] .

The exponential atmosphere is considered to be an adequate solution

to the bending problem for any larger than about 10 milliradians and

all heights above one kilometer.

8. 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 = -l/a (the reciprocal of the earth's radius), then

the equation for t is indeterminate, an expression of the fact that the ray path

remains at a constant height above the earth's surface. This is called
~^NN

Ifuin>3

•\

x - c

II1

"."'A" ' : *• \ L-

>'.•' . -— . .'. *.

V.°: .'•.

.A.- • : #iV' '

-V' • " v.*. ". " 1111 ' \' i\ ... \ \ 8 o o o o o o (7> oo s (o 10 st- o OCO m

Oo

rOO

00

rO> O

rOO10

rOOlO

roO

rOOrorOO

CO

rOO

roO

rOO

en

COooo

COO

COOCD

COOm

COO

COOro

CO

</>

Co• H

CO

CO

<DU

DO

0)

00WoI—

<
-13-

ducting, or trapping of the radio ray. The effect of anomalous initial

N-gradients on ray propagation at elevation angles near zero, and for

gradients less than ducting, (| dN/ dh | < 15 7 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 reference

atmosphere) to account for anomalous initial N-gradients, assuming that

the actual value of the initial gradient is known [Bean and Thayer, 1959b] . The result is

h h s o 100 s o 100 s o where t (N ) = t at height h, for the exponential reference atmosphere

corresponding to NSo, and N * is the N for the exponential reference

^> atmosphere that has the same initial gradient as the observed initial

gradient; f is t at a height of 100 meters.

This procedure has the effect of correcting the predicted bending

by assuming that the observed initial gradient exists throughout a surface layer 100 meters thick, calculating the bending at the top of the

100-meter-thick layer, and then assuming that the atmosphere behaves

according to the exponential reference profile corresponding to the observed value of N for all heights above 100 meters. This approach

s has proved quite successful in predicting t for initial elevation angles

under 10 milliradians, and will, of course, predict trapping when it

occurs

.

9. 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 is given by Bean

and Dutton [ I960] . This method is primarily intended to point out the

difference between actual ray-bending and the average bending that is
-14-

predicted by the exponential N(h) profile and is a powerful method of

identifying air mass refraction effects.

The exponential model described in subsection 7, 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 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 N at any height, h, and the drop in N from the surface value,

N , to the height, h, which is predicted by the exponential profile for the

s given value of N .

Thus

A(N , h) = N(h) + N (l-exp{-c h}). (21)

s s e Thus (2.1) adds to N(h) the average decrease of N 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 , h) for this

profile would be equal to N for all heights. The above analysis shows

that the difference between A(N , h) from N , 6A(N h), is a measure of the

s s s departure of N(h) from the normal, exponential profile:

6A(N ,h) = A(N , h) - N = N(h) - N exp{-c h} . (22)

s s s s e

It seems logical that the application of the A unit to bending would indicate

the departures of bending from normal, in some way, just as it indicates

departures of refractivity, N, from normal. This is indeed the case as

can be seen in figure 4, where for an N = 313. exponential atmosphere,

A(3l3.0,h) is plotted on one set of graphs for various typical air masses,

and the corresponding bending departures from normal are shown in the
00

a:

QC 1 1 1 1 1 ~ LU

0-1-

oz^ 1 1 1 1 1 o

ro

roO

roO

<d"

roO

CM

ro

ro

ro< i r //

CD

auo2

sj"

au

tio CM -4-> n5

•r-l >Q

O ti O oo

<r

CD 2

ti ^r *—

(0

CD i CM h

CO

CO

C(5 i s

ro

O ro <i—1

1- o

a, CJ H i o CO

<D 1—< • iH

CM•

a. i •rH

tii < o

CMI i—t

Li

ro cm

—

(wv|) 1H9I3H
-15 -

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 4 that the similarity exists, although

it is less, for higher initial elevation angles. The similiarity 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 N-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 influenced;

hence a loss of similarity arises at large heights above the earth's

surface.

If (21) is differentiated and substituted into (2) the following equation

results:

T

0,h = TN

(h) + I

- 9,1 1

t AA(Ns^ Xl °~ 6

' (23)

(rad) (ra

Sd) k=0 (rad) frad) k

where

AA(N ) = AN(h) + A[N {l-exp(-c h)} 1 = AN(h) +N c exp(-c h)Ah

s s e see

t (h) is the value of t tabulated for various atmospheres in tables X - s XVIII, and are in milliradians and must be from the N exponential

atmosphere used. AA(N ) is obtained from subtraction of the A value at

s layer level, k, from the value of A at layer, k+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 N [ 1-exp {-ch} ] for the same height which may

be obtained from figure 5. Since t (h) has been calculated only for a few

s of the exponential atmospheres, these being the N - 200.0, 252.9, 289-0,

s 313.0, 344.5, 377.2, 404.9, and 450. atmospheres, one of these
X

CD

LU

CO

(f)

LU>oIX

CDI

(/) om

roOo

roO

LO

1CNJ o

LO

CO

I- o 2 oC\J -z.

IT) 1 1 *—^ w

-C rt o

LO o

Q. 1—1 X h

CD1 . . oo

</> o

w>1 Nl V1H9I3H
-16-

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 XVIII are shown the ranges

of the gradient for the choice of a particular exponential atmosphere.

10. A GRAPHICAL METHOD

Weisbrod and Anderson [ 1959] present a handy graphical method

for computing refraction in the troposphere. Rewriting and enlarging

(11), one obtains

V

N

k+i " Nk

T(mr) = Z 500 (tan 0, Man 9, ,,) ' {M)

where t will be the total bending through n layers. Terms for the denominator can be determined from figure 6. Equation (24) is essentially

Schulkin's result with only the approximation, tan 9 = , for small

angles omitted.

The procedure in using figure 6 is as follows: Enter on the left

margin at the appropriate N - N(h). Proceed horizontally to the proper

height, h, interpolating between curves if necessary. Use the solid

height curves when N - N(h) is positive and the dashed curves when N

5 S

- N(h) is negative. Then proceed vertically to the assumed and read

500 tan along the right margin.

11. SAMPLE CALCULATIONS

The following problem will serve to illustrate the application of

the various methods of calculating bending.
1000 1000

FIGURE 6.
-17-

A particular daily set of RAOB readings from Truk in the Caroline

Islands yields the following data:

height above the N value

surface (km) (N units)

0.000 400.0 = N

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 mr, 52.4 mr (3°) and 261.8 mr (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 ,„_, „T

. ,, Ah"

= O40 —102.9 N units/ km,

and this is a decrease of N per km that is less than the -15 7 N units/ km

required for ducting, no surface duct is present. However, should a

surface duct have been present, it would have been necessary to calculate

the angle of penetration,

9 = N/2[N - N -156.9 (Ah)(in km)] , p s h

J
-18-

to find the smallest initial elevation angle that yields a non-trapped ray.

Any initial elevation angle less than c annot be used in bending calculations .

(a) Schulkin's approach of (13) yields the results shown in table

XIX for mr, table XX for 10 mr, table XXI for 52-.4-mr and table XXII

for 261.8 mr, where 0,,, is determined from (33) in Appendix I using

r. = a + h, , and a is the radius of the earth, k k

a = 6370 km.

It should be remembered that = 0, 10, 52.4, or 261.8 mr only for the

first-level calculation, and that thereafter 0, is equal to the 0, ,, computed

k k+1

for the preceding layer, e.g. , for the second layer of table XIX (0 = Omr),

0. = 6. 15 mr, which is the 0, ,, calculated for the first layer, k k+1 7

(b) The exponential model solution may be found by using tables X

through XVII. Interpolation will usually be necessary for N , , 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 interpolation will not be necessary, thus reducing from 7 to 3 the number of

interpolations necessary. Since in the problem for N = 404. 9, h = 10.

km and =10 mr

o

T n , n ^ ,, n v - l5 - 084 mr,

0, 10. (lOmr)

and at h = 20. 00 km, 0=10 mr.

o

Tn ?r> nnnn \ = 15.946 mr,

0, 10 . 00 ( 10 mr)

and thus by linear interpolation for h = 10.870 km, = 10 mr.

o

T

o, 10.870, no mr, 15 - 084 + < 15 - 946 - i5084 > iornrs-

= 15. 159 mr.
-19-

Similarly for N = 377. 2 in the exponential tables,

1 s T , % = 13. I20mr.

0, 10.870, (10 mr)

Again using linear interpolation, but now between the N = 377. 2 and

s N = 344. 5 atmospheres, the desired value of t at 3. 270 km for N = 360.

s r s and =10 mr is obtained,

o Thus

400.0 - 377.2

T

0, 10.870 (10 mr) = 13 ' 12 °

+ < 15 ' 159 " 13 ' 12 °> 404.9 • 377.2

= 14. 798 mr.

For the G = 0,52.4, and 261.8 mr cases, by similar calculations, using

o linear interpolation:

t„ n n ^ n ln v = 21. 386 mr,

0, 10.870, (0 mr)

t^ ._ n ^„ ,__ . . = 5.816 mr,

0, 10.870, (52.4 mr)

T

0, 10.870,(261.8 mr)

= L270inr '

(c) The initial gradient correction method may be used if one determines the N * which corresponds to the observed initial gradient and

then applies (20). The initial N gradient is -102.9 —

, which, as km

can be seen from table XVIII, corresponds to the N 450.0 exponential

atmosphere. Therefore, using the exponential tables of Bean and Thayer

[ 1959] and (20) to determine the bending for the =0 mr case, one finds

by linear interpolation

T

io, 000 (o) = T

io, coo'

400 ' °- ° mr) + I T

ioo

(45 °- °- 0mr) - T

ioo

(40c '"ml|l

= 21.309 mr + [5.908 - 3.657] mr = 23.560 mr.
-20-

The t (400.0, mr) is determined by linear interpolation between the

404. 9 and 377. 2 atmospheres. At h = 20. km as given in the tables

T

20,000 (Omr) = T20, 000 (4 °°- °" ° mr) + [

100 <«0. 0, mr)-T

1()0

(400. 0, mr)

= 22. 191 + [5.908 - 3.65 7] mr

= 24. 442 mr.

Hence by linear interpolation

o? cLr, , \->A aai i, /nl 10,870 - 10,000

t,« „„^,„ = 23. 560 + [ 24.442 - 23. 560] . . n nnn — nnn 10.870 (Omr) L J 20,000-10,000

= 23.637 mr.

The b endings for 9 = 10, 52.4, and 26 1 . 8 mr are as given below:

V 10;T

10,870(10)

=15053mr

I 6o

= 52 - 4i T

10, 870 (52.4, = 5.864mr

'

e o

= 261 - 8:T

10,870 (261.8)

=»-280mr.

j

(d) To use the departures -from-normal method of determining

bending it is first necessary to know the atmosphere which must be used

for the calculation. In the problem,

dN

dh

I . . . = 102.9 N units/ km,

initial

which is within the range of the N = 450.0 exponential atmosphere, as

can be seen from table XVIII. Thus one will use table XVII to determine

the O's and the t's in the N = 450. exponential atmosphere, or one can

use the exponential atmosphere tables of Bean and Thayer and (33) in

Appendix I.
-21-

For an N = 450. atmosphere:

s ttvt /n ,(10.870) - 30.776 mr, N (0 mr)

s

t .(10.870) = 19.414 mr, N (10 mr)

s

t\t #,= , a ^ ( 10 - 87 °) = 7.024 mr, N (52. 4 mr)

s

tm OAi a \ (10.870) = 1.506 mr. N (26 1 . 8 mr)

s Equation (33) of Appendix I should be used for the 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 interpolation will suffice. Proceeding in table XVII with (33) of

Appendix I for the first layer at h = 0. 340 km and the 9 =0 mr case:

^T 2(r

i - r ] 6

e - e or+ — — x 10 - 2(n si- n)

o

= 6.388 mr. The remaining 9's for the various layers are shown in

table XXIII. To determine the value of A at the bottom and top of the

layer, one makes use of (21) or figure 5. First, however, one must

determine the value of c in (21) to be used. Usually interpolation will

be necessary in table XVIII, but in the N = 450. case it is not possible,

s and thus the straight N = 450.0 exponential atmosphere values are used.

From (21)

A(N ,h) = N(h) + N [ l-exp(-ch)],

s s and for the layer running from h = to h = 0. 340 km, figure 5 yields

377.2 [ l-exp(-cO)] = 0.0,
-22-

and 450. [ l-exp(-c x 0. 340)] = 32. 8,

and therefore,

and

A(450. 0, 0) = 400. + = 400. 0,

A(450.0, 0. 340) = 365 . + 32. 8 = 397. 8,

whence

AA = A(450. 0, 0. 340) - A(450. 0, 0) = 397. 8 - 400. = -2. 2 N units

Therefore, the departure term of (23):

N

-2

k k+1

AA(N ) s

k+1

N,

becomes

0+6. 388

-2.2

365.

400.

= + 0.689 mr.

The remaining calculations are tabulated in table XXIII for the =0 mr

o case, in table XXIV for the 0=10 mr case, in table XXV for the

o =52.4 mr case, and in table XXVI for the = 26 1 . 8 mr case. The

o o sum of the departures for the mr case is

n ?

k^0

k+1

S+i

AA (N ) s

= - 5. 335 mr.

N.
-23-

(e) Determination of the bending is required in part (e) of the

problem by using regression lines. By (10), using table VII and VIII,

it is found for the =0 mr case that at 10.0 km (from table VII) o t = (0. 1149)(400.0)- 18.5627 ± 7.5227

\J j I u • u

= 27. 3973 ± 7.5227 mr,

and at 20.0 km (from table VIII)

T

0, 20.0=(0. 1165)(400.0) - 17.95 73 ± 7.5131

= 28.6427 ± 7.5131 mr.

Thus, by linear interpolation

T

1, 2

= T0,10.87= 27 ' 3973 M28.6427 - 27. 3973) Ij^fif^aT. 5227

T (7.5227- 7.5.31)

10.87-10.00

20. 00 - 10.00

r = 27.5056 ± 7. 5218 mr.

1 , L.

Similarly for the remaining 0's,

t. ,-._ . = 13.9548 ± 0.9701 mr , 1, 2(10 mr)

t, 0/co . = 5. 2186 ± 0.0817 mr,

1, 2(53.4 mr)

T

i 7/7A1 « \ = 1-2695 ± 0.0158 mr.

1 , 2(26 1 . 8 mr)

(f) Determination of the bending by means of the graphical method

of Weisbrod and Anderson yields, from figure 6, for 500 tan for the

first layer:

h(m) At = mr At = 10 mr At =52.4mr At = 261.8 mr

o o Q

0.000 0.0 5.0 26.2 134.0

0.340 3.0 5.8 27.0 134.0
-24-

which yields for the bending in the first layer.

At0 = mr Ate = 10 mr Ate = 52 mr Ate = 26 1 . 8 mr

o o o o 11.67 mr 3.24mr 0.66 mr 0. 13 mr

Similarly, the bending for the entire profile may be obtained, and shown

to be At 6 =0 mr At =10 mr At B = 52.4 mr At 6 =261.8 mr

24.42 14.00 mr 5.32mr *1.18mr

The answers to the several parts of the problem are summarized

in the table which follows on page 25. Bending values for the assumed

profile, from a method which exponentially interpolated layers between

given layers and then integrated between resulting layers, 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 6 =0 mr case and not so much for the = 26 1 . 8 o o mr case is the fact that the accuracy of the regression line method increases

with increasing initial elevation angle, Q . It must be remembered that the

statistical regression technique, like the exponential model, is an adequate

solution to the bending problem for all 6 's larger than about 10 mr, and

all heights above one kilometer.

The reason that the answers in part (f) and part (a) agree more

closely than with any other of the answers is because (24) is, as mentioned

before, Schulkin's result with only the approximation, tan 6 = 9, 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 8 shows

that exponential bending predicts accurately within 1 percent of the average

bending for these five-year means. Figure 7 shows the r.m.s. error in
CVJ

; - <£ *-

1\l .. - — CVJ . Pirvi -cd : o ^

"II |5 — m. ft ! I._ •^ J % ! t ^ i

iiE

£

o > Q5

lit?"

...

(

! ^!*

! -J"

-r-.-f-V-

- - —

..._._ _- —--

Regression

1 1

.... .ft I-

! -J,.L.

—

1 ' %./

j_ i . *< N ........

L._. . i Ph ! I —1— 1 71 ..... i . ill

i s> n

J j i V // V r V •/ &

II —

*ft

, 1

11 -ft—

I ; v. , a & ftjfcl/l £* II-4./-.-4 - y

v

/"*

>

: ST

>ki

2? • / /

Q i H La

r ^ ,s>

S> " ~ £ S-

""

- - .ft ,* - jo ki

\ V / -a

5

/

,_.

.V,

'l\ JS f —

U<- \ y \

i

<

i -••""* v^. <

p

o

CD

8

O

<

<

or

8 ?

o

8

CD

CVJ Si

CVJ

CVJ O CO CD ST CVJ CM — — — —

GO CD CVJ o

(0 -ao

<0

V

uH

>.

Co

-uU

«JM

d)

W>

fl

• iH

»->o

•H

u

5ho

uuW

to

0)y

W

o

SIATU %
CSJ

oo

CO

CVJ

oo </)

CO

OsJ

co

if) ^j

CC (0 LU o

s \- 4J LU <

i—

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t->

C!O

•i-i +-> om0)

a

o

CO

aoU

ooW

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CNJ

LO LO

SNviavumiw ni -*-
-25-

Summary Table of Refraction Results for the Sampl e Computation

Problem Method Bending Bending Bending Bending

Part used inmrat inmrat inmrat inmrat

6 = Omr =10mr G = 52.4mr 9 =261. 8mr

o o o o

a. Schulkin's

Method 24.248 14.008 5.341 1.196

b. ExponentialModel 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

predicting bending at various heights as a per cent of mean bending (not

including super -refraction)

.

In summary, it is recommended that the communications engineer

either use the statistical regression technique or the exponential tables of

Bean and Thayer [ 1959] 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.
-26-

REFERENCES

Anderson, L. J. (1958). Tropospheric bending of radio waves, Trans.

AGU39, 208-212.

Bauer, J. R. , W. C. Mason, and F. A. Wilson ( 1958). Radio refraction

in a cool exponential atmosphere, Tech. Report No. 186, Lincoln

Laboratory, MIT.

Bean, B. R. , G. D. Thayer, andB. A. Cahoon (1959). Methods of predicting the atmospheric bending of radio rays, NBS Tech. Note

No. 44.

Bean, B. R., and G. D. Thayer (1959a). Models of the atmospheric radio

refractive index, Proc. IRE 47, No. 5, 740-755.

Bean, B. R. , and G. D. Thayer (1959b). CRPL exponential reference

atmosphere, NBS Monograph No. 4.

Bean, B. R. , and E. J. Dutton,( I960). On the calculation of non-standard

bending of radio waves, J. Research of NBS 63D, 259-263.

Booker, H. G. , and W. Walkinshaw (1946). The mode theory of tropospheric

refraction and its relation to wave guides and diffraction, pp. 80-

126 of Meteorological Factors in Radio Wave Propagation. Report

of a conference held on 8 April 1946 at The Royal Institution, London,

by The Physical Society and The Royal Meteorological Society.

Kerr, D. E. (1951). Propagation of Short Radio Waves, pp. 9-22 (McGrawHill Book Company, Inc. , New York).

Schelling, J. D., C. R. Burrows, and E. B. Ferrell (1933). Ultra-shortwave propagation, Proc. IRE 21, 429-463.

Schulkin, M. (1952). Average radio-ray refraction in the lower atmosphere,

Proc. IRE 4j0, 554-561.

Smart, W. M. (1931). Spherical Astronomy, Chapter III (Cambridge University Press, London, 1931).

Weisbrod, S., and L. J. Anderson (1959). Simple methods for computing

tropospheric and ionospheric refractive effects on radio waves,

Proc. IRE 47, No. 10, 1770-1777.
-27

APPENDIX I

The approximate relation between 9 and is derived here. This

J. w relation holds for small increments of height and small 0's. The relationship was used in making all sample computations in preceding sections

Since for small 6's

2

cos 6 ^ 1 - —- and cos 6 S 1 - — , (25)

and knowing „ , le . -,

.

6

r - r + Ah (figure 2)

then substituting in (1) yields

2 2

2\ /

6

n.(r. + Ah)( 1 -—> Snr.fl.i ) . (26) 2* 1 'V 2 J 11 I 2

2 2 2

or e

2

9~

9^

Yi +n

z Ah -ViT-n2

Ah T

~ n

i r

i "Vi T ' < 27 >

Dividing by r

Z Z Z n Ah n 9 AU 9, 9, ,2 2 2 Ah 2 ^ 1 n + - — -n — - n - n — . (28)

2 r 2 2 r 2 1 12

2

Ah 9

2

Since the term -n_, —— is small with respect to the other terms of

2 r 2

(28) it may be neglected, and thus:

2 2 n

2

Ah 9

2

G

l n

2 + ~ n2 — " n

l ~ ni T (29)

9

2

9

2 n Ah

or 2 ^ 1 2 , ' n

2 T ~ " n

i T " ~7^ + {n

i - n

2

)

* (30 >
-28-

n r

n

z

If one now divides both sides of equation (30) by n and assumes

n

1 — n -n and —— — 1 (30) may be arranged to yield

1 2 n2

e

2 , e^ + ^-z(vn2 , (31)

Writing (31) in terms of N units,

^

0, (mr) s ©^ + i^- x 10

6 -2(N-N ) 2 1 r 12 (32)

if is in milliradians

.

Generalizing (32) for the kth and the (k+l)th layers,

9

k+i (mr). O^mr) + J^-tlA_. x 10

6 -2^-N^) (33)

Also from the geometry shown in figure 2, a useful relationship

for t _ can be obtained. Tangent lines drawn at A and B will be respec

l , c tively perpendicular to r and r , since r and r describe spheres of

X. C* A. w refractive indices n, and n concentric with o. Therefore,

angle AEC = angle AOB = cj>

;

also, in triangle AEC

angle ACE = 180° - angle CAE - angle AEC = 180° -© $ . (34)

But from triangle DCB

angle ACE = angle DCB = 180 - r -0

1 j —

<

£-, (35)
-29-

Since angle DBC and are vertical angles, (34) and (35) are equal.

Thus

180° - t

1; 2

- e

2

= 180° - b

1 - +, \ 2

= *+(\-e2

)- (36)

Now since <j> in radians = d/a, where d is distance along the earth's

surface:

Tl,2= f

+(9l- e 2»- (37 »

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 9, and (in radians) at the

1 o beginning and end of the layer.

If one considers figure 9, Snell's law in polar coordinates, and the

refraction formula (equation (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 9 let C be the center of the earth,

O 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 O. At P 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 = 0, and angle LPS =

+ d0, and angle Q'PF = i|j. Then, since the radius CP is perpendicular at

* The authors feel that the derivation in the previous edition was not, in

essence, a strong enough derivation of Snell's law in polar co-ordinates.

The authors also feel that more detail is contained in the derivation than

in some other derivations, yet it remains mathematically simple. By

employing the calculus of variations, Snell's law in polar -co-ordinates

can be obtained from Fermat's principle.
EARTH'S

SURFACE

FIGURE 9. Differential Geometry of

Radio Ray Refraction
-30-

P to the bounding surface between layers M and M', by Snell's law we have

(n+dn) sin [90° - (9+d0)] = sin i]j. (38)

Now from the triangle CQP , in which CQ = r and CP = r+dr, and angle

CQP = 90 +0. we have, from the law of sines,

r sin (90° + 0) = r sin (90°- 0) = (r+dr) sin i|i. (39)

Eliminating sin ty from equations (38) and (39), we then have

(n + dn)(r + dr) sin [90° -( 0+d0)] = nrsin (9O°-0),

or

(n + dn)(r+dr) cos(0 + d0) = nrcos 0. (40)

Multiplication of the (n + dn) and (r + dr) terms, ignoring differential

products, yields

(nr + ndr + rdn) cos (0 + d0) = nrcos 0,

or

(nr + ndr + rdn) [cos cos (d0) -sin sin (d0) ] = nrcos 0. (41)

Since cos (d0) = 1, and sin (d0) = d0, another multiplication, again ignoring

products of differentials, yields:

ndr cos + rdn cos - nr sin d0 = 0,

or, dividing all terms by nrcos 0, — + — - tan d0 = 0. (42) r n

Now if (42) is integrated between any two thin layers of refractive indices

n and n , whose radial distances from the earth's center are r and r , * The assumption involved in this triangle is that the path of the ray in M'

is a straight line, 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 (eq. 40) in polar coordinated

can be obtained by the use of the geometry of figure 9, not the finite form

(eq. 43) which has the same appearance.
-31-

and the initial elevation angles of a radio ray entering the layers are

1 and :

r 2 n ®

r r

z

a r

z r

? n-> cos 9 9 dr

+

( dn . \

^

We = in -i + |n -?. + |n J = J r J n J r n cos

r

i n

i e

i

or, taking antilogs of both sides,

r

2 n2

C ° S 9

2

r n cos

whence

n r cos = n r cos , (43)

which is Snell's law for polar co-ordinates, given by (1).

In figure 9, it can be seen that

PF dr tan8 ^oF=^' < 44 > where 4> is the angle at the earth's center between r and COZ. Substituting

(44) in (42)

dn

tan 0d<f> + — - tan 0d0 = 0,

n

or

(d<j> - d0) tan = - — . (45)

n

Since, by considering equation (37) for infinitesimal angles.

dT = d<|> - d0,

or, in (45)

, an

dT tan = - — , n

~ dn .... dT = - cot — . (46)

n

Integration of (46) yields (2).
-32-

APPENDIXII

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 aneie, 0:

4 milliradians

4 mr. < 9 < 100 mr.

o

9 >100 mr.

o Errors in r, € (in milliradians):

± 0.00005 mr.

± 0.000005 mr.

± 0.00004 mr.

nearlyindependent

of N .

N =

s • 450 404.8 377.2 344.5 313 252.9 200

e =0 ±0.001 0.00065 0.0005 0.0004 0,0003 0.0002 0.0002

o0=1° 0.0003 0.00015 0.0001 0.00008 0.00006 0.00005 0.00005

o0=3° 0.00004 0.000025 0.00002 0.000017 0.000015 0.000013 0.000012

o

Errors in R , R, R , or Ah (in meters):

o e

N = 450 404.8 377.2 344.5 313 252.9 200

s

= ± 5.0 2.7

o >1° 0.4 0.3

o

1.8 1.2

0.25 0.2

0.8 0.65 0.6

0.17 0.15 0.14

Assume that error in AR or AR is ± 0.5% or ± 0.1 meters, whichever

e is larger.
33-

Table A-l

N AN •dN

s

200 22,,3317700 0.

e 118399435 23,

o

6798870 1.17769275

210 23, 6125966 0. 119280212 25. 0488444 1. 18991401

220 24. 9668845 0. 120458179 26. 5007993 1.20315637

230 26, 3988468 0. 121916361 28. 0407631 1.21752719

240 27. 9129385 0. 123642065 29. 6740955 1.23314913

250 29. 5138701 0. 125626129 31. 4065323 1.25016295

260 31. 2066224 0. 127862319 33, 2442030 1.26873080

270 32. 9964614 0. 130346887 35. 1936594 1.28904048

280 34, 8889558 0. 1330782 54 37. 2619112 1.31131073

290 36, 8899932 0. 136056720 39. 4564487 1.33579768

300 39, 0057990 0. 139284287 41. 7852861 1.36280330

310 41. 2429556 0. 142764507 44. 2569972 1.39268608

320 43. 6084233 0. 146502381 46. 8807620 1.42587494

330 46. 1095611 0. 150504269 49. 6664087 1.46288731

340 48. 7541501 0. 154777865 52. 6244741 1.50435338

350 51. 5504184 0. 159332141 55. 7662495 1.55104840

360 54. 5070651 0. 164177379 59. 1038565 1.60393724

370 57. 6332884 0. 169325150 62. 6503054 1.66423593

380 60. 9388149 0. 174788368 66. 4195799 1.73349938

390 64. 4339281 0. 180581312 70. 4267116 1.81374807

400 68. 1295015 0. 186719722 74. 6878887 1.90765687

410 72. 0370324 0. 193220834 79. 2205420 2.01884302

420 76. 1686780 0. 200103517 84. 0434770 2. 15232187

430 80. 5372922 0. 207388355 89. 1769927 2.31525447

440 85. 1564647 0. 215097782 94. 6430240 2.51823286

450 90. 0405683 0. 223256247 100. 4653113 2.77761532
-34-

Table A-2

AN N dN

20 180.226277 .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 40.6771452 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
-35-

Table A2

(Continued)

AN N -dN

66 394.305974 . 183181171 72. 2294300 1.85232456

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. 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
-36-

Table A-3

N -AN -dN

1.0 0.0 0.0 0.0 0.0

1.2 217.689023 24.6471681 0.120160519 26.1576259

1.3 275.037959 33.9367000 0.131692114 36.2203304

1.4 312.297111 41.7747176 0.143600133 44.8459068

1.5 339.003316 48.4839018 0.154339490 52.3215987

1.6 359.298283 54.2941700 0. 163827653 58.8629945

1.7 375.341242 59.3759008 0. 172203063 64.6349115

1.8 388.391792 63.8586055 0.179626805 69.7655765

1.9 399.243407 67.8426334 0.186242834 74.3562234

2.0 408.424907 71.4070090 0. 192172034 78.4878451

2.1 416.304322 74.6148487 0. 197514185 82.2260091

2.2 423.146728 77.5171828 0.202351472 85.6243634

2.3 429.148472 80.1557288 0.206751820 88.7272277

2.4 434.458411 82.5649192 0.210771674 91.5715264

2.5 439.191718 84.7734613 0.214458304 94.1883108

2.6 443.438906 86.8054237 0.217851443 96.6038056

2.7 447.272272 88.6811886 0.220984823 98.8403838

2.8 450.750273 90.4181120 0.223887193 100.9172131

00 523.299600 135.5109159 0.299693586 156.829534

4/3 289.036274 36.6922523 0. 135758874 39.2392391
-37-

Tables of Coefficients, Standard Errors of Estimate, and

Correlation Coefficients for use in the Statistical Method

Table I, h - h s = 0. 1km

eo r b a S.E.

0.0 0.2665 0.0479 -8.7011 6.7277

1.0 0.2785 0.0257 -4. 1217 3.4363

2.0 0.2881 0.0162 -2.3732 2.0960

5.0 0.30.48 0.0073 -0.9181 0.8792

10.0 0. 1915 0.0053 -0.6085 1.0551

20.0 0.2070 0.0025 -0.2639 0.4555

52.4 0.2100 0.0009 -0.0973 0. 1688

100.0 0.2105 0.0005 -0.0507 0.0879

200.0 0.2105 0.0002 -0.0250 0.0435

400.0 0.2107 0.0001 -0.0120 0.0208

900.0 0.2108 0.00004 -0.0040 0.0070

Table II, h - hs = 0. 2 km

eo r b

a S.E.

0.0 0.2849 0.05801 -10.4261 7.5726

1.0 0.2979 0.0348 -5.6431 4.3330

2.0 0.3104 0.0239 -3.5707 2.8357

5.0 0.3415 0.0117 -1.5287 1.2512

10.0 0.2306 0.0073 -0.7436 1. 1990

20.0 0.2550 0.0035 -0.3184 0.5122

52.4 0.2604 0.0013 -0. 1162 0. 1890

100.0 0.2610 0.0007 -0.0603 0.0983

200.0 0.2613 0.0003 -0.0299 0.0486

400.0 0.2613 0.0002 -0.0143 0.0233

900.0 0.2604 0.00005 -0.0047 0.0078
-38-

Table HI,h-hs =0.5km

Qo r b a S.E.

0.0 0.3615 0.0769 -14. 6443 7.6170

1.0 0.3997 0. 0510 -9.0567 4.4954

2.0 0.4369 0.0384 -6.5408 3.0395

5.0 0.5205 0.0228 -3.6605 1.4376

10.0 0.3933 0.0140 -1.9055 1.2733

20.0 0.4563 0.0071 -0.8926 0.5365

52.4 0.4731 0.0027 -0.3308 0.1966

100.0 0.4753 0.0014 -0.1721 0.1022

200.0 0.4760 0.0007 -0.0851 0.0505

400.0 0.4761 0.0003 -0.0408 0.0242

900.0 0.4764 0.0001 -0.0137 0.0081

Table IV, h,-hg = 1.0 km

S.E.

0.0 0.3936 0.0840 -15.1802 7.6151

1.0 0.4620 0.0607 -10.3739 4.5217

2.0 0.5238 0.04918 -8. 2066. 3.1040

5.0 0.6348 0.0337 -5.4816 1.5931

10.0 0.5718 0.0224 -3.2378 1.2574

20.0 0.6598 0.0124 -1.6959 0.5531

52.4 0.6823 0.0049 -0.6495 0.2071

100.0 0.6851 0.0026 -0.3388 0.1080

200.0 0.6859 0.0013 -0.1676 0.0534

400.0 0.6860 0.0006 -0.0803 0.0256

900.0 0.6864 0.0002 -0.0270 0.0086
39-

Table V, h - hs =2. Okm

©o r b a S.E.

0.0 0.4524 0.0985 -17.7584 7.5391

1.0 0.5490 0.0752 -12.9451 4.4420

2.0 0.6316 0.0636 -10.7566 3.0277

5.0 0.7707 0.0475 -7.8969 1.5234

10.0 0.7634 0.0345 -5.3712 1.1421

20.0 0.8515 0.02111 -3.1571 0.5086

52.4 0.8668 0.0089 -1.2770 0.2003

100.0 0.8679 0.0047 -0.6705 0.1057

200.0 0.8681 0.0023 -0.3323 0.0524

400.0 0.8682 0.0011 -0.1593 0.0252

900.0 0.8684 0.0004 -0.0535 0.0084

r

Table VI, h -h s

- 5. Okm

eo b a S.E.

0.0 0.4962 .0.1115 -19.1704 7.5676

L0 0.6101 0.0881 -14.3543 4.4401

2.0 0.7030 0.0764 -12.1589 3.0001

5.0 0.8504 0.0601 -9.2514 1.4422

10.0 0.8674 0.0464 -6.6445 1.0420

20.0 0.9484 0.0308 -4.0706 0.4028

52.4 0.9674 0.0139 -1.6236 0.1426

100.0 0.9695 0.0075 -0.8348 0.0739

200.0 0.9701 0.0037 -0.4098 0.0365

400.0 0.9702 0.0018 -0.1960 0.0175

900.0 0.9703 0.0006 -0.0658 0.0059
-40-

Table VII, h-hs = 10.0km

e r b a S.E.

0.0 0.5099 0.1149 -18.5627 7.5227

1.0 0.6290 0.0915 -13.7469 4.3895

2.0 0.7250 0.0799 -11.5514 2.9443

5.0 0.8734 0.0635 -8.6434 1.3733

10.0 0.8950 0.0498 -6.0729 0.9713

20.0 0.9723 0.0338 -3.5012 0.3179

52.4 0.9907 0.0157 -1.1441 0.0844

100.0 0.9927 0.0085 -0.5084 0.0406

200.0 0.9931 0.0043 -0.2310 0.0197

400.0 0.9932 0.0020 -0.1078 0.0094

900.0 0.9932 0.0007 -0.0359 0.0032

Table VIII, h - h s = 20. km

O r b a S.E.

0.0 0.5155 0.1165 -17.9573 7.5131

1.0 0.6367 0.0931 -13.1413 4.3763

2.0 0.7336 0.0814 -10.9463 2.9281

5.0 0.8815 0.0651 -8.0397 1.3521

10.0 0.9028 0.0514 -5.4747 0.9573

20.0 0.9785 0.0353 -2.9228 0.2909

52.4 0.9968 0.0169 -0.6738 0.0535

100.0 0.9984 0.0093 -0.1802 0.0203

200.0 0.9986 0.0047 -0.0467 0.0096

400.0 0.9986 0.0023 -0.0161 0.0046

900.0 0.9986 0.0008 -0.0048 0.0016
-41-

Table IX, h - hs = 70. km

S.E.

0.0 0.5174 0.1170 -17.9071 7.5113

1.0 0.6391 0.0936 -13.0912 4.3738

2.0 0.7361 0.0820 -10.8960 2.9251

5.0 0.8837 0.0656 -7.9895 1.3481

10.0 0.9051 0.0519 -5.4209 0.9539

20.0 0.9797 0.0358 -2.8696 0.2862

52.4 0.9979 0.0173 -0.6246 0.0445

100.0 0.9997 0.0096 -0.1402 0.0095

200.0 1.0000 0.0048 -0.0212 0.0013

400.0 1.0000 0.0024 -0.0027 0.0002

900.0 1.0000 0.0008 -0.0002 0.0001
-42-

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-50-

TABLE XVIII

Initial N Gradients, AN , in the C.R.P.L.

Exponential Reference Atmosphere

Range of AN (N units/km) N(h)

27.55 < 35.33 < 42.13 <

49.52 < 59.68 < 71.10 < 88.65 <

AN

AN

AN

6 AN

AN

€ AN

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27.55

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59.68

71. 10

88.65

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289 exp(-0. 1257h)

313 exp(-0. 1428h)

344.5 exp(-0. 1568h)

377,2 exp(-0. I732h)

404.9 exp(-0. 1898h)

450 exp(-0. 2232h)

* Note height, h, is in kilometers
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TABLE XXIII

]

N(h)

Departures Method Sample C omputation for 9 =0 mr.

h A(N, h) AA e 2

o

departure

term

T(h)NT l,2

s e + e k k+1

0.340

400.0

365.0

400.0

397.8 - 2.2 6.388 0.3131 +0.689

0.950 333.5 419.5 +21.7 11.230 0. 1135 -2.463

3.060 237.0 459.7 +40.0 22.683 0.0590 -2.359

4.340 196.5 475.7 + 16.0 28. 195 0.0393 -0.629

5.090 173.0 478.5 + 2.8 31.383 0.0336 -0.094

5.300 172.0 484. 1 + 5.6 32.191 0.0315 -0. 176

5.940

6.250

155.0

152.0

485.6

490.5

+ 1. 5 + 4.9

34.654

35.811

0.0299

0.0284

-0.045

-0. 139

7.180 134.0 493.4 + 2.9 39. 123 0.0267 -0.077

7.617 125.5 493.3 - 0. 1 40.581 0.0251 +0.003

9.660 98.0 495.9 + 2.6 47.236 0. 0228 -0.059

10.870 85.0 495.2 - 0.7 50.645 0.0204 +0.014 30.776 25.44

I

5. 335

TABLE XXIV

Departures — Method Sample Computation for 9 = lOmr. t o

400.0 400.0 10.000

0.340 365.0 397.8 - 2.2 11.873 0.0914 +0.201

0.950 333.5 419.5 +21.7 15.038 0.0743 -1.613

3.060 237.0 459.7 +40.0 24.789 0.0502 -2.008

4.340 196.5 475.7 +16.0 29.947 0.0365 -0.585

5.090 173.0 478.5 +2.8 32.938 0.0318 -0.089

5.300 172.0 484.1 + 5.6 33.712 0.0300 -0.168

5.940 155.0 485.6 +1.5 36.069 0.0287 -0.043

6.250 152.0 490.5 +4.9 37.181 0.0273 -0.134

7.180 134.0 493.4 +2.9 40.383 0.0258 -0.075

7.617 125.5 493.3 - 0.1 41.801 0.0243 +0.002

9.660 98.0 495.9 + 2.6 48.282 0.0222 -0.058

10.870 85.0 495.2 - 0.7 51.630 0.0200 +0.014 19.414 14.858

r

= -4.556
-56-

TABLE XXV

Departures Method Sample Computation for 9 = 52.4mr (3 )

2

h N(h) A(N , h) AA 9 Q , _ departure T(hL T t, _ k k+1 term s

400.0 400.0 52.4

0.340 365.0 397.8 - 2.2 52.750 0.0190 +0.042

0.950 333.5 419.5 +21.7 53.550 0.0188 -0.408

3.060 237.0 459.7 +40.0 57.061 0.0181 -0.724

4.340 196.5 475.7 +16.0 59.567 0.0171 -0.274

5.090 173.0 478.5 +2.8 61.045 0.0166 -0.046

5.300 172.0 484.1 +5.6 61.477 0.0163 -0.091

5.940 155.0 485.6 + 1.5 62.792 0.0161 -0.024

6.250 152.0 490.5 +4.9 63.433 0.0158 -0.078

7.180 134.0 493.4 +2.9 65.368 0.0155 -0.045

7.617 125.5 493.3 -0.1 66.277 0.0152 +0.002

9.660 98.0 495.9 + 2.6 70.; 512 0.0146 -0.038

10.870 85.0 495.2 - 0.7 72.920 0.0139 +0.010 7.024 5.350

I

= -1.674

TABLE XXVI

Departures Method Sample Computation for 9 = 261. 8 mr (15 )

400.0 400.0 261.8

0.340 365.0 397.8 - 2.2 261.880 0.0038 r0. 008

0.950 333.5 419.5 +21.7 262.035 0.0038 -0.083

3.060 237.0 459.7 +40.0 262.758 0.0038 -0.152

4.340 196.5 475.7 +16.0 263.308 0.0038 -0.061

5.090 173.0 478.5 + 2.8 263.633 0.0038 -0.011

5.300 172.0 484.1 +5.6 263.733 0.0038 -0.021

5.940 155.0 485.6 +1.5 264.034 0.0038 -0.006

6.250 152.0 490.5 +4.9 264.184 0.0038 -0.019

7.180 134.0 493.4 +2.9 264.647 0.0038 -0.011

7.617 125.5 493.3 -0.1 264.872 0.0038 +0.000

9.660 98.0 495.9 + 2.6 265.934 0.0038 -0.010

10.870 85.0 495.2 - 0.7 266.592 0.0038 +0.003 1.506 1.143

2

= -0.363

GPO 843-308