technical riote

Moulder laboratories

PB 161554

92o. 53

CARRIER FREQUENCY DEPENDENCE OF THE

BASIC TRANSMISSION LOSS IN TROPOSPHERIC

FORWARD SCATTER PROPAGATION

BY KENNETH A. NORTON

IS.

DEPARTMENT OF COMMERCE

TIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical <Viote

53

May 12, I960

CARRIER FREQUENCY DEPENDENCE OF THE BASIC

TRANSMISSION LOSS IN TROPOSPHERIC FORWARD

SCATTER PROPAGATION

by

Kenneth A. Norton

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

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FOREWORD

Essentially this same paper, but omitting Tables III and IV,

will be published in the July, 1960 issue of the Journal of

Geophysical Research. The primary purpose of this additional

publication is to provide Tables III and IV.

Carrier Frequency Dependence of the Basic Transmission

Loss in Tropospheric Forward Scatter Propagation

by

Kenneth A. Norton

Summary

A further interpretation is given of certain Lincoln Laboratory

data obtained in an experiment using scaled antennas as presented in a

recent letter to the Proceedings of the I. R. E. from Bolgiano. —' This

paper has four objectives: first, to clarify the significance of these

data from the standpoint of the engineer developing long-range

tropospheric scatter systems; second, to apply a further statistical

analysis to these data; third, to consider their significance as regards

the theory of radio propagation through a turbulent atmosphere; and

fourth, to describe a suitable method for the measurement of the

meteorological parameters entering the theory. Based on this analysis

of the Lincoln Laboratory data, it is concluded that the carrier frequency

dependence of the basic transmission loss cannot be variable from hour

to hour.

1. Significance of the Lincoln Laboratory Experiment

to the Systems Engineer

From the standpoint of the development of a forward scatter

communications system, the radio engineer is usually interested in

the cumulative distribution of the hourly median basic transmission
- 2 -

loss -' (sometimes called path loss) over as long a period of time as

is feasible, and he usually bases the power required for a specified

performance on the expected path antenna power gain, together with

the basic transmission loss which is not exceeded for more than some

large percentage of the hours, say 99% or 99.9%. Fig. 1 shows the

cumulative distributions of basic transmission loss for 417 Mc and

2290 Mc, and these are based on the same 240 pairs of hourly median

values used by Bolgiano for determining the results shown on Fig. 1

of his letter. —' J. H. Chisholm and J. F. Roche of Lincoln Laboratory

supplied the 480 values of available power, P , in decibels above one

watt at the terminals of the receiving antenna, and these were converted

to basic transmission losses by means of the formula

L,btaptr = P, - P +G -L -L where P, is the transmitter power in

t decibels above one watt, G is the path antenna gain, and L, and L

p t r are line losses. On 417 Mc we set P^ = 35.4 db, Lx + L =1.1 db,

t t r G = 56. 2 db, and on 2290 Mc we set P = 40 db, Lx + L =2.1 db,

p t t r G = 61. 2 db; the path antenna gain was estimated by Hartman and

P

3/ Wilkerson's method -J to be 0. 8 db below the sum of the free space

gains for both frequencies.

By examining a large number of cumulative distributions

such as those shown on Fig. 1 for 80 different scatter paths and
-3-

DATA FROM THE LINCOLN LABORATORY EXPERIMENT

February II to July II, 1957

co

_i

UJ

GO

CJ> UJQ

CO

COO

O

CO

CO

CO<

l-o

CO<

DQ

PERCENTAGE OF THE HOURS LOSS IS LESS

THAN THE INDICATED VALUES

Figure I
- 4 -

involving a range of frequencies from 66 Mc to 1046 Mc, Norton, Rice

4/ and Vogler —' found, after correcting by means of a frequency gam

function for the incomplete illumination of the lower part of the common

volume on the lower frequencies, that the basic transmission loss for

the scatter mode of propagation increases with frequency as

(29. 56 ± 2.44) log (f /f ); these results were obtained on the assumpi-U C> X tion that the scattering cross section decreases inversely as the square

of the height above the surface. More recently, Rice, Longley and

5/ Norton —' have found, on the presumably more realistic assumption

that the scattering cross section decreases exponentially with the height

above the surface, that the basic transmission loss for the scatter mode

of propagation increases with frequency, after adjustment by means of

the frequency gain function, as (33.1 ± 5. 55) log (f /f ); this latter

J. U c* X determination involved 105 scatter paths covering the range of

frequencies from 66 Mc to 4090 Mc' Since the frequency dependence

expected on the basis of those mixing -in -gradient hypotheses developed

in several recent papers —'—'—' is 30 log (f /f ) and since this lies well

within the error bands of the above experimental determinations, it has

been common practice for engineers in recent years to use this

theoretical frequency dependence for estimating the basic transmission

loss expected on a scatter path. In this connection it appears that the

A further discussion is given in Appendix III.
- 5 -

formula for the basic transmission loss given by Rice, Longley and

5/ Norton —' for an exponential atmosphere, when used in conjunction

with the Hartman and Wilkerson —' formula for the path antenna gain,

is the most versatile formula presently available for determining the

transmission loss expected on a given path since it makes allowance

for the effects of irregularities in the terrain, and may be used for

essentially any antenna heights, radio frequencies or geographical

locations. The cumulative distributions of basic transmission loss

as predicted by this formula are shown on Fig. 1, together with their

standard errors, and these are evidently in reasonably good agreement with the observed losses; the standard error of predicting the

median using this formula is about 5 db, and the observed medians

indicate less than 6 db more loss than the predicted medians on

both frequencies. A part of this difference may be due to the fact

that our prediction is for an all -day, all-year period, while the data

were obtained in a period from February 11 to July 11, 1957. A more

likely explanation of this small error of prediction is some random

effect of this particular terrain profile since our formula makes use

only of the angular distance in allowing for the effects of the terrain.

Our prediction for this path is evidently not very good for the smaller

losses which occur for small percentages of the hours, and further

work is now in progress on improving our methods of prediction of these
smaller losses since this portion of the cumulative distribution,

although not important for predicting the effective range of systems in

the absence of interference, is important for the prediction of the

interference expected between systems.

2. A Further Statistical Analysis

For percentages of the hours less than about 30%, it appears

that modes of propagation other than, or in combination with, scatter

were probably involved on the Lincoln Laboratory path- -possibly ducts

\

or well defined elevated layers; the smallest observed values exceed the

expected loss in free space by only 7. 3 db on 417 Mc and 13. 6 db on

2290 Mc, and such small losses undoubtedly cannot be attributed

exclusively to the scatter mode of propagation.

Since it appears on Fig. 1 that the stronger fields (smaller

basic transmission losses) probably did not arrive at the receiving

location via the simple scatter mode of propagation, it will be well to

eliminate these from the analysis which will now be made for the

purpose of better understanding the physical nature of the scatter mode

of propagation. This was accomplished somewhat arbitrarily simply

by including only those pairs of hourly medians for which the basic

transmission losses were greater than 192. 3 db on 417 Mc, and

simultaneously greater than 214. 3 db on 2290 Mc; in this way we
-7-

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\- 2

III ^>o

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u_ LU

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- 8 -

obtained a reduced sample of only 140 pairs of hourly medians which

are clearly more nearly representative of the pure scatter mode of

propagation. The cumulative distributions of these values are also

shown on Fig. 1. We next obtained the means and standard deviations

from these samples: L~ (417) = 200. 5 db; o-(417) = 4.65 db;

b

L (2290) = 221. db; cr(2290) = 4. 06 db; p = 0.63; here p denotes the

b

correlation coefficient between the basic scatter transmission losses

for 417 and 2290 Mc. Next we determined the cumulative distribution

of the differences of these 140 pairs of scatter losses

ALl = L (2290) - L (417); this distribution is shown on Fig. 2, and the

mean and standard deviations were found to be: AL = 20. 5 db and

b

tr(AL) = 3. 79 db. Bolgiano's AL was actually not a difference in basic

b

transmission losses, although his label on his Fig. 1 indicates that it

was; however, his wavelength scale is consistent with the assumption

that his AL is the difference in losses relative to free space, and

presumably this is what he plotted. The calculated frequency gain

corresponded to an additional loss of 0. 5 db on 417 Mc and to db on

2290 Mc; after making allowance for this difference, our analysis using

only the above scatter losses appears to indicate a variation in the

wavelength dependence exceeding 34.9 log (2290/417) for 10% of the

hours, exceeding 28. 3 log (2290/417) for 50% of the hours, and

exceeding 22. log (2290/417) for 90% of the hours.
- 9 - We are now in a position to ask the question whether this

apparent variation in the wavelength dependence of the scatter loss

from hour to hour represents a real change in the physical nature of

the atmosphere from hour to hour. In this connection it should be

noted that the hourly medians on closely adjacent paths, such as are

used for diversity reception, and involving the same radio frequency,

9/

are not perfectly correlated —' and the correlation appears to be

substantially smaller when the antennas used on the two adjacent paths

have different radiation characteristics. Thus it appears that the

correlation coefficient is very dependent upon whether the two antennas

9/ illuminate exactly the same portions of the atmosphere. Thus Barsis —

found at 100 Mc that the hourly median values of L, over 644 km scatter

b

paths using Yagi and Rhombic receiving antennas separated normal to

the path by only 18 meters were very poorly correlated; with the transmitter at Fort Carson, the angular distance to the two receiving antennas

was 0. 068 radians, and the correlation between L (Yagi) and L (Rhombic)

b b

was p = 0. 510, whereas with the transmitter at Cheyenne Mountain, the

angular distance to the same two receiving antennas was 0. 0585, and

the correlation p = 0. 331. The observed standard deviations of

AL = L (Yagi) - L (Rhombic) were cr( AL for 0=0. 068) = 1. 61 db and

o-(AL for 9=0. 0585) = 2.48 db. The angular distance for the Lincoln

The definition of the angular distance, 0, is given on Fig. 3.
-10 -

Laboratory path is approximately 8 = 0. 02 71, and it is thus not

surprising that c(AL) was observed on this path to be somewhat

b

larger. Although the free space radiation patterns of the antennas

used for the Lincoln Laboratory experiments were very nearly

identical, the antennas on the two frequencies nevertheless did not

illuminate the same portions of the atmosphere since they were located

at different numbers of wavelength above the ground, and thus had

markedly different vertical plane lobe structures. It is believed that

this may be the correct explanation for at least some if not all of the

observed variance cr(AL ) - 3. 79 db rather than any change in the

b

scattering nature of the atmosphere from hour to hour. It would be

most interesting to repeat the Lincoln Laboratory experiment under

conditions which would ensure that more nearly the same portion of the

atmosphere is illuminated by the antennas on the two frequencies.

Although the writer has been unable to devise an experimental plan

which will ensure that exactly the same portion of the atmosphere is

involved, the following experimental arrangement would appear to

approximate this situation reasonably well: (1) choose a scatter path

with extremely flat terrain (possibly calm fresh water lakes) in the

foregrounds of the antennas so that the vertical plane lobe structures

will be well defined on both frequencies, (2) use parabolic antennas
- 11 -

•with the same diameters expressed in wavelengths, i. e. , with the

same apertures, (3) erect the antennas at approximately the same

heights expressed in wavelengths above the ground in such a way that

the lowest lobes, having regard to the curvature of the earth, are very

nearly coincident on the two frequencies throughout the scattering

volume, (4) keep the spacing between the antennas normal to the path

at a minimum and (5) have the lower frequency antennas to the right of

the higher frequency antennas when facing the far terminal so that the

two propagation paths will cross in the middle of the scattering volume.

With such an experimental arrangement the only significant difference

in the portions of the atmosphere involved in the propagation on the

two frequencies will be found near the transmitting and receiving ends

of the path; to provide a measure of the importance of such effects, a

control path could be added using a slightly different low frequency

with its terminals to the left of the higher frequency antennas. It would

be interesting to carry out such an experiment, and it is anticipated

that the correlation p would then be very much nearer to unity and

ir(AL) substantially smaller.

3. A Theoretical Interpretation

Finally it is of interest, in spite of the above questions raised

as to the appropriateness of these data for this purpose, to assume that

all of the apparent variation in the above -described frequency dependence

does actually represent real physical changes in the characteristics of
- 12 -

the atmosphere from hour to hour and to speculate on the nature of

these changes. Assuming that the scattering occurs in a turbulent

atmosphere, it is well known -' —' that the power scattered per unit

volume is proportional to the magnitude S(k ) of the three dimensional

wave -number spectrum of the variations of the refractive index of the

atmosphere at a fixed instant of time and corresponding to the particular

wave number k = (2ir/i!) k , where the scale length

I - \/2 sin [ (a + (3)/2] ^ W(a + P) and k is a unit vector. Here \

denotes the free space wavelength of the scattered radio waves and--

see Fig. 3--(a + (3) is the scattering angle in the scattering plane, i.e. ,

the plane passing through the transmitting antenna, receiving antenna

and the elementary scattering volume. The wave number spectrum is

determined in each elementary scattering volume for that direction k

which is in the scattering plane and is normal to the line which bisects

the angle (a + P). The total power scattered is obtained by integrating

S[2ir (a + P)A, k _ ] for these appropriate directions k throughout the

scattering volume; k tends to be vertical throughout those portions of

the scattering volume from which the scattering is most intense. Note

that the angular distance, 9, is the minimum value which (a + (3) may

have within the important common volume, i. e. , that portion of the

scattering volume which is within line -of- sight of both the transmitting
-13

Elementary scattering volume

FIG. 3 DEFINITION OF THE ANGULAR DISTANCE 9

AND THE SCATTERING ANGLE (a + /3)
14

and receiving antennas. Table I gives the values of I - \/Q and of ° max

i . = \/0. 2 (assuming that negligible scatter is expected for mm

(a + (3) > 0. 2 radians) for a wide range of frequencies and angular

distances; the distance d = a9 between the horizons of the transmitting

s and receiving antennas is also given on the assumption that the

effective earth's radius, a, is equal to 9, 000 kilometers which corresponds, for a typical CRPL exponential atmosphere, —' to a

surface refractivity of N = (n - 1)- 10 S 316 and this latter value is

s s roughly the average value of N throughout the day and over the

continental United States.

Table I Ranges of Scales Pertinent to the Forward Scatter

of Radio Waves

0= 0. 005 e = 0. 025 0= 0. 05 e= o.i (a +p)= 0.2

d = 45 km

s d = 225 km

s d = 450 km

s d = 900 km

s

fMc i

max

i

max

I max

i

max mm

meters meters meters meters meters

50 1200 240 120 60 30

100 600 120 60 30 15

200 300 60 30 15 7. 5

500 120 24 12 6 3

1000 60 12 6 3 1. 5

2000 30 6 3 1. 5 0. 75

5000 12 2.4 1. 2 0.6 0. 3

10000 6 1.2 0.6 0. 3 1 0. 15
- 15 -

For the Lincoln Laboratory path i = 26. 5 meters on 417 Mc and

max

I = 4. 8 meters on 2290 Mc, and thus only that portion of the wavemax

number spectrum of tropospheric turbulence with wave -numbers k

greater than 2ir/i will be involved in the scattering. Assuming

that the scattering occurs primarily from the common volume lying

above the two surfaces generated by the two horizon rays, it will

occur on the Lincoln Laboratory path at heights above the ground in

2

excess of approximately a9/8 = 0.826 kilometers.

The average scattered power, p , available at a given time

from the receiving antenna for a given power, p , radiated from the trans

mitting antenna is expected to depend on the free space wavelength

in the following way:

(P

s /Pt

) ~ X

~ 4

-§8t 8

r S[2u(a+ P)/\, rx ]dv.\

2

(1)

-4

The first factor \ in (1) may be identified with the Rayleigh law of

uniform scattering in all directions from small scatterers; the second

factor is an integral throughout the scattering volume which properly

corrects the Rayleigh scattering law to allow for the tendency with

large scatterers for the power to be scattered primarily in the forward

2

direction; finally the factor \ allows for the fact that the available

2

power from a receiving antenna is equal to X. g /4ir times the scattered

r
- 16 -

2

power flux density. A factor sin x — 1 which allows for the small

effects of polarization has been omitted under the integral sign in (1);

g and g in (1) denote the effective power gains of the transmitting and

t r receiving antennas at the elementary scattering volume, dv, relative

to an isotropic antenna including the effects of ground reflection,

e.g. , g could in principle be measured as the ratio of the power flux

densities at dv when the actual antenna and the earth are replaced by

an isotropic antenna in free space at the same distance.

The instantaneous received power, p., is Rayleigh distributed

about the mean value < p >:

s

Q(p >y) = exp(-y/<p >)

(2)

Q is the probability that the instantaneous power p. exceeds some

given value y. The Lincoln Laboratory experiment was reported in

terms of hourly median powers, p , and these hourly medians are, on

the assumption of the Rayleigh distribution, related to p by

s pmess = (log 2)<p > where <p > is the time average of p over a period

s

of one hour. Since the values of AL were differences of hourly

medians, we see by (1) that AL is given by:

AL= 10 log

10

Jgt 8

r S[2ir(a+ P)A

X

, ri ]dv>

< \ g+ g S[2w (a+P)A , k ]dv >

*2 r

2

2 l

ZOlog (f

2

/f )

(3)
- 17 -

It appears from (3) that variations of AL. from hour to hour may only

be explained in terms of variations in the above integrations over

space and subsequent averages with respect to time and thus, to the

extent that the transmitting and receiving antennas actually "see" the

same common volume on the two frequencies, i. e. , g = g and

g = g throughout the scattering volume, it appears that AL would

r

l r

2

be expected to vary only if the shape of the wave -number spectrum

varies from hour to hour.

Unfortunately there does not appear to be any exact way to

measure S(k ) directly so that it must be determined indirectly as

the Fourier cosine space transform of a correlation function C( r ):

S(lT) = V \ \ \ d

3

r* cos [IT- r*]C( 7) (4)

< An( R)»An(R + r")> C(r ) = {<[ An(R)] 2 ><[An(R + 7)]

2

>}1/2

(5)

where < > denotes as before a time average over a short period of, say

one hour, An( R) and An( R + r ) denote the measured changes in the

refractive indices of the atmosphere from their mean values <n( R)>

and <n( R + r ) > at the vector locations R and R + r . In what follows

we will assume homogeneous turbulence, and then C( r ) depends only

on the magnitude and direction of r and<[An(R)l > = <[An ( R + r )1>= V . n

The following model correlation function has proved to be useful —' —' —'
-18 -

for characterizing the random deviations of n at least over the range

of r < S. : o C(7)= {2

1 " ^/r(ji)}(r/i

ofK (r/i

o

) (6)

In the above, T denotes a gamma function, K denotes the modified

Bessel function of the second kind and \x is a constant. Some allowance

for the possible effects of anisotropy has been introduced into (6) byallowing the characteristic scale I to vary with the direction of r :

2 —1 cos (b . Z = — + sin *

i i o v p n

2 . 2

COS -y Sin y

i I

(7)

In the above, y and cf> are spherical polar angles; relative to unit

orthogonal vectors a , b and c , where a is somewhat arbitrarily

chosen to lie in the direction of the mean wind velocity and c lies in

the vertical plane through a , the angle cj> is the angle that r makes

with c and y is the angle between a and the projection of r on the

a , b plane. Thus 1 , & , and i are respectively the effective scales

p n v

in the direction of the mean wind, normal to this direction and approximately vertically. It has been found experimentally that the magnitudes

of V and SL depend upon the periods of time over which the averages

n o in (5) are taken and the characteristic scale i , when determined by

o averaging over periods of time of the order of an hour, is of the

order of hundreds of meters in the troposphere. For this form of

correlation function involving scale length ellipsoidal anisotropy, it

is shown in Appendix II that the wave -number spectrum of n is

given by:
S(T) = 8tt nHF {r(u + f)/r(n)}i i i v [1 + (Zui /i)

2

] (8) 2 p n v n k

k = k(cos -y sin <j) a + sin -y sin $ b + cos <j> c ) (9)

L

222222222 = SL cos v sin 6 + SL sm v sin cj> + SL cos cb (10) k p n v

Equation (8) provides a useful model for the spectrum of the turbulence

at least for values of SL < SL . If we set i = \/(a + (3) and assume that

k

this is smaller than SL , then we see by (8) that the wave -number

spectrum S[2ir(a + (3)/\, k ] associated with the scales appropriate to

the forward scatter is proportional to Jt SL i V^lL (a + p)l and,

p n v k

when this is substituted in (3), we obtain:

AL = AL^ = (2jjl + 1) 10 log (f /f ) [Scaled antennas; X <(a+(3)i 1 (11) b 10 2 1 k

If, as in the Lincoln Laboratory experiment, the antenna gains are not

precisely the same and the antenna heights are not scaled, the above

becomes:

AL, = (2\l + 1) 10 log (f /f ) + H (f ) - H (f ) (12)

b 10 c. I o i. • o 1

where the frequency gain function H (2290) = and H (417) = 0. 5 db

o o

in the Lincoln Laboratory experiment. In (11) and (12) we distinguish

between the actual transmission loss L and the basic transmission

loss L, = L + G . It follows from (12) and the data on Fig. 2 that \i b p

exceeds 1. 24 for 10% of the hours, exceeds 0.913 for 50% of the hours

and exceeds 0. 602 for 90% of the hours; note, however, that a 3 db

difference in the effects of the terrain on the two frequencies could

shift these figures over the ranges indicated on Fig. 2. A shift of
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- 22 -

this order of magnitude is not unreasonable to expect in view of the

fact that the observed scatter losses on this particular path (See Fig. 1)

are systematically several decibels greater than is predicted by the

Rice -Liongley -Norton formula for average irregular terrain conditions.

Fig. 4 shows the wave -number spectrum S(k )ji £ I V for several

p n v n values of p. and, if \i varies from hour to hour, it appears that the

slope of S { k ) will vary from hour to hour. Fig. 5 shows C ( r ) for

several values of jx. On the assumption that n is a normally distributed

variable, it is expected that z( r ) = tanh {C( r )} will also be

approximately normally distributed about its true mean value when

C( r ) is determined experimentally. —' —' —' For this reason we

have also shown z( r ) on a linear scale versus x/SL on a logarithmic

o scale on Fig. 5; with these scales the correlation function on Fig. 5

varies linearly for small (r/i ) with a slope dependent on \x just as its

transform S(k) on Fig. 4 varies linearly for large k with its slope also

dependent on jjl. For \x = 0, C ( r ) = 1 for r = and C ( r ) = for

r > 0, and such a function is evidently of no interest for describing

the atmosphere. A further discussion of the slope of the z ( r ) graphs

is given in Section 4.

A more precise idea of the influence of \x and of anisotropy on

the magnitude of the total scattered power may be obtained by using the

particular model ( 8 ) for S ( k ) and integrating this by the methods
- 23 -

given in reference 10 . It is assumed that the magnitude of the

scattering parameter {V I t /l i } has the same value

n p n w v throughout the scattering volume. The transmitting and receiving

antennas are assumed to be at the same height above a smooth earth,

this height being greater than 4X./9 so that g = g =2. for isotropic

t r transmitting and receiving antennas. The antennas are assumed to be

a distance d and angular distance apart. In this case the median

basic scatter transmission loss is expected to be:

I,-, = - 10 log_ < bms e

10

(log 2)|j. {V I t It } e n p n w

(u + i)(2(j. + i) d(2ir i e/x)

2^ 4

" 1 > (13)

In the appendix a general formula for L, is derived for the case bms

where the meteorological scattering parameter {V t t It t } n p n w v varies inversely as h , and the above represents simply the particular

case m = 0. For this derivation it is assumed that the mean wind

direction lies in the plane which passes through the transmitting and

receiving antennas and is normal to the great circle plane; y is the

angle in this plane between the mean wind direction and the great

2 2 2 2 2

circle plane; using this notation i = i sin -v + I cos y. When the

w p n

2u.+ l wind blows along the path, the scattering parameter is V I It , n p v and when the wind blows across the path, the scattering parameter
- 24 -

2u + 1 becomes V SL /I . Note the fact that the characteristic scale i

n n v v

in the vertical direction enters the scattering parameter with a power

Z\x larger than does either i or i . p n

We may re-write the general expression (1-8) for the median

basic scatter transmission loss in the following form:

L = -10 log {(log Z)V i i /! d} bms 10 e n p n w h=h

o + J(n,M,S) + (2^ + 1) 10 log (2-irOi A), , (I-8a)

IU v n= n

o

If we arbitrarily assume that the first term in (I-8a) does not depend

upon |i. and does not vary from hour to hour, then we may determine

the long-term 6 variance of L from the apparent variance of U shown bms Fi^

on Fig. 2, i. e. ,

|J- apparently exceeds 0. 602 for 90% of the time and

apparently exceeds 1. 24 for 10% of the time. For the Lincoln Laboratory

path 8= 0. 0271 and S = 10 log (P /a ) = 0. 7, and it is estimated that

10 o o M = -4 db. Since the magnitude of S. is not known, we will estimate by

v

(I-8a) x the interdecile rangeb L, (10%) - L^ (90%) for the two values bms bm s

Jt - 20 and 200 meters which represent guesses as to the range within

which the median value of i is expected to lie. The following table gives

these calculated values for the two frequencies together with the observed

interdecile ranges from Fig. 1. L^ (10%) - LK (90%) bm s bms Observed Calculated by (I-8a) Calculated by (I-8a)

with i = 20 meters with i = 200 meters

v v 417Mc 11.8 db 8.6 db 21.4 db

2290 Mc 10. 5 db 18.0db 30. 8 db
- 25 -

It is evident from the above table that the observed interdecile ranges

do not agree with those calculated on the assumption that the variations

of |i are responsible for all of the variance of L. ; in particular, the

observed variance is slightly smaller on the higher frequency, and this

is inconsistent with the assumption that any of the variance arises

from variations in u. Furthermore, if the variance of L, is bms

attributed to a variance of \x, then we see by (I-8a) that the long-term

variance of L, would be expected to increase systematically with bms

the angular distance G; for the winter afternoon hours when scatter

would be expected, the observed variance of L, appears to decrease bms

with 9 rather than increasing with 0. This analysis indicates to the

author that \x probably varies, if at all, over a much smaller range

than is indicated on Fig. 2, and that the variance of L from hour to bms

hour must be attributed to a variance in the first term of (I-8a) together

with a variance of Jt rather than to a variance of \x.

v

Further evidence as to the nature of the hour-to-hour variance

of L, is given on Fig. 6. This shows the observed hourly median

b

scatter values of L for both 417 Mc and 2290 Mc as a function of

b

AL. . If a variance of [i were responsible for all of the variance of

L , then L. would be expected on both frequencies to increase linearly

with AL with a larger slope on the higher frequency; instead L has

b b
-26-

SCATTER LOSSES FROM THE

LINCOLN LABORATORY EXPERIMENT

co

_j

Ld

00

O

LdQ

CO

COo

o

CO

CO

<

CO<

QQ

235

230

225

220

215

^ 210

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205

200

195

•

•

1 "T • L

b FOR 2290 MC

x L b FOR 417 MC

•

• •

. .»

•

• • ••

• • •

»

•

•

•

•

•

•

: .• '••

•

• ••• •

• ••

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•

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••• •

• •• • V

*•*

.1

X

XX

•

»

X

X

K

< xx , X X

X

X

c x X X *

X > v X V

x*< x

x

x x

x y

5

X ,X

tf x

c x

i

* X x

x x> 4

lX >°<

X v x xx x

x

x x x

X

X

X

xxx x

<

) * XXX

< X XXx x

X

X

X X

20

Figure 6

25 30 35
. 27 -

little or no dependence on AL , although it appears actually to decrease

b

with increasing AL on 417 Mc.

b

4. Meteorological Measurements Useful

for Evaluating L,

In this section we will consider how meteorological measurements might be made in order to evaluate the magnitude of the

scattering parameter {V S. i /i 4 }. Since direct measure - npnwv

ments of S(k ) cannot be made, it appears to be almost essential,

particularly if the variations of n are anisotropic, to measure C ( r )

for several values of the magnitude and direction of r , use these

measurements to develop an appropriate model for C ( r ) and finally

determine S(k ) as the cosine -space transform (4) of this model C( r ).

The following experiment using a minimum of five refractometers

mounted on a tethered balloon or aircraft is suggested. Three of the

refractometers should be located on a vertical line with locations

r = 0, r , and 3r and the other two located on a horizontal line normal

to the direction of flight and with spacings relative to one of the three

vertically spaced refractometers equal to r and 3r . In this way it will

be possible to measure C(r ) corresponding to spacings r , 3r - r = 2r ,

and 3r . By plotting the resulting six values of C ( r ) on graph paper

1
- 28 -

similar to that of Fig. 5 and shifting this to the right or left until the

points best fit one of the curves of Fig. 5, estimates can be obtained

of |j. and I . If these estimates indicate significantly different values of o either u or I or both for three points obtained from the vertically and

o horizontally spaced refractometers, then it would appear that the variations

of n are anisotropic. If jx is a function of the direction of r , then we

must abandon our model function (8). If jj. is independent of the

direction of r but SL is variable, then the measured value of S. along

o o the vertical line may be identified with I and the measured value of v

I along the horizontal line may be identified with I when the plane

o n

flies with or against the wind, and may be identified with it when the

plane flies normal to the direction of the wind. Assuming that the

measurements are designed to test the forward scatter theory for a

path with angular distance G and on a frequency corresponding to a

free space wavelength X, it will be desirable to choose 3r =\/Q and

then fly the aircraft throughout that part of the common volume most

intensely illuminated by both the transmitting and receiving antennas.

Note that five independent measurements of V will be available and

n

these may be averaged. In order to represent hourly median scattered

fields the balloon measurements should be averaged over an hour, but

the aircraft measurements could be averaged over a shorter period in

the ratio of the mean wind speed to air speed.
-29 -

If the variations of n are found experimentally to be isotropic

for the small scales X/39, \/(3/2)0 and X/6 involved in such an experiment, then we may estimate S(2it/!)» which will now be independent of

the direction k ., as follows:

M-r(p. + -)

S(k) & -— •! <[8n(!)] > (14)

2tt 2m,+

2 T(l - u)

Here 6n(!, r* ) = n(lC) - n(R + ! r ) and <[8n(!)] > is the structure

function for the turbulence; since we are here assuming the turbulence

2

to be isotropic, the expected value of <[6n(!)] > will be independent

of the direction of r and equal to:

<[8n(!)]

2 > = 2V [1 - C(i)] (15)

n

For small values of ! < < SL o

, C (!) = 1 - {r (1 - u)/ T (1 + n)}(!/2!

o^

for [L < I, provided the correlation function has the form (6) and, when

this value is introduced into (15) we obtain

<[6n(!)1 >=2V {T(l - u)/r(l+ |x)}(!/2! ) M'; when this value is in

n o turn substituted in (14), it becomes equal to (8) for the special case of

2

isotropic n variations. Note now that the structure function <[5n(!)] >

can be measured directly in the above -described experiment with

2 2 2

refractometers. If the ratios <[6n(r)] >/<[5n(2r)] >/<[6n(3r )] >

obtained by averaging over a period of an hour are the same from hour

to hour, then it appears that \x is also constant and we must look

elsewhere for the explanation of the observed variations shown on Fig. 2.
- 30 -

Finally it will be useful to consider the nature of the correlation function (6) as shown graphically on Fig. 5. The slope D of those

curves is given by:

9z 1 B[C(r/i

Q

)]

D = B[ln(r/l

o

)] =

1 . [c(l/y] 2'«li»l'« l]

U6)

We will be interested in D for r < < I and we may set o C(r/4 ) = 1 - AC(r/l ) where AC(r/i ) < < 1; thus: o o o

-3[AC(r/i )] -9{in[AC(r/je )]} ~ 2AC(r/4 )8[in(r/l )] 29[in(r/4 )]

V

' o o o

For \i < 1 and r < < I , AC(r/i ) - {r(l - n) / r(l+ \x) } (r /2i ) 2,1 andthus:

o o o

D S -u For < jx < 1 and r < < SL (18) o

The above is illustrated on Fig. 5; for \x > 1 it can be shown that D

approaches -1 as r approaches zero; however, within the scale ranges

shown on Fig. 5, the slope D continues to increase with increasing |jl

between p. = 1 and |i = 1. 5, and we may still use this slope as a means

for determining jjl.

It should be noted that single refractometers have been used to

investigate the frequency spectrum —' —' —' —' of refractivity and the

correlation with time of the refractivity at a single location in the
- 31 ~

atmosphere, but the wave-number spectrum S(k ) and the correlation

function C( r ) may be inferred from such results only on the assumption (essentially Taylor's hypothesis) that all of the variation of n

occurs as a result of the drift of a frozen atmospheric pattern through

the refractometer without any significant change in this pattern due to

the self-motion of the atmosphere during the time occupied by the

19/ measurements. As Gifford —' has pointed out, the connection between

the space -and time -turbulence statistics cannot be said to be well

understood in the case of the free atmosphere as contrasted to the

wind tunnel. Gossard —' claims to have verified Taylor's hypothesis

by approximating a measurement of the wave -number spectrum by

flying rapidly through the atmosphere with a refractometer on an

aircraft and then comparing these results with those obtained indirectly

from the frequency spectrum of refractivity measured with a refractometer on a tethered balloon; however, his aircraft spectra are definitely

steeper (larger p.) than the indirectly determined balloon spectra at the

higher wave numbers, and thus I would interpret his results as being

in disagreement with Taylor's hypothesis.

Aside from the above difficulty arising from a possible failure,

of Taylor's hypothesis, an even more important difficulty with the

spectra determined with single refractometers is the fact that they
- 32 -

measure only the variations of refractivity in the horizontal direction,

and we have seen above that it is the refractivity in the vertical direction that is more important as regards the forward scatter of radio

waves.

For the above reasons it appears most desirable that the

above -described direct measurements of C ( r ) be made using several

refractometers simultaneously with appropriately chosen spacings

along a vertical line and, in the horizontal plane, along two lines

parallel and normal to the average wind velocity.

Acknowledgements: The author wishes to acknowledge the assistance

in the development of this paper which was obtained in several

stimulating discussions with his colleagues E. C. Barrows, P.L. Rice,

W. J. Hartman and R. L. Abbott, as well as with Prof. Ralph Bolgiano

of Cornell University. William V. Mansfield supplied the values in

Table III. Jack C. Stroud assisted with some of the statistical

calculations. Mrs. A. G. Longley carried out the analysis in

Appendix IJJ.
- 33 -

Appendix I

Derivation of a General Formula for the

Median Basic Scatter Transmission Loss

The median basic scatter transmission loss may be

expressed —' —' as follows:

L, = - 10 log,,, bms 6

10

r (log 2)<p >-,

6 S pt

10 log

r log 2 r g g < o- > dv el n

t r

2 HJ , ,2 L 4kr (R R ) f o

(1-1)

The time averages < > are to be taken over the short period of time for

which the median is specified and this period of time must be sufficientlyshort (only a few minutes at 1000 Mc) so that the distribution of p

s within that period may be represented by the Rayleigh distribution. In

the above, k = 2ir/X where X. is the wavelength in free space and the

integration is taken over the common volume. Fig. 7 illustrates the

geometry involved in the evaluation of the scattering integral and

certain approximations made in this evaluation; the justification for

these approximations is given in reference 15 and later in this appendix.

The following derivation is given for the case where the transmitting

and receiving antennas are at the same height over a smooth earth and

sufficiently high (h = h > 4X./0) above the ground so that the sinesquared oscillations of g and g occur within sufficiently small intervals

t r
-34-

R = d/2 + x R = d/2~x

a+/5 =

PLAN VIEW

Figure 7
- 35 -

of i|> and Uj so that the product g" g = 4 and the frequency gain

t r t r function is equal to zero.

We will evaluate (1-1) for an atmosphere characterized by the

correlation function (6) and corresponding to the wave -number spectrum

(8) for which the average scattering cross section < <r > may be

expressed: —' —

< <r> & {2L4sin

2x /NAT}<{r(|JL + |)/r(|JL)} V i i SL q" 2,X " 3 > (1-2)

f

^^v»r-2 npnv

rr 2 2 2 2.2 2 2 J/2

q £ kr (a + (3) {[4 sin y + 4 cos y] sin w + 4 cos w} (1-3)

f p

' n

,J v

The approximation in (1-2) is negligible for q > 2ir. See reference 10

for the derivation of (1-3); the small approximation involved in the

derivation of (1-3) is equivalent to the assumption that the vector k

lies in planes perpendicular to the line joining the antennas. In the

above we have assumed that the mean wind direction lies in the plane

which passes through the transmitting and receiving antennas and is

normal to the great circle plane; and y is the angle in this plane between

the mean wind direction and the great circle plane;

, 2 2 X -1/2

sin go = w(w + h ) . Now make the following substitution:

2 2 2 2 2 , i sin y + i cos v = t (1-4)

p n w

a + (3 t» d(w

2 + h2

) 1/2 /RR (1-5) o

q ^(Ld/RR ){(w! ) 2 + (hi ) 2

}

1/2

(1-6)

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- 37-

d (km) = a 9 + 3>JZb. (meters) + 3NT2h (meters) (1-10)

g

te re

The expression 3 n/21i (meters) in (1-10) for the distance, d , to the

te h

radio horizon from a height h is quite accurate for those antenna

te

heights likely to be involved in tropospheric scatter systems, and may

be corrected for the higher heights in the exponential atmosphere

corresponding to N = 316 by the method of Bean and Thayer; —'• thus

d, (km) = 3 n/Hi (meters) - Ad where Ad = 0. 25 km for h = 500 meters,

h te te

Ad = 1. 82 km for h =2 km, Ad = 6. 44 km for h =5 km and

te te

Ad = 15. 39 km for h =10 km. The distance d in (1-8) may now be

determined from:

d=2(a+h ) sin[d /2a] ^d (1-11)

te gegg

In order to show that the geometrical approximations made in

the integrations leading to (1-8) are negligible, we will obtain the same

solution by a different method for the special case jj. = 0. 5, m = 0,

and \ < < Ql assuming isotropic turbulence i - i = i - SL . The

o p n v o method of integration is the same as that employed in an early paper

17/ 2

by Herbstreit, Norton, Rice, and Schafer. —' We will set sin x

= 1 — — 2 and g g = 4 as before, dv = da dp du(RR ) /d and then we obtain for

t r o

the special case \i = 0. 5 and m = 0:
- 38 -

+ - TT-a

L, = - 10 log bms 10

m y? {V It } da "I

L 4Trdk 3 a {sm(—— ))

ir Pm

(1-11)

If now we replace the spherical earth by a cylinder with its axis

perpendicular to the great circle plane (a very good approximation),

the lower limits a and 6 may be expressed: m m

a = tan (tan a /cos w) m o

(1-12)

6 = tan (tan B /cos oo) m o

(1-13)

All of the integrations in (1-11) may now be performed for our particular

case (a = (3 )and we obtain:

o o

L, = - 10 logn „ bms e

10

(log 2) V i e

J> c a —no o

=r- 1 =. + 8 a

2 1 2

L 6d(2Tri 0/M L tan a

o o

z 1 + sin a

2 /. I °

in -r—: o V 2 sin a

(1-14)

The above may be written:

Li_ = L^ (6<<1) - AL(6) bms bms (1-15)

where L, (0 < < 1) is identical to (13) for [i = — and m = and: bms 2

AL(8) = 10 log

10 I 2

tan a

a /- 1 + sin a

+ 8 a in (

o y 2 sin

(1-16)
- 39 ~

AL( 6) = 0.193 db for 9 = 0.1, AL ( 0) = 0. 3 54 db for 9 = 0.15 and

AL(9) = 0. 530 db for 9 = 0. 2; since this small correction term will be

even smaller for m > 0, it may be considered negligible for practical

applications since m will be large whenever 9 is large.

It appears from the above discussion that the comparatively

simple formula (1-8) will yield precise results in most practical

applications. In practice it will be found that the meteorological

parameter will not vary as (h /h) except over small ranges of h. o However, we may still use (1-8) with quite good accuracy for the more

general case in which the height dependence is given by:

< V

n Vr^w C^

+ ^ h - h

f(h) (I - npnwv 17) n=no

where f (h ) = 1 simply by setting f (h ) = o o

, h N m -i d / o

dh V h

and

h=h

o solving for m:

m = - h f'(h ) (1-18) o o

For example, there are good reasons to believe that the meteorological

parameter varies approximately exponentially with the height, at least

for large heights, in which case f (h) = exp [ - (h - h )/H(h )] where H(h ) o o o

is a scale height which is a very slowly varying function of h ; the

o appropriate value of m to use in (1-8) to approximate this exponential

variation is thus m = h /H (h ). o o
- 40 -

Unfortunately the above approximate method of allowing for

some functional dependence of the meteorological parameter other

than (h /h) will not provide accurate results in many important

cases. For example, a solution for the basic scatter transmission

loss has been obtained by Rice, Longley and Norton —' for the case

(1=1 using an exponential function of h, and the following table gives

the difference in decibels between this exact solution and the approximate solution using (1-8) with m = h /H (h ). o o

Table II m d (km)

g

0.001 88 0.00031

0.01 108 0. 0027

0. 1 208 0.0135

1 530 0.052

10 1470 0. 181

Ll (RLN)-L (1-8) bms bms 0.005

0.087

0.829

6.717

48.86

2.70

3.05

2.91

2. 55

2. 12

The distance d and angular distance in Table II were calculated

g

for the case h, = h = 100 meters and for the particular Bean and

11/

te re

Thayer —' exponential atmosphere corresponding to N = 316.29;

this atmosphere corresponds, with an actual earth's radius of

6370 km, to an effective earth's radius of 9000 km. For this atmosphere H(h ) = 6. 8925 km for all values of h „ We see by Table IE o o that the approximate formula (1-8) seriously underestimates the basic

scatter transmission loss when d is very large. This error can be

g

eliminated by equating the derivatives at some larger height h = Ch , i.e., by setting m = C h /H(Ch ), and the required values of C are

o o given in Table II; however, since the proper value of C to use in any

given instance depends upon the function f (h) assumed, this method of

eliminating the errors in (1-8) for functions f (h) other than (h /Mm

is impracticable.
- 41 -

The formula (1-8) may be readily extended to the case of an

unsymmetrical path (see references 4 or 15 for the geometry) such as

is usually encountered over irregular terrain or over smooth terrain

with unequal antenna heights. For this case the only change in the

above theory is in the term I((j. 3 m) which now depends on the additional

parameter s = (a /(3 ): o o 2^+1 . \ 2u+l

. 2u f m f x dx,l-mfx dx 1

1 «"-—>-(. t iK^i+ ->i' \z^r 3

( L +1 . xr }

(1-19)

For the numerical evaluation of (1-8) it is convenient to tabulate the value

of I(n,m, s) in the form:

J(u,M,S) = - 10 log

io

I(u,m,s) (1-20)

as a function of the parameters n, S = 10 log s and M = 10 log m;

such a tabulation is given in Table III. Note that I(u, m, 1/s) = I(|J., m, s)

and thus the same value J((jl,M,S) may be tabulated for ± S. The values

in Table III were obtained by numerical integration using an IBM 650

computer; the tabulated values are correct to the nearest 0. 001 db, and

it should be possible to interpolate linearly in this table with an

accuracy better than 0. 05 db for any intermediate values of (J., M and S.

When 2u is an integer, I(u,m, s) may be evaluated in closed form as

may be seen from the following expression:
- 42 -

i(ii ' m' s>E5 (s + iKz^i +m) I

s y—~s

+ s J T=

X 1 / x

The above expression was used for |i = 1 as a check on the accuracy of

a few of the values obtained by numerical integration.
-43 -

Table HI

J((i,M,0)

H- = fi = ^= V- = H

= (A = H = ft = |i = Ji = M- = H- = V = M- = M 0.32 0.34 0.36 0.38 0.40 0.45 0.50 0.55 0.60 0.70 0.80 1.0 1.2 1.5

-co 8.303 8.209 8.128 8.056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

-30 8.306 8.213 8.131 8.060 7.997 7.872 7.785 7.725 7.687 7.658 7.674 7.784 7.949 8.241

-25 8.314 8.221 8.139 8.067 8.004 7.879 7.791 7.731 7.693 7.664 7.679 7.788 7.953 8.245

-20 8.339 8.245 8.163 8.090 8.027 7.901 7.812 7.752 7.712 7.682 7.696 7.804 7.967 8.256

-18 8.360 8.266 8.183 8.110 8.047 7.920 7.830 7.769 7.729 7.698 7.711 7.816 7.979 8.266

-16 8.394 8.299 8.215 8.142 8.078 7.950 7.859 7.796 7.755 7.722 7.734 7.837 7.997 8.282

-14 8.447 8.350 8.266 8.192 8.127 7.997 7.904 7.839 7.797 7.761 7.770 7.869 8.026 8.308

-12 8.530 8.432 8.346 8.270 8.204 8.070 7.974 7.907 7.862 7.821 7.827 7.919 8.071 8.347

-10 8.659 8.559 8.471 8.393 8.324 8.185 8.085 8.013 7.965 7.917 7.916 7.999 8.143 8.410

:,-9 8.749 8.647 8.557 8.478 8.407 8.265 8.162 8.087 8.036 7.983 7.979 8.054 8.193 8.454

-8 8.860 8.757 8.665 8.583 8.511 8.365 8.257 8.179 8.125 8.066 8.056 8.124 8.256 8.509

-7 8.999 8.892 8.798 8.714 8.640 8.489 8.376 8.294 8.235 8.169 8.153 8.210 8.334 8.577

-6 9.169 9.060 8.963 8.876 8.799 8.642 8.523 8.436 8.372 8.298 8.274 8.318 8.432 8.663

-5 9.379 9.266 9.166 9.076 8.996 8.830 8.705 8.611 8.541 8.456 8.423 8.452 8.553 8.769

-4 9.636 9.519 9.414 9.320 9.236 9.062 8.929 8.827 8.750 8.652 8.608 8.617 8.703 8.902

-3 9.949 9.826 9.717 9.619 9.530 9.345 9.202 9.091 9.006 8.892 8.834 8.822 8.889 9.066

-2 10.326 10.198 10.083 9.980 9.886 9.689 9.534 9.413 9.317 9.186 9.112 9.072 9.118 9.268

-1 10.779 10.645 10.524 10.414 10.314 10.103 9.935 9.802 9.695 9.543 9.450 9.379 9.398 9.517

11.318 11.177 11.049 10.932 10.826 10.599 10.416 10.268 10.149 9.973 9.858 9.750 9.739 9.821

1 11.954 11.805 11.669 11.544 11.431 11.186 10.987 10.824 10.690 10.487 10.348 10.198 10.152 10.190

2 12.696 12.538 12.394 12.262 12.140 11.877 11.660 11.480 11.329 11.096 10.931 10.733 10.647 10.635

3 13.555 13.388 13.235 13.094 12.964 12.680 12.444 12.245 12.077 11.811 11.616 11.366 11.235 11.168

4 14.537 14.361 14.198 14.048 13.910 13.604 13.348 13.130 12.943 12.642 12.415 12.108 11.928 11.799

5 15.648 15.462 15.290 15.131 14.983 14.656 14.378 14.140 13.934 13.597 13.335 12.968 12.735 12.539

6 16.889 16.693 16.512 16.344 16.187 15.837 15.538 15.279 15.053 14.679 14.383 13.952 13.664 13.395

7 18.258 18.053 17.862 17.685 17.518 17.147 16.826 16.547 16.301 15.889 15.558 15.064 14.71914.375

8 19.747 19.532 19.333 19.146 18.972 18.579 18.238 17.939 17.674 17.225 16.860 16.302 15.900 15.481

9 21.342 21.119 20.912 20.717 20.534 20.122 19.762 19.444 19.161 18.678 18.280 17.661 17.204 16.711

10 23.028 22.798 22.582 22.381 22.191 21.760 21.383 21.049 20.749 20.235 19.806 19.131 18.622 18.059

11 24.786 24.549 24.328 24.120 23.924 23.478 23.086 22.737 22.423 21.881 21.425 20.700 20.144 19.515

12 26.600 26.358 26.132 25.918 25.717 25.259 24.854 24.492 24.166 23.600 23.122 22.352 21.754 21.068

13 28.457 28.211 27.980 27.762 27.557 27.088 26.673 26.301 25.965 25.379 24.881 24.074 23.44122.704
44 -

Table in

J(H,M, ±2)

M- = H

= M- = M- = H = H- -

Y- = V- = V- = H- = M- = V

= M- = H= M 0.32 0.34 0.36 0.38 0.40 0. 45 0. 50 0.55 0.60 0.70 0.80 1.0 1.2 1. 5

-co 8.303 8.209 8. 128 8.056 7.993 7. 869 7. 782 7.722 7.677 7.656 7.672 7. 782 7.947 8.239

-30 8.306 8.213 8. 131 8.060 7.997 7. 872 7.785 7.725 7.687 7.658 7.674 7.784 7.949 8.241

-25 8.315 8.221 8. 139 8. 067 8.005 7. 880 7.792 7.732 7.693 7.664 7.680 7.789 7.954 8.245

-20 8.341 8.247 8. 164 8.092 8.029 7. 903 7.814 7.753 7.714 7.683 7.698 7.805 7.968 8.257

-18 8.363 8.268 8.185 8. 113 8.049 7 922 7.833 7. 771 7.731 7.700 7.713 7.818 7.980 8.268

-16 8.398 8.303 8.219 8. 146 8.082 7. 953 7.862 7.800 7.759 7.725 7.737 7.840 8.000 8.285

-14 8.453 8.356 8.272 8. 198 8.133 8. 002 7.909 7.845 7.802 7.766 7.775 7.874 8.030 8.312

-12 8. 539 8.441 8.355 8.280 8.213 8. 079 7.983 7.916 7.871 7.830 7.835 7.927 8.079 8.354

-10 8.674 8.574 8.486 8.408 8.339 8. 200 8.099 8.027 7.978 7.930 7.929 8.011 8. 155 8.421

-9 8. 768 8.666 8. 576 8.496 8.426 8. 284 8. 180 8. 105 8.053 8.000 7.995 8.070 8.208 8.467

-8 8.884 8.780 8.688 8.606 8.534 8. 388 8.280 8.201 8. 146 8.087 8.077 8. 143 8.274 8. 525

-7 9.028 8.922 8.827 8. 743 8.669 8. 517 8.404 8.321 8.262 8.196 8. 179 8.234 8.357 8.598

-6 9.206 9.096 8.999 8.912 8.835 8. 677 8.558 8.470 8.405 8.330 8.305 8.348 8.460 8.689

-5 9.425 9.312 9.211 9. 121 9.040 8. 874 8.748 8.653 8.583 8.497 8.462 8.489 8.588 8.802

-4 9.692 9.575 9.470 9.376 9.291 9. 116 8.982 8.879 8.801 8.702 8.656 8.663 8.747 8.942

-3 10.018 9.895 9.785 9.686 9.597 9. 411 9.267 9.155 9.069 8.954 8.894 8.878 8.943 9. 115

-2 10.410 10.282 10. 166 10.062 9.968 9. 769 9.614 9.491 9.394 9.261 9.185 9.141 9. 184 9.329

-1 10.881 10.746 10.624 10.514 10.413 10. 201 10.032 9.896 9.788 9.634 9.539 9.462 9.478 9.591 11.440 11.298 11. 169 11.051 10.944 10. 716 10.531 10.382 10.261 10.082 9.965 9.851 9.836 9.911

1 12.097 11.947 11.810 11.685 11.571 11.325 11.124 10.959 10,823 10.616 10.475 10.319 10.267 10.299

2 12.863 12.704 12.559 12.426 12.303 12.038 11.819 11.637 11.484 11.248 11.079 10.875 10.783 10.764

3 13.744 13.576 13.422 13.280 13.150 12.864 12.625 12.425 12.255 11.986 11.787 11.530 11.394 11.318

4 14.746 14.569 14.405 14.255 14.116 13.808 13.550 13.330 13.142 12.838 12.608 12.294 12.109 11.971

5 15.871 15.685 15.512 15.353 15.204 14.876 14.597 14.357 14.150 13.810 13.547 13.174 12.936 12.732

6 17.119 16.923 16.742 16.573 16.416 16.066 15.766 15.506 15.280 14.904 14.607 14.173 13.881 13.606

7 18.484 18.279 18.089 17.911 17.746 17.374 17.055 16.776 16.530 16.118 15.787 15.292 14.945 14.599

8 19.958 19.744 19.546 19.360 19.186 18.795 18.456 18.158 17.895 17.448 17.084 16.529 16.128 15.710

9 21.530 21.308 21.102 20.909 20.727 20.318 19.960 19.645 19.365 18.885 18.491 17.878 17.425 16.936

10 23.187 22.959 22.745 22.545 22.356 21.930 21.556 21.225 20.929 20.420 19.997 19.330 18.828 18.273

11 24.916 24.682 24.462 24.256 24.061 23.620 23.231 22.886 22.576 22.040 21.591 20.875 20.328 19.711

12 26.705 26.465 26.240 26.028 25.828 25.374 24.973 24.615 24.292 23.733 23.261 22.502 21.915 21.241

13 28.540 28.295 28.066 27.849 27.645 27.180 26.768 26.400 26.067 25.487 24.995 24.199 23.576 22.852
45

Table m

J(|4,M, ±4)

H

= V- - V- = M- = H- = |4. = H- = H

= V- = (A = ^= M- = M- = V- = M 0.32 0.34 0.36 0.38 0.40 0.45 0.50 0. 55 0.60 0.70 0.80 1.0 1.2 1.5

-00 8.303 8.209 8. 128 8.056 7.993 7.869 7. 782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

-30 8.307 8.214 8.132 8.060 7.997 7.873 7. 785 7.725 7.687 7.659 7.675 7.784 7.950 8.241

-25 8.316 8.223 8. 141 8.069 8.006 7.881 7.793 7.733 7.695 7.666 7.681 7.790 7.955 8.246

-20 8.345 8.251 8. 169 8.096 8.033 7.907 7.818 7.757 7.718 7.688 7.702 7.808 7.972 8.261

-18 8.370 8.275 8. 193 8.120 8.056 7.929 7.840 7.778 7.738 7.706 7.719 7.824 7.986 8.273

-16 8.409 8.314 8.230 8.157 8.093 7.964 7.873 7.810 7.769 7.736 7. 747 7.849 8.009 8.293

-14 8.471 8.375 8.290 8.216 8.150 8.020 7.927 7.862 7.819 7.782 7.791 7.888 8.044 8.325

-12 8. 568 8.470 8.384 8.308 8.241 8. 107 8.010 7.942 7.897 7.855 7.860 7.950 8. 101 8.375

-10 8.719 8.619 8.530 8.452 8.382 8.243 8. 142 8.069 8.019 7. 970 7.968 8.048 8. 189 8.453

-9 8.824 8.722 8.631 8.551 8.481 8.337 8.232 8.157 8.104 8.050 8.043 8.115 8.251 8.508

-8 8.954 8.850 8.757 8.675 8.603 8.455 8.346 8.266 8.210 8. 149 8.137 8.200 8.328 8.576

-7 9. 116 9.009 8.913 8.829 8.754 8.601 8.486 8.402 8.342 8.273 8.254 8. 305 8.424 8.661

-6 9.315 9.205 9. 106 9.019 8.941 8.781 8.661 8.571 8.505 8.427 8.399 8.436 8.544 8.767

-5 9.560 9.445 9.344 9.253 9. 171 9.003 8.875 8.778 8.706 8.616 8.578 8.599 8.693 8.899

-4 9.859 9.740 9.634 9.539 9.453 9.275 9.139 9.034 8.954 8.850 8. 800 8.799 8.876 9.063

-3 10.222 10.098 9.986 9.886 9.796 9.607 9.460 9.345 9.256 9.136 9.071 9.046 9. 102 9.265

-2 10.659 10.529 10.412 10.306 10.210 10.008 9.849 9.723 9.623 9.483 9.401 9.347 9.379 9. 512

-1 11. 181 11.044 10.920 10.808 10.706 10.489 10.316 10. 177 10.065 9.903 9.801 9.712 9.716 9.815

11.798 11.654 11.523 11.404 11.295 11.061 10.872 10.718 10.593 10.405 10.280 10. 152 10.123 10. 181

1 12.519 12.367 12.228 12.101 11.984 11.732 11.526 11.356 11.215 11.000 10.849 10.677 10.610 10.622

2 13.351 13.190 13.042 12.907 12.782 12.511 12.287 12.099 11.942 11.695 11.517 11.295 11.187 11.146

3 14.296 14.126 13.970 13.826 13.693 13.401 13.158 12.952 12.777 12.498 12.290 12.015 11.86111.763

4 15.354 15.175 15.010 14.857 14.716 14.404 14.141 13.917 13.723 18.411 13.171 12.841 12.639 12.479

5 16.519 16.332 16.158 15.997 15.847 15.515 15.233 14.990 14.779 14.432 14.162 13.775 13.523 13.298

6 17.784 17.588 17.406 17.238 17.080 16.729 16.428 16.167 15.938 15.559 15.257 14.814 14.512 14.222

7 19.139 18.935 18.746 18.569 18.404 18.035 17.716 17.438 17.194 16.783 16.452 15.954 15.603 15.248

8 20.575 20.364 20.168 19.984 19.812 19.426 19.091 18.797 18.537 18.097 17.738 17.188 16.79116.373

9 22.087 21.869 21.666 21.476 21.298 20.896 20.545 20.237 19.962 19.494 19.109 18.511 18.069 17.591

10 23.670 23.446 23.236 23.040 22.856 22.439 22.074 21.752 21.464 20.970 20.560 19.916 19.432 18.897

11 25.320 25.090 24.875 24.673 24.483 24.053 23.675 23.339 23.038 22.520 22.087 21.399 20.876 20.287

12 27.033 26.798 26.578 26.371 26.175 25.732 25.342 24.994 24.682 24.140 23.685 22.958 22.397 21.757

13 28.803 28.563 28.337 28.126 27.926 27.471 27.069 26.710 26.387 25.825 25.350 24.585 23.990 23.302
-46 -

Table HI

J(M.,M,±6)

Ji= n

= n= \>-= ^= H = H- = H- = I1 = ^ = H

= H = M-

H

= M 0.32 0.34 0.36 0.38 0.40 0.45 0.50 0.55 0.60 0.70 0.80 1.0 1.2 1.5

-oo 8.303 8.209 8.128 8.056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

-30 8.308 8.214 8.132 8.061 7.998 7.874 7.786 7.726 7.688 7.659 7.675 7.785 7.950 8.242

-25 8.319 8.225 8.143 8.071 8.008 7.883 7.795 7.735 7.697 7.668 7.683 7.792 7.957 8.248

-20 8.353 8.259 8.176 8.104 8.040 7.914 7.825 7.764 7.725 7.694 7.708 7.815 7.978 8.266

-18 8.382 8.287 8.204 8.132 8.068 7.941 7.851 7.789 7.749 7.717 7.729 7.834 7.995 8.282

-16 8.428 8.332 8.249 8.175 8.111 7.982 7.891 7.828 7.786 7.752 7.763 7.865 8.023 8.307

-14 8.500 8.404 8.319 8.245 8.179 8.048 7.954 7.889 7.846 7.808 7.816 7.913 8.068 8.347

-12 8.614 8.516 8.429 8.353 8.286 8.151 8.054 7.986 7.940 7.897 7.900 7.989 8.137 8.409

-10 8.792 8.691 8.602 8.523 8.454 8.313 8.211 8.137 8.087 8.035 8.031 8.108 8.247 8.507

-9 8.915 8.812 8.721 8.641 8.570 8.425 8.319 8.242 8.188 8.132 8.123 8.191 8.323 8.575

-8 9.068 8.963 8.870 8.787 8.714 8.564 8.454 8.373 8.315 8.251 8.236 8.294 8.418 8.660

-7 9.258 9.150 9.054 8.968 8.893 8.737 8.621 8.535 8.473 8.400 8.378 8.423 8.537 8.766

-6 9.491 9.380 9.281 9.192 9.113 8.951 8.828 8.736 8.667 8.585 8.553 8.583 8.684 8.898

-5 9.778 9.663 9.559 9.467 9.385 9.213 9.082 8.983 8.908 8.812 8.770 8.781 8.866 9.062

-4 10.128 10.008 9.900 9.803 9.716 9.535 9.394 9.286 9.202 9.092 9.036 9.024 9.091 9.264

-3 10.552 10.426 10.313 10.211 10.119 9.925 9.773 9.654 9.561 9.433 9.361 9.322 9.367 9.513

-2 11.061 10.929 10.809 10.701 10.603 10.396 10.231 10.100 9.995 9.846 9.755 9.685 9.703 9.816

-1 11.666 11.526 11.400 11.285 11.181 10.957 10.778 10.633 10.515 10.342 10.230 10.122 10.109 10.184

12.376 12.229 12.094 11.972 11.860 11.619 11.424 11.263 11.131 10.930 10.794 10.644 10.595 10.627

1 13.197 13.041 12.899 12.769 12.649 12.389 12.176 11.998 11.850 11.620 11.456 11.259 11.171 11.153

2 14.132 13.967 13.816 13.677 13.549 13.270 13.037 12.842 12.677 12.416 12.223 11.975 11.843 11.770

3 15.174 15.001 14.84114-694 14.558 14.258 14.007 13.794 13.611 13.317 13.095 12.794 12.616 12.484

4 16.314 16.132 15.965 15.809 15.665 15.347 15.077 14.846 14.646 14.320 14.068 13.713 13.488 13.295

5 17.536 17.347 17.172 17.009 16.857 16.521 16.234 15.986 15.770 15.414 15.133 14.726 14.454 14.201

6 18.825 18.628 18.446 18.277 18.119 17.766 17.463 17.200 16.970 16.585 16.278 15.822 15.507 15.195

7 20.166 19.964 19.775 19.600 19.436 19.069 18.752 18.475 18.231 17.821 17.490 16.990 16.634 16.267

8 21.554 21.346 21.153 20.972 20.802 20.422 20.093 19.804 19.548 19.115 18.762 18.220 17.827 17.410

9 22.989 22.776 22.577 22.391 22.217 21.825 21.483 21.183 20.916 20.461 20.088 19.509 19.081 18.617

10 24.476 24.257 24.054 23.863 23.684 23.279 22.926 22.615 22.337 21.862 21.469 20.854 20.393 19.884

11 26.019 25.795 25.586 25.391 25.207 24.791 24.426 24.104 23.816 23.320 22.908 22.257 21.764 21.212

12 27.622 27.393 27.180 26.979 26.790 26.363 25.987 25.654 25.355 24.839 24.408 23.722 23.197 22.603

13 29.287 29.053 28.835 28.629 28.436 27.997 27.610 27.266 26.957 26.421 25.971 25.250 24.694 24.057
- 47 -

Table III

J(|x,M,±8)

H

= n= (i = ji = fi = h- = M-

I1 = M- = P = M-= H

= t* = M-

M 0.32 0.34 0.36 0.38 0.40 0.45 0.50 0.55 0.60 0.70 0.80 1.0 1.2 1.5

-oo 8.303 8.209 8.128 8.056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

-30 8.309 8.215 8.134 8.062 7.999 7.875 7.787 7.727 7.689 7.660 7.676 7.786 7.951 8.243

-25 8.322 8.228 8.146 8.074 8.011 7.886 7.798 7.738 7.700 7.671 7.686 7.795 7.959 8.250

-20 8.363 8.269 8.186 8.114 8.050 7.924 7.835 7.774 7.734 7.703 7.717 7.823 7.986 8.274

-18 8.398 8.303 8.220 8.147 8.083 7.956 7.866 7.804 7.764 7.731 7.743 7.847 8.008 8.294

-16 8.453 8.357 8.274 8.200 8.135 8.007 7.915 7.851 7.810 7.775 7.785 7.886 8.044 8.326

-14 8.540 8.443 8.358 8.284 8.218 8.086 7.992 7.926 7.883 7.844 7.851 7.946 8.100 8.377

-12 8.677 8.578 8.491 8.415 8.347 8.212 8.114 8.045 7.998 7.954 7.955 8.042 8.188 8.457

-10 8.890 8.789 8.699 8.620 8.550 8.408 8.304 8.230 8.178 8.125 8.118 8.191 8.327 8.582

-9 9.038 8.935 8.843 8.762 8.690 8.544 8.436 8.358 8.303 8.243 8.232 8.295 8.423 8.669

-8 9.222 9.116 9.022 8.938 8.864 8.713 8.600 8.517 8.458 8.391 8.372 8.425 8.543 8.778

-7 9.449 9.340 9.243 9.157 9.080 8.922 8.803 8.715 8.650 8.574 8.547 8.586 8.692 8.913

-6 9.729 9.616 9.516 9.426 9.346 9.180 9.054 8.959 8.889 8.801 8.764 8.785 8.878 9.081

-5 10.073 9.955 9.851 9.757 9.673 9.498 9.363 9.260 9.182 9.080 9.031 9.031 9.106 9.289

-4 10; 491 10.369 10.259 10.160 10.071 9.885 9.740 9.628 9.540 9.422 9.358 9.333 9.388 9.544

-3 10.997 10.868 10.753 10.648 10.554 10.355 10.198 10.074 9.976 9.838 9.757 9.701 9.731 9.857

-2 11.602 11.466 11.344 11.233 11.133 10.919 10.748 10.610 10.500 10.339 10.238 10.147 10.147 10.237

-1 12.317 12.174 12.044 11.926 11.818 11.587 11.401 11.248 11.124 10.937 10.812 10.681 10.647 10.694

13.149 12.998 12.860 12.734 12.619 12.369 12.165 11.996 11.856 11.640 11.489 11.313 11.240 11.239

1 14.100 13.940 13.794 13.660 13.536 13.267 13.044 12.858 12.701 12.454 12.275 12.048 11.933 11.878

2 15.165 14.996 14.841 14.698 14.566 14.277 14.035 13.831 13.656 13.378 13.169 12.889 12.729 12.616

3 16.326 16.149 15.986 15.835 15.695 15.387 15.126 14.904 14.713 14.402 14.164 13.832 13.625 13.453

4 17.563 17.378 17.207 17.049 16.902 16.575 16.298 16.060 15.852 15.512 15.245 14.863 14.611 14.381

5 18.848 18.657 18.480 18.315 18.161 17.820 17.528 17.275 17.054 16.687 16.395 15.967 15.673 15.388

6 20.161 19.964 19.782 19.612 19.453 19.099 18.794 18.529 18.296 17.906 17.593 17.124 16.795 16.460

7 21.489 21.288 21.101 20.926 20.763 20.398 20.083 19.808 19.565 19.156 18.825 18.321 17.960 17.582

8 22.831 22.626 22.435 22.257 22.090 21.715 21.391 2J.. 106 20.855 20.429 20.082 19.548 19. X59 18. 743

9 24.193 23.984 23.789 23.607 23.437 23.053 22.720 22.428 22.168 21.727 21.365 20.804 20.389 19.938

10 25.584 25.371 25.173 24.987 24.813 24.421 24.079 23.779 23.511 23.054 22.678 22.090 21.652 21.168

11 27.016 26.799 26.596 26.407 26.229 25.828 25.477 25.168 24.892 24.420 24.029 23.414 22.95122.436

12 28.497 28.276 28.069 27.876 27.694 27.283 26.923 26.605 26.320 25.832 25.425 24.783 24.295 23.746

13 30.036 29.810 29.599 29.401 29.215 28.794 28.424 28.097 27.803 27.297 26.874 26.202 25.688 25.105
48

Table ni

J((JL,M ±10)

M 0.32

H- = 0.34

H- = 0.36V- = 0.38

K

= 0.40

V- = 0.45

V- = 0.50M- = 0.55

H- = 0.60M- = 0.70

H

= 0.80

H

= 1.0

P- = 1.2

H

= 1.5

-00 8.303 8.209 8. 128 8.056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

-30 8. 310 8.217 8.135 8.063 8.000 7.876 7.788 7.728 7.690 7.662 7.677 7.787 7.952 8.244

-25 8.326 8.232 8. 150 8.078 8. 015 7.890 7.802 7.742 7.703 7.674 7.689 7. 798 7.963 8.253

-20 8.375 8.281 8. 198 8. 126 8.062 7.936 7.847 7.785 7.746 7.715 7.728 7.834 7.996 8.284

-18 8.417 8.322 8.239 8. 166 8.102 7.975 7.884 7.822 7.782 7.749 7. 761 7.864 8.024 8. 310

-16 8.484 8.388 8.304 8.230 8.166 8.036 7.944 7.881 7.839 7.803 7.813 7.912 8.069 8.351

-14 8. 588 8.492 8.406 8.331 8.265 8. 133 8.039 7.973 7.928 7.889 7.895 7.989 8. 140 8.416

-12 8. 753 8.654 8.567 8.490 8.422 8.286 8.187 8. 117 8.070 8.024 8.024 8. 108 8.252 8.518

-10 9.011 8.908 8.818 8.738 8.668 8.525 8.420 8.344 8.291 8.235 8.227 8.296 8.428 8.678

-9 9.189 9.084 8.992 8.910 8.837 8.690 8.580 8.501 8.444 8.382 8.367 8.426 8.549 8.789

-8 9.410 9.303 9.208 9. 123 9.048 8.895 8.780 8.696 8.634 8.564 8.542 8.588 8. 701 8.928

-7 9.683 9.573 9.475 9.388 9.310 9. 149 9.028 8.938 8.871 8.790 8.759 8.789 8.889 9.100

-6 10.020 9.906 9.804 9.713 9.632 9.463 9.334 9.236 9. 162 9.069 9.027 9.038 9. 122 9.314

-5 10.433 10.314 10.208 10. 112 10.027 9.848 9. 710 9.603 9.521 9.412 9.357 9.345 9.409 9.577

-4 10.935 10.811 10.699 10.599 10.508 10.317 10.167 10.050 9.958 9.832 9.760 9.720 9.761 9.900

-3 11. 541 11.410 11.292 11. 185 11.089 10.884 10. 721 10.592 10.488 10.340 10.249 10. 176 10.189 10.294

-2 12.263 12. 125 12.000 11.886 11.782 11. 561 11.384 11.240 11. 123 10.950 10.837 10.725 10.706 10.769

-1 13. Ill 12.965 12.832 12. 710 12.599 12.360 12. 165 J2.005 11.873 11.672 11.534 11.378 11.321 11.338

14.090 13.935 13.793 13.663 13.544 13.285 13.072 12.895 12. 746 12.514 12.348 12.142 12.043 12.007

1 15. 194 15.030 14. 879 14.741 14.613 14.334 14. 101 13.906 13. 739 13.475 13.279 13.021 12.877 12. 783

2 16.406 16.233 16.074 15.927 15.791 15.492 15.240 15.026 14.842 14. 545 14.319 14.007 13.816 13.663

3 17.699 17. 517 17.350 17. 196 17.052 16.734 16.465 16.234 16.034 15.706 15.451 15.087 14.851 14.638

4 19.034 18.847 18.673 18.512 18.362 18.028 17. 743 17.498 17.283 16.928 16.648 16.238 15.960 15.692

5 20.380 20. 187 20.008 19.842 19.687 19.341 19.044 18.786 18.560 18. 183 17.882 17.433 17.119 16.804

6 21.715 21. 518 21.335 21. 165 21.006 20.650 20. 344 20.077 19.842 19.448 19.129 18.649 18.306 17.950

7 23.032 22.832 22.646 22.472 22.310 21.947 21.633 21.359 21.117 20.709 20.377 19.871 19.505 19. 117

8 24.336 24.133 23. 944 23. 768 23.603 23.233 22.913 22.633 22.385 21.965 21.622 21.095 20.710 20.294

9 25.638 25.432 25. 240 25.062 24.894 24.518 24. 192 23.906 23.653 23.222 22.869 22.323 21.920 21.480

10 26.949 26.741 26.547 26.366 26.196 25.813 25.481 25.190 24.930 24. 489 24.126 23. 562 23.141 22. 679

11 28.284 28.072 27.876 27.691 27.519 27.130 26.791 26.493 26.228 25. 776 25.402 24.818 24.381 23.896

12 29.653 29.438 29.238 29.051 28.875 28.479 28.134 27.829 27.557 27.092 26.707 26.102 25.647 25. 137

13 31.067 30.849 30.646 30.455 30.276 29. 871 29.518 29.205 28.927 28.448 28.050 27.423 26.947 26.412
49

Table in

J(H.,M ±12)

M

V- = 0.32M- = 0.34

H- = 0.36H- = 0.38

H

= 0.40

^

= 0.45

H- = 0.50P

= 0.55

V- = 0.60M- = 0.70

M- = 0.80M- = 1.0

H- = 1.2

H- =

1. 5

-co 8.303 8.209 8. 128 8.056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

-30 8.311 8.218 8.136 8.064 8.002 7.877 7.789 7.730 7.691 7.663 7.679 7.788 7.953 8.245

-25 8.330 8.236 8. 154 8.083 8.020 7.895 7.806 7.746 7.708 7.678 7.694 7.802 7.967 8.257

-20 8.389 8.295 8.212 8.140 8.076 7.950 7.860 7.799 7.759 7. 728 7. 741 7.846 8.008 8.295

-18 8.439 8.345 8.261 8. 188 8.124 7.997 7.906 7.844 7.803 7.770 7.782 7.884 8.044 8.328

-16 8.519 8.423 8.339 8.265 8.200 8.071 7.979 7.914 7.872 7.836 7.846 7.944 8. 100 8.380

-14 8.644 8.547 8.462 8.387 8.321 8. 188 8.093 8.026 7.982 7.941 7.947 8.038 8.189 8.462

-12 8.841 8.742 8.654 8. 577 8.509 8.372 8.273 8.202 8.154 8. 106 8.106 8.187 8.329 8.591

-10 9.149 9.047 8.956 8.875 8.804 8.660 8.554 8.477 8.423 8.365 8.355 8.419 8.548 8.793

-9 9.362 9.257 9. 164 9.082 9.008 8.859 8.748 8.667 8.609 8.544 8.527 8.581 8.699 8.933

-8 9.626 9.519 9.423 9.338 9.262 9. 107 8.990 8.904 8.841 8.766 8.741 8.781 8.888 9. 108

-7 9.954 9.842 9.743 9.655 9.576 9.413 9.290 9.197 9.128 9.042 9.007 9.030 9. 123 9.325

-6 10.356 10.241 10. 138 10.046 9.963 9.791 9.659 9.558 9.482 9.383 9.336 9.338 9.413 9.593

-5 10.850 10.729 10.621 10.524 10.437 10.254 10. 113 10.002 9.917 9.801 9.740 9.716 9. 770 9.923

-4 11.449 11.323 11.209 11. 107 11.014 10.818 10.664 10.543 10.446 10.312 10.232 10.178 10.206 10.326

-3 12. 171 12.037 11.917 11.808 11.709 11.498 11.330 11. 195 11.086 10.928 10.828 10.737 10.734 10.817

-2 13.027 12.886 12.758 12.642 12.535 12.307 12.123 11.972 11.849 11.664 11.540 11.406 11.368 11.405

-1 14.028 13.878 13.742 13.617 13.503 13.255 13.053 12.885 12.745 12.530 12.378 12.197 12.117 12.103

15.172 15.013 14.868 14.734 14.611 14.343 14.121 13.935 13.778 13.530 13.348 13.114 12.989 12.917

1 16.445 16.277 16.122 15.980 15.848 15.559 15.317 15.112 14.936 14.655 14.442 14.153 13.980 13.847

2 17.817 17.639 17.476 17.325 17.185 16.876 16.615 16.392 16.199 15.884 15.641 15.298 15.078 14.884

3 19.241 19.056 18.886 18.728 18.580 18.254 17.976 17.737 17.529 17.185 16.914 16.521 16.257 16.006

4 20.671 20.480 20.304 20.140 19.987 19.647 19.356 19.104 18.884 18.516 18.223 17.789 17.487 17.186

5 22.070 21.875 21.695 21.527 21.370 21.020 20.720 20.458 20.228 19.843 19.533 19.067 18.736 18.394

6 23.422 23.225 23.041 22.871 22.711 22.355 22.047 21.779 21.542 21.144 20.822 20.332 19.979 19.606

7 24.730 24.531 24.345 24.173 24.011 23.649 23.336 23.063 22.822 22.414 22.083 21.575 21.205 20.809

8 26.006 25.804 25.617 25.443 25.279 24.913 24.596 24.319 24.073 23.658 23.319 22.797 22.414 21.999

9 27.262 27.059 26.870 26.694 26.529 26.159 25.838 25.557 25.307 24.885 24.539 24.004 23.609 23.178

10 28.514 28.309 28.118 27.940 27.774 27.399 27.074 26.789 26.536 26.106 25.753 25.206 24.800 24.353
- 50

Table ni

J((i,M, ±14)

H

= H= V- = n

= H- = H- = H- = H= P= M- = V- = H

= H

= V- = M 0.32 0.34 0.36 0.38 0.40 0.45 0.50 0.55 0.60 0.70 0.80 1.0 1.2 1.5

-CD 8.303 8.209 8. 128 8.056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

-30 8. 313 8.219 8.138 8.066 8.003 7.879 7.791 7.731 7.693 7.664 7.680 7.789 7.955 8.246

-25 8.335 8.241 8.159 8.087 8.024 7.899 7.811 7.751 7.712 7.683 7.698 7.807 7.971 8.261

-20 8.405 8.310 8.228 8. 155 8.092 7.965 7.875 7.814 7.774 7.743 7.756 7.861 8.022 8.309

-18 8.464 8.369 8.286 8.213 8.149 8.021 7.930 7.868 7.827 7.793 7.805 7.907 8.066 8.350

-16 8. 558 8.462 8.378 8.304 8.239 8.109 8.017 7.952 7.910 7.874 7.882 7.980 8. 135 8.414

-14 8.706 8.609 8.523 8.448 8.382 8.249 8.153 8.086 8.041 8.000 8.005 8.095 8.244 8.516

-12 8.939 8.839 8.752 8.674 8.606 8.468 8.368 8.296 8.247 8.199 8.197 8.276 8.416 8.675

-10 9.303 9.200 9. 108 9.028 8.956 8.811 8.703 8.625 8.570 8.510 8.498 8.560 8.685 8.926

-9 9.555 9.449 9.355 9.272 9.198 9.048 8.936 8.853 8.794 8.726 8.707 8.756 8.871 9.099

-8 9.867 9.758 9.662 9.576 9.500 9.343 9.224 9.136 9.072 8.994 8.966 9.001 9.103 9,315

-7 10.254 10. 142 10.042 9.952 9.873 9.708 9.582 9.487 9.416 9.327 9.288 9.304 9.390 9. 583

-6 10.730 10.614 10. 509 10.416 10.332 10.158 10.023 9.920 9.840 9.737 9.685 9.678 9.745 9.914

-5 11.313 11.191 11.082 10.983 10.895 10.708 10.563 10.449 10.361 10.239 10.172 10.137 10.180 10.320

-4 12.021 11.893 11.777 11.673 11.578 11.378 11.220 11.094 10.994 10.852 10.765 10.697 10.712 10.816

-3 12.871 12.736 12.613 12.502 12.400 12.184 12.011 11.871 11.75, 11.590 11.481 11.373 11.355 11.416

-2 13.878 13.734 13.603 13.484 13.375 13.141 12.950 12.793 12.664 12.468 12.332 12.179 12.122 12.134

-1 15.047 14.894 14.755 14.627 14.509 14.254 14.044 13.870 13.723 13.494 13.330 13.124 13.023 12.979 16.373 16.210 16.061 15.924 15.797 15.521 15.290 15.096 14.930 14.667 14.471 14.210 14.060 13.954

1 17.826 17.654 17.496 17.349 17.214 16.916 16.664 16.451 16.267 15.969 15.740 15.422 15.222 15.053

2 19.360 19.179 19.012 18.857 18.714 18.396 18.126 17.894 17.693 17.363 17.104 16.732 16.485 16.253

3 20.911 20.723 20.550 20.389 20.239 19.905 19.620 19.373 19.158 18.800 18.516 18.097 17.808 17.522

4 22.424 22.232 22.053 21.887 21.733 21.388 21.091 20.834 20.608 20.230 19.927 19.472 19.151 18.820

5 23.866 23.671 23.489 23.320 23.162 22.810 22.506 22.241 22.008 21.617 21.300 20.821 20.476 20.113

6 25.232 25.035 24.851 24.680 24.520 24.163 23.854 23.585 23.347 22.947 22.621 22.125 21.763 21.378
51

Table in

J((i,M, ±16)

M 0.32

H

= 0.34 0.36

^

= 0.38

V- = 0.40V- = 0.45

H

= 0.50 0.55

V- = 0.60

V- = 0.70M- = 0.80M- = 1.0H- = 1.2V- = 1.5

oo 8.303 8.209 8. 128 8.056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239 30 8.315 8.221 8. 139 8.068 8.005 7.880 7.793 7.733 7.695 7.666 7.682 7.791 7.956 8.248

25 8.340 8.247 8.165 8.093 8.030 7.905 7.816 7.756 7.718 7.688 7.703 7.812 7.976 8.266

20 8.422 8.327 8.244 8.172 8.108 7.982 7.892 7.830 7.791 7. 759 7.772 7.876 8.038 8.324

18 8.491 8.396 8.312 8.239 8.175 8.047 7.956 7.894 7.853 7.819 7.830 7.932 8.090 8.373

16 8.600 8.504 8.420 8.346 8.281 8.151 8.058 7.993 7.951 7.914 7.922 8.019 8. 174 8.452

14 8.773 8.675 8.590 8.514 8.448 8.314 8.218 8.151 8.106 8.064 8.068 8.157 8.305 8.575

12 9.044 8.944 8.856 8.779 8.710 8.572 8.471 8.399 8.349 8.300 8.297 8.374 8.512 8.769

10 9.468 9.365 9.273 9.192 9.120 8.974 8.866 8.787 8.731 8.669 8.656 8.714 8.836 9.073

-9 9.762 9.656 9.562 9.478 9.404 9.252 9.139 9.055 8.994 8.925 8.903 8.949 9.060 9.283

-8 10.126 10.017 9. 9*20 9.834 9.757 9.598 9.478 9.389 9.323 9.243 9.212 9.241 9.339 9.545

-7 10.578 10.465 10.364 10.274 10. 193 10.026 9.899 9.802 9.729 9.637 9.595 9.604 9.685 9.870

-6 11.134 11.016 10.911 10.816 10.732 10.555 10.418 10.312 10.231 10.123 10.067 10.052 10.112 10.270

-5 11.814 11.691 11.580 11.481 11.391 11.201 11.053 10.937 10.845 10.718 10.645 10.601 10.635 10.762

-4 12.640 12.510 12.393 12.287 12.191 11.987 11.825 11.696 11.592 11.443 11.350 11.269 11.273 11.362

-3 13.630 13.493 13.368 13.255 13.152 12.931 12.753 12.609 12.490 12.315 12.197 12.074 12.042 12.085

-2 14.800 14.654 14.521 14.399 14.288 14.048 13.851 13.699 13.554 13.348 13.203 13.031 12.956 12.945

-1 16.153 15.997 15.854 15.723 15.603 15.341 15.124 14.943 14.790 14.549 14.372 14.145 14.023 13.952

17.671 17.505 17.352 17.212 17.082 16.798 16.559 16.357 16.185 15.908 15.698 15.412 15.239 15.103

1 19.312 19.136 18.974 18.825 18.686 18.379 18.120 17.898 17.707 17.394 17.151 16.806 16.582 16.380

2 21.007 20.823 20.653 20.495 20.348 20.023 19.745 19.506 19.298 18.953 18.681 18.283 18.012 17.748

3 22.676 22.486 22.310 22.146 21.993 21.653 21.362 21.110 20.888 20.519 20.224 19.783 19.474 19.158

4 24.260 24.065 23.885 23.718 23.562 23.212 22.912 22.650 22.420 22.035 21.723 21.252 20.915 20.562
52 -

Table in

J(H,M, ±18)

V- =

M- = V- = M- = H- = K- = V

= M- = H- = H

= H= V- = M- = H- = M 0.32 0.34 0.36 0.38 0.40 0.45 0.50 0.55 0.60 0. 70 0.80 1.0 1.2 1.5

CD 8.303 8.209 8.128 8. 056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

30 8.316 8.223 8.141 8.069 8.007 7.882 7. 794 7.734 7.696 7.668 7.683 7.793 7.958 8.249

25 8.346 8.252 8.170 8.098 8.035 7.910 7.822 7.762 7.723 7.694 7.709 7.817 7.981 8.271

20 8.439 8.345 8.262 8.190 8. 126 7.999 7.909 7.848 7.808 7.776 7.789 7.893 8.054 8.340

18 8.519 8.424 8.340 8.267 8.203 8.075 7.984 7.921 7.880 7.846 7.857 7.958 8. 117 8.399

16 8.645 8.549 8.464 8.390 8.325 8.195 8.102 8.037 7.994 7.957 7.965 8.061 8.215 8.492

14 8.843 8.745 8.659 8.584 8.517 8. 384 8.287 8.220 8.174 8. 132 8. 135 8.223 8.371 8.639

12 9. 155 9.055 8.966 8.889 8. 820 8.681 8. 580 8.507 8.457 8.407 8.403 8.479 8.615 8.870

10 9.643 9.539 9.447 9.366 9.293 9.146 9.037 8.958 8.901 8.838 8.823 8.879 8.998 9.232

-9 9.981 9.874 9.780 9.696 9.621 9.468 9.354 9.270 9.208 9. 137 9.114 9.156 9.264 9.483

-8 10.400 10.291 10.193 10.106 10.029 9.869 9.748 9.657 9.590 9.508 9.475 9.500 9.594 9.795

-7 10.920 10.807 10.705 10.615 10.533 10.365 10.236 10.138 10.063 9.968 9.923 9.927 10.003 10.181

-6 11.561 11.443 11.336 11.241 11.156 10.977 10.838 10.730 10.647 10.535 10.476 10.454 10.508 10.658

-5 12.345 12.221 12.109 12.009 11.918 11.726 11.575 11.456 11.362 11.231 11.154 11.101 11.127 11.243

-4 13.297 13.166 13.047 12.940 12.843 12.636 12.471 12.338 12.231 12.076 11.977 11.886 11.881 11.955

-3 14.438 14.299 14.172 14.057 13.952 13.728 13.546 13.397 13.275 13.092 12.967 12.831 12.787 12.812

-2 15.782 15.634 15.498 15.375 15.261 15.016 14.814 14.647 14.508 14.292 14.138 13.950 13.860 13.829

-1 17.329 17.170 17.025 16.892 16.769 16.501 16.278 16.090 15.932 15.680 15.493 15.246 15.106 15.010

19.050 18.881 18.726 18.582 18.450 18.158 17.913 17.704 17.525 17.235 17.014 16.705 16.512 16.. 347

1 20.883 20.704 20.539 20.386 20.245 19.931 19.664 19.436 19.237 18.911 18.656 18.288 18.042 17.810

2 22.734 22.547 22.375 22.214 22.065 21.733 21.449 21.203 20.989 20.632 20.349 19.929 19.637 19.345

3 24.510 24.317 24.139 23.974 23.819 23.474 23.178 22.921 22.695 22.316 22.012 21.553 21.227 20.887

4 26.151 25.955 25.774 25.606 25.448 25.096 24.792 24.528 24.295 23.903 23.585 23.102 22.753 22.382
- 53 -

Table 01

J(|i,M, t20)

M

V- = 0.32M- = 0.34M- = 0.36M- = 0.38^

= 0.40

V- = 0.45V- = 0.50H

= 0.55

V- = 0.60H

= 0.70

V- = 0.80H

= 1.0

V- = 1.2

H- = 1.5

00 8.303 8.209 8. 128 8.056 7.993 7.869 7.782 7.722 7.677 7.656 7.672 7.782 7.947 8.239

30 8.318 8.225 8.143 8.071 8.009 7. 884 7.796 7.736 7.698 7.669 7.685 7.794 7.960 8.251

25 8.352 8.258 8.176 8.104 8.041 7.916 7.828 7.767 7.729 7.699 7.714 7.822 7.987 8.277

20 8.458 8.363 8.280 8.208 8.144 8.017 7.928 7.866 7.826 7.794 7.807 7.911 8.072 8.358

18 8.548 8.453 8.369 8.296 8.232 8.104 8.013 7.950 7.909 7.875 7.885 7.986 8. 144 8.426

16 8.691 8.595 8.510 8.436 8.371 8.240 8.147 8.082 8.039 8.002 8.010 8.105 8.259 8.53 5 14 8.916 8.818 8.732 8.656 8.590 8.456 8.359 8.292 8.246 8.203 8.206 8.293 8.440 8.707

12 9.270 9. 170 9.081 9.003 8.934 8.795 8.694 8.621 8.570 8. 520 8.515 8.589 8.724 8.978

10 9.824 9.720 9.628 9. 546 9.474 9.326 9.217 9.137 9.079 9.015 8.999 9.053 9-171 9.402

-9 10.208 10.402 10.007 9.923 9.848 9.694 9.579 9.494 9.432 9.359 9.334 9.374 9.480 9.695

-8 10.686 10.576 10.478 10.391 10.313 10. 152 10.030 9.938 9.870 9.786 9.752 9.774 9.864 10.060

-7 11.278 11.163 11.061 10.970 10.889 10.719 10.589 10.489 10.414 10.316 10.269 10.269 10.341 10.513

-6 12.007 11.888 11.781 11.685 11.599 11.418 11.278 11.169 11.084 10.969 10.907 10.880 10.929 11.072

-5 12.900 12.775 12.663 12.561 12.469 12.275 12.123 12.002 11.906 11.770 11.690 11.630 11.650 11.757

-4 13.985 13.852 13.733 13.625 13.526 13.316 13.149 13.013 12.904 12.744 12.640 12.540 12.526 12.590

-3 15.284 15.143 15.016 14.899 14.793 14.564 14.379 14.227 14.102 13.912 13.782 13.634 13.579 13.590

-2 16.813 16.662 16.525 16.400 16.284 16.034 15.829 15.657 15.514 15.290 15.128 14.926 14.823 14.774

-1 18.564 18.403 18.256 18.120 17.995 17.721 17.493 17.300 17.137 16.875 16.679 16.415 16.259 16.141 20.496 20.325 20.167 20.021 19.885 19.588 19.336 19.122 18.937 18.636 18.404 18.076 17.864 17.675

1 22.523 22.341 22.174 22.018 21.874 21.554 21.281 21.046 20.842 2Q. 505 20.239 19.850 19.586 19.328

2 24.524 24.335 24.160 23.998 23.846 23.508 23.219 22.968 22.749 22.382 22.089 21.651 21.342 21.026

3 26.393 26.199 26.019 25.852 25.696 25.347 25.047 24.786 24.556 24.171 23.859 23.387 23.047 22.689

4 28.079 27.883 27.701 27.532 27.373 27.019 26.713 26.446 26.211 25.815 25.493 25.001 24.643 24.259
54 -

Appendix II

Derivation of the Spectrum S(k ) Corresponding

to the Correlation Function {2 " ^/r(|J.) }(r /! f" K (r/i ) o u o

In this appendix it will be shown that S( k ) is given by (8) for

a correlation function having the form (6). By definition + co S(lT) = \ \ \ cos ( k"- ~r) C(r/i ) dx dy dz (H-l)

CO

where r = xa+yb+zc = rr (H-2)

k = k a+k b+k c =kk, (H-3)

p n v 1

The characteristic scale SL is defined by (7) or by the following:

p* - r

* + f~^ + t

^ = (r/V ^r p ^i (n " 4)

p n v

It will be convenient also to define a vector q

q=ika+ikb+ik c =1 kq, (II-5) pp nn vv kl

With this notation

k • r = q • p = xk + yk + zk (H-6)

p n v

Let p = x/i , p = y/i and p = z/i ; then (II-l) may be expressed:
- 55 -

+ oo S(k) = i i

n

iv j jj cos U*7) C(p) d

Pl

'dp

2

dp

3

(H-7)

-oo

If we now choose the spherical polar coordinate system, p, (J>, with

q in the direction of the polar axis so that q • p = cos <J), then

(H-7) becomes

co -rr 2-rr S(lT) = | 4

n

*v J J J

cos (qp cos <j>) C(p) p sin <j>ded<j>dp (II-8) o o o When we note that

1 d

sin <j> cos (q p cos <j>) = - — — sin (p q cos <j>)

we may perform the integrations with respect to 9 and <)>:

I I J?

S(k )

: r n V

j

p sin(pq) C(p) dp (H-9)

4iri i i

o

With C(p) defined by (6) the above integration may be performed and

the result is given by (8).

The reader should be cautioned that this particular form of

correlation function is not necessarily the correct function for

describing the characteristics of the atmosphere. However, measured

atmospheric spectra do tend to behave qualitatively, i. e. , decrease inversely

as some power of k, at least for large wave numbers, in the manner

illustrated on Fig. 4. The author is not aware of any information

which would lead one to believe that anisotropy, if it exists at all, may
- 56 -

be described by the particular ellipsoidal scale length anisotropy

described by (6), (7), (8) and (9); in particular it seems not

unlikely that the slope [ - (2u + 3)] of the spectrum S(k ) may depend

on the direction of k . Nevertheless it is believed that this particular

model is useful for describing one possible way in which the slope of

the spectrum S(k ) and anisotropy may enter into the various measurable

quantities involved in the theory. It seems particularly desirable to

have a model in view of the difficulty of forming any intuitive feeling —

for the nature of S ( k ) and the practical impossibility of directly

measuring its magnitude. The nearest approach to a direct measurement of S ( k ) involves measurements of the structure function--see

(14) --but this is only valid over a limited range of k and then only

for isotropic media; furthermore the derivation of (14) depends upon

assuming the particular form (6) for the correlation function, and the

function of |Ji in (14) derives entirely from this form of correlation

function.

An interesting discussion of the fundamental differences

between one dimensional and three dimensional spectra is given in

20/

a recent note by Gifford. —

Table IV is a tabulation of C (r/i ) for several values of [i o which may prove to be useful:
- 57 -

Table IV

Short Table of the Correlation Function

C(~) ee {2

1 "

li /r(u)}(r/i

o

)

hi K^(r/i

o

)

(r la ) o

M-= 1/3 H = 1/2 u = 2/3 HL= 1 (J. = 3/2

0. 001 0.99044762 0.99900050 0.99988298 0.99999624 0.99999950

0. 002 0.98483745 0.99800200 0.99970625 0.99998634 0.99999800

0. 003 0.98013282 0.99700450 0.99949720 0.99997109 0.99999551

0. 005 0.97207684 0.99501248 0.99901185 0.99992607 0.99998754

0. 007 0.96506163 0.99302444 0.99845978 0.99986334 0.99997561

0. 01 0.95569674 0.99004983 0.99753775 0.99973894 0.99995033

0. 018 0.93450651 0.98216103 0.99468727 0.99924936 0.99983993

0. 02 0.92975947 0.98019867 0.99390624 0.99909436 0.99980265

0. 03 0.90809119 0.97044553 0.98969568 0. 99814463 0.99955890

0. 05 0.87122615 0.95122942 0.98017441 0.99548372 0.99879090

0. 07 0.83943571 0.93239382 0.96967707 0.99196974 0.99766139

0. 1 0. 79755899 0.90483742 0.95276233 0.98538448 0.99532116

0. 2 0.68589016 0.81873075 0.89154336 0.95519450 0.98247690

0. 3 0. 59862209 0. 74081822 0.82835332 0.91679760 0.96306369

0. 5 0.46514732 0.60653066 0. 70684216 0. 82822055 0.90979599

0. 7 0.36646271 0.49658530 0. 59766997 0. 73519848 0. 84419502

1. 0.25979142 0.36787944 0.46007775 0.60190723 0.73575888
- 58 -

For numerical work, the following expressions for the correlation

function C(p, p) = {2/r (p) }(p/2) K (p) are useful:

M

1

C(p, 0. 5) = exp[-p]

C(p, 1. 5) = ( 1+ p) exp[-p]

(H-10)

(11-11)

C(p,p) = r(l -M-)(p/2)

fl {l (p)-I(p)}

-p p

(n-12)

where I (p) is the modified Bessel function of the first kind.

co up) = y

(p/2)

p + 2r

r ! T(l + p + r) r = o

co C(p,p) =

( P /2)

2r

r(i - p)

(n-13)

i

_ r(i - p+ r) /p\

r ! r (1 - p + r) |_ T (1 + p + r) V 2 /

2p -,

r = o

When < p < 2 and p ^ 1, the leading terms in (11-14) are:

^)--^}(ff+ r^(f)

2

(H-14)

r(i -n)

N 2p + 2

(i+ p) r(i+ p) V 2 +

2(1 - p)(2 - p) V 2

When p = 1 the following series may be used:

(11-15)

C(p, 1) = 1+ 2[y - 0.5 + in(p/2)](p/2)

2 + [y -1.25 + in (p/2)] (p /2)

4 + .

where 7=0. 5772156649.

(n-16)
- 59 -

Bessel function tables which have been found useful in this application

are:

(1) British Association Mathematical Tables

Vol. VI Bessel Functions Part I, Functions of

Orders Zero and Unity (1937)

and Vol. X Bessel Functions Part II, Functions of

Positive Integer Order 2 to 20 (1952)

Published for the Royal Society at the Cambridge

University Press

(2) Tables of Bessel Functions of Fractional Order

Volumes I and II, Prepared by the Computation

Laboratory of the National Applied

Mathematics Laboratories of the National

Bureau of Standards, Columbia University

Press, New York, 1948

In particular, K (p) is tabulated in reference (1), Vol. VI and

I -1/3^ P ^' ^l^^' I -2/3^ P ^ andI+2/3^ are tabulated in reference 2,

Vol. II.
- 60 -

Appendix III

Carrier Frequency Dependence of the Basic

Transmission Loss by Regression Analysis

By using first the 1955 —' and then the 1958-/ prediction

formulas, the basic transmission loss was calculated for 98 transmission paths for which reliable winter afternoon observations were

available, and A. = L, (Calculated) - L, (Observed) was

i bms bms

determined for each of these paths. These 98 values of A, were then

i

fitted by least squares to the following formula:

A=a+blOg

10

fMc < III -1 >

These 98 paths covered a frequency range from 65.8 to 4090 Mc.

Using the 1955 formula this regression analysis indicated

b = - 6. 01 ± 5. 77, whereas the 1958 formula resulted in

b = - 2. 50 ± 5. 53. Since the 1955 and 1958 prediction formulas were

both based on the assumption that |A = 1, it follows from the above

analysis that (2(1+ 1)10 = 36. 01 ± 5. 77 using the 1955 formula, and that

(2JT + 1)10 = 32. 50 ± 5. 53 using the 1958 formula, i. e.,

jl = 1.3005 ± 0.2885 or jl = 1.125 ± 0.2765, respectively.

Fig. 8 compares the basic transmission losses predicted by

these two formulas, and it is clear from this figure that the 1955
-61-

co

00 CO Q <

9*

CO m ^

CD

<

—

Q

O

<

o

LU

>-

DQ

Ll

O if)

o

<

a.

o

o

o

ho

Q

UJ

(T

lOtr

LU

hLU

O

ro

tn

< 3

lo en c: lu

I- lu

lu ^

LU

Ll. O

LO

n

CD*

a>\-

r lu

" LU

o

O

ro O ii

ii 9

i

i

, , i *—

1 — o ii »—«* i

i

i o X LO ^ —

,*—»» UJ Ul O ii o

z Z i UJ uJ -C oV £V or or _j-/ / — Ul Ul ui. i u.

Ul ro u

u

i

i /

1 z

c I

ro > ^ ro ~»

i

i /

i o ^ 2 ^ i h- 5 !- =: 1

1

5

i /

//

Ul <= LU c // DC ^H q:: ^ ij

Q_ a- ll a. LO ^. 00

^ /l // //

LO

//

I //

1 ' 1

1 1

1 1 / / / / / / / / /

/ / / / / /

//

/

// // /

s^ssy

=rS*^

o

c=> co

c=> oo

CO

CT

LU

hLU_ 00

Z en

LUO

<

hCO

CD

CD CD

LO CO GO CO

CVJ cm

CD CD CD CD

cm ro «=3" LO

cvj M cm rj

s~i3ai03a ni

SS01 NOISSIIAISNVdl 3ISV9 NVIQ3IAI
- 62 -

formula indicates much less loss at distances greater than 800 km

than the 1958 formula. Since the data agree somewhat better,

particularly at these larger distances, with the 1958 formula, the

above regression analyses were confined to the 98 paths for which

d(miles) G(radians) < 35.

It appears from this analysis and the fact that

(2jT + 1)10 = 33.1 ± 5. 55 when all 105 paths are used with the 1958

prediction formula, that the higher frequencies may be subject to a

somewhat greater loss than is predicted by using |i = 1. Some, if not

all, of this additional loss may be attributed to the fact that our theory

postulates that ground reflection at the transmitting and receiving ends

of the path quadruples ( g g = 4) the available scattered power at the

t r receiver. It seems likely that g g will be less than 4 and may

t r approach 2 or less at the higher frequencies, at least for propagation

over land paths.
-63-

References

1. Ralph Bolgiano, "Wavelength dependence in transhorizon

propagation, " Letter to the Editor, Proc. I. R.E. 47, 331 (1959).

2. K. A. Norton, "System loss in radio wave propagation, " J. Res. NBS 63D, 53 (1959).

3. W. J. Hartman and R. E. Wilkerson, "Path antenna gain in

an exponential atmosphere, " J. Res. NBS 63D , 273 (1959).

4. K. A. Norton, P. L. Rice and L. E. Vogler, "The use of

angular distance in estimating transmission loss and fading

range for propagation through a turbulent atmosphere over

irregular terrain, " Proc. I. R.E. 43, 1488 (1955).

5. P. L. Rice, A. G. Longley and K. A. Norton, "Prediction

of the cumulative distribution with time of ground wave and

tropospheric wave transmission loss, " NBS Technical Note

No. 15, Available at a cost of $1. 50 from the Office of

Technical Services, U.S. Department of Commerce,

Washington, D.C.

6. F. Villars and V. F. Weisskopf, "On the scattering of radio

waves by turbulent fluctuations of the atmosphere, " Proc. I. R.E.

43, 1232 (1955).

7. A. D. Wheelon, "Spectrum of turbulent fluctuations produced

by convective mixing of gradients, " Phys. Rev. 105 , 1706

(1957).

8. A. D. Wheelon, "Radio-wave scattering by tropospheric

irregularities, " J. Res. NBS63D, 205(1959).

9. A. P. Barsis, "Comparative 100 Mc measurements at

distances far beyond the radio horizon, " Convention Record

I. R.E. 2 , Part 1 - Antennas and Propagation (1954).

10 K. A. Norton, "Point-to-point radio relaying via the scatter

mode of tropospheric propagation, " I. R.E. Trans, on

Commun. Systems CS-4 , no. 1, 39 (1956).
-64-

11. B. R. Bean and G. D. Thayer, "On models of the atmospheric

radio refractive index, " Proc. I. R.E. 47, 740 (1959).

12. R. A. Fisher, "On the probable error of a coefficient

of correlation deduced from a small sample, " Metron 1, no. 4 (1921). ~

13. A. Hald, "Statistical theory with engineering applications, " John Wiley and Sons, 1952.

14. K. A. Norton, "Recent experimental evidence favoring the

pK (p) correlation function for describing the turbulence of

refractivity in the troposphere and stratosphere, M Report

given at the Joint Commissions on the Ionosphere and on

Radio Meteorology held under the auspices of the International

Scientific Radio Union at New York University, August, 1957;

J. Atmospheric and Terrest. Phys. 15 , 206 (1959).

15. K. A. Norton et al, "Radio Studies of Atmospheric Turbulence, " Vol. I and Vol. II, to be published as an NBS Technical Note.

16. M. C. Thompson, H. B. Janes, and A. W. Kirkpatrick, "An

analysis of time variations in atmospheric refractive index

and apparent radio path length, " J. Geophys. Res. 65 , 193

(1960).

17. J. W. Herbstreit, K. A. Norton, P. L. Rice and G. E.

Schafer, "Radio Wave Scattering in Tropo spheric Propagation, "

I. R.E. Convention Record, Part 2, Antennas and Communications,

85 (1953).

18. Earl E. Gossard, "Atmospheric spectra and trans-horizon

microwave fields, " Proc. 7th Weather Radar Conference,

Miami Beach, Florida, Nov. 17-20, 1958.

19. Frank Gifford, Jr. , "The relation between space and time

correlations in the atmosphere," J. Meteorol. 13, 289 (1956).

20. Frank Gifford, "The interpretation of meteorological spectra

and correlations, n J. Meteorol. 16, 343(1959).
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