PB 151403

^Boulder laboratories

TABLES FOR THE STATISTICAL PREDICTION

OF RADIO RAY BENDING AND

ELEVATION ANGLE ERROR USING

SURFACE VALUES OF THE REFRACTIVE INDEX

BY B.R. BEAN, B.A. CAHOON AND G.D. THAYER

:*'S C? : ..

:

: : '•''

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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technical ^iote

44

March 16, I960

TABLES FOR THE STATISTICAL

PREDICTION OF RADIO RAY BENDING AND ELEVATION ANGLE

ERROR USING SURFACE VALUES OF THE REFRACTIVE INDEX

by

B. R. Bean, B. A. Cahoon and G. D. Thayer

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TABLES FOR THE STATISTICAL

PREDICTION OF RADIO RAY BENDING AND ELEVATION ANGLE

ERROR USING SURFACE VALUES OF THE REFRACTIVE INDEX

by

B. R. Bean, B. A. Cahoon and G. D. Thayer

ABSTRACT

Radio ray bending, t, and elevation angle error, € , have been

calculated for a wide range of meteorological conditions at 13 climatically diverse U. S. radiosonde stations. The parameters in the observed linear regression equations of t and € upon the surface value

of the refractive index are given for heights of 0.1 to 70 kilometers

and initial elevation angles of the ray from to 900 milliradians.

TABLES FOR THE STATISTICAL

PREDICTION OF RADIO RAY BENDING AND ELEVATION ANGLE

ERROR USING SURFACE VALUES OF THE REFRACTIVE INDEX

by

B. R. Bean, B. A. Cahoon and G. D. Thayer

Consider Figure 1 where the geometry of radio ray refraction

is shown. The angular bending of the radio ray, t, is obtained from

[1.2]:

n

2 t = _coL0_ dn (1)

C cot

1 where n is the radio refractive index and 6 is the local elevation angle.

It is customary to evaluate (1) by assuming that n is spherically stratified. Snell's law is used to determine from

n r cos 6 = nr cos 9, (2) o o o

, n r cos 9 e = co S -(^ 2.) ,3)

where r is the radial distance from the center of the earth to the point

under consideration. The zero subscript denotes the initial conditions

of n, r, and 9; usually taken as those at the earth's surface. It is

seen that the integral for t is a function of the initial conditions and the

distribution of n over the radio ray path. Even though T is determined by

the n structure over the entire radio path, the integral (1) becomes

relatively insensitive to n-profile characteristics when 9 is suitably

large.
-2-

Schulkin [ 2] has shown that r may be written

cot e f °

10 6 cot 9 - \ Nd (cot 9)- 10

(cot 9)T

t = N • 10 b cot 9_ - \ Nd (cot 9)- 10 b (4)

'N=o

where N = (n - 1) 106

, for the case of a radio ray passing completely through the earth's atmosphere. The notation N is used to denote the value of N at the

s earth's surface. For 9 > 10 the integral expression on the right

hand side of (4) amounts to only a few per cent of the total, and thus

the bending is expressed as a linear function of the initial conditions.

This is a very useful observation since it means that t may be determined from ordinary surface weather observations of temperature,

pressure and humidity which are needed to determine n. This possibility has been explored recently by the present authors [3] with the

conclusion that the total bending of a radio ray passing completely

through the earth's atmosphere may be usefully expressed as a linear

function of N for 9 as small as 1 mr. The present report extends

s o

this conclusion by deriving the regression parameters for additional

heights. Regression parameters are also derived for the prediction

of elevation angle errors from N . s In this work a large sample of data was assembled, and regression

lines of t and € upon N were determined. The "goodness -of-fit" of the

3 s data to the regression line was determined in each case by comparing the

standard deviation of the raw data, say s(t), (which is taken about the

mean r of the sample) with the standard deviation of the same data about

the regression line. This last quantity is called the standard error of

estimate, S.E. The percentage reduction in the uncertainty of estimating
-3-

t by using the regression line rather than the mean of the raw data is

thus given by

S(t) - S E

100

S(t)

As a general observation we note that a 50% reduction in error is

accomplished by this method only when the bending is taken over at

least 5 km and is equal to at least 20. mr. For the case of total

o

atmospheric bending of the ray (h = 70 km) a 50% reduction is accomo plished at of 5 mr (~ 0. 3 ) for both t and e . o The details of the n-profile sample used to derive the regression

parameters are given in Appendix A while the regression parameters

for bending are given in Appendix I and the regression parameters

for elevation angles are given in Appendix II.

,

APPENDIX A

Characteristics of the Profile Sample

The following table is a listing of the 77 refractive index profiles used in the refraction study, giving pertinent profile characteristics,

by stations. The three more important quantities are listed: the surface value of N, N ; the initial gradient of N (with respect to height)

s to the first significant radiosonde level, (dN/ dh) (per km), and the drop in

o N over the first kilometer above the surface, AN.

Each station is listed, along with the Weather Bureau code

number, and the characteristics of six profiles are given. The six profiles fall into six mutually exclusive profile types, as follows:

Type 1. Modified Ground Layer (MGL)

Superrefractive profiles having a gradient in excess of 100

N-units per km, but less than 156.9 N-units per km, in the initial layer, but no elevated layer with a gradient larger than 156.9

N-units per km.

Type 2. Maximum Surface (Max).

Profile with the highest value of N found for that station

s over period of record, but not having an initial gradient in excess of 100 N-units per km, nor any elevated layer with a gradient in excess of 156.9 N-units per km.

Type 3. Minimum Linear (Lin).

Profiles having a minimum value of N for that station, but

s not having any layer with a gradient in excess of 60 N-units per

km.
A-2

Type 4. Elevated Duct (ED).

Profiles having at least one elevated layer with a gradient

of over 156.9 N-units per km, but with an initial layer having a

gradient less than 100 N-units per km.

Type 5. Surface Duct (Duct).

Profiles having an initial layer with a gradient in excess of

156.9 N-units per km.

Type 6. Combined Gradient (Comb).

Profiles having an initial layer with a gradient between 100

and 156.9 N-units per km, and at least one elevated layer with a

gradient in excess of 156.9 N-units per km. A combination of

super refraction (type 1) and elevated duct (type 4).

Note that no profile fulfilling the requirements of the type 6 definition could be found for Ely, Nev. Also, the definitions for types 2, 3 and

4 had to be somewhat modified in order to include Truk, Caroline Islands

(the modification being to allow initial gradients of above 100 for type 2

and 4 and gradients up to 80 for type 3).
A-3

Station Type N (dN/dh) AN

Bismarck, N. D. MGL 322.5 -132 -58

24011 Max 350.5 -59 -44

Lin 295.0 -36 -35

Duct 341.5 -715 -82

ED 314.5 -39 -40

Comb 318.0 -141 -48

Columbia, Mo. MGL 367.0 -145 -56

13983 Max 383.0 -38 -73

Lin 309.5 -37 -36

Duct 365.0 -472 -74

ED 307.5 -41 -34

Comb 346.0 -141 -56

Denver, Colorado MGL 266.0 -135 -44

23062 Max 289.5 -39 -39

Lin 237.0 -22 -23

Duct 254.0 -197 -46

ED 278.0 -58 -65

Comb 286.0 -134 -56

Ely, Nevada MGL 280.0 -127 -38

23154 Max 280.0 -46 -42

Lin 249.5 -28 -30

Duct 267.0 -246 -49

ED 255.0 -47 -30

Comb None found

Joliet, Illinois MGL 390.5 -129 -78

14834 Max 395.0 -58 -97

Lin 309.5 -36 -36

Duct 337.0 -451 -54

ED 320.0 -32 -48

Comb 340.0 -142 -60
A-4

N

Station Type

MGL

s (dN/dh)

-150

AN

Miami, Florida 376.5 -59

12839 Max 391.5 -37 -60

Lin 333.6 -24 -39

Duct 379.5 -194 -66

ED 360.0 -65 -54

Comb 358.5 -134 -55

Portland, Maine MGL 357.5 -146 -80

14764 Max 375.0 -64 -75

Lin 315.0 -37 -39

Duct 347.5 -322 -68

ED 345.0 -68 -56

Comb 342.0 -133 -46

San Antonio, Texas MGL 366.0 -150 -65

12921 Max 377.5 -53 -54

Lin 301.5 -42 -36

Duct 375.5 -181 -61

ED 335.5 -21 -37

Comb 359.0 -136 -44

Santa Maria, Calif. MGL 340.0 -145 -52

23236 Max 343.0 -37 -51

Lin 323.5 -45 -42

Duct 337.0 -197 -48

ED 339.5 -48 -74

Comb 330.0 -129 -79

Tatoosh, Wash. MGL 336.5 -128 -68

24240 Max 343.0 -45 -46

Lin 315.0 -38 -38

Duct 337.0 -253 -70

ED 326.0 -42 -51

Comb 323.0 -132 -57
A-5

Station Type

Washington, D. C. MGL

13743 Max

Lin

Duct

ED

Comb

Truk Island MGL

99999 Max

Lin

Duct

ED

Comb

Fairbanks, Ala,ska MGL

26411 Max

Lin

Duct

ED

Comb

N

(dN/dh) AN

344.0 -144 -48

391.0 -82 -68

297.5 -29 -30

336.0 -378 -65

350.0 -57 -51

370.5 -144 -74

388.5 -138 -62

400.0 -103 -70

383.5 -75 -64

402.5 -330 -72

373.5 -115 -52

393.5 -142 -64

313. 1 -140 -47

343.5 -48 -48

291.0 -35 -34

305.0 -400 -34

328.0 -43 -62

307.0 -147 -34

1-1

TABLE I

Values of b and a in the regression equation t = bN + a,

s where:

t is the total bending in milliradians to the height h,

N the value of the radio refractivity at the earth's surface,

s b the change in total bending per unit change in N , s a is the zero intercept,

r is the correlation coefficient,

S E is the standard error of prediction using the regression line,

S is the standard deviation of the calculated values of the total

T

bending,

S - S E

P = 100 T
1-2

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

9 OSN t r b a S.E S P

T

oTo 334.0 7. 3077 0.2665 0.0479 -8.7011 6.7277 6.9801 3.6160

1.0 334.0 4.4555 0.2785 0.0257 -4.1217 3.4363 3.5778 3.9549

2.0 334.0 3.0547 0.2881 0.0162 -2.3732 2.0960 2.1888 4.2398

5.0 334.0 1.5038 0.3048 0.0073 -0.9181 0.8792 0.9231 4.7557

10.0 334.6 1.1519 0.1915 0.0053 -0.6085 1.0551 1.0749 1.8420

20.0 334.6 0.5604 0.2070 0.0025 -0.2639 0.4555 0.4656 2.1692

52.4 334.6 0.2129 0.2100 0.0009 -0.0973 0.1688 0.1727 2.2583

100.0 334.6 0.1111 0.2105 0.0005 -0.0507 0.0879 0.0899 2.2247

200.0 334.6 0.0550 0.2105 0.0002 -0.0250 0.0435 0.0445 2.2472

400.0 334.6 0.0264 0.2107 0.0001 -0.0120 0.0208 0.0213 2.3474

900.0 334.6 0.0088 0.2108 0.00004 -0.0040 0.0070 0.0072 2.7778

Table I-B, h = 0. 2 km.

6 OSN t r b a S.E S p T

0.0 334.0 8.9492 0.2849 0.05801-10.4261 7.5726 7.9001 4.1455

1.0 334.0 5.9957 0.2979 0.0348 -5.6431 4.3330 4.5391 4.5405

2.0 334.0 4.3994 0.3104 0.0239 -3.5707 2.8357 2.9831 4.9412

5.0 334.0 2.3844 0.3415 0.0117 -1.5287 1.2512 1.3312 6.0096

10.0 334.6 1.6872 0.2306 0.0073 -0.7436 1.1990 1.2322 2.6944

20.0 334.6 0.8368 0.2550 0.0035 -0.3184 0.5122 0.5297 3.3038

52.4 334.6 0.3198 0.2604 0.0013 -0.1162 0.1890 0.1958 3.4729

100.0 334.6 0.1671 0.2610 0.0007 -0.0603 0.0983 0.1019 3.5329

200.0 334.6 0.0827 0.2613 0.0003 -0.0299 0.0486 0.0504 3.5714

400.0 334.6 0.0397 0.2613 0.0002 -0.0143 0.0233 0.0241 3.3195

900.0 334.6 0.0133 0.2604 0.00005 -0.0047 0.0078 0.0081 3.7037
1-3

Table I-C, h = 0.5 km.

G N t r b a S.E t P

o s

0.0 333.1 10.9591 0.3615 0.0769 -14.6443 7.6170 8.1694 6.7618

1.0 333.1 7.9386 0.3997 0.0510 -9.0567 4.4954 4.9042 8.3357

2.0 333.1 6.2579 0.4369 0.0384 -6.5408 3.0395 3.3790 10.0474

5.0 333.1 3.9386 0.5205 0.0228 -3.6605 1.4376 1.6837 14.6166

10.0 333.8 2.7795 0.3933 0.0140 -1.9055 1.2733 1.3850 8.0650

20.0 333.8 1.4731 0.4563 0.0071 -0.8926 0.5365 0.6029 11.0134

52.4 333.8 0.5573 0.4731 0.0027 -0.3308 0.1966 0.2232 11.9176

100.0 333.8 0.3026 0.4753 0.0014 -0.1721 0.1022 0.1161 11.9724

200.0 333.8 0.1499 0.4760 0.0007 -0.0851 0.0505 0.0574 12.0209

400.0 333.8 0.0719 0.4761 0.0003 -0.0408 0.0242 0.0275 12.0000

900.0 333.8 0.0241 0.4764 0.0001 -0.0137 0.0081 0.0092 11.9565

Table I-D, h = 1.0 km

e N

s t r b a S.E S

T P

0.0 334.0 12.8881 0.3936 0.0840 -15.1802 7.6151 8.2839 8.0735

1.0 334.0 9.9030 0.4620 0.0607 -10.3739 4.5217 5.0985 11.3131

2.0 334.0 8.2213 0.5238 0.04918 -8.2066 3.1040 3.6438 14.8142

5.0 334.0 5.7851 0.6348 0.0337 -5.4816 1.5931 2.0619 22.7363

10.0 334.6 4.2571 0.5718 0.0224 -3.2378 1.2574 1.5326 17.9564

20.0 334.6 2.4573 0.6598 0.0124 -1.6959 0.5531 0.7360 24.8505

52.4 334.6 1.0040 0.6823 0.0049 -0.6495 0.2071 0.2833 26.8973

100.0 334.6 0.5297 0.6851 0.0026 -0.3388 0.1080 0.1482 27.1255

200.0 334.6 0.2629 0.6859 0.0013 -0.1676 0.0534 0.0734 27.2480

400.0 334.6 0.1262 0.6860 0.0006 -0.0803 0.0256 0.0352 27.2727

900.0 334.6 0.0423 0.6864 0.0002 -0.0270 0.0086 0.0118 27.1186
1-4

Table I-E, h = 2.0

N S.E

0.0 334.0 15. 1568 0.4524 0.0985 -17.7584 7.5391 8.45 35 10.8168

1.0 334.0 12. 1678 0.5490 0.0752 -12.9451 4.4420 5.3145 16.4173

2.0 334.0 10.4745 0.6316 0.0636 -10. 7566 3.0277 3.9053 22.4720

5.0 334.0 7.9624 0.7707 0.0475 -7.8969 1.5234 2.3907 36.2781

10.0 334.6 6. 1731 0.7634 0.0345 -5.3712 1.1421 1.7681 35.4052

20.0 334.6 3.9063 0.8515 0.02111 -3.1571 0.5086 0.9699 47.5616

52.4 334.6 1.7023 0.8668 0.0089 -1.2770 0.2003 0.4018 50. 1493

100.0 334.6 0.9089 0.8679 0.0047 -0.6705 0. 1057 0.2128 50.3289

200.0 334.6 0.4529 0.8681 0.0023 -0.3323 0.0524 0.1057 50.4257

400.0 334.6 0.2175 0.8682 0.0011 -0. 1593 0.0252 0.0507 50.2959

900.0 334.6 0.0730 0.8684 0.0004 -0.0535 0.0084 0.0170 50.5882

Table I-F, h = 5.0

N S.E

0.0 334.0 18.0620 0.4962 0. 1115 -19. 1704 7.5676 8.7166 13.1817

1.0 334.0 15.0710 0.6101 0.0881 -14.3543 4.4401 5.6037 20.7649

2.0 334.0 13.3714 0.7030 0.0764 -12. 1589 •3.0001 4.2194 28.8975

5.0 334.0 10.8162 0.85 04 0.0601 -9.2514 1.4422 2.7415 47.3938

10.0 334.6 8.8939 0.8674 0.0464 -6.6445 1.0420 2.0943 50.2459

20.0 334.6 6.2299 0.9484 0.0308 -4.0706 0.4028 1.2699 68.2810

52.4 334.6 3.0324 0.9674 0.0139 -1.6236 0. 1426 0.5627 74.6579

100.0 , 334.6 1.6670 0.9695 0.0075 -0.8348 0.0739 0. 3017 75.5 055

200.0 334.6 0.8388 0.9701 0.0037 -0.4098 0.0365 0. 1505 75.7475

400.0 334.6 0.4039 0.9702 0.0018 -0. I960 0.0175 0.0723 75.7953

900.0 334.6 0. 1357 0.9703 0.0006 -0.0658 0.0059 0.0243 75.7202
1-5

Table I-G, h = 10.0 km

N S.E

0.0 334.0 19.8159 0.5099 0. 1149 -18.5627 7.5227 8.7448 13.9752

1.0 334.0 16.8244 0.6290 0.0915 -13.7469 4.3895 5.6464 22.2602

2.0 334.0 15. 1232 0.7250 0.0799 -11.5514 2.9443 4.2746 31. 1210

5.0 334.0 12.5574 0.8734 0.0635 -8.6434 1.3733 2.8201 51.3031

10.0 334.6 10.5936 0.8950 0.0498 -6.0729 0.9713 2. 1773 55.3897

20.0 334.6 7.8054 0.9723 0.0338 -3.5012 0.3179 1.3596 76.6181

52.4 334.6 4. 1243 0.9907 0.0157 -1. 1441 0. 0844 0.6217 86.4243

100.0 334.6 2.3435 0.9927 0.008b -0.5084 0.0406 0.3359 87.9131

200.0 334.6 1. 1950 0.9931 0.0043 -0.2310 0.0197 0. 1679 88.2668

400.0 334.6 0.5776 0.9932 0.0020 -0. 1078 0.0094 0.0807 88.3519

900.0 334.6 0. 1942 0.9932 0.0007 -0.0359 0.0032 0.0271 88. 1919

Table I-H, h = 20. km

N S.E

0.0 334.0 20.9521 0.5155 0.1165 -17.9573 7.5131 8.7682 14.3142

1.0 334.0 17.9604 0.6367 0.0931 -13. 1413 4.3763 5.6752 22.8873

2.0 334.0 16.2588 0.7336 0.0814 -10.9463 2.9281 4.3085 32.0390

5.0 334.0 13.6897 0.8815 0.0651 -8.0397 1.3521 2.8638 52. 7865

10.0 334.6 11.7166 0.9028 0.0514 -5.4747 0.9573 2.2263 57.0004

20.0 334.6 8.8856 0.9785 0.0353 -2.9228 0.2909 1.4109 79.3820

52.4 334.6 4.9843 0.9968 0.0169 -0.6738 0.05 35 0.6637 91.9391

100.0 334.6 2.9329 0.9984 0.0093 -0. 1802 0.0203 0.3645 94.4307

200.0 334.6 1.5226 0.9986 0.0047 -0.0467 0.0096 0. 1837 94.7741

400.0 334.6 0.7401 0.9986 0.0023 -0.0161 0.0046 0.0885 94.8023

900.0 334.6 0. 2492 0.9986 0.0008 -0.0048 0.0016 0.0297 94.6128
1-6

Table I-I, h = 70.0 km

e o N

s T r b a S.E ST

P

0.0 334.0 21. 1795 0.5174 0. 1170 -17.9071 7.5113 8.7773 14.4236

1.0 334.0 18. 1878 0.6391 0.0936 -13.0912 4.3738 5.6864 23.0831

2.0 334.0 16.4861 0.7361 0.0820 -10.8960 2.9251 4.3216 32.3144

5.0 334.0 13.9167 0.8837 0.6558 -7.9895 1.3481 2.8800 53.1910

10.0 334.6 11.9426 0.9051 0.0519 -5.4209 0.9539 2.2430 57.4721

20.0 334.6 9. 1072 0.9797 0.0358 -2.8696 0.2862 1.4292 79.9748

52.4 334.6 5. 1787 0.9979 0.0173 -0.6246 0.0445 0.6800 93.4559

100.0 334.6 3.0817 0.9997 0.0096 -0. 1402 0.0095 0.3768 97.4788

200.0 334.6 1.6126 1.0000 0.0048 -0.0212 0.0013 0. 1910 99.3194

400.0 334.6 0.7862 1.0000 0.0024 -0.0027 0.0002 0.0922 99.7831

900.0 334.6 0.2650 1.0000 0.0008 -0.0002 0.0001 0.0310 99.6774
II1

TABLE II

Values of m and Si in the regression equation € = mN + 1, s where:

6 is the elevation angle error in milli radians to the height h»

N the value of the radio refractivity at the earth's surface,

s m the change in elevation angle error per unit change in N , s

t is the zero intercept,

r is the correlation coefficient,

S E is the standard error of prediction using the regression line,

S is the standard deviation of the calculated values of the elevation

€ angle error

P = 100

S

T

- SE

~~

S

T
11-2

Table II-A, h = 0. 1 km

9 N 7 r m i S.E S P

o s t 576 334.0 3. 7064 0.2637 0.0243 -4.3954 3.4429 3.5692 3.5386

1.0 334.0 2.2716 0,2760 0.0131 -2.1055 1.7712 1.8428 3.8854

2.0 334.0 1.5598 0.2855 0.0083 -1.2185 1.0837 1.1307 4.1567

5.0 334.0 0.7682 0.3017 0.0037 -0.4741 0.4561 0.4784 4.6614

10.0 334.6 0.6233 0.1609 0.0027 -0.2834 0.6501 0.6587 1.3056

20.0 334.6 0.3005 0.1777 0.0012 -0.1175 0.2706 0.2750 1.6000

52.4 334.6 0.1140 0.1805 0.0005 -0.0426 0.0998 0.1014 1.5779

100.0 334.6 0.0595 0.1824 0.0002 -0.0227 0.0518 0.0527 1.7078

200.0 334.6 0.0294 0.1815 0.0001 -0.0110 0.0256 0.0260 1.5385

400.0 334.6 0.0142 0.1844 0.0001 -0.0055 0.0123 0.0125 1.6000

Table II-B, h = 0. 2 km

e Q

N

g 7 r m | s.E S^ P

0.0 334.0 4.8189 0.2739 0.0310 -5.5364 4.2249 4.3929 3.8244

1.0 334.0 3.2877 0.2838 0.0191 -3.0776 2.4984 2.6055 4.1105

2.0 334.0 2.4324 0.2927 0.0132 -1.9685 1.6702 1.7467 4.3797

5.0 334.0 1.3252 0.3146 0.0065 -0.8506 0.7628 0.8036 5.0772

10.0 334.6 1.0273 0.1919 0.0044 -0.4623 0.8906 0.9075 1.8623

20.0 334.6 0.5048 0.2125 0.0021 -0.1941 0.3757 0.3845 2.2887

52.4 334.6 0.1925 0.2168 0.0008 -0.0705 0.1385 0.1419 2.3961

100.0 334.6 0.1006 0.2175 0.0004 -0.0367 0.0721 0.0739 2.4357

200. J 334.6 0.0498 0.2170 0.0002 -0.0179 0.0356 0.0365 2.4658

400.0 334.6 0.0239 0.2170 0.0001 -0.0085 0.0170 0.0174 2.2989
11-3

Table II-C, h = 0.5 km

9 N T r mi S.E S P

o s e 0.0 333.1 6.2802 0.3255 0.0430 -8.0534 4.8023 5.0789 5.4461

1.0 333.1 4.6053 0.3457 0.0289 -5.0198 3.0131 3.2111 6.1661

2.0 333.1 3.6281 0.3661 0.0216 -3.5807 2.1131 2.2708 6.9447

5.0 333.1 2.2499 0.4197 0.0125 -1.9306 1.0427 1.1488 9.2357

10.0 333.8 1.6990 0.2887 0.0083 -1.0741 1.0693 1.1168 4.2532

20.0 333.8 0.8853 0.3304 0.0041 -0.4899 0.4568 0.4840 5.6198

52.4 333.8 0.3449 0.3418 0.0016 -0.1803 0.1679 0.1787 6.0436

100.0 333.8 0.1806 0.3430 0.0008 -0.0934 0.0872 0.0929 6.1356

200.0 333.8 0.0894 0.3439 0.0004 -0.0463 0.0431 0.0459 6.1002

400.0 333.8 0.0429 0.3408 0.0002 -0.0218 0.0208 0.0221 5.8824

Table II-D, h = 1.0 km

9OSN 7 r m t S.E S P

€ 0.0 334.0 7.5072 0.3371 0.0472 -8.2579 5.1154 5.4334 5.8527

1.0 334.0 5.7577 0.3772 0.0339 -5.5583 3.2281 3.4856 7.3875

2.0 334.0 4.7279 0.4180 0.0270 -4.2916 2.2775 2.5070 9.1544

5.0 334.0 3.2343 0.5150 0.0179 -2.7591 1.1590 1.3521 14.2815

10.0 334.6 2.4855 0.3983 0.0123 -1.6158 1.1044 1.2041 8.2800

20.0 334.6 1.4036 0.4831 0.0066 -0.8154 0.4703 0.5371 12.4372

52.4 334.6 0.5689 0.5129 0.0026 -0.3070 0.1714 0.1997 14.1713

100.0 334.6 0.2998 0.5167 0.0014 -0.1595 0.0890 0.1039 14.3407

200.0 334.6 0.1488 0.5177 0.0007 -0.0787 0.0440 0.0514 14.3969

400.0 334.6 0.0714 0.5180 0.0003 -0.0379 0.0211 0.0247 14.5749
II-4

Table II-E, h = 2.0 km

N 7 r m i S.E S p

O S 6 0.0 334.0 9.0572 0.3795 0.0567 -9.8908 5.3659 5.7999 7.4829

1.0 334.0 7.1992 0.4413 0.0427 -7.0696 3.3704 3.7560 10.2662

2.0 334.0 6.1064 0.5020 0.0355 -5.7509 2.3735 2.7443 13.5116

5.0 334.0 4.5071 0.6351 0.0258 -4.1251 1.2198 1.5791 22.7535

10.0 334.6 3.5774 0.5642 0.0188 -2.7184 1.0773 1.3048 17.4356

20.0 334.6 2.2144 0.7007 0.0113 -1.5650 0.4499 0.6306 28.6552

52.4 334.6 0.9557 0.7531 0.0047 -0.6256 0.1615 0.2455 34.2159

100.0 334.6 0.5095 0.7608 0.0025 -0.3279 0.0835 0.1287 35.1204

200.0 334.6 0.2538 0.7633 0.0012 -0.1627 0.0412 0.0638 35.4232

400.0 334.6 0.1219 0.7643 0.0006 -0.0784 0.0198 0.0306 35.2941

Table II-F, h = 5.0 km

ON 7 r m I S.E S P

O S € 0.0 334.0 11.4413 0.4324 0.0704 -12.0857 5.6994 6.3209 9.8325

1.0 334.0 9.4276 0.5186 0.0554 -9.0682 3.5425 4.1432 14.4985

2.0 334.0 8.2509 0.5983 0.0477 -7.6679 2.4766 3.0909 19.8745

5.0 334.0 6.5231 0.7543 0.0372 -5.9085 1.2570 1.9147 34.3500

10.0 334.6 5.4083 0.7424 0.0290 -4.3100 1.0255 1.5306 33.0001

20.0 334.6 3.7306 0.8789 0.0194 -2.7570 0.4118 0.8631 52.2883

52.4 334.6 1.8030 0.9248 0.0089 -1.1721 0.1431 0.3761 61.9516

100.0 334.6 0.9898 0.9316 0.0048 -0.6154 0.0733 0.2015 63.6228

200.0 '334.6 0.4978 0.9336 0.0024 -0.3043 0.0360 0.1004 64.1434

400.0 334.6 0.2396 0.9339 0.0012 -0.1460 0.0173 0.0483 64.1822
II-5

Table II-G, h = 10.0 km

N S.E

0.0 334.0 13.3409 0.4617 0.0797 -13.2953 5.9448 6.7020 11.2981

1.0 334.0 11. 1993 0.5602 0.0637 -10.0839 3.6560 4.4137 17. 1670

2.0 334.0 9.9541 0.6475 0.0555 -8.5935 2.5360 3.3278 23.7935

5o0 334.0 8. 1208 0.8057 0.0443 -6.6912 1.2652 2. 1359 40.7650

10.0 334.6 6.8842 0.8125 0.0353 -4.9338 0.9913 1.7006 41.7088

20.0 334.6 5.0188 0.9301 0.0245 -3. 1637 0.3779 1.0286 63.2607

52.4 334.6 2.6400 0.9679 0.0118 -1.3002 0. 1195 0.4759 74.8897

100.0 334.6 1.4988 0.9742 0.0065 -0.6607 0.0584 0.2592 77.4691

200.0 334.6 0.7641 0.9763 0.0032 -0.3208 0.0281 0. 1299 78. 3680

400.0 334.6 0.3693 0.9766 0.0016 -0. 1533 0.0135 0.0626 78.4345

Table II-H, h = 20. km

N m S.E

0.0 334.0 15. 1600 0.4805 0.0874 -14.0318 6. 1895 7.0577 12.3015

1.0 334.0 12.8858 0.5866 0.0703 -10.5942 3.7658 4.6500 19.0151

2.0 334.0 11.5687 0.6780 0.0616 -8.9972 2.5905 3.5241 26.4919

5.0 334.0 9.6224 0.8351 0. 0495 -6.9273 1.2665 2.3024 44.9922

10.0 334.6 8.2658 0.8497 0.0397 -5.0189 0.9637 1.8279 47.2783

20.0 334.6 6.2329 0.9538 0.0278 -3.0650 0.3423 1. 1397 69.9658

52.4 334.6 3.4978 0.9861 0.0136 -1.0682 0.0899 0.5414 83.3949

100.0 334.6 2.0613 0.9917 0.0076 -0.4672 0.0384 0.2981 87.1184

200.0 334.6 1.0712 0.9934 0.0038 -0.2067 0.0172 0. 1504 88.5638

400.0 334.6 0.5209 0.9939 0.0018 -0.0947 0.0080 0.0724 88.9503
n-6

Table II-I, h = 70. km

N m S.E

0.0 334.0 17.6781 0.4999 0.0985 -15.2202 6.6214 7.6453 13.3925

1.0 334.0 15. 1699 0.6137 0.0794 -11.3649 3.9661 5.0233 21.0459

2.0 334.0 13.7249 0. 7083 0.0698 -9.5814 2.6984 3.8227 29.4111

5.0 334.0 11.5706 0.8617 0.0563 -7.2363 1.2867 2.5357 49.2566

10.0 334.6 10.0018 0.8811 0.0451 -5.0975 0.9476 2.0036 52.7051

20.0 334.6 7.6843 0.9701 0.0317 -2.9121 0.3102 1.2772 75.7125

52.4 334.6 4.4705 0.9948 0.0158 -0.8032 0.0630 0.6198 89.8354

100.0 334.6 2.7082 0.9983 0.0089 -0.2595 0.0202 0.3476 94. 1887

200.0 334.6 1.4354 0.9992 0.0045 -0.0822 0.0070 0. 1776 96.0586

400.0 334.6 0.7035 0.9995 0.0022 -0.0319 0.0028 0.0860 96.7442
REFERENCES

[ 1] W. M, Smart, "Spherical Astronomy, " Chapter 3, Cambridge

University Press, London, (1931).

[ 2] M. Schulkin, "Average Radio-Ray Refraction in the Lower

Atmosphere," Proc. IRE, 40, 554-561, (1952).

[3] B. R. and B. A. Cahoon, "The Use of Surface Weather Observations to Predict the Total Atmospheric Bending of Radio Waves

at Small Angles, " Proc. IRE, 45, 1545-1546, (1957).
RADIO RAY

Figure 1
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