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Atmospheric Refraction Z w o :::> l­ I­ ~ <x: '----1-'-1----- WAVE FR ON T Z = -rnX + Zal rn = tan e '----------------------X HORIZONTAL POSITION de = ~~ = (~~) tan8dz v FIG. 1. Origin of atmospheric refraction. DR. SIDNEY BERTRAM The Bunker-Ramo Corp. Canoga Park, Calif. Atmospheric Refraction INTRODUCTION it is believed that the treatment presented herein represents a new and interesting ap­ T IS WELL KNOWN that the geometry of I aerial photographs may be appreciably proach to the problem. distorted by refraction in the atmosphere at THEORETICAL DISCUSSION the time of exposure, and that to obtain maximum mapping accuracy it is necessary The origin of atmospheric refraction can to compensate this distortion as much as be seen by reference to Figure 1. A light ray knowledge permits. This paper presents a L is shown at an angle (J to the vertical in a solution of the refraction problem, including medium in which the velocity of light v varies a hand calculation for the ARDC Model 1 A. H. Faulds and Robert H. Brock, Jr., "Atmo­ Atmosphere, 1959. The effect of the curva­ spheric Refraction and its Distortion of Aerial ture of the earth on the refraction problem is Photography," PSOTOGRAMMETRIC ENGINEERING analyzed separately and shown to be negli­ Vol. XXX, No.2, March 1964. 2 H. H. Schmid, "A General Analytical Solution gible, except for rays approaching the hori­ to the Problem of Photogrammetry," Ballistic Re­ zontal. The problem of determining the re­ search Laboratories Report No. 1065, July 1959 fraction for a practical situation is also dis­ (ASTIA 230349). cussed. 3 D. C. Brown, "A Treatment of Analytical Photogrammetry," RCA Data Reduction Technical While atmospheric refraction has been dis­ Report No. 39, AFMTC-TR-57-22, 20 August cuss~c! in the photogrammetric literature,1,2·3 1957 (ASTIA 124144). 76 ATMOSPHERIC REFRACTION 77 with the altitude Z. The light ray is shown fol­ I t is also useful to express this in terms of the lowing a line Z=(l/m) X+Zo, where m=tan horizontal position X from the object point 0; the wave front, perpendicular to the ray on the ground. This is given by path, is then given by Z = -mX+Zo'. At a distance W along the wave front the velocity is higher because of the increase in altitude; 1 JXo (::) 00 = - X --_. tan 0 dX. (5) this results in a tipping of the wave front and, X o 0 v hence, a cur vature of the ray path. If v=v(Z) expresses the variation of the Equation 5 is used in the analysis of the effects velocity as a function of altitude, then of the curvature of the earth on the refrac­ tion, because X is more nearly independent of v(Z') = v(Z) + (:;) W sin 0 (1) the curvature than is Z. Equation 4 may be considerably simplified where (dv/dZ) is the rate of change of \'elocity by observing that both v and tan 0 are essen­ with altitude and W sin 0 is the vertical dis- tially constant and. hence, may be taken out- ABSTRACT: The distortion of aerial photographs caused by atmospheric re­ fraction is shown to be readily calculable in terms of an intl:'gral involving the variation with altitude of the velocity of light. The integral is evaluated numeric­ ally for the ARDC Model Atmosphere, 1959, using an acapted relationship be­ tween velocity and atmospheric density. The effect of local atmospheric condi­ tions is explored by calculating the refraction for the extreme cases of an arctic winter and for the tropics. These are shown to yield refractions of comparable magnitude /0 that found for the A RDC model. It is concluded that adequate cor­ rections for the distortion can usually be made using the data given. However, more precise corrections can be made using the technique described whenever it is practical to determine the variation of (tir density as a function of altitude. placement between P (at altitude Z) and P' side the integral (v varies less than 0.1 % (at altitude Z'). The ray at P travels a dis­ while 50 is less than 10-4 tan 0 radians). The tance d in time (d/v); in this same time the resulting equation is ray at P' travels a distance tanoJZo {)()=- Z (dV)- dZ. (6) (2) d' = [v + (:;) W sin 0] ~ vZo 0 dZ leading to a curvature of the path with a de­ viation angle dO given by d' - d (:}) do=---= ---d sin 0 W v (~}) = --- tan 0 dZ. (3) v If the element producing the refraction is at a height Z, the bending, dO, will produce a tangential displacement (Z/cos 0) dO at the ground; for an observation point at a height Zo this will be seen as an angular deviation of (Z/Zo)dO. The total refraction as viewed from Zo is given by the integral of (Z/Zo)dO; thus 1 JZo (:;) 00 = - Z--- tan 0 dZ. (4) Zo 0 v DR. SIDNEY BERTRAM 78 PHOTOGRAMMETRIC ENGINEERING 100 ,.... 0 1/ 1'\ 0 \ I II 0 <D f­ 0 / '\ -0: Vl ::::z I ~ o 50 -0: a::: o I a::: 0 I u "'- ~ ~ r--...... z 3 2 0/ f­ U 20 -0: a::: lJ.. UJ a::: 10 10 20 30 40 50 60 70 80 AL TITUDE (KILOMETERS) NOTE: For angles other than 45° multiply values read from curve by tan €I FIG. 2. Variation of refraction with altitude---ARDC Standard Atmosphere. CALCULATIONS separation of 1,000 meters, estimated to yield Equation 6 can be used to evaluate the re­ a tolerable error (a sample calculation, using 200-meter intervals, yielded refraction values fraction if the nature of the variation of if with altitude is known. It is useful to relate within one per cent of those given here). The the velocity at a given position in the atmo­ particular values of Z were selected so that sphere to the density at the position using density values, p, for values on either side of the formula4 fJ. = 1+0.000226p so that the entry could be read directly from the ARDC tables. Thus, for the first entry at v 1 - = - "'" 1 - O.OOO226p (7) Z=500, density values were obtained for c !L Z.=0(p=1.225) and Z=I,000(p=1.112). The where fJ. is the index of refraction at a point dIfference between these values is taken to be where the density is p in kg./m.3 and c is the the rate of change of density at Z=500, i.e., 3 velocity of light in free space; here vic "" 1, p'=(1.225-1.112)=0.113 in kg./m. /1,OOO but dv/dZ= -0.000226 (dp/dZ)c. Values for meters. As shown by Equation 7, this is p as a function of altitude have been tabu­ multiplied by 0.000226 to obtain the velocity lated for the ARDC Standard Atmosphere, change for the I,OOO-meter interval; this 19595 and have been used with Equation 6 to multiplication is performed in the last column. calculate the refract:on characteristic shown The integration is carried out using the in Figure 2. Since the procedure may be used trapezoidal rule. Thus, for a given interval, whenever the density as a function of alti­ the contribution to the integral is tude is known, it is shown in the detailed (p'Z)n + (P'Z)n+l dZ (8) calculations of Tables 1 and 2; an explanation 2 t:,Z of the tables follows. The various entries in the table are at a Z where (p'Z)n is the product of p' (as deter­ mined at Zn) and Zn, and (p'Z)n+1 is the corre­ 4 D. C. Brown, op. cit., p. 36. sponding product for the next point. The two 5 R. A. Mizner, KSW Champion, and H. L. differentials dZ and ~Z, have both been P~nd, "The ARDC Model Atmosphere, 1959," dZ/~Z=I; Atr Force Surveys tn Geophysics No. 115 (ASTIA made 1,000 meters so the third 229482). column is, therefore, p'Z/2. The fourth ATMOSPHERIC REFRACTIO 79 TABLE 1 REFRACTION FOR ARDC MODEL ATMOSPHERE (lOOO-meter calculation in terval) 226 1 -2: Z p p' p'Z/2 -2: Z Z (rnicroradians) 1.225 500 0.113 28 14 0.028 6.4 1.112 1,500 0.105 79 121 0.081 18.3 1.007 2,500 0.098 121 321 0.129 29.1 0.909 3,500 0.090 157 599 0.172 38.8 0.819 4,500 0.083 186 942 0.210 47.5 0.736 5,500 0.076 210 1,338 0.244 55.1 0.660 6,500 0.070 228 1,776 0.274 61.9 0.590 7,500 0.064 241 2,245 0.300 67.8 0.526 8,500 0.059 249 2,735 0.322 72.8 0.467 9,500 0.054 255 3,239 0.341 77.1 0.413 10,500 0.049 256 3,750 0.357 80.7 0.365 11,500 0.053 304 4,310 0.375 84.8 0.312 12,500 0.045 283 4,897 0.392 88.6 0.267 13,500 0.039 262 5,442 0.403 91.1 0.228 14,500 0.033 240 5,944 0.410 92.7 0.195 15,500 0.028 218 6,402 0.413 93.3 0.166 16,500 0.024 200 6,820 0.413 93.3 0.142 17,500 0.021 181 7,201 0.412 93.1 0.122 18,500 0.018 163 7,545 0.408 92 .2 0.104 19,500 0.015 147 7,855 0.403 91.1 0.089 20,500 0.013 132 8,137 0.397 89.7 0.076 (~= column IS the current sum.
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