ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR ATMOSPHERIC REFRACTION FOR OBLIQUE PHOTOGRAPHY (Supplement to report of June 20, 1951, by J. C. Rowley) Eldon Schmidt Approved by Edward' Young Project 1699-1 DEPARTMENT OF THE AIR FORCE WRIGHT AIR DEVELOPMENT CENTER WRIGHT-PATTERSON AIR FORCE BASE, OHIO CONTRACT NO. W33(038)ac-15518 PROJECT NO. 54-650A-460 August 1955

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ABSTRACT In an effort to extend the investigations of refraction to high-tilt or oblique photography,. it was not possible to assume a "flat" earth as in the previous report since in oblique photographs the nadir point is appreciably displaced from the principal point. The following report gives the angular. displacement due to refraction as a function of altitude and vertical angle of sight. OBJECTIVE The objective of this project is to determine methods of photogralm.metric reductions for aerial-defense purposes. ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The purpose of this supplement is to expand the considerations of refraction to high-tilt or oblique photography. This problem is basically the same as in the above report, except that it is not possible in this case to assume a "flat" earth. Since in high-tilt photographs the nadir point is appreciably displaced from the principal point, one cannot simply ratio the refraction displacement to the photograph and assume it to be radial in direction We are able in this case to give only the refraction displacement in angular component with direction radial to the ground nadir point. In this way the angular displacement Aa is a function of altitude H and vertical angle of sight a (Fig. 1). The correction on the photograph requires the position of the nadir point, which may not be known. In such a case it is necessary for tilt computation either to estimate the nadir point or to use some iterative procedure to incorporate the refraction correction. CAX R H' >_ s ~PATH OF RAY,-""^^^.1 r>E~RUE OBJECT POINT SURFACE EARTH Ro CENTER OF EARTH Figure 1 1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN As before, the angular distance G0 on the surface of the earth is JRo+H H /2 (1) where K =,(Ro + H) sin c and Ro is the mean radius of the earth. This can be expressed generally, 0 = f(Hc,). (2) From the geometry we have R sin Go' = tan -1 0 0 tan - H + 2Ro sin2 e6/2' (3) and, consequently, we get Aa = C - a = F(H,a). (4) If we can find the function f(H,c) in Equation (2), then the function F(H,a) is established and we can find Ato for any combination of H and a. To find f(H,a), it is necessary to integrate Equation (1). To do this we first find an approximate ((R)2 in the form (R)2 A B C ( = -+-+ C. (5) B R Using Humphrey's data as before, we get A = 1,895,-272,657 B = -592,945.4535, and C = 47.57652247 * This is for R varying between Ro and Ro+20 km. Ro is taken to be 6370.9 km. The integral now becomes BRo+H KdR = J R(A + BR + CR2 - K2)1/2 * (6) 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Integrating this we get [P(0O) Q l]/ +x _ In Ro + ___rdn. (7) cot a + x -_Ro + H Where we have introduced the notation P(H) = A + B(Ro + H) + C(Ro + H)2, Q = [P(H) sin2 ac], and k = (AQ - 1)1/2 Xc = BQ/2X. As these functions are somewhat complex to enumerate, the calculations were done on an electronic computer with the results given in Table I and the curves in Figs. 2, 3, and 4. The incorporation of the curved surface introduces the concept of horizon. The visible horizon occurs when the refracted ray becomes tangential to the earth's surface. In Equation (7) this is the condition P(O) Q = 1 (8) or sin a = [ ](H' 1/2 (9) The geometric horizon for altitude H imposes the condition sin t' = R (10) R + H and we have for Aca Aa = a - a'. (11) ---------------------- 5

TABLE I. ATMOSPHERIC REFRACTION IN SECONDS OF ARC H km 1 2 3 4 5 6 7 8 9 1 C AU As Aa AC Aa A a a a AA as m 5.225.430.621.795.960 1.101 1.238 1.360 1.471 1.566 10.447.867 1.250 1.604 1.935 2.219 2.492 2.742 2.967 3.155 C 15.679 1.314 1.891 2.442 2.941 3.373 3.786 4.166 4.504 4.791? 20.970 1.787 2.582 3.313 3.994 4.586 5.148 5.663 6.123 6.516 m 25 1.183 2.313 35.12 4.255 5.124 5.874 6.594 7.248 7.843 8.544 m 30 1.466 2.804 4.075 5.257 6.328 7.271 8.170 8.981 9.711 10.339 ~ 35 1.755 3 390 4.932 6.366 7.684 8.810 9.900 10.883 11.777 12. 54 40 2.138 4.121 5.949 7.640 9.196 10.564 11.874 13.054 14.116 15.023 45 2.595 4.899 7.084 9.099 10.971 12.597 14.163 15.559 16.833 17.907 Z 50 2.998 5.790 8.432 10.842 13.058 15.015 16.879 18.551 20.060 21.347 ^ 55 3.574 6.964 10.083 12.992 15.651 18.008 20.246 22.244 24.048 25.598 ~ - 60 4.523 8.462 12.238 15.741 18.991 21.836 24.557 26.999 29.189 31.068 -d 65 5.400 10.445 15.171 19.498 23.511 27.080 50.457 53-455 36.170 38.504 70 6.970 13.577 19.405 24.991 30.158 54.745 39.064 42.943 46.438 49.450 72.5 8.039 15.488 22.427 28.868 34.845 40.162 45.151 49.647 53.700 57-195 c 75 9.415 18.220 26.408 34.012 41.057 47.336 53.247 58.560 63.361 67.512 Z 77.5 11-433 22.054 31.935 41.193 49.738 57.380 64.563 71.060 76.929 82.019 < 80 14.364 27.795 40.324 51.996 62.828 72.488 81.667 89.977 97.547 104.126 82.5 20.001 38.235 55.400 71.453 84.703* 98.313* 110.167* 124.408* 133.601* 142.405 85 28.695 56.255 82.510 107.456 131.113 153.662 174.719 194.520 213.021 230.350 - 87.5 59.422 120.171 185.528 251.803 329.746 430.239 676.390 O These are corrected values. It was necessary to carry more than the ten significant figures carried by the automatic computer in order to have these critical points fall on a smooth curve. Corrections were computed C by J. M. Vukovich. I Z

m TABLE I (concluded) G H km 11 12 13 14 15 16 17 18 19 20 m a ha A ca Act Act Ac MA AU Ua A C A G 5 1.647 1.712 1.765 1.8o6 1.828 1.842 1.839 1.824 1.796 1.753 m 10 3.317 3.452 3.559 3.641 35685 3.712 3.707 3 677 3.620 3.5533 15 5.040 5.242 5.406 5.532 5.602 5.638 5-633 5.585 5.500 5.367 20 6.847 7.129 7.348 7.519 7.609 7.664 7.655 7.591 7.475 7.294 n 25 8.779 9.126 9.412 9.629 9.752 9.817 9.807 9.727 9-578 9.545 30 10.865 1 0 1105 11.657 11.924 12.074 12.158 12.145 12.044 11.858 11.576 Z 35 13.183 13.710 14.135 14.462 14.643 14.747 14.733 14.611 14.385 14.039 - 40 15-797 16.436 16.945 17.341 17.55880 17 664 17.515 17.247 16.854' 45 18.828 19.603 20.204 20.672 20.930 21.086 21.067 20.888 20.567 20.077 - 50 22.450 23.359 24.087 24.643 24.959 25.142 25.122 24.909 24.526 23.944 m 55 26.912 28.013 28.883 29.550 29.936 30.154 30.134 29.887 29.427 28.734 60 32.673 54.002 35.069 35.878 36.555 36.627 36.599 36.303 35-753 34.918 e 65 40.505 42.165 43.491 44.507 45.113 45.450 45.429 45.074 44.402 43.372 Z 70 52.029 54.181 55.904 57.219 58.029 58.482 58.481 58.049 57.211 55.914 < 72.5 60.198 62.697 64.709 66.255 67.211 67.761 67.782 67.307 66.362 64.894 - 75 71-077 74.069 76.473 78.337 79.508 80.196 80.269 79.758 78.688 76.999 n 77.5 86.407 90.102 93.097 95.444 96.944 97.875 98.063 975539 96.347 94.400 -< 80 109.839 114.688 118.667 121.862 123.932 125.365 125.835 125.424 124.194 121.990 o 82.5 150.705 157.912 164.037 168.645 173.078 175.505 177.039 177.415 176.515 174.576 " 85 246.470 261.420 275.239 287.961 299.759 310.600 320.716 330.286 339.586 349.069 3 87-5 (I z)

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 38 36 34 32 30 28 26, 24 a 10 z6 14 12 lO 10,o00 20,000 30,000 40,000 j0,00 6o,ooo H, Elevation Above Sea Level in feet 6

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN This gives the horizon deflection which is shown in Fig. 5. The angle shows the apparent elevation of the horizon. As this is of interest to the photogrammetrist, the angle 90~-a or the dip of the horizon is shown in Fig. 6. The relief consideration is for all practical purposes the same as in the body of the report. For a ground point of relief h and a camera at altitude H, the angular deflection is ACH-h = ACH - ACh. (12) These values can be taken from the curves in Figs. 2, 3, and 4. In the case of the horizon dip and deflection, a relief correction can be found. Equations (9) and (10) then become sin = P(h) 1/2 (13) and sin a = Ro + h (14) Ro + H Here, h is the relief of the horizon. In general, the relief at the horizon is a difficult concept as, with the exception of water horizons, the irregular terrain imposes conditions that are in general insoluble with photographic data. ERROR CONSIDERATIONS In the same manner as with the flat terrain, we find for variations in p. from normal conditions the approximate relation Ro0AO' H [tan c sec 2 (1 R 0 0 (2 ( (15) The error in Ace, which we shall call dca, becomes 1 da - 2 tan acz sec a cos(@O + a)A(2. (16) 9

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN As Go is a very small angle, this becomes 1 tan OjbL2 da _ - tan cAp2. (17) This relation is valid for a almost up to the horizon. Useful values of a in photogrammetry extend only up to around 70~, where it becomes difficult to distinguish ground points. This, of course, varies with atmospheric conditions, targets, etc. At 700 we have da _ 1.4A2. (18) From the gas laws we find for typical maximum variations Ap2 = 0.000033. (19) This gives, in Equation (18), da - 4.7 x 105 rdn. (20) This is about 0.007 mm in the focal plane of a six-inch lens and certainly can be considered negligible when compared to measuring accuracy in a point at that elevation. Near the horizon a + Go nears 90~ and relation (16) indicates the error da becomes very small and even vanishes at the horizon itself. The above error considerations are only valid for uniform variations in Aj. This is sufficient for all angles that are not near the horizon elevation. There are actually many local variations in p. due to thermal currents in the atmosphere. These effects are more extreme at low altitudes. When the path of the light ray passes through the lower levels of the atmosphere, it suffers many variations of different magnitudes. The longer this path, the worse the variations. In the case of small a, this path is short and the deviations are less important. But when a approaches the horizon, this path becomes quite long and the effect becomes important. As noted in the main report, for vertical photographs this effect can be neglected. A well-known example of these effects is the shimmering effect on a hot day. Unfortunately, this effect cannot be treated accurately by theoretical methods as the conditions very too widely. Since the photogrammetrist can 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN only use the horizon itself in this region, the effect is not very troublesome. This results from the fact that the horizon on a photograph has a wide expanse, causing these local variations to mean out over the photograph. 15