The Effects of Earth's Curvature andRefraction on the Mensurationof Vertical Photographs*

J. W. SMITH,

N. Y. College of Forestry,Syracuse University, Syracuse, N. Y.

COMPARED to errors from other sources,the errors introduced into most conven­

tional photogrammetric techniques byearth's curvature and atmospheric refrac­tion are of such small magnitude as to benegligible. However, analytical control ex­tension, for relatively long strips of photo­graphs, is made less tedious and more eco­nomically feasible by high-speed electroniccomputing methods. Accordingly, such fac­tors as earth's curvature, atmospheric re­fraction and meridian convergence requireconsideration. Utilization of the geocentriccoordi nate system, devised by ProfessorEarl Church [1] in 1951 and more recentlydescribed by Dr. C. S. Sh u and ProfessorA. ]. McNair, [3] effectively eliminateserrors due to earth's curvature, and com­pensates for the effect of meridian converg­ence.

Because of the longer image rays and thelarger areas covered, it might be supposedthat high-altitude photography would besubject to significantly greater errors fromearth's curvature and atmospheric refrac­tion. For example, a photograph taken at analti tude of 15,000 feet by a 9" X9" formatcamera with a 6" focal-length lens coversapproximately 18 square miles; if the samecamera is used at 50,000 feet, the areacovered by each photograph is approxi­mately 200 square miles. From this observa­tion it may logically be concluded that ifthere is willingness to expend the sameenergy areawise, eleven times as much ef­fort, either instrumentally or analytically,may be concentrated on the high-altitudephotograph. The small scale and resultingloss of detail are likely to cancel this super­ficial advantage and to restrict the use ofhigh-altitude photography, for the presentat least, to the military.

The purpose of this paper is not, how­ever, to analyze or even discuss the meritsof one or another type of photography.Rather, it is an investigation into the natureand magnitude of curvature and refractionerrors as they may be expected to occur inaerial photographs. Also included are somesuggestions for the analytical eliminationor compensation of such errors when thenumber of photographs being dealt withdoes not justify the use of geocentric co­ordinates.

CURVATURE

Figure 1 illustrates the relationships,greatly exaggerated, between a verticalaerial photograph and the earth's surface.In which

L = exposure stationf = focal-length of lens0= principal point of the vertical

photographe = photographic image of ground­

point Coe=m=radial distance to point e

H=Aying height above a sea-leveldatum, assumed coincident withthe earth's surface

V=ground nadir pointE' = intersection of ray LeC with a

horizon tal plane through VVE'=M

V1=interesection of LV extendedand a horizontal plane throughC

V1C= M 1

VV1=DC=hc=a correction to scale dataor flying height for earth's cur­vature.

OV=OC=R=radiusof earth (mean valueof semi-major and semi-minor

* In the 1957 contest for the Bausch & Lamb Photogrammetric Award, this paper was awardedone of the two Second Prizes.

751

---~-------

752 PHOTOGRAMMETRIC ENGINEERING

o nt e

H

R

o

R

he

~I--------------''''!{

v

FIG. 1. Error caused by earth's curvature.

Substituting the value of R in (6):

dh e = Md20.88 X 106 ·dM, (7)

From equation (7) it is evident that theremust be a considerable change in the valueof M 1 to effect a small change in the com­puted value of he. From this observation itis concluded that the computed value of hewill be sufficiently precise if the assumption

axes of Clark's 1866 spheroid is20.888 X 106 feet)

0= center of the earth8=angle subtended at the earth's

center by arc VCEC = E. C. = elevation error.

From the diagram, it is evident that:

R2 = (R - he)2 + M I 2 (1)

Expanding:

R2 = R2 - 2Rhe + h; + M I 2 (2)

h; - 2Rhe + M I 2 = 0 (3)

To circumvent the necessity of a quadrat­ic solution the assumption can be made thathe

2 is a negligible quantity, as indeed it iswhen compared to 2Rhe and M 12.

Equation (3) then becomes:

he ==MN2R

Taking the first derivative of (4):

dhe = 2MI /2R·dM,

Reducing:

(4)

(5)

(6)

EARTH'S CURVATURE AND VERTICAL PHOTOGRAPHS 753

TABLE 1 8 = Mt7r/R sin 1"· (3,600) ·180 (13)

is made that M is equal to MI.Then, substituting M for M I in equation

(4):

This () is converted to degrees by dividi ng by3,600"jdegree. () may then be converted toradians by factoring by 7rj180 giving:

~·· 20~

0

5

"0"""'~~·'" 10

Reducing, assuming the arc 1" equals4.848Xl0-6

}

Ra.dial Dlstwlce (lD) inches

I

//

//

/

Arc VC = M'1r/3.141504 (16)

Equating the values of () given by (14) and(15) :

8 = M'1r/R3.141504 (14)

8",,,;,,", = Arc VC/R (15)

FIG. 2. Height error (he) versusradial distance (m).

The value in the denominator of (16) is sonearly equal to the value of 7r, that for allexcept large values of VC or M I , the assump­tion that they are equal results in a negligi­ble error.

It can be seen from Figure 1 that if it iswished to establish the relative ground-co­ordinates of point C from the photo-co­ordinates of its corresponding image pointe, the procedure should be as follows:

Assuming, as aforementioned, that thephotograph is truly vertical and that, forthe moment, point C lies on the horizontaldatum:

Letting Xc and Ye and Xe and Ye equal theground-coordinates and photo-coordinatesof point C and its image point e, respec­tively, the relationships between them,

(8)

(9)

(12)

33.7'26.5'20.4'15.0'10.4'6.7'3.7'1. 7'0.4'0.0'

he in feet

he = (H/f)2· (m)2/2R

4.5"4.0"3.5"3.0"2.5"2.0"1.5"1.0"0.5"0.0"

Radial distance (m)in inches from

photograph

I t is observed in the diagram that R isvery nearly equal to (R-he ), and as a con­sequence the cosine () is almost unity, and ()is an extremely small angle. This leads tothe observation that E. C. is approximatelyequal to he. The computed value for he canthen be considered equal to the height error,E.C., for point C caused by the curvature ofthe earth.

Since M = Hjf- m, equation (8) may berewri tten :

M, = (H + heW' 111

The computed distance M, may be assumedequal to arc distance VC for any values withwhich there is concern. That this is true canbe seen from the following equations:

sin 8 = M,/R (11)

Since () is a very small angle:

I f a camera focal-length of 6" is assu medand a value of 50,000 feet for the flyingheight (H), Table 1 gives the relationshipbetween various radial distances corre­sponding to m and the computed value of he.

The information and relationship shownin Table 1 are further illustrated in Figure 2.

The following equation, in addition tothose already mentioned, is evident fromFigure 1:

754 PHOTOGRAMMETRIC ENGINEERING

neglecting the curvature of the earth (i.e.assuming that a horizontal plane through Vis coincident with the earth's surface), are:

Xc = Hlj·xc and Y c = Hlj·yc (17)

As is usually the case, point C does notusually lie on the datum plane, and if theelevation of point C above the datum isgiven by h, formulas (17) are modified tocompensate for this so that:

Xc = (H - h)lf·xc and Y c = (H - h)lj·yc (18)

From the foregoing, it is evident that tocorrect for the error due to earth's curva­ture, there may be introduced another term(viz. he), further modifying (18) to:

Xc = (II - h + hc)lf·xc andY c = (II - h + hc)lf' Yc (19)

Of course, factoring the rectangularphoto-coordinates by any factor is exactlyequivalent to factoring the radial distancefrom the origin to the point, since:

If m is factored by some factor F:

lIlF = v(Fxc)2 + (FYc)2 (21)

Thus, the error due to earth's curvaturemight be considered a radial displacementtoward the center of the photograph. Re­ferring to Figure 3, it is apparent that thephotographic image displacement ee' iscaused by the curvature of the earth. Thedisplacement ee' is derived as follows:

ee' = oe' - oe (22)

oe' = !/II·M (23)

oe=jl(H+hc)·M (24-)

Substituting (23) and (24) in (22)

ee' = MjhclH(H + he) (25)

It is intuitively and mathematically ap­parent that the magnitude of this displace­ment ee' is directly proportional to the radialdistance oe of the image point e from theprincipal point of the photograph. Since oeis directly proportional to H and inverselyproportional to j, it follows that, otherthings being constant, an increase in theflyi ng heigh t (H) or a decrease in the focallength (j) will result in a greater apparentimage displacement due to earth's curva­ture. For any given combination of j, H, M,and computed he, the error ee' may be com­puted from formula (25) and if it is of meas­urable magnitude, correction may be made.

.__---.-Jt!.L ----"o

o

FIG. 3. Apparent photographic image displace­ment caused by earth's curvature.

REFRACTION

Figure 4 illustrates the image displace­ment on an aerial photograph caused byat­mospheric refraction. In which

L = exposure stationj = focal-length0= principal-point of photographV=ground nadir-pointA and B = two ground-pointsb = apparent image registration on the

photograph of point A on the grounde = ab =image displacement on photo­

graph caused by refractionR = angle of refraction or the angle be­

tween the tangent LbB to the re­fracted ray and LaA the ray posi tionif there were no refraction

l=the difference in the inclination of atangent to the ray at A and a tangentto the ray at L, therefore the changein inclination of the ray due to re­fraction (N.B. 1/2 =R)

H = elevation of exposure station abovethe datum, represented by VAB.

The exact determination of the amountof atmospheric refraction corresponding toany altitude is a problem that cannot becompletely solved, since the refraction de­pends on the temperature and density of theair, not only at the surface of the earth, but

EARTH'S CURVATURE AND VERTICAL PHOTOGRAPHS

A

FIG. 4. Apparent photographic image displacement on a verticalphotograph caused by atmospheric refraction.

755

As before indicated, the refraction cor­rection may be considered negligible, but ifit does in some unusual circumstances war­rant consideration, values of hr can be com­puted and tabulated for various values of

I t is apparent on the photograph thatpoint A on the ground which should registerat point a on the photograph actually regis­ters at point b, thus being radially displacedby an amount e. By introducing a scalefactor the correct radial ground distance VAcan be computed from the radial photo dis­tance ob from the following formula:

also along the whole path of the ray.Examination of Bessel's table of refrac­

tions for astronomic use indicate that at­mospheric refraction for the ray inclinationsencountered in photogrammetric tech­niques using vertical aerial photographs isso small as to be negligible. For example:

tan oLb = oblJ (26)

tan oLa = oalJ (27)

ab = e = J(tan oLb - tan oLa) (28)

Since

oLa = oLb - R, (29)

e = J(tan oLb - tan (oLb - R)) (30)

M = (ll - h,)IJ·ob (31)

756 PHOTOGRAMMETRIC ENGINEERING

f, H, ob, and R. So calculated, it may be in­serted as a scale factor in formula 19, th uscorrecting for earth's curvature and atmos­pheric refraction simultaneously. Equa­tions (19) then become:

X e = (H - h + he - hr)/j,xe

Y e = (H - h+ he - Ilr)/f·yc (32)

I t is apparent that the effect of atmos­pheric refraction is to radially displace theimage-point from the principal-point of thevertical photograph.

CONCLUSIONS

Apparently the effect of atmospheric re­fraction is to partially offset the effect ofearth's curvature. It is, however, nearlynegligible, and difficult to accurately eval­uate quantitatively.

Because of the difficulty of theoreticallyevaluating the effect of atmospheric refrac­tion, the possibility of making empiricaldeterminations of atmospheric refractionand earth curvature in combination suggestsusing balloon photography over an areawi th su fficien t con trol to permit the precisedetermination of image point displacements

incidental to photography taken at variousaltitudes with various types of cameras.

The possibility of correcting for earth'scurvature and refraction by equations (32)on isolated photographs taken at high alti­tudes appears to be practical, since the rela­tive ground coordinates so established canbe used to determine scalar quantities andareas. The effect of tilt on this techniqueremains to be investigated.

BIBLIOGRAPHY

[1]. Church, Earl, "Coordinate Determination ofGround Points on Aerial Photographs," (AirForce Problem 48), Problem Report No. 25,June 15, 1951.

[2J. Kelsey, Ray A., "Effects of Earth Curvatureand Atmospheric Refraction upon Measure­ments Made in Stereoscopic Models of HighAltitude Photography," Project 8-35-08-001,Engineer Research and Development Labor­atories, Report 1242, July 9, 1952.

[3J. Shu, C. S. and McNair, A. J., "Determina­tion of Geographic Coordinates, FlightHeights, and True Orientation for Exten­sions of Strips of Aerial Photographs," PHO­TOGRAMMETRIC ENGINEERING, September1956, Volume XXH, No.4, pp. 637-643.

Photogrammetric Applications ofRadar-Scope Photographs*

PAMELA R. HOFFMAN,

Aeronautical Chart & Information Center,St. Louis, Missouri

ABSTRACT: The utilization of radar-scope photographs as standard photo­grammetric source material has been studied for over ten years. Almost theentire bulk of completed work is experimental in nature but has providedvaluable information. It has led to the development of special compilationprocedures, and special compilation equipment, as well as the capabilityof applying these developments in production compilation. Improvementof the basic radar equipment will permit even greater success in this fieldin the future. This paper describes procedures and techniques for apply­ing a specialized kind of photogrammetric knowledge in extractingtopographic information from radar-scope photographs.

* Presented at the Society's 24th Annual Meeting, Hotel Shoreham, Washington, D. C.,March 27, 1958.