Some Factors Causing Vertical Exaggeration and …...SOME FACTORS CAUSING VERTICAL EXAGGERATION AND...

SOME FACTORS CAUSING VERTICAL EXAGGERATIONAND SLOPE DISTORTION ON AERIAL PHOTOGRAPHS*

Victor C. Miller,l Miller-McCulloch, Photogeologists, Calgary, Alberta

ABSTRACT

When two overlapping aerial photographs are viewed stereoscopically, the verticalscale of the terrain model seen is rarely the same as the horizontal scale; the model isvertically exaggerated, usually with the vertical scale exceeding the horizontal. In addition, all topographic features are not ~entally displaced vertically alone, but also in ahorizontal direction, depending on the position of the stereoscope. This is termed model"distortion." .

Vertical exaggeration is considered the product of the photographic variablesfocal length, camera height, air base, relief, tilt, and optical imperfections; vertical exaggeration also varies with the stereoscopic factors-viewing distance, photograph separation, eye'base, and photograph rotation. Image distortion is largely dependent uponthe position of the stereoscope. The entire paper treats the qualitative aspects of thesevariables. Though not substantiated by quantitative data, it is felt that the generalmagnitude of vertical exaggeration is largely dependent on the photographic variabfesand that the stereoscopic variables introduce minor differences in exaggeration.

All accompanying illustrations are diagrammatic. At the end of the paper the authorpresents several suggestions for improved stereoscopic examination of aerial photographs.He also includes a: new graph relating vertical exaggeration, true ground slope, apparent(stereoscopic model) slope; this graph is intended primarily for the photogeologist whomust estimate dips seen on aerial photographs.

INTRODUCTION

I N A recent paper (Miller, 1950) theauthor presented a method for obtain

ing an accurate yet rapid estimate of slopeor dip, through the estimation of apparentslope or dip, as seen stereoscopically onaerial photo'graphs, and the application ofa correction scale. The amount of verticalexaggeration, assumed to be constant fora given pair or set of aerial photographs,determined the amount of correction necessary in a given case.

Since the time of that publication considerable thought has been given to thefactors which influence vertical exaggeration. The results of this later considerationare presented in this paper.

Slope distortion is closely related to, bu tin some ways independent of, verticalexaggeration. If all image exaggerationswere vertical only, there would be a simple tangent relation between slope andexaggeration. This, essentially, is the assumption on which the author's previous

paper was based, but the variables considered below show that no such simplerelation exists. Many photogeologists, or atleast those geologists who only on occasionlook at aerial photographs, may not beaware of the errors introduced by the improper (though common) use of the stereoscope.

WHAT IS VERTICAL EXAGGERATION?

When one looks at overlapping verticalaerial photographs through a refractionstereoscope, he sees a three-dimensionalmodel whose vertical scale exceeds thehorizontal. Such a model, or mental image,is said to be vertically exaggerated. Quantitatively, vertical exaggeration may bedefined as the ratio of vertical scale tohorizontal scale.

Vertical exaggeration has been discussed or mentioned in many articles andbooks on. photogrammetry and photogeology. When illustrated graphically, thesubject is usually depicted by diagrams

* Paper read at Nineteenth Annual Meeting of the Society, Hotel Shoreham, WashingtonD. c., January 14 to 16, 1953. It was a part of the Report of the Photo Interpretation Committee.

1 The author is greatly indebted to Mr. Vincent F. Kotschar, assistant in Geology at Columbiaand Cartographer with the Geographical Press, for having drafted the illustrations appearing inthis paper.

592

VERTICAL EXAGGERATION AND SLOPE DISTORTION 593

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similar to Figure 1. As Figures la, b, c andd, stand, they are geometrically sound, butone point must be made; in Figures lb, cand d, the eyes of the viewer are picturedas being directly above the principal points(centers) of the photographs. If this werethe case in actual practice, problems of

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Under such viewing conditions, exaggeration would simply be a function of eyeto-photograph distance, and if this distance be equal to the focal length of thecamera that took the photographs, vertical

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FIG. 1. An illustration of the commonly used method of depicting overlapping aerial photography and various geometric relations associated with stereoscopic examination of aerial photographs.

A) Two cameras, located at L1 and L 2 , are shown photographing a 100-foot pole and a 100-footground distance.

B) The photographs taken in Figure lA are viewed stereoscopically from eye positions E 1 andE2• Vertical exaggeration is 1.50.

C) The photographs taken in Figure lA are viewed stereoscopically from eye positions E 1 and&. Vertical exaggeration is 0.65. "

D) The photographs are shown here again being viewed from the same distance as in C, butnow the separation of the photographs is greater. Vertical exaggeration is still 0.65.

594 PHOTOGRAMMETRIC ENGINEERING

•scale and horizontal scale would be thesame; the solid model seen would havetrue three-dimensional scale. By movingthe eyes farther from the photographs, apositively exaggerated image of the modelwould be obtained. (Figure lb; VerticalExaggeration 1.50), and by decreasingeye-to-photograph distance, a negativeexaggeration would be produced. (FigureIe; Vertical Exaggeration 0.65.)

Under these conditions the separationof the photographs would have no bearingon vertical exaggeration. Figure ld showsthe geometric relations existing when eyeto-photograph distance is the same as inFigure lc, but with greater photographseparation. It will be noted that verticalexaggeration is the same in both instances(0.65). This is true because in both lc andld all angular relations are the same, andall corresponding triangles involved aresimilar.

I t has been the author's observationthat in photogeologic study the refraction,or lens, stereoscope is the type most commonly used. It is unfortunate that theabove relations, invitingly simple thoughthey may be, do not apply when the refraction stereoscope is employed. Twoaerial photographs (usual dimensions: 9X9 inches or 9X 7 inches) cannot beviewed through a refraction stereoscope insuch a way that the eyepieces stand directly over the photograph centers.

However, by the lIse of a mirror stereoscope, it is possible to look directly downon the centers of the photographs, but inpractice the photographs or the instrumentare generally placed so that the observation point is at some chosen portion of thephotographs, regardless of image positionrelative to the centers. Therefore, evenwhen a mirror stereoscope is used, a photograph is rarely viewed from a positionover the photo centers, and as a result thegeometry of Figures lb, c and d, againdoes not apply.

The proper method of viewing a stereopair with a refraction stereoscope consistsof setting the lenses of the stereoscope at aproper separation (determined by measuring one's eye base) and placing the instrument over the photographs, which arespaced at an optimum separation. Thephotographs are not side by side, but partially overlapping, with identical imagesseparated about 2 inches. This separationmay not be optimum for all persons or for

all stereoscopes, but many measurementsmade by the author, using the FairchildModel C-2 stereoscope, vary inconsiderably from the 2-inch figure.

Figure 2 shows the relation betweencamera lenses at the time of photographyand eye positions at the time of stereoscopic examination. (Note should be madeof the fact that in this figure, and all thatfollow, actual conditions or dimensionsof stereoscopy or photography are not at-

, tempted, but merely the various geometricprinciples involved.) The height AB, ofthe object photographed, is the same asthe ground distanceAC. PointX is anotherground point on the level of points A and

At C,

FIG. 2. An illustration of the usual relation between camera lenses and stereoscopeposition. In this figure, vertical exaggerationis 2.65.

C. These points are photographed fromlens positions Ll, and L 2, a-nd appear onthe resulting prints at points al, bl, Cl, Xl,and a2, b2, C2, and X2, respectively. Whenthese photographs are arranged for stereoscopic study, the instrument is placed overthem so that the viewer's effective eyepositions are at E I , and E2• The left eye, atEl, sees images ai, bl, Cl, and Xl, while theright eye, at E 2, sees images a2, b2, C2, andX2. The projected model, identified bypoints Al, B lJ Cl, ·and Xl, is seen.

The unnatural vertical elongation of theobject, AlBl, is striking. To determine vertical exaggeration AB is not comparedwith AIBI, but AlBI is compared with


AlGI. These two distances were originallyequal, and vertical exaggeration, as described above, i!5 the ratio of vertical scaleto horizontal scale. Therefore, AIBI andAlGI, representing identical distances,may be compared directly to determinevertical exaggeration. AB measures 2.65times as large as AlGI, therefore verticalexaggeration in this case is 2.65.

The point X was photographed from theleft from LI, and from the right from L 2•

With the stereoscope in the position shown,however, XI and X2 are both seen from theleft hand side. The author believes thisfact to be highly significant because theterrain at and around X cannot be correctly restituted in the stereo image if oneof the viewer's eyes looks at it from a direction opposite to that from whIch it wasphotographed. This incorrect viewing position is the cause of distortion, as distinctfrom vertical exaggeration. Actually thevertical exaggeration at and around Xwould be the same as that in the vicinityof the object AB, because the purely vertical element is a function of parallax displacement, but a stereoscopic model alsodepends on the manner in which the photographs are viewed. While E I and E 2

• roughly approximate directionally L I andL 2 with respect to the photograph imagesof A, Band G, they are out of relationshipwith respect to X. Therefore, if X were tobe correctly viewed it would be necessaryto move the stereoscope to the right, andthus duplicate the original directional relations from L I and L 2 to X.

VARIABLES

Aerial photographs are subject tomany variable factors which directly influence the three-dimensional impressionwhen the terrain is studied stereoscopically.. Some of these variables inv,olve thegeometry existing at the time the photographs were exposed; others are concernedwith the manner in which the photographsare viewed. These variables may be called,respectively, "Photographic" and "Stereoscopic."

Photographic variables include:

1) Scale; this in turn is a function ofa) Camera focal length, andb) Camera height, at the time of

photography.2) Air Base; the horizontal distance be-

tween two consecutive photographexposures.

3) Relief; both over-all relief and differences in relief from one area' to another.

4) Tilt; a complex and complicatingfactor which, due to the limited scopeof this paper, is omitted in the following discussion.

5) Optics; optical imperfections of modern lenses are so minor that theireffect on vertical exaggeration wouldbe too slight to be given consideration here.

Stereoscopic variables include:

1) Eye-to-photograph distance; the actualdistance, if the photographs areviewed without a stereoscope; the effective distance if lenses are used.

2) Photograph separation; the photographs may be moved closer togetheror farther apart.

3) Photograph rotation; the rotation ofone or both photographs a'fay fromproper alignment.

4) Movement of the stereoscope; a changein the position of the stereoscope,or one's eyes, in a plane parallel tothat of the photographs.

5) Eye base; the separation of one'seyes. (This of course is constant foranyone individual, but different fordifferent persqns.)

DISCUSSION OF VARIABLES

Let us.consider these various factors 1ndi~idually, and attempt to picture diagrammatically hO\y stereoscopy is affectedby each.

To be understood fully, scale must beconsidered to be a function of focal lengthand camera height, both of which are variable. Figure 3 (a and b) shows how photograph scale may be kept constant thoughboth focal length (!I and h) and cameraheight (HI and H 2) are changed. Air bases,LIL2 and LaL4, are equal, and the pyramidphotographed, AB G, is the same in bothcases. The sides of the pyramid slope 30degrees. It will be noted that alCI, the image of the pyramid base as photographedfrom camera position LI, is equal to aaCa,that taken from La. Images a2C2 and a4C4are also of equal size. However, albl, bICI,a2b2, and b2c2 are not equal in size to the corresponding images, aaba, baca, a4b4 and b4c4.In either words, there is a difference in the


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FIG. 3. A) Pyramid ABC is photographed from camera positions L1 and L2 •

B) The same pyramid is photographed from camera positions L 3 and L.. Compare withFigure 3A.

parallax displacements on the two sets ofphotogra hs, and as a result they will notgive the same three-dimensional imagewhen viewed beneath the stereoscope.

Figure 4 (a and b) shows how verticalexaggeration is influenced by the abovenoted differences in the two sets of photographs, both having the same scale. For aproper comparison, the conditions ofstereoscopy are kept constant for bothstereo pairs; eye-to-photograph distance,eye base, and photograph separation areth same. Model A.B.C. (V.E. 1.79) appears considerably more exaggerated vertically than does Model ABC (V.E. 1.39).I t is therefore apparent that if scale be keptconstant, by the main tenance of a constan tf/H ratio, vertical exaggeration will varyas an inverse function of camera height andfocal length.

A change in camera height, or focallength, if not accompanied by a corresponding change in the other, will of coursecause a change in photograph scale. Figure5 illustrates photography in which thefocal length, f, is the same as that shownin Figure 3a, while the camera height, H,is equal to that shown in Figure 3b. Thescale of the photographs in Figure 5 issmaller than that of Figures 3a and 3b. Thegeometry of Figure 5, however, is similarto Figure 3b; the ratio of asbs to bscs, for example, is the same as the ratio of a3b3 tob3C3. A change in scale due to a change infocal length, therefore, produces the same

results as does photographic enlargementor reduction of a single negative, whereasa change in scale due to a change in cameraheight produces different relative parallaxdisplacements which cannot be duplicatedby projection in the photographic laboratory.

When the photographs taken in Figure5 are placed beneath the stereoscope theyappear as shown in Figure 6; again theconditions of stereoscopy ar~ the same asbefore. If Figure 6 (V.E. 1.66) is comparedwith Figure 4b (V.E. 1.39), it will be notedthat, other factors being equal, verticalexaggeration is greater when photographscale, due to smaller focal length, is smal1er.Vertical exaggeration, then, varies inversely with photograph scale, when scale isa function of focal length.

Though there is but a slight differencein the vertical exaggerations of Figures 6(V.E. 1.66) and 4a (V.E. 1.79), it shouldbe noted that if a greater difference incamera height had been chosen for illustration, it would be much more evidentthat with a decrease in scale, due to an increase in camera height, there is a corresponding decrease in vertical exaggeration.In other words, vertical exaggeration varies directly with scale when scale depends onthe camera height. This is true becausewith increased camera height there is adecrease of relative parallax displacemen tcaused by relief.

Figure 7 shows the conditions of photog-


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stereoscopically from eye positions E, and E 2 • Vertical exaggeration is 1.79. This figure should becompared with Figure 4B.

B) The photographs taken from camera positions L 3 and L. in Figure 3B are viewed stereoscopically from eye positions E 3 and E•. Vertical exaggeration is 1.39.

raphy when there is a smaller air basethan that shown in the previous figures.This will result in greater overlap, whenthe same photographic equipment is used.This figure may be compared with Figure3a, because both have the same focallength and camera height, and hence thesame scale. Both cameras, L 7 and La, inFigure 7 "look down" more directly on thepyramid than did the cameras shown inFigure 3a. When these new photographsare viewed, again under the same conditions of stereoscopy as before, the modelseen is much less exaggerated vertically(Figure 8). Compare Figures 8 (V.E. 1.22)and 4a (V.£. 1.79). It then follows thatvertical exaggeration varies as a direct function of air base. (Or inversely with overlap.)

R. F. Thurrell, ] r. 2 is at present carrying on a quantitative investigation of theinterrelation of base-height ratio and vertical exaggeration. In his work, Thurrell

2 Geophoto Services, Denver, Colorado.

considers the actual air base, focal length,camera height, and overlap encountered inaerial photography. His conclusions, beingof a quantitative nature, are expected tosupplement the qualitative aspect of thepresent paper.3

The relative amount of relief in an areais also a factor that must be considered.In Figure 7, points X and Y stand at equalvertical intervals above point B. Theseintervals are equal to the heigh t of B abovethe base of the pyramid.

In Figure 8 the stereoscopic images ofpoints X and Y (X., Y.) appear to be almost equally spaced, but actually thevertical image B.-X. is shorter than theheight of the pyramid and longer than theimage X 4-Y4•

It might be concluded from the abovethat the higher elevations shown on anygiven stereo-pair will produce a sligh tlysmaller vertical exaggeration than will

3 See page 579.


FIG. 6. The· photographs exposed in Figure 5are shown being viewedfrom eye positions Es andEs under stereoscopic conditions identical to thoseshown in Figure 4. Verticalexaggeration is 1.66..

FIG. 5. The photographs shown here arebeing taken with a camera whose focal lengthis f1 (the same as in Figure 3A), from aheight of H 2 (the same as in Figure 3B).


FIG. 7. In this figure air base is •smaller than that shown in Figure3A; focal length and camera heightare the same. Points X and Yareequally spaced vertically abovepoint B; the vertical intervals BXand XY are equal to the height ofthe pyramid.

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FIG. 9. Photograph images X 7, Y7, X s,and Y s are shown being viewed from eye positions E 7 and E s, as in Figure 8. Image separation X.,-X s is the same as a.,-a,s in Figure 8.

tho'se points at lower elevations. This willbe true if both high and low topographicpoints are studied stereoscopically withthe photo separation being the same at alltimes.

Figure 9 shows the viewing of points Xand Y with the photos separated to aposition in which the interval from X7 toXs is equal to the original distance a,-as

• shown in Figure 8. Since other conditionsof stereoscopy have not been changed between Figures 8 and 9, a comparison between the stereoscopic image of the heightof the pyramid in Figure 8 and imageX4- Y4

in Figure 9 will show the relative verticalexaggerations produced when high andlow topographic points are individuallystudied with identical image separation.

I t will be noted that the stereoscopicimage distance X 4- Y4 in Figure 9 is almosttwice the height of the image of the pyramid in Figure 8. It may therefore be concluded from this illustration that the vertical exaggeration of high topographicpoints is greater than the vertical exagger-

ation of low topographic points providedthat the high and low areas studied stereoscopically are each viewed with identicalphoto-image separation.

The eye base (distance between E g andE 10) in Figure 10 is greater than the eyebase in Figure 8, even though all otherconditions of stereoscopy, including thephotographs beneath the stereoscope, arethe same. The vertical exaggeration (0.86)of the model seen in the case of Figure 10,is considerably less than that shown inFigure 8 (V.E. 1.22), where the eye base issmaller. To some this comparison may notseem sound, as it is difficultfor one individual to comprehend the optical perceptionof another individual. However, the geometry of the comparison should be obvious. An individual with a relatively smalleye base does not see a solid object inexactly the same way as another, with alarger'eye base, In this instance the twoindividuals see images from two differentsets of perspective points, and it would beexpected that their mental images orstereoscopic models should not be identical.Recent quantitative tests, carried on bythe author, substantiate the above statements; anaglyphs were used in the testingof relative vertical exaggerations seen by

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persons with different eye bases. Eyebases of from 57 mm. to 69.5 mm. weretested, and results were very satisfying.

Focal length, camera height, air baseand relief, are all variables, but at the timea photograph is exposed, the results ofthese factors are "frozen" in the photographic images. After exposure, they are nolonger variable. Eye base of course is constant for anyone individual.

There now remain to be consideredthose factors which may be varied by theobserver, with or without his consciousknowledge. When a refraction stereoscopeis used a constant eye-to-photograph distance is maintained. However, when different stereoscopes are used or the photographs are viewed without a stereoscope,as is frequently the practice in the field,the distance from the eyes to the photographs may be increased or decreasedwhile a stereoscopic model is still main-.tained in the mind.

There is some disagreement as to howthe vertical dimension of the modelchanges when this viewing distance ischanged. Some state that with increasedviewing distance there is an increase invertical exaggeration. On the other handthere are those who maintain that changesin viewing distance have no such effect.Salzman (1950), for example, has takenthis latter view. The author believes thatthere is very definitely a direct relation between viewing distance and vertical. exaggeration. With increased distance bothvertical and horizontal "model dimensionschange, and the ratio of vertical scale tohorizontal scale, which is the expression ofvertical exaggeration, is increased. Compare Figure 11 with Figure 8. Both showthe same eye base, photo separation, andthe same photo images. The stereoscopicmodel produced in Figure 11, when viewing distance is greater, is much more exaggerated vertically. (Vertical exaggerationin Figure 11 is 1.87; in Figure 8 it is 1.22.)

Photograph separation has been mentioned several times in this paper. Howdoes this factor influence the stereoscopicmodel? In Figure 12 (V.E. 1.39) the photographs of Figure 11 (V.E. 1.87) are shownbeing viewed with a smaller photographseparation.

It will be noted that the greater the separation, the greater the vertical exaggeration.(Figures lc and Id, which are not representative of actual viewing conditions,

would give the false impression thatchanges in separation are of no 'consequence.)

This particular fact bears strongly onthe practice of estimating slope and dipwithout resorting to photogrammetricmeasurements. Since photograph separation affects the viewer's impression of vertical exaggeration, it is important to keepseparation constant in order to maintainconstant exaggeration. With practice, itcan be ascertained that there is a definitephotograph separation that "fits" theviewer's eyes best. At this separation, thestereoscopic model has a minimum, or lackof, eye strain. Once this ptage has beenreached a few times, it will become almostautomatic to arrange the photographs atthis separation. For the beginner, it mightbe advisable to devise some sort of template or convenient scale that might easilybe slipped over the photographs to checkon the consistency of separation.

Another closely associated type of photograph movement is that of rotation.Many times it is possible to see a stereoscopic model even when one or both photographs are sligh tly rotated out of properalignment. Most texts on photogeologyand many on elementary photogrammetricmethods treat the proper method ofmounting photographs for stereoscopicstudy, so a discussion of that particularsubject will not be included here. What issignificant in this respect is that, in addition to the actual change in orientation,there is a variance in photograph separation from point to point, and therefore achange in vertical exaggeration within thesame stereo pair. This change may not bevery great; but is nevertheless worthy ofmention.

The final variable, and it is of prime importance, is the position of the stereoscopeover the photographs.• When a stereo pairis studied the viewer "looks down" on theterrain of the three dimensional model. Itis common practice to move the stereoscope over the photographs and to vieweach locality from a position directly overhead, rather than leaving the stereoscopein one position and seeing the entire areaof overlap by oblique viewing.

The direction in which a certain featureis viewed determines the form of the modelwhich is seen. By moving the stereoscopefrom side to side, after obtaining a stereoscopic model, it is possible to note the


FIG, 11, This figure should be compared with Figure 8; both are identicalexcept for the factor of viewing distance, Vertical exaggeration is 1.87.

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changes which take place in the model;slopes facing the direction of movementwill become steeper, while those facingaway from the direction of movement willbecome more elong~teand gentle. Hill topsand ridge crests will "follow" the stereoscope as it is moved.

Similar distortion occurs when the stereoscope is moved toward or away from theobserver.

Compare Figure 13 with Figure 12.They are identical in that they representan observer with a given eye base, at aconstant distance from the photographs,looking at the same photographs which inboth cases have the same separation. Thedifference is that in Figure 13 the eyes hadbeen moved to the right. Whereas Figure

"12 produced a symmetric model form,Figure 13 makes the hill quite asymmetric.

The great majority of objects photographed must appear in positions latevallymarginal from the central portion of theoverlap zone, toward the far or near edgesof the photographs. Figure 14 illustratesthe photography of the pyramid by aplane which is flying not parallel to thepage, but normal to it, toward or awayfrom the reader. Therefore two consecutivecamera positions, L 21, and L 22 , are designated by the same point, and the twoprints, converging rays, etc., are all indicated by the single diagram.

Figure 15 diagrammatically shows theposition of such images on overlapping


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. FIG. 13. The photographs depicted in Figure 12 are here shown being viewed fromdifferent eye positions (Eyes have been moved to the right).

prints. When such areas are being viewed,it is not possible to merely put the stereo-

. scope over this part of the photographsand "look down," and still expect to get arealistic mental model of the terrain. Thecamera was "looking out" at this area,and therefore the viewer must move hisstereoscope (or his eyes) a little nearer thecentral portion of the photograph; he must"back off" to get a more nearly true mental picture, and look toward the marginalzone.

Figure 16a shows the effect of lookingstraight down on the images, and Figure16b shows how the model is improved bylooking at this part of the photographfrom an oblique position. The verticalexaggeration mentioned earlier will ofcourse not be eliminated by moving theeyes or stereoscope about in this manner,

but at least the severe distortion of slopesthat may be superimposed on exaggerationmay thereby be minimized or completelyremoved.

SUMMARY OF THE EFFECT OF THEINDEPENDENT VARIABLES

The foregoing illustrations and comparisons have brought out the followinggeneral relationships:

Vertical exaggeration varies;

a) inversely with camera height, whenscale is constan t

b) inversely with focal length, whenscale is constant

c) inversely with scale, when focallength varies

d) directly with scale, when cameraheight ,varies


FIG. 14. The air base in this figure is normal to the page. ote that the pyramid isnear the margin of photographic coverage.

e) directly with air basef) inversely with eye baseg) inversely with ground elevation,

when constant photo separation ismaintained; directly with groundelevation, when constant image separation is maintained.

h) directly with viewing distancei) directly with photograph separation

SUGGESTIONS FOR BETTER STEREOSCOPIC

STUDY

The author suggests the following stereoscope practices:

1. By whatever method the individualfinds best for his work, he should maintaina constant image separation.

2. He should prevent photograph rota-

FIG. 15. The photographs taken in Figure 14 are here shown arranged for stereoscopic examination.

VERTICAL EXAGGERATI N AND SLOPE DISTORTION 605

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FIG. 16. A) The photographs taken in Figure 14 are shown viewed stereoscopically, with theobserver's eyes directly over the images of the pyramid. Compare with Figure 16B.

B) The same photographs are viewed from a position farther to the left, that is nearer thecenters of the photographs.

tion by following sound standard methodsof photograph ·alignment.

3. The use of a single stereoscope removes the possibility of a change in viewing distance. If he uses different instruments, or works part time without astereoscope, he should be well aware of theeffect of changing his viewing distance.

4. When viewing ground objects whichwere apparently midway between thetwo camera stations, he should place thestereoscope so that each eye "looks in,"about equally, toward the photographicimages.

5. When viewing ground objects whichappear near the center of, say, the lefthand photograph, and extreme left marginal zone of the right hand photograph,he should place his stereoscope so that hisleft eye "looks down," while the right'eye"looks in" at a definite angle. This maynot be most comfortable. but it will helpto eliminate slope distortion. I.f exact slopeor dip determination is not a prime requisite, he may not go too far wrong in lookingdown from the most convenient position tostudy the topographic or geologic details inthis area.

6. When studying ground objects which

appear on laterally marginal portions ofboth photographs, he should keep hisstereoscope a bit nearer the central stripof the photographs, and look obliquely atthese marginal areas. During this sametime, he should note whether or not thoseobjects are midway between photographcenters, or more nearly opposite the oneor the other, and then modify his directionof viewing as in #5 above.

These practices should effectively cu tdown much of one's "self-inflicted" distortions of slopes and topography. Aspointed out above, vertical exaggerationcannot in this way be eliminated, but ifdistortion is done away with, essentially,then the desired tangent relation betweenvertical exaggeration and slope and dipexaggeration will more nearly be broughtabout, and the principle proposed in theauthor's previous paper will be much moreapplicable.

An improvement has been made on theoriginal graph presented in that earlierpaper. The new grapq, Figure 17 is morelegible and much more workable. It ishere presented in the hope that it will beof some use to those who are not in a position to apply photogrammetry in their


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1..!::.,~ ...L-_~_----=-::-_7;-_~O

1.5 2.0 2.5 3.0EXAGGERATIO N'

FIG. 17. This graph is essentially similar to the one published in the November 1950 AAPGBulletin. The author feels, however, that it is much more useful and accurate, especially in thelow dip (or slope) range.

Vertical exaggeration is shown by the concentric arcs, which represent exaggerations of 1.0,1.5, 2.0, 2.5, and 3.0. The apparent slope or dip, that seen through the stereoscope, is shown on theuniformly divided inner scale. The true dips or slopes appear as the curved lines which intersectthe arcs representing exaggeration.

The graph may be read in two ways:1) If the amount of vertical exaggeration is known to be 2.5, for example, and a certain slope

appears to be 60°, the radial line from the 60° point on the inner scale is traced outward until itintersects the 2.5 exaggeration line. The position of this point of intersection is then noted withrespect to the curved true slope lines. In this case, a value of a little over 34° is found to be thetrue slope.

2) If a known slope of 26° appears as one of say 44° on the photographs, the radial 44° line istraced to its intersection with the curved 26° line. The position of the point determined by thisintersection is noted with respect to the arcs.representing exaggeration, in this case 2.0.

APPROXIMATING THE VERTICAL EXAGGERATION RATIO 607

geological study of aerial photographs, butwho wish a rapid, yet simple, way torectify slope and dip estimation.

In conclusion, it is well to repeat thatthe illustrations accompanying this paperare intended to depict the principles, andnot actual conditions of photography andstereoscopy; also, the magnitude of variation in vertical exaggeration and distortionproduced by each variable discussed mayor may not be significant. Each variable

must be subjected to an intensive quantitative test before its relative importancecan be known.

REFERENCES CITEDMiller, Victor c., "Rapid Dip Estimation in

Photo-Geological Reconnaissance." BulletinAmer. Assoc. Pet. Geol., Vol. 34, No.8 (August, 1950).

Salzman, M. H., "Note on Stereoscopy."PHOTOGRAMMETRIC ENGINEERING, Vol. XVI,No.3 (1950).

AN EQUATION FOR APPROXIMATING THE VERTICALEXAGGERATION RATIO OF A STEREOSCOPIC VIEW

E. R. Goodale, Staff Photogrammetrist, Creole Petroleum Corporation,Caracas, Venezuela

AT THE VII International Congress ofr1.. Photogrammetry, attention was calledto the interesting subject of the stereoscopic estimation of dip-slope angles bythe photogeologist. One of the importantfactors involved in estimating dip anglesis the amount any given stereoscope exaggerates the vertical scale over the horizontal.

This vertical exaggeration of the stereoscopic view need only be determined approximately. However, up to the presenttime, the author has not seen any mathematical expression of the vertical exaggeration ratio, although several articles onthe subject have appeared recently inPHOTOGRAMMETRIC ENGINEERING.

The purpose of this paper is to offer suchan equation and, at the same time, to substantiate it with sufficient proofs to at leastinvite its further discussion.

The geometry in this paper is limited tothat concerned with truly vertical airphotographs; also, for simplicity and convenience, ground relationships are represented in the positive counterpart of thelens-negative plane.

For use of the equation, the only information needed is the focal length of theair-camera lens. All other terms can bemeasured on a stereo-pair of contactprints. The eye base and viewing distancewill correspond to the individual observerand the stereoscope used, and can easily bemeasured.

DEFINITIONS

At this point some of the terms to be

used in the discussion will be defined. Noclaims are made for these definitions ofterms other than to clarify their meaningwithin the text of this paper. Photogrammetric terms of general usage have thesame definitions as given in chapter XIXof the MANUAL OF PHOTOGRAMMETRY.

1) Vertical Exaggeration

Exaggerated impression of depth overimage size when viewed stereoscopically.

2) Photo Base

The distance measured between thecenter points of two photographs of astereo-pair, when the images of all common ground points which lie at the datumelevation are in coincidence.

3) Photographic Datum Plane

Theoretical plane at the elevation ofany selected photographic image point (inthis paper a low point of the stereo-view),from which image displacements aremeasured, and to which the measurementof the photo-base is referred.

4) Object Ray

Ray which has passed through the aircamera lens and by which the photographicimpression of a ground object has beenmade.

5) Image Ray

Ray reflected from the photographicimage of a ground object to the eye or tothe stereoscope eyepiece.

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