Determination of 

Quantitative Geologic 

Data with Stereometer 

Type Instruments

By WILLIAM R. HEMPHILL

PROCEDURES AND STUDIES IN PHOTOGEOLOGY

GEOLOGICAL SURVEY BULLETIN 1043-C

A discussion of the applications 

and limitations of Stereometertype instruments as applied to 

geologic problems

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1958
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U. S. Government Printing Office 

Washington 25, D. G. - Price 35 cents (paper cover)
CONTENTS

Page

Abstract___. .________ . _ _______________ 35

Introduction. _ __________ _ _ ._ ._ _______ 35

Working principle of stereometer-type instruments__ ____.___-_ 36

Formulas used to convert change in parallax to change in elevation._____ 39

Tilt._____-.._ __-.-._..__ ._... .. _.._...__- . . 46

Measurement of strike and dip_________-___________________________ 49

Effect of surface expression of the strata___.__ ___-_-___-,__ _ 49

Effect of photograph scale.___ _ ___-..----.__--. 50

Effect of relief displacement_____________________________ 50

Correction for relief displacement..__ ... ________ ____ 52

Limitation of relief displacement correction ___________________ 53

Measurement of stratigraphic intervals __ _ ______ 53

Summary __ _____________________________________________________ 55

Literature cited.____________ ____ ___ -____-____._-.______ 56

ILLUSTRATIONS

Page 

PLATE 1. Stereometers used by the Geological Survey._____.Facing__ 38

2. Relief displacement of dip slope.. _ __ ________ ____ Facing __ 52

FIGURE 13. Diagram showing difference in elevation (Aft) on ground expressed as difference in parallax (Ap) on photographs______ 37

14. Properly alined stereoscopic pair.________________________ 38

15. Graph showing V varying as a function of H' and (a&+Ap)_. 43

16. Graphs showing effect of tilt on low-dippingbeds___________ 48

17. Diagram showing image displacement of a flagpole due to

the height of the flagpole-_-___-___-_-_---_-------_---_ 51

in

PROCEDURES AND STUDIES IN PHOTOGEOLOGY

DETERMINATION OF QUANTITATIVE GEOLOGIC DATA 

WITH STEREOMETER-TYPE INSTRUMENTS

By WILLIAM K. HEMPHILL

ABSTRACT

A stereometer-type instrument is a photogrammetric measuring device used to 

determine differences in elevation between features as seen on vertical aerial 

photographs. The operation of stereometer-type instruments and some of their 

uses in obtaining quantitative geologic data are described. Measurement of 

stratigraphic thickness and strike and dip are emphasized because such measurements not only demonstrate general stereometer procedures but also are basic in 

many geologic field-mapping programs. Necessary photogrammetric terms, 

normally not familiar to many geologists, are defined and their use explained.

INTRODUCTION

Quantitative geologic data often may be obtained quickly and 

reliably with stereometer-type instruments from overlapping vertical 

aerial photographs. Where evidence of rock units is discernible on 

aerial photographs, these instruments may aid in determining strike 

and dip, in obtaining amount of fault throw, in measuring stratigraphic thickness, and in obtaining the difference in elevation between 

selected features of topographic significance.

The purpose of this paper is to discuss the procedures for applying 

stereometer-type instruments to geologic problems. The advantages 

as well as the limitations of these instruments are discussed in some 

detail. Only vertical aerial photographs are considered.

Photogrammetric terms as used in this paper are defined as follows:

Parallax: Apparent displacement of an object with respect to a 

point of reference owing to a shift in the point of observation (Am. 

Soc. Photogrammetry, 1952, p. 829).

Stereoscopic pair: "Two photographs of the same area taken from 

different camera stations so as to afford stereoscopic vision" (idem, 

p. 838).

Stereoscope: "An optical instrument for assisting the observer to 

view two * * * photographs" in order "to obtain the mental impression of a three dimensional model" (idem).

Stereoscopic model: The impression of three dimensions received 

when viewing a stereoscopic pair beneath a stereoscope.

35
36 PROCEDURES AND STUDIES IN PHOTOGEOLOGY

Image point: Point at which the image of an object on the ground 

appears on one photograph of a stereoscopic pair. Images of the 

same object on both photographs making up the stereoscopic pair are 

referred to as conjugate images or conjugate image points.

Stereoscopic image point: Point at which the image of an object 

on the ground appears in the stereoscopic model.

Fiducial mark: Reference mark imprinted midway along each 

margin, of an aerial photograph at the time of exposure.

Photograph center: Point located at the intersection of lines drawn 

between fiducial marks on opposite margins of the photograph. For 

the purposes of this paper, the photograph center may be considered 

to be coincident with the plumb point.

Transferred center: Point on a photograph at which the photograph 

center of the adjacent photograph of the stereoscopic pair appears:

Photobase: Distance on one photograph between the photograph 

center and transferred center.

Camera focal length: Distance measured along the lens axis between 

the nodal point of the lens and the plane of best average definition 

(positive plane or negative plane) (idem, p. 820).

General photograph scale: Photograph scale as stated on the 

photograph or photograph index.

Applicable photograph scale: Scale at a particular datum in thestereoscopic model as determined from a reliable base map or from 

field control.

Tilt: Angle of departure, expressed in degrees, of the camera axis 

from true vertical.

Relief exaggeration: Apparent exaggeration of the vertical scale 

with respect to the horizontal scale in the stereoscopic model.

Relief displacement: The horizontal linear distance an image point 

is displaced on the photograph radially from the plumb point, owing 

both to the difference in elevation between that image point and the 

plumb point and to the horizontal distance between that image point 

and the plumb point. For the purposes of this paper, the plumb 

point may be considered coincident with the photograph center.

WORKING PRINCIPLE OF STEREOMETER-TYPE 

INSTRUMENTS

A stereometer-type instrument is used to measure the difference 

between the absolute parallax of two or more stereoscopic image 

points of different elevations. Absolute parallax of a stereoscopic 

image point may be defined as the algebraic difference of the distances 

measured parallel to the photobase between the two image points 

and their respective photograph centers. The distance measured to 

the right of the photograph center is positive; the distance measured 

to the left of the photograph center is negative. This definition
GEOLOGIC DATA DETERMINED WITH STEREOMETER INSTRUMENTS 37

assumes that the photographs making up the stereoscopic pair are 

truly vertical and identical in scale. Tewinkel (1952, p. 325) expresses 

absolute parallax algebraically:

p=x x'

where p is the absolute parallax, x is the distance from the photograph 

center to the image point on the left photograph, and x' is the corresponding distance on the right photograph; both distances are measured parallel to the photobase. For example, in figure 13

FIOURE 13. Diagram showing difference in elevation ((Aft) on ground expressed as difference in parallax 

(Ap) on photographs.

U Topofflagpoleinnature. 

u Image of the top of the flagpole on the photographs.

A Distance between conjugate image points /. 

B Distance between conjugate Image pointsji.

ci Left photograph center. 

£3 Right, photograph center. 

/ Camera focal length. 

0 Camera lens or theoretical position of eyes when

viewing stereoscopic model. 

L Bottom of flagpole in nature. 

I Image of the bottom of the flagpole on the photographs.

The difference between distances A and B is the change in parallax (Ap) between the top and bottom of 

the flagpole.
38 PROCEDURES AND STUDIES IN PHOTOGEOLOGY

absolute parallax (p=x x') of the top of the flagpole (U) can be 

shown :

Similarly, the absolute parallax of the bottom of the flagpole can 

shown:

p=Cil ( c2l)

Determining differences in absolute parallax with stereometer-type< 

instruments is based on the so-called "floating-dot" principle. A. 

target consisting of a dot, cross, or small circle scribed on transparent 

plastic or glass is mounted on each end of a supporting bar as shown 

on plate 1. The horizontal distance between the dots can be varied 

by manipulating a reading drum graduated to either hundredths 

millimeter or thousandths inch. When the horizontal distance 

between the dots equals the distance between conjugate image points 

on a properly alined stereoscopic pah- (fig. 14), the dots will fuse and

Right photograph center

Transferred right center

Transferred left center

Fiducial mark

Left photograph center

FIGURE 14. Properly alined stereoscopic pair.
GEOLOGICAL SURVEY BULLETIN 1043 PLATE 1

STEREOMETERS USED BY THE GEOLOGICAL SURVEY
GEOLOGIC DATA DETERMINED WITH STEREQMETER INSTRUMENTS 39

appear to "float" in the stereoscopic model at the elevation of the 

conjugate image points or stereoscopic image point in question (distance A, fig. 13). The reading on the drum is noted and the procedure may be repeated at a stereoscopic image point of different elevation (distance B, fig. 13). The difference between these readings is 

the change in absolute parallax or simply the parallax change (Ap) 

between the two stereoscopic image points. Thus,

&p= (cil-\-c2l) (ciM+c2/i) = A B.

A mathematical relationship exists between this parallax change 

(A#») and change hi elevation (A/&) between the two points on the 

ground. This relationship is not affected by relief exaggeration; 

that is, the amount of measured parallax change (Ap) between stereoscopic image points of different elevation is not exaggerated with 

respect to the change in elevation (A/t) on the ground.

FORMULAS USED TO CONVERT CHANGE IN PARALLAX 

TO CHANGE IN ELEVATION

A simple formula for converting change in parallax to change hi 

elevation is as follows :

AA=yAp (1)

where A/z, is the change in value of h or the difference in elevation 

between two stereoscopic image points, b is the average of the photobases of the left and right photographs, and H is the height of the airplane above the ground based upon camera focal length and general 

photograph scale. This formula is commonly incorrectly applied. 

It wih1 provide satisfactory measure of Ah in areas of low relief; if 

applied without discrimination to" all types of topography, considerable 

error may result because the numerator, H, and denominator, b, as 

used in the formula are only approximations.

The numerator, H, is commonly found by dividing camera focal 

length by general photograph scale. This scale may be locally inaccurate; for example, photographs of an area in the plateau country 

of the Western United States with a general photograph scale of 

1 : 20, 000 have been found to have an applicable photograph scale of 

1 : 29,000 at a particular datum in the stereoscopic model, owing either 

to variations in the flying height of the airplane, to extreme topographic relief of the photographed area, or to a combination of both. 

If a 6-inch focal-length lens were used with a general photograph 

scale of 1 : 20,000, H would have been computed to be 10,000 feet. 

The applicable photograph scale of 1 : 29,000 should be used, in which 

case H would be computed correctly at 14,500 feet. The 30-percent

449043 58 2
40 PROCEDURES! AND STUDIES! DT PHOTOGEOLQGY

error in H in. this example is extreme, but 10-percent error in using H 

as the numerator is common and will cause similar error in the resulting value of Ah. Therefore, H should be based upon the applicable 

photograph scale as determined from a reliable base map or from 

field control.

Reliable base maps from which to determine applicable photograph 

scale at a particular datum may not be available. This poses no 

problem for some types of quantitative geologic determinations. 

A strike and dip, for example, does not require absolute differences in 

elevation; any convenient scale can be used as long as this same scale 

is used to calculate both the vertical and horizontal components of the 

tangent of dip. Also, where relative differences in elevation are 

required, such as for comparison of relative stratigraphic thickness, 

and base maps or field control are not available, appropriate steps 

may be taken to obtain the most reliable results. These steps are 

discussed on page 54.

The other factor that can be responsible for appreciable error in 

Ah is 6, the average of the photobases of the left and right photographs.

If the elevation of the lower of the two stereoscopic image points, 

between which the difference in elevation is being measured, is approximately midway between the elevations of the left and right 

photograph centers, an average of the photobases will be adequate for 

most geologic measurements. Likewise, in areas of low relief the 

error incurred will generally be insignificant if the lower point is at an 

elevation other than approximately midway between the elevations 

of the two photograph centers. However, in areas of extreme relief 

the error in using the average photobase (6) as the denominator commonly exceeds 5 percent and will cause an approximately equivalent 

percentage of error in Ah.

Thus, formula (1) should be used to provide rapid computation of 

Ah for areas of low topographic relief only; significant error may result 

when this formula is applied to rugged terrain. To compensate for 

the effect of high relief on the measurement of Ah, the following formula (modified from Tewinkel, 1952, p. 325-326) should be used:

Ah= ,*?. Ap (2) ab+Ap *

where H' is the height of the camera above the lower of the two 

stereoscopic image points between which an elevation difference is 

desired, and ab is the photobase adjusted to the photograph scale at 

the lower stereoscopic image point. Factor H' is found by applying 

the following formula:

#'= (3)
GEOLOGIC DATA DETERMINED WITH STEREOMETER INSTRUMENTS 41

where / is the camera focal length in feet and S is the applicable 

photograph scale determined from a reliable base map or field control 

at or near the elevation of the lower stereoscopic image point. 

Adjusted photobase (ab) may be found by the following method: l

1. Orient the photographs (fig. 14).

2. With a millimeter scale, measure the distance between the two 

photograph centers.

3. Similarly, measure the distance between the conjugate images of 

the lower of the two stereoscopic image points between which an 

elevation difference is desired.

4. Subtract the distance obtained in step 3 from the figure obtained 

in step 2 (round off to nearest millimeter). The difference is the 

photobase adjusted to the photograph scale at, or the elevation 

of, the lower stereoscopic image point.

Where several differences in elevation are to be determined in the 

same stereoscopic model, the necessity of measuring with a scale, for 

each calculation, the distance between the lower conjugate image 

points in step 3 above may be omitted by using the following procedure :

1. Orient the photographs (fig. 14).

2. With a millimeter scale, measure either of the two photobases of the 

stereoscopic pair to the nearest half millimeter.

3. With a stereometer, measure A# to the nearest half millimeter between the lower stereoscopic image point and the photograph 

center of the photograph whose photobase was not measured in 

step 2.

4. If the lower stereoscopic image point is below the photograph center 

used in step 3, subtract (and round off to nearest half millimeter) 

the figure obtained in step 3 from the figure obtained in step 2. 

If the lower stereoscopic image point is above the photograph 

center used in step 3, add (and round off to nearest half millimeter) the figure obtained in step 3 to the figure obtained in 

step 2. The result is the photobase adjusted to the elevation of 

the lower stereoscopic image point.2

Both the photobase obtained in step 2 and the stereometer reading on 

the photograph center in step 3 should be recorded and used in determining adjusted photobase for all additional computations of difference 

in elevation in the same stereoscopic model. Because the stereometer 

reading at the lower stereoscopic image point in step 3 is made in the 

course of determining Ap between the lower and upper stereoscopic 

image points, it does not represent an additional measurement in 

obtaining the adjusted photobase.

i Modified from Instruction manual for the stereo elevation meter, 1954, Photogrammetry, Inc., Silver 

Spring, Md., p. 2. 

> This method for obtaining adjusted photobase Is similar to a method described by Elliot (1052, p. 10).
42 PROCEDURES! AND STUDIES IN" PHOTOGEQLOGY

An adjusted photobase need not be computed where the lower of the 

two stereoscopic image points appears at or near the elevation of one of 

the photograph centers; the photobase of the adjacent photograph of 

the stereoscopic pair will suffice for the adjusted photobase. Similarly, 

where the lower of the two stereoscopic image points appears at a 

position approximately midway between the elevations of the two 

photograph centers, the average of the two photobases may be used 

for the adjusted photobase.

It will be noted that factor Ap has been added to the denominator 

in formula (2). Ap is generally very small compared to factor ab; 

therefore, the error in AA, is not generally appreciable if &p is omitted. 

Nevertheless, the ease of computing factor (ab-\-kp) to the nearest half 

millimeter makes including it desirable. If A# is comparatively large, 

say more than 2 millimeters, A# should be included in the denominator 

if the most reliable results are to be ob tamed.3

To facilitate computation of AA, part of this formula has been 

reduced to a variable called V so that

TTf

TT ab+Ap

As

(5)

V may be determined rapidly by referring to a graph (fig. 15) so 

constructed that V varies as a function of Hr and (ab+Ap). V as 

determined from this graph represents the number of feet on the 

ground for each millimeter of parallax change (A#) in the stereoscopic 

model. Some of the flight altitudes (Hf) for photograph scales of 

1 : 12,000 to 1 : 50,000 are included. These are scales of photographs 

commonly used by geologists. Additional flight altitudes may be 

added when other photograph scales are met. A similar graph may 

be constructed when (afr+A^i) and V are required in terms of inches. 

An inexpensive photogrammetric computer, currently manufactured by a commercial company, may also be used to determine V. 

This computer is particularly advantageous because it may be operated like a circular slide rule to multiply change in parallax (Ap) by 

the feet per unit of parallax (V). In addition, the computer may be 

used as an aid in determining photograph scale ($), aerial camera 

altitude (H), and aerial camera focal length (/) 

' Inasmuch as factor (afi+Ap) is equivalent to photobase adjusted to the upper stereoscopic image point, 

a small amount of calculation could be avoided by adjusting photobase to the upper stereoscope image point 

Instead of the lower. However, in keeping with normal practice, adjusted photobase is computed in the 

present paper with respect to the lower stereoscopic image point.
GEOLOGIC DATA DETERMINED WITH STEREOMETER INSTRUMENTS 43

V, IN FEET

obt^p, IN MILLIMETERS

FIGURE 15. Graph showing V varying as a function of //' and (u6+Ap).
44 PROCEDURES AND STUDIES EST PHOTOGEOLOGY

To illustrate the use of the formula (2) in an area of high relief, 

a sample problem is presented below. An extreme example was 

purposely chosen in order to emphasize the fact that large errors may 

result from ^discriminate use of formula (1). 

Problem:

A stereoscopic image point has been located on each of two flatlying beds cropping out in an area of rugged relief. The 

photobase of the right photograph is 92.5 millimeters; the 

photobase of the left photograph is 97.5 millimeters. When 

the dot is "floated" on the lower and upper stereoscopic image 

points and the left photograph center, the stereometer reads 

30.50, 28.30, and 21.30 millimeters, respectively. (Both 

stereoscopic image points are at lower elevation than the 

left photograph center.) Applicable photograph scale at the 

lower stereoscopic image point is 1:23,400, and the photographs were taken with a 6-inch focal-length lens. What is 

the thickness of section between the two beds? 

Solution:

Applying formula (3) to find Hr :

23,400 

= 11,700.

Adjusted photobase is found by subtracting the difference in parallax 

(Ap) between the left photograph center and the lower stereoscopic image point from the photobase of the right photograph. 

(Adjusted photobase could also be found by subtracting the 

difference in parallax (Ap) between the right photograph center 

and the lower stereoscopic image point from the photobase of 

the left photograph.)

92.5 mm Photobase of right photograph 

9.2 mm Ap between left photograph center and lower stereoscopic image point

83.3 mm Adjusted photobase (adjusted 

to lower stereoscopic image 

point).
GEOLOGIC DATA DETERMINED WITH STEREOMETER INSTRUMENTS 45

Applying formula (2) to find thickness of the stratigraphic section

/ 11,700 \ 2 , 

V83.3+2.2/

/ll^OOX 

V 85.5 /

From the reference graph (fig. 15) :

F=137

(5)

= 137X2.2 mm 

=301 ft.

For comparison, formula (1) is applied to the same problem. General photograph scale is 1 : 20,000. 

Applying formula (3) to find H:

 rr_f_ (Here, S is the general photograph 

~S scale.)

0.5

20,000 

= 10,000.

Average photobase: 95 mm. 

Applying formula (1) to find A/&:

A^=-r-

=231 ft.

Errors in both the numerator (H) and the denominator (6) are not 

necessarily mutually compensating. Neither is the sum of their 

errors necessarily the percentage of error to be found in the result; 

for example, a 10-percent error in both numerator and denominator 

may not mean a 20-percent error in AA. In application of formula (1) 

to the problem above, a 15-percent error in the numerator (H) and 

an 11-percent error in the denominator (6) caused a 23-percent error 

in the result (AA) .
46 PROCEDURES. AND STUDIES EST PHOTO GEOLOGY

TILT

Tewinkel (1952, p. 319) estimates that with modern-day methods, 

90 percent of vertical aerial photographs are tilted less than 2° and 

50 percent are tilted less than 1°.

The method of detection of such small amounts of tilt must be commensurate with the accuracy of the instrument used to measure 

parallax change (Ap). Double-projection stereoplotting instruments 

are constructed to correct mechanically for tilt; stereometers, however, are not so constructed and depend solely upon the alertness and 

understanding of the operator to notice manifestations of tilt as they 

occur in the stereoscopic model.

Tilt manifests itself in the stereoscopic model in several ways. 

If applicable photograph scales of several image points, all at the 

same elevation, are computed from a topographic base map, the scale 

throughout the tilted stereoscopic model will be different for points 

of equal altitude. Also, streams may appear to flow uphill or their 

gradients may appear too steep or not steep enough. Meandering 

rivers or swamps, features that should be level or nearly level, may 

appear inclined in the stereoscopic model. Strike of low-dipping 

strata may appear to be different when the same area is viewed on 

adjacent stereopairs. These and additional geomorphic manifestations 

of tilt are more easily detected and interpreted with the aid of a 

mirror-type stereoscope than with a lens-type stereoscope. The 

mirror type has a large field of view and local evidence of tilt can be 

viewed with respect to the stereoscopic model as a whole. The lens 

type has a comparatively small field of view. Also, stereoscope 

distortion of mirror-type stereoscopes is negligible, whereas, with 

some lens-type stereoscopes, distortion produces a noticeable "bowllike 

effect" in the stereoscopic model. For example, flat terrain would 

appear inclined inward toward the center of the field of view, thus 

complicating and perhaps obscuring evidence of any tilt that may 

be present.

Tilt of two photographs comprising a stereoscopic pair is seldom 

the same in either amount or direction; therefore, tilt in a stereoscopic 

pan- must be considered in terms of two independently tilted photographs whose net effect in the stereoscopic model produces (1) an 

inclined surface or datum and (2) an unevenness or warpage of this 

inclined surface or datum. For example, different points on the 

shoreline of a large lake may have several hundredths millimeter of 

parallax change in the stereoscopic model owing to the warping effect 

of tilt, in addition to possibly several millimeters of parallax change 

owing to tilt's physical inclination of the shoreline.

Usually, when computing stratigraphic thickness, these effects of 

tilt can be sufficiently minimized by choosing positions of stereoscopic
GEOLOGIC DATA DETERMINED WITH STEREOMETER INSTRUMENTS 47

image points not more than 1 inch apart on photographs of say, 

1:20,000 scale (a correspondingly shorter distance should be used on 

photographs of smaller scale). In this manner the effect of tilt will 

be much the same on all points involved. Stratigraphic thicknesses 

should not be computed between stereoscopic image points located 

on opposite sides of the stereoscopic model unless control is available 

or the vertical distance to be measured is large and only an approximate 

value is required. Only 1° of tilt in photographs of 1:20,000 scale 

could result in an error of as much as 280 feet if the upper and lower 

stereoscopic image points were located on opposite sides of the stereoscopic model.

When determining strike and dip, the most serious effect of tilt is 

on the azimuth of true dip of strata dipping less than 6° or 7°. Tilt 

in these circumstances can swing the azimuth of true dip in the stereoscopic model, the amount of swing depending upon the direction and 

amount of tilt relative to the direction and amount of true dip (fig. 16). 

By being aware of this effect of tilt when working with low-dipping 

strata, one can take the precaution of carefully inspecting the stereoscopic model for whatever geomorphic manifestations of tilt may be 

present. Where low-dipping strata crop out as dip slopes, their true 

dip may be nearly the same as the direction and gradient of intermittent streams flowing on their surfaces. Where this is so, one 

should suspect presence of tilt if the direction of true dip appears to 

be transverse or opposite to the direction of streamflow. The direction 

of true dip, when recorded on the photograph or base map, should be 

adjusted with respect to the stream or other geomorphic indications 

of tilt that may be present. Where stream gradients are steep or 

where meandering rivers, lakes, or swamps are absent, it is difficult 

to detect evidence of tilt in the stereoscopic model; consequently, 

azimuth of true dip of low-dipping beds in these areas should be 

regarded with suspicion. The effect of tilt on strata dipping more 

than 10° probably is not significant for most geologic problems.

Tilt may be inadvertently introduced into the stereoscopic model 

by careless alinement of the stereoscopic pair. To demonstrate this 

point, it is suggested that two photographs of a stereoscopic pair be 

rotated or skewed in opposite directions beneath a stereoscope, 

preferably of mirror type, with the photograph centers functioning 

as the axes of rotation. The stereoscopic model will be observed to 

tilt alternately away from, then toward the viewer, as the direction 

of rotation is changed. Because of this tilting effect, it is important 

that the photograph centers and the transferred centers be positioned 

accurately. The transferred centers should be located stereoscopically and not with respect to surrounding conjugate images observed 

with the naked eye. The photographs should be alined in such a
CHANGE IN'AZIMUTH OF TRUE DIP, IN DEGREES 

g'g S S 8 3 S 8

_. 5- 2. CD

CHANGE IN AZIMUTH OF TRUE DIP, IN DEGREES 

, 5 S S § S S S

3 ? S- ^

I 3 S'

5 2 p

o 3 01 -

? § O

= ~ m 

Q. -; 3) MS'
GEOLOGIC DATA DETERMINED WITH STEREOMETER INSTRUMENTS 49

manner that the two centers and two transferred centers are in a 

straight line (fig. 14). The largest amount of image displacement 

due to tilt is in the extreme corners of the photographs; consequently, 

determination of quantitative data should be avoided, if possible, 

in the corners of the stereoscopic model.

Camera-lens distortion and stereoscope-lens distortion probably 

have no significant effect on stereometer measurements. The effect 

of differential shrinkage of film and photograph print paper may be 

minimized somewhat by positioning stereoscopic image points, 

between which differences in elevation are being measured, as close 

together as possible. Vertical exaggeration and magnification have 

no distorting effect on differences in elevation (A^) as determined from 

stereometer measurements.

To determine the attitude of a bed, field geologists often use the 

"three-point method." The procedure for determining the attitude 

of a bed on a vertical aerial photograph is basically the same as this 

field method except that a stereometer, rather than a barometer or 

alidade and stadia rod, is used to obtain the differences in elevation 

(A/fc). This procedure takes into account three factors influencing 

strike and dip determinations on aerial photographs: (1) effect of 

surface expression of the strata, (2) effect of photograph scale, and 

(3) effect of relief displacement.

EFFECT OF SURFACE EXPRESSION OF THE STRATA

Where individual beds crop out as dip slopes, strike and dip determinations can be based upon elevations at only two stereoscopic 

image points located on the azimuth of true dip. The "threepoint method" is unnecessary because the azimuth of true dip can 

be determined by stereoscopic inspection. For example, where 

strata are well exposed, attitudes of low-dipping beds are easily detected in the stereoscopic model because of an apparent exaggeration 

of the model's vertical scale relative to its horizontal scale. A dip of 

6° may appear to be as much as 20° or more. Moreover, evidence of 

dip is indicated in the stereoscopic model by the three-dimensional 

image of each discernible bedding plane both in the local area in 

which a strike and dip determination is desired and in nearby areas. 

Stream patterns also may reflect either strike or dip of the strata. 

In short, azimuth of true dip can be determined by stereoscopic inspection because vertical aerial photographs provide an overall 

picture of an area and indicate the direction of dip at many places in 

the stereoscopic model. However, where evidence of bedding is only 

along canyon walls or hillsides, the three-point solution must be used.

To simplify the discussion of the effect of scale and relief displace-
50

merit on strike and dip determinations, reference will be made to 

only two points on a bedding plane chosen along the azimuth of true 

dip. These points in the stereoscopic model will be referred to as the 

upper (u) and lower (I) stereoscopic image points.

EFFECT OF PHOTOGRAPH SCALE

Formula (2) should be used for strike and dip determinations in 

areas of high relief and the photobase should be adjusted to the lower 

of the two stereoscopic image points on the bedding plane. However, 

it is not necessary to know the applicable photograph scale at the 

lower stereoscopic image point; instead, any convenient scale may be 

chosen. In spite of the fact that the scale may be erroneous, its effect 

upon the length of the vertical and horizontal distances will be proportional; that is, the percentage of error in the vertical distance 

(A/i) will be compensated by an equal percentage of error in the horizontal distance. The tangent of the dip in question will remain the 

same.

EFFECT OF BELIEF DISPLACEMENT

A photograph differs from a map in that horizontal distances cannot 

be reliably measured directly from the photograph with a rule because 

the position of any image point does not necessarily correspond to its 

correct orthographic position. Instead, a point on the photograph 

may be displaced from its orthographic position, the amount of displacement depending upon its vertical and horizontal distance from 

the plumb point or point directly beneath the camera lens. (For the 

purposes of the present paper, the plumb point may be considered as 

the photograph center.) In figure 17, A-A' is a horizontal reference 

datum at the elevation of the photograph center (c). Because point 

E on the flagpole lies on reference datum A-A', its image (e) will 

appear on the photograph in its correct orthographic position with 

respect to that datum. Points U and L do not lie on datum A-A'; 

therefore, their images u and I do not appear on the photograph in 

their correct orthographic positions. Thus, although points U and E 

occupy the same geographic position on the ground, their images are 

separated by the distance eu; similarly, although E and L are in the 

same geographic position, their images are separated by the distance el. These distances are image displacements due to relief of 

points U and L relative to datum A-A'.

The amount of relief displacement of an image point is directly 

proportional to the horizontal and vertical distance between that 

point and the photograph center and inversely proportional to the 

height of the camera above the center. With reference to a horizontal 

datum at the elevation of the photograph center, any other point at 

lower elevation will be displaced toward the photograph center; any
GEOLOGIC DATA DETERMINED WITH STEKEOMETER INSTRUMENTS 51

Negative plane'

FIGURE 17. Diagram showing image displacement of a flagpole due to the height of the flagpole.

c Photograph center. 

I Image point of L. 

u Image point of (7. 

e Image point of E.

A-A' Horizontal datum at elevation of C. 

L Base of flagpole. 

U Top of flagpole. 

E Point on ground on datum A-A'. 

C Point on ground on datum A-A' and appearing on the photograph at c.

Image point e is the orthographic position of U and L on the photograph relative to datum A-A'. Distance eu is the relief displacement of image point u away from the photograph center, and distance el is the 

relief displacement of image point I toward the photograph center.

point at higher elevation will be displaced away from the photograph 

center.

Thus, an image point will appear on the photograph in its correct 

orthographic position with respect to a datum passing through the 

photograph center only if its elevation or its location is coincident 

with the photograph center. Where an image point is located at any 

other position, relief displacement, if appreciable, will render any
52 PROCEDURES! AND STUDIES IN PHOTOGEOLOGY

horizontal distances measured relative to that point invalid. Dip 

measurements based on these horizontal distances will be unreliable.

CORRECTION FOR RELIEF DISPI/ACEMENT

On plate 2A upper and lower points U and L of dip angle 0 appear 

on the left and right photographs at lu and II and ru and rl, respectively 

(pi. 2A and B). The relief displacement of the upper point U relative 

to the lower point L is clearly seen if the discrepancy between the 

photograph distances lu-ll and ru-rl is noted and compared with the 

corrected or true horizontal photograph distance represented by id 

on plate 20. A method for correcting this displacement and obtaining 

true horizontal distance between points U and L is as follows:

1. Label and mark with a fine dot on each photograph of the stereoscopic pan: the following image points: the center (ci and c2 ), 

transferred center (lie and rtc), upper points (lu and ru), and 

lower image points (II and rl). Connect the photograph center 

and transferred center of each photograph with a line (pi. 2B).

2. Determine the photobase adjusted to the elevation of the lower 

point L (pi. 2A\ by following the procedure outlined in the discussion of formula (2).

3. Mark a line, the length of which is equal to the length of the adjusted photobase, on transparent overlay material (such as dyrite, 

acetate, or tracing paper). Label the left terminus of this line ct 

and the right terminus c2 .

4. Position the overlay on the right photograph in such a manner that 

.point c2 on the overlay is coincident with point c2 on the photo- 

' graph and line c\-c2 on the overlay is coincident with c2-rtc on 

the photograph.

5. On the overlay, plot point rl and label it /. Draw a fine line on 

the overlay from c2 through ru on the photograph beneath (pi. 

2B and 2(7).

6. Position the overlay over the left photograph in the same manner 

as in step 4. If no tilt is present, the position of the point U on 

the photograph should be coincident with the position of I 

plotted on the overlay in step 5. If tilt is present, point U on 

the photograph will appear a short distance from point I on the 

overlay. The overlay should then be shifted in such a manner 

that point I becomes coincident with image point U while., line 

Ci~c2 is maintained parallel to the photobase of the left photograph. The error introduced by this shift is seldom significant.

7. Draw a fine line from Ci through lu (pi. 2B and 0} as in step 5.

8. The intersection of lines c2 ru and c\ lu (pi. 20) is the corrected 

position of the upper stereoscopic image point. Label that intersection u. Distance ul is the corrected horizontal photograph
53

distance ;on the overlay between ground points U and L. Horir 

zontal distance as it is to be used in determination of dip may be 

obtained by dividing the photograph distance ul by the same 

photograph scale used in measurement of the vertical distance 

(see "Effect of photograph scale," page 50). Similarly, horizontal ground distance between points U and L may be determined by dividing photograph distance vl by the applicable 

photograph scale at the lower point (rl or II).

Where three stereoscopic image points on the bedding plane are 

necessary, relief displacement of each of the upper two points is 

determined relative to the lower point. A small amount of calculation 

may be omitted if two lower stereoscopic image points can be posi-. 

tioned at the same elevation; this allows the relief displacement to be 

determined relative to both lower points in only one operation. .... .

The overlay procedure may be eliminated under certain conditions. 

Where beds are either low dipping or located near one of the photograph centers, the amount of relief displacement of the upper image 

point relative to the lower is negligible. Also, where strike is parallel 

to a radial line connecting the photograph center and the lower image 

point, the effect of relief displacement upon horizontal distance is 

insignificant. Under any of these conditions, horizontal distance may 

be measured directly from the photograph.

LIMITATION OF RELIEF DISPLACEMENT CORRECTION

To minimize error in computing the tangent of the angle of dip, 

corrected horizontal distance should probably measure at least 0.2 

inch on the overlay. This horizontal distance requirement will vary, 

depending upon clarity of rock outcrop and quality of the photographs 

and stereometer. It will vary also with the magnitude of dip; dips 

less than 5° require a longer horizontal distance so that any error iri 

differences in elevation (AA) between stereoscopic image points will 

be minimized. Dips more than 50° also require a longer horizontal 

distance if accuracy is to be maintained, because the change in the 

tangent function per degree of dip is greater at this magnitude. Dips 

more than 70° should probably not be measured with stereometers 

because the change in tangent per degree of dip is too great. In fact, 

beds dipping over 60° or 65° rarely crop out with enough areal extent 

in the direction of true dip to provide sufficiently accurate measurement of horizontal distance from aerial photographs.

MEASUREMENT OF STRATIGRAPHIC INTERVALS

Measurement of stratigraphic intervals or differences in elevation 

between topographic features involves the same "floating-dot" technique described on page 38. The photobase is adjusted to the level of
54

the lower point, and formula (2) is applied to convert change in parallax (Ap) to change in elevation (AA,).

When measuring stratigraphic intervals, variations of procedure 

are often necessary, depending on the geologic and photogrammetric 

characteristics of the individual problem. In areas where strata are 

flat lying it is not necessary to correct for relief displacement. However, in areas where dip is significant it is necessary first to correct 

for relief displacement and determine strike and dip, and then correct 

for the effect of dip on the stratigraphic thickness.

Although general photograph scale or any other convenient scale 

can be used in determining strike and dip, an error which may be 

significant, is commonly incurred by using general photograph scale 

in measuring stratigraphic intervals. For example, if scale is erroneous in a strike and dip determination, the percentage of error in the 

vertical distance will be compensated by an equal percentage of error 

in the horizontal distance. In measuring stratigraphic intervals 

there is no such compensating effect; the percentage of error in an 

erroneous scale will cause an equal percentage of error both in the 

height of the camera (H) as used hi the formula and in AA, the stratigraphic interval in question. Therefore, it is essential for accurate 

measurement of stratigraphic intervals that applicable photograph 

scale at the lower point be computed from reliable base maps or 

field control.

If only relative thicknesses of a rock unit are desired in an area 

where base maps are not available, a value can be assigned to Hr. 

This value is obtained by arbitrarily selecting a horizontal datum 

within the stereoscopic model (the datum chosen is usually at or near 

mean relief), and then relating the general photograph scale and 

H' to that datum. For example, suppose it is necessary to obtain 

the rate of thinning along the outcrop of a flat-lying stratigraphic 

unit: the photographs on which this rock unit appear have a general 

photograph scale of 1:20,000 and were taken with a 6-inch focallength camera. First, a datum is chosen (in this example, the datum 

happens to be below the outcrop in question) and from the general 

photograph scale it is assumed that the camera is 10,000 feet above 

that datum. Second, the photobase is adjusted to the datum and 

found to be 100 millimeters. Next, difference in parallax (A^>) between the datum and the lower bed, measured with a stereometer, 

is found to be 4.25 millimeters. From the graph it is noted that 

17=100; therefore, the lower rock unit appears about 425 feet above 

the datum, or approximately 425 feet nearer the camera. For determining relative stratigraphic thickness of the beds in question, 9,575 

feet (which may be rounded off to 9,600 feet) will be assumed to 

be the value of H'. H' assigned for additional stratigraphic thick-
GEOLOGIC DATA DETERMINED WITH STEKEOMETEK INSTRUMENTS 55

ness measurements in the area must be determined with respect to 

the same datum.

It is not recommended that this method of assigning a value to 

Hf be applied to an area larger than that covered by one stereoscopic 

model because considerable variance in scale, due to fluctuation in 

height of the airplane above the terrane or to tilt in the photographs, 

may be present between photographs of the same and adjacent 

flight strips. It is not practical to compensate for differences in scale 

in this method of assigning a value to H'.

SUMMARY

The generalized procedures for obtaining quantitative geologic 

data with stereometer-type instruments are summarized as follows:

1. Locate and mark the photograph center and transferred center on 

each photograph of the stereoscopic pair. Carefully aline the 

photographs in such a manner that a straight line may be drawn 

between the photograph centers intersecting the transferred 

centers (fig. 14).

2. Locate and mark on the photographs the stereoscopic image points 

between which differences in elevation are to be measured. Tilt 

may be present in some stereoscopic models, and in order to 

minimize its effect, the points should be positioned as close 

together as possible, particularly when measuring stratigraphic 

intervals.

3. Compute adjusted photobase (aft) at the elevation of the lower of 

the two stereoscopic image points. If formula (1) can be used, 

an average of the photobases of each photograph of the stereoscopic pah* is sufficient.

4. Obtain applicable photograph scale at the lower stereoscopic image 

point. This step is very important when measuring stratigraphic 

intervals or differences in elevation. Any assumed scale is 

acceptable for determining strike and dip.

5. Determine Hf from the photograph scale obtained in step 4.

6. Measure Aj? with the stereometer and convert to AA, by applying 

formula (1) if the area is of low relief or formula (2) if the area 

is of high relief.

7. In determining strike and dip, correct for relief displacement and 

obtain the correct horizontal distance along the azimuth of 

true dip. The effect of the tilt of azimuth of true dip should 

be considered when measuring dips less than 6° or 7° (fig. 16).

8. Compute the angle of dip.

Unless compensatory steps are taken, two geometrical relationships 

of vertical aerial photographs may prevent obtaining quantitative 

data with stereometer-type instruments:
56 PROCEDURES AND STUDIES IN PHOTOGEOLdGT - ;;

1. Relief displacement: Relief displacement aff'ects horizontal distance in dip measurements.

2'. Tilt: The most serious effect of tilt is on the azimuth of true dip of 

low-dipping beds. The amount of image displacement due totilt is comparatively large in the corners of the stereoscopic 

model. 

Stereometers may be used to:

1. Obtain reliable differences in elevation between geologic features 

when applicable photograph scale is known.

2. Obtain approximate or relative differences in elevation between 

geologic features when only general photograph scale is known.

3. Determine the attitude of beds dipping less than 70°, regardless

of whether or not applicable photograph scale is known. 

Stereometers normally should not be used to:

1. Obtain reliable measurement of differences in elevation between 

geologic and topographic features when applicable photograph 

scale is not known.

2. Determine strike and dip when the horizontal distance along the 

azimuth of true dip is of insufficient length on the photograph. 

3- Measure dips greater than 70°. 

. Utilization of Stereometers is advantageous to the geologist because:

1. Stereometers are simple in construction and usage.

2. Stereometers are portable they may be set up and used in any 

field camp.

3. Stereometer techniques, if properly applied, provide reliable data.

4. Stereometric data can often be used to supplement certain phases

of fieldwork in the office.

The stereometer procedures described in the present paper apply toa wide variety of geologic and photogrammetric conditions. Modifications of these procedures generally may be made to accommodate 

the photogrammetric conditions and the geologic aspects of the 

individual problem.

LITERATURE CITED

American Society of Photogrammetry, 1952, Manual of photogrammetry:

Washington, 2d ed., 876 p. 

Elliot, D. H., 1952, Photogeologic interpretation using photogrammetric dip

calculations: California Div. Mines Spec. Kept. 15, 21 p. 

Tewinkel, G. C., 1952, Basic mathematics of photogrammetry, in American.

Society of Photogrammetry, Manual of photogrammetry: Washington, 2d

ed., p. 309-380.

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