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9NINIWd3i3Q AUVlBNVld S3d01S 
A Photometric Technique for 

Determining Planetary Slopes 

From Orbital Photographs

By W. J. BONNER and R. A. SCHMALL

CONTRIBUTIONS TO ASTROGEOLOGY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 812-A

Prepared on behalf of the National Aeronautics and 

Space Administration

A closed form, general solution for photometric slope 

determination has been derived by use of vector 

mathematics. The technique can be applied equally 

well to vertical and to oblique photography

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1973
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress catalog-card No. 73-600144

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

Washington, D.C. 20402 - Price 40 cents (paper cover)

Stock Number 2401-02409
CONTENTS

Abstract._______________________________________

Introduction __....___________.._..__...____.__._

Photometric function--_-_-__-______.____________

Derivation of coordinate systems________________

Coordinate transformation matrix._._.......__..

Summary of the procedure to compute the slope 

Conclusions.--._______ _ _________________________

References....-.___.___..__.____..._._____..____

Page

Al

1

2

3

10

14

15

16

ILLUSTRATIONS

FIGURE 1 .

2.

3. 

4-12.

Page

Diagram showing geometry for photometric slope determination ______-__-___----_----_--_-_------------------ A2

Diagram showing phase-plane nomenclature depicting relations between brightness longitude and phase

--.----------------------------------------------------------- 2

Graph showing lunar photometric functions------------------------------------------------------------------ -------------------------------------------------------------- 3

Diagrams showing:

4. Moon-center coordinate system_-------_--___---_--_--_------------------------------------------------ 4

5. Right-handed orthogonal system-_-------------------__----__ --------------------------------------- 5

6. Parameters for determining the phase angle g.. .....................-.-..---.----.----.---.----.----- 5

7. Relationships between the moon-center coordinate system and the moon-surface coordinate system. . 7

8. Image-plane coordinate system and the definition of the angle £-. ------------------------------------ 8

9. Procedure for evaluating the third image-plane coordinate Z ....-....--..--.--.-.-- .---.-----.---- 9

10. Image-plane coordinate systems------------------.--------------------------------------------------- 10

11. Geometric quantities in the plane described by the points S, O, and the center of the moon__-_---..-. 11

12. Relationships between the e'z vector and the e vector------------------------------------------------- 12

SYMBOLOGY

Greek Symbols

a Brightness longitude.

X Longitude coordinate on the lunar surface.

X,, Longitude of the optic-axis intercept at lunar surface

(X. = X T ).

X r Longitude of a point north of optic-axis intercept. 

X, Longitude of the spacecraft. 

X 0 Longitude of the subsolar point. 

/3 Latitude coordinate on the lunar surface. 

/3,, Latitude of the optic-axis intercept at lunar surface

(/8» = /8 r ).

/3r Latitude of a point north of optic-axis intercept. 

/3» Latitude of the spacecraft.

/30 Latitude of the subsolar point.

<& Photometric function.

t|/ Slope of the lunar surface measured in the phase plane. 

£ North deviation angle in the image plane. 

T) Coordinate on the lunar surface (positive toward the east). 

V Image-plane coordinate of the lunar surface. 

£ Coordinate on the lunar surface (positive toward the north). 

£' Image-plane coordinate of a point on the lunar surface. 

o> Angle between the mean lunar surface and thee z> direc- 

	tion. 

T Point due north of optic-axis intercept (O).

in
IV CONTENTS

Vectors

e Unit vectors denning a coordinate system,

ei Base vector in the X or (17 + £) direction.

e-2 Base vector in F or (17 + £) direction.

e3 Base vector in the Z direction.

e. Base vector in the Z direction.

v Vector.

v,t Vector from the moon's center to the optic-axis intercept.

v0 Vector from the moon's center to the subsolar point.

vs Vector from the moon's center to the spacecraft.

v,, Vector from the moon's center to an arbitrary point on

	the lunar surface.

v,,s Vector from the arbitrary point p to the spacecraft.

v Unit vector as denned above.

VH Unit vector as denned above.

v0 Unit vector as denned above.

vs Unit vector as denned above.

v.,, Unit vector as denned above.

vl>s Unit vector as denned above.

vt Unit vector tangent to the local surface in the phase

	plane (also the slope of the lunar surface). 

xt Vector component in the direction of e, vector, where i can

	take the value 1, 2, or 3.

Coordinates

X, Y, Z Lunar-centered coordinates:

X is through the mean libration center.

Z is north pointing (collinear with the axis of

rotation).

Y completes the right-hand system. 

Z Vertical coordinate on the lunar surface. 

Z' Image-plane coordinate along the optic axis. 

7), £, Z Moon-surface coordinates. 

7)', t,\Z' Image-plane coordinates.

Other Symbols

A Altitude of the spacecraft above the lunar surface.

E Emitted ray (vector) to the camera.

/ Focal length of the lens.

g Phase angle.

7 Incident ray (vector) from the sun.

w Magnification factor.

R,n Radius of moon.

Rs Slant range along optic axis to optic-axis intercept.

S Spacecraft.

O Optic-axis intercept with the lunar surface.

SS Subspacecraft point on lunar surface.

Ti Coordinate transformation matrix.
CONTRIBUTIONS TO ASTROGEOLOGY

A PHOTOMETRIC TECHNIQUE FOR DETERMINING 

PLANETARY SLOPES FROM ORBITAL PHOTOGRAPHS

By W. J. BONNER and R. A. SCHMALL

ABSTRACT

Interest has been recurrent in applying photoclinometry 

(photometric technique for slope determination) to determine 

lunar slopes and heights where stereoscopic coverage was not 

available. Some scientists involved in the planetary program 

foresee the eventual application of the technique to such 

planets as Mars. As a result, a new approach to the problem 

by exclusive use of vector mathematics has been developed. 

The solution is generalized and can be applied equally well to 

extract slope information from either vertical or oblique 

photography. A summary of the procedure used for computing 

the slope is given for use in adapting the technique to computer 

programing.

INTRODUCTION

Photoclinometry is the process of relating the 

measured scene brightness, the viewing and 

lighting geometry, and the photometric function 

of a surface to obtain slope information. Historically, the technique was developed by van 

Diggelen (1951) to study mare ridges. Dale (1962) 

extended the technique to a generalized study of 

the topography of the maria. Wilhelms (1963) 

refined the technique and was first to apply it 

successfully to other types of lunar terrain. 

McCauley (1965) expanded the work to produce a 

detailed quantitative terrain map of the lunar 

equatorial belt. Finally, Watson (1968) established a rigorous mathematical foundation for 

the process and extended the technique to an 

analysis of normal photography from spacecraft 

imagery. The present paper expands the work to a 

generalized solution for determining roughness 

of terrain from spacecraft imagery.

The method is based on the assumption that the 

brightness of an element of the lunar surface is 

a function of the normal albedo, one component 

of the surface, and the sun's angle of elevation. 

Normal albedo is defined as the ratio of the emergent light to the incident light observed at zero 

phase angle. If the point-by-point variation oi 

the normal albedo on the lunar surface is eliminated, the brightness of an element then depends 

only on the slope and the sun's angle of elevation, 

which are expressible as differences in lunar longitude near the equator. Figure 1 is an illustration 

of the relationship. The figure gives a polar view 

of the moon. The surface elements P and Q have 

the same albedo, and as a result of their similar 

angular relationship to the sun, they have the 

same brightness.

The angles POQ and DPE are equal, and the 

angle DPE is equal to the angle ABC, which is the 

east-west component of the slope for the element 

at P. The angle ABC can be expressed in terms of 

the difference in lunar longitude between the elements P and Q, or the difference in the sun's angle 

of elevation for measurements made near the 

equator. Thus, if two surface elements, separated 

in an east-west direction, have the same albedo 

and the same measured brightness, the slope of 

the surface at P is equal to the difference in lunar 

longitude or the difference in the angle of sun 

elevation between the elements (McCauley, 1965). 

Stated another way, if the brightness of the ridge 

at P is equal to the brightness of the sphere at a 

given distance from P (for example, Q), then the 

slope of the surface at P with respect to the sphere 

at P is equal to the angle i/» = POQ. The same is

Al
A2 CONTRIBUTIONS TO ASTROGEOLOGY

FIGURE 1. Geometry for photometric slope determination. P and Q, surface elements.

true if P and Q are points of equal photographic 

transparency (van Diggelen, 1951).

PHOTOMETRIC FUNCTION

Use of the technique of photoclinometry to determine the roughness of the lunar surface requires that the reflection characteristics of the 

surface be taken into consideration', for the reliability of the data obtained from photoclinometry 

is primarily dependent on the precision to which 

the lunar photometric function is known.

From the work of Minnaert (1961) it is known 

that the functional form of the lunar photometric 

function has an interesting property (to within a 

precision of a few tens of percent): the brightness 

at any point, corrected for the normal albedo, is 

solely dependent on the brightness longitude and 

the phase angle.

Figure 2 represents the intersection of the 

phase plane with the lunar surface. The figure 

shows the path of an incident ray from the sun 

(/) and the ray emitted to the spacecraft the 

emergent ray (E\ The angle between the incident 

and emergent rays, measured in the plane containing both rays, is called the phase angle, g. 

Note that when the plane is tilted about the surface point, the phase angle remains constant. The 

angle measured in the phase plane between the

emergent ray and the normal to the surface is denned as the brightness longitude, a. (The brightness longitude is denned positive when the projection of the normal into the phase plane is on the 

opposite side of the observer line from the location 

of the sun line, and negative when the projection 

lies between them the only two possibilities.)

If the plane is rotated about the line of intersection with the surface, the brightness longitude 

remains constant. However, if the plane is tilted 

along an axis at a right angle to the line of intersection, the brightness longitude varies. Thus, the 

brightness at any point is a function of the slope 

component in the phase plane, the normal albedo, 

and the phase angle (Watson, 1968).

According to Watson (1968), two accepted photometric functions have been derived from earthbased observations. The first is that of Hapke 

(1963, 1966), which was derived by parametric 

fitting from theoretical considerations of a scattering model of the lunar surface. The second photometric function is based entirely on lunar photographic data and was derived entirely empirically 

by Herriman, Washburn, and Willingham (1963) 

and was later revised by Willingham (1964). 

Figure 3 shows the two functions in graphical 

form. In the figure, brightness is plotted as a 

function of the brightness longitude, a, and the 

phase angle, g. As Watson (1968) pointed out, a 

major limitation in constructing a photometric 

function from terrestrial observations is that a 

particular lunar feature cannot be observed for a 

wide variety of earth-angle observations. The

Incident ray from sun 

(I)

Normal

/ Projection of normal 

in phase plane

LUNAR SURFACE

FIGURE 2. Phase-plane nomenclature depicting relations 

between brightness longitude (a) and phase angle (g).
PHOTOMETRIC TECHNIQUE FOR DETERMINING PLANETARY SLOPES A3

problem is primarily that a libration in longitude 

of the moon of only ±8° makes data on a single 

feature available only over a narrow range of 

brightness longitudes. Another serious problem 

with obtaining the photometric function terrestrially is that scientific data are not available on 

its dispersion characteristics. R. L. Wildey (unpub. 

data) has devised an ingenious method for determining these dispersion properties, and it is hoped 

that this experiment will eventually be carried 

out.

DERIVATION OF COORDINATE SYSTEMS

In presenting this mathematical process of 

using photometric data to evaluate the slope, it is

1.4

1.2

EXPLANATION 

Hapke (1963) 

Hapke (1966)

Herriman, Washburn, and 

Willingham (1963)

Willingham (1964)

0=0°

0.8

0.6

0.4

0.2

90 60 30 0 30 

BRIGHTNESS LONGITUDE (<*), IN DEGREES

60 90

FIGURE 3.-Lunar photometric functions. Brightness (normalized to unity at#=0°) is plotted as a function of 

brightness longitude (a) and phase angle (#). From Watson (1968).
A4 CONTRIBUTIONS TO ASTROGEOLOGY

convenient to define several coordinate systems. 

To minimize confusion, each coordinate system 

will be defined when it is needed in the mathematical development. In addition, the nomenclature has been selected to eliminate ambiguity.

The physical phenomenon that forms the basis 

for photoclinometric terrain analysis is that a 

functional relationship exists between the brightness, the phase angle, and the brightness longitude. Thus, if any two of these quantities are 

known, the third is also known. Figure 2 depicts 

the phase plane geometry. As shown in the 

figure, if the phase angle g and brightness 3? are 

known, the brightness longitude a can be evaluated. The scene brightness can be determined by 

relating the measured film density to brightness 

through the film H and D curve. The phase angle 

is obtained from the position geometry of the point 

of interest on the film, the lunar time, and the 

season. The problem is thus reduced to evaluating 

g from the image information to permit the evaluation of a. Then a and the image information are 

used to evaluate the slope of the surface.

To determine the phase angle, use was made of 

a coordinate system whose lines intercept at the 

center of the moon. Figure 4 shows this mooncenter coordinate system. The Z axis goes through 

the north pole, the X axis is located through the 

mean libration plane, and the Y axis completes

FIGURE 4. Moon-center coordinate system.

a right-handed orthogonal coordinate system. 

(See fig. 5.)

Figure 6 illustrates the geometry of the phase 

plane projected onto the plane containing the 

origin (center of the moon), the subsolar point, 

and an arbitrary point of interest, p, on the lunar 

surface.

The following subscripts are used to denote 

position vectors (these vectors originate at the 

origin of the coordinate system):

Subscript

o

Position vector

Subsolar point (on the lunar surface). 

Arbitrary point (on the lunar surface).

Spacecraft (at an altitude A above lunar 

surface).

In addition to the above notation, an overscribed 

caret represents a unit vector and an overscribed 

line denotes a vector; thus 

v = unit vector. 

v= vector.

From figure 6, the phase angle is found from the 

vector relationships

(1)

where

Also,

and

A Vn v = -zr

Vps

VpS = Vs Vp.

(2)

From figure 4, the latitude and longitude are 

used to locate positions on the lunar surface. The 

vector representation in rectangular coordinates

is

vp = cos /3p cos \p i 4- cos ftp sin \p j 4- sin /3P k, (3)

where i, 7, and k are unit vectors in the X, Y, and

Z directions.

Similarly,
PHOTOMETRIC TECHNIQUE FOR DETERMINING PLANETARY SLOPES 

Z

A5

FIGURE 5. Right-handed orthogonal system. A righthanded system means that if one rotates OX into 

OY, then OZ will lie in the direction in which a righthand screw would advance. The lines OX, OY, OZ 

indicate the positive directions of the coordinate

v0 = cos /30 cos X 0 i + cos /3G sin X0j + sin /30 k (4) 

= cos ps cos Xs i + cos ps sin \s j + sin /3s A\ (5)

^o 

Vs

We note that

Vps = Vs Vp = (R m + A) Vs R m Vp. (6)

Substituting equations 3 and 5 in equation 6, we 

obtain

VpS = Rm\ ( 1 +B~ ) COS /3S COS \s ~ COS /3p COS Xp \i

Ml +77~

L \ R m

cos /3« sin \s cos ftp sn

J

sn ^ -

Also,

 V

A\ I 2 

- j cos /3s cos Xs cos pp cos Xp

A \ I 2 

+ | ( I+T; ) cos/3s sin X s cos/3P sin Xp

-ftm/ J

2-i 1/2

[cos /3scos /3Pcos (Xs Xp) + sin /3ssin (8)

Using equations 4, 7, 8 in 1, we now determine 

the phase angle.

Vps

cos g = p_

:. cos r =

/ A \ (

1 1 H ) \ [cos/30cos /3S cos (Xs XQ) + sin /3e sin /3S ]

V R m ' l

- [cos /3 0cos /3Pcos (Xp Xo) + sin /3Q sin /3P]

}

cos (X« Xp) + sin /3S sin /3p] [

1/2

(9)

Incident ray

0 [Origin of moon-centered coordinate system]

FIGURE 6. Parameters for determining the phase angle g. 

p, arbitrary point on the lunar surface.
A6 

where

CONTRIBUTIONS TO ASTROGEOLOGY

/30 , X0 are the latitude and longitude of the

subsolar point. 

/3S , Xs are the latitude and longitude of the

subspacecraft point, (also latitude

and longitude of the spacecraft). 

/3P , Xp are the latitude and longitude of an

arbitrary point of interest on 'the

lunar surface. 

R m is the radius of the moon.

The quantities X0 , /30 , Xs, /8S can all be determined 

from the mission support data. R m is a constant 

over the area of interest, and /3P, Xp can be determined from the location of R m in the image plane. 

The method by which /3p, Xp are determined will 

be discussed later.

Referring to figure 2 the vector vt represents the 

local tangent. Therefore, the slope of vt is the 

local slope of the terrain. Since vp is a unit vector 

in the radial direction (that is, points to the local 

zenith, see fig. 6), the slope of the terrain can be 

found from the relationship

cos (90 i//) = Vt ' vp , (10) 

where <|/ = the slope of the terrain.

The unit vector vt is determined as follows. By 

defining vg as a unit vector normal to the phase 

plane, then

_V0 X Vps _ V0 X Vt

Also,

V

X

sin g sin (90 a + g) '

X Vvs) _ V0 X X

sin g sin (90 a g)' (12)

Expansion of equation 12 by means of the triple 

vector product identity yields

(Vo ' Vus) Vo ( Vo ' Vo) Vys _ (^o ' Vt) Vo ~ (^o ' ^o) Vt

sin g sin (90 a g)

(13)

Note that

v0 * vPs = cos g (which is equation 1)

vt = cos (90 a + g) = - sin (g a).

Substituting the above set of equations into equation 13 and solving for vt, we obtain

(g a)]

Vps\L

cos (9 - a)1 : .

sm g J

(14)

The slope of the terrain may now be determined 

from equation 10. We have

sin i// = cos (90 i/;) = vp (15)

Substituting equation 14 into equation 15 we 

obtain

sin i// = [sin (g - a) + cot g cos (g a)] v0 vp

cos (g a) I 

sing J

. ""' 1"- <16>

From equations 3 and 4 we note

v0 -vp = cos /30 cos /3p cos(X - Xp) + sin /30 sin (3P .

From equations 2 and 3 and 6 we have

(17)

\VpS \IRr

 Vu =

\V,ts\IR

(18)

From equations 3 and 5 we have"

vp vs = cos /3p cos /3S cos (Xs Xp) + sin pp sin /3S .

(19)
PHOTOMETRIC TECHNIQUE FOR DETERMINING PLANETARY SLOPES A7

Substitution of equations 8 and 19 in equation 18 

yields

vps - vp =

I l

cos &p cos &s cos ^' ~~

-I- sin j3P sin /3S 1 1

cos (\s \p)+ sin sin /3P |

1/2

(20)

.-. The slope of the surface may be elevated by 

substituting equations 17 and 20 into equation 16. 

Note, however, that the quantities /3P and \p are 

still undetermined.

In order to evaluate /3P and \p a second coordinate system is now defined. Figure 7 defines a 

curvilinear coordinate system which is centered 

at the camera optic axis intercept with the surface. The coordinates 17 and £ are curvilinear and 

coincide with lines on constant longitude (north 

pointing positive £) and constant latitude (east 

pointing positive 17). The Z coordinate is in the 

direction of the local vertical with positive up. 

(Note that we are dealing with two different coordinate systems as defined in the nomenclature. 

Also note that this is the second instance that the 

letter Z has been used in defining a system 

coordinate.)

Assume that the distances involved are small 

compared to the moon's radius. Then the arbitrary 

point p can be approximated as follows:

 'ro = m cos

Also,

Rearranging the two equations and noting that 

170 and £0 are defined as the center of the coordinate 

system and are therefore equal to zero, we have

Rm cos (21)

and

R, (22)

where /30 , X0 = optic axis intercept with the lunar 

surface. Figure 8 represents the image plane coordinate system. The origin is located at the intersection of the optic axis with the image plane. In 

other words, 170 and & is the image of the optic axis 

intercept with the lunar surface. The third coordinate Z' is parallel to the optic axis and is positive 

in the direction toward the moon.

Figure 8 also defines the angle £. This angle is 

between the image north (the positive £' axis) and 

the image of north pointing ft0 (the north-pointing 

longitude through the optic axis intercept with the 

lunar surface).

Therefore, if the r)'p , & coordinates of the image 

plane can be transformed into the T)P, £P coordinates 

of the lunar surface, the local terrain slope can be 

evaluated as follows:

1. Equations 21 and 22 are evaluated for (/3P, XP).

2. With this information, equation 9 is evaluated

FIGURE 7. Relationships between the moon-center coordinate 

system and the moon-surface coordinate system, p, arbitrary 

point.
A8 CONTRIBUTIONS TO ASTROGEOLOGY

3. a is now determined from the photometric 

function.

4. Using this information and equations 17 and 20 

in equation 16, the slope can be determined.

The position vector vop in the image plane coordinate system (17', £',Z') is

(23)

where

R 

m= magnification factor = jRs = slant range from the camera to the optic 

axis intercept with the lunar surface. 

/= camera focal length.

Equation 23 permits the evaluation of the 

vector, VOP, in the image plane coordinates (that 

is, except for Z'p which will be discussed later). 

Therefore, a transformation matrix between the 

lunar surface coordinate system and the image 

plane coordinate system will solve the problem.

Since a vector is an invariant, a vector (v) may 

be represented in two coordinate systems. (Recall 

that an invariant is an expression involving the 

coefficients of an algebraic function which remains constant when a transformation, such as 

translation or rotation of coordinate axes, is 

made.)

where x\ is the vector component in the direction 

ei and i can take the values 1, 2, or 3.

To lunar north

FIGURE 8. Image-plane coordinate system and the definition 

of the angle £

Dotting ei into the above equation we obtain

Solving equation 24 for 

1, we have

and noting that et & =

Solving the above equation for Ttj, we obtain

Tij =ei -e'j . (25)

The elements of the matrix Tt j are the cosines 

of the angles between the coordinate axes. That 

is, 1*33 is the cosine of the angle between the Z 

and Z' axes, and Tis is the cosine of the angle 

between the 17 and Z' axes, and so forth.

The transformation matrix is now defined and 

some generalizations are possible with respect 

to equation 23. These are

1. The distance of concern on the lunar surface 

is small compared to the slant range.

2. The lunar surface is assumed to be flat over the 

distances involved.

In figure 9, the two coordinate systems under 

consideration have been drawn with a common 

origin. From this illustration the following can be 

seen:

.

i an (t) 

(e3 Xe3 )]

(26)

Expanding equation 26 using the triple vectorproduct identity (summary of the relationships 

used in this report given in Coburn (1955) and 

Kaplan (1959)), we obtain

tan <o =

(e3 -

(27)

where

7*33 = 63 63-

Noting that el ej = 0 for i ^ j and performing the 

vector dot product, we have

tan w =

 7*33 (Tsi TJp + 7^32 £p ~t~ Tss Zp) + Zp

(28)
PHOTOMETRIC TECHNIQUE FOR DETERMINING PLANETARY SLOPES A9

plane

Intersection of the optic-axis 

and lunar-surface coordinate 

systems origins

FIGURE 9.-Procedure for evaluating the third image plane coordinate Z'. 

Note: The plane of the figure is the ez ez< plane.

From figure 9, we note

cos (cos- 1 T33 - 90) - sin (cos- 1 r33 )

tan &> =

sin (cos-^33-90) cos (cos~l T33 )

or

tano» = (29)

Substituting equation 29 into equation 28 and 

solving for Zp, we obtain

Let,

It is now possible to evaluate TJP and ^ by knowing 

-' and .

^= w [r,nj;+ r,2 g+ r23z;] (sib)

Substituting equation 30 into equation 31a and 

31b, we obtain

g] (32)

^] (33)

Equations 32 and 33 can be substituted into 

equations 21 and 22 to obtain /3P , kp . ft,, and \,, can 

then be used to determine g in equation 9. With this 

information the brightness longitude («) can be determined from the photometric function. And then
A10 CONTRIBUTIONS TO ASTROGEOLOGY

using equations 16,17, and 20, the slope of the terrain can be determined. The last step remaining 

then is the evaluation of the transformation matrix (Tij) which is determined in the following 

section.

COORDINATE TRANSFORMATION MATRIX

To facilitate the evaluation of the coefficients of 

the transformation matrix, this matrix will be 

evaluated in two steps. Recall that if three coordinate systems exist such that

then,

where

(34)

jk

Therefore, defining an image plane coordinate 

system (V, £', Z') such that £ = 0 (see fig. 10) also 

defines a transformation matrix such that

i }j I ijXj, (35)

where

That is,

Z=T'31 7)"+T32C'+T'33Z'.

(36a) 

(36b) 

(36c)

Since £ = 0, the £"> Z' and £ axes lie in the same 

plane and the 17" axis is perpendicular to the £" and 

Z' plane and is therefore perpendicular to the £ 

axis; also

(37a)

f

FIGURE 10. Image-plane coordinate systems.

T' L =0 21 v. (37b)

Now,

Referring to figure 11 and applying the law of 

cosines, we have

Solving for TM, we obtain

(38>

From figure 12, we note that

XT

Vso ' Vr

_ _ 

Vio Vro

(39)

where r is defined as a point due north of O 

(XT =Xo).

NOTE. Recall that single subscripted vectors 

refer to position vectors relative to the center of 

the moon. 

As previously defined, we have

v 0 = Rm [cos /So cos K0ex + cos /80 sin \0 eu + sin /30e*] 

vr = Rm [cos /8T cos \T ex + cos /8T sin \^ey + sin /8Te^]
PHOTOMETRIC TECHNIQUE FOR DETERMINING PLANETARY SLOPES All

\vro \ = Rm(Pr-po), for (ft-ft,) small. (43) 

We note the following relationships

vg = (Rm + A) [cos ft cos X,ex

+ cos ps sin Xs ey + sin

 ' V«o = [ (A + #m ) COS ft COS kg Rm COS ft, COS X0] «x

+ [ (A + Rm ) cos @s sin X, flm cos j80 sin \0 ] e tf

+ [(A + #m ) sin ft-#m sin ft,] e* (40)

VTO = Rm [ (COS ft COS ft,) COS X0ex

+ (cos ft cos j80 ) sin \0&v + (sin ft sin ft,) 02 ] (41)

Also

\Vso\ = Rg (42)

Center of the moon

FIGURE 11. Geometric quantities in the plane described by 

the points S, O, and the center of the moon.

sin (0T - 0«) =

(cos ft cos Po) = cos [ft + (ft ft) ] cos ft *

= cos ft cos (ft ft,) sin ft 

= sin (ft Po) cos Po 

that is,

ftlimft, (cos ft-cos PO ) = -(PT-PO) sin ft,. (44) 

Similarly we have

lim , (sin ft-sin ft) = (ft-ft) cos ft,. (45)

Substitution of equation 40 and 41 into equation 

39 yields

TM = lim \ _ 1 {[ (A + Rm ) cos ft cos X,

 Rm cos Po cos X0 ] cos X0 (cos ft cos ft)

+ [(A + Rm) cos ft sin \s Rm cos ft, sin X0 ]

sin X0 (cos ft cos ft) + [(A + Rm ) sin ft

 Rm sin ft] (sin ft sin ft)}

In the limit (and from equations 44 and 45) this 

becomes

23

[cos ft, cos (X« X0 ) sin ft sin ft cos ft,]. (46)

Equations 37, 38, and 46 provide three (7^, 

TL, TL) of the nine elements of the transforma- oo' it> '

tion matrix. Properties of the elements of the 

transformation matrix are

T" 2 _|_ T" 2 _|_ T" 2 = 1

1 11 ~ l 12 ~ l 13 A (47)
A12 CONTRIBUTIONS TO ASTROGEOLOGY

FIGURE 12. Relationships between the e, vector and the

vector.

T" 2 _1_ T" 2 4- T" 2 = 1

J 21 ' * 22 ' * 23 x

T" 2 _|_ T" 2 _L T" 2 = 1

-* 31 ~ * 32 ~ J 33 x

r

; /TV _i_ rni /TV _i_ /TV /TV f\ 

II 1 12 ~r '£ 21'£ 22~r ^ 31^32 "

r

' T" -L T" T" 4- T" T" 0 II 1 13 ~" '*21't 23~r * 31*33 "

r

/ /TV _i_ rni rru _ _ /Tff /Tf> _ n 

12'£ 13~r '*22't 23~r '£ 32'£ 33 "

(48)

(49)

(50)

(51)

(52)

Therefore, having evaluated three of the elements previously and having six independent 

equations defining the properties of the elements 

of the matrix, it is now possible to evaluate all 

nine elements of the transformation matrix.

From equation 48, since

TZI = 0 (equation 37) 

then

(53)

NOTE. It is possible to express equations 47 

through 49 in the form

This is not an independent equation but is implicit from equations 47 through 49. Therefore,

r

' x/1 _ T' 2_ T' 2 13 VI J. 33 J. 23 (54)

Dividing equation 50 by equation 51 and noting 

that

T'2l = 0 (equation 37)

then

/TV rni rni rni

* 11 * 12 _ * 31 * 32 

rni rni rni rni

 * 11-* 1Q * Q1 * QQ

rni __ rni 32

1 12 ~~ -* 13 T'

J. 

(55)

33

Substitution of equation 55 in equation 52 yields

(56)

__ /TV /TV

T" _ L 22 1 23 _ * »»

13

Using equations 54 and 56 in equation 55,

jf72

-J ____J .33 /Ttf .

1 1 _ T" 2 * 23

A J 23

From equation 47, we obtain

(57)

r

' = \/1 T" 2 T" 2 11 vj. 1 12 J. 13 ,

which using equations 54 and 57 can be written as

rni

L 33

r11 ' = (58)

Using equations 49 and 56, this equation becomes

V

/Tt/ 2 

1 _ -t 33 

-r' (59)

Thus, all nine elements of the T£ transformation 

matrix have now been evaluated. The elements 

are given by equations 37, 38, 46, 53, 54,56,57,58, 

and 59. It is now possible to determine Xi from the 

X coordinates.
PHOTOMETRIC TECHNIQUE FOR DETERMINING PLANETARY SLOPES A13

Recall from equation 35

Therefore, if the coordinates x-' (for £=0) can be 

transformed from the #4 coordinates (^ and 4') the 

problem is finished. Referring to figure 10, we note 

that

Dotting e'J into the equation above, we have

2 *; <*;*;>= 2 tf-^n)-

Noting that e". e".= 1, we obtain

«; = 20; -IK-S

k k

or

Tjk = e]-el (60) 

Note the following: 

1. In the £=0 and x' systems, Z' is coincident,

2. Since V' and £" are perpendicular to the Z' axis, 

Hi = n2 = 0 (62a) 

and

r" T" ft fA9Vk\ 13 * 23 " \O£D)

Again referring to figure 10, we have

T"2 = cos (90 4- ^)=-

(63)

(64)

(65)

(66)

In matrix notation, we have

cos £ sin £ 0

sin £ cos £ 0

0 01

Then from definition,

jk

or

and

__ ^ri rjjt frtff

which in matrix notation is

rifc=

r

' 11

T'

 *21

«,

r

f Ffl t 

12 -« 13

rrti rrif

-t 22 -» 23

r3'2 «,

cosf -sinf

sin ^ cos £

0 0

0

0

1

Listing the elements we have 

!Tii = cos £7Vi + siE

/TT _ fWJ t

T2l = cos£T2l + sin£T22 

T22 - - sin £ r2', 4- cos £ T2'2

32

(67)

(68)
A14 CONTRIBUTIONS TO ASTROGEOLOGY

T3Z = - sin £ 7*3, -I- cos £ T&

1 33 = 1 33

Where the 7y elements are given in equations 

37, 38, 46, 53, 54, 56, 57, 58, and 59. Making these 

substitutions, we obtain

- cos f r!!3

_ rjji 2 

 -I 23

_ rjjt 2 

 -I 23

33

7*23 = 7*23

(69)

(70)

(71)

(72)

(73)

(74)

(75)

_ rrii \ 2 __ TI

± 23? L 3J

7" T'

 * 23-1 33

l T'i A -'ss

7*33 

33

(76)

(77)

It is now possible to compute TI V and £p using 

equations 32 and 33 and thus the slope. A summary 

of the equations to be used in calculating the slope 

follows.

SUMMARY OF THE PROCEDURE TO 

COMPUTE THE SLOPE

This section is presented to encourage the eventual programing of the technique developed in this 

paper. The summary is therefore arranged in the

order in which a computer program would be 

written.

1. From the support data we have the following 

parameters:

R m Radius of moon, constant.

Rs Slant range along optic axis to intercept 

with surface.

A Spacecraft altitude. 

ps , As Latitude and longitude of spacecraft. 

j8u, A Latitude and longitude of subsolar point. 

p0 , A0 Latitude and longitude of optic axis intercept with surface.

2. Determine the values of 7T23 and T^> 

From equation 46,

7*2 = [cos fts cos (As - A0) sin #>- sin ps cos p0]

and from equation 38,

3. Determine £, 17^, & from the photograph.

4. Generate the transformation matrix

whose elements are given in equations 

69 through 77:

= cos

7*12 = - sin

7" 2

1 33

_ T" 2

-* 23

1-

T" 2 1 33

1 _ T" 2

A * 23

= cos

^23 =
PHOTOMETRIC TECHNIQUE FOR DETERMINING PLANETARY SLOPES A15

TSI = COS 1 -

"" 2 

33

- sn

/TT; /TT>

 * 23 * 33

23

T' T'

1 23 L 33

d-^23)2

7T33 = r33

5. Determine the brightness from the density data 

and the H and D curve.

6. Using j]'lt and £p determine #, a, and i|/.

(a) To determine g, we use equation 9.

cos a =

+--5 ) Ucos fi cos fis cos (As A ) + sin fi- sin Bs]

K m / [

 [cos ft cos /3,, cos (Ap A )+ sin (3 sin )8/J |

Rm / \ Rm

[cos /3S cos /3,, cos (X s \,,)+ sin )8 S sin )8 ,j| 1/2

where /3/>, X/j are the latitude and longitude of an arbitrary point of interest on 

the lunar surface and are determined 

from equations 21 and 22.

Rm COS /3,,

T),, and £,, can be determined from equations 32 and 33.

m[(Ttl + 

where

_ T-2 ( 1 _ T«2 \

J :{:» I l * :\:v

(b) To determine a, we use g and the brightness value.

(c) Using a, we now determine the slope 

from equation 16.

sin ty = [sin (g a) + cot g cos (g a)]

cos -

where v0 vp and v 

equations 17 and 20.

are determined from

cos cos (X 0 \p) sn

I ( 1 4- -p- ) cos & cos )8 S cos (As Ap)

4- sin 8P sin s \ 1 1

cos (AS . sn sn 1/2

Using the equations summarized in this section, 

it is now possible to compute the slope component 

of the surface in the phase plane.

CONCLUSIONS

A new generalized derivation of an operational 

technique for extracting terrain slope information from planetary surfaces photometrically has 

been developed. The solution is in a general form 

and can be applied to an analysis of either vertical 

or oblique photography. Although the solution is 

intended primarily for deriving topographic data 

for the lunar surface, it can also be applied to 

obtain information about other planetary surfaces 

if used within the limitations of the basic 

assumptions.

During the derivation the authors made two 

assumptions: (1) the mean lunar surface is flat
A16 CONTRIBUTIONS TO ASTROGEOLOGY

rather than spherical over the area of interest, 

and (2) distances on lunar surface were small as 

compared to the radius of the moon.

Finally, the summary of equations should serve 

as a useful tool to those interested in adapting 

the technique to a digital computer.

SELECTED REFERENCES

Coburn, Nathaniel, 1955, Vector and tensor analysis [1st ed.]: 

N.Y., Macmillan Co., 341 p.

Dale, E. D., 1962, The application of the van Diggelen method 

of slope analysis to lunar domes and wrinkle ridges: Manchester, England, University of Manchester, Thesis, p. 46.

Diggelen, J. van, 1951, A photometric investigation of the 

slopes and heights of the ranges of hills in the maria of 

the moon: Netherland Astron. Inst. Bull., v. 11, p. 283-289.

 1959, Photometric properties of lunar crater floors: 

Utrecht Observatory Recherches Astronomy, v. 14, no. 2, 

114 p.

Hapke, B. W., 1963, A theoretical photometric function for the 

lunar surface: Jour. Geophys. Research, v. 68, no. 15, 

4571-4586.

 1966, An improved theoretical lunar photometric function: Astron. Jour., v. 71, no. 5, p. 33-339.

Herriman, A., Washburn, H., and Willingham, D., 1963, Ranger 

preflight analysis and the lunar photometric model: 

California Inst. Technology Jet Propulsion Lab. Tech. 

Report 32-384.

Kaplan, Wilfred, 1959, Advanced calculus [5th ed.]: Reading, 

Mass., Addison-Wesley Pub. Co., Inc.

McCauley, J. F., 1965, Terrain analysis of the lunar equatorial 

belt: U.S. Geol. Survey open-file report, 44 p.

Minnaert, M., 1961, Photometry of the Moon, in Kuiper, G. P., 

and Middlehurst, B. M., eds., Planets and satellites, v. 3 

of The solar system: Chicago Univ. Press, p. 213-245.

Watson, Kenneth, 1968, Photoclinometry from spacecraft 

images: U.S. Geol. Survey Prof. Paper 599-B, 10 p.

Wildey, R. L., and Pohn, H. A., 1964, Detailed photoelectric 

photometry of the Moon: Astron. Jour., v. 69, no. 8, p. 

619-634.

Wilhelms, D. E., 1963, A photometric technique for measurement of lunar slopes: Studies for space flight program: 

Astrogeol. Studies Ann. Progress Report, August 1962 

to July 1963, U.S. Geol. Survey open-file report, p. 1-12.

Willingham, D., 1964, The lunar reflectivity model for Ranger 

block III analysis: California Inst. Technology Jet Propulsion Lab. Tech. Report 32-664.

*U.S. GOVERNMENT PRINTING OFFICE: 1973 543 576/12