Lunar Heights From Shadows AutomaticallyBYRON Vv. RUFFIN*

A eronautical Chart and Information CenterSt. Louis, Missouri 63118

A~STRACT: In lltnar charting, the largest portion of information for relativel~e~ghts ~f features comes from Ihe reduclion of the measured lengths of shadow~mages ~n lunar ~ho~ographs. The magnitude of this reduction program is seenu;. ~ new perspe~twe ~n terms of automatic measuring instruments and computers.7 Ins program ~s executed through solutions using analog, digital, and hybridsystems.

INTRODUCTION

THE PROCEDURES of charting the surface ofthe Moon are complicated by the fact

that most of the observation data must becollected from a distance of almost a quarterof a million miles. This problem is partictl­larly acu te in regard to the topographic de­scription of individual lunar features where itis necessary to develop accurate contours forthe delineation of relief. The key solution tothis problem has been the exploitation ofoblique illumination phenomena, particularlythe shadows cast by features at low Sunangles. Astronomers have measured thiseffect for over 100 years to estimate theheights of lunar mountains and the depths ofcraters.

The current program of production of lunarcharts by the Air Force Aeronautical Chartand Information Center has demanded thatthese shadow reduction techniques be refinedboth in accuracy and efficiency. Most of thisrefmement has taken place through improvedmeasurement and reduction of lunar photo­graphs. Since an en tire hemisphere of theMoon can be observed from a station onEarth, a lunar image may contain quite ala rge area wi th a wide range of perspec ti veforeshortening and a variety of local Su nangles. New instrumentation for photo­graphic scanning, and modern data reductionequipment, make it possible to compute rela­tive height data automatically in a mannerwhich is without precedent in lunar or terres­trial cartography.

THE BASIC PROBLE~I

The concept of shadow reduction is asimple one, but the execution of the procedure

on a large scale can be qui te tedious and su b­ject to error. In its most basic form, the prob­lem can be described as follows.

Inasmuch as the Moon has no atmosphere,sunlight is not scattered in the sky as isexperienced on Earth. For this reason, theSun is the only intense source of light abovethe lunar horizon, and shadows cast by ob­jects have extreme con trast.

After sunrise, and before sunset, \I,hen theangle of elevation of the Sun is small, there isan emphasis of relief since shadow lengthsare greater than the height of the feature,The shadow/height ratio is equal to thecotangent of the solar elevation.

The shadow edges are not completelyabrupt changes in light intensity, but aremodified by a penumbral zone of partialshadow. The width of this zone correspondsto the angle subtended by the Sun's diam­eter, approximately one-half degree. Theshadow edge is defined as the half-intensitypoint in this zone from which point thecenter of the Sun's disk would appear to beon the horizon.

The relative positions of the Earth, Sun,and Moon are known for any given time.Therefore, the observer is able to compu te hisown position, and that of the Sun, with re­spect to a feature casting a shadow.

The shadow ima!!:e is measured in the direc­tion of the solar azimuth in the photograph,and the true shadow length is computed frol1lthe photo scale and foreshortening angle.From the lunar coordinates of the feature, thesolar ele\'ation is determined. I t is then nec­essary to apply only a simple trigon;)metricsolution to establish the relati\'e height of thepoint casting the shadow above the point atwhich it ends.

* Paper Presented at Annual Convention of the Society at \\'ashington, D. c., March 1965.

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742 PHOTOGRAMMETRIC ENGINEERING

In practice, the lunar coordinates of a par­ticular shadow, as well as the phase andlibration variables, are computed for thephotograph being measured. After the shadowimage has been measured, the photo scalefactor and perspective corrections are appliedto obtain the true length of the shadow on thelunar surface. The local solar elevation angleis used for the relative height determination.

CALIBRATION OF MEASURES

The main difficulties in conducting ashadow measuremen t program come underthe category of calibration. This is a necessaryprogram which must achieve two goals: themeasures must be tied to the image pointscorresponding to the shadow edge; and themeasured heights must be related to a lunarcoordinate system. If the shadow images aremeasured on a linear comparator, the opera­tor must make a decision as to the properplacement of the cross-hair for the best mea­surement of the shadow edge. Because theedge is diffuse in the photo image, this rangeof judgment may correspond to a consider­able error in the heigh t being derived.

To improve the identification of the shadowedge in the measurement process, it is possibleto scan across the area with a microdensitom­eter, using a preselected density level in theimage as the effective limit of the shadow.This technique is valid if the exact relation isknown between the intensity of the lightcoming from the Moon and the density of theprocessed photograph-requiring continualcalibration of exposure, processing, and theintensity-transfer function of the telescope.

Where shadow measures are made on amicrodensitometer, it is necessary to relatethe posi tion of the scan line to some coordi­nate system at the lunar surface. The usualreference systems are based on the posi tionsof craters distributed across the Moon'svisible disk, but frequently the feature beingmeasured is a considerable distance from oneof these reference points, and the scan isdifficult to locate and interpret. Ideally, themeasures should be made on a microdensi tom­eter which has the capabili ty to relate scandata to physical distances and directions inthe photographic image.

AUTOMATED MEASUREMENT AND

REDUCTION

A new program of shadow reduction hasbeen developed at ACIC, designed aroundthe measuring capability of the Micro­Analyzer, built by the David W. Mann Com-

pany and Data Corporation. This instrumentis a sophisticated microdensitometer with ameasuring stage built upon a precision x,y­linear comparator. The comparator axes aredigitally encoded, such that the image coordi­nates of the microdensitometer spot may berecorded simultaneously with the densityvalue. Although it is possible to use theMicro-Analyzer in a point-by-point measur­ing mode, its most effective use is realized,with it is operated in a scanning modewhereby density values and coordinates arerecorded on the fly as the densitometer spottraces out a preset pattern in the photograph.

MEASURING PROCEDURE

A photograph is selected for measurementon the basis of scale, resolution and phasesui table for coverage of the area of in terest.After the photometric calibration data havebeen established, the photograph is placed onthe measuring stage of the Micro-Analyzer.

The first phase of the measuring programconcerns the relation of the photograph tocoordinates in a lunar control network. Fea­tures having known lunar coordinates areidentified and their projected positions aremeasured in the photograph. The origin ofmeasurement and orientation of the com­parator axes are held constant throughou tthis and all succeeding phases of the measure­ment.

After the control point coordinates havebeen recorded, the boundaries of the area ofinterest are established. This permits settingthe limits of X and V-travel for the scanningmode. For simplicity in scanning and reduc­tion, the areas are chosen as rectangularblocks within the image. The microdensitom­eter spot is positioned at one of the corners ofthe rectangle, and the other scan parametersare selected.

I t is possible to vary the size of the micro­densitometer spot over a wide range throughcom bi nations of fixed and variable aperturesand a series of optics having different focallengths. Thus, the spot size may be chosen tocorrespond to a desired fraction of the resolu­tion of the photograph. For shadow measuresin a typical lunar photograph, a spot of 15 to20 microns is used. The spacing of the indi­vidual scan lines, and the frequency of sam­pling along a scan, are also left to the optionof the investigator. For the most detailedstudies, a complete raster tracing is madewith the scan and sample separation corre­sponding to the dimensions of the micro­densitometer spot.

The last portion of the measurement pro-

LlIN,\R IIEIGHTS FROM SHADOWS AUTOMATICALLY 743

gram-\\' here the shadows are actuallyscanned-can be done completely auto­matically. After the image area limits, scanpattern, and sampling frequency have beenestablished, the j\1icro- \nalyzer can carryout the entire scan program without furtherinterference by the operator. As each sampleis taken, digi tal values of the densi ty and theX- and Y-coordinates of the spot are recorded.

DATA REDliCTIO:-l

The data reduction program fulfills threebasic tasks: the photograph coordinates aretransformed into the perspective coordinatesof the control points measured; the individualscans arc related to posi tions on the surface ofthe Moon; the relath'e heights are computedfor objects casting shado\\·s. Tn the past, thisreduction has been completed on a point-by­point basis for indi"idual measures. The na­ture of the scanned densitometric data is suchthat the reduction phase can be handled com­pletely in an electronic computer without thenecessity of a manual interpretation stepbetween measurement and computation.

The first stage of the reduction relates thepositions of indi"idual points \\'ithin thescanned data to physical locations on the1\100n. Si nce the ti me and date are kno\\'n forthe particular photograph, it is possible tocompute the direction of perspective for theobservation. The known coordinates of thelunar control features are mathematicallyrotated through the angle of perspective, andthe posi tions are made to fi t wi th the mea­sured photo coordinates of the points througha least squares transformation. Through in­terpolation it is then possible to execute acontrol extension procedure to compute lunarlatitude and longitude values for any pointwithin the scan area of the photograph.

The next step is built around the densito­metric data that constitute the output fromthe :\licro-.-\nalayzer scan. A matrix is estab­lished from the recorded data, \\·ith rows andcolumns corresponding to respective x- and y­positions. The elements of the matrix are thenumerical values of density measured for thecorrespondi ng positions.

The densitometric values are then trans­formed into light value readings through theintensity transfer function obtained fromemulsion calibration data. Thus the matrixbecomes a digital plot of light values in theplane of perspecti"e of the photograph.

Shadow edges arc located within the lightvalue array through numerical differentiationalong the matrix rows (measured along thesolar azimuth direction projected into thephoto plane). A preselected value is used forthe half-intensity point \\'ithin the shadowedge, and the coordinates of this point arefound through interpolation within the lightdifferential function. Along the individualscan lines, these positions arc "tagged" andbecome the output of a new matrix whoseelements describe the pattern of shadowedges and indicate whether the edge is begin­ning or ending the shadow.

The photographic coordinates of theshado\\' extremities are transformed intolunar latitudes and longitudes through thepreviously established control computations.The reduction is then performed in a mannersimilar to pt'esent relative height calculationswhere the local solar elevation is computedand shadow lengths are related to the heightsof features on the 1\[oon.

For the output of this information, thecomputer prepares a data tape for an elee­tmnic plotter. From this tape, the plotterdrafts a hard-copy manuscript that locatesthe scanned features on a lunar latitude­longitude grid. The beginning and end pointsof the shadow are iden ti fied, together wi th avalue of the relative height bet\\'een the twolocations.

COKCLUSIONS

The measurement and reduction programdescribed has been developed within thelunar charting acti\'ity at ACTC. Although nosingle concept or procedure within the pt'ocesscan be described a completely original orunique, the combination of instrumentationand data-handling represent a distinct stepfOf\\'ard in the achie"ement of accuracy,speed, and versatility in fields of photogram­metrv. lunar astronomy and cartography.