Sources of Shape Variation in Lunar Impact Craters: Fourier Shape Analysis

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Sources of shape variation in lunar impact craters: Fourier shape analysis

DUANE T. EPPLER Polar Oceanography Programs, Naval Ocean Research and Development Activity, NSTL Station, Mississippi 39529 ROBERT EHRLICH Department of Geology, University of South Carolina, Columbia, South Carolina 29208 DAG NUMMEDAL Department of Geology, Louisiana State University, Baton Rouge, Louisiana 70803 PETER H. SCHULTZ The Lunar and Planetary Institute, 3303 NASA Road One, Houston, Texas 77058

ABSTRACT outline of Tsiolkovsky crater is tectonically controlled. Shoemaker (1960) and Roddy (1978) show that the quadrate shape of Meteor R-mode factor analysis of Fourier harmonics that describe the Crater in Arizona is related directly to the orientation of regional shape-in-plan-view of 716 large (diameter > 15 km) nearside lunar faults and joints in Colorado Plateau rocks. craters shows that two factors explain 84.3% of shape variance Impact crater shape could be used to indicate structural pat- observed in the sample. Factor 1 accounts for 68.2% of the sample terns in heavily cratered terrane but has not received wide use as a variance and describes moderate-scale roughness defined by har- supplement to conventional sources of geologic structural data. In monics 7 through 10. Shape variation described by these harmonics part, this is due to previous absence of shape descriptors with which is related to surficial lunar processes of degradation that modify shape features that are related to structural variables can be dis- crater shape-in-plan. Dominant among these processes are ejecta criminated from those related to nonstructural variables. Although scour from large impact events and ongoing aging. Factor II past investigators used circularity indices to measure relative polyg- accounts for 16.1% of the observed shape variance and describes onality and circularity (Ronca and Salisbury, 1966; Murray and polygonal shape elements related to harmonics 2, 3, 4, and 6. Varia- Guest, 1970; Pike, 1974, 1977), such techniques characterize shape tion in these harmonics is tied to variables that distort the spherical incompletely. They carry no information regarding the number, symmetry of crater-forming processes. The dominant contributor location, magnitude, or orientation of deviations from the funda- among these variables is the nature of geologic structural patterns mental circular crater shape. Furthermore, each method preselects in impacted material. Unlike transient features described by factor the scale of noncircularity to be measured. One method looks only I, polygonal shape elements described by factor II do not change at deviations in the gross outline, another only at scalloping. A appreciably with time. The permanence of these features and their method that captures the complete range of crater-shape informa- relation to lunar geologic structure suggest that the shape of old tion is needed to determine first, whether systematic deviations in craters carries the imprint of geologic structural relationships pres- shape occur, and second, if they occur, the extent to which such ent in early lunar crust. deviations reflect geologic structural relationships of the impacted target. INTRODUCTION In this study, we measure shape using a descriptor that quanti- fies crater shape completely. We have chosen Fourier analysis in Ejected debris that covers the moon's surface obscures lunar closed form (Ehrlich and Weinberg, 1970), a technique that does bedrock and makes detailed determination of underlying geologic not merely estimate shape, but rather describes shape to whatever structures over broad regions of the satellite difficult. Inferences degree of precision is necessary. This paper reports the results of regarding geologic structural trends come principally from analysis applying this technique to shape analysis of large craters ( > 15 km of surface lineations such as volcanic alignments, graben, and ridges in diameter) from across the lunar nearside. (Fielder, 1963; Strom, 1964; Mason and others, 1976; Lucchitta and The objectives in performing this analysis are fourfold: (1) to Watkins, 1978; Fagin and others, 1978). Effective use of these fea- identify variables that contribute to shape variability among lunar tures is confined to those regions of low crater density in which craters, (2) to define the scale of shape variation that can be attrib- surface topography reflects geologic structural trends clearly. Their uted to each variable, (3) to define the relative contribution each use is limited in heavily cratered terrane. variable makes to total shape variation within the crater sample, Past work suggests that polygonal elements of impact crater and (4) to assess the potential utility of crater shape as an indicator shape1 also reflect geologic structural relationships present in of geologic structural relationships in impacted crust. impacted material. Scott and others (1977) show that straight wall METHODOLOGY segments of farside lunar craters parallel the trend of local structural lineaments. Murray and Guest (1970) suggest that the polygonal The Sample

'The term "shape" as used throughout this paper refers to the two- The sample consists of 716 nearside lunar craters larger than 15 dimensional crater outline as it appears in plan view. km in diameter. Crater outlines are defined by the break-in-slope

Geological Society of America Bulletin, v. 94, p. 274-291, 20 figs., 1 table, February 1983.

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Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/94/2/274/3434528/i0016-7606-94-2-274.pdf by guest on 23 September 2021 Figure 1. Graphic representation of selected Fourier harmonics. Points used to draw each shape were computed by incrementing equation 1 at half-degree intervals of d for discrete values of n. Thus, shape two represents equation 1 evaluated for n = 2, shape three n = 3 and so forth. Plus and minus signs on even-numbered harmonics indicate that alternating lobes (-) are defined by negative radii and are subtracted when shape components are summed.

along the upper rim of the crater wall and were traced directly from scribed by Ronca and Salisbury (1966), Murray and Guest (1970), contact prints of Lunar Orbiter IV images. The lower size limit (15 and by Pike (1974): (1) all aspects of shape are described com- km) represents the smallest crater that could be traced accurately pletely, objectively, and accurately by the series, (2) the series con- from Lunar Orbiter IV photographs. This size coincides roughly verges rapidly when rounded forms are described such that fewer with the transition from simple to complex craters (Quaide and than 20 terms are required to describe crater shape, and (3) shape others, 1965; Pike 1967, 1974). Craters that exhibit complex mor- components are related easily in a visual sense to shape characteris- phology thus predominate in the sample, although a limited tics of the composite form being described. number of simple craters also are included. Craters in which Closed-form Fourier analysis partitions shape into a series of younger, overlapping impacts destroy a significant portion of the shape components (harmonics) according to equation 1: crater wall were eliminated from the sample. Previous work suggests that ~85% of the crater outline must be intact to obtain valid R(0) = Ao + ! ,Ancos(nM>„) (1) shape data (Eppler, 1980). Certain Lunar Orbiter frames contain n = I considerable distortion due both to the curvature of the moon and where R is the radius vector measured from the shape centroid to a to the obliqueness of view. An algorithm based on Schmid's (1962) point on the periphery in a polar direction 0, where Ao is the mean mapping equations for the "general oblique projective projection" radius of the shape, and where An is the amplitude and 4>n the phase was used to rectify crater outlines (Eppler, 1980). Imagery of areas angle of the "nth" term of the series. Shape information is present in in latitudes higher than 50° typically is distorted in complex ways both the amplitudes and phase angles. In general, the amplitude of that the program did not correct fully. These craters are excluded the nth harmonic represents the contribution that an "n-leaved from the sample. Approximately 25% of the remaining craters clover" makes to over-all shape (Figs. 1 and 2). The amplitude of located in latitudes lower than 50° were found to contain minor the fourth harmonic represents the contribution of a "four-leaved distortions and were rectified. All craters not eliminated on the clover." Thus, the fourth harmonic measures quadrature. The basis of these criteria are included. This gives a sample of 716 amplitude of the second harmonic represents the contribution of a craters, which represents, as closely as possible, the total nearside "figure eight" and is thus a measure of elongation. Phase angles crater population in terms of age, morphology, and both physio- carry information regarding the orientation of each shape compo- graphic and geographic location within the size range investigated. nent with respect to a reference point on the shape periphery. Expanded discussions of the Fourier technique are presented else- Shape Analysis where (Ehrlich and Weinberg, 1970; Ehrlich and others, 1980; Eppler and Meloy, 1980). The method selected to characterize shape is Fourier analysis in closed form. Fourier descriptors quantify two-dimensional shape R-Mode Factor Analysis to any degree of precision that is required (Ehrlich and Weinberg, 1970). Use of the Fourier technique for crater-shape analysis has Harmonic amplitudes that describe shape typically are not sta- several advantages over conventional shape estimators that are de- tistically independent of each other. Fundamental properties of

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100

Figure 2. Imprint of selected harmonics (Fig. 1) on a circle (zero harmonic). Note that the number of lobes produced by any given shape component is the same as the harmonic number. Low-order harmonics (2 through 6) modify gross shape. Higher order harmonics produce increasingly fine-scaled perturbations in shape. Amplitude of the zero harmonic is 1.0; that of all other harmonics is 0.02. Phase angle for all harmonics is 0°.

symmetry inherent in any shape dictate that a certain degree of TABLE 1. FACTOR LOADINGS FOR AMPLITUDES dependence will exist between all odd harmonics, and between all 2 THROUGH 10 FOR THE FIRST FOUR FACTORS even harmonics. In terms of data analysis, this means that many harmonics carry similar or related information. To determine the FACTOR nature of such underlying relationships and eliminate redundancy I II III IV in the data, an R-mode factor analysis employing varimax rotation HARMONIC (Davis, 1973; Kim, 1975) was performed on the first ten amplitudes 2 0.07850 0.54868 0.14652 -0.03690 of all craters sampled. 3 0.12032 0.47522 0.03247 0.26601 Of the total shape variation in the crater sample, 94% is 4 0.15776 0.50252 0.25146 0.09961 explained by four factors (Table 1). That is, factor analysis elimi- 5 0.22109 0.06823 0.14568 0.41421 6 0.23510 0.27917 0.45886 0.17979 nates redundancy in the original amplitudes such that four factors 7 0.46351 0.13536 0.11099 0.14619 carry essentially as much information as ten harmonic amplitudes. 8 0.30820 0.17649 0.39004 0.15076 Each factor describes a different portion of the shape spectrum 9 0.55361 0.07634 0.15654 0.12391 (Table 1). Factor I accounts for 68% of the total sample variance 10 0.44417 0.13404 0.15526 0.15196 and describes intermediate-scale deviations from circularity related Percent variation: 68.2 16.1 5.1 4.6 to the 7th, 8th, 9th, and 10th harmonics. Factor II accounts for 16% Cumulative percent: 68.2 84.3 89.4 94.0 of the variance and describes polygonal departures from circularity related to the 2nd, 3rd, 4th, and 6th harmonics. Factors III and IV Note: High loadings (greater than 0.25) for a given factor are italicized each account for only 5% of the variance and are not discussed here. and indicate harmonics that contribute strongly to shape variation described The wide range of shape variation described by factors I and II by that factor. For example, factor II describes variation in gross shape as is shown in Figures 3 and 4, respectively. Difference in shape expressed by harmonics 2, 3, 4, and 6. between high- and low-end members is gradational for both factors.

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<-1.0 Figure 3. Variation in scores of factor I as a function of crater shape. Scores of the first factor that describe craters in each row are shown in the center of the -0.5 figure. Outlines of the same craters are shown on both sides of these factor scores. Craters to the left have been drawn by combining (summing) only the 0.0 7th, 8th, 9th, and 10th harmonics. It is or X X ' shape variation in these harmonics that factor I describes. Shapes of these same craters were redrawn using harmonics 1 through 20 and are shown on the right 0.5 half of the figure. Shapes plotted from harmonics 1 through 20 essentially represent the raw input shape of crater outlines used in the analysis. Crater shapes 1.0 were plotted using program DRAW- SHAP (Eppler, 1980). >1.0

HARMONICS 7 8.910 HARMONICS 1 THRU 20

FACTOR H SCORES

Figure 4. Variation in scores of factor II as a function of crater shape. Scores of the second factor that describe craters in each row are shown in the center of the figure. Craters shown to the left of these factor scores have been summed using only the 2nd, 3rd, 4th, and 6th harmonics. Factor II describes shape variation in these harmonics. Shapes of these same craters were computed using harmonics 1 through 20 and are shown on the right half of the figure.

HARMONICS 2,3,4,6 HARMONICS 1 THRU 20

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Craters that score low on either factor (less than -1.0) are nearly Wedekind (1978) predict the shape of laboratory-size craters circular. Craters with intermediate scores of factor 1 (0.0 to 0.5) formed by oblique impact. Gault and his co-workers show that show moderate small-scale irregularities (Fig. 3). Craters that score hypervelocity projectiles encountering unconsolidated quartz sand in the intermediate range with respect to factor II (0.0 to 0.5) are or pumice dust substrate at angles of incidence higher than 30° polygonal (Fig. 4). Craters that score high on either factor (greater from horizontal produce craters that are circular in plan view. Pro- than 1.0) are irregularly shaped, but craters that score high on jectiles that impact unconsolidated targets at angles between 30° factor II are less rough and more elongate than their factor I and 10° typically are circular but may be slightly elongate in a counterparts. direction perpendicular to the trajectory path (Gault and Wede- kind, 1978). Projectiles that impact unconsolidated targets at angles ORIGINS OF NONCIRCULAR CRATER SHAPE less than 10° produce craters that, before slumps occur, are elongate in a direction parallel to the direction of travel. Gault and Departures from circularity of lunar crater rim shapes (Figs. 3 Wedekind's (1978) data show that oblique impacts into granite pro- and 4) reflect either (1) deviations from circular symmetry of the duce circular craters for angles greater than or equal to 30°, and crater-forming processes, or (2) postformational processes of modi- elongate craters for angles of 15°. Data for angles between 30°.and fication that alter original crater morphology. This is not to suggest 15° are not presented. Thus, the exact angle below which obliquity that shape variability arising from either source necessarily corre- of the impact affects crater shape is not defined for lithified targets. sponds to variation described by one factor or the other. However, The probability of a body impacting at or below an angle of the nature of shape variation that arises from a specific source (for incidence /' is given by sin2i (Gilbert, 1893; Shoemaker, 1962). If 15° example, oblique impact, structured target, or topography) might is used as the angle below which oblique impact affects shape-in- be unique and correspond to a specific factor, a specific harmonic, plan, ~6.7% of the total lunar crater population is expected to show or a specific group of harmonics. The discussion that follows pre- elongation attributed to low-angle impact. If 30° is used as the sents an analysis of sources of variation in crater shape, the scale critical angle, 25% of all lunar craters will be so affected. The and nature of variation that each source produces, and the relative second harmonic measures elongation and thus is the principal dominance of each source in determining crater shape-in-plan. shape component that oblique impact will affect. Thus, between 48 and 179 of the 716 craters analyzed here are expected, in theory, to Variability of the Crater-Forming Process show inflated amplitude values of the second harmonic as a conse- quence of oblique impact. Theoretical work by Opik (1969) demonstrated that the shape The actual number of craters that show higher than normal of a crater produced by normal (vertical) impact of a spherical amplitude values might be lower than these theoretical estimates object into a uniform substrate will be circular in plan. Departures indicate. Low-angle oblique impacts (< 5°) that produce highly from circularity related to the impact event arise from deviations elongate craters comparable to Messier or Schiller represent less from Opik's model constraints: (1) angle of impact, (2) velocity of than 1% of the complex lunar crater population. Craters produced the impacting body (that is, hypervelocity primary impact versus by higher angle oblique impacts (10°-30°), although more common lower velocity secondary impact), (3) size and shape of the impact- in theory, also are more difficult to identify. Elongate-shape com- ing body, and (4) character of the impacted target. ponents become less pronounced at higher impact angles and are Angle of Impact. Experiments reported by Gault and others easily masked by wall slumps. Ejecta about Proclus, for example, (1968), Fechtig and others (1972), Gault (1973), and Gault and are arrayed in patterns that indicate oblique impact. Extensive

HARMONIC

CRATERS OVERLAPPED BY YOUNGER CRATERS vs. AO. CRATERS ON OLDER B CRATER RIMS vs. A.O.

"7" MODIFIED vs. UNMODIFIED CRATERS

CRATERS IN GROOVED vs. D UNGROOVED TERRAIN PREGROOVE VS. EPOSTGROOVE CRATERS

r CRATERS IN DEEP MARIA r vs. SHALLOW MARIA

G MARE vs. HIGHLAND

SIZE: 15 TO 30 vs. 30 TO 50 vs. H 50 TO 75 vs. LARGER THAN 75

Figure 5. Results of chi-square analyses of contingency tables (Maxwell, 1971; Everitt, 1977) for comparisons between various crater groups. Darkened squares indicate harmonics at which significant differences are detected for different groups of craters at the 0.05 level (Kimball, 19S4).

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slumps along Proclus' walls, however, produce a scalloped, equant involved in secondary crater formation are less likely to override outline that lacks measurable elongation. Masking or obliteration such local effects than are higher energies associated with primary of elongate shape components thus effectively lowers the angle impacts. In general, then, we expect secondary craters to be more below which complex craters formed by oblique impact can be irregular in shape than primary craters (Pike and Wilhelms, 1978). identified on the basis of shape. Probably only basin-related secondaries from Imbrium and Orien- Impact Velocity (Primary or Secondary Origin). Differences in tale are included in the sample used here. However, the precise both the magnitude and scale of shape irregularities attributed to contribution that these secondary impacts make to total shape vari- secondary -primary origin are difficult to define, particularly for ability of the crater sample is undetermined. craters in the size range encompassed by this investigation. It is Size and Shape of the Impacting Body. Projectile Shape. difficult to distinguish, with certainty, primaries in the lower range Theoretical work by Ahrens and O'Keefe (1977) and experimental of sizes investigated here from large secondaries of comparable size. work by Thomsen and others (1979) shows that crater shape is Although somewhat reliable criteria have been established for dis- independent of projectile shape for hypervelocity impacts. The pro- tinguishing small (diameter < 15 km) secondaries from small prim- jectile velocity in such events is sufficiently large to assume that aries (Schultz, 1976; Wilhelms and others, 1978), larger secondaries transfer of energy from projectile to target is instantaneous at a are difficult to identify in this way for several reasons. First, ejecta that surround large lunar craters typically are poorly defined. Characteristic herringbone patterns of ejecta that occur about clusters of small secondaries are not always evident about large secondaries. By virtue of their size, large secondary craters survive longer than smaller secondary craters and so are more likely to outlive ephemeral patterns in their ejecta than their smaller counterparts. Second, clusters and chains in which smaller secondary craters commonly are arrayed may not be typical of large secondaries. Theoretical work that involves comminution of rock materials sug- LU _J gests that the abundance of largle ejecta blocks will be lower than that of smaller blocks by a significant margin (Von Rittenger, 1867; velocity impact energies (Meloy and Faust, 1965). Comminution Li. tests of terrestrial rock materials support Von Rittenger's theoreti- O cal work and demonstrate that the amount of coarse material produced by a given impact event is inversely proportional to the uZi u amount of energy used to break the rock (Berlioz and Fuerstenau, œ LU 1967). Primary impact events, then, are likely to produce small Q. ejecta blocks in great abundance but only small numbers of large blocks. If production of large blocks is assumed to be random with respect to direction about the point of primary impact, then the probability of ejecting clusters of large blocks (that would form chains) from any given point is less than that for smaller ejecta. Thus, the abundance of chains of large secondary craters and associated large-crater clusters that result from large, single blocks is expected to be lower than the abundance of small secondary crater Figure 6. Shape-fre- chains. quency histograms of Recent work by Schultz and Mendenhall (1979) suggests that amplitudes of the 17th clustered systems of ejecta swarms composed of assemblages of harmonic for craters small blocks can produce large secondary craters. The abundance grouped by size. Ampli- of such swarm-generated secondary craters is not yet known; tude increases to the neither is their geographic distribution with respect to their parent right. Left-most modes impact understood. If it can be shown that swarm-generated secon- represent nearly circular dary craters commonly are arrayed in recognizable patterns (for craters; modes to the example, characteristic chains and clusters), then it might be possi- right represent highly ir- ble to identify this class of secondaries with a high degree of regular shapes. Irregular certainty. craters increase in abun- The lack of definitive criteria with which large secondary cra- dance over more circular ters can be distinguished from primaries of comparable size shapes with increasing precludes accurate identification of secondaries that might be size, as shown by pro- embedded in the crater sample. The shape of the impacting body as gressive enlargement of well as local variation in topography, regolith, and bedrock charac- the right-hand mode at teristics, however, are more likely to influence the shape of secon- the expense of the left- dary craters than the shape of primary craters. Lower energy levels hand mode. (HARMONIC 17)

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point. The initial configuration of the shock front so induced is from 30 to 75 km, however, follow no apparent pattern over this spherical and is independent of projectile shape. interval, except to remain between bounds set by the largest and Low-velocity, secondary impacts are more likely to show the smallest crater groups. Below the 6th harmonic, systematic rela- effects of projectile shape, particularly where highly irregular pro- tionships between the four size groups are not evident. jectiles are concerned. Nonspinning projectiles or ejecta clouds These data indicate that variation in projectile size primarily (Schultz and Mendenhall, 1979) probably produce secondary cra- affects high-order harmonics in complex craters. Large projectiles ters with shape characteristics that mimic the shape of the are more likely to produce craters with small-scale irregularities impacting body or cloud as projected in the direction of travel. The than are small projectiles. This relationship is defined well only shape of craters produced by spinning projectiles is less easily pre- above the 9th harmonic. Intermediate-scale features described by dicted but is likely to be distinctly irregular as well. As noted above, factor I are involved only marginally. Polygonal aspects of shape some large secondary craters from the Imbrium and Orientale (factor II) are largely unaffected. events are likely to be included in the sample analyzed here. How- Substrate Character. Deviations in crater shape that result ever, the contribution that they make to overall shape variation from nonuniformity of the impacted substrate arise from variation displayed by the crater sample is not believed to occur consistently in either of three substrate attributes: (1) geologic structure in the at specific harmonics. Rather, effects of projectile shape are more upper crust as manifest in faults, fractures, and other zones of likely to occur randomly through the shape spectrum. crustal weakness; (2) layering in the upper crust and regolith; and Projectile Size. Changes in projectile size produce concomitant (3) surface topography. changes in crater size. Shape features that vary with projectile size, Geologic Structure. The possible existence of well-developed then, can be identified by comparing craters that are grouped into structural trends in the upper portion of the moon's crust may be discrete size categories. Craters sampled were divided into four implied by the northeast- and northwest-trending conjugate set of groups: diameter 15 to 30 km, 30 to 50 km, 50 to 75 km, and dominant surface lineations referred to as the lunar grid (Fielder, diameter>75 km. A chi-square test (Maxwell, 1971; Everitt, 1977) 1963; Strom, 1964). Superimposed on this large-scale grid system, of the 4 x c contingency table formed by these groups shows that and in some cases dominating it, are lineations radial and concen- significant shape differences occur in harmonics 10 through 20 at tric to major impact basins (Gilbert, 1893; Hackman and Mason, the 0.05 level of significance (Fig. 5, H). Histograms of amplitude- 1961; Mason and others, 1976). Structural features related to frequency distributions indicate that the abundance of craters with smaller impacts also exist. Within the maria, graben and ridge sys- small-scale shape irregularities in each group increases progres- tems reflect past (and perhaps present) tectonic regimes (McGill, sively with increasing crater size (Fig. 6). Plots of amplitude spectra 1971; Muehlberger, 1974; Mason and others, 1976; Fagin and oth- that characterize craters in each size group reflect these same differ- ers, 1978; Lucchitta and Watkins, 1978). Once such structural ences and suggest further that a limited size-shape effect is present trends are established, it is likely that they will persist as zones of in intermediate-scale harmonics (6 through 9) as well. weakness throughout lunar history. Terrestrial Precambrian faults Figure 7 shows log-mean amplitude plotted as a function of commonly are the site of repeated movement through time even frequency (harmonic) for each of the four size groups. Samples with though they may be annealed repeatedly by secondary mineraliza- high mean amplitudes contain an abundance of irregular craters; tion and remain dormant for eons. In this regard, the structural lower mean amplitudes generally indicate fewer irregularly shaped state of lunar crust probably is similar to terrestrial crust. Schultz craters. Spectra of large craters fall consistently above spectra of (1976) notes that the original structural imprint of pre-Imbrian smaller craters for all harmonics above the 9th. This same relation- lunar basins persists to the present. It is likely that other ancient ship is observed between the largest and smallest size groups (>75 zones of weakness persist in lunar crust. km, 15 to 30 km) for harmonics 6 through 9. Craters ranging in size Lunar crater shape is related to zones of crustal weakness asso-

~~Figure 7. Log-mean amplitude plotted as a function of frequency (harmonic) for each of four size groups. Arrows mark harmonics in which significant differences occur at the 0.05 level.~~

~~HARMONIC (frequency)~~

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~~ELONGATION QUADRATURE~~

///////// Figure 8. Structural control of /////// crater shape Model 1: excavation of the crater cavity will occur preferentially in // ' '///// directions parallel to directions of weak- / / / / / / ness produced by faults, fractures, or joints.

~~/al/ ' ' / ' « / i!«/ ' ' / ' ' /VO/ > ' / i'' '•W / / / ' ' / / /£•//<£> / >'/ >'/ / / / / /' / <0 ! ! ! / / / ! ! / / / / / / / / /~~

~~ELONGATION QUADRATURE~~

A / i Figure 9. Structural control of crater shape Model 2: slumping of the crater wall at the time of crater formation occurs along pre-existing faults and /• Vw / fractures, and the crater cavity is enlarged perpendicular to structural trends.

~~,4 / / / / / / / ' /V~~

ciated with these structural trends. Scott and others (1977) show Model 2. Upon completion of the excavation phase, walls fail that linear wall segments of lunar farside craters commonly parallel along pre-existing faults, fractures, and joints in lunar crust. The structural lineations. Guest and Murray (1969) suggest that the transient cavity is enlarged in directions perpendicular to trends of polygonal outline of Tsiolkovsky may be tectonically controlled. crustal structure (Schultz, 1976; Scott and others, 1977). Linear Fielder (1961) and Fielder and Jordan (1962) show that the long- segments of crater walls parallel trends of crustal weakness. Phase axis orientation of many noncircular craters parallels the prominent angles correspond to bisectors of the conjugate fracture sets trend of the lunar grid. If shape-in-plan indeed is related to geologic (Fig. 9). structure, as these studies suggest, then the orientation of certain Lunar and terrestrial evidence suggests that shape deviations shape components (harmonics) in craters so affected should be arise from both mechanisms. Square craters were produced in related to structural elements in surrounding lunar crust. More laboratory experiments by Gault and others (1968) when projectiles specifically, a systematic relationship should exist between low- were fired at hypervelocities into cemented sand targets. The targets order harmonics described principally by factor II (2, 3, 4, and 6) contained perpendicular fracture sets that passed through the and the orientation of prominent structural lineations (rimae, corners of the crater that formed. In a similar fashion, diagonals ridges, graben, scarps, volcanic alignments). The nature of this rela- drawn through the corners of Meteor Crater parallel regional con- tionship is suggested by alternative models that describe the manner jugate joint sets in the manner shown in Figure 8. Continuity of in which departures from circularity might arise from structural both the crater rim and adjacent hummocky terrain formed by relationships in impacted crust at the time of crater formation: ejected debris, and lack of excessive accumulation of alluvium on Model 1. Excavation of the crater cavity proceeds preferen- the crater floor adjacent to crater corners, indicate that the quad- tially along directions of crustal weakness. The cavity is enlarged in rate shape is not a result of preferential, postformational fluvial directions parallel to trends of crustal structure (Shoemaker, 1960; erosion along these joint sets (Roddy, 1978). Rather, the observed Gault and others, 1968; Roddy and Davis, 1977; Roddy and others, coincidence of crater shape with geologic structure suggests 1975; Roddy, 1978). The trend of linear segments of crater walls strongly that excavation of Meteor Crater occurred preferentially in bisects the angle between conjugate fracture sets. Phase angles of directions parallel to zones of crustal weakness (Model 1). large-amplitude harmonics correspond directly to fracture trends Evidence that this same mechanism operates in forming lunar (Fig. 8). craters is provided by the relationship between linear structural

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features on the lunar surface (rimae, ridges, graben, scarps) and the Although the imprint of crustal structure can be placed on shape of adjacent craters. In some areas, crater elongation as mea- crater shape by either of the two mechanisms outlined above, crater sured by the 2nd harmonic phase angle parallels the trend of prom- size will determine, in large part, which mechanism predominates. inent rimae (Eppler and others, 1978). The phase vector of the 2nd The shape of small, simple craters, in which large wall slumps are harmonic parallels major structural lineations in each region. At the not common, conforms principally to Model 1. Larger complex southwest margin of Mare Fecunditatis, for example, the trend of craters, in which wall slumps typically are well developed, behave rimae Goclenius and Gutenberg parallels the long axis of adjacent according to Model 2. The shape of some craters might reflect both craters, especially Goclenius (Fig. 10). Although Goclenius predates mechanisms. That is, although polygonal outlines are imposed on adjacent rimae, terrestrial examples of such lineaments commonly craters during excavation of the transient cavity, slumping along follow older, pre-existing structural trends. fault traces during the modification phase of crater formation can Evidence in support of the alternative model for structurally reorient initial polygonality such that linear wall segments parallel induced shape effects (Model 2) is given by polygonal lunar craters fault traces. such as Ritter, Sabine, Cavalerius, and Palmieri. Such relationships Layering of the Impacted Medium. The effect of layering on are displayed particularly well by the Sirsalis series of craters crater shape and morphology has been studied on a small scale in (Fig. 11). In these craters, alignment of linear wall segments laboratory experiments for both low-velocity impacts (Quaide and parallels structural lineations expressed in topography of surround- Oberbeck, 1968) and hypervelocity impacts (Gault and others, ing lunar terrain. 1968), and for explosion craters (Piekutowski, 1977). On a much In other areas, straight wall segments of 17 of 30 marginal larger scale, characteristics of craters produced by high-yield, out- mare craters in areas of well-defined surface lineations parallel the door explosions have been described (Jones, 1976; Roddy, 1976, trend of adjacent mare graben and ridges. These craters typically 1977). Gault's and Piekutowski's experiments show that the mor- are distinctly polygonal. The shape of three other marginal craters phology and shape of impact and explosion craters that form in shows possible relation to local structure. Only 10 of the 30 mar- layered substrate in which competent strata overlie incompetent ginal craters examined show no apparent relation to local structure. material is more complex than the shape and morphology displayed Craters in which shape does not appear to be related to geologic by craters formed in unlayered material. Observations of Roddy structure typically are more circular than craters that show evidence (1976, 1977) and Jones (1976) support these conclusions and show of structural imprint. further that the complexity of craters that form in layered material Azimuth-frequency distributions presented by Scott and oth- of this nature increases with increasing yield of the explosive charge ers, (1977) show that straight wall-segments of farside craters paral- used to create the crater. Gault's and Piekutowski's experiments lel the orientation of farside faults, ridges, crater chains, and also suggest that crater shape-in-plan is not affected by impact lineaments in this same way. Parallelism between crater wall seg- through surficial unconsolidated debris that overlies consolidated ments and geologic structural elements suggests that these linear substrate. This implies that regolith alone does not contribute to segments of crater walls and rims might reflect deep-seated faults or shape variability, even though morphologic effects of the regolith- fractures. substrate interface are well documented (Oberbeck and Quaide,

Figure 10. Parallelism between the orientation of crater long Figure 11. Orientation of crater walls parallels local structural axes and local structural trends. Long axes of the craters Goclenius, trends. The quadrate shape of Sirsalis and Sirsalis A, D, and E Gutenberg, and Magelhaens, as defined by the phase angle of the parallels local conjugate sets of rimae (heavy lines). This relation- second harmonic, parallel prominant rimae (heavy lines). This rela- ship conforms to that shown schematically in Figure 9. Base map is tionship conforms to that shown schematically in Figure 8. Base from McCauley (1973). map is from Elston (1972).

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1967; Quaide and Oberbeck, 1968; Schultz, 1976). Although explo- (Fagin and others, 1978) than in central maria. Structural models sive processes that form craters studied by Piekutowski, Roddy, that take into account these relationships predict that structural and Jones differ radicially from processes active in hypervelocity relationships also are more complex at mare margins than in central impact events studied by Gault and others (1968), morphologic maria (Lucchitta, 1976; Fagin and others, 1978; Maxwell, 1978). characteristics of craters formed by explosive and impact processes Irregularity of craters at mare margins could reflect the imprint of bear great similarity to each other (Roddy, 1977). In light of these geologic structure as well as (or instead of) the effect of layering. experiments, layering in the outer few kilometres of lunar crust and The observed shape effect might also reflect the interaction of struc- regolith is a possible contributor to crater shape irregularity. ture and layering. That is to say that layering might enhance the In order to test this hypothesis, mare craters at mare margins imprint of structural relationships present in the uppermost compe- were compared with craters from basin interiors where basalt fill is tent layer. thicker. Mare basalt and underlying debris on the basin floor Differences in irregularity between the shape of central and beneath constitute a layered system. If shape is affected by layering, marginal mare craters thus appear to result as much from geologic craters formed in basalt near mare margins where basalt is thin structure of the impacted lunar surface as from stratigraphic layer- should be more irregular in shape than craters located in central ing. Central and marginal craters differ at harmonics 4, 5, 9, and 12 maria where basalt is thicker. The method of chi-square analysis of (Fig. 5, F), with harmonics 4 and 5 being the dominant descriptors contingency tables (Maxwell, 1971; Everitt, 1977) was used to com- of polygonality. Harmonics 9 and 12 measure smaller-scale devia- pare frequency distributions formed by amplitudes of each of the tions from circularity that could arise from aspects of layered sub- first 20 harmonics for craters in central and marginal mare regions. strate that are unrelated to geologic structure. The nature of the Chi-square analysis shows that shape differences occur in harmon- geometric relationship between the higher and lower order harmonics 4, 5, 9, and 12 (Fig. 5, F). Visual comparison between marginal ics (if a relationship exists at all) is not clear. The orientation of and central mare craters shows that marginal craters are more these higher order harmonics is not in phase with respect to har- polygonal in plan than their central mare counterparts (Fig. 12). monics 4 and 5 for some craters. This suggests that variation in Observed shape irregularities that are characteristic of marginal harmonics 9 and 12 might be seated at least in part in variables craters thus might reflect the contact between mare basalt that over- other than geologic structure. However, the meager contribution lies pre-mare regolith on the basin floor. The contact between these that the higher harmonics make to total crater shape suggests that two units represents an abrupt change in both density and cohesion the imprint of nonstructural variables on crater shape might be of the impacted substrate, factors that Piekutowski (1977) found minimal with respect to mare craters. to affect crater morphology in laboratory experiments. Circular Topography. Topographic irregularities of the lunar surface craters in central maria pierce greater thicknesses of basalt before clearly affect impact crater shape. Impacted scarps, graben, and encountering the basin floor beneath. Lithologic and stratigraphic rims of older craters all are reflected in crater morphology and homogeneity exhibited by the thicker mare basalt might produce shape. Aspects of shape that define the polygonal outline of craters craters that are more circular in plan. such as Romer, Calippus, and Stiborius coincide in part with topo- An alternative explanation for shape differences observed graphic features that these craters overlap. Two comparisons were between craters in central and marginal mare locations concerns made to test for the influence of topography on shape. differences in structural characteristics of impacted basalt. At mare First, mare craters were compared with highland craters using margins, graben are observed to be far more numerous (Lucchitta a chi-square test of contingency tables. Significant differences occur and Watkins, 1978) and ridge assemblages are more complex in harmonics 3, 4, 10, 12, 15, and 18 (Fig. 5, G). Histograms of

HARMONICS 1 THRU 20 HARMONICS 4,5,9*12 Figure 12. The effect of substrate structure and layering on crater shape. The shape of randomly selected craters in mare regions of thick and thin basalt fill were reconstructed (a) using all 20 harmonics and (b) using only statistically significant harmonics (4, 5, 9, and 12). Craters formed in shallow areas of mare basins are consistently more polygonal than their counterparts formed in areas of thicker basalt fill. From left to right and top to bottom, craters shown are: Deep Maria: Pierce, Picard, Bessel, Lambert, Pytheas, LeVerrier, Helicon, Diophantus, Reiner, Herman, Galilaei, and Schiaparelli. Shallow Maria: Briggs B, Dawes, Harpalus, Plinius, Daniell, Madler, Cassini A, Thaetetus, Mosting, Flamsteed, Konig, and Protagoras.

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amplitude distributions (shape frequency distributions) that characterize mare and highland craters (Fig. 13) show that both populations contain craters that represent the same range of amplitudes; the same range of shapes is present in each population. Differences between the histograms occur primarily in the abundance of craters in each tail. The highland population contains a higher percentage (-20% more) of irregular craters (high amplitudes) and a lower Mare (n=i24) percentage of nearly circular craters (low amplitudes) than does the mare population. Observed differences between mare and highland shapes probably arise principally from differences in topography. However, structural dissimilarity between undeformed flood basalt and older, more complexly deformed highland crust might also contribute to observed shape variation, particularly at low-order harmonics (3 u and 4). Similarities in the range and shape of distributions that a* underlie both populations (Fig. 13) suggest that a large portion of craters in the highland samples ("•-80%) either are unaffected by topography or are affected to only a limited extent. A second comparison was made between craters that overlap older craters, and craters that show no apparent overlap to define further the influence of topography on shape. Among the most clearly defined and most pronounced topographic features on the lunar surface are crater rims. The internal morphology of craters that postdate and straddle the rim of older craters typically reflects the influence of the underlying, older crater rim. As such, crater rims might represent a dominant topographic contributor to crater shape variability. Chi-square analysis shows that shape characteristics displayed by the two groups of craters are virtually identical. Significant differences occur only in the tenth harmonic at the 0.05 level of significance (Fig. 5, B). The observed similarity in shape suggests that whereas impacted rims affect internal crater morphol- AMPLITUDE ogy, their effect on the outline of overlapping crater rims is minimal. (HARMONIC 4) Figure 13. Shape-frequency histograms of amplitudes of the Postformational Modification of Crater Morphology 4th harmonic for craters grouped by location. The highland population contains a greater percentage of quadrate craters (high ampli- The slow but continuous action of micrometeorite bombard- tude, right tail) than the mare population. The relative abundance ment, flooding of the crater floor by upwelling magma, the forma- of noncircular craters in the highlands is attributed to irregular tion of adjacent and overlapping impact craters, scour of the crater topography and complex structural relationships that are character- rim by debris ejected from other impacts, and significant regional- istic of the highland target. scale tectonism each holds potential to alter crater shape. The nature and extent of shape alteration varies significantly among Fielder and Jordan, 1962; Ronca and Salisbury, 1966; Adler and degradational processes. Salisbury, 1969), specific shape components (harmonics) that General Degradation and Aging of Crater Rims. Ongoing sur- change with time have not been defined. ficial lunar processes progressively degrade crater morphology. An estimate of the extent to which degradational processes Crater rims, walls, terraces, and peaks are eroded by meteorite and alter crater shape-in-plan was made in two comparisons of factor micrometeorite bombardment and become mantled by debris scores descriptive (1) of craters at various stages of degradation, ejected by these and other impacts. Sporadic catastrophic events and (2) of craters of various age. In the first comparison, the classi- (ejecta scour from basin-forming impacts, flooding of the crater fication system of Arthur and others (1963) was used to divide the cavity) also leave their mark. The over-all effect is one of aging. crater sample into degradational classes based on the completeness Dominant features that initially are sharp become softened and of crater rims. "The craters with complete and sharply defined rims appear less distinct. Ephemeral features such as hummocky depos- are classed as 1, while craters whose rims are either blurred or its of ejecta are effaced completely. broken are classed as 2 and 3. Objects which are usually described Such morphologic changes were well documented by Schultz as ruins are classed as 4, while class 5 covers objects that are so (1976) and form the basis for classification schemes that link crater battered or fragmentary that they are not easily recognized as morphology to relative crater age (Baldwin, 1949; Arthur and oth- former craters" (Arthur and others, 1963, p. 76). Only a small ers, 1963; Pohn and Offield, 1970). The extent to which changes in number of craters investigated here are classed as type 5. Thus, morphology affect crater shape-in-plan has not been defined with comparisons presented are made between the first four classes only. equal clarity. Although pre-Apollo work suggests that old craters The percentage of irregularly shaped craters in each degrada- are less circular than their younger counterparts (Fielder, 1961; tional class is plotted for both factors in Figure 14. Irregular craters

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are defined as craters that have positive factor scores for the particular factor considered. Negative scores characterize smoother 50- craters. The elapsed time between consecutive degradational classes œ -Factor I < is not known. Thus, degradational classes are plotted at equal inter- -Factor II vals for lack of a substantive time-scale model. 3 40- Data presented in Figure 14 corroborate the findings of pre- O vious workers that suggest an increase in shape irregularity with increasing crater age (Fielder, 1961; Fielder and Jordan, 1962; SÉ 30. Ronca and Salisbury, 1966; Eppler and others, 1977). The relationship between the two curves in Figure 14 indicates that the effect is more pronounced for moderate-scale roughness (factor I harmon- 20- ics) than for shape components that describe polygonality and elongation (factor II harmonics). This result probably reflects the 10 fact that more energy is required to efface and create large-scale —I— CO ER IM Pl features than small-scale features. Thus, fundamental shape characteristics described by factor II are likely to persist relatively RELATIVE AGE unchanged by degradational processes longer than factor I harmon- Figure 15. Percentage of irregular craters plotted as a function ics that describe smaller scale roughness. of relative crater age. In general, old craters are more irregular than The second comparison is between factor scores of craters of young craters. However, the Copernican population contains a high different age. Craters were assigned to one of four time- percentage of irregular craters. stratigraphic categories (Shoemaker and Hackman, 1962) on the basis of their relative age as indicated on U.S. Geological Survey higher than would be predicted from the trend of decreasing irregu- Geologic Maps (Copernican, Eratosthenian, Imbrian, and pre- larity with decreasing age established by pre-Imbrian, Imbrian, and Imbrian). Then the percentage of irregular craters (factor score Eratosthenian craters (Fig. 15). The inordinately high percentage of >0.0) was computed for each age group and plotted as a function of irregular Copernican craters could arise in one of three ways. First, age (Fig. 15). In general, the age comparison shows the same trend the results could imply that morphologic changes that result from as the degradational class comparison—crater irregularity increases progressive degradation occur two phases—one characterized by with crater age. A higher percentage of old craters (pre-Imbrian and increasing circularity (Copernican to Eratosthenian) and the other Imbrian) are irregular in plan than are young craters (Eratosthenian by increasing irregularity (Eratosthenian to pre-Imbrian). During and Copernican) for both factors. Factor I harmonics also are sub- the first phase, circularity might increase as perched slumps collapse ject to greater modification with time than factor II harmonics. and other unstable portions of crater walls fail. Then, as ongoing The relationship shown in Figure 15 differs from that in Figure surficial processes further degrade the rim in phase two, new irregu- 14 in that the percentage of Copernican craters that are irregular is larities would develop. The Copernican difference in irregularity also could arise from a fundamental change in the way lunar crust responds to impact • Factor I processes. For example, progressive gardening of the upper crust Q£ < Factor II through time could alter the structural state of impacted material. —I 3 60- Or a decrease in lunar heat flow could affect physical characteristics O of crustal material. However, such changes would be more likely to LU OC occur earlier in lunar history than during the Eratosthenian- — 50. Copernican interval and so be expressed in the shape of older craters. Finally, the observed relationship could result if key irregular U 40 Copernican and Eratosthenian craters were misclassified with respect to age. Although assignment of relative ages is interpretive and not absolute, an error of this magnitude is not likely. Estab- lished geologic mapping techniques were used to determine crater ages according to well-defined criteria (Shoemaker and Hackman, 20' -1- 1962; Pohn and Offield, 1970; Wilhelms and McCauley, 1971). I III IV Modified and Lava-Flooded Craters. Upwelling magma that Pristine Degraded floods crater cavities through fractures in and beneath crater floors CLASS modifies crater shape by initiating slumps along crater walls. Such Figure 14. Percentage of irregular craters plotted as ai function modification is common in those craters that predate mare flooding of degradational class. Definition of degradational classes is based and that are located either adjacent to or in the maria. The degree to on rim completeness as defined by Arthur and others (1963)i Irregu- which craters investigated here are flooded varies. Highly modified lar craters are defined as craters that exhibit a positive score on the craters such as Daguerre and Taruntius are completely flooded. particular factor considered. Craters become more irregular in They are characterized by shallow floors of low relief that exhibit shape as the severity of degradation increases. However, factor I ring or moat structures (Schultz, 1976). Floors of less severely mod- harmonics show more substantial modification than factor II ified craters such as Ritter and Sabine typically are deeper, show harmonics. greater relief, and are characterized by abundant fractures. Frac-

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SLIGHT MODIFICATION indicate that slumping typically occurs primarily at the time the crater is created (Schultz, 1976). Craters in which slumping clearly postdates crater formation exist but are not common (Dawes, Mosting). Such postformational slumps occur soon after crater formation as evidenced by the absence of fresh slumps on old crater floors. Postformational slumps typically are small and do not involve large portions of the crater rim. As such, only mid- to high-order harmonics are likely to show the effect of late-stage slumps. Small craters that occur on the rim of older craters represent a more significant problem in that they destroy information regard- EXTENSIVE MODIFICATION ing the shape of the older crater. Previous work suggests that shape can be described accurately at a scale defined by factors I and II Figure 16. Comparison of slightly modified and extensively (Figs. 3 and 4) with as much as 20% of the crater outline obliterated or obscured (Eppler, 1980). Craters with greater than 15% of the modified craters. Shape similarity between the two crater groups periphery missing are excluded from the sample used in this study. suggests that mare flooding produces negligible increases in circu- Fifty craters included here show overlapping impacts that destroy larity of crater outlines. From left to right, craters shown are: slight up to 15% of the crater wall. The shape of these 50 craters was modification: Posidonius, Damoiseau, Ritter, Lavoisier A. and compared with the shape of all other craters sampled using a chi- Davy; extensive modification: Pitatus, Cassendi, Encke, Daguerre, square test to verify the accuracy of the 20% figure. Shape differ- and Briggs. ences were found to exist at the 11th, 15th, and 19th harmonics (Fig. 5, A), or beyond the scale of shape variation measured by tures in such craters commonly are arrayed, at least in part, in factors I and II (harmonics 2 through 10). Therefore, overlapping concentric rings (Schultz, 1976). impacts do not make significant contributions to shape variability Visual comparison of modified and highly modified craters among craters studied here. defined on this basis suggests that highly modified craters are no Ejecta Scour. Scour by ejecta that impact and flow outward less irregular in plan than their less severely modified counterparts from major basin-forming impacts such as Imbrium and Orientale (Fig. 16). Use of chi-square techniques to verify that differences do modifies surficial lunar features. Evidence of the destructive nature not exist is precluded by the small sample of highly modified cra- of ejecta emplacement is recorded 1,500 km from the center of the ters. The small sample is due in part to the fact that many seveiely Imbrium basin in the northern portion of the southern highlands. modified craters lack complete rims and are excluded from the sample studied here. However, when all modified craters are com- MODIFIED UN-MODIFIED bined in a single sample, statistical comparison of modified and unmodified craters becomes possible. Such comparison shows that shape differences occur only at the 3rd and 10th harmonics (Fig. 5, C). This suggests that although flooding might modify the shape of some craters located adjacent to or in the maria, it is not a source of o o o ooo substantial shape alteration in a large number of craters investigated. Insufficient harmonics are affected for extensive shape changes to arise from flooding. Visual comparison of the outlines of o o o ooo a random selection of modified and unmodified craters supports this conclusion (Fig. 17). Nonetheless, significance of the 3rd harmonic results from a slight excess of circular craters in the modified-crater sample relative to the unmodified sample. Thus, o o o ooo flooding does increase circularity. Shape relationships defined by the 10th harmonic are less clear. Differences in the 10th harmonic between the two crater groups arise from differences in intermediate crater shapes rather than from the relative abundance of circular or o o o ooo irregular craters. Adjacent and Overlapping Craters. Formation of younger, Figure 17. Comparison of the shape of modified and unmodi- adjacent or overlapping impact craters can modify shape either by fied mare craters. Craters shown were selected at random from obliterating significant portions of the original crater wall or by subsamples of modified and unmodified mare craters. The similar- initiating wall slumps along pre-existing faults and fractures. Major ity shown between the two groups supports the conclusion based on wall slumps in existing craters probably are not common, although chi-square analyses that negligible shape modification results from failure of smaller perched slumps clearly occurs. Theoretical work -flooding of the crater floor. From left to right and top to bottom by Melosh (1977) and McKinnon and Melosh (1978) suggests craters shown are: modified: Bohnenberger, Daguerre, Sabine, strongly that major slumps in large lunar craters occur principally Archimedes, Keis, Plato, Vitello, Enke, Puiseux, Billy, Briggs, and at the time of the initial crater-forming event, not at a later time in Kopoff; unmodified: Lichtenberg, Harpalus, Kraft, Arago, Jansen response to younger events. In most craters, age relationships dis- B, Maskelyne, Bullialdus B, Birt, Triesnecker, Delisle, Timocharis, played by wall slumps and slumped material on the crater floor also and Helicon.

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In this region, Imbrium secondary craters and ejecta flow incised craters), were compared with craters that formed prior to the scour- parallel sets of linear grooves into the highland surface. Portions of ing event and thus were modified (46 craters). Significant shape crater rims are sculpted so as to be parallel to the direction of ejecta differences exist at the 6th, 8th, 9th, and 10th harmonics (Fig. 5, E), scour. Julius Caesar is a type example of the effects of this process or at a scale described principally by factor I (Fig. 3). Shape fre- carried to an extreme. quency histograms of these two crater groups are shown for each Two independent comparisons were made to determine the significant harmonic in Figure 18. At each harmonic, pregroove scale at which shape modifications attributed to ejecta scour occur. distributions are skewed strongly toward high amplitude values, First, the shape of all craters sampled from grooved terrain (80 indicating that abundant irregular craters are present. Postgroove craters) was compared with the shape of craters in groove-free ter- distributions are skewed just as strongly toward low amplitudes, rane (636 craters). These two groups differ only at the 15th har- indicating an abundance of craters that are more circular in plan. monic (Fig. 5, D), which indicates that shape variation among To obtain a graphic representation of these differences, the shape of craters in areas affected by ejecta scour is not significantly different selected craters from each group was reconstructed using only sta- from shape variation displayed by other lunar craters in the sample. tistically significant harmonics (6, 8, 9, and 10) (Fig. 19). The circu- That is, although the shape of specific craters (for example, Julius lar shape of postscour craters and the irregular shape of prescour Caesar) is affected by ejecta scour, the nature of shape variation craters is evident. displayed by craters in scoured terrane is the same as that displayed Major Regional Tectonism. Expansion or contraction of lunar by lunar craters in regions that have not been altered visibly by the crust that occurs on a regional scale could induce broad zones in effects of ejecta for the scale of shape features tested. This result which deformed craters would be measurably elongate in a com- could indicate that most nearside regions have been subjected to mon direction. Fielder (1961) and Fielder and Jordan (1962) show scour from impacts at some time in the past. Although grooves that that crater long-axes in the southern highlands and in the Hippar- are indicative of past episodes of widespread scour are not now chus region of the moon typically parallel lineations that define the visible, younger impacts could well hide or efface ancient grooved lunar grid system of Fielder (1963) and Strom (1964). Fielder (1961) terrane. and Fielder and Jordan (1962) suggest that these distorted craters The second comparison was made between two subgroups of record past episodes of crustal compression and extension. Ronca craters in grooved terrane. Craters in grooved terrane that postdate and Salisbury (1966) and Adler and Salisbury (1969) observed that grooving, and thus were not modified directly by ejecta scour (34 younger craters are more circular than old craters. Ronca, Adler,

~~HARMONIC 6 HARMONIC 8 HARMONIC 9 HARMONIC 10~~

~~AMPLITUDE~~

Figure 18. Shape-frequency histograms of pregroove and postgroove craters at harmonics in which chi-square analysis shows significant shape differences to be present. Pregroove histograms are skewed consistently toward higher amplitude values with respect to their postgroove counterparts. This indicates that craters modified by ejecta scour typically are more irregular in plan than craters formed on the same surface after grooves have been incised by ejecta scour.

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~~HARMONICS 6,8,9,10 HARMONICS 1 THRU 20~~

Figure 19. Shape differences between prescour and postscour craters developed on the same substrate. The shape of craters randomly selected from each group was reconstructed (a) using all 20 harmonics and (b) using only statistically significant harmonics (6, 8, 9, and 10). The higher degree of irregularity displayed by prescour craters with respect to the postscour group is evident. From left to right and top to bottom, craters shown are: prescour: Snellius A, Taylor, Euctemon, Saunder, Parrot C, Sporer, Schickard, Vieta A, Piazzi C, Inghirami, Pingre, and Vasco de Gam- ma; postscour: Stevinus, Neander, Al- fraganus, Theon Junior, Agrippa, Mani- lius, Conon, Sirsalis, Damoiseau, Byr- gius R, Riccioli H, and Krasnov.

and Salisbury also interpret this finding as evidence that significant shape variation are marked by solid blocks. Shaded blocks indicate compression and extension of the lunar crust occurred at some time variables that make secondary contributions to over-all shape vari- in the past. ability. Dominant shape contributors are defined as variables Data presented here (Figs. 14 and 15) corroborate Ronca and whose imprint is reflected clearly in the shape of most, if not all, Salisbury's (1966) and Adler and Salisbury's (1969) finding that craters in all regions of the lunar nearside. Dominant contributors young craters typically are more circular than old craters. However, affect multiple harmonics and define fundamental shape features. our data do not suggest that the observed age-shape relation is indicative of postimpact crustal deformation for several reasons. First, craters that are distinctly elongate fail to cluster within well- defined areas as would be expected if lunar tectonism was a con- FACTOR II FACTOR I tributing cause of elongation. Second, although elongate craters oc- 2 3 4 6 7 8 9 10 cur in the crater sample, observed deviations of crater outlines from Oblique Impact circularity are predominantly polygonal, not elongate. Finally, Projectile Shape & Size O although crater elongation does appear to be related to geologic • 30 Projectile Velocity structural patterns in lunar crust (Figs. 10 and 11), observed rela- >2 tionships between structure and elongation can be explained by Geologic Structure H O mechanisms other than postimpact deformation of the crust. Mod- Layering ? • o z els presented here (Figs. 8 and 9) place the imprint of geologic Topography ? ? structure on crater shape at the time of formation. Experimental evidence (Gault and others, 1968) and field evidence (Shoemaker, Flooding 1960; Scott and others, 1977; Roddy, 1978) exists to support these S Adjacent Impacts o models. g Overlapping Impacts -n r> Ejecta Scour > DISCUSSION Early Degradation zO Figure 20 summarizes the effects on shape components by Continuing Degradation variables considered in this study. Variables are divided into the general groups discussed in the text. Variables of formation alter crater-forming processes and so determine initial crater shape. Var- Figure 20. Summary table of shape-affecting variables. Solid iables of modification are active throughout the life of a crater and blocks mark dominant sources of shape variation. Secondary con- alter shape features determined originally by formational variables. tributors are indicated by shaded blocks. Question marks indicate Harmonics that carry the imprint of dominant contributors to possible sources of shape variation. See text for discussion.

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Secondary contributors affect isolated harmonics scattered through 1. Factor I measures transient shape features that reflect surfi- the shape spectrum. Their effect is less pronounced and is present in cial lunar processes of degradation and aging. a relatively small number of craters. 2. Low-order harmonics measured by factor II (2, 3, 4, and 6) Geologic structure, ejecta scour, and degradational processes primarily reflect variables that affect the crater-forming process. that alter shape through time are the dominant contributors to Polygonal shape features that are described by factor II are not crater shape variation. Each affects a major portion of the shape transient but persist for the life of the crater. The dominant con- spectrum. Furthermore, their effect is widespread, if not universal, tributor to factor II variation is structural variation in the target. throughout craters sampled. Oblique impact, topography, layered 3. Large projectiles are more likely to produce craters with substrate, and mare flooding make relatively minor contributions small-scale irregularities than are small projectiles. This size-shape to crater shape. Oblique impact and topography each affect fewer relationship is defined well only above the 9th harmonic. than 25% of all craters. Modification by mare flooding is confined Intermediate-scale features described by factor I are involved only largely to the maria and affects only 5% of the craters sampled here. marginally by projectile size. Polygonal aspects of shape (factor II) Although the effect of layered substrate is not well defined, data are largely unaffected. presented above suggest that at most it contributes to variation in 4. Craters in grooved terrane that postdate grooving (and thus scattered harmonics. Harmonics described by factor I reflect pro- were not modified directly by ejecta scour) are more circular than cesses of shape modification almost exclusively. Ongoing surficial craters in the same area that formed prior to scouring (and thus processes of degradation and aging, and catastrophic episodes of were modified). Shape differences between these groups exist in the ejecta scour and emplacement, are the principal contributors to 6th, 8th, 9th, and 10th harmonics, or at a scale described principally variance described by factor I. As such, factor I measures transient by factor I. features that change over the life span of the crater. The high por- 5. The range of shape variability among complex craters that tion of variance explained by factor I (68%) reflects this transiency. are located in regions modified by Imbrium and Orientale ejecta Polygonal shape features described by factor II tend not to be scour is the same as that characteristic of craters in unscoured transient. The dominant contributors to factor II are geologic struc- nearside regions. This might indicate that the shape of most old, ture and degradational processes that act soon after crater forma- nearside craters shows the effect of ejecta emplaced by basin- tion. The influence of geologic structure is cast in crater shape at the forming events that predate the Imbrium event. time of crater formation. Degradational processes that act soon 6. Degradational processes increase crater irregularity with after formation may alter polygonal shape elements somewhat, but increasing age. The effect is more pronounced for moderate-scale they are not likely to efface the imprint of structure completely. The shape irregularities (factor I harmonics) than for polygonal shape large scale of polygonal shape features insures that they will survive. components (factor II harmonics). Thus, fundamental, large-scale In some instances, polygonal features might be enhanced by degra- characteristics of crater shape are likely to persist through time in a dation as unstable wall segments slump along structural planes of relatively unchanged, undegraded state. weakness (Dawes, Mosting). Thus, factor II measures fundamental 7. Flooding of crater cavities by upwelling magma makes slight shape features that remain largely unchanged over the life of the modifications in crater shape. Minor changes that arise occur in the crater in spite of continued degradation of smaller-scale features. 3rd and 10th harmonics and result in slight increases in crater The permanence of polygonal features that reflect structural circularity. variables suggests that the shape of old craters is indicative of struc- 8. Shape irregularities that result from oblique impact occur tural patterns that existed in early lunar crust. That is, old craters principally as slight increases in amplitudes of the 2nd harmonic. could carry vestiges of lunar features effaced long ago by younger Between 6.7% and 25% of all lunar craters are affected according to impacts. The shape of younger craters would reflect remnants of calculations from theory. these same old patterns, if they persist, as well as younger features 9. Geologic structural relationships in lunar crust (faults, frac- imposed by more recent events. Polygonal aspects of crater shape, tures, joints) are a principal source of crater polygonality. Geologic when evaluated in this light, could define past episodes in the struc- structure affects crater shape according to two independent mecha- tural evolution of cratered surfaces. nisms: (1) during formation of the transient crater cavity, excavation occurs preferentially in directions parallel to trends of CONCLUSIONS structural weakness, and (2) in the course of modifications that occur to the crater cavity, walls fail and slumps develop along R-mode factor analysis of the first ten harmonic amplitudes trends of structural weakness. that describe the shape of 716 complex (diameter > 15 km) nearside 10. Topographic irregularities of the lunar surface affect the lunar craters shows that two factors describe 84.3% of the total shape of ~ 20% of all complex highland craters. shape variance observed within the sample. Factor I accounts for 11. Shape features related to adjacent and overlapping 68.2% of the observed shape variance and describes moderate-scale impacts, although undeniably present in craters sampled, are not irregularities (harmonics 7, 8, 9, and 10). Factor II accounts for consistently manifest in specific harmonics. Rather, they contribute 16.1% of the observed crater shape variance and describes crater' to noise in the shape spectrum. polygonality and elongation (harmonics 2, 3, 4, and 6). Systematic Results of this study demonstrate that abundant information analysis of the 716 lunar craters sampled permits the following regarding impact processes of crater formation and ongoing surfi- conclusions to be drawn regarding sources of shape variability de- cial processes of crater modification is stored in the rim-crest shape scribed by the first two factors: of impact craters. Much of this information is carried in discrete,

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nonrandom elements of shape. Fourier shape analysis is a powerful leum Engineers, p. 462-480. tool with which to define these elements quantitatively and to Eppler, D. T., Nummedal, D., and Ehrlich, R., 1977, Fourier analysis of lunar crater shape—Possible guide to impact history and lunar geology, extract the information they contain. The fact that shape features in Roddy, D. J., Pepin, R. O., and Merrill, R. B., eds., Impact and related to geologic structure can be so described by a single variable explosion cratering: New York, Pergamon Press, p. 511-526. (factor II) is of singular importance. It permits structural informa- 1978, Structural implications of lunar crater shape: Fourier crater tion contained in shape to be isolated from a field of conflicting shape analysis [abs.]: Houston, Texas, Lunar Science 9, The Lunar and variables and used with more conventional tools to define events in Planetary Institute, p. 294-296. Everitt, B. S., 1977, The analysis of contingency tables: London, Chapman the evolution of lunar and planetary surfaces. and Hall, 128 p. Fagin, S. W., Worrall, D. M., and Muehlberger, W. R„ 1978, Lunar ACKNOWLEDGMENTS mare ridge orientations: Implications for lunar tectonic models: Lunar Planetary Science Conference, 9th, Houston, Proceedings, p. 3473-3479. Work reported here was funded under NASA grant NSG-7076 F'echtig, H., Gault, D. E., Neukum, G., and Schneider, E., 1972, Laboratory from the Office of Space Science, Planetary Geology Program. simulation of lunar craters: Die Naturwissenschaften, v. 59, p. 151-157. Acquisition and analysis of data were supported in part by grants Fielder, G., 1961, The contraction and expansion of the moon: Planetary from the Visiting Scientist and Visiting Graduate Student programs and Space Science, v. 8, p. 1-8. of the Lunar and Planetary Institute. Lunar Orbiter imagery was 1963, Lunar tectonics: Geological Society of London Quarterly Jour- nal, v. 119, no. 473, pt. a, p. 65-69. supplied by the National Space Science Data Center. NASA, the Fielder, G., and Jordan, C., 1962, Selenological implications drawn from U.S. Geological Survey, and the Defense Mapping Agency pro- the distortions of craters in the Hipparchus region of the moon: Plane- vided Lunar Topographic Orthophoto Maps. H. J. Melosh, tary and Space Science, v. 9, p. 3-9. Richard Pike, and David Roddy provided constructive reviews that Gaudin, A. M., and Meloy, T. P., 1962, Model and a comminution distribu- improved the manuscript. Bonnie Head prepared typescript copies tion equation for repeated fracture: Society of Mining Engineers Tran- sactions, v. 223, p. 243-250. with consummate skill. A significant portion of this work formed Gault, D. E., 1973, Impact craters, in Greeley, R., and Schultz, P., eds., A the basis of a Ph. 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