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 structur- al 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. 274 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 sug- gests 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 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 276 EPPLER AND OTHERS 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.