Maps of Hectometer- and Kilometer-scale Topographic Roughness of the Moon

Lunar topographic roughness maps from Lunar Orbiter Laser Altimeter (LOLA) data: Scale dependence and correlation with geologic features and units Mikhail A. Kreslavsky a, ⇑ , James W. Head b , Gregory A. Neumann c , Margaret A. Rosenburg d , Oded Aharonson e , David E. Smith f , Maria T. Zuber f a Earth and Planetary Sciences, University of California, Santa Cruz, CA 95064, USA b Department of Geological Sciences, Brown University, Providence, RI 02912, USA c Solar System Exploration Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA d Division of Geological and Planetary Sciences, Caltech, Pasadena, CA 91125, USA e Center for Planetary Science, Weizmann Institute of Science, Rehovot, 76100, Israel f Department of Earth, Atmospheric and Planetary Sciences, MIT, Cambridge, MA 02139, USA article info Article history: Received 2 November 2012 Revised 8 March 2013 Accepted 20 April 2013 Available online 3 May 2013 Keywords: Moon, surface Regoliths Data reduction techniques Geological processes abstract We present maps of the topographic roughness of the Moon at hectometer and kilometer scales. The maps are derived from range profiles obtained by the Lunar Orbiter Laser Altimeter (LOLA) instrument onboard the Lunar Reconnaissance Orbiter (LRO) spacecraft. As roughness measures, we used the interquartile range of profile curvature at several baselines, from 115 m to 1.8 km, and plotted these in a global map format. The maps provide a synoptic overview of variations of typical topographic textures and utilize the exceptional ranging precision of the LOLA instrument. We found that hectometer-scale roughness poorly correlates with kilometer-scale roughness, because they reflect different sets of processes and time scales. Hectometer-scale roughness is controlled by regolith accumulation and modification processes and affected by the most recent events, primarily, geologically recent (1–2 Ga) meteoritic impacts. Kilometer-scale roughness reflects major geological (impact, volcanic and tectonic) events in earlier geological history. Young large impact craters are rough, and their roughness decreases with age. The global roughness maps revealed a few unusually dense clusters of hectometerand decameter-size impact craters that differ in their morphology and settings from typical secondary crater clusters and chains; the origin of these features is enigmatic. The maps can assist in the geological mapping of the lunar maria by revealing contacts between volcanic plain units. The global roughness maps also clearly reveal cryptomaria, old volcanic plains superposed by younger materials, primarily crater and basin ejecta. Ó2013 Elsevier Inc. All rights reserved. 1. Introduction Laser altimeter instruments onboard orbital planetary missions significantly advanced our knowledge of the Moon, Mars, Mercury. Topographic maps derived from the laser altimeter data are widely used in geological studies. In addition to the topographic maps, synoptic maps of topographic roughness can be useful. There are a few reasons, why in some circumstances the use of roughness maps is essential for geologic studies. First, roughness maps provide a convenient large-scale overview of small-scale textures. To map regional variations of textures solely with topographic maps, a geologist constantly needs to switch from large scale to small scales, which is time-consuming and inconvenient. Roughness maps give a generalized overview of texture variations at large scale. Second, roughness maps help to focus on typical topography rather than on peculiar features. When we look at topographic maps and images, our eyes see the most prominent features and often miss background textures. For example, when we look at the lunar highlands, we see the distinctive impact craters, and it is very difficult to ignore craters and focus on intercrater textures. Properly designed roughness maps display the most typical topographic textures and ignore rare features. Finally, roughness maps utilize the exceptional internal precision of laser altimeter data. The precision of the range determination along each spacecraft orbit is much higher than the accuracy of orbit knowledge, and the accuracy of the topographic maps is much worse than the internal precision of the original measurements. In addition, the gaps between orbit tracks are often wider than the distance between elevation measurements along the orbit, and the effective resolution of the topographic maps is worse 0019-1035/$ - see front matter Ó2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.icarus.2013.04.027 ⇑ Corresponding author. Address: Earth and Planetary Sciences, University of California–Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA. Fax: +1 831 459 3074. E-mail address: [email protected] (M.A. Kreslavsky). Icarus 226 (2013) 52–66 Contents lists available at SciVerse ScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarus than measurement spacing along each orbit track. Roughness maps rely on and utilize the exceptional internal precision and available along-orbit spacing of the orbital laser altimeter data. Roughness maps have proven to be useful in planetary geology. Kilometer-scale topographic roughness maps of Mars (Kreslavsky and Head, 2000) generated with Mars Orbiter Laser Altimeter (MOLA) data (Smith et al., 2001) clearly showed many geomorphologic units on Mars, for example, patches of heavily cratered Noachian-age volcanic plains are well distinguished from surrounding heavily cratered highlands. The maps revealed a latitudinal trend in topographic roughness, which was important in understanding the nature of recent climate change on Mars (Head et al., 2003). Cord et al. (2007) used stereo-derived digital elevation models to make decameterand hectometer-scale roughness maps of some martian terrains. Kreslavsky (2010) produced maps of topographic roughness of the Moon using data from laser altimeter LALT onboard Kaguya mission (Araki et al., 2008, 2009). These maps revealed the uniqueness of Orientale basin ejecta (Hevelius Formation) in comparison to other impact basin ejecta deposits on the Moon. The use of LALT data, however, is limited because of long distances between measurements along the orbit and a small total amount of data. Due to its exceptional vertical precision, short along-orbit spacing, and large volume of data, the Lunar Orbiter Laser Altimeter (LOLA) (Smith et al., 2010a,b) onboard LRO is an excellent data source for roughness mapping. The first roughness maps of the Moon with LOLA data were produced by Rosenburg et al. (2011). Kreslavsky and Head (2012) used LOLA-derived roughness maps to analyze the uniqueness of the Hevelius Formation roughness signature. Whitten et al. (2012) use LOLA-derived roughness maps in their analysis of cryptomaria. In this paper we (1) consider the technique of roughness mapping with LOLA data, (2) discuss the rationale for our choice of the statistical measure of roughness, (3) present global roughness maps of the Moon, (4) analyze the appearance of major geological features in these maps, and (5) discuss primary inferences about surface-shaping and modifying processes. 2. Mapping roughness In this section we first describe the algorithm we used to produce the maps of roughness. Then we present the rationale for our particular choice of the measure of roughness we use and the other algorithmic details. 2.1. Data processing We started with the whole set of topographic profiles along LRO orbits from LOLA Reduced Data Records (RDRs) available from the NASA Planetary Data System (PDS). Every time LOLA probes the surface (every ‘‘shot’’), it obtains range (and inferred elevation) measurements in 5 small (5 m) ‘‘spots’’; however, because of the LOLA anomaly (see Smith et al., 2010b for details), only spots #3 and #4 yield good range measurements on the night-side half of the orbits, and near-terminator data are mostly missing. We limited our data processing with a single spot #3: for this spot, the number of good measurements is the greatest. Some isolated data points inside ‘‘good’’ segments of the spot #3 profiles are also missing or marked as ‘‘bad’’ in the data set and discarded. In addition, there are rare cases where elevation measurements are incorrect, but are not marked as ‘‘bad’’. Maps presented here are made with the entire set of LOLA data obtained during LRO operations in the circular orbit from September 2009 to December 2011 (LRO orbits 1005–11,403), in total about 1.50 10 9 good shots. A typical distance between orbit tracks at low latitudes is on the order of 0.8 km, however, the distance varies widely, and there are gaps as wide as 4 km. Topographic roughness depends on spatial scale; this dependence bears essential information, as we will see later in the paper. To characterize it, we map roughness for a set of baselines l. Consecutive shots along each orbit are separated by 57.4 m; this distance is conveniently constant within ±3% through the whole data set. Our roughness measure uses baselines with an even number of shot-to-shot steps. Thus, the shortest baseline we can use is 2 shot-to-shot steps, that is l= 115 m. In addition to this, we also present maps of roughness at baselines of 8, 16, and 32 shot-toshot steps, that is l= 0.46 km, 0.92 km, and 1.8 km. For each shot, we calculated a proxy for the second derivative (‘‘curvature’’), c, of along-orbit topographic profiles, according to the equation: c¼ðh þ þh  2hÞ=l 2 ;ð1Þ where h,h + , and h  are surface elevations at the given laser shot, and shots a half-baseline ahead and a half-baseline behind, respectively. If any of h,h + , and h  were missing or marked as bad in the LOLA data set, we discarded such a curvature value. In total, about 6% of data points were discarded in this way. For lin (1) we used actual horizontal distance between the shot locations at h + , and h  . As we noted above, its difference between this actual distance and the ‘‘nominal’’ baseline of 115 m, 0.46 km, 0.92 km, and 1.8 km does not exceed 3% and causes no bias. We built the global maps in different map projections; nominally, we used the scale that correspond to 8 pixels per degree sampling, in other words, 3.8 3.8 km 2 pixel size at the projection’s standard point or line. For each map pixel, we found all LOLA shots located within the distance R pix from the center of the pixel. The minimal reasonable R pix is p2/2 of the nominal pixel side: for shorter R pix some data would be unused. We used this minimal R pix = 2.7 km for all maps, except the longest baseline of l= 1.8 km, for which we used R pix = 3.8 km. For the minimal R pix the majority of shots belong only to one pixel, but some of them belong to 2 pixels. Typically, 2–8 orbits contribute to a pixel (except high latitudes); each orbit typically adds a few tens of data points. Some pixels (1%, depending on the map projection used) have no data. We also considered pixels having too few points (less than 20) as having no data. Curvatures cwithin the R pix -vicinity of each pixel center are scattered around zero; for rougher surfaces, typical absolute values are higher, and hence the scattering is wider. For each pixel we considered the frequency distribution of the curvature c, and calculated the quartiles c 1/4 ,c 3/4 of this distribution. We use the interquartile range of this distribution, c 3/4 c 1/4 , as a measure of the distribution width and thus a measure of roughness. The numerical values of c 3/4 c 1/4 are not intuitive and difficult to conceptualize, and we normalized them by a typical value for typical highlands. Thus, we defined roughness rat each pixel as: r¼ðc 3=4 c 1=4 Þ=r 0 ;ð2Þ where r 0 = 2.3 10 4 m 1 , 6.6 10 5 m 1 , 4.4 10 5 m 1 , and 2.9 10 5 m 1 for l= 115 m, 0.48 km, 0.96 km, and 1.8 km baselines, respectively. The pixels with no data were filled using the roughness of their neighbors. The mean roughness calculated over all pixels with data in some vicinity of the pixel without data was used to fill it; the vicinity was progressively expanded starting from the smallest possible (four neighbors) until the pixel is filled. Data processing for a single global map takes a few hours on a high-end general-purpose personal computer. The resulting global maps (Figs. 1–3) are available on the Brown University Planetary M.A. Kreslavsky et al. / Icarus 226 (2013) 52–66 53 Geosciences web site http://www.planetary.brown.edu/ at http:// www.planetary.brown.edu/html_pages/data.htm. 2.2. Roughness maps Figs. 1–3 present roughness maps at baselines of 1.8 km, 0.48 km, and 115 m in Lambert azimuthal equal-area projection in a grayscale rendition, with brighter shades denoting higher roughness. The maps have a rather high dynamical range: by changing the stretch of the maps it is possible to reveal some details not readily seen in the figures. For example, white (rough) Copernican-age craters in Figs. 2 and 3 are saturated; a proper stretch reveals details, as shown in Fig. 9. The proper stretch of the 1.8 km baseline map in Fig. 1 can reveal significant roughness variations in the maria. Figs. 4 and 5 present RGB composites of roughness maps at 1.8 km, 0.96 km, and 0.48 km for red, green, and blue channels, respectively; stretch in each channel was chosen individually to optimize the visual perception of the maps. A greater intensity in each channel denotes higher roughness, which means that again, generally brighter shades correspond to generally rougher surfaces. Color variations characterize the scale dependence of roughness. Quantitatively the scale dependence of roughness can be characterized by so-called Hurst exponent H(see Rosenburg Fig. 1. Maps of kilometer-scale roughness of the Moon at two different baselines: top, 1.8 km and bottom, 0.46 km. Lambert azimuthal equal-area projections centered at the centers of the nearside (left) and farside (right). Latitude/longitude grid is 30°30°. Brighter shades denote rougher surfaces. Dimensionless absolute roughness values are defined according to Eq. (2). 54 M.A. Kreslavsky et al. / Icarus 226 (2013) 52–66 et al., 2011 and references therein), a dimensionless number between 0 6H61. Warm (reddish) tints in the maps (Figs. 4 and 5) mean relatively higher intensity in the red channel and the prevalence of large-scale roughness, which means higher H. Analogously, cold (bluish) tints mean the prevalence of small-scale roughness and lower H. Color variations are subtle because the range of baselines used is rather narrow. 2.3. Rationale for the data processing algorithm Our experience shows that the success of geological interpretation of roughness maps strongly depends on a good choice of roughness measure and an optimal choice of mapping technology. A good statistical measure of roughness should possess the following properties. (1) It should correspond to an intuitive perception of ‘‘roughness’’: if, in an image taken at the proper resolution, geological unit A looks distinctively rougher than unit B, then the roughness map at the proper baseline should reflect this difference. (2) The tilt of the surface as a whole should not change roughness values. For example, the variance of elevation at a given baseline is not a good measure of roughness, because it is sensitive to regional slopes. (3) Roughness should characterize a typical surface rather than its prominent features (as discussed in Section 1). For example, the RMS slope at a given baseline, a popular statistic of topography, is not a good measure of roughness, because it is very sensitive to the presence of a small proportion of very steep slopes in the sample; for the Moon, RMS slope would primarily reflect steep features (for example, crater walls). (4) Roughness should characterize topography at a well-defined scale, because the nature of its scale-dependence is an important characteristic of geological surfaces. (5) The statistical measure of roughness should be stable: if there is a homogeneous geological unit, its roughness calculated over a large data set and over a small (but representative) subset of the same data should be similar. For example, RMS slope is often not stable: natural surfaces often have heavy tails of slope–frequency distributions, and the occurrence of rare very steep slopes offset the RMS slope significantly. For many natural terrains the tails of the slope–frequency distribution are so heavy that the RMS slope calculated over a smaller data subset is systematically lower than that calculated over the whole data set. (6) The selected measure of roughness should be tolerant to individual peculiarities of the source data set used. Mapping of roughness (rather than just characterizing different areas) adds two more requirements. (7) Low noise (which is partly, but not exactly, the same as the stability requirement above). (8) Visual sharpness of resulting maps. Our experience shows that the latter is extremely important for successful geological interpretation of the maps. From this ‘‘definition’’, it is clear that there is no universally good measure of topographic roughness. First, ‘‘intuitive perception of roughness’’ in requirement (1) is quite subjective, and different researchers may have different personal preferences. Second, different data sets have individual problems and peculiarities, Fig. 2. Map of hectometer-scale roughness of the Moon (115 m baseline). Lambert azimuthal equal-area projection centered at the center of the nearside. Latitude/longitude grid is 30°30°. Brighter shades denote rougher surfaces. Dimensionless absolute roughness values are defined according to Eq. (2). M.A. Kreslavsky et al. / Icarus 226 (2013) 52–66 55 and requirement (6) dictates that different measures are used with different data sets. Requirements for a ‘‘good’’ measure of roughness are often contradictory, and the choice involves multiple trade-offs. In the case of LOLA data, the high regularity and precision along the orbit tracks and lower precision and irregular gaps between the orbits suggest the use of along-track statistics for characterization of the surface topography. Since all LRO orbit tracks are in the north–south direction, this means that the measure of roughness used is anisotropic, which is not consistent with our intuitive perception of ‘‘roughness’’. The spurious effect of the anisotropic roughness measure is noticeable, for example, on walls of large (resolved) craters in the longer-baseline maps: the northern and southern walls of the craters are rougher than the western and eastern walls. Topographic data derived from stereo images (e.g., Scholten et al., 2012) are (almost) free from such inherent anisotropy; their use for roughness mapping, however, is limited due to the lower vertical precision in comparison to LOLA. The ‘‘full’’ LOLA shots with good range measurements for all five spots could give a much more isotropic roughness measure, but only for a single baseline of 50 m. Since we aimed to study the scale-dependence of roughness and wanted the same roughness measure at different baselines, we restricted ourselves to bare one-spot along-track profiles. To meet requirement (2) we followed Kreslavsky and Head (2002) and chose to use the second derivative (‘‘curvature’’) of topographic profiles. Another possibility to meet requirement (2) is the differential slope used by Kreslavsky and Head (2000) and Rosenburg et al. (2011). The use of the differential slope instead of the ‘‘curvature’’ yields very similar-looking maps. Calculation of the ‘‘curvature’’ requires at least three data points (Eq. (1)), while calculation of the differential slope requires at least four; because of this, the ‘‘curvature’’ can be a little more tolerant of missing and bad measurements and this is why we prefer to use it here. A popular alternative way to fit requirement (2) is to use slopes (the first derivative) after some detrending procedure that removes large-scale tilts. We preferred not to use this approach because this method gives a worse scale separation (requirement 4) and introduces new subjectivity in the choice of the detrending scale and procedure. The shortcoming of our choice is that the numerical values of our roughness measure are not intuitive. Calculation of the second derivative at a given baseline is actually an application of some linear filter: a sequence of discrete points representing a profile is convolved with some kernel. This kernel can be chosen in different ways. We chose the minimal kernel defined by Eq. (1): it contains the fewest possible number of points (three) and has the shortest possible support range for a given baseline. The former gives us the best tolerance to missing and bad points (requirement 6); the latter enhances the visual sharpness (requirement 8). The minimal kernel, however, is far from optimal from other points of view: including more points in the kernel would reduce noise (especially, for long baselines) (requirement 7) and Fig. 3. Map of hectometer-scale roughness of the Moon (115 m baseline). Lambert azimuthal equal-area projection centered at the center of the farside. Latitude/longitude grid is 30°30°. Brighter shades denote rougher surfaces. Dimensionless absolute roughness values are defined according to Eq. (2). 56 M.A. Kreslavsky et al. / Icarus 226 (2013) 52–66 better separate scales (requirement 4). Fig. 6 compares our minimal kernel with the popular ‘‘Mexican hat’’ kernel in the spatial frequency domain. The spectra shown in Fig. 6 are ideal power spectra of the result of application of the filters to Brownian motion profiles; similar spectra are expected for real surfaces with a Hurst exponent of H0.5. It is clearly seen that in our case there is some leak of the high-frequency (short baseline) topography, while the ‘‘Mexican hat’’ kernel performs better. The use of the ‘‘Mexican hat’’ kernel, however, is not practical, because at longer baselines it involves many points. If we discarded all sliding windows containing at least one missing point, too few windows would survive. If we somehow replaced missed or bad points using their neighbors, we actually would have different kernels in these cases, and the statistics of the missing/bad points could affect the result: the algorithm could produce artifacts. We decided to sacrifice all advantages of the ‘‘Mexican hat’’ and similar kernels to be sure that our results are free of any artifacts related to missing and bad points. Our choice of the interquartile range (Eq. (2)) as an estimator for the distribution width works perfectly with respect to the requirements of typicality (3), stability (5), tolerance to unmarked bad measurements (6) and visual sharpness (8). We tried different percentile points a and 1  a instead of quartiles a = 0.25 and 1 a = 0.75. The resulting maps were almost indistinguishable with minor variations of the noise level. The lowest noise was for a close to 0.25, thus quartiles are about optimal with respect to the noise requirement (7). The choice of map pixel size (8 pixels per degree) was dictated by the data point density. The choice of R pix is a trade-off between visual sharpness of the maps (that requires shorter R pix ) and better statistics/lower noise (that requires longer R pix ). We chose R pix on the basis of several trials. When maps are analyzed visually as grayscale or color images and compared with images, mosaics of images, and geological maps, a rather high noise level is tolerable, but visual sharpness is highly desirable. To maximize the visual sharpness we used the minimal R pix = 2.7 km for all maps, except the longest baseline of l= 1.8 km. The latter baseline becomes comparable to the pixel size of 3.8 km; this causes a strong increase in noise; actually, we start seeing the presence/absence of individual sharp topographic features in the footprint rather than typical roughness. Because of this we decided to use a slightly wider vicinity R pix = 3.8 km for this baseline. 3. Roughness signature of resurfacing processes and time scales The most obvious feature of the kilometer-scale (0.48 km, 0.96 km, and 1.8 km baselines) roughness maps (Figs. 1, 4, and 5) is the dichotomy between smooth (dark) maria and rough (bright) highlands. At 0.48 km baseline, typical values of highland roughFig. 4. Kilometer-scale roughness of the Moon rendered as an RGB composite of roughness maps at 1.8 km, 0.96 km, and 0.48 km for red, green, and blue channels, respectively; higher intensity in each channel denotes higher roughness. Lambert azimuthal equal-area projection centered at the center of the nearside. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) M.A. Kreslavsky et al. / Icarus 226 (2013) 52–66 57 ness are within the range of 0.9–1.1, and that of maria are within the range of 0.4–0.5; in other words, the systematic difference between maria and highlands (a factor of 2) is much greater than the typical roughness variations over maria and over highlands (20% in both cases). For longer baselines the systematic difference between maria and highlands increases and reaches a factor of 10 at 1.8 km baseline. At this baseline, typical short-range variations of highland roughness are significant (a factor of 3) due to an effective noise increase when the baseline becomes comparable to R pix . However, typical values over larger areas vary within 30% for both maria and highlands and are minuscule in comparison to the systematic difference between maria and highlands. The maps of hectometer-scale roughness (115 m baseline, Figs. 2 and 3) look strikingly different, despite the fact that the baselines differ only by two octaves (a factor of four) from 0.48 km. At the hectometer scale, there is no significant difference between maria and highlands. In many locations the mare/highland boundary is not associated with any roughness contrast. All typical roughness contrasts are minor (within 40%, compare gray scales in Figs. 1 and 2), except that of young craters (Section 4.1). Although lowered roughness values tend to be associated with maria, some mare surfaces are rougher than typical highlands, for example, the central part of Mare Crisium has r1.1. We interpret this striking difference to be due to the principal difference in the processes that shape the surface at hectometer and kilometer scales. The hectometer scale is dominated by formation of small (hectometer-scale and smaller) craters and regolith gardening by even smaller impacts. Fig. 7 illustrates that the difference between typical mare and highland surfaces, considered at the resolution relevant to the hectometer-scale roughness, is not very strong. These LRO Lunar Reconnaissance Orbiter Camera (LROC) images were specially selected to minimize the difference between illumination conditions, so that their direct comparison is most useful. The typical values of dimensional roughness, r 0 210 4 m 1 at l= 115 m baseline correspond to a characteristic vertical scale of r 0 l 2 3 m, less than a typical regolith thickness. This is consistent with regolith processes being solely responsible for the roughness signatures at the hectometer scales. Both on maria and highlands the population of hectometer and smaller craters is in an equilibrium state, when ongoing emplacement of small craters is balanced by ongoing obliteration of old craters by regolith gardening. Progressively degrading hectometer-scale craters form the background topography in a similar manner on the mare and highland surfaces. Formation of larger, kilometer-scale craters is not equilibrated with crater obliteration, and there is no uniform equilibrated background topography at such scales. At the longest baseline, l= 1.8 km, the characteristic vertical scale, r 0 l 2 90 m, is greater than the typical regolith thickness. This means that regolith Fig. 5. Kilometer-scale roughness of the Moon rendered as an RGB composite of roughness maps at 1.8 km, 0.96 km, and 0.48 km for red, green, and blue channels, respectively; higher intensity in each channel denotes higher roughness. Lambert azimuthal equal-area projections centered at the center of the farside. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 58 M.A. Kreslavsky et al. / Icarus 226 (2013) 52–66 processes are not responsible for the roughness values at this scale (though they may slightly affect it). At the scale of kilometers, roughness reflects ‘‘bedrock geology’’: the surface is shaped by volcanic flooding of maria, large impacts, and tectonics. Exact thickness of regolith layer in highlands is poorly measured and perhaps poorly defined (due to the absence of uniform consolidated substrate); indirect estimates (e.g., Shkuratov and Bondarenko, 2001; Fa and Jin, 2011) yielded the typical highland regolith thickness of 10 m; recent measurements based on morphology of small craters by Bart et al. (2011) gave estimates within 6–8 m. As the characteristic vertical scale of roughness increases with horizontal baseline, the typical highland regolith thickness intersects this trend at 1 km baseline, indicating that roughness at scales below this value is controlled by regolith processes while roughness at larger scales is controlled by bedrock topography. Rosenburg et al. (2011) found that the Hurst exponent within the lunar highlands commonly transitions from a high value near 1 to a lower value of 0.8, with the transition occurring at approximately the 1-km scale, in good agreement with the roughness calculations presented here. This change in slope of the structure function is consistent with a transition between roughness regimes governed by different surface processes. Formation and degradation of hectometer and smaller craters, and formation and gardening of regolith, are universal processes that occur everywhere on the Moon, to a first order, in a uniform way. Due to this, the difference between hectometer and kilometer scales also reflects different time scales: the hectometer-scale roughness maps show features younger than the characteristic time scale of reworking of the uppermost layer of the regolith, on the order of a meter. Thus, this is a feature of Copernican and, at least partly, Eratosthenian age, while the kilometer-scale roughFig. 7. High-resolution images of typical mare (A) and highland (B) surfaces taken under similar solar incidence angles of 45°. Scale bars are similar to the baseline used for the roughness map in Figs. 2 and 3. Portions of LROC NAC images M137685293L (A) and M136572848R (B). Fig. 6. Left panels, three examples of spatial kernels of several linear filters for characterization of along-profile roughness at a given baseline l, top to bottom: 4-point differential slope (used by Kreslavsky and Head (2000) and Rosenburg et al. (2011)), 3-point curvature used in this paper and defined by Eq. (1), and ‘‘Mexican hat’’. Vertical scale is arbitrary. A baseline equal to 8 shot-to-shot distances is chosen in this example. Right panels, spatial power spectra of the result of the application of these linear filters to an idealized Brownian walk profile (‘‘brown noise’’). Spectra are normalized by the total power (that is, the areas below the curve are the same). M.A. Kreslavsky et al. / Icarus 226 (2013) 52–66 59 ness shows older features. From another point of view, this distinction represents the time scale of the equilibration of the small crater population. This clear separation of processes and ages causes the ‘‘roughness units’’ on hectometer-scale and kilometer-scale roughness maps to be different: boundaries outlining units of different roughness on kilometer-scale and hectometer-scale roughness maps often do not coincide. This situation differs significantly from Mars, where, despite a very wide diversity of scale-dependence of roughness for different terrains, the unit boundaries were the same over 4 octaves (factor of 16) of scales, and the color composite of three roughness maps separated by 2 octaves made a nice synoptic overview of martian geomorphologic diversity (Kreslavsky and Head, 2000). On the Moon combining hectometer-scale and kilometerscale roughness in a single composite does not produce easily interpretable maps. In the rest of the paper we consider scales separately: first, hectometer-scale roughness maps (Figs. 2 and 3)in Section 4and then kilometer-scale roughness (Figs. 1, 4, and 5) in Section 5. 4. Hectometer-scale roughness 4.1. Craters Both the roughest and the smoothest terrains on the Moon are associated with large, young impact craters. While the typical variations of roughness over maria and highlands are within the 0.7–1.3 interval, the roughness of the youngest craters and their vicinities varies over a much wider interval, 0.4–5. This observation is well in line with conclusions from Section 3: large young craters are the only resolved features young enough so that the accumulated regolith layer is thin, and regolith gardening processes have not brought the roughness values to equilibrium. The most prominent, unusually smooth crater-associated terrain is a large sheet of impact melt just outside the Copernicanage crater Rutherfurd (60.9°S 12.1°W, Fig. 8). The median roughness for the smooth area is 0.6, its smoothest part is 0.4. A similar but smaller melt sheet occurs just to the north of Copernican-age crater Glushko (8.4°N 77.6°W, former Olbers A). A few very smooth pixels are associated with large melt pools on the southeastern outer wall of Tycho and just to the north of King. With these exceptions, all other parts of the youngest craters are rough. Because the maps in Figs. 2 and 3 are stretched to show minor variations of roughness over maria and highlands, the rough young craters there are saturated white. Fig. 9 shows local maps for the largest (D> 70 km) Copernican-age craters; the grayscale is stretched to show crater details; the stretch is the same for all maps. Here, following Wilhelms (1987) and McEwen et al. (1997), we consider craters with prominent bright rays as Copernican; this is not consistent with the redefinition of the Copernican period by Stöffler and Ryder (2001). It is seen that there are significant variations of roughness both within the craters and between them. It is interesting that the inner walls, where they are resolved in the map, are smoother than the other parts of the craters (see the southern walls of Rutherfurd in Fig. 8, the northern walls of Tycho and southern walls of Hayn in Fig. 9). Unlike the smooth melt sheets outside the craters, the melt pools on the crater floors are rough. Craters in Fig. 9 are arranged according to their general decrease in roughness. Among the largest (D> 70 km) Copernican-age craters, the median roughness calculated over the whole crater is the highest for Tycho and Jackson, 3.4; it decreases to 2.3 for Copernicus and further decreases to 1.9 for Stevinus and Vavilov. Fig. 8. Crater Rutherfurd (D47 km) in the lower right part of the image and impact melt sheets outside the crater. Portion of LROC WAC mosaics centered at 60°S 13.5°W, local equirectangular projection, north at the top. Insert shows the hectometer-scale roughness (115 m baseline) for the same area; brighter shades denote rougher surface. This map was created in local equirectangular projection with 16 pixels per degree sampling, R pix = 1.9 km. 60 M.A. Kreslavsky et al. / Icarus 226 (2013) 52–66