Morphometric Characterization and Reconstruction Effect Among Lunar Impact Craters

Home , Apianus (crater), Barnard (lunar crater), Byrgius (crater), Cannon (crater), Cavalerius (crater), Cavendish (crater)

Earth Moon Planets (2014) 111:139–155 DOI 10.1007/s11038-014-9431-0

Weiming Cheng • Jiao Wang • Cong Wan

Received: 2 January 2014 / Accepted: 11 March 2014 / Published online: 19 March 2014 Ó Springer Science+Business Media Dordrecht 2014

Abstract Impact craters on the lunar surface have a variety of morphometric characteristics that are very useful in understanding the evolutionary history of lunar landscape morphologies. Based on digital elevation model data and photographs from China’s Chang’E-1 lunar orbiter, we develop morphologic parameters and quantitative methods for presenting the morphometric characteristics of impact craters, analyzing their relational distribution, and estimating the relative order of their formation. We also analyze features in proﬁle where craters show signs of having formed on the edge of previously existing craters to show that superimposed impacts affect morphologic reconstructions. As a result, impact craters have signiﬁcant effects on the reconstruction of ancient topography and the estimation of relative formation ages.

Keywords Morphometric characterization Position relationship Relative construction age Chang’E-1

1 Introduction

The Earth’s Moon, it’s only natural satellite, has a potentially complete record of the 4.5- billion year evolutionary history of the solar system (Ronca 1966; Ouyang 2005). Impact craters are the most obvious and typical geomorphologic units (Ronca 1969; Neukum and Ivanov 1994; Neukum et al. 1975); they form when a planetary body (meteoro, comet, etc.) impact against the surface (King 1976). The diameters of impact craters on the lunar

W. Cheng (&) J. Wang C. Wan State Key Laboratory of Resources and Environmental Information Systems, Institute of Geographic and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China e-mail: [email protected]

J. Wang C. Wan University of Chinese Academy of Sciences, Beijing 100049, China 123 140 W. Cheng et al. surface range widely from only a few centimeters to hundreds of kilometers (Peter 1999). For most craters, their rims are higher than the center; as the diameters increase, the morphologic characteristics of specific impact craters appear to become more complex (Oberbeck et al. 1974). The morphometric characteristics of impact craters can provide much information regarding impact characteristics and processes (Hartmann et al. 1981; Cameron 1984). Asteroids or comets striking planetary surfaces generally produce circular impact craters, even when those objects hit at angles substantially off vertical (Melosh 1996). Elliptical impact craters are produced when impacting angles are low relative to the horizontal, with the transition between circular and elongated craters occurring somewhere near 10° (Bottke et al. 2000). Such results have been confirmed by laboratory experiments con- sisting of small aluminum and Pyrex spheres shot at several km/s into sand or aluminum targets (Burchell and Mackay 1998). Burchell et al. (2010) further used laboratory methods to imitate the impacting process, with the same speed and angle of incidence, combined with extrapolating to the correct size scale to match the SMART-1 impact; this predicts a highly asymmetric crater approximately 5.5–26 m long, 1.9–9 m wide, 0.23–1.5 m deep with a volume of 0.71–6.9 m3. Craters constitute rays and impact units formed by small objects (Melosh 1996; Neukum and Ivanov 1994) and this information can be used to estimate the relative geologic ages of a given lunar region (Shoemaker 1965). The National Aeronautics and Space Administration (NASA 1969) used the ratio of impact crater depth to diameter as a determination of relative age, and divided the craters into four categories: fresh, young, mature and aged. Methods of determining an estimate ages for cratered planetary surface units is to find a size-frequency distribution for observed craters of a given surface unit with a known crater production function, and to use the crater frequency for certain crater sizes together with a calibrating chronology function to obtain an estimate formation age for the impact craters (Oberbeck et al. 1974; Hartmann et al. 1981; Neukum 1984; Neukum and Ivanov 1994; Hartmann and Neukum 2001; Ivanov 2001). Neukum et al. (1975) found that the lunar impact crater size distribution is largely constant in the size range 0.3 km B D B 20 km for regions with formation ages between *3 and C4 Ga, based on a calibration size distribution curve. The relationship between cumulative crater frequency N and crater diameter D can be expressed as a logarithmic function. Moutsoulas and Preka (1982) used the ratio intervals from small- and medium-scale craters for morphological descriptions. On different planets or over different terrains of the same planet, a change in the fractal dimensions of craters appears to indicate different geological settings related to the geological age. Ouyang (2005) established a relationship between the distribution density of impact craters with diameters [4 km and lunar relative geological age. As mentioned earlier, many investigations have focused on craters size-frequency distribution to derive estimates of relative formation age. However, there have been far fewer studies assessing crater statistics to include complex morphologic characteristics such as excavation shape and to explore profile morphologies and reconstruction effects for edge-intersected craters. Thus, in this paper, based on digital elevation model (DEM) data and photographs from Chang’E-1, China’s first lunar probe, we address this shortfall by developing a set of parameters for the morphometric characterization of impact craters, establishing a method to estimate the positional relationships and relative impacting orders of impact craters, and characterizing the profile morphologies of two edge-intersected impact craters.

123 Lunar Impact Craters 141

2 Morphometric Characteristics of Lunar Impact Craters

The Center for Lunar Science and Exploration at the Lunar and Planetary Institute, published a database of impact craters, which was initially created as part of the Lunar Exploration Summer Intern Program at the Lunar and Planetary Institute in 2008 (http:// www.lpi.usra.edu/lunar/surface/). The database consists of 8,713 craters, of which about 1,644 have constrained ages. In this database, 51 physical characteristics of the craters are listed. The main attributes include: name, diameter, depth, location, volume, central peak height and diameter, width of central peak, ray length, and age. Many of these properties can be expanded in detail, e.g., the diameter can include simple surface diameter, crater instant diameter, complex crater instant diameter, floor diameter, etc. Of these attributes, only the crater diameter, depth, positional coordinates, central peak height and age of strata are obtained from maps and imagery; other indices are calculated by theoretical predic- tions. Of the 8,713 impact craters identified, only 1,644 have an age attribute assigned and in some cases there are multiple geological ages proposed for the same impact crater. Thus, the statistical properties of these named craters can be used to characterize the basic features of lunar craters (Fig. 1). From Fig. 1, we find that 36 % of lunar crater diameters are \10 km, and that as diameters increase the abundance of lunar craters gradually decreases. Against crater depth, with the enlargement of impact crater diameter crater, average depth of craters in each level increase gradually. This rule is also a more prominent appearance in central peak height and crater diameter. The central peak height of craters is generally\1 km and the percent of average central height in craters [60 km almost reaches to 30 %. The statistics above show that the larger the craters are, the deeper and the higher central peak the craters own (Head 1976). Based on the above analysis, this study selects the morphological indicators (including diameter and depth) of bowl-shaped impact craters to determine the specific numerical attributes from Chang’E-1 DEM data. This work can provide basic data for further study.

40%

35%

30%

25%

20% Percent 15%

10%

0% <10 10-20 20-30 30-40 40-50 50-60 >60 Diameter (km)

Percent of crater in different diameter class Percent of average depth in different diameter class Percent of average height of central peak in different diameter class

Fig. 1 Histogram of lunar crater attributes 123 142 W. Cheng et al.

Fig. 2 Horizontal projection and schematic proﬁle diagram of bowl-shaped craters (after Grosse et al. 2012)

Various complex morphologic relationships exist between the various impact craters on the lunar surface. Based on the morphometric parameters of the impact craters, the typical bowl-shaped crater models can be used to estimate the positional relationship and relative impact order between pairs of impact craters. A schematic diagram showing the vertical proﬁles and horizontal projections of two impact craters is presented in Fig. 2. Most of the impact craters on the lunar surface have a raised rim and low depression, whose dimensions depend on size, velocity from impacting object and age (Head 1976). A crater rim is deﬁned simply as the crater boundary in this paper. Based on DEM data, 11 morphometric parameters, including radius (R), depth (Dp), area (A), perimeter (P), roundness (r), elevation (E), diameter (d), length (L), rays (Rs), latitude (Lat) and longitude (Lon) are selected and calculated. The basic attribute indices of impact craters are presented in Table 1.

3 Relative Impacting Order Between Two Craters

3.1 Positional Relationship Between Two Impact Craters

Relationships between two polygons can be generalized into four types: contained, intersected, tangent and disjoint. Thus, the positional relationships between two impact craters can be characterized by these same type descriptions by comparing their centroid distance values and radius additions. The method for assessing this positional relationship is expressed in Table 2. Statistical analysis of impact carters from remote sensing imagery and DEM data indicates that there are few crater relationships on the lunar surface that can be characterized as tangent. As a result, in this study, the basic position relationships between two impact craters are simpliﬁed to disjoint and intersected types. Tangent types are regarded as disjoint and contained types are regarded as intersected (see Fig. 3).

123 Lunar Impact Craters 143

Table 1 Lists of morphometric parameters and their descriptions for the characterization of impact craters Parameter (unit) Description

Radius (R) Recorded as the radius of same area circle Depth (Dp) The elevation difference between crater rim and ﬂoor Area (A) Bounded by crater rim in the horizontal direction Perimeter (P) Perimeter in the horizontal direction Roundness (r) An assessment of the shape of the crater rim, expressed by P=ðÞ2 SQRT ðÞpA , standing for ratio of crater perimeter to the perimeter of a circle of equal area Elevation (E) Elevation of crater ﬂoor Diameter (d) Diameter of crater Length (L) Length of centroid of two craters Rays (Rs) Status of rays, with ray is 1, and without ray is 0 Latitude (Lat) Latitude of centroid of impact crater Longitude (Lon) Longitude of centroid of impact crater

Table 2 Method for deducing Position relationship Deduction method the positional relationship between two impact craters Disjoint L [ Ra ? Rb Ra, Rb are the radii of the two Tangent L = Ra ? Rb craters, respectively; L is the Intersected Ra - Rb \ L \ Ra ? Rb centroid distance of the two Contained L \ Ra - Rb impact craters

a b a b a b a b

Disjoint Tangent Intersected Contained Not superposed Superposed

Disjoint Intersected

III

Fig. 3 Positional relationships between two simple impact craters (I disjoint types; II intersected types)

3.2 Relative Impacting Order Between Two Craters

Impacting has a great influence on lunar surface morphology. Additionally, the ages of craters, which provide the timing of impacts, can be used to estimate a history of lunar morphological change. Over the course of lunar evolution, the lunar surface has been modified by many endogenic and exogenetic factors such as surface lava flows, blanketing by ejecta, superposition of craters, infilling and abrasion and mass wasting (Neukum and Ivanov 1994). Here, we focus on edge-intersected craters in order to constrain the relative 123 144 W. Cheng et al. impact order between two craters with a simple positional relationship. Such character- izations provide basic information for studies of lunar chronology. The edge-intersected method can estimate typical combinations between two impact craters. This information reveals whether or not the edge of a certain impact crater has been overlain by a younger crater, and whether or not the original crater edge was destroyed. Because of erosion and reconstruction, the rims of complex and relative older craters appear polygonal or irregular in shape. In contrast, relative younger craters often have rays and are round or have arc-shaped rims that are steep inside and shallow outside. Ronca (1969) classified the craters on the nearside of the Moon into five classes based on their degree of erosion, i.e., very sharp and fresh-looking craters, craters with blurred rims, craters with more extensively broken rims, craters usually described as ruins, and ghost craters that are so fragmentary that they are not easily recognized. With regard to interactions with secondary craters and calderas, the primary and newly formed craters can be distinguished from a number of ways; we can classify them based on four characteristics. The first one is their hue in remote sensing imagery. Relative older and/or younger craters can be distinguished by means of light or dark hues in a relative given small region. Relative younger craters appear brighter than their surroundings, showing that unweathered materials maybe be excavated and scattered around the impact area. However, in some highland-mare borders, the opposite process may occur because of aluminous materials from the highlands may lighten the optical appearance of the crater’s ejecta, it should be taken into consideration in the future. The second one is the appearance of rays. Lunar rayed craters are the relatively youngest features on the Moon, superposed to any other morphology (Shoemaker and Hackman 1962). Their visibility depends mainly on the state of optical maturity of the ray material; if the composition of the ray deposits differs from the surrounding terrain, they can remain visible longer, particularly over mare units (Hawke et al. 2004). The third one is the depth of the impact crater. According to the statistic in Fig. 1, we found that the deeper the crater is, the larger diameter the crater is. The fourth one is that older craters become more weathered and flat floor with time. The longer a crater is exposed on the lunar surface, the more likely the floor of the crater may be flat. Therefore, craters with floors containing a large number of small craters are relatively older. Younger craters can retain their original impact morphology for longer time as the rate of impacts decreases comparing to older ones from the perspective of long time span (i.e., billion of years). Synthesizing the above morphometric parameters, positional relationships and impact orders enables the establishment of a framework for estimating relative impact orders between craters. The framework of this method is shown in Fig. 4.

4 Case study: Positional Relationship and Relative Impact Timing of Two Craters

4.1 Data Source

During China’s ﬁrst lunar exploration mission, the ﬁrst map of the lunar surface was produced using a stereo charge-coupled device (CCD) camera on Chang’E-1 that was orbiting the Moon at an elevation of about 200 km. The CCD camera uses the linear push- broom imagery technique, with a scan track width of 60 km and a pixel resolution of 120 m. The DEM data from Chang’E-1 were determined from digital photogrammetric image data. The altitude of this DEM is based on a spherical surface within a radius of 1,737.4 km from the center of mass of the Moon; the spatial resolution is 500 m. The 123 Lunar Impact Craters 145

Crater attribute index Crater position relationship

Rays Roundness Depth-radius Intersected Disjoint (Rs) (r) ratio(D/R)

Ray older 0 1 Smaller superposed superposed Ray younger 1 1 Larger superposition superposition

Crater geological age estimation

Fig. 4 Framework flowchart for assessing relative impact order for impact craters horizontal error is 192 m and vertical error is 120 m (Li et al. 2010a, b). The Chang’E-1 first full coverage data are useful for lunar studies across multiple fields, and can make a significant contribution to the development of lunar science and future lunar exploration missions (Li et al. 2010a, b). In comparison, the Chang’E-1 laser altimeter DEM model is similar to the United States ULCN2005 model in terms of accuracy and resolution, and has almost the same accuracy and resolution as Japan’s SELENE laser altimeter DEM model. The highest point in the Chang’E-1 model is similar to that found in the SELENE model; however, the lowest point in the Chang’E-1 model is lower than that found in the SELENE model (Li et al. 2010a, b).

4.2 Setting of the Study Area

The study area is located in the middle-low latitude area on the near side of the Moon at 90°W–90°E, 45°N–45°S (Fig. 5). The terrain of the research area is complex, and ring-like structures are plentiful. In addition, the time span represented by impact craters is very long, with both the oldest pre-Nectarian strata and the younger Copernican craters present (Wilhelms 1987). Based on the DEM map in Fig. 5, the range of elevations present in the

Fig. 5 Digital elevation model (DEM) of study area from Chang’E-1 123 146 W. Cheng et al. study area is -6,658 to 4,450 m. About a total of 1,764 impact craters that have been identiﬁed (http://www.lpi.usra.edu/lunar/surface/). Although there are numerous impact craters on the farside of the Moon, their superimposed relationships are very complex. Therefore, it is not possible to analyze the positional relationships and relative impact order for all lunar impact craters. As a result, this paper focuses on relationships between pairs of impact craters to develop a method that can be extended to multiple crater analyses in the future.

4.3 Relative Impacts

In this study, 39 groups of impact craters from the study area are selected and presented to highlight edge-overlaid morphometric characteristics (Table 3) and 18 groups of impact craters were selected to illustrate the relative impact timing of pairs of craters (Table 4). After manually extracting the rim lines of impact craters, various parameters were determined by the methods presented in Table 1 and Fig. 2. The above mentioned information assists in estimating relative age of impact craters. We summarize a method of scientific inference used to deduce particular relative impact timing between craters. This inference process is described as follows. (1) The roundness of the crater rim can be expressed by r ¼ P=ðÞ2 SQRT ðÞpA , which stands for the ratio of the crater perimeter to the perimeter of a circle with the same area. When the r value is close to 1, the crater is more round. When r value is far from 1, the shape of the crater is more irregular, most likely because of the partial destruction of the ancient crater rim. (2) The floor shape is expressed by the ratio of the crater depth to its radius Dp/R. Because the floors of ancient craters are flat, the value of Dp/R is smaller for older craters. (3) Younger craters maybe have bright rays, while ancient ones have none. (4) The relative position of craters can also provide evidence for timing. For craters with an intersecting relationship, the upper crater is the younger. For craters with disjoint relationships, if both craters have rays, then ray superposition can be used to determine which one is younger. The relative order of impact for a pair of craters was inferred by the method laid out in Table 2 and Figs. 3 and 4. The DEM and image maps of these craters are shown in Table 3. Estimating relative impact order is shown in Table 4. From Table 4, we can exploit various derived ratios to extract trends concerning the relative age of impact craters. In study area, we arranged the Dp/R and r (degree of roundness) of 36 impact craters in ascending order (Fig. 6) and found that most craters are relatively older, formed prior to the Eratosthenian when Dp/R B 0.207 and r C 1.032. In contrast, craters that formed mostly in Eratosthenian and Copernicus times when Dp/ R [ 0.207 and r \ 1.032 are relatively younger. Therefore, Dp/R B 0.207 and r C 1.032 can be treated as a relative age boundary for the impact craters. In series of older impact craters, there are just four younger impact craters, which is similar to that presented in series of younger impact craters. This phenomenon is likely the result of determining relative age based on a set of superposition criteria that do not include all 36 of the impact craters in Table 4. According to the geological age released by the IAU, these eight craters are still regarded as the corresponding series divided by the critical point Dp/R is 0.207, where r value is 1.032.

4.4 Proﬁle Analysis of Edge-Intersected Craters

Edge-intersected relationships between impact craters are the main types of combined craters being found. They can provide much information for use with determinations of 123 Lunar Impact Craters 147

Table 3 Morphometric characteristics of 39 groups of edge-intersected impact craters in study area Group Name Position R (m) E(m) L(m) Ra/Rb L/Ra Ea/Eb

1 Lavoisier B Up (a) 15,417.90 -4,015 41,342.40 0.50 1.33 1.03 Lavoisier E Down (b) 31,087.70 -3,913 2 Cepheus A Up (a) 7,664.27 -4,555 22,813.60 0.32 0.95 0.88 Cepheus Down (b) 23,978.80 -5,189 3 Tralles Up (a) 23,048.70 -2,990 79,202.70 0.36 1.22 0.51 Cleomedes Down (b) 64,834.70 -5,893 4 Hooke D Up (a) 11,126.90 -4,205 28,202.20 0.53 1.35 1.46 Hooke Down (b) 20,832.80 -2,882 5 Hahn F Up (a) 12,096.80 -5,617 36,543.40 0.28 0.84 1.09 Hahn Down (b) 43,247.90 -5,155 6 Beals Up (a) 27,959.10 -4,431 73,810.00 0.41 1.08 0.84 Riemann Down (b) 68,300.50 -5,302 7 Gauss A Up (a) 10,925.40 -3,769 97,331.20 0.11 1.01 0.64 Gauss Down (b) 96,517.80 -5,872 8 unknown Up (a) 8,051.93 -4,757 11,828.90 0.75 1.10 1.25 Zeno F Down (b) 10,758.80 -3,794 9 unknown Up (a) 10,524.10 -3,068 25,588.50 0.56 1.37 0.67 Cannon B Down (b) 18,708.40 -4,604 10 Macrobius C Up (a) 5,998.07 -2,154 27,902.90 0.20 0.92 0.53 Macrobius Down (b) 30,279.50 -4,070 11 Cameron Up (a) 4,988.06 -3,093 24,355.20 0.19 0.94 1.16 Taruntius Down (b) 25,845.40 -2,675 12 Vasco da Gama S Up (a) 13,005.80 -3,907 41,164.50 0.30 0.94 1.14 Vasco da Gama Down (b) 43,953.00 -3,416 13 Glushko Up (a) 18,050.20 -4,903 47,546.90 0.53 1.39 1.19 Olbers Down (b) 34,285.00 -4,109 14 Sirsalis Up (a) 19,020.70 -4,033 24,227.30 0.85 1.08 1.46 Sirsalis A Down (b) 22,400.50 -2,770 15 Gassendi A Up (a) 15,491.90 -3,889 53,375.60 0.31 1.07 1.28 Gassendi Down (b) 50,095.70 -3,050 16 Davy A Up (a) 5,955.08 -3,963 14,883.10 0.38 0.96 1.43 Davy Down (b) 15,526.70 -2,769 17 Thebit A Up (a) 9,380.86 -4,369 27,711.40 0.36 1.06 1.30 Thebit Down (b) 26,116.10 -3,354 18 Klein Up (a) 21,062.40 -3,176 43,817.10 0.36 0.75 0.88 Albategnius Down (b) 58,316.60 -3,601 19 Theophilus Up (a) 44,423.70 -5,478 76,664.10 1.02 1.75 1.61 Cyrillus Down (b) 43,691.20 -3,402 20 Colombo A Up (a) 19,111.80 -3,024 48,739.00 0.50 1.29 1.11 Colombo Down (b) 37,883.80 -2,733 21 Byrgius A Up (a) 9,868.42 -2,783 39,907.20 0.22 0.90 1.21 Byrgius Down (b) 44,169.70 -2,295 22 Vieta A Up (a) 16,583.00 -2,651 23,320.50 0.83 1.17 1.43

123 148 W. Cheng et al.

Table 3 continued

Group Name Position R (m) E(m) L(m) Ra/Rb L/Ra Ea/Eb

Vieta B Down (b) 19,886.10 -1,855 23 Lacroix J Up (a) 10,260.20 -3,581 15,293.60 0.50 0.74 1.36 Lacroix Down (b) 20,554.10 -2,639 24 Cavendish E Up (a) 11,991.70 -2,973 26,799.30 0.44 0.98 1.04 Cavendish Down (b) 27,231.20 -2,865 25 Fourier B Up (a) 6,043.98 -1,924 24,579.60 0.23 0.92 0.56 Fourier Down (b) 26,574.20 -3,461 26 Clausius BA Up (a) 9,898.66 -2,535 11,601.50 0.71 0.84 1.43 Clausius B Down (b) 13,885.90 -1,770 27 Cichus C Up (a) 6,373.65 -1,917 20,944.50 0.31 1.03 0.48 Cichus Down (b) 20,290.80 -3,995 28 Orontius D Up (a) 8,772.12 -3,243 31,125.20 0.34 1.19 1.41 Sasserides A Down (b) 26,128.50 -2,298 29 Zagut E Up (a) 19,358.30 -1,125 34,732.90 0.47 0.84 0.56 Zagut Down (b) 41,564.80 -2,013 30 Reichenbach A Up (a) 15,864.10 -2,897 23,402.90 0.78 1.16 1.38 Reichenbach B Down (b) 20,252.40 -2,102 31 Reichenbach F Up (a) 7,554.49 -2,430 31,060.10 0.23 0.94 0.70 Reichenbach Down (b) 32,900.80 -3,481 32 Fraunhofer C Up (a) 21,863.20 -3,476 41,348.50 0.63 1.18 0.95 Fraunhofer J Down (b) 34,974.40 -3,671 33 Fraunhofer U Up (a) 12,890.80 -3,212 21,643.80 0.96 1.61 1.29 Fraunhofer M Down (b) 13,458.80 -2,488 34 Furnerius S Up (a) 10,150.10 -3,059 22,849.00 0.57 1.28 0.98 Furnerius Q Down (b) 17,893.60 -3,133 35 Adams P Up (a) 12,839.40 -4,091 16,435.00 1.84 2.36 1.83 unknown Down (b) 6,974.97 -2,232 36 unknown Up (a) 29,374.40 -6,157 54,016.70 0.53 0.98 1.17 Barnard Down (b) 54,911.30 -5,246 37 Van Biesbroeck Up (a) 5,385.51 -3,333 7,978.85 0.47 0.70 1.39 Krieger Down (b) 11,411.90 -2,397 38 Seneca A Up (a) 9,244.86 -1,996 14,695.50 0.80 1.27 1.06 Seneca C Down (b) 11,576.50 -1,875 39 Apianus B Up (a) 5,122.48 -1,582 31,739.50 0.16 1.01 0.68 Apianus Down (b) 31,580.40 -2,321

Ra refers to the radius of the younger, superjacent craters; Rb refers to the radius of the older, nether craters; Ra/Rb refers to the radius ratio of the two superimposed craters; L refers to central distance of the two superimposed craters; L/Ra refers to positional relations of the two edge-overlaid craters; Ea and Eb refer to ﬂoor elevation of the younger and older edge-overlaid craters, respectively; Ea/Eb refers to the ﬂoor elevation ratio of the two craters relative impact order. Because younger craters superimpose on older impact craters in these cases, the edges of the older ones are destroyed. From the DEM and remote sensing map, the size, shape and superposition relationship of two craters can be clearly shown; 123 ua matCraters Impact Lunar Table 4 Results of relative impact order estimations for impact crater pairs Position relationship Name Rs r Dp (m) A (km2) D/R Relative Reference impacting order age (IAU 2011)

Disjoint (group 1) Messala (a) 0 1.069 4,313 12,351 0.069 Older pNc Bernoulli (b) 0 1.061 5,117 1,890.22 0.209 Younger Ic1 Disjoint (group 2) Glushko (a) 1 1.032 4,068 1,362.52 0.195 Younger Cc Cardanus (b) 0 1.034 3,944 1,860.43 0.162 Older Ic2 Disjoint (group 3) Kepler (a) 1 1.028 3,402 708.9 0.226 Younger Cc2 Copernicus (b) 1 1.048 4,421 6,935.28 0.094 Older Cc1 Disjoint (group 4) Hevelius (a) 0 1.048 5,143 11,043.9 0.087 Older Ip Cavalerius (b) 0 1.049 4,731 2,859.51 0.157 Younger Ec Disjoint (group 5) Menelaus (a) 0 1.017 887 31.736 0.279 Younger Cc1 Manilius (b) 0 1.030 460 26.916 0.157 Older Ec Disjoint (group 6) Aristarchus (a) 1 1.024 4,433 1,233.67 0.224 Younger Cc2 Herodotus (b) 0 1.027 2,263 920.94 0.132 Older Im Disjoint (group 7) Maury A (a) 0 1.019 2,705 344.88 0.258 Older Ic2 Maury B (b) 0 1.022 1,625 60.10 0.372 Younger Cc Disjoint (group 8) Archimedes (a) 0 1.046 3,348 5,148.41 0.083 Older Im Autolycus (b) 0 1.019 4,142 1,184.19 0.213 Younger Cc1 Disjoint (group 9) Cichus C (a) 0 1.019 2,784 90.71 0.518 Younger Cc Cichus A (b) 0 1.017 2,455 307.01 0.248 Older Ic Intersected (group 10) Albategnius (a) 0 1.021 6,293 17,398.4 0.084 Older pIc2 Klein (b) 0 1.026 3,754 1,599.14 0.166 Younger pIc Intersected (group 11) Reichenbach B (a) 0 1.071 3,552 1,246.3 0.178 Older pIc3 Reichenbach A (b) 0 1.023 3,789 882.97 0.226 Younger Ic1 123 Intersected (group 12) Krieger (a) 0 1.031 2,369 391.05 0.212 Older Ic2 Van Biesbroeck (b) 0 1.023 2,287 66.97 0.495 Younger Ec 149 150 123 Table 4 continued

Position relationship Name Rs r Dp (m) A (km2) D/R Relative Reference impacting order age (IAU 2011)

Intersected (group 13) Gauss (a) 0 1.079 6,142 22,545.5 0.073 Older Nc Gauss A (b) 0 1.085 3,866 259.32 0.426 Younger Nt Intersected (group 14) Cavendish (a) 0 1.039 3,948 2,307.63 0.146 Older pIc3 Cavendish E (b) 0 1.038 3,910 411.55 0.342 Younger Ic2 Intersected (group 15) Krafft (a) 0 1.032 5,495 2,132.79 0.211 Older Ic2 Krafft C (b) 0 1.026 2,484 120.54 0.401 Younger Ic2 Intersected (group 16) Furnerius S (a) 0 1.047 3,039 235.66 0.351 Younger Nt Furnerius Q (b) 0 1.061 3,262 783.68 0.207 Older Nc Intersected (group 17) Gassendi (a) 0 1.030 4,435 9,840.16 0.079 Older pIc3 Gassendi A (b) 0 1.038 4,324 838.20 0.265 Younger Cc1 Intersected (group 18) Theophilus (a) 0 1.037 5,887 7,811.10 0.118 Older Cc1 Cyrillus (b) 0 1.029 6,228 7,891.51 0.124 Younger pIc3 .Cege al. et Cheng W. Lunar Impact Craters 151

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0.2

0 ooooooooyoooyoyoyoyooyyyyooyyyyyyyyy

Dp/R r Relative Age (o: older; y: younger)

Fig. 6 Relationship between relative age and Dp/R, r

Fig. 7 Profile analyses of typical edge-intersected impact craters. Notes: I Colombo (15.1°S, 45.8°E); II Albategnius (11.7°S, 4.3°E); III Thebit (22.0°S, 4.0°W); IV Cleomedes (27.7°N, 56°E) however, floor elevation and the extent of morphologic reconstruction for a pair of craters cannot be acquired directly. Therefore, terrain analysis, including elevation profiling can provide useful constraints for geomorphologic research (Grosse et al. 2012). From study area, four groups of typical edge-intersected impact craters were chosen for profile analysis (Fig. 7). The left side of each profile is the younger crater and the right side is the older one. It can be seen from DEM data that there are great differences in size, radius ratio and overlaid forms among the four groups. The younger crater is superimposed on the external rim of the older crater in groups I and III. However, the younger one is superimposed on the inside rim of the older one in group II, and the younger one is almost superimposed on the rim of the older one in group IV. For the profile of craters in group I, the eroded extent of the older crater rim is about 1,300 m, enabling it to still present some of the original morphology. As for the group II profile, the eroded extent of the older crater rim is about 2,500 m, and the overall shape of the older crater has changed significantly. Some of group III has an eroded extent, which is at about 2,500 m, and the depth of the two craters shows a great difference. In group IV, the radius ratio of the younger to older craters was small. The eroded extent of the rim of the older one is about 500 m, and the depth of the younger impact craters is shallow. 123 152 W. Cheng et al.

Fig. 8 Terrain estimating recovery for Thebit crater

5 Discussion

5.1 Inferring the Relative Impact Order of Craters

Taking the geological ages of the craters published by IAU (http://www.lpi.usra.edu/lunar/ surface/) as standard, the inferred positional relationship results and relative impact timing for many groups of craters are found to agree with the geologic record. Comparative results (Table 4) show that the inferring method in this paper is feasible. Similarly, relative age orders for craters on a geologic setting can be determined. The universal parameters and supplementary methods mentioned above can be used to infer the relative order of impact for most ‘overlapping’ impact craters, but a few anomalous impacts could not be assessed. These include situations where the impacts occurred close together in time and also where there are external influences or interference factors; e.g., diversity of lunar soil and rock composition, or where depth and rays cannot be distinguished. Furthermore, owing to basalt fill, tectogenesis, isostatic compensation and other impact effects of the lunar surface, thus, shape, size and depth of older craters would gradually change with time of exposure. Infilled by ejecta is one of the mechanisms of crater destruction which leads to a reduction in the depth of these craters. Mass movement caused by structural factors is widespread on the Moon, and it is an important geological effect to consider when reconstructing impact craters and other landform morphologic types (Muthch 1972), thus in these cases, the relative impact order of craters was difficult to infer.

5.2 Characteristics of Edge-Intersected Impact Craters on Impact Development

Taking Thebit crater (22.0°S, 4.0°W) as an example, the profile map shows the morphometric characteristics of the crater with a small crater superimposed on its left side in Fig. 8 left. The floor of the superimposed small impact crater is deeper than that of the larger one. Using the centroid of the floor of Thebit crater as a central point, a symmetric line to the right profile was drawn. This line, the dotted one in Fig. 8 right, can be regarded as the original left side of the profile from the older crater. It can be seen that the small superimposed crater impacted at the rim of the bigger crater, and the height of the rim was altered by about 4,500 m. Furthermore, two-third of the left part of the Thebit crater is eroded, which implies that the depth of the secondary impact craters may have close relations with the crater size, impacting velocity, change of rock density after the first impact and degree of erosion. This aspect is worth further investigation.

123 ua matCraters Impact Lunar

Table 5 Results of correlation analysis among Ra/Rb, L/Rb and Ea/Eb

Coefﬁcient R square Ajusted R square Std. error of the estimate

5–1 Model summary 0.643 0.413 0.380 0.28206

Sum of squares df Mean square F Sig.

5–2 Anova Regression 2.014 2 1.007 12.657 0.000 Residual 2.864 36 0.080 Total 4.878 38

Unstandardized coefficients Standardized coefficients t Sig. 95 % Confident interval Collinearity statistics

B Std. error Beta Lower Upper Tolerance VIF bound bound

5–3 Coefﬁcients Constant 1.249 0.194 6.424 0.000 0.854 1.643 L/Rb -0.581 0.237 -0.496 -2.445 0.019 -1.062 -0.099 0.397 2.520 Ra/Rb 1.074 0.230 0.947 4.669 0.000 0.608 1.541 0.397 2.520 123 153 154 W. Cheng et al.

From the above profile analysis, the superposition of different edge-intersected types is seen to make a big difference in lunar morphology transformations. Such differences should be connected to the overlaid position and diameters ratios of two craters. Also, the results may be affected by the impacting angle, the thickness of the lunar regolith, and/or the age gap between two impact craters. The characteristics of superimposed craters can help quantitatively understand the effects of impact on crater development based on the morphologic parameters of edge-intersected craters as previously mentioned. We use parameters such as Ra/Rb, L/Rb and Ea/Eb to quantitatively analyze the cor- relations among them. According to Table 3, Ra/Rb, L/Rb as a variable, and Ea/Eb as a dependent variable can be used as part of a multiple linear regression analysis procedure supported by SPSS software. The outputs are a statistical model, an analysis of variance models and a determination of regression coefficients. In Table 5, the correlation coefficient is 0.64 and the coefficient of determination is 0.413. We can find that Ra/Rb, L/Rb and Ea/Eb have a certain linear correlation. The coefficient of determination is not so high, which confirms that the Ea/Eb size is also affected by the impacting angle and other significant factors. In F test, F is 12.657, with a concomitant probability sig is 0.000 \ 0.05, which means that sig passes through a significant level as it is obvious in the linear relationship of the regression equation. Then we select the least squares method to determine the regression coefficient and confirm the regression coefficient through a t test of the significance. The sig (L/Rb) = 0.019 \ 0.05 and sig (Ra/Rb) is 0.000 \ 0.05, which explains the two regression coefficients to be remarkable. The regression equation is therefore Y ¼ 1:249 þ 1:074x1 0:581x2. Finally, a test for multi-collinearity shows that there is no collinearity between variables; VIF (Variance Inflation Factor) is 2.520. Ee can therefore conclude that there is a linear correlation between Ra/Rb, L/Rb and Ea/Eb for the superposition of impact craters.

6 Conclusions

By using CCD and DEM data from the Chang’E-1 lunar orbiter, this study has established a set of morphologic parameters for the characterization of impact craters. We have analyzed methods for estimating the relative order of impacts based on different positional relationships and successfully implemented them, which shows that this inferring method is feasible. The morphologic characteristics of edge-intersected impact craters were analyzed, the effect of crater impacts on the edge of pre-existing craters and surrounding regions shows that proﬁle analysis of impact craters is reliable. Determinations of superposition relationships and morphologic characteristics, and proﬁle analysis of edge-intersecting impact craters, provide multiple ways to estimate the relative order of impact and chronological characteristics of impact craters. Several issues need to be resolved in the future. The geological ages of some impact craters remain unresolved, given that morphologic characteristics of many regions of the lunar surface are complex. One example of this is in locations where numerous impact craters are overlapping. However, the methods established here to analyze pairs of simple craters should provide some assistance in developing techniques to decipher the complex morphologic characteristics of large groups of impact craters in the future.

Acknowledgments The Chang’E-1 data were provided by the National Astronomical Observatories and Chinese Academy of Sciences. This work was supported by the National Natural Science Foundation of China (Grant No. 41171332), Key Project for the Strategic Science Plan in IGSNRR, CAS (Grant No.

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2012ZD009), and the Knowledge Innovation Project of the Institute of Geographic and Natural Resources Research, the Chinese Academy of Sciences (Grant No. 201001005). The Editor-in-Chief and anonymous referees are also greatly appreciated for their critical comments and for the time they spent on reviewing and improving this paper.

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