3.1. Secondary Cratering
 The contribution of basin secondary cratering has the potential to affect the crater size-frequency distribution of individual basins [e.g, Wilhelms, 1976; Wilhelms et al., 1978]. However, most secondary craters are smaller than 20 km, even for the largest basins such as Imbrium and Orientale, because of the steep size-frequency distribution of secondary craters [Wilhelms et al., 1978]. Thus, measured crater densities are much less contaminated by secondaries at the scales we consider here (craters ≥20 km in diameter) than would be the case if we included smaller craters.
 There are also apparently substantial differences in the number of large secondaries produced from a given large basin. Wilhelms et al.  classified 58 craters ≥20 km as secondaries from the Imbrium basin in an area of 4.165 × 106 km2 (equivalent to N(20) = 14 ± 2 in their count region), but only one crater ≥20 km as a secondary from the smaller Orientale basin in an area of 1.751 × 106 km2 (equivalent to N(20) = 0.6 ± 0.6). The inferred contribution of secondaries from Imbrium is likely a maximum estimate for the density of secondaries that a D ∼ 1000 km basin will typically produce, since this measurement was taken where the density of secondaries was highest, and Imbrium produced far more secondaries than Orientale. The contributions of secondaries from later basins to the crater statistics of earlier basins such as Nectaris is likely to be less than 20%, although the precise contribution of secondaries is dependent on the age of the basin and its proximity to later large basins.
 Given these factors, we interpret the superposed crater size-frequency distributions of lunar basins as being generally controlled by primary cratering for ≥20 km craters. This view is bolstered by (1) the lack of detection of abundant secondaries ≥20 km surrounding large young basins at appropriate ranges based on LOLA data [Head et al., 2010], (2) the consistency of stratigraphic and crater counting sequence determinations, which suggests that secondary cratering does not significantly contaminate and affect these measurements, and (3) the reasonable agreement of the basin sequence based on craters of larger size (≥64 km) with what is found at N(20); secondary craters ≥64 km in diameter are unlikely to exist.
3.2. Evolution of the Lunar Crater Population and Implications for the Late Heavy Bombardment
 A major question in lunar science is whether the crater population or size distribution of impacting bodies that impacted the Moon was stable over time, even though the flux was changing. A long-standing and important hypothesis is that the lunar highlands were impacted by a distinct, early population of impactors that differs from what has affected the Moon since the time of the emplacement of the lunar maria [e.g., Whitaker and Strom, 1976; Wilhelms et al., 1978; Strom, 1987; Strom et al., 2005; Head et al., 2010]. This idea has been disputed by workers who have argued that the entire visible crater record can be explained by a single impactor population, [e.g., Neukum and Ivanov, 1994; Hartmann, 1995; Neukum et al., 2001], and that any observed differences in the crater record can be attributed to geological resurfacing and difficulty in finding a terrain that is an unmodified sample of the early impact record.
 The basis for the idea that impact populations on the Moon have changed over time is that the observed crater size-frequency distributions of ancient highland terrains are distinct from those that are observed on the maria (e.g., Figure 4) [see also Strom et al., 2005; Head et al., 2010]. This is manifested by a having a lower ratio of ∼20–40 km craters to ∼80–100 km craters in the highlands than in the mare (in other words, there is an excess of these larger craters in the highlands). That the maria and highland have differently shaped crater size-frequency distributions is statistically significant when applying the two-sample Kolmogorov-Smirnov test to their empirical cumulative densities (Figure 5a). Ćuk et al. [2010, 2011] also demonstrated that the size-frequency distribution of Imbrian craters on the Moon are statistically distinguishable from the highlands curve, using data from both Wilhelms et al.  and the fresh crater distribution (of class 1 craters) from Strom et al. .
Figure 4. R-Plot showing South Pole-Aitken (SPA) basin, as well as the highlands (excluding SPA, Orientale, and regions covered by mare), and the mare from Head et al. . These data illustrate the difference in population that affected the lunar highlands and SPA compared to the lunar mare. Note that we follow the convention of Strom et al. , who term the crater size-frequency distribution of the highlands ‘Population 1’ and that of younger units like the mare ‘Population 2’.
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Figure 5. Empirical cumulative density functions (cdfs) for a variety of different terrains on the Moon along with the relevant Kolmogorov-Smirnov statistic (Kstat) for two distributions. Kstat is a statistical metric based on the maximum difference between two cdfs and provides a test of whether two populations are different. (a) Comparison of the mare and highlands excluding SPA [see also Strom et al., 2005; Ćuk et al., 2010, 2011; Head et al., 2010]. These terrains (Figure 4) are from statistically significant different populations of craters. (b) Comparison of SPA and the highlands excluding SPA (Figure 4); these distributions are consistent with being from the same population (not significantly different), though SPA has a modestly fewer craters in the D = 20–64 km size range. (c) Comparison of Imbrian basins and the mare, which are not significantly different. (d) Comparison of Nectarian basins and the mare, which are not significantly different. (e) Comparison of Imbrian and Nectarian basins (see also Figure 6), which are not significantly different. (f) Comparison of Nectarian and Pre-Nectarian basins (see also Figure 6). These are different at 94% confidence. (g) Comparison of Nectarian and Pre-Nectarian basins for craters larger than 40 km; these are different at 83% confidence. (h) Comparison of Pre-Nectarian basins and the highlands excluding SPA for craters larger than 40 km. These are not distinguishable in this size range. At smaller sizes, the ‘average’ highlands excluding SPA have fewer craters than these Pre-Nectarian basins (compare Figures 4 and 6). This difference at small sizes may be a result of moderate deficiency in 20–40 km craters in the highlands curve (due to crater removal and modest incompleteness) (see section 3.2).
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 As noted above, this observation does not guarantee that the difference in observed crater size-frequency distribution is a direct result of a shift in the impactor population that affected the Moon in its early history relative to more recent times. A viable alternative hypothesis is that early surfaces were bombarded by a crater population similar to the population that impacted the maria, but that geologic processes such as volcanism or repeated cratering preferentially removed small craters and thus resulted in the observed change [Hartmann, 1984, 1995]. This is plausible because both later impacts and volcanic infilling affect smaller craters more easily than larger craters, which extend across more area and have greater initial relief.
 There are several reasons, however, to question this resurfacing explanation: (1) A variety of ancient terrains with distinct crater density and geological histories, such as inside South Pole-Aitken (SPA) and in the heavily cratered highlands, have similar shapes to their crater size-frequency distributions [Head et al., 2010] (Figure 4 and 5b). Despite these different histories, both distributions are statistically distinguishable from the mare and indistinguishable from each other. (2) A similar distinction exists between old and young terrains on Mercury and Mars as well as the Moon [Strom et al., 2005; Fassett et al., 2011a]. There is no reason that crater removal processes on each planet should be manifested in a similar manner in their crater statistics, while an inner solar system-wide change in the impactor population can readily explain the same change directly. (3) A process-oriented explanation [Strom et al., 2005] exists for such a change in impactor population, since the early population (what Strom and colleagues call ‘Population 1’) has a shape that matches well with a collisionally evolved population like the Main Asteroid Belt, and the younger population (‘Population 2’) may be a result of size-selective processes that preferentially transport smaller asteroids to the inner solar system [e.g., Morbidelli and Vokrouhlický, 2003]. Because of this size-selection, the resulting population has a ‘flatter’ shape on an R-plot (Figure 4). The Population 2 shape also matches well with the inferred crater population that would be expected to be produced from the present Near Earth Object distribution [Strom et al., 2005].
 Using our new database of the global crater population ≥20 km from LOLA data, we found support for the presence of the two populations proposed by Strom et al.  [Head et al., 2010; Kadish et al., 2011], and the results of our new analysis continue to support this interpretation. We thus provisionally accept the hypothesis that different impactor populations affected the Moon, and hence basins on its surface, as a function of time. With our new data, we can examine when the hypothesized transition between populations occurred. In practice, it is difficult to address this question on a basin by basin basis, because counting statistics are insufficient to demonstrate significant changes over these short time intervals, particularly when relying on the large craters that we consider most reliable for assessing basin ages. This challenge is demonstrated by the ongoing arguments about whether Orientale has a crater size-frequency distribution reflecting Population 1 or 2 [Strom, 1977; Woronow et al., 1982; Hartmann, 1984; Head et al., 2010; Ćuk et al., 2010, 2011]. Orientale's ≥20 km crater population cannot be statistically distinguished from either population with confidence.
 We choose to address this counting statistics problem by aggregating the statistics from individual basins into average crater size-frequency distributions for basins of a given period (Figures 5 and 6). We combine the crater counts and areas for the Imbrian basins (including Imbrium), the Nectarian basins (including Nectaris), and the Pre-Nectarian basins (excluding South Pole-Aitken to avoid it dominating the statistics).
Figure 6. R-Plot showing the integrated crater size-frequency distributions for Pre-Nectarian-aged basins (excluding SPA), Nectarian-aged basins (including Nectaris), and Imbrian-aged basins (including Imbrium). Nectarian basins have a flat distribution on an R-Plot, consistent with the mare-like Population 2 crater distribution. The Pre-Nectarian basins are more similar to Population 1. This suggests that the transition from terrains with highlands-like Population 1 to Population 2 happened by the mid-Nectarian, as the Nectarian basins primarily accumulated craters from Population 2.
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 No single basin unduly influences any of these aggregate crater size-frequency distributions. For the Pre-Nectarian, with aggregate N(20) = 188 ± 7, no basin represent more than ∼20% of the aggregated data; the largest two contributors are Smythii (20%) and Nubium (16%), and the other 13 basins each have less than a 10% influence on the sum total. Crisium and Nectaris each represent ∼20% of the Nectarian curve, and the other 9 basins represent the remaining ∼60%. The aggregate N(20) for the Nectarian average is 110 ± 5. In the Imbrian, with few basins, Imbrium and Orientale contribute 43% and 47% of the craters respectively, and Schrodinger represents the other 10%. The aggregate N(20) for the Imbrium data is 22 ± 3. In total, the Imbrian, Nectarian, and Pre-Nectarian basin distributions are a sample of 8%, 10%, and 9% of the lunar surface area.
 These aggregated data imply that both the Imbrian-aged basins and the Nectarian-aged basins are consistent with the flatter Population 2 (more mare-like) shapes (see Figures 4, 5c, 5d, and 6). The Pre-Nectarian and Nectarian-aged basins are distinct from each other at the 94% confidence level (Figure 5f and 6); the Nectarian-aged basins also differ significantly from the highlands population (Population 1), assuming that the non-SPA highlands data set from Head et al.  are representative of this population.
 These observations are surprising, since the transition from Population 1 to Population 2 has previously been linked to the transition away from the Late Heavy Bombardment population of impactors to the modern population [Strom et al., 2005], and the basins formed during the Nectarian are commonly assumed to be part of the Late Heavy Bombardment. For this reason, we might expect that these Nectarian basins were witness to the putative Late Heavy Bombardment population of impactors (Pop. 1) and would have a superposed crater population different from that of the maria. Moreover, because of the high crater flux during this time period, 65–75% of the craters that we measure on the Nectarian basins actually formed during the Nectarian period (Table 1; compare the N(20) of Nectarian basins with Imbrium). So if Population 1 dominated the impactors during this period, we would expect to see its signature in the Nectarian basin curve.
 Instead, our data show a difference between the crater size-frequency distributions of the Nectarian-aged and the Pre-Nectarian-aged basins (Figure 5f and 5g), and a close similarity between the Nectarian-aged basin population and the population of impactors recorded on the maria or Imbrian basins (Pop. 2) (Figures 5d and 5e). These observations are not consistent with a mid-Nectarian-to-Imbrian Late Heavy Bombardment dominated by a size-independent delivery of impactors from a collisionally evolved, Main Asteroid Belt-like population of impactors (Population 1) [Gomes et al., 2005; Strom et al., 2005]. Based on different data, Ćuk et al. [2010, 2011] reached a similar interpretation that later, Imbrian-aged impactors that may have been part of the Late Heavy Bombardment also lack a Population 1 shape. Instead, the impactor size-frequency distribution by the mid-Nectarian is consistent with that of the lunar mare, despite the high flux during this period, rather than with the size-frequency distribution characteristic of the lunar highlands or SPA.
 The Pre-Nectarian-aged basins have a size-frequency distribution of superposed craters that is qualitatively closer in shape to the highlands than Nectarian-aged basins (Figure 4, 5f, and 6), although there is still some observed difference in these distributions between D = 20 and 40 km. (This difference may arise from the fact that the pre-Nectarian basins size-frequency distribution is more representative of the original impactor population than the lunar highlands curve. The highlands data may be a modest underestimate of the crater frequency characteristic of this population between 20 and 40 km because of removal of craters at these size ranges [e.g., Hartmann, 1995]. It is also likely that there is modest incompleteness in the highlands data set in this size range (estimated to be <20%)).
 If craters D < 40 km are excluded, the Pre-Nectarian-aged basins and highlands are similar (Figure 5h), but Nectarian-aged and Pre-Nectarian-aged basins are different with 83% confidence (Figure 5g; see also Figure 5f). Taken at face value, these data are consistent with a scenario in which the population of impactors was evolving over time, starting with Population 1 primarily recorded on the ancient cratered highlands and Pre-Nectarian-aged basin surfaces, with a transition to predominantly Population 2 impactors by the mid-Nectarian.
 These data would suggest that the transition observed in the lunar impact crater population occurred earlier than has been previously suggested. Although it is difficult to ascertain whether the transition between the two impactor populations was gradual or abrupt., Population 1 cannot have remained the predominant source of lunar impacts as late as Imbrium. If this were the case the majority of craters on the Nectarian basins would be expected to be from Population 1, which would have resulted in a size-frequency distribution distinct from younger surfaces, unlike what we observe. Given the high impactor flux during the Nectarian, this shift in population also appears to have occurred before the flux of impactors rapidly declined; this transition may be before the end of the hypothesized Late Heavy Bombardment. The lunar impactor population then appears not to have varied significantly over the last ∼3.9–4 Gyr, as all younger surfaces have a Population 2-like crater size-frequency distribution.
3.4. Saturation Equilibrium and the Basin Impact Crater Record
 The lunar highlands have crater densities for craters with a diameter D ≥20km that are close to, and likely to be at, the densities expected for saturation equilibrium, the condition where the formation of a new crater on average erases enough pre-existing craters that the overall density of crater ceases to increase with time [Gault, 1970; Marcus, 1970, Hartmann, 1984; Chapman and McKinnon, 1986; Richardson, 2009; Head et al., 2010].
 Modeling by Chapman and McKinnon  and more recently by Richardson  reveals two important elements concerning the way surfaces behave when shallow-sloped impact crater size-frequency distributions approach saturation. First, for these distributions, there is no single ‘saturation’ value or characteristic distribution; instead, crater densities oscillate in a range that depends on how recently the infrequent formation of a large crater erased previous craters over a large area. An approximate estimate for the R-values where this commonly occurs is ∼0.1 to 0.3, although this is dependent on model parameters. Second, these models indicate that the shape of the production population can be preserved even on saturated surfaces. This helps bolster the argument that the transition between Population 1 and 2 that we describe above is a robust determination and not simply a result of some currently not well-understood saturation behavior.
 Are any of the basins that we observe cratered to saturation? If the lunar highlands are in fact saturated, it is very likely that SPA is also saturated given that it is characterized by higher crater densities at large crater sizes than the highlands (Figure 4). Indeed, the deviation of its N(20) value from the highlands may result from a saturation phenomenon where it was affected by an unusual concentration of large late basins (presumably as a function of chance).
 Many other Pre-Nectarian basins, perhaps excepting Apollo and Freundlich-Sharonov, are characterized by densities that likely imply that they also reached saturation. This means that their age and sequence may be imperfectly tied to the density of craters that are superposed on their surface (Table 1). Most Pre-Nectarian basins are clustered at N(20)∼165 to 265 and N(64)∼30 to 48; equivalent to R20∼0.1–0.15 and R64∼0.25–0.4 with a highlands-like crater size-frequency distribution. If all of these basins are saturated, differences in degradation state and topography hint at how long the basin has been in this condition, although this relationship should be size-dependent. For basins the size of SPA, later cratering is almost certainly ineffective as a process to completely erase basin topographic signatures. However, for small 300–500 km diameter basins, this is potentially a far more efficient process.
 The apparent saturation of early surfaces on the Moon that our measurements support weakens the evidence for forms of the Late Heavy Bombardment hypothesis that postulate a lower impact flux before Nectaris, because the cratering record of the Pre-Nectarian must be incomplete [see also Hartmann, 1975; Chapman et al., 2007]. Since this early impact record is missing, the large number of Nectarian and younger basins (at least 13; Table 1) need not require an anomalously high basin-forming flux compared to the preceding pre-Nectarian period, although the impact flux in early periods was far higher than that during later times.
3.5. Degraded and Uncertain Basins
 In addition to the basins documented and discussed here, there are numerous less well-defined basins [e.g., Wilhelms, 1987; Frey, 2011; Wood, Impact basin database, 2004] that have been suggested to exist on the Moon. A number of these suggested basins do not appear to have a clear signature in LOLA altimetry data. In other instances, some evidence exists for basins that are now simply too ambiguous to be confidently identified using our criteria. The list we provide here (Table 1) is conservative in the sense that we are confident that all of the basins that we measure have a very high probability of being impact basins. The vast majority of the remaining highly degraded, ambiguous and uncertain basins are Pre-Nectarian in age, as discussed further below, and thus do not affect the observations and conclusions discussed above.
 Frey  evaluated the basins compiled by Wilhelms  using the Unified Lunar Control Network topography of Archinal et al. . Additional efforts to analyze the degraded basins with LOLA topography are ongoing, so we do not dwell on this issue here and address only a few points. First, there is significant agreement between the Frey  judgments of the Wilhelms  basin list and our independent evaluations, although we additionally exclude as doubtful Keeler-Heaviside, Fecundatitis, Mutus-Vlacq, Lomonosov-Fleming, and Tsiolkovsky-Stark (as well as Balmer-Kapteyn and Bailly, which we assess as having smaller main ring diameters than our size cutoff).
 We also examined the additional suggested topographic basins of both Wood (Impact basin database, 2004) and Frey . A few of these meet our criteria as probable-to-certain basins; in the Frey  naming scheme these are TOPO-30 (Cruger-Sirsalis) [Spudis et al., 1994; Cook et al., 2002], TOPO-24 (Dirichlet-Jackson) [Cook et al., 2000], and TOPO-41 (Fitzgerald-Jackson) [Cook et al., 2000]. Conversely, in a few instances, we think ‘positive evidence’ exists against some suggest candidates being actual impact basins, such as TOPO-38, which is entirely inside Imbrium and demarcated by wrinkle ridge ring. We interpret this as an inner ring of the Imbrium basin rather than a separate basin. Again, most of the suggested basins that we do not include here are ambiguous; some evidence would argue for their existence and some against.
 What do we know about the age and crater statistics of these ambiguous basins? In general, the vast majority must be Pre-Nectarian in age. We have made measurements which support this view in regions that have traditionally been suggested to have one or more basins by various authors, such as Australe, Lomonosov-Fleming, and Werner-Airy. All have superposed crater densities of N(20) > 180 and N(64) > 35, indicative of Pre-Nectarian ages (see Table 1). As discussed above, these densities are at levels consistent with saturation equilibrium, which also would explain the very highly degraded state of the purported basins.
 Younger impact basins also should obviously not have high crater densities or saturated surfaces, nor have subtle basin topographic signatures (excepting any that were erased by direct superposition of later, larger basins). For these reasons, the Nectarian and younger basins in Table 1 are likely to be a nearly complete representation of the basins that actually formed on the Moon during this period.
3.6. Calibrating Ages and Sampling Suggestions
 Better understanding of the absolute ages of various basins on the Moon is important for understanding lunar geology as well as for understanding the impact record across the inner solar system. This problem is highly convolved with the provenance of the lunar samples, and making progress in this area may require additional in situ fieldwork and/or robotic sample return.
 When considering how to calibrate the translation of measured crater size-frequency distributions into absolute ages, one issue is that the most commonly applied models for lunar crater statistics [e.g., Neukum et al., 2001] do not yet account for the fact that the impactor population on the Moon appears to have evolved with time [Strom et al., 2005]. Recent work by Marchi et al.  has made some progress in considering this problem, but the detailed nature of the transition between the two populations has been uncertain, which complicates any attempt at incorporating this into absolute age models. Our work supports the conclusion that basins from the mid-Nectarian onwards were dominated by Population 2 impactors. The radiogenic dates derived for the relatively late basins and the mare, and their associated crater frequencies, represent only the younger population and no confident calibration of a surface with the earlier population presently exists.
 At present, suggested ages exist for Imbrium, Orientale, Crisium, Nectaris and Serenitatis [see Stöffler et al., 2006], although the youthful ages that have been interpreted to represent Serenitatis are certainly inconsistent with our preferred interpretation of its stratigraphy (see section 3.3.2 [see also Spudis et al., 2011]). It remains plausible that many of the absolute ages for basins that have been assigned on the basis of samples actually relate to Imbrium because of its potential for having played a dominant role in the sample collection [Haskin et al., 1998]. In general, it is highly desirable to have further measurement of pre-Imbrian basins on the Moon to allow firmer translation of the relative frequencies that we derive (e.g., Table 1).
 What sample return sites would be best to visit to get additional calibration of the absolute timescale for the lunar surface? Although more samples are undoubtedly better, one candidate of interest for future sampling is the Freundlich-Sharonov basin. It is one of the oldest basins with a well-preserved topographic signature and only moderate resurfacing, and it does not appear to have been cratered to saturation equilibrium. It also has crater statistics that are quite similar to Nectaris, so it would potentially provide a ‘second check’ on ages derived on the lunar nearside. Along with clarifying the ages of nearside basins, future lunar exploration should seek to expand the sample collection to the lunar farside and deep into the impact basin record. Samples from within South Pole-Aitken that could address its absolute age, as well as potentially provide dates for other basins that superpose it, would also provide new calibration of the early lunar cratering record [see also Norman, 2009; Joliff et al., 2010].