Icarus 250 (2015) 438–450

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Icarus

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The resurfacing history of Venus: Constraints from buffered craterdensities

http://dx.doi.org/10.1016/j.icarus.2014.12.0240019-1035/� 2015 Elsevier Inc. All rights reserved.

⇑ Corresponding author at: Earth and Planetary Sciences, University of Californiaat Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA.

E-mail address: mkreslav@ucsc.edu (M.A. Kreslavsky).

Mikhail A. Kreslavsky a,⇑, Mikhail A. Ivanov b, James W. Head c

a Earth and Planetary Sciences, University of California at Santa Cruz, Santa Cruz, CA 95064, USAb V.I. Vernadsky Institute, Moscow, Russiac Department of Earth, Environmental and Planetary Sciences, Brown University, Providence, RI 02912-1846, USA

a r t i c l e i n f o

Article history:Received 28 July 2014Revised 4 December 2014Accepted 21 December 2014Available online 30 December 2014

Keywords:Venus, surfaceGeological processesCrateringVolcanismTectonics

a b s t r a c t

Because of atmospheric shielding and endogenic resurfacing, the population of impact craters on Venus issmall (about a thousand) and consists of large craters. This population has been used in numerous studieswith the goal of deciphering the geologic and geodynamic history of Venus, but the nearly spatially ran-dom nature of the crater population has complicated efforts to understand this history. Here we utilizethe recent 1:15 M-scale global geological map of Venus (Ivanov, M.A., Head, J.W. [2011]. Planet. SpaceSci. 59, 1559–1600) to help address this problem. The global geological map provides a stratigraphicsequence of units, and known areas where each unit is exposed on the planet. For each crater on Venuswe identify the specific geological units predating and postdating the crater. We perform a statisticalanalysis of this set of observations with a buffered crater density approach, which rigorously and consis-tently takes into account the large size of craters and the fact that many craters are known to predateand/or postdate more than one unit. In this analysis we consider crater emplacement as random andresurfacing history as determined (although unknown). We obtain formal confidence intervals for themean ages of geological units and the mean age differences between the pairs of units at the unit bound-aries. We find that (1) size–frequency distributions of craters superposed on each unit are consistent witheach other; (2) regional plains and stratigraphically older units have similar crater retention ages; (3)stratigraphically younger units have a mean crater retention age significantly younger than the regionalplains. These findings are readily and consistently explained by global resurfacing scenarios and are dif-ficult to reconcile with equilibrium resurfacing scenarios. Our analysis also shows that the latest recordedpart of intensive resurfacing period lasted on the order of 10% of the mean surface age (tens of millions ofyears). The termination of intensive resurfacing may or may not be synchronous over the planet. Ourresults also indicate that there are extended deposits associated with large craters that are almost indis-cernible in the radar images, but obscure radar contrasts between predating lava flows. We do not seeevidence for any significant and prolonged change of atmospheric pressure following the terminationof the intensive resurfacing epoch.

� 2015 Elsevier Inc. All rights reserved.

1. Introduction

The density of impact craters is widely used in planetary sci-ence to study relative and absolute surface ages and the natureof resurfacing on planets. Impact craters on the surface of Venuswere first studied in Venera 15–16 radar mosaics, which covereda quarter of the surface of the planet in the northern hemisphere.Analysis of the population of two hundred craters identified inthose radar images (Ivanov et al., 1986; Bazilevskii et al., 1987;

Ivanov, 1990) led to a number of significant results: (1) it wasunderstood that Venus lacks old heavily cratered crust in thisregion, (2) a typical crater retention age of the surface in this areais within the range of 0.2–1 Ga, and (3) the atmosphere preventsformation of small (kilometer and smaller) craters. Additional cra-ters were seen in Arecibo radar images (Campbell et al., 1990).Finally, Magellan global coverage with radar images allowed thecataloguing of almost all impact craters on the planet (e.g.,Schaber et al., 1992; Herrick et al., 1997). This global populationof somewhat less than a thousand craters has been the subject ofcareful scrutiny in almost a hundred papers (reviewed below),and a number of these were devoted to statistical analysis of thissmall population. Despite the fact that the power of statistical

M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450 439

methods is limited by the small total number of impact craters,these latter works resulted in a significant advance in the under-standing of the geological history of the planet.

The global spatial distribution of craters turned out to be statis-tically indistinguishable from a uniformly random distribution on asphere, also referred to as complete spatial randomness (Schaberet al., 1992; Phillips et al., 1992; Bullock et al., 1993; Strom et al.,1994; Hauck et al., 1998; Turcotte et al., 1999). This fact inspireda series of stochastic Monte-Carlo models of Venus resurfacing(Phillips et al., 1992; Schaber et al., 1992; Bullock et al., 1993;Izenberg et al., 1994; Strom et al., 1994, 1995; Kreslavsky,1996a,b; Hauck et al., 1998; Bjonnes et al., 2008; Ivanov, 2009;Romeo and Turcotte, 2010; Romeo, 2013; Ivanov and Head, inpreparation-b). In each of these papers, the authors modeled resur-facing on Venus as a time-varying random sequence of more or lessrealistically schematized volcanic and/or tectonic events that oblit-erate and/or modify impact craters concurrently with theiremplacement. The results were used to constrain the possible evo-lution of the resurfacing rate on Venus.

Unfortunately, the fact that the spatial distribution of craters isstatistically indistinguishable from uniformly random means thatthe mapping of crater density alone cannot be used as a tool in geo-logical analysis: any observed variations in local crater density maybe attributed to random fluctuations due to the stochastic nature ofcrater emplacement. However, if some other parameters of the cra-ters are correlated against the crater density, some limited infer-ences are possible (Phillips and Izenberg, 1995). The situation isbetter, however, if we involve additional a priori information inthe crater density analysis. For example, if we outline some geolog-ical units on the basis of their morphology (independent of anyinformation about craters), the crater densities on such units cangive important constraints on the mean ages of the units (Ivanovand Basilevsky, 1993; Namiki and Solomon, 1994; Price andSuppe, 1994, 1995; Price et al., 1996; Gilmore et al., 1997; Haucket al., 1998; Ivanov and Head, in preparation-a). Studies of this kindshowed that the craters are not random with respect to geology,despite their apparent spatial randomness. Under this approachthe geological history is considered as determined and only crater-ing is random, unlike the Monte-Carlo simulations mentioned in theprevious paragraph. In the present paper we pursue this approach:we consider the deterministic geology and stochastic cratering.

Another approach to the incorporation of a priori geologic infor-mation into statistical inferences from craters makes use of the factthat the majority of craters on Venus are large, and for many ofthem it is possible to distinguish several geological events that pre-date and postdate crater emplacement. In this way statistics cangive additional constraints on ages and durations of geologicalevents (Gilmore et al., 1997; Collins et al., 1999; Basilevsky et al.,1999, 2003; Basilevsky and Head, 2002a,b, 2006; Ivanov andHead, 2013, in preparation-a). Finally, some crater properties cangive information about the ages of individual craters, and thisagain allows better constraints from statistics. On Venus, extendedcrater-related diffuse radar-dark deposits have been used as suchan age indicator (Izenberg et al., 1994; Basilevsky and Head,2002a,b; Basilevsky et al., 2003). All of these contributions alsoimplicitly treated the geological history in a deterministic manner,and cratering as random.

In this contribution we use a straightforward approach ofobtaining age constraints from crater densities in mapped geolog-ical units. In comparison to Price et al. (1996), who used a similarapproach, the present work is more robust in several ways due toadvances in the geological study of Venus. First, we take advantageof the much more detailed 1:15 M scale global geological map ofVenus (Ivanov and Head, 2011). Second, we distinguish geologicalunits that predate and postdate each individual crater. Third, thehigh resolution of the map gives us a way to rigorously account

for the large size of craters, which increases the accuracy of the sta-tistical inferences.

In this contribution we first briefly describe the geologicalinformation that we incorporate in the analysis. Then, in Section3 we describe the buffered density technique, the rigorous statisti-cal approach that we utilize in the formal derivation of age infor-mation for large craters. In Section 4 we describe the practicalapplication of this formal approach to Venus data sets. In Section5 we outline the primary results of our statistical analysis and dis-cuss possible caveats and biases. Finally, in Section 6 we discussthe implication of these results for Venus, especially for its resur-facing history.

2. Source data

The unique advantage of the global geological map of Venus(Ivanov and Head, 2011) is that the unit definitions and theapproach to their identification are consistent over the entireplanet.

The map contains the following geomorphologic units, in gen-eral stratigraphic sequence, oldest to youngest (see Ivanov andHead, 2011) for detailed descriptions of each:

� t, tessera (Fortuna Formation);� pdl, densely lineated plains (Atropos Formation) dissected by

numerous subparallel narrow and short lineaments;� pr, ridged plains (Lavinia Formation) comprising elongated belts

of ridges;� mt, mountain belts (Akna Formation) around Lakshmi Planum;� gb, groove belts (Agrona Formation), plain material contempo-

raneous or predating regional plains and deformed by groovebelts;� psh, shield plains (Accruva Formation) having numerous small

volcanic edifices and locally predating regional plains;� rp1, regional plains, lower unit (Rusalka Formation), mostly uni-

formly radar-dark, deformed by wrinkle ridges;� rp2, regional plains, upper unit (Ituana Formation), radar-bright

plains superposed on rp1 and deformed by wrinkle ridges;� sc, shield clusters (Boala Formation), morphologically similar to

psh but occurring as small patches that postdate regionalplains;� ps, smooth plains (Gunda Formation) of uniformly low radar

brightness occurring near impact craters and at distinct volca-nic centers;� pl, lobate plains (Bell Formation), fields of lava flows that typi-

cally are not deformed by tectonic structures and are associatedwith major volcanic centers;� rz, rift zones (Devana Formation).

The map also contains impact craters and their ejecta, as well ascrater outflows. More detailed descriptions of these units, numer-ous type examples, the relation to units from other geologicalmaps, details of their stratigraphic relationships, etc. are given byIvanov and Head (2011). Fig. 1 depicts the observed stratigraphicrelationships between the units (Ivanov and Head, 2011) togetherwith interpretation in terms of a succession of volcanic and tec-tonic styles (Ivanov and Head, 2013, in preparation-a). Fig. 2 showsthe percentage of the mapped area covered by each unit, cratersexcluded.

In the list above, the units are arranged in a general strati-graphic order, from locally older to younger. Not all pairs of unitshave well-established pervasive stratigraphic relationships witheach other.

For each crater from the USGS crater database (Schaber andStrom, 1999), one of us, M.A.I., registered unit(s) superposed bythe crater and its continuous ejecta (that is units that predate the

Fig. 1. Stratigraphic chart showing the observed stratigraphic relationships between the units mapped by Ivanov and Head (2011), their relation to Venus geochronology byBasilevsky and Head (2000), and interpretation in terms of resurfacing regimes sequence according to Ivanov and Head (2013, in preparation-a).

440 M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450

crater) and unit(s) that embay the crater (postdate it). This is donewith the Magellan radar mosaics, as illustrated with an example inFig. 3. This assessment is done uniformly and consistently withunit definitions with the geological map. Of the 965 craters inthe catalog, 948 have sufficient Magellan image coverage and areused in this contribution; for these craters, a total of 1439 superpo-sition relationships and 67 embayment relationships are regis-tered. This information is used for the statistical analysis in thepresent contribution.

3. Statistical approach

3.1. Background: cratering as a Poisson process

Although crater statistics is widely applied in planetary science,some publications on the topic contain vague or mathematicallyincorrect statements; because of this, we outline the mathematical

approach that underlies the traditional statistical studies of craterpopulations.

Let us consider a geological unit u of area Au bearing Mu craters.In this subsection we think about craters as points, that is cratersare smaller than any spatial dimension of the unit. The estimationof ages utilizing crater counts is based on the assumption that cra-ter emplacement is well described with a mathematical model of aPoisson process (e.g., Snyder and Miller, 1991). This means thateach impact occurs randomly, ‘‘not knowing about’’ the site andtime of previous impacts. Within this mathematical model, thesmall probability, DP, that a crater of a diameter D in a range fromDmin to Dmax is emplaced in a unit u during a short time Dt is:

DP ¼ RAuDt ¼ RV ðFðDminÞ � FðDmaxÞÞAuDt; ð1Þ

where R (km�2 Ma�1) is the impact rate in the given diameter range,which is assumed to be known and constant. In the case of Venus,where small craters are absent due to atmospheric shielding, R

Fig. 2. Pie diagram of areas of geological units on Venus as mapped by Ivanov andHead (2011).

M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450 441

can be expressed as R = RV(F(Dmin) � F(Dmax)), where RV is a totalimpact rate on Venus, and F(D) is the cumulative size–frequencydistribution of newly forming craters defined as the proportion ofnewly forming craters larger than D among all newly forming cra-ters. The product RVF(D) is often called the production function ofcraters. The factor RAu in (1) is called the rate of the Poisson process.The validity of the Poisson process assumption is a special questionand is discussed in Section 5.4.

Mathematical statistics demonstrate that under the Poissonprocess assumption, the observed number of craters in the areagives an unbiased estimate Mu/(RAu) of the average crater retentionage tu of the unit. This mathematical statement means that if wehad many planets, each with an identical geological history (thesame Au), and with a randomly different cratering history (differentMu), the value of Mu/(RAu) averaged over all such planets wouldtend to tu, when the number of planets tends to infinity. Sincewe have only a single planet, Mu/(RAu) differs from the true agetu due to random fluctuations in Mu. If the number of craters Mu

is small, the quantity Mu/(RAu) can differ from the true age tu sostrongly that its use as a proxy for tu makes no sense. However,even in this case, the number of craters still bears some informa-tion about age. This information can be utilized through consider-

Fig. 3. (a) Magellan radar mosaic of crater Gautier (26.3�N, 42.8�E) and vicinity Radar lookPortion of the global geological map of Venus (Ivanov and Head, 2011) for the same area.is intact and hummocky ejecta are clearly superposed on a dark patch of regional plemplacement of these plains. All other parts of the rim are embayed and covered by lobalobate plains (at least those in contact with the rim remnants) postdate the impact.

ing the confidence interval for the mean age tu (e.g., Snyder andMiller, 1991):

F�1C ð1� p; MuÞ

RAu< tu <

F�1C ðp; Mu þ 1Þ

RAu; ð2Þ

where p is the chosen confidence level, for example, 0.9 or 0.95 or0.99, and F�1

C ðp; kÞ is the inverse cumulative gamma distribution.Obviously, for Mu = 0 only the upper boundary in (2) is meaningful;the lower boundary in this case should be 0. The function F�1

C ðp; kÞ isincluded in the majority of statistical software.

For a large number of craters Mu, in practice, for Mu > 10, theconfidence interval (2) is well approximated by the traditionallyused

ffiffiffiffiffiMp

error bars:

Mu � F�1n ðpÞ

ffiffiffiffiffiffiffiMup

RAu< t <

Mu þ F�1n ðpÞ

ffiffiffiffiffiffiffiMup

RAu; ð3Þ

where F�1n ðpÞ is the inverse cumulative standard normal distribu-

tion. It is equal to 1.3, 1.6, and 2.3 for p = 0.9, 0.95, and 0.99, respec-tively. In turn, ‘‘one-sigma’’, ‘‘two-sigma’’ and ‘‘three-sigma’’ errorbars (F�1

n ðpÞ = 1, 2, and 3) correspond to confidence levels p of0.84, 0.98, and 0.999, respectively.

We would like to reiterate that (3) is valid only for large Mu; ifthere are only a few craters, it is essential to use (2) rather than (3).On the other hand, all assumptions behind (2) are the same asbehind (3), and the applicability of constraints (2) from a few (oreven absent) craters is as well grounded as for large crater popula-tions and can be applied with the same certainty. Of course, for asmall number of craters the confidence intervals are wide.

Note that the crater retention age can vary over the outlinedunit area, and the confidence intervals are for the arithmetic aver-age of the age. The number of craters alone cannot say anythingabout the range of the ages inside the unit. The crater retentionage may depend on crater size, because resurfacing may obliteratesmall craters more effectively than large ones.

3.2. Buffered crater density for units

Craters on Venus are large and often overlap several differentunits; the dimensions of unit patches are often comparable in sizeto craters. In this case, the classic approach to cratering statisticsoutlined above should be modified to account for the non-negligi-ble size of craters. This can be done in a consistent and

is from the left, which means that topography appears illuminated from the left. (b)The crater diameter is about 60 km. The south–southwest segment of the crater rimains (unit rp2) with bright wrinkle ridges. Thus, the impact event postdates thete plains (unit pl), which also completely fill the crater interior. Emplacement of the

442 M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450

mathematically rigorous way by the use of buffered crater density.Some ideas on this technique were used by Namiki and Solomon(1994) and applied to dating linear features on Mars by Fassettand Head (2008).

If the center of a newly emplaced crater is just a little outside aunit u, the fact that it postdates the unit can be recognized due tothe finite size of the crater and its proximal ejecta. This means thatthe target area where a projectile should hit is somewhat largerthan Au (Fig. 4). On the other hand, if the projectile hits a smallisland of the unit, it would be completely destroyed by the newlyformed crater and the age relationship of the crater with the unitwould not be recognizable. In brief, for each crater diameter D,there is a certain buffer area Au

buf(D), where a projectile should hitto make it recognizable that the newly formed crater postdatesunit u. Area Au

buf(D) is usually an increasing function of D.For each narrow range of diameters [D, D + DD], the formation

of such craters is a Poisson process; for this case, in (1) we needto use Au

buf(D) instead of Au:

DP ¼ RV ðFðDÞ � FðDþ DDÞÞAbufu ðDÞDt:

If we sum all such contributions from all narrow diameterranges within a wide [Dmin, Dmax] range, we obtain the small prob-ability, DP, that a crater superposed over unit u is formed during asmall interval Dt:

DP ¼Z Dmin

D¼Dmax

RV Abufu ðDÞdFðDÞDt ¼ RAeff

u Dt; ð4Þ

where

Aeffu ¼

R DminD¼Dmax

Abufu ðDÞdFðDÞ

FðDminÞ � FðDmaxÞ: ð5Þ

Integration in (4) and (5) is formally written from the largest tosmallest diameter (from Dmax to Dmin), because F(D) is defined as adecreasing function of D.

It is seen from comparison of (4) and (1) that formation ofcraters superposed on unit u is a Poisson process in time with arate of RAeff

u . This means that Aueff defined by (5) plays the role of

an effective target area for unit u. This also means that the formu-lae for the confidence interval (2) are valid, if Au is substituted withthe effective buffer area Au

eff:

F�1C ð1� p; MuÞ

RAeffu

< t <F�1

C ðp; Mu þ 1ÞRAeff

u

: ð6Þ

The buffer areas Aubuf(D) should reasonably represent the target

area for craters of size D. The choice of these functions is notarbitrary; it is constrained by the following condition of self-consistency. Let us consider two units, a and b, and consider a

Fig. 4. Sketch explaining the calculation of buffer area for craters. (A) A piece of aschematic geological map with units a, b, and d, and a crater c of diameter Dc; thecrater is superposed on units a and b and its stratigraphic relationship with unit d isundefined. (B) The same piece of the map with lined area showing buffer areaAa

buf(Dc) for unit a with respect to a crater of diameter Dc.

formally merged unit a [ b; this merging is formal; the units donot need to have any geological similarity. The mean craterretention age of the merged unit can be estimated in two ways.First, we can apply (6) immediately to the merged unit:

ta[b � Ma[b=RAeffa[b: ð7Þ

Second, the same age can be estimated as the properly weightedaverage of ages of the individual units:

ta[b ¼Aa

Aa þ Abta þ

Ab

Aa þ Abtb �

Aa

Aa þ Ab

Ma

RAeffa

þ Ab

Aa þ Ab

Mb

RAeffb

: ð8Þ

Combining (7) and (8), we obtain a constraint:

Ma[b

Aeffa[b

� Aa

Aa þ Ab

Ma

Aeffa

þ Ab

Aa þ Ab

Mb

Aeffb

: ð9Þ

The approximate equality means that both sides should fit eachother within the confidence intervals allowed for randomfluctuations; in other words, (9) is the brief notation of a cumber-some system of inequalities analogous to (6). The constraints anal-ogous to (9) should hold for all combinations of units, not only pairs,but also multiple combinations. Thus, we have a large number ofindependent constraints. On the other hand, units with small num-bers of craters do not give good constraints because the confidenceintervals are wide. In any case, the choice of the set of functionsAu

buf(D) should comply with constraints (9). In particular, this require-ment should hold for the all units merged together, which means:

M �X

u

Au

Aeffu

Mu; ð10Þ

where M is the total number of craters in the considered diameterrange [Dmin, Dmax].

3.3. Size–frequency distributions

A necessary condition for the use of craters in some diameterrange [Dmin, Dmax] for age estimations is that the size–frequencydistribution of craters in this diameter range is similar to thesize–frequency distribution of the production function. If we donot know the production function, we still may meaningfully com-pare ages of units by comparing superposed crater densities, butfor this, again, the size distributions should be similar. The verifica-tion of the distributions is an essential component of the analysisof crater populations.

Since the target area for each unit depends on crater size, andthis dependence is different for different units, the comparison ofcrater size–frequency distributions is not as straightforward asfor the ‘‘standard’’ case of large units and small craters. We usethe following approach to this. If, again, F(D) is the productionsize–frequency distribution of craters on Venus, the productionsize–frequency distribution Fu(D) for craters superposed over agiven unit u can be calculated as

FuðDÞ ¼R D

D0¼1 Abufu ðD

0ÞdFðD0ÞR 0D0¼1 Abuf

u ðD0ÞdFðD0Þ

: ð11Þ

Thus, the observed size–frequency distribution for craterssuperposed over unit u should be similar to Fu(D), and not toF(D), as in the ‘‘standard’’ case of small craters.

If some unit is distributed as small and especially elongatedpatches, Au

buf(D) steeply increases with D, and the distributionFu(D) is significantly skewed toward large crater size in comparisonto the true production distribution. Thus, an apparent excess of

M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450 443

large superposed craters and a deficiency of small craters are a nat-ural consequence of large crater size and small unit size and notnecessarily related to the selective obliteration of small craters,as for the ‘‘standard’’ case of large units and small craters.

3.4. Buffered crater density for boundaries

Craters that are superposed on unit a and are embayed by unit bcan be used to put formal constraints on the mean time intervalbetween emplacement of units a and b. This mean time intervalis not necessarily equal to the difference between the mean agesof units a and b, because the actual ages can vary over the unit. Cra-ters that are superposed on a and are embayed by b sample onlythe vicinity of the contact between units a and b, and not theirwhole areas. In principle, the local time interval between unitemplacement may differ. For example, we can imagine that forma-tion of some volcanic complex starts with massive long flows thatform peripheral parts of the complex-associated lavas, while latereruptions produce much shorter flows; in this case the mean ageof the peripheral part of this hypothetical unit, where partly emb-ayed craters could be found would be older than the mean age ofthe whole unit.

Formal statistical constraints on the mean time interval Dta|b

between units can be obtained in the same way as for unit ages.We consider the number Ma|b of craters that are superposed on unita and are embayed by unit b, and a D-dependent buffer, the locusof crater centers, where it would be possible to identify cratersuperposition over unit a and embayment of unit b, whose areais Aa|b

buf(D). Then the effective area Aa|beff is calculated with (5),

where Aa|bbuf(D) is used for Au

buf(D), and, analogously to (6),

F�1C ð1� p; MajbÞ

RAeffajb

< Dtajb <F�1

C ðp; Majb þ 1ÞRAeff

ajb

: ð12Þ

3.5. Alternative approaches

In the procedure outlined above, if the number of craters islarge, the age estimate is:

tu � Mu=ðRAeffu Þ: ð13Þ

Fassett and Head (2008) used another version of the bufferedcrater count, namely, instead of (13) they applied estimate

tu �1R

XMu

i¼1

1

Abufu ðDiÞ

; ð14Þ

where the sum is taken over all craters superposing unit u. The dif-ference is that in (13) a single effective area Au

eff is used, while in (14)each crater is counted with its own buffer area. It is clear that in anideal case, for a very large number Mu of craters, (13) and (14) givethe same result. Intuitively, for a moderately large Mu, (14) gives abetter estimate, because it involves more information, namely, theinformation about actual craters sizes, while (13) uses their totalnumber only. However, calculation of supposedly narrower confi-dence intervals for (14) cannot be done in the framework of the rig-orous classic statistical approach. This can be done, however, in theframework of Bayesian inference (e.g., von Toussaint, 2011 and ref-erences therein), a popular modern approach under which themathematical probability theory is applied to natural problems ina principally, philosophically different way than in the classic‘‘frequentist’’ statistics. A good elementary-level comparison of‘‘frequentist’’ and Bayesian inferences is given by Cowan (2007).Being able to involve more information into statistical inferences,the Bayesian inference, however, relies on some additional poorlyconstrained assumptions about ‘‘prior’’ information, and the results

depend on these assumptions. For these reasons, in the presentpaper we consistently use the classic, ‘‘frequentist’’ approach.

4. Analysis procedure

4.1. Choice of the production function

The practical application of the theory outlined above requires aknowledge of the crater production size distribution F(D) in orderto perform calculations according to (5) and (11). We choose theglobal size–frequency distribution of all craters on Venus as aproxy for F(D). This choice ignores the probable preferentialobliteration of small craters, which possibly skews the actual dis-tribution toward larger craters in comparison to the true produc-tion function. This, however, is unlikely to bias the results, asfurther discussed in detail in Section 5.4. With our choice of F(D),for the practical calculation of (5) we have:

Aeffu ¼

1M

XM

i¼1

Abufu ðDiÞ; ð15Þ

where the sum is taken over all M craters on Venus in the selecteddiameter range [Dmin, Dmax]. In practice, we used the whole range ofcrater sizes on Venus, and thus M is the total number of craters onVenus.

Similarly, for practical calculations of (11), the expected produc-tion distributions are

FuðDÞ ¼P

i:DiPDAbufu ðDiÞP

jAbufu ðDjÞ

¼P

i:DiPDAbufu ðDiÞ

MAeffu

; ð16Þ

where the sums are taken over all craters and craters larger than Din the denominator and numerator, respectively.

4.2. Choice of the buffer area for units

Calculation of Aubuf(D) is the most cumbersome, computationally

expensive, and least certain part of the analysis procedure. Fig. 4illustrates the basic idea behind its estimation: the superpositionrelationship between a crater of diameter D and some unit u is dis-cernible if the unit material is present within a ring with outer andinner radii Rmax(D) and Rmin(D) around the center. Outside Rmax, cra-ter ejecta do not reach the unit, and superposition cannot be estab-lished. Inside Rmin everything is erased by the crater and itsproximal ejecta.

We chose Rmin and Rmax proportional to the crater radius D/2:

RmaxðDÞ ¼ 3ðD=2Þ; ð17Þ

and

RminðDÞ ¼ 1:5ðD=2Þ: ð18Þ

The Venus Crater Database (Herrick et al., 1997) available athttp://www.lpi.usra.edu/resources/vc/ contains the maximumextent of continuous ejecta for almost every crater on Venus. Weuse this quantity from the catalog to deduce the ratio of the max-imum radial distance of continuous ejecta from the crater center tothe crater radius. The median of this ratio is 3.0. This is the basis forour choice of factor 3 in (17). Herrick et al. (1997) have proposed anonlinear dependence of ejecta extent on crater size; however, thenatural scattering is huge, and we do not think that the observedrelatively wider ejecta for the smallest craters adequately reflectthe ability to discern superposition relationships. Thus, we choosea simple linear function (17). Our choice of factor 1.5 in (18) issomewhat arbitrary; however, the results are not sensitive to it.

We find the locus of such potential centers of craters of radiusD/2 in which there is unit u in the ring with outer and inner radii(17) and (18) around the center. Its area is the target area Au

t (D)

Fig. 5. (A) True areas Au (black bars) and effective areas Aueff (gray bars) for all units.

Note the logarithmic scale of the vertical axis: due to this the length of the graysection of each bar illustrates the ratio Au

eff/Au. (B) 90% confidence intervals for themean age of each unit. The vertical age axis is linear and is in the ‘‘stratigraphic’’direction: the present day is at the top, the past is downward.

444 M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450

for given unit u. Areas covered by actual impact craters, their prox-imal ejecta and outflows, were considered in this calculations asgaps in map coverage. Their total area is negligibly small and doesnot affect the results.

Our source data were a raster version of the global geologic mapin simple cylindrical projection with 156.5 pixels per degree sam-pling, which is equivalent to 0.675 km per pixel at the equator (thesame pixel size as C2-MIDR Magellan radar mosaics). For each unitand a number of their formal combinations, we map the target areain the same projection and the same sampling. Of course, for thecalculations we use true distances on a sphere, and latitude-depen-dent pixel area, so that the distortion of the projection does notaffect the results. We performed these calculations for a set of fivecrater diameters D = 5, 10, 20, 50, 100 km, and then interpolated/extrapolated the results for any D with the followingapproximation:

AtuðDÞ ¼ Auð1þ D=duÞcu ; ð19Þ

where Au is the true area of unit u, while du > 0 and cu are fittingparameters. Such an approximation automatically provides forAu

t (0) = Au (what we expect in the ‘‘standard’’ case of small craters).Two parameters turn out to be sufficient to span the variability ofdependences: fitting five calculated points with two parameterscan be done accurately.

When we use the calculated target area directly:

Abufu ðDÞ ¼ At

uðDÞ; ð20Þ

and then calculate Aueff with (15), we find that constraints (9) and

(10) are not satisfied: the calculated Aueff are way too large. Our

attempts to use a smaller sensitivity ring (17), (18) or use a nonlin-ear ring size according to Herrick et al. (1997), did not give muchimprovement, unless the sensitivity ring becomes smaller thanthe crater, which is unrealistic. The reason for this apparent discrep-ancy is well understood: the assumption that any superpositionwithin the sensitivity ring can be discerned is too optimistic (seeFig. 4, unit d for an illustration). To account for this, instead of(20) we choose:

Abufu ðDÞ ¼ nAt

uðDÞ þ ð1� nÞAu; ð21Þ

where the balance factor n is chosen so that constraint (10) is satis-fied precisely. When we use the whole set of craters on Venus (notlimited to any diameter range), we find n = 0.59. When we mergeunits rp1 and rp2 into a single unit rp (see Section 5.1), we obtainn = 0.60. With such a choice of Au

buf(D) and n, the other constraints(9) are satisfied within the formal statistical limits.

The values of Aubuf(D) obtained are essentially different for differ-

ent units. For units forming small outcrops, especially, elongatedones, for example, pdl, Au

buf(D) increases steeply with D, while forunits in larger contiguous isomeric patches, for example, sc, thisdependence is less steep. For the examples mentioned, Apdl

buf

(50 km)/Apdl = 5.9, while Ascbuf(50 km)/Asc = 2.8. Fig. 5A shows the

relative excess of the effective areas Aueff over true unit areas

Au for each unit.Although our choice of Au

buf(D) is an educated guess supportedby constraints (9), it captures the essential geometric differencesbetween units and is much more accurate than any attempt at a‘‘blind’’ definition of Au

eff that does not take the actual geologicmap into account.

4.3. Choice of the buffer area for boundaries

For calculating Aa|bbuf(D) we used the same procedure as

described above. We find a target as the locus of such potentialcenters of craters that there are both units a and b in the same sen-

sitivity ring (17) and (18). We approximated the target area Aa|bt (D)

with a two-parametric increasing function providing for Aa|bt (0) = 0:

AtajbðDÞ ¼ ðD=dajbÞcajb ; ð22Þ

where da|b > 0 and ca|b > 0 are fitting parameters. Then, analogouslyto (21), we define

Abufajb ðDÞ ¼ nAt

ajbðDÞ; ð23Þ

with the same value of n as in (21).Our educated guess about Aa|b

buf(D) is not supported by anyadditional constraints and is based only on analogy with the betterconstrained case for units, however, our definition of Aa|b

buf(D) cer-tainly captures the essential geometric features of unit boundariesand their spatial distribution.

4.4. Absolute ages

The absolute cratering rate RV on Venus is poorly known; conse-quently, the absolute ages are also poorly constrained. To avoidthis uncertainty, we express all ages in terms of T, the mean craterretention age of Venus. Practically, for T we use its estimate

T ¼ M=RV AV ; ð24Þ

where AV is the total imaged area of Venus.With this estimate of T,the age constraints (6) can be written for practical use in the follow-ing form:

M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450 445

F�1C ð1� p; MuÞ

MAV

Aeffu

T < tu <F�1

C ðp; Mu þ 1ÞM

AV

Aeffu

T: ð25Þ

An analogous formula can be used for (12).According to cratering rate estimates, the absolute value of T is

bracketed between 0.2 Ga and 1 Ga (see McKinnon et al., 1997, fordiscussion).

4.5. Processing summary

To summarize, data processing includes the following steps:

� The map in raster format is used to calculate target areas Aut (D)

for each unit and several of their combinations, and target areasAa|b

t (D) for each pair of units using a sensitivity ring with innerand outer radii defined by (17) and (18). All these calculationsare done with true distances and areas on a sphere (rather thandistorted distances and areas in the map-projected raster). Thecalculations are repeated for the set of five diameters D.� For each unit u and each pair a|b, two fitting parameters d and c

of approximations (19) for Aut (D) and (22) for Aa|b

t (D) are calcu-lated to fit five points for five diameters. The substitution of(19) and (22) with the d and c found into (21) and (23) yieldsAu

buf(D) and Aa|bbuf(D), respectively.

� Expected effective production size–frequency distributionsFu(D) are calculated with (16) and compared against actualcumulative size–frequency distributions.� Effective areas Au

eff and Aa|beff are calculated with (15).

� 90%-confidence intervals for ages tu and age intervals Dta|b inunits of T are calculated with (25), assuming p = 0.9.

5. Primary results

5.1. Size–frequency distributions

Fig. 6 compares the actual size–frequency distributions of thesuperposed craters with the predicted effective production distri-bution defined by Eq. (11) and calculated according to Eq. (18).To assess the significance of the observed differences, we formallyapply the Kolmogorov–Smirnov test. If the curves in Fig. 6 arewithin the corridors around the diagonals, the deflection is not sta-tistically significant, it can just be a random fluctuation; if a curvebreaks out of the corridor, it means that the deflection is signifi-cant: randomness is formally rejected (at 90% confidence).Although the use of the Kolmogorov–Smirnov test for the ‘‘buf-fered’’ distributions is not strictly mathematically grounded, itgives a correct idea about significant and not significant deflection,except in marginal cases.

Comparison of the shapes of the size distributions shows thatfor each unit (with a single exception) the actual shape of the sizedistribution of superposed craters is consistent or marginally con-sistent with the shape of the distribution for the global craterpopulation.

The single exception is the younger regional plains subunit rp2

(Fig. 6). It has a significant relative excess of smaller craters anddeficiency of larger craters in comparison to the whole population(the curve in Fig. 6 breaks upward out of the corridor). Therandomness of this deflection is formally rejected by the Kolmogo-rov–Smirnov test at 99.3% confidence. Resurfacing can selectivelyobliterate smaller craters, but cannot preferentially remove largercraters. Thus, a straightforward explanation would be that rp2 rep-resents a rather pristine population, while all other units under-went some small-scale resurfacing and preferential obliterationof smaller craters. This explanation is consistent with the geologi-cal nature of rp2: this unit is a younger part of the regional plainsboth stratigraphically and according to large-crater density.

However, this straightforward explanation has significant diffi-culties. First, it is difficult to explain why the preferential oblitera-tion of small craters produces the same result for such differentprocesses as small-scale volcanic resurfacing of different styles(pl, ps), tectonic deformation of different styles (t, rz, pdl),unknown process of regional plains aging (rp1), etc. Second, thedeflection (Fig. 6) occurs for larger craters (upper right part ofthe plot), and not for small craters (lower left part) as would bepredicted by preferential obliteration of small craters. In a sense,we observe not an excess of small craters for rp2, but a deficiencyof large ones. Third, the actual size–frequency distribution for thelargest craters is unrealistically steep; for D > 40 km the power-law fit has an exponent of �3, steeper than for comparable craterson the Moon and Mars, and steeper than inferred from the present-day population of potential impactors (McKinnon et al., 1997). Thisis why we prefer another explanation.

We suggest that large craters and identification of rp2 are notindependent: in the broad vicinity of some large superposed cra-ters, the rp2 cannot be identified as such and is mapped as rp1

instead. This leads to an apparent deficiency of large craters onrp2. A possible mechanism of this is the following: A large impactevent ejects a significant volume of fine-grained material formingextensive deposits known (for the youngest craters) as radar-darkparabolas and haloes. With time the radar-dark signature of thehaloes and parabolas fades away by some mechanism(s), whilethe deposit itself remains in place and obscures pre-impact radarsignatures. This prevents identification of rp2 in some areas whererp2 would be visible if the crater-related deposits were absent. Thesurficial deposits obscuring radar signatures can be redistributedby winds. For our explanation to be viable, they only need toremain in some spatial association with their parent craters for ageologically significant time.

The unusual crater size–frequency distribution for rp2 makesage inference for this unit impossible. To mitigate this problemwe do not consider rp2 separately, and consider a single mergedregional plain unit rp that comprises both rp1 and rp2. Not surpris-ingly, the crater size–frequency distribution for this merged unit isconsistent with all others.

The major plains units (rp, psh, pl) have remarkably similar dis-tributions after the correction for buffer area. The only unit thathas only marginally consistent crater size–frequency distributionis gb (Fig. 6); it shows a relative excess of small craters and adeficiency of large ones. It is possible that this is an effect of anincorrectly estimated Agb

buf(D). Even heavily tectonized units(t, rz) are within the statistical consistency margins. Our primaryinterpretation of this fact is that overall, small-scale resurfacingon Venus is minor, and selective obliteration of small craters isminor. This result justifies our choice of the global population asa proxy for the production distribution. The opposite scenario,although not completely excluded, seems unlikely: it is difficultto imagine how preferential obliteration of small craters can befine-tuned to produce a similar outcome for units of differentresurfacing styles and ages.

Heavily tectonized units, t and rz have some deficit of smallcraters, although it is not statistically significant (see deflectionin the lower left part of the plot for t in Fig. 6). A deficiency ofsmall craters on tesserae (t) has been found by Ivanov andBasilevsky (1993) who suggested that some small craters are hid-den, and thus lost, amid the radar-bright tectonic fabric of theseunits. Instead, we see here that at least partly this is due to thelarge sizes of craters on Venus: the correction for buffered targetarea reduces the apparent deficiency of small craters significantly.The effective tessera target area increases with the increase incrater size, and this is why the distribution of superposed craterson tessera is skewed toward larger sizes in comparison to the glo-bal population.

Fig. 6. Comparison of actual and production size–frequency distributions of craters superposed over six selected units. For each point on the curve, its abscissa is the expectedproduction distribution Fu(D) calculated according to (16), while its ordinate is the actual distribution for the same crater diameter D. If the distributions were identical, thecurve would coincide with the diagonal. Deflection upward from the diagonal means a relative excess of small craters and a deficiency of large craters in the actualdistribution in comparison to the production one; deflection downward means the opposite. Two lines parallel to the diagonal outline a formal 90% confidence corridoraccording to the Kolmogorov theorem. If the curve deflects out of the corridor, the consistency of the distributions is rejected at 90% confidence with the Kolmogorov–Smirnov test. Small craters are in the lower left corner of each plot, large craters being in the upper right corner.

446 M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450

5.2. Mean crater retention ages

Fig. 5B shows 90% confidence intervals for the mean craterretention ages of all units. Ages are given in terms of T, the meancrater retention age of the entire observed population. It is seenthat there is a group of older units (t, pdl, pr, mt, gb, psh, rp) anda group of significantly younger units (sc, pl, rz), with ps occupyingan intermediate position.

The observed mean crater retention ages, taking the 90% confi-dence intervals into account, do not contradict the documentedlocal, regional and global stratigraphic relationships between theunits (Basilevsky and Head, 1998; Ivanov and Head, 2001, 2011).The difference between the mean age of any young unit (sc, pl, rz)and any old unit is (t, pdl, pr, mt, gb, psh, rp) is statistically signifi-cant and consistent with the observed stratigraphic relationships.This difference is also consistent with previously reported youngcrater retention ages of rift zones (Namiki and Solomon, 1994)and lobate plains (Price et al., 1996). Within each of the two unitgroups the age differences are not statistically significant and thusdo not contradict the observed stratigraphic relationships.

Within the group of older units, the mean ages of pdl and prappear shifted toward the younger side in comparison to the regio-nal plains rp, contrary to well-documented stratigraphic relation-ships, although this difference is not statistically significant (90%confidence intervals overlap). These units have the highest Au

eff/Au

ratio (Fig. 5A), namely Apdleff/Apdl = 4.5, Arp

eff/Arp = 2.9; hence theage estimates are the most sensitive to the accuracy of the estimateof Au

buf(D). The younger age of sc with respect to ps does not contra-dict the observed stratigraphic sequence: the units are rarely incontact and no pervasive stratigraphic relationship can be estab-lished from observations (Ivanov and Head, 2011).

In Fig. 7A all units stratigraphically older than the regionalplains (t, pdl, pr, mt, gb, psh) are merged together, and the meancrater retention age of the combined unit that now has a narrowconfidence interval is compared to regional plains. The same isdone with a set of younger units (ps, sc, pl); rift zones (rz) areexcluded from this calculation as a deformational rather thanmaterial unit. The stratigraphically older units taken together are

shifted toward the older side from regional plains (rp), however,the crater retention age difference is not statistically significant.

Similarly, Fig. 7B shows the results for units merged accordingto their general relationships of embayment and crosscutting.These unit groups represent specific regimes of resurfacing accord-ing to Ivanov and Head (in preparation-a) (Fig. 1): (i) tectonicregime when tectonic deformation dominated (t, pdl, pr, mb, andgb), (ii) volcanic regime when vast and mildly deformed plainswere emplaced (psh, rp1 and rp2), and (iii) network rifting-volca-nism regime when both volcanism and tectonism played aboutequally important roles in resurfacing (sc, ps, pl, rz). The 90%-con-fidence intervals for the tectonic and volcanic regimes almost com-pletely overlap each other indicating that their ages are notdistinguishable on the basis of crater densities.

5.3. Mean age intervals

Craters superposed on one unit and embayed by material ofanother unit provide constraints on the mean interval betweenlocal emplacement of the units, as discussed in Section 3.4. Forexample, 11 craters superposed on psh and embayed by rp indicatethat the mean interval Dtpsh|rp between local psh and rp emplace-ment is bracketed between 0.03T and 0.12T (90% confidence). Sincethe mean ages of psh and rp are not statistically distinguishable,the lower boundary of Dtpsh|rp indeed places an important con-straint on geological history: emplacement of rp does not alwaysoccur immediately after emplacement of psh.

For each pair of units a and b, the number of craters superposedon a and embayed by b is always rather small; therefore the con-fidence intervals for the mean age intervals are wide. Not surpris-ingly, with a single exception, all mean intervals are consistentwith the difference in the mean ages discussed in Section 5.2, tak-ing the wide confidence intervals into consideration.

The only exception is again related to the rp2 unit (cf. Section5.1). There are no craters superposed on rp1 and embayed by rp2.This puts an upper boundary of Dtrp1|rp2 < 0.07T (90% confidence)for the mean interval between their emplacement. This is not con-sistent with the formal mean crater retention age brackets

Fig. 7. (A) Bold lines show the 90% confidence intervals for the mean age of regional plains (rp) and two merged units: ‘‘old’’ are merged t, pdl, pr, mt, gb, psh; ‘‘young’’ aremerged sc, ps, pl. ‘‘Old’’ line is repeated twice. Pairs of thin lines show the 90% confidence intervals for the mean time difference between the pairs of (merged) units. Theirvertical location is arbitrary, only their length is informative: they should be considered as freely sliding up and down. The vertical age axis is linear and is in the‘‘stratigraphic’’ direction: the present is at the top, the past is downward. (B) The same plot, except that the units are merged according to resurfacing regimes followingIvanov and Head (in preparation-a) (Fig. 1): T, tectonic regime, merged t, pdl, pr, mt, gb; V, volcanic regime, merged psh and rp; N, network rifting-volcanism regime, mergedsc, ps, pl, rz.

M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450 447

trp1 > 1.07T and trp2 < 0.86T. Of course, because of the unusual size–frequency distribution, the formal ages for rp2 are not meaningful,as was discussed in Section 5.1. The observed absence of rp1|rp2 cra-ters provides additional insight into the nature of the size–fre-quency distribution anomaly. It contradicts the ‘‘straightforward’’explanation with a truly young rp2, an explanation that we alreadyrejected above. On the other hand, it is consistent with our preferredexplanation, where the true mean ages of rp1 and rp2 are similar.

In order to reduce the high formal uncertainties in the mean ageintervals, we considered boundaries between merged units. Thereare 19 craters superposed over at least one ‘‘older’’ unit (t, pdl, pr,mt, gb, psh) and embayed by regional plains (rp); 13 craters aresuperposed over rp and embayed by at least one ‘‘younger’’ unit(ps, sc, pl); finally, 30 craters are superposed over at least one‘‘older’’ unit and embayed by at least one ‘‘younger’’ unit. Threepairs of thin lines in Fig. 7A show formal lower and upper bound-aries for the mean interval derived from these three numbers.Within the 90%-confidence intervals, the mean intervals are con-sistent with the differences between mean ages.

Similarly, Fig. 7B shows constraints on the mean age intervalsfor different resurfacing regimes according to Ivanov and Head(in preparation). Although the mean ages of the tectonic (t, pdl,pr, mb, and gb) and volcanic (psh, rp) regimes are indistinguish-able, 15 craters superposed over tectonic units and embayed byvolcanic units constrain the mean age interval between thembetween 0.06T and 0.12T.

5.4. Biases and caveats

Our quantitative inferences are based on the assumption thatcratering is well described by the mathematical model of the Pois-son process. For Venus this assumption is more well-justified thanfor other bodies. Secondary craters, the main violator on otherbodies, play no role on Venus due to its thick atmosphere. Doubleasteroids violate the assumption in principal due to correlation ofimpacts in space and time. However, all distances between craterson Venus are longer than a characteristic size of unit outcrops;therefore the spatial correlation does not affect our results at all.Correlation in time makes actual confidence intervals a little wider,which we ignore.

The latitudinal dependence of impact rate on Venus, unlike onthe Moon, Earth and Mars, is absent, according to Le Feuvre andWieczorek (2008). Their results ignore the collimating effect ofthe atmosphere; however, taking this effect into account wouldnot make latitudinal dependence significant.

Atmospheric shielding affects the impact flux; therefore, inprinciple the flux and production size–frequency distributiondepend on elevation, which is ignored in our calculations. Forplains units this effect is negligible: the elevation amplitude ofplains is tiny in comparison to the atmospheric scale height of�16 km. For tesserae (t) the effect might be non-negligible, how-ever, since we do not observe an excess of small craters on tes-serae, which would result from a thinner shielding layer, thiseffect is (over)compensated by preferential obliteration of smallcraters.

The assumption of stability of the impactor flux in time is notwarranted. However, modest variations of the flux simply meanmodest nonlinear rescaling of the age axis and do not affect qualita-tive results. Much more significant variations could affect theresults. For example, the similarity of the size–frequency distribu-tions and concentration of the inferred unit ages at 1.2–1.0T in prin-ciple could be explained by a low flux during formation of the ‘‘old’’units (so that apparent ages 1.2–1.0T reflect a long physical timespan) followed by a short intensive impact shower which createda significant part of the observed present-day crater populationrather recently. Such huge long-term variations of the impactorflux, if they indeed occurred, would occur over the entire inner SolarSystem and be obvious in the terrestrial geological record.

A change in the areas of units due to global tectonic extensionor contraction (formation of fractures or wrinkle ridges) after cra-ter emplacement is ignorable: its effect is well below a percentlevel (e.g., Kreslavsky and Basilevsky, 1998).

Another assumption in our quantitative statistical inference isthe use of the actual global venusian size–frequency distributionas the production distribution in Eqs. (5) and (11). Since thesize–frequency distributions for all units are actually consistent(Section 5.1), the validity of this assumption does not affect theresults. We suggest that this consistency results from the fact thatobliteration of craters on Venus is minor. Even if this suggestion iswrong, and there is a significant preferential obliteration of small

448 M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450

craters that is for some unusual reason consistent over all units,this would lead effectively to renormalization of absolute ages (abias in T), but will not affect the relative age relationships.

The most uncertain and potentially damaging assumption is thechoice of functions Au

buf(D) and Aa|bbuf(D) for each unit or unit pair. As

we already discussed, our choice is a mere educated estimate sup-ported by constraints (9). Our choice certainly captures the essen-tial geometric differences between units. For larger units, theeffective area Au

eff is dominated by the true unit area Au; for exam-ple, for regional plains (rp) Arp

eff exceeds Arp by 47% (Fig. 5A). Sinceour choice cannot be very bad, the uncertainty in Au

eff for the largeunits is perhaps at a level of a few percent and is not important. Forsmall units, where Au

eff exceeds Au by a factor of 3 and more, tens ofpercent bias can be anticipated. We probably see this bias for unitspdl and pr in Fig. 5B (see Section 5.2 for discussion).

Since there are no constraints like (9) for the effective unitboundary buffer area Aa|b

eff, they are less reliable than Aueff. We can-

not exclude tens of percent uncertainties in inferences from emb-ayed craters (Section 5.3).

6. Interpretation and discussion

6.1. Resurfacing style: good consistency with global resurfacingscenario

The key question that many researchers have tried to addresswith crater statistics is the history of Venus resurfacing (Schaberet al., 1992; Phillips et al., 1992; Basilevsky et al., 1997). At firstglance, Occam’s razor dictates the so-called equilibrium resurfacingor stationary resurfacing history (e.g., Phillips et al., 1992; Guestand Stofan, 1999), under which the surface is gradually renewedby volcanism and tectonics that have the same style and rate (onaverage), and old craters are obliterated at the same rate as newcraters are formed. This is similar to the ‘‘non-directional’’ resurfac-ing history of Venus proposed by Guest and Stofan (1999). A num-ber of observations, however, suggest another, more complexresurfacing scenario involving a significant change of styles andrates of tectonics and volcanism in time, so-called global resurfac-ing. Under this scenario (Basilevsky et al., 1997; Ivanov and Head,2011, 2013, in preparation-a), an earlier epoch of intensive tecton-ics and volcanism, when all pre-existing impact craters were oblit-erated, is followed by a modern epoch with modest localizedvolcanism and rifting, during which craters are accumulating. Insummary, the observations that suggest global resurfacing are:generally uniform spatial distribution of craters on the whole pla-net, a large proportion of pristine craters in the crater population,and the same stratigraphic sequence of tectonic textures and volca-nic styles observed in many localities over the whole planet. Thepresent work adds one more independent observation to this list:similarity of size–frequency distributions of craters superposedon different units of different morphologies and ages.

Crater densities that we analyze in this paper give informationabout the mean ages of the units and the mean age differencesbetween the units, but cannot say anything about age variationswithin units. Therefore these data alone cannot provide an ulti-mate answer about the resurfacing style. However, some infer-ences are possible.

Our analysis reveals three robust facts about the relationshipsbetween craters and morphological units: (i) all size–frequencydistributions are consistent with each other; (ii) all units strati-graphically older than regional plains have a mean age close to thatof regional plains; (iii) all units stratigraphically younger thanregional plains have a mean crater retention age significantlyyounger than the regional plains. (The latter two facts have beenalready established in a preliminary analysis by Ivanov and Head,in preparation-a.) These three facts are naturally and consistently

explained in the framework of the global resurfacing scenario: (i)After �T ago the overall resurfacing was minor, the total numberof obliterated craters was minor, therefore preferential removalof small craters by resurfacing was negligible, and all size–fre-quency distributions are consistent with the production distribu-tion. (ii) Intensive tectonic and volcanic resurfacing before �Tago is responsible for similar mean ages of the regional plains(rp) and all pre-rp units. (iii) Formation of the post-rp unitsoccurred slowly through the whole geological history after �Tago, which produces ages close to �0.5T, which is actuallyobserved.

In the context of this scenario, our results provide valuableinformation on the duration of global resurfacing, at least for itslatest stages that left the readable record at the surface. The pres-ence of craters embayed by rp and older materials rejects the mostradical, catastrophic version of global resurfacing (Schaber et al.,1992), under which resurfacing occurs geologically instantly. Nine-teen (19) craters embayed by rp give a mean age differencebetween rp and pre-rp material within 0.08–0.15T (Fig. 7); themean age constraints give a difference within 0–0.20T. Two craterssuperposed over tesserae (t) and embayed by shield plains (psh)give 0.02–0.18T mean age interval between t and psh, etc. In sum-mary, the observable ‘‘wake’’ of the intensive resurfacing lasted onthe order of �0.1T, tens of millions of years.

It is important, however, that these numbers constrain the localduration of the early intensive events, however, they do not con-strain how synchronous the intensive resurfacing was over the pla-net. Our data alone cannot address synchronicity. The absence ofinverted stratigraphic sequences and the even spatial distributionof craters exclude extremely wide age differences.

More detailed analysis of the stratigraphic sequences has ledIvanov and Head (in preparation-a) to a more detailed scenarioof directional resurfacing history of Venus, where the followingthree regimes occurred consecutively: (1) tectonic regime, whentectonic deformation dominated (units t, pdl, pr, mb, and gb), (2)volcanic regime, when vast and mildly deformed plains wereemplaced (psh, rp1 and rp2), and (3) network rifting-volcanismregime, when both volcanism and tectonism played about equallyimportant roles in resurfacing (sc, ps, pl, rz). The 90%-confidenceintervals for the mean age of tectonic and volcanic regimes almostcompletely overlap each other indicating that the regimes are notdistinguishable by the crater statistics. Stratigraphically, however,the vast plains are always younger. This implies that the global-scale change from the tectonic to the volcanic regime was fasterthan the temporal ‘‘resolution’’ of the crater statistics. The meanage of units of the network rifting-volcanism regime is much youn-ger. The interpretations of relationships of impact craters and themost important features of this regime (flows of lobate plainsand graben of rift zones) suggest that both lobate plains and riftzones formed during a prolonged time interval, perhaps duringthe entire Atlian period of the geologic history of Venus(Basilevsky and Head, 2000).

6.2. Resurfacing style: poor consistency with equilibrium resurfacingscenario

We now discuss our observations in the context of the station-ary or equilibrium resurfacing concept. At first glance, in equilib-rium, all units should have approximately the same mean age,which is not observed. However, definition of units is affected bytheir stratigraphic relationships. For example, shield clusters (sc)are fields of small shield volcanoes that postdate regional plains.Therefore, in the equilibrium case we would expect their agesomewhere in the middle between rp age and the present, that is�0.5T, which is consistent with observations (Fig. 5B). In an analo-gous manner, we would expect shield plains (psh), fields of volca-

M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450 449

nic shields embayed by regional-plain-forming lavas, to be �0.5Tolder than rp (�1.5T), which contradicts the inferred upper limitof 0.12T for the mean interval between psh and rp. To avoid thisdiscrepancy we need to involve some geophysical model thatwould predict that plains-forming flood lavas would quickly followshield formation locally.

According to the equilibrium resurfacing concept, the ‘‘old’’tectonized units (pdl, pr) are locally older regional plains that accu-mulated more tectonic deformation and then were embayed bylocal younger flood lavas. In this case, we would expect a signifi-cant mean age difference along their boundaries with rp (the timeneeded for accumulation of tectonic deformation), which is notsupported by the observations. The observed short mean timeintervals between tectonized plains and regional plains requireanother specific geophysical explanation to make them consistentwith the equilibrium concept.

The young crater retention age of lobate plains (pl) cannot beexplained within the equilibrium concept with the ‘‘stratigraphicbias’’ in the same way as it was successfully done above for shieldclusters (sc): there are no lobate plains embayed by regional plains.According to Guest and Stofan (1999), the young age of lobateplains is explained by fading of radar contrasts with time: olderlobate plains are invisible and make an inherent part of regionalplains. In this case we would anticipate the same distortion ofsize–frequency distribution for pl, as we observe for rp2 (see Sec-tion 5.1). Such a distortion, however, is not observed (Fig. 6), whichis difficult to explain in the framework of the equilibrium concept.

In summary, adhering to the equilibrium resurfacing conceptrequires numerous additional assumptions to make the obtainedconstraints consistent with the concept; virtually each unit requiresindividual explanations and individual set of assumptions. Mean-while, the global resurfacing scenario naturally and consistentlyexplains the observations without any additional assumptions.Thus, the Occam’s razor logic together with our observations makesglobal resurfacing scenario much more favorable.

6.3. Surficial deposits

Evidence for extensive surficial deposits associated with largeold craters comes from the anomalous size–frequency distributionof craters superposed over regional plain subunit rp2 (a significantdeficiency of large craters, Section 5.1) and lack of craters embayedby it (Section 5.3). These observations are explained through thepresence of surficial deposits created by granulated materialejected by large-scale impact events and reworked and redistrib-uted by winds. Such deposits have been hypothesized byBasilevsky et al. (2004, 2007) and Bondarenko and Head (2004).Bondarenko and Head (2009) showed some morphological obser-vations in radar images that could be explained by such deposits.Here we see that crater statistics supports their existence.

6.4. Constraints on climate change

The production size–frequency distribution of craters is con-trolled by atmospheric shielding. If the atmospheric pressure weresignificantly different during the early half of the recorded geolog-ical history, we would see a significant difference in crater size–fre-quency distributions between older and younger units. Theobserved remarkable similarity of the size–frequency distributionsof younger lobate plains (pl) and older regional plains (rp) andother units indicates that the atmospheric pressure was about con-stant during �T. Additional modeling is required to quantify thisstatement.

The observed absence of atmospheric pressure changes is notsurprising. In the frame of the global resurfacing scenario, it is rea-sonable to expect significant outgassing and pressure increase dur-

ing the period of active volcanism, about �1.1T ago (and possiblyearlier). The oldest units have a comparable mean age and theircrater populations therefore cannot record a possible earlier lowerpressure atmosphere.

7. Conclusions

We analyze quantitatively the population of craters on Venusand their stratigraphic relationships with geological units thathave been globally and uniformly mapped by Ivanov and Head(2011). Our analysis systematically takes the large sizes of cratersinto account. Constraints on the mean ages of the units are sum-marized in Figs. 5B and 7. We found the following:

� Size–frequency distributions of craters superposed on eachstratigraphic unit are consistent with each other, if the regionalplain subunits are merged together.� Regional plains and stratigraphically older units have similar

crater retention ages.� Stratigraphically younger units have a mean crater retention

age significantly younger than the regional plains.

These findings are naturally and consistently explained by adirectional scenario of resurfacing history, with global intensiveresurfacing followed by prolonged (0.2–1 Ga) slow spatially lim-ited rifting and volcanism (e.g., Basilevsky et al., 1997; Basilevskyand Head, 2002a,b; Ivanov and Head, 2011, 2013, in preparation-a). A remaining question not addressable in this study is the dura-tion of the intensive resurfacing back into time; for example, both asingle geologically short resurfacing spike and a one-time transi-tion from intensive to sluggish resurfacing are equally consistentwith crater statistics. Our analysis, however, shows that the latestpart of intensive resurfacing lasted on the order of 10% of the meanage of the global resurfacing event (tens of millions of years). Ouranalysis cannot address the question of whether the termination ofthe intensive resurfacing was synchronous over the whole planetor not. Our observations are difficult to reconcile with the equilib-rium (stationary) resurfacing scenario.

Analysis of the population of craters superposed over two sub-units of regional plains suggests that there are extended depositsassociated with large craters that are almost indiscernible in theradar images, but obscure radar contrasts with predating lavaflows. We do not see evidence for any significant and prolongedchange of atmospheric pressure after the global resurfacing event.

Our comprehensive analysis mostly exhausts what can be under-stood on the basis of the classical, ‘‘frequentist’’ statistical approach.Application of Bayesian inference with the same data set andassumptions could lead to slightly narrower formal confidenceintervals of ages at the expense of a less robust statistical treatment;however, this is unlikely to add any essential knowledge aboutVenus. Bayesian inference does allow, however, the involvementof some additional, possibly reasonable assumptions; these include,for example, that a stratigraphically younger unit always has ayounger mean age, the mean age difference at the unit boundaryis equal to the difference between the mean unit ages, any craterwith a parabola is younger than any crater of the same size withouta parabola, etc. Such an exercise in Bayesian inference might havesome potential for new insights, but it is limited by the inherentinaccuracy of our knowledge of the target buffer area functions.

Acknowledgments

This work was partly supported by NASA grant NNX11AQ46Gto MAK and by the grant of the Russian Academy of Sciences Pre-sidium Program 22: ‘‘Fundamental problems of research andexploration of the Solar System’’ to MAI. JWH gratefully acknowl-

450 M.A. Kreslavsky et al. / Icarus 250 (2015) 438–450

edges earlier support from the NASA Planetary Geology and Geo-physics Program which supported numerous Venus mapping andanalysis efforts that helped lay the groundwork for this synthesis.

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