1.1 Impacts and Crustal Magnetization
 Large impacts on Mars (which, for our purposes, form craters > ~300 km in diameter), alter the magnetization of the entire depth of crust over a geographic area comparable to the final size of the resulting crater [Hood et al., 2003; Shahnas and Arkani-Hamed, 2007]. Excavation removes and reorients magnetized material within the transient cavity [Melosh, 1989]. Further, shock heating causes thermal demagnetization [Mohit and Arkani-Hamed, 2004]. Following the impact, as the new crust cools, the melt sheet and any other crustal minerals heated above their Curie point can acquire a new thermoremanent magnetization (TRM) with a magnitude proportional to the strength of the local ambient magnetic field and the capacity of the rock to carry thermoremanence.
 In addition, shock from the impact can add or remove net magnetization, depending on this local magnetic field and prior magnetization state of the crust. Unmagnetized materials can be magnetized in an external magnetic field through shock remanent magnetization (SRM), and existing magnetization can be reduced or erased if the minerals are shocked in an ambient field too weak to induce a sufficient SRM. In addition, SRM may not be stable over geologic timescales [Cisowski and Fuller, 1978, Gattacceca et al., 2010].
 Brecciation and fluid circulation can combine to produce post-impact hydrothermal systems which can lead to the acquisition by crustal rocks of chemical remanent magnetization (CRM), the strength of which is controlled primarily by oxygen fugacity which controls the minerals that form and cooling speed which affects grain size [Grant, 1985]. It is important to note that essentially all magnetization in Martian impact structures is TRM, SRM, or CRM, comprising what is commonly referred to as natural remanent magnetization (NRM). This is in contrast to the case of terrestrial impact structures where a substantial component of magnetization induced by the geomagnetic field can account for anywhere from ~5% to >90% of total magnetization [Ugalde et al., 2005]. Mars' lack of a global magnetic field, and hence induced magnetization, thus removes a substantial complication from the interpretation of impact crater magnetic signatures.
 TRM tends to be the strongest and most stable of the types of NRM for the iron-bearing minerals likely responsible for Mars' remanent magnetism [Dunlop and Arkani-Hamed, 2005] and, if there exists a sufficiently strong ambient field, is virtually certain to occur in a substantial portion of large craters [Shahnas and Arkani-Hamed, 2007]. Although the magnetic mineral fraction in post-impact crust can be strongly affected by CRM processes like serpentinization [Quesnel et al., 2009], one can nonetheless argue that the magnetization contained within an impact crater is a reasonable proxy for the strength of the ambient magnetic field averaged over the crater cooling time (less than several million years [Ivanov, 2004]). Impact craters are thus useful in piecing together the history of Mars' ambient magnetic field.
1.2 The History of the Martian Dynamo
 Mars does not currently possess a global dynamo-driven magnetic field, but evidence of strong crustal magnetization implies that such a field is all but certain to have existed in the planet's early history [e.g., Acuña et al., 1999]. The dynamo may have started immediately following accretion/differentiation [Williams and Nimmo, 2004] or alternatively may have been inhibited for up to ~100 Ma (i.e., past initial crustal formation) by thermal stratification of the core resulting from collisions with large planetary embryos. This latter hypothesis is supported by the large extent of likely nonmagnetic primordial crust in the southern hemisphere [Arkani-Hamed and Boutin, 2012]. We have no solid evidence tying the start of the dynamo to any particular time before the formation of the oldest detectable impact basins (see below).
 Although the dynamo may or may not have started when the first Martian crust formed, the crustal strong magnetic fields detected by the Mars Global Surveyor (MGS) magnetometer (MAG), primarily over the heavily cratered crust of the southern highlands, can only be explained by large, coherently magnetized regions of crust (at least) hundreds of kilometers in scale [Acuña et al., 1999; Connerney et al., 2001], which in turn can only be adequately explained by the past presence of a dynamo-driven global magnetic field comparable in strength to that at the Earth's surface (i.e., ~10s of μT).
 Large-crater counts performed with crustal thickness and topographic data place younger bounds on ~20 impact basins in regions of moderate to strong crustal magnetic field [Frey, 2008, 2010; Lillis et al., 2008a], each larger than 1000 km in diameter, i.e., sufficiently large that magnetic field measured over the center of the crater is assuredly an adequate proxy for crustal magnetization [Lillis et al., 2010] and that they were formed when Mars had a global dynamo magnetic field. These magnetized basins are, by cratering densities, the oldest detectable structures, with N(300) crater densities greater than 2.6 (i.e., more than 2.6 craters greater than 300 km per 106 km2). The conversion from N(300) to model age suffers from systematic uncertainties in the lunar cratering rate earlier than 3.9 Ga and in the extrapolation of the cratering function from lunar maria to Martian conditions (discussed in sections 2.2 and 5.1). With these substantial caveats in converting to absolute model age, these magnetized basins were deemed to have model ages between 4.1 and 4.3 Ga.
 The only definitive thermochronometric data placing a younger bound on the life of the Martian dynamo comes from studies of Martian meteorite ALH84001. Its NRM resides primarily in single-domain magnetite- and pyrrhotite-bearing carbonates [Antretter et al., 2003; Rochette et al., 2005; Weiss et al., 2002]. The non-carbonate-hosted TRM was acquired at 4.1 Ga, while the carbonate-hosted TRM was acquired either at the same time (4.1 Ga) or possibly as late as 3.9 Ga [Shuster and Weiss, 2005; Weiss et al., 2002], in a paleomagnetic field with a magnitude of ~50 μT [Weiss et al., 2008].
 Thus, it is not seriously debated that the Martian dynamo was active sometime before 4.1 Ga, possibly for several hundred million years. Lillis et al. [2008a] noted that the large impact craters Utopia, Hellas, Isidis, and Argyre have diameters >1000 km, model ages ≤ 4.1 Ga and extremely weak crustal magnetic fields above them. Lillis et al. [2008a] thus concluded that Martian crustal magnetization underwent a rapid decrease around 4.1 Ga and that this was likely due to the cessation of the Martian dynamo. Some recent work on the magnetic field signature of Martian volcanic terranes with surface flow features younger than 4.0 Ga has suggested the possibility that the dynamo may have been subsequently active [Lillis et al., 2006, Hood et al., 2010; Milbury et al., 2012], though as discussed in the following section, we believe these interpretations to be insufficiently supported.
1.3 The Limitations of Using Volcanic Features to Constrain the Magnetic History
 The magnetic signatures of volcanic features offer another opportunity to further constrain dynamo history, in particular, the time period after the large craters formed. Hood et al.  examined positive magnetic anomalies over Apollinaris Mons and Lucus Planum, concluding that substantial magnetization exists in the crust beneath both. Milbury et al.  jointly used gravity and magnetic field data to separately model 29 magnetic sources in the Syrtis Major and Tyrrhenus Mons region. In both these cases, inferred magnetization beneath crust with a surface age in the Late Noachian/Early Hesperian (~3.7 Ga) was taken to support the idea of an active dynamo at this later time.
 However, for this line of reasoning to be valid, two assumptions must be true: (1) the surface age must represent the age at which all or most of the magma beneath the surface (which acquires a new magnetization when it cools below the magnetic blocking temperature(s) of its primary magnetic mineral(s)) was emplaced, i.e., no substantially earlier magmatic episodes can have occurred at that location which either did not produce significant intrusions and surface lavas or whose lavas were covered by later episodes; (2) the magnetization over the great majority of the magnetizable depth of crust (30–60 km [Voorhies, 2008; Dunlop and Arkani-Hamed, 2005]) and over an area large enough to influence orbital magnetic field data (i.e., > ~2 × 105 km2) must be reset at approximately the same time as the emplacement of the oldest visible lava flows on the surface, i.e., requiring magma chambers, sills and, dikes with cumulative thickness much greater than the 2.8–7.8 km inferred by Kiefer [2004a, 2004b] for several highland volcanoes (hydrothermal circulation does not significantly expand the region that experiences thermal demagnetization [Ogawa and Manga, 2007]). In other words, volcanoes we wish to utilize for constraining Martian dynamo history must be built and must reset the magnetization of ~10 million cubic kilometers of crust, both in a relatively short time (say, 100 Ma). This is not the case for any of the Martian highland volcanoes, many of which are too small (see Plescia  for their sizes) and for which we cannot know the eruption history further back in time than the oldest visible flows. Criterion (2), however, may be satisfied by the largest Tharsis and Elysium volcanoes [Lillis et al., 2009], which have likely been completely thermally demagnetized by prolonged, pervasive magmatism, starting in the Noachian [Johnson and Phillips, 2005] and continuing well into the most recent Amazonian epoch [Robbins et al., 2011].
 For the reasons mentioned above, we argue there is a weak and uncertain link between the oldest visible surface age of a volcanic edifice and the age of the magnetization of the underlying crust. Therefore, the orbital magnetic field signatures of volcanoes are in general poorly suited to constrain dynamo history, and it is unwise to conclude from magnetic fields measured over highland volcanoes with surface ages <4 Ga that the Martian dynamo was active at these times. This is in contrast to the situation for large impact craters (the topic of this paper), where both the entire depth of magnetization and the surface crater retention age are reset almost simultaneously (in geologic terms). Impact craters (over volcanic features) offer a more definitive and reliable probe of the evolution of ambient magnetic field conditions over Martian history.
1.4 Is a Given Impact Crater Demagnetized?
 Section 4 of Lillis et al.  demonstrated that magnetic field at 185 km altitude is a robust proxy for magnetization in the inner regions of craters greater than 1000 km in diameter. Examination of only the ~30 detected impact craters larger than 1000 km gives a consistent history of the Martian dynamo: cessation around a model age of 4.1 Ga, as mentioned earlier. However, none of these craters is younger than ~4.0 Ga in absolute model age. In order to investigate Mars' dynamo history over an extended time period, in particular to ages < 4.0 Ga, we must rely on smaller craters. However, as crater diameter decreases, the altitude of observation increasingly masks the magnetic signature of a demagnetized crater (which can be a local minimum or local maximum depending on the observation altitude, crater size, and coherence scale of the magnetization). To address this uncertainty, in this paper, we develop new statistical tools to calculate the probability distribution function of fractional crater magnetization for a crater of a given size and magnetic field signature. As an example, we intend to be able to answer questions like the following: if a 375 km crater displays a mean magnetic field magnitude at 400 km altitude in its central regions that is 75% of the mean field magnitude between 1.25 and 2 crater radii, what are the relative likelihoods of that crater being 90%, 50%, and 25% demagnetized, etc.? Coupled with improved crater retention ages, these tools should enable a more confident and comprehensive examination of the history of the Martian dynamo.