The Age and Erosion Rate of Young Sedimentary Rock on Mars

We focused on 14 sites within the MFF and other nearby sites that are believed to be young sedimentary rock (Figure 4). The craters for each site were counted by one of us using Mars Reconnaissance Orbiter Context Camera (CTX) imagery. Figure 5 provides an example of two counted craters. Obvious secondary craters were excluded during crater counting, such as highly elliptical craters, very clustered craters, and secondary craters concentrated in rays coming from primary craters (Robbins & Hynek 2014).

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We initially defined our site boundaries based on geological units of Mars (Tanaka et al. 2014) and, where available, further distinguished units into subcomponents, such as "a," "b," and "c," by using a combination of radar reflections, age estimates from previous works (e.g., Kite et al. 2015), and clear visual differences in crater densities. For Central Medusae Fossae 2 and the Far East Medusae Fossae, we used only the global geologic map (Tanaka et al. 2014) to separate distinct geologic units. For Central Medusae Fossae 1, sites 1a–1c (Figure 4(b)), known as Lucus Planum, our sites 1a–1c correspond to the Amazonian and Hesperian transition undivided unit (AHtu) in Tanaka et al. (2014). We do not include the area to the left of sites 1a–1c (which is the focus of Mittelholz et al. 2020 ) because it corresponds to the older Hesperian transition undivided unit (Htu; Tanaka et al. 2014). Orosei et al. (2017) identified distinct areas within Lucus Planum based on radargrams of the subsurface. Since these areas with distinct radargrams also match variations in surface morphology (Orosei et al. 2017), we decided to assess these areas independently from each other for the age and erosion rate. Central Medusae Fossae 1 C (site 1c) has similar boundaries to a large lobe northeast of Memnonia Sulci, labeled "C" in Figure 4 of Orosei et al. (2017). Similarly, we defined Central Medusae Fossae 2 B (site 4b) based on lobe "B" in Figure 4 of Orosei et al. (2017). By taking radargrams of Lucus Planum, Orosei et al. (2017) found "B" and "C" as areas with concentrated radar reflectors, unlike the central section of Lucus Planum. As a result, both "B" and "C" indicate differences within the subsurface material at Lucus Planum relative to areas of no radar signal (Orosei et al. 2017). For Zephyria Planum, since most of Aeolis Dorsa has a relatively old age (Figure S1 in Kite et al. 2014; Zimbelman & Scheidt 2012; Burr et al. 2021), we restricted our count to the relatively young Zephyria Planum (unit "Y" in Kite et al. 2015). Sites 6a and 6b, East MFF, were separated based on visual inspection of the crater density.

By plotting the logarithm of the number of counted craters in the data set for each given minimum diameter, we found the beginning of the diameter rolloff to be 0.510 km. This rollover at approximately 0.5 km represents where the crater counts are affected by survey incompleteness. As a result, we set 0.5 km as the minimum bin diameter for the smallest crater bin. We use manually counted craters within each of the 14 sites and divide the craters into bins, based on crater diameters 0.5–128 km, for a total of 16 bins that are spaced by a factor of about 2^−1/2 between each consecutive minimum diameter (Michael 2013). The Hartmann isochrons in Table 1 of Michael (2013) are defined as the size distribution of craters for a surface with a specified age and without erosion processes, based on a widely used Mars crater chronology model (Hartmann 2005).

Previous models typically only consider crater count data in order to estimate the absolute age of the sedimentary areas. The next step is to create a two-parameter model that addresses both erosion rate and age, which is the route that we explore in this study. The erosion rate is set as a fixed rate (after a single episode of deposition) without interruption because this is the simplest treatment of erosion. The reality is likely more complicated (Section 1). Our data set includes a total of 2119 craters, too few to merit a more complicated model. However, future work incorporating larger areas and populations of craters could investigate using an erosion rate that varies with time. We assume that there was a single episode of sedimentation, corresponding to the age of the mound when it was initially deposited. As discussed in Section 2.1, our crater data set is limited to 0.5 km, due to incompleteness. However, this larger diameter allows this work to focus on the bulk age of the MFF, since smaller craters would only be sensitive to the uppermost MFF. In our model, we assume that craters are generated randomly as a Poisson process, given by the equation

$$\begin{eqnarray}&&P(n)=\displaystyle \frac{{e}^{-\mu }{\mu }^{n}}{n!},\end{eqnarray} \tag{ 1 }$$

where P(n) is the Poisson probability of observing n craters and n is the observed number of craters, as determined by the crater counts. Our model fits an expected number of craters, μ, and estimates μ without erosion and with erosion. The model then chooses the smaller μ between the no-erosion and erosion cases. The expected number of craters for no erosion is

$$\begin{eqnarray}&&\mu =Hat,\end{eqnarray} \tag{ 2 }$$

where H is the expected number of craters for each crater diameter bin (craters km⁻² Gyr⁻¹), taken from Table 1 in Michael (2013), which is based on the Hartmann production function (Hartmann 2005). a is the area of the crater count site (km²) and t is the age (Gyr).

The expected number of craters with erosion is:

$$\begin{eqnarray}&&\mu =Had/\beta .\end{eqnarray} \tag{ 3 }$$

H is still the expected number of craters for each crater diameter bin (craters km⁻² Gyr⁻¹) and a is the area of the crater count site (km²), as mentioned above for the no-erosion case. β is the inputted trial erosion rate (nm yr⁻¹). d is the logarithmic mid-depth, based on the minimum and maximum diameter for each specific crater bin. The depth is related to diameter as follows:

$$\begin{eqnarray}d=\left\{\begin{array}{ll}0.2D & \mathrm{if}\ D\lt 2.82\,\mathrm{km}\\ 0.323{D}^{0.538} & \mathrm{if}\ D\geqslant 2.82\,\mathrm{km}\end{array}\right..\end{eqnarray} \tag{ 4 }$$

D is the logarithmic mid-diameter of the crater bin in meters:

$$\begin{eqnarray}&&D=\sqrt{{D}_{\min }{D}_{\max }}.\end{eqnarray} \tag{ 5 }$$

We adapted pre-existing depth-to-diameter equations to better fit the observed depth-to-diameter relationships (such as the idealized cases mentioned in Gabasova & Kite 2018 and Watters et al. 2015). For small craters (D < 2.82 km), we used d = 0.2D, which was first determined by Pike (1980) and reconfirmed by Watters et al. (2015). This depth-to-diameter relationship was also used by both Palucis et al. (2020) and Smith et al. (2008). For large craters (D ≥ 2.82 km), the component 0.323D^0.538 in Equation (4) is from the Mars data set of Tornabene et al. (2018). Tornabene et al. (2018) use this depth–diameter scaling relationship for diameters larger than 12 km. We chose 2.82 km as the transition diameter because it is the intersection between the two equations d = 0.2D and d = 0.323D^0.538. As a result, we adjusted the cutoff to 2.82 km, to make sure the depth estimates were continuous between the two equations and monotonically increasing. In previous work, Pike (1980) used a transition of D = 1 km, while Palucis et al. (2020) and Smith et al. (2008) used the 0.2D relationship with a transition at D = 5.8 km.

The μ for both the erosion and no-erosion cases are also multiplied by a correction factor to account for the increased impact flux early in Martian history. Based on Table 1 from Michael (2013), N is the expected number of craters in the last column (craters km⁻² Gyr⁻¹). Taking the cumulative density N at 1 km from Table 1, we have N_steady = 5.84 × 10⁻⁴ t, which incorrectly assumes that the modern impact flux over the past 10 million yr has always held. Empirical estimates of the modern impact flux are based on satellite observations of newly appeared craters (Daubar et al. 2013), and vary from model predictions by about a factor of 4 (Daubar et al. 2014). This observed present-day crater flux is uncertain due to the spatial nonrandomness of the observations (Daubar et al. 2013), since the true flux is expected to be almost spatially random (Le Feuvre & Wieczorek 2008; Kite & Mayer 2017). In addition, satellite observations may not reflect the recent changes in flux throughout the different orbital cycles of Mars over the past 10 million yr (Kite & Mayer 2017). As a result, we focus on the Michael (2013) flux, instead of a flux derived from empirical observations.

Adapting Equation (3) from Michael (2013) states that N = 3.79 × 10⁻¹⁴(e^6.93t − 1) + 5.84 × 10⁻⁴ t. This gives the correction factor

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{N}{{N}_{{\rm{steady}}}}=\displaystyle \frac{3.79\times {10}^{-14}({e}^{6.93t}-1)+5.84\times {10}^{-4}t}{5.84\times {10}^{-4}t}\end{array}\end{eqnarray} \tag{ 6 }$$

for the case without erosion. For the case with erosion, t is replaced by d/β, where d is again the logarithmic mid-depth (m) for a given crater bin and β is again the erosion rate (nm yr⁻¹). The uncertainties in this correction factor are discussed in Section 4.2.

For our model, the range of ages considered were 1 s to 4.5 Gyr, with 1000 ages linearly spaced in between. Similarly, β ranged from 1 nm yr⁻¹ to 2000 nm yr⁻¹, with 1000 βs linearly spaced in between (higher values are considered for the fastest-eroding sites). Taking each array of P(n) values from the individual crater diameter bins, we find the total probability as the product of all 16 bins:

$$\begin{eqnarray}&&{P}_{\mathrm{total}}(t,\beta )=\prod _{n=1}^{16}P(n).\end{eqnarray} \tag{ 7 }$$

In addition to evaluating the two-parameter probability distribution, we also collapsed the two-parameter probability array, by summing along one dimension (either β or age) to find the normalized one-parameter probability array for age and for β. Then, for the one-parameter cumulative probability array, we found the cumulative sum of the probability array and normalized it to a maximum cumulative probability of one. The median of the cumulative distribution (for either age or β) corresponds to the best fit for the one-parameter model.

From the two-parameter model, we calculated the crater density predicted by the best fit, which corresponds to the age and β pair that has the highest predicted probability from our model. Using the best-fit age and β, we create a model-generated CSFD to compare with the real data's CSFD. For the model-generated CSFD, we found the expected crater density per km² for each crater diameter bin by using Equations (2) and (3). The error bars for the crater densities came from the 1σ standard deviation from the center of the cumulative distribution, in which n, the observed number of craters, is equal to μ, the expected number of craters for the crater count data. The model Python scripts for generating the figures are available via the link in the Appendix.

To assess whether our model is accurate and to examine the uncertainty of our model's predictions, we generate synthetic crater distributions based on Table 1 of Michael (2013) for a given age and erosion rate. We then apply our model to estimate the best fit for age and erosion rate, and compare our model's result with the known inputs.

For test 1 (Figure 6(a)), we first found the expected number of craters per bin using Table 1 of Michael (2013; based on Hartmann 2005) and Equations (2) and (3). The input parameters for test 1 were age = 4 Gyr, erosion rate = 1 nm yr⁻¹, and area =10⁵ km². The expected craters per bin (corresponding to the Table 1 bin indices −2–13 in Michael 2013) rounded to the nearest whole number were [30124, 8618, 2308, 959, 513, 275, 148, 79, 43, 23, 12, 7, 3, 2, 1, and 0]. From Figure 6(a), the 1σ range of ages was narrow between 3.99 and 4.01 Gyr, whereas the 1σ, 2σ, and 3σ uncertainty regions for β almost reached 30 nm yr⁻¹, larger by almost a factor of 30 than the input β = 1 nm yr⁻¹. This is because small but non-negligible erosion rates cannot be ruled out without data for D < 500 m craters, which were not included in the test. In Figure 6(a), the yellow region of highest probability contained the blue star, corresponding to the input parameters, and the red circle, the model's best-fit age and erosion rate, accurately predicted the input parameters at 4 Gyr and 1 nm yr⁻¹.

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The input parameters for test 2 (Figure 6(b)) were age = 4 Gyr, erosion rate = 4000 nm yr⁻¹, and area = 10⁶ km². The expected craters per bin were [120, 48, 18, 11, 8, 6, 4, 2, 2, 1, 1, 0, 0, 0, 0, and 0]. From Figure 6(b), we see that age, as expected, was not well constrained. The yellow region of highest probability, with the red circle for the best-fit age of 0.31 Gyr and erosion rate of 3990 nm yr⁻¹, was offset from the blue star representing the input parameters. Nevertheless, the 1σ region, which enclosed 68% of the total probability, did contain the blue star.

The input parameters for test 3 (Figure 6(c)) were age = 60 Gyr, erosion rate = 4000 nm yr⁻¹, and area = 10⁶ km², corresponding to an unrealistically old surface, and assuming the same relationship for impact flux as described in Section 2.2. The expected craters per bin were [120, 48, 18, 11, 8, 6, 4, 2, 2, 1, 1, 0, 0, 0, 0, and 0]. Holding the erosion rate and area constant, but changing the age in comparison with test 2, the expected craters per bin were still the same as those in test 2. This is because the high erosion rate of 4000 nm yr⁻¹ prevents the older surface in test 3 from having more craters than test 2. From Figure 6(c), we see that age was not constrained at all, as expected. While the best-fit erosion rate was 3980 nm yr⁻¹, and corresponded well to the input erosion rate of 4000 nm yr⁻¹, the best-fit age was only 0.30 Gyr. However, the 1σ uncertainty region contained the blue star, corresponding to the input parameters.

The input parameters for test 4 (Figure 6(d)) were age = 2 Gyr, erosion rate = 200 nm yr⁻¹, and area = 10³ km², corresponding to a small area, which is the approximate area of site 7, Upper Mount Sharp. The expected craters per bin (corresponding to Michael 2013 Table 1 bin indices −2 to 13) were [2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. From Figure 6(d), we see that age was not well constrained, ranging from 0 to 4.5 Gyr for the 3σ region. This is likely due to the minimal number of craters. Maximum β for 1σ versus 3σ varied greatly, from 710 nm yr⁻¹ to 4730 nm yr⁻¹. The yellow region of highest probability with a best-fit age of 0.58 Gyr did not include the input parameters, but the 1σ region did. In addition, the best-fit erosion rate of 240 nm yr⁻¹ did approximate the input erosion rate of 200 nm yr⁻¹.

The input parameters for test 5 (Figure 6(e)) were age = 2 Gyr, erosion rate = 200 nm yr⁻¹, and area = 10⁵ km², corresponding to the approximate area of Central Medusae Fossae 1 and the age and erosion rate predicted for the site (Figure 6(e). The expected craters per bin (corresponding to the Table 1 bin indices −2 to 13 in Michael 2013) were [239, 97, 37, 21, 14, 7, 4, 2, 1, 1, 0, 0, 0, 0, 0, and 0]. The 1σ region had an age above 1.48 Gyr and below 2.67 Gyr, with an erosion rate between 190 and 220 nm yr⁻¹. The 3σ region estimated an age between 1.08 and 3.43 Gyr, with a similar erosion rate to the 1σ region between 170 and 240 nm yr⁻¹. The region of highest probability contained the blue star, and the best-fit age of 1.96 Gyr was very close to the input age of 2 Gyr and the best-fit erosion rate was the same as the input.

The tests reveal a few reasons as to why age might be unconstrained. For example, tests 2 and 3, which had the same inputs for area and erosion rate, but different ages, resulted in the same amount of expected craters per bin and, correspondingly, wide error ranges for age. This reveals that in cases of high erosion rates, such as 4000 nm yr⁻¹ in tests 2 and 3, it is difficult for the model to tightly constrain the age. In addition, test 4 is an example of a small area with few craters. Due to the small amount of craters, it is a challenge for the model to pinpoint age, and the size of the error regions of the erosion rate also increases. As a result, the tests show that the erosion rate is more tightly constrained than age in cases of small areas and/or high erosion rates. Nevertheless, since the input parameters for both age and erosion rate (the blue star) in Figure 6 were always contained within the σ error regions, these tests demonstrate that the model is reliable in constraining age and erosion rate within these error regions.

After collapsing the two-parameter probability contour plot along β to form a one-parameter cumulative distribution function (CDF) of age in Figure 7(a), we compared the age probability distributions for each region. The best-fit absolute model ages for 13 of the 14 regions fall below 2.5 Gyr, and 11 of the 14 sites have 1σ ages that are younger than 3 Gyr. Four of the sites have 1σ error regions that are completely constrained below 2.5 Gyr. Eastern Candor (3), in particular, is predicted to be young because its 1σ error region is constrained below 1 Gyr. All six Central Medusae Fossae sites (1a–1c and 4a–4c) have lower 1σ error bounds that are older than 1 Gyr and upper 1σ error bounds that are between 2.0 and 3.2 Gyr, which makes the Central Medusae Fossae sites the oldest among the sedimentary mounds examined in this study.

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In Figure 7(b), well-constrained cumulative probability curves for β (with tight 1σ error bounds) include all sites, except Eastern Candor (3) and Upper Mount Sharp (7). Most sites (excluding Eastern Candor and Upper Mount Sharp) require large erosion rates, at least greater than 10 nm yr⁻¹. It is likely that Eastern Candor and Upper Mount Sharp also require erosion rates >10 nm yr⁻¹ because although their lower 1σ are not bound above 10 nm yr⁻¹, both the median and upper 1σ are bound above 10 nm yr⁻¹. As a result, with 1σ certainty, we can say that 12 of the 14 sites have non-negligible erosion rates.

All two-parameter contour plots are presented in Figure 8 and the model predictions are summarized in Table 2. The site-specific results are discussed below.

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Central Medusae Fossae 1 (A: t = ${1.43}_{-0.43}^{+1.45}$ Gyr, β = 210 ±30 nm/yr; B: t = ${1.89}_{-0.94}^{+1.79}$ Gyr, $\beta ={560}_{-70}^{+80}$ nm yr⁻¹; C: t = ${1.67}_{-0.54}^{+2.15}$ Gyr, β = 650 ± 80 nm yr⁻¹) (Figures 8(a)–(c))

• 1.  
  The estimated age ranges are similar between these three sites.
• 2.  
  The β values vary greatly: sites B and C have β two or more times greater than that of site A.

Zephyria Planum (t = ${0.47}_{-0.35}^{+4.03}$ Gyr, $\beta ={1910}_{-1910}^{+560}$ nm yr⁻¹) (Figure 8(d))

• 1.  
  While the 1σ region for β was predicted to be between 0 and 2470 nm yr⁻¹, age was not constrained.
• 2.  
  With only 36 craters (Table 1), the paucity of craters is likely one reason why age and β were not well constrained.

Eastern Candor (t = ${0.42}_{-0.35}^{+4.08}$ Gyr, $\beta ={1140}_{-1140}^{+1330}$ nm yr⁻¹) (Figure 8(e))

• 1.  
  Similar to Zephyria Planum, at ages younger than 0.5 Gyr, the predicted β values drop down to include very low values. In fact, all three error regions drop down to low β values.
• 2.  
  In the entire Eastern Candor site studied, there were only five craters, with four craters in the smallest crater bin diameter size. This paucity of craters likely resulted in no useful age constraint.

Central Medusae Fossae 2 (A: t = ${1.81}_{-0.79}^{+1.61}$ Gyr, $\beta ={390}_{-40}^{+60}$ nm yr⁻¹; B: t = ${2.59}_{-1.12}^{+1.17}$ Gyr, $\beta ={500}_{-80}^{+80}$ nm yr⁻¹; C: t =${3.41}_{-0.15}^{+0.03}$ Gyr, $\beta ={90}_{-10}^{+0}$ nm yr⁻¹) (Figures 8(f)–(h))

• 1.  
  Site C of Central Medusae Fossae 2 had the most tightly constrained age and β values of all the sites, with all three of the error ranges falling between 2.98 and 3.51 Gyr for age, and a best-fit prediction for β between 80 and 100 nm yr⁻¹.

Far East Medusae Fossae (A: t = ${0.27}_{-0.25}^{+4.23}$ Gyr, $\beta ={4010}_{-4010}^{+2400}$ nm yr⁻¹; B: t = ${0.68}_{-0.22}^{+0.52}$ Gyr, $\beta ={750}_{-90}^{+100}$ nm yr⁻¹; C: t = ${0.37}_{-0.13}^{+4.13}$ Gyr, $\beta ={1690}_{-240}^{+290}$ nm yr⁻¹) (Figures 8(i)–(k))

• 1.  
  While the best-fit age is young, we see that the 2σ uncertainty range is very wide, spanning a range of almost 4 billion yr.
• 2.  
  Site C of Far East Medusae Fossae is similar to site B with a younger age estimated within the 1σ region, but a right tail permitting higher ages for larger-σ uncertainty regions.
• 3.  
  Site A has only 12 craters, so age and β were not well constrained (Table 1).

East Medusae Fossae (A: t = ${0.73}_{-0.24}^{+0.68}$ Gyr, $\beta \,={660}_{-80}^{+90}$ nm yr⁻¹; B: t = ${0.42}_{-0.33}^{+4.08}$ Gyr, $\beta ={2590}_{-410}^{+420}$ nm yr⁻¹) (Figures 8(l)–(m))

• 1.  
  Both the Far East Medusae Fossae and East Medusae Fossae sites have younger best-fit ages, with older ages permitted at 2σ.
• 2.  
  Despite being next to each other, site A and site B have very different predicted erosion rates (580–750 nm yr⁻¹ versus 2180–3010 nm yr⁻¹).

Upper Mount Sharp (t = ${2.04}_{-1.49}^{+1.91}$ Gyr, $\beta ={170}_{-170}^{+190}$ nm yr⁻¹) (Figure 8(n))

• 1.  
  The erosion rate estimate is broad. We also see that on the left-hand side of the plot, the uncertainty regions drop to include 1 nm yr⁻¹ for the erosion rate, indicating that the lowest erosion rate values only occur for the youngest ages. However, unlike previous sites, with the lowest β values the youngest ages permitted here reach values as high as 2 Gyr (2σ).
• 2.  
  In the entire Upper Mount Sharp site studied, there were only five craters (Table 1), with two craters in the two smallest crater bin diameter sizes. The paucity of craters accounts for the wide age range.

All 14 sites aggregated (t = ${1.37}_{-0.34}^{+0.24}$ Gyr, β = 520 ±20 nm/yr) (Figure 8(o))

• 1.  
  When combining the crater count data for all 14 sites, age and especially β are tightly constrained within the framework of the model assumptions: ignoring regional variation (which is not a safe assumption), young sedimentary rocks on Mars are ${1.4}_{-0.3}^{+0.2}$ Gyr, with an erosion rate of 520 ± 20 nm yr⁻¹).

Figure 9 compares the observed crater distributions with the model's best-fit age and erosion rate for each site. From Figures 9(a) through 9(n), the sites with best-fit model predictions that were the most consistent with the crater data included Central Medusae Fossae 1 A (1a) and B (1b), Central Medusae Fossae 2 A (4a) and C (4c), Far East Medusae Fossae B (5b) and C (5c), East Medusae Fossae A (6a), and the aggregation of 14 sites. It is unclear whether the model was a good fit for sites such as Eastern Candor (3) in Figure 9(e), where there were relatively few filled crater bins to compare with the model.

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3.4.1. Central Medusae Fossae

3.4.2. Far East Medusae Fossae

3.4.3. East Medusae Fossae

3.4.4. Zephyria Planum and Eastern Candor

3.4.5. Upper Mount Sharp

The map of the one-parameter model's best-fit age and erosion rates (Figure 10) demonstrates trends in age and erosion rate by location. Older ages greater than 1.5 Gyr dominate Central Medusae Fossae. East Medusae Fossae, Far East Medusae Fossae, Zephyria Planum, and Eastern Candor all have young best-fit ages, below 1 Gyr. Upper Mount Sharp, which is hypothesized to have similar material to the MFF (Zimbelman & Scheidt 2012; Lewis & Aharonson 2014; Edgett et al. 2020), also has a younger age, close to 0.9 Gyr. However, Zephyria Planum (2), Far East Medusae Fossae A (5a) and C (5c), East Medusae Fossae B (6b), and Upper Mount Sharp (7) all have 1σ uncertainties reaching into the next age increment (corresponding to the color bar in Figure 10) beyond the age increment of the best-fit age. Overall, most of Central Medusae Fossae has older ages and lower erosion rates, while most of East Medusae Fossae, most of Far East Medusae Fossae, and Zephyria Planum have younger ages and higher erosion rates. Eastern Candor and Upper Mount Sharp are exceptions, as they have younger ages and relatively low erosion rates.

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Figure 11 shows the two-parameter model predictions for the eight young sites aggregated together, the six older sites aggregated together, and all 14 sites aggregated together. The younger sites are predicted to have an age of 0.5 ± 0.2 Gyr. In contrast, while the aggregation of the older sites had a better constrained erosion rate of around 250 nm yr⁻¹, the age for the older sites was less well constrained, with a 1σ range spanning from 2.2 to 3.3 Gyr.

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The uncertainties in adopting the Hartmann isochrons (Hartmann 2005; Michael 2013) include crater identification, terrain selection, and secondary crater populations, as well as the application to Mars of a crater flux model that was originally based on radiometric age dating of lunar samples and comparisons to lunar crater counts (Hartmann 2005). Uncertainties relating to crater identification are inherent, due to the limited image resolution and crater-counting fatigue (Robbins et al. 2014). As mentioned in Section 2.1, the rolloff for our observed CSFD indicating survey incompleteness occurred at about 0.5 km, so we only included craters with diameters of 0.5 km and larger for our model.

For the terrain selection, other researchers might define the site boundaries differently from ours, especially sites 1, 4, 5, and 6, which have subregions within the boundaries based on the geological units defined in Tanaka et al. (2014). To analyze the effects of different terrain boundaries, we compared model estimates for age and erosion rate for sites 1, 4, 5, and 6 before and after dividing them into subregions. The age and erosion rate estimates for the sites as a whole were broadly consistent with their subregions, with the ages toward the older side if one subregion was comparatively older with more craters than the other subregions.

To account for secondary craters, as mentioned in Section 2.1, apparent secondary craters (such as craters demonstrating high ellipticity, tight clustering, or with defining rays coming from the primary craters) were not included in the data set. Secondary craters that look like primary craters do not affect the CSFD model estimates, unless a large fraction of the craters come from a single event (Hartmann 2005). This is unlikely, since the MFF has wind-eroded rocks with different stratigraphic layers, so the craters were likely produced over a longer period of time.

The known problem of the overestimation of small craters by crater production functions (Hartmann 2005; Kite & Mayer 2017) is addressed by the inclusion of the erosion rate in our model (Figure 2). The probabilistic model does not take into account variable erosion rates, burial rates, and the obliteration of craters by impacts. Recent work has proposed a statistical model, based on crater size and depth frequency distribution, to determine obliteration rates as a function of time (Breton et al. 2022). While Breton et al. (2022) do not solve for age, their method does investigate how obliteration changes over time. Future work could expand upon our work that used a fixed erosion rate and age, and combine it with the changing erosion rate of Breton et al. (2022) by solving for the time-variable erosion rate and age of young sedimentary rock.

To test whether our two-parameter model is preferable to a simpler one-parameter model, we performed minimum χ² tests—an extension of χ² goodness-of-fit tests (Wall & Jenkins2012)—for the best fit based on the two-parameter model for age and β, the best fit along the x-axis for age, and the best fit along the y-axis for β, for all 14 sites grouped together (Table 3). All three tests were significant at the 99% confidence level, since the p-values were <0.01, so we can reject the null hypothesis that the crater distributions from the three models are the same as the crater count data.

The χ² of 82.1 for the two-parameter model was the smallest. However, the χ² exceeds the number of bins minus 1 then multiplied by 2, since 82.1 > 20. As a result, we reject the null hypothesis that the two-parameter model's distribution was exactly the same as the crater count's distribution. The crater distribution predicted by the two-parameter model ("observed") was the most similar to the craters in the data ("expected"), in comparison to both one-parameter models. On the other hand, the χ² values were much larger for both one-parameter models, which suggests that the values predicted by the one-parameter model are quite different from the data. The two-parameter model performs better than the one-parameter models when compared to the crater distribution data. For a distribution generated by a two-parameter model to better emulate the observed CSFD, we recommend future work that uses a time-varying erosion rate instead of a fixed erosion rate.

The subsurface ice table is the predicted location or depth of the boundary between icy ground and ice-free ground in the soil. The equatorial climate of Mars is extremely dry and relatively warm, which makes water ice unstable (Mellon & Jakosky 1995). As a result, the water ice at the equator can sublimate, forming water vapor that is then lost by diffusion through empty soil pore space, escaping to the dry atmosphere (ultimately going to the North Polar ice cap; Mellon & Jakosky 1995). With erosion rates of less than 1000 nm yr⁻¹, the ice can swiftly retreat to depths too deep to be detected by GRS or NS, which only probe around 1 m down (Boynton et al. 2002; Feldman et al. 2002; Mitrofanov et al. 2002; Feldman et al. 2004). By contrast, if the erosion rate is above 1000 nm yr⁻¹, the ice table can persist at shallow depths (∼1 m) in steady state because the erosion rate equals the ice table retreat rate under these conditions (Hudson et al. 2007). We obtain the erosion rate value of 1000 nm yr⁻¹ necessary for a shallow subsurface ice table from Figure 12 of Hudson et al. (2007). The right dashed line in this figure corresponds to retreat depth versus time for a higher value of the diffusion coefficient in the lag-forming case, with an initial barrier of 1 mm. The lag-forming case occurs when ice fills a low-porosity regolith, so that the removal of ice leaves behind a lag that increases in thickness (Hudson et al. 2007). From Figure 12 of Hudson et al. (2007), the right dashed line shows a retreat depth to 1 m in 10⁶ yr, which is equivalent to 1000 nm yr⁻¹ and corresponds to our white dashed line in Figure 8. We infer that if β is greater than 1000 nm yr⁻¹, the depth of the buried ice can be less than 1 m, if there is any left.

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The water ice would still be too deep for detection by cameras or IR spectroscopy, but it would be shallow enough to be detected by GRS/NS. As seen in Figure 12, Zephyria Planum (2), Far East Medusae Fossae A (5a), Far East Medusae Fossae C (5c), and East Medusae Fossae B (6b) are the four sites that have β values in the 1σ confidence region above 1000 nm yr⁻¹. All four sites are part of the young sites in Figure 11, which are expected to have erosion rates higher than 1000 nm yr⁻¹ when all eight young sites are aggregated together. Although the best-fit erosion rate based on the two-parameter model for Eastern Candor is predicted to be above 1000 nm yr⁻¹, the 1σ range is not confined to above 1000 nm yr⁻¹. The nine remaining sites, including Central Medusae Fossae (1a–1c and 4a–4c), Far East Medusae Fossae B (5b), East Medusae Fossae A (6a), and Upper Mount Sharp (7), all have slower erosion rates. As a result, we expect Zephyria Planum, Far East Medusae Fossae A, Far East Medusae Fossae C, and East Medusae Fossae B to have the highest likelihood for water ice detection by GRS/NS.

Campbell et al. (2021) proposed a two-layer model for the MFF, with the top 300–600 m as a fine-grained, self-compacting layer, and the bottom 2 km as a thicker layer of possible water ice. If the self-compacting layer proposed by Campbell et al. (2021) did in fact extend to shallow depths (<10 m) as well, then it is unlikely that the material would be very rich in ice, though this is difficult to reconcile with the GRS and NS data (e.g., Wilson et al. 2018).

While two of the three sites in Campbell et al. (2021) are adjacent to, but do not overlap with, our study regions, for the region that does overlap at Zephyria Planum, Wilson et al. (2018) found a relatively high water equivalent hydrogen (WEH) content of >10% for the upper meter. Other western MFF locations, like Lucus Planum (referred to as Central Medusae Fossae in this paper) had >10% WEH as well, and Aeolis Planum had >40% WEH, which is too high to be explained by hydrated silicates, and instead strongly indicative of water ice (Wilson et al. 2018). Using the Fine Resolution Epithermal Neutron Detector neutron telescope on board the ExoMars Trace Gas Orbiter, Mitrofanov et al. (2022) recently reported high hydrogen abundances in central Valles Marineris that are consistent with the presence of water ice in the top 1 m of the subsurface. This motivates monitoring of the sites that we identified that are also in the equatorial region of Mars for small new impacts that might temporarily expose subsurface ice (Byrne et al. 2009; Dundas et al. 2014).

The potential presence of subsurface water ice at the equator of Mars has implications for in situ resource utilization (ISRU) as a source of liquid water and propellant. Liquid water is necessary for humans to survive and for life as we know it. While the poles of Mars have large amounts of water ice (Byrne 2009; Smith et al.2009), this is less relevant to ISRU because early human missions are unlikely at those high latitudes (Starr & Muscatello 2020). Readily accessible water ice at minimal depths (on the meter scale) in the equatorial region of Mars would be invaluable for ISRU (Starr & Muscatello 2020; Bar-Cohen & Zacny 2021). In addition to liquid water, through the reaction CO₂ + 2 H₂O $\to$ CH₄ + 2 O₂, water would also be useful for producing methane propellant. Producing propellant on Mars will be important in order to return a crew from the surface of Mars, since it is unlikely that the crew would be able to carry all of the propellant that they need with them from Earth (Ash et al. 1978; Rapp 2013; Heldmann et al. 2021). As a result, further investigations of potential subsurface ice tables in the equatorial and midlatitude regions of Mars will be key for future human missions (e.g., Morgan et al. 2021).

The β estimations from our model were consistent with previous work. Our estimated erosion rates for all sites combined (Figure 8(o)), Central Medusae Fossae 1 B and C (1b and 1c; Figures 8(b)–(c)), and Central Medusae Fossae 2 A and B (4a and 4b; Figures 8(f)–(g)) of 450–580 nm yr⁻¹ (Figure 8(o)) were all consistent with the Dunning (2019) estimated deflation rate of 400–600 nm yr⁻¹. Similarly, Grindrod & Warner (2014) state that erosion rates are 300–800 nm yr⁻¹, after converting horizontal retreat rates to vertical erosion, and Kite & Mayer (2017) found the crater obliteration rate of light-toned layered sedimentary rock to be 10² nm yr⁻¹. The combined erosion rate from our model is 6.4 × 10⁻⁴ km³ yr, which is a 4.6 nm yr⁻¹ global equivalent. According to Ojha et al. (2018), the MFF may be the single largest source of dust on Mars, and they estimate a total volume loss of 3 × 10⁵–1.8 × 10⁶ km³ from the MFF, or a 2–12 m global layer of dust. By summing up all of the dust in the North Polar layered deposit (NPLD), the South Polar layered deposit (SPLD), Arabia, Tharsis, Amazonis, Elysium, and other dusty regions, they estimate a global equivalent of around 3 m of dust (Ojha et al. 2018). By applying our erosion rate of 6.4 × 10⁻⁴ km³ yr, and treating the erosion rate as constant over long timescales, eroding the range of volumes estimated by Ojha et al. (2018) would require timescales of 0.47–2.8 Gyr. When we compare our estimated global equivalent erosion rate to the 3 m of dust from sources like the NPLD and SPLD, we would expect timescales of around 0.65 Gyr. Thus, our erosion rate estimates demonstrate that the MFF could in fact be a sustainable source of dust for the NPLD and SPLD.

According to Zimbelman & Scheidt (2012), the stratigraphically upper part of the lower member of the MFF is of early Amazonian age (Aml2), which is similar to layers near the top of Mount Sharp. This is in agreement with our collapsed one-parameter model results, which predict an age close to 0.9 Gyr for Upper Mount Sharp, which is similar in age to the young Amazonian ages that we predict for East and Far East Medusae Fossae. On the other hand, the other component of the lower member of the MFF analyzed by Zimbelman & Scheidt (2012) has an age near the Amazonian–Hesperian boundary (AHml1), which is consistent with the cratering ages of Mount Sharp as a whole (Thomson et al. 2011). These older ages are consistent with our one-parameter model estimations for Central Medusae Fossae. Previous work analyzing Mount Sharp within Gale Crater and the MFF has found that the dominant bedding scales are similar but not identical for the two sites (0.22 ± 0.02 m⁻¹ at Gale versus 0.35 ± 0.04 m⁻¹ and 0.29 ± 0.06 m⁻¹ for the MFF), and suggested a possible analogous formation link (Lewis & Aharonson 2014). After the initial formation of the MFF, Kerber & Head (2010) predict sediment recycling at Aeolis Mons, and, in fact, extraformational sediment recycling, the process of sedimentary rock being repeatedly buried then exposed by erosion, is known to occur at the top of Aeolis Mons (Edgett et al. 2020). Due to the similarities between Aeolis Mons and the MFF, it is possible that similar processes have occurred at the MFF.

The purpose of this study was to provide constraints on the absolute ages and erosion rates of the MFF and not its formation mechanism. However, our results are consistent with the leading hypothesis of initial emplacement in the Hesperian, with later reworking in the Amazonian (e.g., Kerber & Head 2010; Kerber et al. 2011). In a recycling hypothesis, the formation age that we calculate represents the time of the most recent deposition. The Central Medusae Fossae (Lucus Planum) was predicted to have overall older ages (1.5–4.5 Gyr) and lower erosion rates (<650 nm yr⁻¹), while Zephyria Planum, Far East Medusae Fossae, East Medusae Fossae, Eastern Candor, and Aeolis Mons were predicted to have younger ages with large erosion rates (>650 nm yr⁻¹; Figure 10). This suggests that Zephyria Planum, East Medusae Fossae, and Far East Medusae Fossae most likely have young Amazonian ages, consistent with the most sediment recycling, while reworking plays less of a role for the older terrains of Central Medusae Fossae. While Kerber & Head (2010) also predicted sediment recycling at Aeolis Mons, we predict both Aeolis Mons and Easter Candor to have young ages, with relatively lower rates of erosion, perhaps indicating slower reworking than Zephyria Planum, East Medusae Fossae, and Far East Medusae Fossae.

In the future, this model may also be applied to additional sedimentary locations on Mars. In particular, the model can provide more precise age and erosion rate estimations for regions that contain larger amounts of craters. For these 14 sites, the model has been able to estimate a much more precise erosion rate than age, as seen in Figure 10. In addition, the geologic context of a location may provide further constraints, to enable the model to find either the erosion rate or age. For example, if the geologic context of a location suggests very minimal erosion, then one could use this constraint and the one-parameter model to find the allowed error regions for age. As a result, future work should also focus on methods for better constraining age.