We have identified the locations and diameters of confirmed craters, crater candidates, confirmed positive relief features, and positive relief feature candidates through the analysis discussed above. Assuming that the cm to meter-scale topography on Phobos mostly results from craters and boulders distributions, we might evaluate the topographic roughness by creating an artificial terrain model (DTM) based on such information.
To artificially develop a DTM, we define two-dimensional grid arrays of 10,000 × 10,000 elements for simulating a region of 1 × 1 km, which means the resolution of this model is 10 cm/element. Each element has its topographic height value, whose initial value is 0.0 m (i.e., a reference level). We assume that each crater is composed of a circular uplifted rim around a bowl-shaped depression and their morphology corresponds to the one given by a standard crater gravity scaling law. We put such craters in the terrain model by subtracting the height values corresponding to the area of the bowl-shaped depression and adding for the circular uplifted rim. We place the craters in descending order; larger craters are placed earlier.
By using Geospatial Data Abstraction Library (GDAL) version 3.1.2 and the ArcGIS’s Spatial Analyst tool, the resulting artificial DTMs were converted to shaded relief images with shadows, which are artificially illuminated at an azimuth of 270° (from the left of the figure to the right) and an altitude of 15° (i.e., the roughly similar illumination condition of the target region (Fig. 6a) in the MOC image SP2-55103). Figure 6b shows the shaded relief image of the resulting artificial DTM with confirmed craters, whose d/Ds are randomly given between 0.01–0.2 according to the simple uniform distribution. Note that, even though we used both crater locations and diameters from mapping results, the resulting DTM is not similar to the original MOC image. This result indicates the importance of evaluating the d/D (depth to diameter) of craters.
C
rater
depth-to-diameter ratio
The topographic characteristics of craters below 100 m in diameter are difficult to evaluate from the numerical shape (e.g., Willner et al. 2014). Basilevsky et al. (2014) partially overcame this issue by carefully evaluating images to obtain craters’ 2D profiles. They classified craters into three morphologic classes: Class 1 for those with > 0.1 in d/D (depth to diameter) with steepest inner slopes (> 20°), Class 2 for those with 0.05–0.1 in d/D with shallower inner slopes (10–20°), and Class 3 for those with < 0.05 in d/D with shallow inner slopes (< 10°). Such variations may be the results of degradations through many possible processes.
Following this idea, we assume that each crater initially has a traditional simple bowl shape. The simple crater’s rim-to-floor depth is about 1/5 of its diameter, and its sharp-crest rim stands about 4% of the crater diameter above the surrounding crater (Richardson 2009). The change in elevation for the crater interior and exterior is assumed to follow the following equations (O’Brien and Byrne 2020):
$$\Delta h = \left\{ {\begin{array}{*{20}c} \begin{gathered} \left( \frac{r}{R} \right)^{2} \left( {h_{r} + d} \right) - d \;\left( {r \le R} \right) \hfill \\ h_{r} \left( \frac{r}{R} \right)^{ - 3} - \frac{{h_{r} }}{{\eta^{3} \left( {\eta - 1} \right)}}\left( {\frac{r}{R} - 1} \right) \;\left( {R < r \le \eta R} \right) \hfill \\ \end{gathered} \\ {} \\ \end{array} } \right.$$
where r is the radial range from the crater’s center, R is the rim crest radius of the crater, \({h}_{r}\) is the rim height, \(d\) is the depth of the crater, and \(\eta\) is the radius of the continuous ejecta blanket (in our model, \(\eta =4\) as the continuous ejecta blanket extends to 4 crater radii). A degradation process is simulated with the following two-dimensional diffusion equation (e.g., Richardson et al. 2020):
$$\frac{\partial z}{{\partial t}} = \kappa \nabla^{2} z$$
where \(\kappa\) is diffusivity. The degradation process is simulated by fixing the coefficient and setting the appropriate diffusion time t. As a result, we obtain 20 profiles of the crater’s morphology with its d/D of 0.01–0.2. The spline interpolation method is used to keep the continuous shape of the resulting crater morphology (Fig. 7).
We modified the d/D of the confirmed craters to make the appearance of DTM similar to the MOC image. Note that this procedure is similar to the shape-from-shading method in some sense, but is also different because we empirically assume the shapes of craters and neglect any possible brightness difference due to surface textures. Thus, we can focus on each shadow length of each crater to adjust d/D. For a deep crater with its shadow cast from its wall to its interior, we selected the profile (d/D) of a crater that shows the same shadow length as the one observed in the MOC image. For a shallow crater without any interior shadow, we used the profile of the deepest shadowless crater at a given solar incidence angle at its location because the worst-case crater should be considered for the assessment of future landing operation. Figure 6c shows the result of DTM with modified d/D for all confirmed craters. As for crater candidates, d/D should generally be very small because otherwise we can identify them as a confirmed crater. Thus, we chose 0.01–0.02 for d/D of the candidate craters and placed the craters in Fig. 6c. The shaded relief image of the resultant DTM is shown in Fig. 6e.
Note that we excluded craters smaller than 82.7 cm in diameter in our artificial models. This comes from the aim of this study, i.e., the assessment of landing safety, most of which is based on Rodgers et al. (2016). For the landing performance of the lander footpads, one small boulder is more hazardous than one small crater. A footpad placed on a single acuate (convex-upward) boulder with a certain height is very unstable; however, a footpad over/inside a small (1–2 × footpad-wide) crater is in a (quasi-)stable state because of the convex-downward geometry of a crater profile. Besides, little is known about the spatial/size-frequency distributions of sub-meter craters on small bodies, compared to those of sub-meter boulders on small bodies (e.g., asteroid Eros, Itokawa, Ryugu, etc.). For the reasons above mentioned, we focused on the distributions of positive relief features only for the sub-meter-scale elements of our artificial DTMs.
S
hapes
of positive relief features
We put positive relief features on the DTM as bowl-shaped uplifts. Similar to craters, shapes of positive relief features are difficult to define. For simplicity, we use an ellipsoid to represent a positive relief feature. Thomas et al. (2000) measured the boulders’ width/height ratio by measuring the heights from lengths of shadows, which showed that the height/width ratios follow a normal distribution with its mean 0.25 and standard deviation 0.17. Thus, we distributed confirmed positive relief features with the randomly selected height/width values according to the Gaussian with a mean of 0.25 and a standard deviation of 0.17. The height/width values of all candidate positive relief features were assumed to be 0.1.
E
xtrapolation
of positive relief feature’s
SFD
The resultant DTM (Fig. 6f) shows a consistent appearance to Region A’s original MOC image. Because the illumination condition of Fig. 6e is carefully arranged to be the same as that of the original image, the overall similarities may indicate that we can evaluate surface irregularities if we can appropriately estimate the distributions of smaller blocks below the resolution image. However, extrapolation of an SFD is very challenging because the observed SFDs on small bodies appear to saturate at different sizes, which may reflect some nature of granular materials. However, these could be biased by resolution limits. Thus, to make the discussion simple, we simply extrapolated our SFDs from the analysis discussed above, knowing that it overestimates the numbers of smaller particles. We consider this is still justified because (1) our primary purpose is for engineering safety evaluations, and (2) the number of particles is always overestimated, which always gives the worst case.
We prepared two types of model terrains, Models 1 and 2, which correspond with two target regions, Regions A and B, respectively. Small positive relief features, down to 35 cm, were placed spatially randomly. The extrapolations of their SFDs of Models 1 and 2 were made using the measured boulder SFD of Region A at a diameter of 6.0 m and the one of Region B at a diameter of 4.5 m, respectively (Fig. 8).
