Comparison of Rain-Driven Erosion and Accumulation Modelling of Zafit Basin on Earth and Tinto-B Valley on Mars

Abstract

While fluvial features are plentiful on Mars and offer valuable insights into past surface conditions, the climatic conditions inferred from these valleys, like precipitation and surface runoff discharges, remain the subject of debate. Model-based estimations have already been applied to several Martian valleys, but exploration of the related numerical estimations has been limited. This work applies an improved precipitation-based, steady-state erosion/accumulation model to a Martian valley and compares it to a terrestrial Mars analogue dessert catchment area. The simulations are based on a previously observed precipitation event and estimate the fluvial-related hydrological parameters, like flow depth, velocity, and erosion/accumulation processes in two different but morphologically similar watersheds. Moderate differences were observed in the erosion/accumulation results (0.13/−0.06 kg/m²/s for Zafit (Earth) and 0.01/−0.007 for Tinto B (Mars)). The difference is probably related to the lower areal ratio of surface on Mars where the shield factor is enough to trigger sediment movement, while in the Zafit basin, there is a larger area of undulating surface. The model could be applied to the whole surface of Mars. Using grain size estimation from the global THEMIS dataset, the grain size value artificially increased above that observed, and decreased hypothetic target rock density tests demonstrated that the model works according to theoretical expectations and is useful for further development. The findings of this work indicate the necessity of further testing of similar models on Mars and a better general analysis of the background geomorphological understanding of surface evolution regarding slope angles.

1. Introduction

Although fluvial features are abundant on Mars and provide useful indications of former surface conditions, there is debate and uncertainty with regard to the exact climatic conditions inferred from these valleys and the values of related specific precipitation events and produced surface runoff-related discharges. Several aspects of how fluvial erosion and deposition occur on Mars are poorly understood, for which established background knowledge exists with respect to the Earth. Model-based estimations have already been applied to several Martian valleys [1,2,3]; however, few attempts have been made to numerically estimate fluvial activity-produced surface erosion and deposition. Fluvial-dominated surface material transport models for Mars have also been applied to understand precipitation, infiltration and runoff [4], various aspects of fluvial systems [5,6], sediment deformation [7], and sediment deposition. Despite little evidence firmly indicating that rain has occurred on Mars in the past, the melting of surface ice or snow might also produce runoff, as well as related erosion and sedimentation. In this work, the rain-driven approach was applied; however, in the future, related models should be expanded to consider possible melting-driven fluvial processes.

Existing models aim to reconstruct river transport with a focus on discharge [8], like paleohydraulic conditions in the Ebro Basin [9], rainfall estimation [10], and general paleoclimatic evolution [11]. Using Earth-based depositional fans, paleoflow discharges [12] under cold conditions at the Black Mountain in the Aklavik Range, Canada, exhibited a runoff range of 0.005–0.2 mm/h with sediment fluxes of ~0.04 m³/h. However, in extreme cases, 14 mm/h and fluxes up to ~550 m³/h might occasionally emerge. Morphology-based analysis of five craters at the north of Hellas basin provided discharges of 60–400 m³/s and a supply runoff of about 1–20 mm/h [13], where the upper range of values came from rainstorms or fast melt runoff by rain-on-snow events. Although we cannot be certain whether rain has, in fact, occurred on Mars instead of simply local ice melting, in this work, we assumed a former precipitation event and focused on the aspects of accumulation/erosion. However, it might also be useful (with some modifications) for studying ice-melting-related runoff in the future. Channel widths and wavelengths were also used as proxies for paleodischarge and indicated a >(1–3) mm/h rate; thus, it is unlikely that the observed discharge was due to the melting of snow and ice, with fast runoff being more probable. Alluvial fans, channel-cross sections, and woody debris along the Hilina Pali in Hawaii indicated discharges of 1.6–11.4 m³/s, implying the presence of precipitation up to 4 cm/h [14]. Evaluating cross-sectional dimensions of fluvial channels and watershed topography on Mars, runoff production is estimated on the scale of 1 cm/day in general, while meander dimensions of inverted paleochannels indicate floods (200–400 m³/s) [15]. Evaluating several fluvial systems around the Noachian/Hesperian transition, we observed a runoff between 1 and 10 cm/day [16]. Using these methods, discharge values have been calculated in many studies, while sediment transport rates have rarely been estimated—this is an area in which the model proposed in this work provides new results.

Exploring further aspects that might also influence erosion/accumulation, the following contextual information should be considered: the composition of the regolith might have changed along with the planetary geological history, with fewer oxidized mineral components in the Noachian and a growing proportion of oxidized components in more recent periods, including many hygroscopic salts (sulphates, chlorides, perchlorates) with increased H₂O molecule bonding capacity. Although the latter components might increase the potential water content of the regolith (decreasing bulk density compared to that of dry rock but increasing the total regolith mass by filling the pores), it is probable that a general desiccation trend was present during planetary evolution. The occurrence of a cryosphere (especially regarding the depth of the cryosphere’s top) might also be important, as the erodibility could have decreased to a large extent due to the decreasing ice content. The decreased atmospheric pressure might have influenced the erodibility of the regolith due to the rate and intensity of global atmospheric circulation and related H₂O transport; how this occurred is not completely known, but these and related erosion/accumulation rates could have been increased in the earlier phase of the planet’s geological history. It is worth mentioning that the reconstruction of fluvial processes on Mars plays a key role in better understanding the “whole” geological history and interaction between various factors like climate and surface evolution trends [17,18]. The modelling aspects could provide input and suggestions to better optimize future colonization and human settlement localization for the following decades [19,20].

This work aims to apply the Earth-based model to Martian valleys, namely the Tinto-B valley [21], comparing it with the almost vegetation and soil-free Zafit basin in the Negev desert of Israel, from which the data on the reference precipitation event were collected. The rationale for applying this model approach to Mars is that surface runoff produces erosion, and deposition could be used to reconstruct ancient otherwise unknown precipitation rates and, eventually, paleoclimatic conditions. An important aspect of this work is that while most earlier publications focused on the morphological similarity between Earth- and Mars-based fluvial systems, here, the comparison is focused on model-based numerical calculations to adapt Earth-based equations to Martian conditions. One of the main differences between this and earlier studies is that the modeling of the fluvial process is based on erosion/accumulation-related calculations, going beyond solely estimating discharge.

This work is relevant to Martian research because it aims to estimate ancient precipitation-driven runoff, related sediment transport, and early surface evolution in general. This provides new constraints for the reconstruction of poorly understood ancient climates and surface conditions on Mars. The novelty of this work is that it targets sediment transport estimation, whereas most earlier models focused only on discharge estimation.

2. Materials and Methods

This section provides an overview of the two target areas on Earth and Mars, followed by the numerical approach based on the SIMWE model and its further developed version with the formula and calculated parameter types.

In the case of the terrestrial sample area, the initial topography model was a Shuttle Radar Topographic Mission (SRTM) Digital Terrain Model (DTM) [22] with a spatial resolution of 30 meters per pixel (m/px). In contrast, the initial DTM used for the Martian sample area was a High Resolution Stereo Camera (HRSC) [23] with a spatial resolution ranging from 50 to 250 m/px. In this study, the spatial resolution of the DTM employed was 50 m/px.

The Tinto B sample area is situated in the southern hemisphere of Mars, close to its equator (2°55’ S, 111°53’ E). The name is derived from Tinto Vallis, a valley located west of the area in question (Figure 1). The sample area comprises a network of main and tributary valleys. The main valley measures approximately 81 km in length, with an average width of 1.85 km and an average depth of 250 m. It originates in a higher crater to the south and terminates in a lower crater to the north. The elevation difference between the headwaters and the estuary is 1419 m. Two competing theories have been used to explain the formation of this Martian region. The initial hypothesis posits that Tinto Vallis was formed as a consequence of a catastrophic flood-like event [24], which also shaped the surrounding environment, including the Tinto B valley area. The second hypothesis posits that the valley is of volcanic origin [25] and that the surrounding valleys were formed by water released from the permafrost melted by volcanic activity. This hypothesis is corroborated by the results of the absolute age estimation based on crater statistics [26] in Tinto B [21]. The results suggest that the age of the study area corresponds to the period of the flood channels [27], which were common at the boundary of the Martian Hesperian and Amazonian periods.

The terrestrial sample area for comparison constituted a sub-catchment of the Zafit sub-catchment of the Zin Basin in Israel. The sample area is situated in the Negev Desert, to the south of the Dead Sea (31°1’ N, 35°12’ E) (Figure 2). This region is characterized by aridity, with an average annual rainfall of approximately 90 mm, a substantial proportion of which occurs during the winter months (December to February) and a smaller proportion during spring [28]. The average annual potential evapotranspiration is approximately 2600 mm, with a gradient that increases from west to east. The sample area comprises a primary valley and a number of secondary valleys. When selecting the terrestrial sample area, careful evaluation was made to minimize the number of morphological features that could be attributed to human intervention, as these anthropogenic forms can significantly influence the modelled values. The bedrock of the sample area is composed of Tertiary marine sedimentary deposits of Late Cretaceous age, predominantly dolomite and marl.

All calculations employed the open-source QGIS program (version 3.38.1.), which integrates the tools for calculating the slope angle, exposure, and flow accumulation. The model can be freely displayed in this software using its model designer function, whereby the user can customize the parameters and modify the calculation process in a graphical interface.

During the research project, the maximum accumulated water depth, velocity, and discharge rates were determined based on precipitation data from the terrestrial sample area. This approach was applied because there are highly uncertain precipitation estimations for Mars, while this Earth-based area is also dry but well-monitored, in contrast to most other deserts. During the project, the area received rainfall with an average intensity of 18 mm/h for 3 h, during a period that had already been the subject of study. The parameters used in the calculation and their abbreviations and values are listed in the following table (Table 1).

In the initial stage of the process, the model was used to ascertain the value of flow accumulation. This value indicates the number of pixels situated above (at higher elevation) each pixel in the direction of the steepest slope [29].

(1)$A={\tan }^{-1}\left({\displaystyle \frac{\mathsf{\partial }z/\mathsf{\partial }x}{\mathsf{\partial }z/\mathsf{\partial }y}}\right).$

Accordingly, the following formula can be employed to ascertain the accumulated maximum water depth within the study area:

(2)$h={\left({\displaystyle \frac{\frac{\frac{\left(I{A}_w\right)}{3600}}{px}n}{S^{0.5}}}\right)}^{0.6}.$

In the subsequent stage, the model determines the flow velocity values for each pixel based on the previously calculated water depth and slope using the Darcy–Weisbach method [30], which requires the specification of a friction factor. To calculate the friction factor, it is first necessary to define the D50 (median) sediment size within the model. In this case, a median sediment size of 1 mm was specified in the test [31]. The selection of the median sediment size has been reported by several studies. The analysis of Martian sediment sizes primarily relies on imagery captured by rovers and landers [31,32] and satellites within the visible spectrum. These observations mostly underrepresent the smaller (silt or clay) sediment sizes. To address this limitation, the THEMIS TI [33,34] data were used to determine the different sediment size distributions of the Martian surface [35,36]. For Tinto B, the median sediment size from the entire watershed was 1 mm [21]. To ensure comparability of the results, this median sediment size was also employed for the terrestrial area of interest.

(3)$F_v={\left({\displaystyle \frac{8ghS}{F{F}_w}}\right)}^{0.5}$

(4)$F{F}_w={{\displaystyle \frac{8}{\left(2.2\left({\frac{h}{D_{50}}}^{-0.055}\right)\left(S^{-0.275}\right)\right)}}}^2.$

The widths of the valleys and gullies in the model are identical to the spatial resolution of the DTM. This is because natural watercourses undergo continuous morphological change within a study range. In the majority of cases, valleys are assumed to be rectangular for simplicity, as this geometry is the most straightforward method for determining the area and avoidance.

The following section outlines the parameters required for surface erosional and depositional development studies. Various models have been developed to investigate and simulate short- or long-term surface modification. The method presented in this paper is based on stream power because the stream power index (SPI) is a highly effective predictor in surface erosion and accumulation models [37].

(5)$TC=K_{t}G{\rho }_w{h}^m{S}^n,$

where K_t is the transport coefficient (SVTR in this research) of the soil, and the variables m and n represent the water depth and slope weights, respectively. The definition of K_t is not entirely straightforward. Based on several sources [38], K_t is the product of the universal soil loss equation (USLE) [39], K (soil erodibility index), P (land use and land cover change factor), and C (land cover index). In the present study, the coefficient K_t was derived using a method other than the USLE equation, as no relevant data were available for the calculation of the C and p factors for the study area. Furthermore, the model presented here attempts to avoid the inclusion of non-physical parameters.

The variables m and n can be given a constant value [37], suggesting that they should be scaled. The scaling is based on the value of flow accumulation in pixels for the variable m and the value of slope in degrees for the variable n. Values of m are generally less than 2 and should be scaled inversely with increasing flow depth between two or more boundaries, as described in the above literature. The n values, like the m values, are also inversely proportional, although not to the depth of flow, but to the angle of slope. Tc is calculated using the Grass GIS raster calculator (r.mapcalc), where the scaling is specified by graph(x, threshold, value1, threshold2, value2, …).

(6)$SVTR={\displaystyle \frac{1}{1-\lambda }}{q}_{nondim}{\left({\rho }_{submerged}G\right)}^{0.5}{D}_{50}^{1.5}.$

The presented method allows for the determination of K_t from the specific volumetric transport rate (SVTR). The K_t value can be obtained from the SVTR value by dividing the SVTR value by the spatial resolution value. To determine SVTR (6) [30,31], the shear stress and the derived Shield parameter are required.

(7)$\tau ={\rho }_{water}GhS$

(8)${\tau }_{crit}^{\ast }={\displaystyle \frac{\tau }{\left({\rho }_{sediment}-{\rho }_{water}\right)G{D}_{50}}}.$

The dimensionless shear stress can be determined from the shear stress, i.e., the Shield parameter (8), which is a prerequisite for the determination of a dimensionless coefficient (q_nondim) from which the SVTR can be derived.

(9)$q_{nondim}=8{\left({\tau }^{\ast }-{\tau }_{crit}^{\ast }\right)}^{1.5,}$

where ${\tau }_{crit}^{\ast }$ is a critical shield parameter. Above the critical shield parameter, particles are set in motion and transported away. In this research, a value of 0.03 $ta{u}_{crit}^{\ast }$ was used [30,40]. Before defining the erosion/accumulation values, a water network exposure map must be defined, which can be calculated using the x and y-axis partial derivatives of the flow accumulation values in pixels or determined using the r.slope.aspect tool. The cosine and sine values of this exposure are equal to the flow direction. The change in sediment transport capacity is given and can be expressed by a directional function [41] (11). The erosion/deposition rate (ED) (10) indicates the amount of erosion or deposition, i.e., the accumulation, in the area under consideration.

(10)$ED={\displaystyle \frac{\nabla T{c}_x}{\nabla x}}+{\displaystyle \frac{\nabla T{c}_y}{\nabla y}},$

where $T{c}_x$ and $T{c}_y$ can be calculated with the following:

(11)$T{c}_x=Tc\ast cos\left(Aspect\right) \&\& \mathrm{T}c\_y=Tc \ast sin\left(Aspect\right).$

The ED rate can be calculated by adding $T{c}_x$ and $T{c}_y$ together.

3. Results

To make a comparison of Earth- and Mars-based modelling possible, identical parameters were employed at both study areas, with the exception of the planet- and area-specific parameters, as outlined in Table 1, including the difference in surface gravity, sediment density, and spatial resolution. The hydrological analyses for Mars were also based on parameters determined from real precipitation event intensities observed in the terrestrial sample area, what was maximum 18 mm/h for three hours.

The above listed parameters and conditions of the terrestrial study area resulted in a maximum water level of 3.45 m by accumulation of surface runoff with an average of 0.02 m. The maximum flow velocity was 1.8 m/s, with an average of 0.2 m/s in the terrestrial area, while for Mars the corresponding values were a maximum velocity of flow of 1.86 m/s, with an average value of 0.32 m/s (Table 2).

The peak discharge for the terrestrial sample area is 35.78 m³/s, as calculated using Manning’s equation. For the total sample area, this equates to 1.0 m³/s/km². In [28], the peak discharge for the same catchment was 1.3 m³/s/km², while the modelled peak discharge was 1.1 m³/s/km². The discrepancy between the values found in the present study and those measured and modelled in [28] is 0.3 and 0.1 m³/s/km², respectively.

With respect to the model for Earth surface topography development by erosion and accumulation, the maximum transport capacity was calculated to be 3.12 kg/m²/s, with an average of 0.001 kg/m²/s. Regarding the erosion/accumulation end-result raster values, positive values indicate the extent of accumulation, whereas negative values indicate the extent of erosion. The maximum value of accumulation in the terrestrial study area is 0.13 kg/m²/s, while the maximum value of erosion is −0.06 kg/m²/s. For the total rainfall event, the maximal accumulation rate is 1432 kg/m²/s, and the erosion rate is −667 kg/m²/s.

In the area designated for the Mars sample, the maximum depth of flow accumulated by a precipitation event of the same intensity and duration as on Earth is 9.73 meters, with an average depth of 0.06 meters (Table 2).

The maximum value of transport capacity produced by the rainfall event in the Martian study area is 0.22 kg/m²/s, with an average value of 0.000098 kg/m²/s. The maximum rate of accumulation is 0.01 kg/m²/s, while the maximum rate of erosion is −0.007 kg/m²/s. The total rainfall event results in a deposition rate of 683 kg/m²/s, with a corresponding erosion rate of −490 kg/m²/s. The summarized values are listed in Table 2.

The maximum accumulation rate in the terrestrial sample area is 1.3 times that of the Martian sample area, considering percentage values. Regarding erosion, the maximum value in the terrestrial sample area is 8.57 times higher than that in the Martian case. With respect to transport capacity, the value in the terrestrial sample area is 14.18 times higher than the value modeled in the Martian sample area.

It is challenging to make a direct comparison between the Earth and Martian sample areas; however, normalizing each value by area facilitates a more nuanced understanding of the precise differences and similarities between the two catchments. The slope angle (expressed in degrees) and flow depth result raster values were categorized, and the area of each category was subsequently normalized by the total area of the watershed (see Figure 3 for diagrams).

The proportion of flat areas (0° and 5° degrees) is 2.3 times greater (Figure 3, inset a) for the slope angle data than at the sample area. Only the terrestrial sample area exhibited steep slopes (45° and above).

The normalized flow depth distribution (Figure 3, inset b) reveals that, for the category not exceeding 0.01 m, there are 6.2 times more pixels in the ground sample area than in the Martian sample area. Conversely, for the higher classes, the proportion of pixels in the Martian sample area was larger. Please also note that the first column in inset a (flat areas) does not necessarily correspond to the first column in inset b (smallest flow depth), but the flat areas present somewhat larger flow depth values, for example., the first column I inset a provides values in the second, third, etc., columns for the diagram in inset b.

Below, the performance of model runs using modified input parameters were tested. The rationale for such testing is partly to see the differences emerging in the modified parameters to better understand the modelling process itself and to see whether the changes occurred according to the expectations or not. In the following paragraphs, two parameters have been varied and tested: the role of decreased bulk density (which comes from heavily weathered surface and shallow surface rock material, although the exact realization of the weathering process and produced state is poorly known on Mars but should definitely be smaller and weaker than on Earth); and the average grain size was also modified (which might be smaller than the THEMIS-based values suggested by field experiences from Earth-based analogues; however, the exact values are unknown for Mars).

The model’s default value of density, the upper limit of the density of Martian basalt, was derived from theoretical argumentation and measurements of meteorites from the planet [42]. The density of the Martian surface varies between 2700 and 3100 kg/m³ [43,44,45] owing to the variable porosity. Consequently, the changed value assumes more porous rocks in the target region of Tinto B on Mars with a value of 2840 kg/m³, which is the same as that reported in the Zafit basin study, to increase the comparability of the calculations between the two planets. This value corresponds to an average of 2700–3100 kg/m³ reported for the Zafit basin. Within the model, the flow depth and flow velocity values remain almost constant at the modified density; however, substantial changes are observed for transport capacity and erosion/accumulation rate. The transport capacity values changed from a maximum of 0.22 kg/m²/s to a maximum of 6.93 kg/m²/s, a 31.5-fold increase for the Martian intrasample area and a 2.2-fold increase from the maximum for the terrestrial sample area. The average value for the Martian sample area is an order of magnitude larger for the value calculated with a more porous density, but is still an order of magnitude smaller than the average value for the terrestrial sample area. The erosion/accumulation rate values for the lower density model of the Martian sample area are an order of magnitude higher than those of the default value model. For the terrestrial sample area, the maximum accumulation rate on Mars modeled at the lower density is 0.03 kg/m²/s, while the erosion rate is 0.01 kg/m²/s (Table 2).

The model was also tested with different sediment grain size ranges (from 0.00001 to 0.1 mm). No significant differences in the erosion/accumulation rate results were observed for the different sediment grain size ranges. However, a small, gradual modification was observed. In the case of the Martian study area, the maximum accumulation rates exhibited a decrease in magnitude with increasing sediment grain size, from 0.00001 m to 0.1 m; a difference of one order of magnitude between the maximum and minimum values of accumulation emerged. A similar trend was observed in the maximum erosion rates, with the decreased 0.00001 m sediment size demonstrating the highest erosion rate, while the increased 0.1 m sediment size exhibited the lowest rate (Table 3). Analogous conclusions can be drawn for the terrestrial study area as for the Martian study area; the smallest median sediment size exhibits the highest maximum erosion and accumulation rates, and the smallest rate for the largest median sediment size (Table 3).

4. Discussion

The identical precipitation parameters and calculated inferred values permit the model to be employed for the comparison of precipitation-induced alterations to the surface at the two catchments on the two planets with analogous morphological attributes.

When evaluating the values calculated for the Earth- and Mars-based areas, the maximal and average flow depths were both larger for Mars, while the flow velocity was moderately the same. The areal distribution of flow depth is visualized in Figure 4, where the expected hierarchical increase in the flow depth is visible. The notable discrepancy between the flow depth values of the terrestrial (Figure 4, inset b) and Martian sample areas (Figure 4, inset a) can be partly attributed to the disparate sizes of the two areas under investigation. The Martian sample terrain (4758 km²) has a considerably (133 times) larger drainage basin area than the terrestrial one (35.78 km²), which permits the accumulation of significantly greater quantities of water in the deeper regions, producing two-to-three times larger flow depth (see the second column in Table 2).

The intensity of precipitation during the early wet/wet periods on Mars remains unclear. However, several studies [10] indicate that if precipitation did occur, it may have occurred from the middle of the Noachian period until the beginning of the late Hesperian period (~4.0 Ga–3.5 Ga). Additionally, the majority of the flood channels may have been formed during this period, supported by volcanic activity [21,46].

A number of studies have successfully estimated the maximum flow discharge for various Martian valleys that formed and were active during the wet period of Mars. In these studies, the estimation of flow discharge, flow depth, and other outcomes was based exclusively on morphological factors [31,40,47]. Accordingly, the potential maximum flow depth, flow discharge, and velocity values can be calculated to characterize a given valley.

The model presented in this study was used to determine the flow depth, flow discharge, and flow velocity values based on precipitation data from a terrestrial Mars analogue field. It should be noted that the results of the hydrological parameterization by rainfall data do not represent the theoretical maximum of the entire watershed, but rather the values of runoff and accumulation during a rainfall event. Consequently, given the potential differences in magnitude between the two different methods, the data presented in this study are not directly comparable. Our calculated values roughly fit the already published range. However, the corresponding range values for Mars are much larger than the Earth-based ones.

In the case of the flow velocity and accumulated deposits, there is not a significant discrepancy between the values for the two planets. Given that the Martian gravity is approximately one-third of that of the Earth, it can be anticipated that runoff will be relatively slower at the same water or flow depth. However, in the case of the steep valley walls in the Tinto B area (Figure 5, inset a), along the valley at right running vertically in the image), the flow velocity may potentially accelerate with sufficient time or precipitation accumulation from a large area, which could in turn lead to roughly the same scale of erosion as for Earth, although this is relevant only for the steep walls. The main valley of Tinto B has a gradient of up to 41° in the steepest side wall parts. Considering the Earth, in the lower part of the main valley, the Zafit catchment area (Figure 5, inset b) also has a steep valley wall with a maximum slope angle value of approximately 56°. Although most of the valley walls in the terrestrial sample area are, on average, gentler, the slope angle distribution in the two planetary areas are roughly similar if this consideration is based only on the flow velocity results. However, weathering produced loose soil that is expected to be thinner on Mars because of the lack of vegetation [48], introducing a further unknown factor to this consideration.

A significant discrepancy was observed in the erosion/deposition values between the terrestrial and the Martian area, which can be attributed partly to the dissimilarities in the underlying bedrock density. In the case of the Martian sample area, the average density of the basalt was taken to be 3650 kg/m³ [42], whereas in the case of the terrestrial sample area, the calcareous sedimentary rock has a density of 2840 kg/m³ [49], as reported by Wieler et al. In the Martian case, a larger area is known to have been affected by the simulated precipitation event, resulting in higher accumulated water levels. This elevated flow thickness seems to counterbalance the weaker gravity and produces roughly the same flow velocity in Table 2. Based on the flow depth results, it can be concluded that the area with higher water levels experiences increased flow velocity, which is proportional to the slope but is also decreased by the reduced Martian gravity. In the case of the Martian sample area, despite having a steeper slope and higher flow depth than the terrestrial area, the flow velocity is increased almost to the same level as on the Earth, despite the gravity being only approximately one-third as strong.

The discrepancies between the erosion and accumulation values can be elucidated in a manner analogous to that employed for the transport capacity, as the erosion and accumulation rates can be derived directly from the estimated transport capacity (Figure 6). The study site-specific element is the direction of the steepest slope, which, as demonstrated in the calculations presented in the Methods section, influence the erosion and accumulation rates. The much-reduced erosion and deposition masses are mainly influenced by the lower density of terrestrial basement rock (please note, at the targeted desert area no observed vegetation and soil were present, which would have even further decreased the erosion/deposition masses) Figure 7.

It is important to note that the model results did not provide a definitive indication of the processes occurring in the main valleys of the study areas. Typically, prior to the application of an accumulation model, the depressions of the digital terrain model (DTM) must be “filled” by some method to remove these depressions and unevenness that might impede free runoff. This was not carried out in the method presented here, as preliminary tests of the model demonstrated that during recharge in the Martian sample area, small craters are filled by the recharge algorithm, and flow-through accumulation and pounding were generated. Given the specific topography of Mars, this results in an inadequate accumulation picture and, consequently, erroneous results.

The notable discrepancies in the transport capacity and, consequently, the erosion/accumulation scores of the two areas can be attributed to the Shield parameter result matrix, in addition to the aforementioned reasons. The Shield parameter played a pivotal role in the calculation of the final transport capacity. In the model, if the calculated Shield parameter does not reach the previously defined threshold value of 0.03, then zeros are taken up, indicating the absence of sediment transport. For Tinto B, the morphological peculiarities exhibit a higher prevalence of pixels where the Shield parameter is zero or precisely 0.03 in comparison to the Zafit catchment.

The above argumentation indicates there is a larger area on Mars where the terrain is not steep enough to trigger sediment movement according to the Shield parameter, while in the case of Earth, a larger area allows movement. This difference does not emerge in the maximal slope angles and maximal flow speed; even the average flow speed on Mars produced a larger area of low slope and a smaller area of very high speed at the steep sideway slopes of Tinto-B valley. As a result, the erosion and accumulation rates were much smaller on Mars, considering the whole analyzed watershed. This difference might be related to the generally weaker surface modification on Mars, which produces only occasionally steep terrains, while the whole relief and undulation is smaller than on Earth. These interesting findings are awaiting a detailed analysis of geomorphology-related surface evolution schema on Mars compared to those on Earth.

The elevated flow depth values observed in the Martian catchment can be attributed to the larger area and the associated higher flow accumulation values. Conversely, the Martian area exhibited a higher proportion of flat terrain, which is associated with elevated flow depth values. Similarly, the lower transport capacity values observed on Mars may be attributed to the prevalence of flat terrain (0° to 5° ). In these flat areas, the elevated water height and the reduced waterpower generally result in diminished transport capacity.

Considering the application of the model for climate and environmental reconstruction of past Martian conditions, especially for the Noachian and Noachian/Hesperian transition where waterflows and related transport were widespread, the following are relevant. Better information on erosion and sedimentation could help to constrain the ancient paleo-discharge values as the mass of transported sediment and the volume of material that have been eroded away from channel depressions could be measured. If the erosion process could be better characterized in the future by improved modeling, the maximal discharge estimation could support the separation of the temporal emergence of water source types: sudden rainfall, continuous “light” raining events, or even gradual ice melting. In the case of more moderately sudden ice melting (like hot ejecta emplacement on a formerly deposited snow or ice sheet), it could be also better reconstructed and understood.

Feasibility and Limitations of the Model

Testing the model’s performance for different input parameters, the role of density and sediment size change were evaluated in detail, especially the transport capacity and erosion/accumulation rate values accordingly. For grain size permutation, no significant difference was found, but a minor change happened according to expectations: the decreasing grain size caused increased erosion/accumulation rates as smaller grains are more easily transported. These findings suggest the model is realistic, and if in the future better grain size estimation is available using remote sensing measurements correlated with in situ data, improved modelling would be available for the whole Martian surface.

Regarding the change of density of target rocks for the Martian sample area (Tinto B), decreased density produced a substantial increase in transport capacity, which is typical of more porous rocks. This significant increase also coincides with expectations, further confirming the reality of the model approach used. This also indicates that in the future, the weathering state and related density of target rocks, if better known for Mars, could be better implemented in the modeling, supporting the emergence of better erosion/accumulation rates.

Limitations and uncertainties of the model come from two groups of sources. 1. The accuracy of input parameters is limited by the observational possibilities and currently available instruments for Mars. Improved remote sensing data are expected to produce better DTMs soon, while the improvement of target density knowledge is less expected, as a global dataset for improved thermal inertia analysis requires both higher spatial resolution and temporal density of measurements, which are difficult to obtain. Besides these aspects, many further remote sensing data–ground truth measurement pairs by landers are also difficult to obtain as it is not yet realistic to land on a wide range of surface types. 2. Uncertainty comes from the poorly known micro-scale physical processes, like shield parameters, role of flocculation (grain grouping) by any unique water chemistry, or specific vortex behavior under reduced gravity, which could be improved by laboratory tests on the Earth. However, a great problem regarding Mars’s relevance is to estimate the corresponding values under the reduced Martian gravity, but theoretical argumentation might help.

5. Conclusions

In this work, an improved fluvial surface erosion and deposition model was adapted and applied to Mars for the ancient Tinto-B valley in order to better understand the surface evolution of the area. For comparison, an Earth-based area that provided the numerical values of the precipitation event from the Zafit basin is the Negev desert area of Israel. During the model application, a 180 min-long 18 mm/h precipitation event was supposed to happen on Mars, which has been observed and numerically measured in the field of the Earth-based location.

The model was able to provide flow depth, flow speed, and erosion versus accumulation areal distribution patterns that are expected using the available data. There is a notable discrepancy between the estimated transport capacity and the observed erosion and accumulation rates. In addition to the aforementioned reasons, the use of different rock densities in the study also affects these two results. In this study, we have aimed to utilize the most accurate data available for the areas under study and therefore have not generalized parameters such as rock density and gravity. Although there were differences between the two planets (weaker gravity on Mars, smaller bedrock density on Earth), as well as similarities (desert areas without soil on both planets), the comparative approach helps in evaluating the model, which should be further improved.

The above calculations indicated a precipitation event on Earth and extrapolated it to Mars, finding a similar scale of maximal flow speed, while the larger flow depth (3.45 m for Zafit on Earth and 9.73 m for Tinto B on Mars) provided by the larger watershed area in the second case was counterbalanced by the smaller local gravity, thus roughly the same flow speed values emerged. However, the erosion and deposition-influenced masses were much larger for Earth (0.13/−0.06 kg/m²/s for Zafit (Earth) and 0.01/−0.007 for Tinto B (Mars)), partly because of the higher density of bedrock on Mars.

The importance of the model approach presented above lies in the fact that the calculations used on Mars have been validated to the extent that this preliminary test provides information, indicating that further development on different terrains and targets on Mars is warranted in order to gain a deeper understanding of the rate of erosion and deposition. This, in turn, will support the targeting of the next surface mission.

The advantage of the model over other existing models is that it is straightforward to parameterize and can be applied effectively not only to terrestrial but also to Martian conditions, given the limited information available. It is capable of simulating the results of the so-called steady-state condition in its current state. The long-term objective is to be able to perform unsteady-state calculations, taking into account the specific Martian terrain (many overprinting craters, potential interpolation errors in the digital terrain model).

At the present stage, the model is only capable of considering so-called steady-state conditions, which can be utilized to simulate one-off, short-timescale events. Long-term improvements include the development of a model that can also investigate unsteady conditions, i.e., conditions that are in a state of constant change over the simulated time. The implementation of such studies will facilitate the investigation of the formation of Martian river valleys and the direct study of paleoclimatic conditions on the red planet. Unsteady models possess the capacity to study long-term (up to several thousand years) changes in a dynamic manner. It is envisaged that the development of the model will not only enable the study of precipitation-induced erosion but also the erosion of ice melted during Martian volcanic activity and subsequent run-off in the form of flash floods. These future developments will facilitate the acquisition of a more precise depiction of the evolution of the Martian surface and could also contribute to the study of the morphology of the Earth’s watersheds.

Author Contributions

The research model was constructed on the basis of the hypotheses put forth by V.S., who selected the sample areas to be analyzed on the strength of preliminary studies. Á.K. and V.S. then proceeded to analyze the results of the model and to draw conclusions. English proofreading was carried out by Á.K. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

All the used data and the model is available after personal contacting.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Overview of the analyzed Martian area on CTX composite images. Insets b and c provide a closer look at the analyzed area. In insets b and c, the red line indicates the valley of Tinto B (the analyzed valley), and the blue line indicates the Tinto Vallis.

Figure 2. Overview of the terrestrial study area for comparison. The small inset in the upper right corner shows the location of the Zin catchment on the Arabian Peninsula, while the main image shows the outline of the sub-catchment of the Zafit catchment.

Figure 3. Diagrams of the area normalized slope (with inset (a) and flow depth (inset (b) distributions of the catchment areas of interest. The red color shows the Tinto B (Martian) catchment, and the blue color represents the Zafit (terrestrial) catchment. The x-axis shows the categories; the y-axis shows the normalized distribution.

Figure 4. The result maps of the flow depth estimation, where the darker shade of blue color marks larger accumulated water thickness, e.g., flow depth. Inset (a) shows the Martian study area, and inset (b) shows the terrestrial area.

Figure 5. Result maps of the flow velocity estimation. Inset (a) shows the Martian study area, and inset (b) shows the terrestrial area. For the visualization, the cumulative count cut procedure was used. For this reason, the show values can be different from the values listed in Table 2.

Figure 6. Result maps of the transport capacity estimation. Inset (a) shows the Martian study area, and inset (b) shows the terrestrial area. For the visualization, the cumulative count cut procedure was used. For this reason, the shown values can be different from the values listed in Table 2.

Figure 7. Result maps of the accumulation and erosion rate estimations. Inset (a) shows the Martian study area, and inset (b) is the terrestrial area. For the visualization, the cumulative count cut procedure was used. For this reason, the shown values can be somewhat different from the values listed in Table 2.

Table 1

List and determination of the parameters used. The parameters in bold lettering are the input parameters.

Parameters Name	Acronym	Used Value (Earth/Mars)	Unit of Measurements
Resolution	r	30/50	m
Gravity	G	9.807/3.72	m/s2
ϱ water	ϱ water	1000/1000	kg/m3
ϱ sediment	ϱ sediment	2840/3560	kg/m3
Rainfall Intensity	I	18	mm/h
Duration of the Storm	t	180	Min
D50 Sediment Size	D50	0.001	m
Porosity	p	0.2
Manning Roughness	n	0.0545
Flow Accumulation	A		m2
Submerged Density	ϱsb		kg/m3
Flow DEPTH	h		m
Friction Factor for Flow Velocity	FFW
Flow Velocity	v		m/s
Shear Stress	τ		Pa
Shields Parameter	τ *
Nondimensional Transport Rate	qnondim
Specific Volumetric Transport Rate	SVTR		m2/s
Transport Capacity	Tc		kg/m2/s
Erosion/Deposition Rate	ED		kg/m2/s

Table 2

Summarization of the model result values with the two input parameter options on the right: larger density in the third and smaller in the fourth column.

	Zafit (Earth)	Tinto B (Mars, Basic Density)	Tinto B (Mars, Decreased Density
Flow Depth (m)	Max: 3.45	Max: 9.73	Max: 9.73
	Avg: 0.02	Avg: 0.06	Avg: 0.06
Flow Velocity (m/s)	Max: 1.8	Max:1.86	Max:1.86
	Avg: 0.2	Avg: 0.32	Avg: 0.32
Transport Capacity (Max/Average kg/m2/s)	Max: 3.12	Max: 0.22	Max: 6.93
	Avg: 0.001	Avg: 0.00009	Avg: 0.0001
Maximum rate of the Erosion/Accumulation Rate (Accumulation/Erosion) kg/m2/s	Accumulation: 0.13	Accumulation: 0.01	Accumulation: 0.10
	Erosion: −0.06	Erosion: −0.007	Erosion: −0.07

Table 3

Comparison of the accumulation (positive values) and erosion (negative values) modeled for the different median sediment sizes.

Median Sediment Size (m)	Zafit (Earth)	Tinto B (Mars)
0.1	Accumulation: 1.13 × 10–1	Accumulation: 9.43 × 10–3
	Erosion: –5.09 × 10–2	Erosion: –6.61 × 10–3
0.01	Accumulation: 1.13 × 10–1	Accumulation: 1.01 × 10–2
	Erosion: –6.08 × 10–2	Erosion: –7.06 × 10–3
0.001	Accumulation: 1.33 × 10–1	Accumulation: 1.02 × 10–2
	Erosion: –6.18 × 10–2	Erosion: –7.11 × 10–3
0.0001	Accumulation: 1.33 × 10–1	Accumulation: 1.02 × 10–2
	Erosion: –6.19 × 10–2	Erosion: –7.11 × 10–3
0.00001	Accumulation: 1.33 × 10–1	Accumulation: 1.02 × 10–2
	Erosion: –6.19 × 10–2	Erosion: –7.11 × 10–3

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