A Case Study of an Application of Fractal Theory to Gully's Alcove on Mars

Introduction: Two main theories of gully formation on Mars exist depending on the source of water. One holds that water was released from the subsurface [1]. The other proposes that water is deposited as nearsurface ice or snow from the atmosphere and is subsequently melted by insolation [2-4]. In this paper we applied fractal theory to study a gully's head alcove in different orientations of a small crater (~5km) within the 30°–45°S latitude band, and to support the hypothesis that gully formation within the 30°–45°S latitude band is related to snow and ice deposition and melt (mostly sublimation) due to climatic processes. Methods:Just as with coastlines, mountain ridges, rivers, etc. on Earth, gullies on Mars take on the features of scale invariance and self-similarity, which can be analyzed from the non-linear characteristics by using fractal theory to reveal its inherent laws directly. A box-dimension method is a simple, objective and universally-applied method. As to a gully's alcove, we can use squares with the side length of r to cover it, thus the number of the boxes required is N(r). The total number of lattices, N(r), will vary with r, the measurement scale. N(r) is a function of r. According to the definition of fractal dimension, the relation between r and N(r) can be presented as equation (1) [5]: N ( r) ∝ r -D (1) where r refers to the side-length of a square lattice while N(r) represents the number of lattices covered by the gully and D is the fractal dimension. If we applied logarithm to both sides of equation (1), we get: ln N ( r) ∝ D ln r (2) In the whole process of covering, we can obtain a group of (r, N(r)) data, thus we can draw a double logarithmic chart and then draw a straight line using the least squares technique on the map of lgN(r)-lgr. ln N(r) = ln C D ln r (3) The fractal dimension D refers to the size of the slope, which reflects the complexity of the distribution of a gully. Data and Results:In order to reduce the control of the difference of the latitude, elevation, material and slope, a small crater with similar environments at the slopes where the gullies exist is needed, so we representatively selected the gullies at an unnamed crater (37.7°S,192.9°E) in the Sirenum region as an example. This crater is suborbicular, 5km in diameter, with gullies in every orientations of the crater wall which is not common on Mars (Fig 1-A). The gullies on the northern wall of the crater are more mature with the length of alcoves ranging from 200m to 350m, which may contribute to rapid upslope propagation of the landform. On the contrary, the gullies on the southern wall are less developed, with the length of alcoves only ranging from 140m to 200m.