A statistical parameterization of supraglacial melt pond area and depth on fractal ice sheet surfaces from percolation theory

In recent years, the extent and intensity of surface melt formation has increased on Antarctic ice shelves. Such increasing surface melt is spatially redistributed by surface hydrological processes, and ultimately has potentially important influences on ice shelf stability. In the terrestrial hydrology community, the fractal (or self-similar) roughness of geomorphic surface has been used previously to predict the statistical properties of freshwater bodies over a wide range of spatial scales. Similarly, in this study, we use ice elevation data from ICESat-2 to show that over length scales of tens to thousands of meters (i.e., the typical scale of supraglacial melt ponds), ice shelf surfaces are also fractal. We then used algorithm that realistically fills depressions in randomly generated fractal surfaces to show that simple empirical relationships can be used to relate the roughness statistics of ice sheet surfaces to the lake statistics, i.e., average lake depth and area. This numerically derived parameterization extends previous results from percolation theory in statistical physics, relating the size of connected clusters in random media (i.e., Ising model). We plan future work to validate these findings with observations by calculating comparisons between LandSat-derived melt pond size and the predicted melt pond sizes from our parameterization. We also plan to use this parameterization to improve the calculation for albedo over an ice sheet and to drive hydrofracture parameterization in ice sheet models.