Discrete differential geometry of fluvial landscapes

Abstract

Abstract. Geomorphology as a discipline is defined by the use of topographic form to understand surface processes on Earth and other planets. In practice this requires drawing connections between quantitative metrics of surface geometry and rates of erosion and deformation, to understand the spatial partitioning of different erosion processes and the feedback between them. Curvature, perhaps the most fundamental way to measure and categorize surfaces of any kind, also appears explicitly in many erosion models and is therefore of significance to geomorphology. However, there is ambiguity in how curvature of discretely sampled topographic surfaces such as digital elevation models is defined and calculated. In this study we use a formal surface theory approach to compute intrinsic and extrinsic curvature metrics, and associated shape-class distributions, of approximate steady-state fluvial topography of the Oregon Coast Range, USA. We develop a workflow, including careful spectral filtering to isolate wavelengths of interest, that provides a nuanced view of landscape geometry that is consistent and accurate across steep landscape regions. Two invariants of the curvature tensor – the mean and Gaussian curvatures – reveal systematic structure of topographic geometry in channel and ridge networks that captures transitions between hillslope, debris flow, and fluvial process regimes. Mean curvature and associated shape classes are equipartitioned between concave-down and concave-up elements, forming complementary branching structures that span the landscape. These results suggest that formal surface theory approaches could prove valuable in improving process regime identification from digital elevation data in fluvial landscapes.