Hölder-conditioned hypsometry: A refinement to a classical approach for the characterization of topography

©2020. American Geophysical Union. All Rights Reserved. The effective characterization of topographic surfaces is a central tenet of geomorphology. Differences in land surface properties reveal variations in structural controls and the nature and efficacy of Earth-shaping processes. In this paper, we employ the Hölder exponents, α, characterizing the local scaling behavior of topography and commonly used in the study of the (multi)fractal properties of landscapes and show that the joint probability distribution of the area of the terrain with a given elevation and α contains a wealth of information on topographic structure. The conditional distributions of the hypsometric integrals as a function of α, that is, Ihyp|α, are shown to capture this structure. A multivariate analysis reveals three metrics that summarize these conditional distributions: Strahler's original hypsometric integral, the standard deviation of the Ihyp|α, and the nature of any trend of the Ihyp|α against α. An analysis of five digital elevation models (DEMs) from different regions of the United States shows that only one is truly described by the hypsometric integral (Mettman Ridge from central Oregon). In the other cases, the new metrics clearly discriminate between instances where topographic roughness is more clearly a function of elevation, as captured by the conditional variables. In a final example, we artificially sharpen the ridges and valleys of one DEM to show that while the hypsometric integral and standard deviation of Ihyp|α are invariant to the change, the trend of Ihyp|α against α captures the changes in topography.