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Hölder‐Conditioned Hypsometry: A Refinement to a Classical Approach for the Characterization of Topography
Author(s) -
Keylock Christopher J.,
Singh Arvind,
Passalacqua Paola,
FoufoulaGeorgiou Efi
Publication year - 2020
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2019wr025412
Subject(s) - digital elevation model , geology , standard deviation , elevation (ballistics) , terrain , geomorphology , scaling , fractal , ridge , geometry , cartography , remote sensing , mathematics , geography , statistics , mathematical analysis , paleontology
Abstract 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, I h y p | 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 I h y p | α , and the nature of any trend of the I h y p | 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 I h y p | α are invariant to the change, the trend of I h y p | α against α captures the changes in topography.