TY - JOUR
T1 - On the theory of drainage area for regular and non-regular points
AU - Bonetti, S.
AU - Bragg, A. D.
AU - Porporato, Amilcare Michele M.
N1 - Funding Information:
S.B. and A.P. acknowledge support from the US National Science Foundation (FESD EAR-1338694). A.P. also acknowledges support from the USDA Agricultural Research Service cooperative agreement 58-6408-3-027; National Science Foundation (NSF) grant nos. CBET-1033467, EAR-1331846 and EAR-1316258; and the Duke WISeNet grant no. DGE-1068871.
Funding Information:
Data accessibility. The Calhoun DEM [61] was made available by the OpenTopography Facility with support from the National Science Foundation under NSF Award nos. 1226353 and 1225810. Authors’ contributions. All the authors contributed equally to conceiving and designing the study. S.B. performed the analyses and wrote an initial draft of the paper, to which all the authors contributed edits at all stages. All the authors helped to interpret the results. Competing interests. We have no competing interests. Funding. S.B. and A.P. acknowledge support from the US National Science Foundation (FESD EAR-1338694). A.P. also acknowledges support from the USDA Agricultural Research Service cooperative agreement 58-6408-3-027; National Science Foundation (NSF) grant nos. CBET-1033467, EAR-1331846 and EAR-1316258; and the Duke WISeNet grant no. DGE-1068871. Acknowledgements. We thank Stefano Orlandini and John Gallant for their valuable comments and suggestions. We also thank John Gallant for pointing out [56].
Publisher Copyright:
© 2018 The Author(s) Published by the Royal Society. All rights reserved.
PY - 2018/3
Y1 - 2018/3
N2 - The drainage area is an important, non-local property of a landscape, which controls surface and subsurface hydrological fluxes. Its role in numerous ecohydrological and geomorphological applications has given rise to several numerical methods for its computation. However, its theoretical analysis has lagged behind. Only recently, an analytical definition for the specific catchment area was proposed (Gallant & Hutchinson. 2011 Water Resour. Res. 47, W05535. (doi:10.1029/2009WR008540)), with the derivation of a differential equation whose validity is limited to regular points of the watershed. Here, we show that such a differential equation can be derived from a continuity equation (Chen et al. 2014 Geomorphology 219, 68-86. (doi:10.1016/j.geomorph.2014.04.037)) and extend the theory to critical and singular points both by applying Gauss's theorem and by means of a dynamical systems approach to define basins of attraction of local surface minima. Simple analytical examples as well as applications to more complex topographic surfaces are examined. The theoretical description of topographic features and properties, such as the drainage area, channel lines and watershed divides, can be broadly adopted to develop and test the numerical algorithms currently used in digital terrain analysis for the computation of the drainage area, as well as for the theoretical analysis of landscape evolution and stability.
AB - The drainage area is an important, non-local property of a landscape, which controls surface and subsurface hydrological fluxes. Its role in numerous ecohydrological and geomorphological applications has given rise to several numerical methods for its computation. However, its theoretical analysis has lagged behind. Only recently, an analytical definition for the specific catchment area was proposed (Gallant & Hutchinson. 2011 Water Resour. Res. 47, W05535. (doi:10.1029/2009WR008540)), with the derivation of a differential equation whose validity is limited to regular points of the watershed. Here, we show that such a differential equation can be derived from a continuity equation (Chen et al. 2014 Geomorphology 219, 68-86. (doi:10.1016/j.geomorph.2014.04.037)) and extend the theory to critical and singular points both by applying Gauss's theorem and by means of a dynamical systems approach to define basins of attraction of local surface minima. Simple analytical examples as well as applications to more complex topographic surfaces are examined. The theoretical description of topographic features and properties, such as the drainage area, channel lines and watershed divides, can be broadly adopted to develop and test the numerical algorithms currently used in digital terrain analysis for the computation of the drainage area, as well as for the theoretical analysis of landscape evolution and stability.
KW - Digital elevation model
KW - Drainage area
KW - Geomorphology
KW - Gradient fow
KW - Landscape evolution
KW - Topography
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U2 - 10.1098/rspa.2017.0693
DO - 10.1098/rspa.2017.0693
M3 - Article
C2 - 29662340
AN - SCOPUS:85045505119
SN - 1364-5021
VL - 474
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
IS - 2211
M1 - 20170693
ER -