A representation of an instantaneous unit hydrograph from geomorphology

TLDR

River basin channel networks and overland flow obey Horton's geomorphologic laws, but the formal analogy to linear reservoir models does not hold physically. The study employs this setting to derive an explicit mathematical representation of the instantaneous unit hydrograph at the basin outlet. The authors develop two kinetic‑theoretic examples yielding explicit iuh formulas, parameterized by Horton's ratios and mean lag time, and validate them against runoff hydrographs from three Illinois basins. The theory matches observed runoff for two ~1100‑mi² basins but underestimates peak flow for a 300‑mi² basin, suggesting the linearity assumption in rainfall‑runoff transformation may be invalid for smaller basins.

Abstract

The channel network and the overland flow regions in a river basin satisfy Horton's empirical geo‐morphologic laws when ordered according to the Strahler ordering scheme. This setting is presently employed in a kinetic theoretic framework for obtaining an explicit mathematical representation for the instantaneous unit hydrograph (iuh) at the basin outlet. Two examples are developed which lead to explicit formulae for the iuh. These examples are formally analogous to the solutions that would result if a basin is represented in terms of linear reservoirs and channels, respectively, in series and in parallel. However, this analogy is only formal, and it does not carry through physically. All but one of the parameters appearing in the iuh formulae are obtained in terms of Horton's bifurcation ratio, stream length ratio, and stream area ratio. The one unknown parameter is obtained through specifying the basin mean lag time independently. Three basins from Illinois are selected to check the theoretical results with the observed direct surface runoff hydrographs. The theory provided excellent agreement for two basins with areas of the order of 1100 mi 2 (1770 km 2 ) but underestimates the peak flow for the smaller basin with 300‐mi 2 (483‐km 2 ) area. This relative lack of agreement for the smaller basin may be used to question the validity of the linearity assumption in the rainfall runoff transformation which is embedded in the above development.