Two-dimensional differential form of distributed Xinanjiang model

Abstract. The distributed hydrologic models (DHMs) evolved from lumped hydrologic models, inheriting their modeling philosophy along with persistent numerical-error issues. Historically, these models tend to use established one-dimensional (1D) methods for slope concentration, which often struggle to effectively represent complex terrains. In this study, we formulated a purely differential form of mathematical equations for the distributed Xinanjiang model and developed a fully coupled numerical solution framework. We also introduced two-dimensional (2D) diffusion wave equations for surface slope concentration and derived 2D linear reservoir equations for subsurface slope concentration to replace their 1D counterparts. This culminated in the development of a two-dimensional differential form of the distributed Xinanjiang (TDD-XAJ) model. Two numerical experiments and the application of the TDD-XAJ model in a humid watershed were conducted to demonstrate the model. Our results suggested that (a) numerical errors in the existing distributed Xinanjiang model are significant and may be exacerbated by a potential terrain amplification effect, which could be effectively controlled by the fully coupled numerical framework within the TDD-XAJ model; (b) the 2D slope concentration methods showed enhanced terrain capture ability and eliminated the reliance on the flow direction algorithms used in 1D methods; and (c) the TDD-XAJ model exhibited improved simulation capabilities compared to the existing model when applied in the Tunxi watershed, particularly for flood volume. This study emphasizes the need to revisit DHMs which stem from lumped hydrological models, focusing on model equations and numerical implementations, which could enhance model performance and benefit the hydrological modeling community.