General Unit Hydrograph from Chow’s Linear Theory of Hydrologic Systems and Its Applications

This research solves Chow’s linear hydrologic systems equations thoroughly to result in a theoretical instantaneous unit hydrograph (UH), which is a superposition of many ($M$) negative exponential functions. This implies that the instantaneous UH can be imagined as a superposition of many linear reservoirs in parallel. Mathematically, at $M\to \infty$, the theoretical UH (in terms of Taylor series) converges to the writer’s general UH that is a simple analytic expression derived inductively from empiricism. Therefore, this research turns the recent conceptual general UH to a theoretical law that approximates real-world watershed processes as a time-invariant linear hydrologic system. Specifically, we first review Chow’s linear hydrologic systems model and apply it to a conceptual watershed with an instantaneous storm, which results in a theoretical instantaneous UH and an S-hydrograph in the superposition of many negative exponential functions. The resulting S-hydrograph then is shown mathematically to be identical to the writer’s general UH at $M\to \infty$. Finally, the general theoretical UH is applied to 10 real-world watersheds for 19 rainfall-runoff simulations. It is noteworthy that the proposed method has two advantages: (1) it is general for storms with different rainfall durations, and (2) it does not require to define excess rainfall and direct runoff in advance because rainfall losses and baseflow can be a part of the solution process. It is expected that this research provides a deeper understanding of the general UH and thus helps promote its applications in practice.