Rainfall elasticity functions explain divergent runoff sensitivity to rainfall errors in hydrological models

Despite numerous past and ongoing efforts towards characterizing the propagation of rainfall estimation uncertainties in rainfall-runoff hydrologic models, modelers struggle to identify the main features that impact how rainfall errors are transmitted to simulated runoff. With this work, we introduce the concept of the rainfall elasticity function, i.e. the measure of how responsive the simulated event runoff is to a change in rainfall. We analytically derive the functions for two well-known runoff generation model types: the Probability Distributed Model (PDM), where the Pareto distribution is used to describe the distribution of soil-moisture storage capacity, and the Soil Conservation Service – Curve Number (SCS-CN) model. These functions are explored to examine the propagation of rainfall errors through the two models. It is shown that the two models are characterized by very different elasticity functions, which results in diverging propagation features of the rainfall errors. For the PDM case, increasing the precipitation depth, or reducing the storage capacity, results in the elasticity growing from 1 to a peak whose value and location depend on the model parameters, and then asymptotically decreases again to 1. For the SCS-CN model, increasing the precipitation depth, or decreasing the maximum potential retention, makes the elasticity decrease from infinity to 1. The capability of the elasticity functions to describe the propagation of rainfall errors through the models is illustrated by using the data from the Vaia 2018 flood in the Eastern Italian Alps. Owing to the very dry initial conditions and the extremely high precipitation depth associated with the event, application to this case allows exploring the whole range of hydrological conditions characterizing the elasticity functions. It is shown that the analytical functions closely resemble the results obtained by forcing the models with the actual distribution of rainfall errors, thus paving the way for the practical application of this approach, such as in hydrological model calibration, and the use of multi-model ensemble for flood forecasting.