Describing the catchment-averaged precipitation as a stochastic process improves parameter and input estimation

Rainfall input uncertainty is one of the major concerns in hydrological modeling. Unfortunately, during inference, input errors are usually neglected, which can lead to biased parameters and implausible predictions. Rainfall multipliers can reduce this problem but still fail when the observed input (precipitation) has a different temporal pattern from the true one or if the true nonzero input is not detected. In this study, we propose an improved input error model which is able to overcome these challenges and to assess and reduce input uncertainty. We formulate the average precipitation over the watershed as a stochastic input process (SIP) and, together with a model of the hydrosystem, include it in the likelihood function. During statistical inference, we use "noisy'' input (rainfall) and output (runoff) data to learn about the "true'' rainfall, model parameters, and runoff. We test the methodology with the rainfall-discharge dynamics of a small urban catchment. To assess its advantages, we compare SIP with simpler methods of describing uncertainty within statistical inference: (i) standard least squares (LS), (ii) bias description (BD), and (iii) rainfall multipliers (RM). We also compare two scenarios: accurate versus inaccurate forcing data. Results show that when inferring the input with SIP and using inaccurate forcing data, the whole-catchment precipitation can still be realistically estimated and thus physical parameters can be "protected'' from the corrupting impact of input errors. While correcting the output rather than the input, BD inferred similarly unbiased parameters. This is not the case with LS and RM. During validation, SIP also delivers realistic uncertainty intervals for both rainfall and runoff. Thus, the technique presented is a significant step toward better quantifying input uncertainty in hydrological inference. As a next step, SIP will have to be combined with a technique addressing model structure uncertainty.