Signature-Domain Calibration of Hydrological Models Using Approximate Bayesian Computation: Theory and Comparison to Existing Applications

This study considers Bayesian calibration of hydrological models using streamflow signatures and its implementation using Approximate Bayesian Computation (ABC). If the modeling objective is to predict streamflow time series and associated uncertainty, a probabilistic model of streamflow must be specified but the inference equations must be developed in the signature domain. However, even starting from simple probabilistic models of streamflow time series, working in the signature domain makes the likelihood function difficult or impractical to evaluate (in particular, as it is unavailable in closed form). This challenge can be tackled using ABC, a general class of numerical algorithms for sampling from conditional distributions, such as (but not limited to) Bayesian posteriors given any calibration data. Using ABC does not avoid the requirement of Bayesian inference to specify a probability model of the data, but rather exchanges the requirement to evaluate the pdf of this model (needed to evaluate the likelihood function) by the requirement to sample model output realizations. For this reason ABC is attractive for inference in the signature domain. We clarify poorly understood aspects of ABC in the hydrological literature, including similarities and differences between ABC and GLUE, and comment on previous applications of ABC in hydrology. An error analysis of ABC approximation errors and their dependence on the tolerance is presented. An empirical case study is used to illustrate the impact of omitting the specification of a probabilistic model (and instead using a deterministic model within the ABC algorithm), and the impact of a coarse ABC tolerance.