Parameter estimation of hydrologic models using a likelihood function for censored and binary observations

Observations of a hydrologic system response are needed to accurately model system behaviour. Nevertheless, often very few monitoring stations are operated because collecting such reference data adequately and accurately is laborious and costly. It has been recently suggested to use observations not only from dedicated flow meters but also from simpler sensors, such as level or event detectors, which are available more frequently but only provide censored information. Binary observations can be considered as extreme censoring. It is still unclear, however, how to use censored observations most effectively to learn about model parameters. To this end, we suggest a formal likelihood function that incorporates censored observations, while accounting for model structure deficits and uncertainty in input data. Using this likelihood function, the parameter inference is performed within the Bayesian framework. We demonstrate the implementation of our methodology on a case study of an urban catchment, where we estimate the parameters of a hydrodynamic rainfall-runoff model from binary observations of combined sewer overflows. Our results show, first, that censored observations make it possible to learn about model parameters, with an average decrease of 45% in parameter standard deviation from prior to posterior. Second, the inference substantially improves model predictions, providing higher Nash-Sutcliffe efficiency. Third, the gain in information largely depends on the experimental design, i.e. sensor placement. Given the advent of Internet of Things, we foresee that the plethora of censored data promised to be available can be used for parameter estimation within a formal Bayesian framework.