A Hybrid Empirical–Bayesian Artificial Neural Network Model of Salinity in the San Francisco Bay–Delta Estuary

With the growing maturity of artificial neural network (ANN) applications in the environmental literature, it has become clear that the “black-box” model relationship between inputs and outputs embodied in ANNs may not adequately represent the physical system being modeled. Thus, a trained and validated ANN model may fit the aggregate response to multiple inputs well, even though the sensitivity to a specific input is not physically meaningful, or in some cases, not physically plausible. The condition of representing inputs and outputs in a manner that is physically plausible, given an a priori understanding of a system, is termed “structural” validity, and is needed for developing robust environmental models. This paper reports the refinement of a published empirical model of salinity in the San Francisco Bay-Delta estuary by integration with a Bayesian ANN model and incorporation of additional inputs. Performance goals established for the resulting hybrid model are based on the quality of fit to observed data (replicative and predictive validation) as well as sensitivity when compared with a priori knowledge of system behavior (structural validation). ANN model parameters were constrained to provide plausible sensitivity to coastal water level, a key input introduced in the hybrid formulation. In addition to representing observed data better than the underlying empirical model while meeting structural validation goals, the hybrid model allows for characterization of prediction uncertainty. This work demonstrates a real-world application of a general approach--integration of a preexisting model with a Bayesian ANN constrained by knowledge of system behavior--that has broad application for environmental modeling.