Rainfall-Runoff Forecast and Model Parameter Estimation: a Dynamic Bayesian Networks Approach

The suggested climate-driven non-stationarities and intrinsic uncertainties of hydrological processes such as precipitation (P) and runoff (R), represent a fruitful context to develop new methods that may be able to detect parametric variations in time series and incorporate them into forecasts. In this research, we developed a method to forecast runoff from precipitation time series based on Dynamic Bayesian Networks (DBN). The purpose of the research was to determine an appropriate structure of the DBN and the optimal lengths of hydrological time series required to establish statistical parameters (i.e., first two moments) of P and optimal fits of forecasted R at daily and weekly intervals. A DBN can be briefly interpreted as a set of nodes (representing conditional probabilistic variables) connected by arrows that establish a causal, time-oriented, relationship among them. A DBN is defined by two components: a static network (structure) and a transition probability matrix between consecutive stages. Similarly to neural networks, DBN must be trained in order to learn about the subjacent process and make useful predictions. To determine the ability of the DBN to forecast R from P we initially generated long synthetic P series and run a deterministic model (HEC-HMS) to generate R. The DBN were then trained with different lengths of these synthetic series to forecast R (using smoothing and filtering methods). Two structures were considered: 1) DBN with P(t), P(t-1) and R(t-1) and 2) DBN with P(t), P(t-1), R(t-1) and ΔR=[R(t-1)-R(t-2)]. Both smoothing and filtering methods were appropriate to make predictions on a daily and weekly basis (filtration performing better). Setting the complexity (number of states of the random variables) in a DBN proves to be a critical issue, since an increase in the number of states, which implies larger training sets, does not always mean an improvement in the prediction. We found that acceptable results could be obtained from DBN models with a number of states between 20 and 40 and training sets between six and 10 years, achieving the best outcomes at 30 states per node and eight years of daily training series. We also found that the inclusion of ΔR in the structure noticeably improved the fit to extremely low and high R, improving the estimation of the transition matrix in the DBN and enhancing its prediction skills. Similarly, we found that the DBN can be used to identify the optimal parameters of the hydrological model that best fit the R series, which implies the possibility of improving the interpretation of hydrological models in terms of factors that are generally difficult to estimate like antecedent soil moisture conditions or the effect of land uses. We obtained also similar good fitting results when we trained the DBN with historical data from a well instrumented basin with more than 40 years of daily records. Further developments of DBN may address not only seasonal changes in time series but also nonstationarities in hydrological variables as expected with climate change predictions. Acknowledgements: authors thank the financial support of Universidad de Antioquia through its Sustainability Program 2011-2012.