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1] In this paper, a Bayesian learning approach is introduced to train a multilayer feedforward network for daily river flow and reservoir inflow simulation in a cold region river basin in Canada. In Bayesian approach, uncertainty about the relationship between inputs and outputs is initially taken care of by an assumed prior distribution of parameters (weights and biases). This prior distribution is updated to posterior distribution using a likelihood function following Bayes' theorem while data are observed. This posterior distribution is called the objective function of a network in the Bayesian learning approach. The objective function is maximized using a suitable optimization technique. Once the network is trained, the predictive distribution of the network outputs is obtained by integrating over the posterior distribution of weights. In this study, Gaussian prior distribution and a Gaussian noise model are used in defining posterior distribution. The network has been optimized using a scaled conjugate gradient technique. Posterior distribution of weights is approximated to Gaussian during prediction. Prediction performance of the Bayesian neural network (BNN) is compared with the results obtained from a standard artificial neural network (ANN) model and a widely used conceptual rainfall-runoff model, namely, HBV-96. The BNN model outperformed the conceptual model and slightly outperformed the standard ANN model in simulating mean, peak, and low river flows and reservoir inflows. The significant contribution of the Bayesian method over the conventional ANN approach, among others, is the uncertainty estimation of the outputs in the form of confidence intervals which are particularly needed in practical water resources applications. Prediction confidence limits (or intervals) indicate the extent to which one can rely on predictions for decision making. It is shown that the BNN can provide reliable streamflow and reservoir inflow forecasts without a loss in model prediction accuracy as compared to standard ANN and conceptual model HBV. Another significant advantage of BNN approach is that the overfitting and underfitting problems are automatically taken care of by the Bayesian learning algorithm, which conversely remain serious problems with conventional ANN learning algorithm.Citation: Khan, M. S., and P. Coulibaly (2006), Bayesian neural network for rainfall-runoff modeling, Water Resour. Res., 42, W07409,

Abstract:Three downscaling models, namely the Statistical Down-Scaling Model (SDSM), the Long Ashton Research Station Weather Generator (LARS-WG) model and an artificial neural network (ANN) model, have been compared in terms of various uncertainty attributes exhibited in their downscaled results of daily precipitation, daily maximum and minimum temperature. The uncertainty attributes are described by the model errors and the 95% confidence intervals in the estimates of means and variances of downscaled data. The significance of those errors has been examined by suitable statistical tests at the 95% confidence level. The 95% confidence intervals in the estimates of means and variances of downscaled data have been estimated using the bootstrapping method and compared with the observed data. The study has been carried out using 40 years of observed and downscaled daily precipitation data and daily maximum and minimum temperature data, starting from 1961 to 2000. In all the downscaling experiments, the simulated predictors of the Canadian Global Climate Model (CGCM1) have been used. The uncertainty assessment results indicate that, in daily precipitation downscaling, the LARS-WG model errors are significant at the 95% confidence level only in a very few months, the SDSM errors are significant in some months, and the ANN model errors are significant in almost all months of the year. In downscaling daily maximum and minimum temperature, the performance of all three models is similar in terms of model errors evaluation at the 95% confidence level. But, according to the evaluation of variability and uncertainty in the estimates of means and variances of downscaled precipitation and temperature, the performances of the LARS-WG model and the SDSM are almost similar, whereas the ANN model performance is found to be poor in that consideration. Further assessment of those models, in terms of skewness and average dry-spell length comparison between observed and downscaled daily precipitation, indicates that the downscaled daily precipitation skewness and average dry-spell lengths of the LARS-WG model and the SDSM are closer to the observed data, whereas the ANN model downscaled precipitation underestimated those statistics in all months.

A major challenge in assessing the hydrologic effect of climate change remains the estimation of uncertainties associated with different sources, such as the global climate models, emission scenarios, downscaling methods, and hydrologic models. There is a demand for an efficient and easy-to-use rainfall-runoff modeling tool that can capture the different sources of uncertainties to generate future flow simulations that can be used for decision making. To manage the large range of uncertainties in the climate change impact study on water resources, a neural network-based rainfall-runoff model-namely, Bayesian neural network (BNN)-is proposed. The BNN model is used with Canadian Centre for Climate Modelling and Analysis Coupled GCM, versions 1 and 2 (CGCM1 and CGCM2, respectively) with two emission scenarios, Intergovernmental Panel on Climate Change (IPCC) IS92a and Special Report on Emissions Scenarios (SRES) B2. One widely used statistical downscaling model (SDSM) is used in the analysis. The study is undertaken to simulate daily river flow and daily reservoir inflow in the Serpent and the Chute-du-Diable watersheds, respectively, in northeastern Canada. It is found that the uncertainty bands of the mean ensemble flow (i.e., flow simulated using the mean of the ensemble members of downscaled meteorological variables) is able to mostly encompass all other flows simulated with various individual downscaled meteorological ensemble members whichever CGCM or emission scenario is used. In addition, the uncertainty bands are also able to typically encompass most of the flows simulated with another rainfall-runoff model, namely, Hydrologiska Byrå ns Vattenbalansavdelning (HBV). The study results suggest that the BNN model could be used as an effective hydrological modeling tool in assessing the hydrologic effect of climate change with uncertainty estimates in the form of confidence intervals. It could be a good alternative method where resources are not available to implement the general multimodel ensembles approach. The BNN approach makes the climate change impact study on water resources with uncertainty estimate relatively simple, cost effective, and time efficient.