K P Sudheer

The quantification of uncertainty in the simulations from complex physically based distributed hydrologic models is important for developing reliable applications. The generalized likelihood uncertainty estimation method (GLUE) is one of the most commonly used methods in the field of hydrology. The GLUE helps reduce the parametric uncertainty by deriving the probability distribution function of parameters, and help analyze the uncertainty in model output. In the GLUE, the uncertainty of model output is analyzed through Monte Carlo simulations, which require large number of model runs. This induces high computational demand for the GLUE to characterize multi-dimensional parameter space, especially in the case of complex hydrologic models with large number of parameters. While there are a lot of variants of GLUE that derive the probability distribution of parameters, none of them have addressed the computational requirement in the analysis. A method to reduce such computational requirement for GLUE is proposed in this study. It is envisaged that conditional sampling, while generating ensembles for the GLUE, can help reduce the number of model simulations. The mutual relationship between the parameters was used for conditional sampling in this study. The method is illustrated using a case study of Soil and Water Assessment Tool (SWAT) model on a watershed in the USA. The number of simulations required for the uncertainty analysis was reduced by 90 % in the proposed method compared to existing methods. The proposed method also resulted in an uncertainty reduction in terms of reduced average band width and high containing ratio.

The application of artificial neural network (ANN) has gained significant interest while modeling various hydrologic processes. The main reason is the ANN models have produced promising results without the detailed information of watershed characteristics as required in physics based models. Still, the uncertainty in ANN models is a major issue that cannot be ignored. There could be different forms to represent model uncertainty, in which quantification of prediction interval for the model output has been mostly reported. In this paper, three different methods [i.e. Bootstrap method, Bayesian Approach and Prediction Interval (PI) method] were employed for quantifying the prediction interval in ANN models. The modeling procedure presented in this paper, is illustrated through river flow forecasting using the data collected from Kolar basin, India. The prediction interval was quantified using the measures such as percentage of coverage and average width. The comparison between these methods indicated that PI method has resulted in relatively less prediction and parameter uncertainty, besides the improved model performance. In addition, the PI method produced accurate prediction of hydrograph peak, which is a general concern in ANN models.

The physically based distributed hydrological models are ideal for hydrological simulations; however most of such models do not use the basic equations pertaining to mass, energy and momentum conservation, to represent the physics of the process. This is plausibly due to the lack of complete understanding of the hydrological process. The soil and water assessment tool (SWAT) is one such widely accepted semi-distributed, conceptual hydrological model used for water resources planning. However, the over-parameterization, difficulty in its calibration process and the uncertainty associated with predictions make its applications skeptical. This study considers assessing the predictive uncertainty associated with distributed hydrological models. The existing methods for uncertainty estimation demand high computational time and therefore make them challenging to apply on complex hydrological models. The proposed approach employs the concepts of generalized likelihood uncertainty estimation (GLUE) in an iterative procedure by starting with an assumed prior probability distribution of parameters, and by using mutual information (MI) index for sampling the behavioral parameter set. The distributions are conditioned on the observed information through successive cycles of simulations. During each cycle of simulation, MI is used in conjunction with Markov Chain Monte Carlo procedure to sample the parameter sets so as to increase the number of behavioral sets, which in turn helps reduce the number of cycles/simulations for the analysis. The method is demonstrated through a case study of SWAT model in Illinois River basin in the USA. A comparison of the proposed method with GLUE indicates that the computational requirement of uncertainty analysis is considerably reduced in the proposed approach. It is also noted that the model prediction band, derived using the proposed method, is more effective compared to that derived using the other methods considered in this study.

Application of artificial neural network (ANN) models has been reported to solve variety of water resources and environmental related problems including prediction, forecasting and classification, over the last two decades. Though numerous research studies have witnessed the improved estimate of ANN models, the practical applications are sometimes limited. The black box nature of ANN models and their parameters hardly convey the physical meaning of catchment characteristics, which result in lack of transparency. In addition, it is perceived that the point prediction provided by ANN models does not explain any information about the prediction uncertainty, which reduce the reliability. Thus, there is an increasing consensus among researchers for developing methods to quantify the uncertainty of ANN models, and a comprehensive evaluation of uncertainty methods applied in ANN models is an emerging field that calls for further improvements. In this paper, methods used for quantifying the prediction uncertainty of ANN based hydrologic models are reviewed based on the research articles published from the year 2002 to 2015, which focused on modeling streamflow forecast/prediction. While the flood forecasting along with uncertainty quantification has been frequently reported in applications other than ANN in the literature, the uncertainty quantification in ANN model is a recent progress in the field, emerged from the year 2002. Based on the review, it is found that methods for best way of incorporating various aspects of uncertainty in ANN modeling require further investigation. Though model inputs, parameters and structure uncertainty are mainly considered as the source of uncertainty, information of their mutual interaction is still lacking while estimating the total prediction uncertainty. The network topology including number of layers, nodes, activation function and training algorithm has often been optimized for the model accuracy, however not in terms of model uncertainty. Finally, the effective use of various uncertainty evaluation indices should be encouraged for the meaningful quantification of uncertainty. This review article also discusses the effectiveness and drawbacks of each method and suggests recommendations for further improvement.

Accurate and reliable forecasting of reservoir inflows is crucial for efficient reservoir operation to decide the quantity of the water to be released for various purposes. In this paper, an artificial neural network (ANN) model has been developed to forecast the weekly reservoir inflows along with its uncertainty, which was quantified through accounting the model’s input and parameter uncertainties. Further, to investigate how the effect of uncertainty is translated in the process of decision making, an integrated simulation–optimization framework that consists of (i) inflow forecasting model; (ii) reservoir operation model; and (iii) crop simulation model was developed to assess the impacts of uncertainty in forecasted inflow on the irrigation scheduling and total crop yield from the irrigation system. A genetic algorithm was used to derive the optimal reservoir releases for irrigation and the area of irrigation. The proposed modeling framework has been demonstrated through a case example, Chittar river basin, India. The upper, lower, and mean of forecasted inflow from the ANN model were used to arrive at the prediction interval of the depth of irrigation, total crop yield, and area of irrigation. From the analysis, the ANN model forecast error of ± 69% to the mean inflow was estimated. However, the error to mean value of simulation for total irrigation, total yield, and area of irrigation was ± 13.3%, ± 6.5%, and ± 4.6%, respectively. The optimizer mainly contributed to the reduction in the errors (i.e., maximizing the total production with the optimal water releases from the reservoir irrespective of inflow to the reservoir). The results from this study suggested that the information on the uncertainty quantification helps in better understanding the reliability of the systems and for effective decision making.

The research towards improving the prediction and forecasting of artificial neural network (ANN) based models has gained significant interest while solving various engineering problems. Consequently, different approaches for the development of ANN models have been proposed. However, the point estimation of ANN forecasts seldom explains the actual mechanism that brings the relationship among modeled variables. This raises the question on the model output while making decisions due to the inherent variability or uncertainty associated. The standard procedure though available for the quantification of uncertainty, their applications in ANN model are still limited. In this chapter, commonly employed uncertainty methods such as bootstrap and Bayesian are applied in ANN and demonstrated through a case example of flood forecasting models. It also discusses the merits and limitations of bootstrap ANN (BTANN) and Bayesian ANN (BANN) models in terms of convergence of parameter and quality of prediction interval evaluated using uncertainty indices.