عنوان مقاله [English]
نویسندگان [English]چکیده [English]
Estimation of uncertainty is an important basis of decision-making in modern water resources management. There is no absolutely accurate model that could predict every component of hydrologic cycle, due to the high complexity in the nature. Because of this relative failure, hydrological modeling is always subject to uncertainty. The subjects that are well-known produce certainty; while, those are not known to us make uncertainty. Probabilistic framework can be applied for representing uncertainty in a mathematical formal fashion. There are two classes of approaches for coping with uncertainty. The first class, considers uncertainty as lumped, such as Generalized Likelihood Uncertainty Estimation (GLUE) method. The second class considers uncertainty in different parts of a model such as inputs, states, parameters and outputs. Data assimilation can be classified as the second approach, allowing the estimation of different uncertainty sources. Kalman Filter (KF), a state-estimation method, is one of the efficient data assimilation techniques that provide a probabilistic framework of assessing uncertainty in separate model elements. The original KF was designed for linear models, whereas most hydrological models are nonlinear. Extended Kalman Filter (EKF) was developed for nonlinear models by using Taylor series of a model. It was not efficient because of approximation and heavy computation cost. Ensemble Kalman Filter (ENKF) is an alternative method for nonlinear models by generating ensemble of states. Joint Ensemble Kalman Filter (JEnKF), is a modified version of ENKF for simultaneous parameter and state estimation that we applied in this study.
In the current study, JEnKF has been implemented for estimating of uncertainty in a conceptual rainfall-runoff model of Behesht-Abad catchment in Chaharmahal and Bakhtiari Province, during 2001-2006. The catchment area was 3860 km2 on the upstream of Karun River. One advantage of this implementation, in addition to its probabilistic nature for considering uncertainty, is its ability of simultaneous state parameter estimation without need of an extra global optimization algorithm for parameter calibration. The JEnKF has been applied on a modified version of HYMOD rainfall-runoff model. HYMOD has five reservoirs including: soil moisture reservoir; three linear reservoirs in series representing the fast runoff component; and one slow reservoir. Due to the mountainous landscape of the catchment, we added an extra reservoir for considering snow. The dimension of parameter space increased from five to seven due to this modification. It has five parameters representing the maximum storage capacity of watershed, the spatial variability of soil moisture capacity, the partitioning between fast reservoirs and slow reservoir, as well as the residence time of the fast and slow reservoirs. The two parameters of the snow reservoir are threshold temperature and the degree-day factor, which defines the melted water per day per Celsius degree above threshold temperature. The key point of ENKF is the application of Monte Carlo to generate ensemble of state variable of the model and sequentially propagate ensembles in time as a filter, by updating the ensembles whenever a new observation is provided. JEnKF has two major steps: first, forecast step that the parameters are kept invariant, while the states are integrated in time through the dynamical model; Second, analysis step in which once a new observation is available, both states and parameters are updated according to Kalman Filter equation.
Parameter uncertainty has been indicated in probability density function forms. After sampling the prior distribution of parameters from a uniform distribution, the posterior marginal probability density was obtained by applying the filter. The dispersion of the distribution around Maximum a Priori (MAP) of the parameter was relatively narrow which showed the parameter identifiability achieved well-defined parameters. The efficiency of the method in prediction of runoff in the catchment was also investigated by comparing its results with SCE global optimization algorithm by RMSE statistic in four discharge classes. It showed state updating decreased the errors in different discharge classes, especially base flows. Comparing JEnKF with SCE showed 18 to 63 percent error reduction in calibration period, while 7 to 73 percent in validation period. Finally, predictive uncertainty was presented for both three-year calibration and two-year validation periods, which confirmed the importance of considering other sources of uncertainty rather than just parameter uncertainty. The predictive uncertainty caused by parameter uncertainty covered 32% of the discharge, while total uncertainty, the mixture of all uncertainty sources, covered 95% of the measured discharge.