Real-time groundwater flow modeling with the Ensemble Kalman Filter: Joint estimation of states and parameters and the filter inbreeding problem

Real-time groundwater flow modeling with filter methods is interesting for dynamical groundwater flow systems, for which measurement data in real-time are available. The Ensemble Kalman Filter (EnKF) approach is used here to update states together with parameters by adopting an augmented state vector approach. The performance of EnKF is investigated in a synthetic study with a two-dimensional transient groundwater flow model where (1) only the recharge rate is spatiotemporally variable, (2) only transmissivity is spatially variable with sigma(2)(lnT) = 1.0 or (3) with sigma(2)(lnT) = 2.7, and (4) both recharge rate and transmissivity are uncertain (a combination of (1) and (3)). The performance of EnKF for simultaneous state and parameter estimation in saturated groundwater flow problems is investigated in dependence of the number of stochastic realizations, the updating frequency and updating intensity of log-transmissivity, the amount of measurements in space and time, and the method (iterative versus noniterative EnKF), among others. Satisfactory results were also obtained if both transmissivity and recharge rate were uncertain. However, it was found that filter inbreeding is much more severe if hydraulic heads and transmissivities are jointly updated than if only hydraulic heads are updated. The filter inbreeding problem was investigated in more detail and could be strongly reduced with help of a damping parameter, which limits the intensity of the perturbation of the log-transmissivity field. An additional reduction of filter inbreeding could be achieved by combining two measures: (1) inflating the elements of the predicted state covariance matrix on the basis of a comparison between the model uncertainty and the observed errors at the measurement points and (2) starting the flow simulations with a very large number of realizations and then sampling the desired number of realizations after one simulation time step by minimizing the differences between the local cpdfs (and bivariate cpdfs) of hydraulic head for the large ensemble and the corresponding cpdfs for the reduced ensemble. The two measures, which cause very limited CPU costs, allowed making 100 stochastic realizations for the reproduction of the states as efficient as 200-500 untreated stochastic realizations.