An approach to handling non-Gaussianity of parameters and state variables in ensemble Kalman filtering

The ensemble Kalman filter (EnKF) is a commonly used real-time data assimilation algorithm in various disciplines. Here, the EnKF is applied, in a hydrogeological context, to condition log-conductivity realizations on log-conductivity and transient piezometric head data. In this case, the state vector is made up of log-conductivities and piezometric heads over a discretized aquifer domain, the forecast model is a groundwater flow numerical model, and the transient piezometric head data are sequentially assimilated to update the state vector. It is well known that all Kalman filters perform optimally for linear forecast models and a multiGaussian-distributed state vector. Of the different Kalman filters, the EnKF provides a robust solution to address non–linearities; however, it does not handle well non-Gaussian state-vector distributions. In the standard EnKF, as time passes and more state observations are assimilated, the distributions become closer to Gaussian, even if the initial ones are clearly non-Gaussian. A new method is proposed that transforms the original state vector into a new vector that is univariate Gaussian at all times. Back transforming the vector after the filtering ensures that the initial non-Gaussian univariate distributions of the state-vector components are preserved throughout. The proposed method is based in normal-score transforming each variable for all locations and all time steps. This new method, termed the normal-score ensemble Kalman filter (NS-EnKF), is demonstrated in a synthetic bimodal aquifer resembling a fluvial deposit, and it is compared to the standard EnKF. The proposed method performs better than the standard EnKF in all aspects analyzed (log-conductivity characterization and flow and transport predictions).