Estimating the full anisotropy of the covariance function in geostatistical inversion using the pilot-point ensemble Kalman filter

In geostatistical inversion, good prior knowledge about the covariance function is important in estimating hydraulic conductivity from hydraulic-head observations, but may be hampered by poor knowledge about anisotropy. In this study we propose an extension of the pilot-point ensemble Kalman filter (PP-EnKF) that can infer the full anisotropy of the covariance function based on attainable, initially random knowledge. We address the periodicity of rotation by incorporating the unique elements of the covariance transformation matrix into the set of parameters to be estimated. The filter is further modified by generating conditional realizations in each assimilation step, increasing the inherent variance of the ensemble and counteracting filter inbreeding. We demonstrate the methodology in a synthetic study of a 2-D groundwater-flow model where we estimate the full anisotropy of the covariance function and the hydraulic conductivity at pilot points via the assimilation of hydraulic-head data. The success of this method depends more on the configuration of pilot points than on the quality of prior knowledge, as ensembles initialized with faulty random priors successfully estimated the correct parameters of the covariance function, as well as the log-hydraulic conductivity values at the pilot points. The resulting parameter fields enabled accurate predictions of hydraulic heads during a verification period, with normalized root mean square errors reduced by up to 66% compared to ensembles with isotropic covariance functions. The methodology presented in this study mitigates the importance of informative prior knowledge of the covariance function in geostatistical parameter-inference methods, especially in highly anisotropic settings.