Estimating the Full Anisotropy of the Covariance Function in Geostatistical Inversion using the Pilot-Point Ensemble Kalman Filter

The covariance function is a powerful tool for characterizing random processes and is fully parameterized by (anisotropic) correlation lengths and angles of rotation. While correlation lengths have been successfully estimated, the periodicity of rotation has posed challenges in determining unique covariance function parameterizations. Good prior knowledge of the covariance function has been shown to greatly improve the results of parameter inference methods. However, knowledge of the full anisotropy of the covariance function is difficult to obtain. Therefore, we propose an extension to the pilot point ensemble Kalman filter (PPEnKF) that is capable of estimating 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 PPEnKF. Based on the estimates of the covariance function, we further modify the filter by generating conditional field realizations in each assimilation step, increasing the inherent ensemble variance and preventing filter inbreeding. We demonstrate the methodology by estimating the covariance function of a field of hydraulic conductivity in a synthetic study of a 2D groundwater model. The full anisotropy of the covariance function and the hydraulic conductivity at pilot points are estimated 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 knowledge successfully estimated the correct parameterization of the covariance function, as well as the corresponding parameter values at the pilot points. The resulting parameter fields enabled accurate predictions of groundwater head levels during a verification period, with normalized root mean square errors reduced by 77 - 97 % compared to ensembles excluded from the parameter update. The methodology presented in this study mitigates the importance of informative prior knowledge of the covariance function in parameter inference methods, showcasing the effectiveness of random processes in achieving robust parameter field estimations, especially in highly anisotropic settings.