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Institut für Wasser- und Umweltsystemmodellierung - IWS



"Best Unbiased Ensemble Linearizaion and Quasi-Linear Ensemble Cokriging"

Linearized representations of the stochastic groundwater and transport equations have been heavily used in hydrogeology, e.g., for geostatistical inverse modeling and generating conditional realizations. In such applications, linearizations are commonly defined via Jacobians (numerical sensitivity matrices). This study will show that Jacobian-based linearizations are biased with non-minimal error variance in the ensemble sense. A Best Unbiased Ensemble Linearization technique will be derived from the principles of unbiasedness and minimum error variance. The resulting paradigm prefers empirical cross-covariances from Monte-Carlo analyses over those from linearized error propagation, and points towards methods like the Ensemble Kalman Filter (EnFK). Unlike conditional simulation in geostatistical applications, EnKFs condition transient state variables rather than geostatistical parameter fields. Recently, modifications towards geostatistical applications have been tested and used. This study completes the transformation of EnKFs to geostatistical conditioning tools, based on Best Unbiased Ensemble Linearization. To distinguish it from the original EnKF, the new method is called Ensemble Cokriging (EC). EC combines the computational efficiency of linearized methods with the robustness of EnKFs, while drawing on the advantages of conditional simulation over conditional estimation. Sequential updating, acceptance/rejection sampling, successive linearization and a Levenberg-Marquardt formalism are added to increase robustness and accuracy. The new context of Best Unbiased Ensemble Linearization provides an additional theoretical foundation to EnKF-like methods (such as EC). As proof of concept, a large-scale numerical test case with 100 synthetic sets of flow and tracer data is conducted and analyzed.