A modified Levenberg–Marquardt algorithm for quasi-linear geostatistical inversing

The Quasi-Linear Geostatistical Approach is a method of inverse modeling to identify parameter fields, such as the hydraulic conductivity in heterogeneous aquifers, given observations of related quantities like hydraulic heads or arrival times of tracers. Derived in the Bayesian framework, it allows to rigorously quantify the uncertainty of the identified parameter field. Since inverse modeling in subsurface flow is in general a non-linear problem, the Quasi-Linear Geostatistical Approach employs an algorithm for non-linear optimization. Up to presence, this algorithm has been similar to the Gauss–Newton algorithm for least-squares fitting and fails in cases of strong non-linearity. In this study, we present and discuss a new modified Levenberg–Marquardt algorithm for the Quasi-Linear Geostatistical Approach. Compared to the original method, the new algorithm offers increased stability and is more robust, allowing for stronger non-linearity and higher variability of the parameter field to be identified. We demonstrate its efficiency and improved convergence compared to the original version in several test cases. The new algorithm is designed for the general case of an uncertain mean of the parameter field, which includes the cases of completely known and entirely unknown mean as special cases.