The sustainable management of groundwater demands a faithful characterization of the subsurface. This, in turn, requires information which is generally not readily available. To bridge the gap between data need and availability, numerical models are often used to synthesize plausible scenarios not only from direct information but also additional, indirect data. Unfortunately, the resulting system characterizations will rarely be unique. This poses a challenge for practical parameter inference: Computational limitations often force modelers to resort to methods based on questionable assumptions of Gaussianity, which do not reproduce important facets of ambiguity such as Pareto fronts or multi-modality. In search of a remedy, an alternative could be found in Stein Variational Gradient Descent, a recent development in the field of statistics. This ensemble-based method iteratively transforms a set of arbitrary particles into samples of a potentially non-Gaussian posterior, provided the latter is sufficiently smooth. A prerequisite for this method is knowledge of the Jacobian, which is usually exceptionally expensive to evaluate. To address this issue, we propose an ensemble-based, localized approximation of the Jacobian. We demonstrate the performance of the resulting algorithm in two cases: a simple, bimodal synthetic scenario, and a complex numerical model based on a real-world, pre-alpine catchment. Promising results in both cases - even when the ensemble size is smaller than the number of parameters - suggest that Stein Variational Gradient Descent can be a valuable addition to hydrogeological parameter inference.

Uncertainty estimation is an important part of practical hydrogeology. With most of the subsurface unobservable, attempts at system characterization will invariably be incomplete. Uncertainty estimation, then, must quantify the influence of unknown parameters, forcings, and structural deficiencies. In this endeavour, numerical modeling frameworks support an unparalleled degree of subsurface complexity and its associated uncertainty. When boundary uncertainty is concerned, however, the numerical framework can be restrictive. The interdependence of grid discretization and the enclosing boundaries make exploring uncertainties in their extent or nature difficult. The Analytic Element Method (AEM) may be an interesting complement, as it is computationally efficient, economic with its parameter count, and does not require enclosure through finite boundaries. These properties make AEM well-suited for comprehensive uncertainty estimation, particularly in data-scarce settings or exploratory studies. In this study, we explore the use of AEM for flow field uncertainty estimation, with a particular focus on boundary uncertainty. To induce versatile, uncertain regional flow more easily, we propose a new element based on conformal mapping. We then include this element in a simple Python-based AEM toolbox and benchmark it against MODFLOW. Coupling AEM with a Markov Chain Monte Carlo (MCMC) routine using adaptive proposals, we explore its use in a synthetic case study. We find that AEM permits efficient uncertainty estimation for groundwater flow fields, and its analytical nature readily permits continuing analyses which can support Lagrangian transport modelling or the placement of numerical model boundaries.