A representer-based inverse method for groundwater flow and transport applications

Groundwater inverse problems are concerned with the estimation of uncertain model parameters, such as hydraulic conductivity, from field or laboratory measurements. In practice, model and measurement errors compromise the ability of inverse procedures to provide accurate results. It is important to account for such errors in order to determine the proper weight to give to each source of information. Probabilistic descriptions of model and measurement errors can be incorporated into classical variational inverse procedures, but the computational demands are excessive if the model errors vary over time. An alternative approach based on representer expansions is able to efficiently accommodate time-dependent errors for large problems. In the representer approach, unknown variables are expanded in finite series which depend on unknown functions called representers. Each representer quantifies the influence of a given measurement on the estimate of a particular variable. This procedure replaces the original inverse problem by an equivalent problem where the number of independent unknowns is proportional to the number of measurements. The representer approach is especially advantageous in groundwater problems, where the total number of measurements is often small. This approach is illustrated with a synthetic flow and transport example that includes time-dependent model errors. The representer algorithm is able to provide good estimates of a spatially variable hydraulic conductivity field and good predictions of solute concentration. Its computational demands are reasonable, and it is relatively easy to implement. The example reveals that it is beneficial to account for model errors even when they are difficult to estimate.