The direct inverse problem in aquifers

Solution of the inverse problem for aquifers requires identification of a distributed parameter system. In addition to ill‐posedness, the inverse problem may be underdetermined if recharge is treated as a spatially and temporally variable distribution parameter. Galerkin's method is applied to a weak formulation of the direct inverse problem, leading to a (possibly nonlinear) weighted least squares problem. Consideration of generic errors suggests that if the estimate for some derivative of head contains a significant error on one element for all time, then significant estimation errors are possible for all nodes and for all reservoir parameters. In contrast, if some derivative of head is small in some spatial region, for all time, causing the matrix to be inverted in the least squares algorithm to be nearly singular, then significant estimation errors can occur in the corresponding reservoir parameters in that spatial region; although it is still possible that other reservoir parameters in that region, as well as all reservoir parameters elsewhere, may be satisfactorily estimated.