A continuous‐time inverse operator for groundwater and contaminant transport modeling: Deterministic case

The value of time‐dependent data in the calibration of numerical models of groundwater flow has been recently highlighted in the literature. In this paper the background for a new development in the inverse problem is placed, and a new operator for inversion of the spatially discretized diffusion equation is introduced. The motivation for the development of an inverse problem solution procedure which uses information in transient data is illustrated with a focused review of recent literature. Such an approach which implicitly incorporates the continuous‐time dependency of the flow model is then presented. This approach is based upon the new concept of inverting the model after the (for example, finite element) spatial discretization but before the (typically) finite difference temporal discretization of the model, not afterward, as has been done in the past. The Laplace transform is used to replace discretization in time of the system of ordinary differential equations to discretization in the Laplace variable of the algebraic system in Laplace space. The existence of the deterministic solution in the Laplace transform space does not require knowledge of the eigenvalues of the diffusion matrix but only the specification of the Laplace variables near zero. A simple numerical example from the literature is used to generate a set of three simplified test cases, upon which the proposed method is tested. The implications of the theoretical investigation and the results of the numerical experiments are discussed with a view toward prospects of practical implementation.