A BRIEF REVIEW OF THE PROPERTIES OF FORWARD AND INVERSE PROBLEMS IN GROUNDWATER HYDROLOGY

Groundwater flow and transport models are based on the solutions of equations of mathematical physics with appropriate initial and boundary conditions. Physically based models apply mass conservation equations, closed with constitutive equations derived from phenomenological laws. This procedure results in a set of partial differential equations that relate the physical quantities that describe the state of the system with mathematical parameters and forcing terms. The problem of solving these equations under appropriate initial and boundary conditions to determine the state variables is known as the forward problem (FP) and is the core of forecasting models. The FP can be solved if physical parameters and initial/boundary conditions are known; since available measurements are usually relevant to scales different from those involved in the FP, it is often necessary to estimate physical parameters with model calibration, i.e. by solving an inverse problem (IP) that provides estimates of model parameters that give rise to a matching in some sense between simulated and observed state. The IP is often ill posed, thus its solution cannot be guaranteed, for which uniqueness is secured only in special cases and stability cannot be achieved, because the solutions to IP depend on the gradient of state variables and the differential operators are not continuous. These concepts and the results for discrete models are even much more complex. This is partly due to the great variety of permitted choices: numerical method, discretisation scales, availability/use of measured quantities; solution strategy (direct, indirect, etc.) and approach, etc. Existence of a solution to the discrete IP is usually admitted on the basis of physical principles. Uniqueness and stability however are often nowhere to be found and their absence typically confounds the groundwater IP in practice. Solutions of the groundwater IP are ubiquitous and widely reported, and reviews on solution techniques appear with regularity; however less study has been made on the fundamental mathematical characteristics of existence and uniqueness in the IP. The goal of this communication is to clarify some of these concepts and to provide a comprehensive conceptual framework for characterization of uniqueness and stability to modellers.