Inverse Problems in the Earth Sciences

Most mathematical models of fluid flow are of the "forward" type; that is, the relevant properties of the aquifer or reservoir are assumed known, as well as the initial and boundary conditions. A model then predicts the resultant flow. This is typically the approach taken in sensitivity studies, which are quite useful, and can show what the most important features or processes are likely to be for a site. However, in the field, we generally do not know the full spatial distribution of important properties such as permeability and saturations. Instead, we may have sparse and noisy measurements of pressure, flow rates and concentration at a set of wells, and an incomplete knowledge of the subsurface geology, obtained from cores and seismic soundings. From this information, we need to resolve the spatial distribution of properties such as permeability and saturation and concentration to adequately assess the aquifer or reservoir. Interpretations of this kind typically constitute what are called inverse problems. Finding solutions of inverse problems is a particularly difficult task because of the non-uniqueness difficulties that arise. Non-uniqeness means in effect that the true solution cannot be selected from among a large set of possible solutions without further constraints imposed. This undesirable behavior is due to noise in the measurements, and insufficient number of measurements. Many areas of geophysics, including atmospheric science, oceanography, geomagnetism and remote electromagnetic sensing, as well as hydrology and reservoir engineering, have developed methods for solving inverse problems. All the methods attempt to remove non-uniqueness by using a priori information as constraints. These constraints generally involve imposing smoothness on the unknown solution or its derivatives, or positivity, or maximum entropy or some other very general property.