Inverse modeling of aquifers usually involves identification of effective parameters such as transmissivities over a finite number of subregions, or zones. Theoretical restrictions on the maximum size of a zone for which such effective transmissivities can be properly defined and the desire to obtain a good resolution of the spatial variability of transmissivity may suggest that the aquifer be divided into numerous small zones. Considerations of parameter identifiability, on the other hand, may require that the number of unknown transmissivities be limited. To satisfy both requirements, an inverse approach has been developed in which the number of zones can be as large as deemed necessary on the basis of hydrogeological considerations. However, instead of trying to estimate a similar number of transmissivities, a smaller number of surrogate parameters, which are defined as linear combinations of the original log transmissivities, is estimated. The optimum number and definition of the surrogate parameters are determined through a singular value decomposition of a matrix arising from the linearization of the inverse problem. A "resolution matrix" and an "information density matrix" can also be obtained from the singular value decomposition. The resolution matrix is indicative of parameter identifiability and is valuable in deciding whether specific log-transmissivity zones should be lumped with their neighbors or left intact. The information density matrix shows how well the model can reproduce each measured hydraulic head value and may be used to determine the relative worth of each datum point for parameter estimation. This, in turn, may suggest discontinuing the collection of certain data and/or starting to collect data at other points in the aquifer. The methodology is illustrated by using data from the Avra Valley aquifer of southern Arizona.

In this work we develop and analyze algorithms motivated by the parameter estimation problem corresponding to a multilayer aquifer/interbed groundwater flow model. The parameter estimation problem is formulated as an optimization problem, then addressed with algorithms based on adjoint equations, quasi-Newton schemes, and multilevel optimization. In addition to the parameter estimation problem, we consider properties of the parameter to solution map. This includes invertibility (known as identifiability) and differentiability properties of the map. For differentiability, we expand existing results on Fréchet sensitivity analysis to convection diffusion equations and groundwater flow equations. This is achieved by proving that the Fréchet  derivative of the solution operator is Hilbert-Schmidt, under smoothness assumptions for the parameter space. In addition, we approximate this operator by time dependent matrices, where their singular values and singular vectors converge to their infinite dimension peers. This decomposition proves to be very useful as it provides vital information as to which perturbations in the distributed parameters lead to the most significant changes in the solutions, as well as applications to uncertainty quantification. Numerical results complement our theoretical findings. / Ph. D.

It has been said that watersheds and aquifers ignore political boundaries. This phenomenon is often the reason for extensive regulation of surface -water and ground -water resources which are shared by two or more political entities. Regulation is often the result of years of litigation over who really owns the water, how much is owned, and how much is available for future use. Groundwater models are sometimes used as quantitative tools which aid in the decision making process regarding appropriation and regulation of these scarce, shared, water resources. The following few paragraphs detail the occurrences in the Lower Rio Grande Basin which led to the current ground -water modeling effort. New Mexico, Texas and Mexico have wrestled forever over the rights to the Lower Rio Grande and the aquifers of the Rio Grande Basin (Figure 1). As early as 1867, due to a flood event on the Rio Grande, Texas and Mexico were disputing the new border created by the migrating Rio Grande. During the 1890's, the users upstream from the Mesilla and El Paso Valleys were diverting and applying so much of the Rio Grande that the Mesilla and El Paso valley farmers litigated in order to apportion and guarantee the supply. In the recent past, disputes over who may use the ground -water resources of the region and the effect of surface- water uses on aquifer water levels resulted in litigation between El Paso, Texas, and New Mexico.