## 1. Introduction

[2] Simulating the dynamics of real-world groundwater systems requires the use of accurate numerical models. Even though these models may be based on the underlying physical processes of a system, intrinsic model parameters must be identified in order for the model response to be sufficiently accurate. A multitude of algorithms exist whose purpose is to adjust the parameter values of a model such that the model output matches its associated measured values as closely as possible. This type of problem is commonly referred to as the inverse problem. *Yeh* [1986], *Sun* [1994], and *Oliver and Chen* [2011] provide comprehensive reviews on the inverse problem as it applies to groundwater hydrology. Currently, the most popular methods are based on the output error criterion, where a starting estimate of the parameter vector is updated such that the norm of the difference between observed states and their corresponding model-predicted values is minimized. *Cooley* [1985] provides a comparison of four different nonlinear regression methods of parameter identification; the most efficient methods were found to be the Marquardt [*Marquardt*, 1963] and quasilinearization [*Yeh and Tauxe*, 1971a, 1971b] methods. Some current, popular software include PEST [*Doherty*, 2002] and UCODE [*Poeter et al.*, 2005]. These software applications employ algorithms that are largely based on the Gauss-Marquardt-Levenberg methods [*Levenberg*, 1944; *Marquardt*, 1963].

[3] Methods based on the output error criterion require a significant number of model runs in order to evaluate parameter updates from one iteration to the next. Therefore, the computational demand associated with a forward run of the numerical model has a large impact on the overall CPU requirement of the parameter estimation algorithm. Reducing the computational demand associated with the numerical model can significantly reduce the computational demand of the parameter estimation algorithm. A method known as proper orthogonal decomposition (POD) has become very popular recently for achieving significant model-order reduction [*Cazemier et al.*, 1998; *Willcox and Peraire*, 2002; *Vermeulen et al.*, 2004; *McPhee and Yeh*, 2008; *Siade et al.*, 2010]. This method of model reduction essentially projects the original full-model solution from the space of functions where it resides into a subspace (e.g., via Galerkin projection) generated from only a few model runs, such that the number of equations that need to be solved is greatly reduced. The resulting loss in accuracy remains small or controllable. This is achieved by developing a specific set of basis functions such that time-varying linear combinations of these basis functions can adequately approximate the original full-model solution for all times and with any forcing. Since the number of POD basis functions is much smaller than the number of computational nodes, the magnitude of model reduction can be very significant. *Siade et al.* [2010] reduced a basin-scale groundwater model, using POD, resulting in a reduced model that ran approximately 1000 times faster than its corresponding original full model with a negligible loss in accuracy.

[4] The accuracy of the reduced model via POD is dependent on the quality of the basis functions that span the reduced model subspace. *Siade et al.* [2010] provides a methodology for evaluating a good set of basis functions when considering changing values of forcing, e.g., groundwater extraction/injection rates. However, when one is changing the values of the parameters, such as hydraulic conductivity, these basis functions begin to lose accuracy. This loss of accuracy is due to the nonlinear relationship between model parameters and model states. This presents a problem for reduced-order parameter estimation, which requires iterative updates of the parameter values. *Park et al.* [1998] and *Vermeulen et al.* [2005] present methodologies for dealing with this issue. In both articles, the authors use the method of snapshots to develop the basis functions that span the reduced model space. Snapshots are samples of the original full-model state variable at specified instants in time. A snapshot set is collected for each well, individually, given a constant unit forcing and a specific set of parameter values. Snapshot sets are collected over a specific range of parameter values that adequately cap-ture parameter variability around their current estimates. Throughout the parameter estimation algorithm, the current estimate of the parameters may “move” outside this range, requiring the re-evaluation of the reduced model using a new range of parameter values. However, many snapshot sets are needed in order to adequately capture all possible combinations of parameter ranges each time the reduced model is evaluated. For example, a snapshot set for each extraction/injection well is needed when one of the parameters is at the upper end of its range and the others are at their lower ends. Additional snapshot sets are required for each of these combinations at each extraction/injection well. In particular, in the case of one well and two parameters, four snapshot sets are needed; in the case of two wells and three parameters, 16 snapshot sets are needed, etc. Additionally, snapshot sets may be required for parameter values within their ranges rather than at the upper and lower bounds only. Each snapshot set requires an original full-model run. Therefore, for highly parameterized systems with a large number of extraction wells, the computational gain of the model reduction is overcome by the computational burden of developing snapshot sets.

[5] In this paper, we propose a methodology that no longer requires the development of a “moving” parameter range when developing snapshots. The reduced model must be developed once only; the resulting basis functions are accurate for the entire parameter estimation procedure. The parameters under investigation are zonal hydraulic conductivity values. The parameter estimation procedure employed is based on quasilinearization and quadratic programming. *Bellman and Kalaba* [1965] originally developed quasilinearization for parameter identification in a system of nonlinear ordinary differential equations. It involves solving a series of linearized initial value problems such that the sequence of solutions converges to the solution of the original nonlinear problem. *Yeh and Tauxe* [1971a, 1971b] applied quasilinearization to parameter estimation in groundwater modeling while *Park et al.* [1998] applied it to flow reactor modeling. *Yeh* [1975] combined quasilinearization and quadratic programming for parameter estimation in a partial differential equation. The algorithm essentially consisted of solving a series of sequential quadratic programming (QP) problems. However, in practice this algorithm suffers from the fact that each QP problem is so large that the computational burden of solving it is near the same magnitude as that of current Gauss-Newton type approaches. In this study, we show that POD model reduction can dramatically reduce the computational requirement of the individual QP problems, resulting in a drastic increase in overall inversion efficiency. The method requires the evaluation of one snapshot set for each hydraulic conductivity zone in order to build the reduced model. Snapshots are collected from the linearized full model (where changes in conductivity become the forcing term) rather than the original full model (where groundwater extraction/injection is the forcing term). The proposed method can handle highly parameterized systems with a large number of extraction/injection wells and still achieve significant reductions in CPU time.