## 1. Introduction

[2] Sensitivity analysis of large-scale systems governed by differential equations has continued to be of importance in groundwater modeling and parameter estimation [*Carter et al.*, 1974; *Sun and Yeh*, 1985; *Yeh*, 1986; *Carrera and Neuman*, 1986; *Yeh and Sun*, 1990; *Sun and Yeh*, 1990a, 1990b]. Applications of such analysis cover a wide spectrum, including optimization, optimal control, model reparameterization, uncertainty analysis, and experimental design. However, the cost of computing sensitivity coefficients often poses a challenge. This cost may be central to determining the choice of optimization method to use in parameter estimation. When the computational overhead of sensitivity calculation becomes prohibitively high, methods such as the conjugate gradient and quasi-Newton, which avoid this computation, are often used in place of gradient-based methods such as Gauss-Newton and Levenberg-Marquardt.

[3] *Jacquard and Jain* [1965] presented a procedure for numerically computing the sensitivity coefficients for history matching and applied the procedure to estimate permeability in a two-dimensional reservoir from pressure data. Subsequently, *Carter et al.* [1974] presented a derivation of the method to compute the sensitivity coefficients for two-dimensional single-phase flow problems. *Chen et al.* [1974] and *Chavent et al.* [1975] independently proposed the optimal control method to calculate the gradient of the objective function with respect to model parameters for single-phase flow. *Wasserman et al.* [1975] extended the optimal control theory to automatic history matching in a multiphase reservoir but only computed the adjoint variables for the overall pressure equation and used an objective function based on only the pressure mismatch term. Later, *Carrera and Neuman* [1986] and *Sun and Yeh* [1990a] used the optimal control theory to solve the parameter identification problem for groundwater flow. A detailed review of the parameter identification procedures in groundwater hydrology was done by *Yeh* [1986]. *Wu et al.* [1999] later derived the adjoint equations for multiphase flow in a hydrocarbon reservoir, but the computational cost is still very high when the number of data is large.

[4] Efforts have been vested in finding cheaper methods of computing the sensitivity matrix without compromising the accuracy of the solution. One method, the forward sensitivity analysis (also known as the gradient simulator method), is very efficient when the model space is small [*Yeh*, 1986; *Tang et al.*, 1989; *Landa*, 1997]. This is because this method requires the solution of a linear system with multiple right-hand-side vectors. The number of right-hand-side vectors is exactly equal to the number of model parameters. Moreover, in the forward sensitivity method, the sensitivities of all grid block variables are computed. This is inefficient because only the sensitivities of variables at measurement locations are required. For high-dimensional problems, this method becomes prohibitively expensive. Model space reduction via reparameterization [*Oldenburg et al.*, 1993; *Reynolds et al.*, 1996; *Lu and Horne*, 2000; *Sahni and Horne*, 2006; *Sarma et al.*, 2008] is often coupled with the forward sensitivity method to stabilize the algorithm and speed up the computation of sensitivity coefficients.

[5] Another method, the adjoint approach [*Shah et al.*, 1978; *Anterion et al.*, 1989; *Plessix*, 2006; *Michalak and Kitanidis*, 2004; *Li and Petzold*, 2004], is commonly used to compute the sensitivity coefficients and the gradient of the objective function. The adjoint method of sensitivity computation is particularly useful when the number of data is relatively small. This method is also based on solving a linear system with multiple right-hand-side vectors. However, the number of right-hand-side vectors in this case is equal to the number of data for which sensitivities are to be calculated. The number is therefore independent of the number of parameters. This approach is preferred to the forward sensitivity method when the number of data to match is significantly smaller than the number of parameters. However, there are several instances in which the data space and the model space are both of high dimensions. In such instances, the cost of computing sensitivity coefficients can be very large.

[6] In this paper, we propose the application of a linear transformation of the data space to reduce the associated cost of adjoint state sensitivity computation. First, we review wavelet analysis, wavelet reparameterization of the data space, inverse modeling, and the conventional adjoint approach to calculating sensitivities in steady state linear systems. We subsequently derive a wavelet approach to adjoint sensitivity computation. The approach uses the data compression capability of the wavelet transform to reduce the size of the adjoint equations for sensitivity computations. Finally, we verify the approach using numerical examples applied to spatially sampled hydraulic head data. The examples involve finding the maximum likelihood estimates of reservoir parameters by matching a reduced set of wavelets of the measured data.