## 1. Introduction

[2] Detailed spatial distributions of hydraulic parameters are imperative to improve our ability to predict water and solute movement in the subsurface [e.g., *Yeh*, 1992, 1998]. Traditional aquifer tests like pumping tests and slug tests only yield hydraulic parameters integrated over a large volume of geologic media [e.g., *Butler and Liu*, 1993; *Beckie and Harvey*, 2002]. On the other hand, *Wu et al.* [2005] reported that the classical analysis for aquifer tests yields spurious transmissivity estimates and storage coefficient estimates that reflect local geology. For characterizing detailed spatial distributions of hydraulic parameters, a new method, hydraulic tomography [*Gottlieb and Dietrich*, 1995; *Renshaw*, 1996; *Yeh and Liu*, 2000; *Liu et al.*, 2002; *McDermott et al.*, 2003], which evolved from the CAT (computerized axial tomography) scan concept of medical sciences and geophysics, appears to be a viable technology.

[3] Hydraulic tomography is, in the most simplified terms, a series of cross-well interference tests. In other words, an aquifer is stressed by pumping water from or injecting water into a well, and monitoring the aquifer's response at other wells. A set of stress/response yields an independent set of equations. Sequentially switching the pumping or injection location, without installing additional wells, results in a large number of aquifer responses caused by stresses at different locations and, in turn, a large number of independent sets of equations. This large number of sets of equations makes the inverse problem (i.e., using aquifer stress and response relation to estimate the spatial distribution of hydraulic parameters) better posed, and the subsequent estimate approaches reality.

[4] Interpreting data from hydraulic tomography presents a challenge, however. The abundance of data generated during tomography can lead to information overload, and cause substantial computational burdens and numerical instabilities [*Yeh*, 1986; *Hughson and Yeh*, 2000]. Moreover, the interpretation can be nonunique. *Yeh and Liu* [2000] developed a sequential successive linear estimator (SSLE) to overcome these difficulties. The SSLE approach eases the computational burdens by sequentially including information obtained from different pumping tests; it resolves the nonuniqueness issue by providing the best unbiased conditional mean estimate. That is, it conceptualizes hydraulic parameter fields as spatial stochastic processes and seeks their mean distributions conditioned on the information obtained from hydraulic tomography, as well as directly measured parameter values (such as from slug tests, or core samples). Using sand box experiments, *Liu et al.* [2002] demonstrated that the combination of hydraulic tomography and SSLE is a propitious, cost-effective technique for delineating heterogeneity using a limited number of invasive observations. The work by *Yeh and Liu* [2000], nonetheless, is limited to steady state flow conditions, which may occur only under special field conditions. Because of this restriction, their method ignores transient head data before flow reaches steady state conditions. Transient head data, although influenced by both hydraulic conductivity and specific storage, are less likely to be affected by uncertainty in boundary conditions. The development of a new estimation procedure thus becomes essential so that all data sets collected during hydraulic tomography surveys can be fully exploited.

[5] Few researchers have investigated transient hydraulic tomography. *Bohling et al.* [2002] exploited the steady shape flow regime of transient flow data to interpret tomographic surveys. Under steady shape conditions at late time of a pumping test before boundary effects take place, the hydraulic gradient changes little with time, a situation where sensitivity of head to the specific storage is small. As a consequence, the steady shape method is useful for estimating hydraulic conductivity but not specific storage.

[6] Their steady shape method relies on the classical least squares optimization method and the Levenberg-Marquardt algorithm [*Marquardt*, 1963] for controlling convergence issues [see *Nowak and Cirpka*, 2004]. This optimization method is known to suffer from nonuniqueness of the solutions if the inverse problem is ill posed and regularization [*Tikhonov and Arsenin*, 1977] or prior covariance of parameters [*Nowak and Cirpka*, 2004] is not used. The least squares approach is also computationally inefficient if every element in the solution domain (in particular, three-dimensional aquifers with multiple, randomly distributed parameters) is to be estimated. This inefficiency augments if the sensitivity matrices required by the optimization are not evaluated using an efficient algorithm, such as the adjoint state approach.

[7] These shortcomings may be the reasons that test cases of *Bohling et al.* [2002] were restricted to unrealistic, perfectly stratified aquifers, where the heterogeneity has no angular variations, and specific storage is constant and known a priori. The assumption of a spatially constant and known specific storage value for the entire aquifer makes the inverse problem almost the same as the steady hydraulic tomography as explored by *Yeh and Liu* [2000]. Perhaps inversion of the transient tomography by *Bohling et al.* [2002] is less affected by unknown in boundary conditions. Nonetheless, for perfectly horizontal layered aquifers, many traditional hydraulic test methods, without resorting to hydraulic tomography, can easily estimate hydraulic properties of each layer using just one borehole.

[8] Similar to *Vasco et al.* [2000], *Brauchler et al.* [2003] developed a method that uses the travel time of a pneumatic pressure pulse to estimate air diffusivity of fractured rocks. Similar to X-ray tomography, their approach relies on the assumption that the pressure pulse travels along a straight line or a curve path. Thus an analytical solution can be derived for the propagation of the pressure pulse between a source and a pressure sensor. Many pairs of sources and sensors yield a system of one-dimensional analytical equations. A least squares based inverse procedure developed for seismic tomography can then be applied to the system of equations to estimate the diffusivity distribution. The ray approach avoids complications involved in numerical formulation of the three-dimensional forward and inverse problems, but it ignores interaction between adjacent ray paths and possible boundary effects. Consequently, their method requires an extensive number of iterations and pairs of source/sensor data to achieve a comparable resolution to that achieved from inverting a three-dimensional model. *Vesselinov et al.* [2001] applied an optimization technique and geostatistics to pneumatic cross-borehole tests in fractured rocks. Because of the baseline of the pneumatic properties is unknown, it is difficult to assess the accuracy of their results.

[9] To our knowledge, few researchers have developed an inverse method for transient hydraulic tomography to estimate both hydraulic conductivity and specific storage of aquifers. For general groundwater inverse problems other than hydraulic tomography, *Sun and Yeh* [1992] assumed a specific storage field that was homogeneous and known a priori. They then developed a stochastic inverse method to estimate the spatial distribution of transmissivity using only transient head information. For transient hydraulic tomography, *Vasco et al.* [2000] and *Brauchler et al.* [2003] estimated diffusivity, the ratio of hydraulic conductivity to specific storage, without any attempt to separate the two parameters.

[10] In this paper, we extended the SSLE developed by *Yeh and Liu* [2000] to transient hydraulic tomography for estimating randomly distributed hydraulic conductivity and specific storage in 3-D aquifers. This paper begins with the derivation of the SSLE for use with transient hydraulic heads. We introduce a loop iteration scheme to improve the accuracy of sequential usage of head data. We then verify our new approach by applying it to a synthetic one-dimensional heterogeneous aquifer. During this one-dimensional test, temporal variation of cross correlation between transient heads and parameters, as well as temporal correlation of transient heads, is investigated. Results of this investigation lead to a better understanding of effects of conditioning using head measurements on estimates of hydraulic conductivity and specific storage, and an effective sampling strategy, as opposed to utilizing an entire drawdown time history, for efficient inversion of the transient hydraulic tomography data. Finally, the new SSLE is applied to a hypothetical three-dimensional, heterogeneous aquifer to demonstrate the robustness of our new approach.