## 1. Introduction

[2] Multiscale heterogeneity of geologic media is a rule rather than the exception. The knowledge of detailed spatial distributions of hydraulic properties is imperative to predict water and solute movement in the subsurface at high resolution [e.g., *Yeh*, 1992, 1998]. Traditional aquifer tests (e.g., pumping and slug tests) have been widely employed for estimating hydraulic properties of the subsurface for the last few decades. Besides their costly installation and invasive natures, *Beckie and Harvey* [2002] reported that slug tests can yield dubious estimates of the storage coefficient of aquifers. Validity of classical analysis for aquifer tests was also questioned by *Wu et al.* [2005]. They reported that the storage coefficient, *S*, value obtained from the traditional Theis analysis primarily represents a weighted average of *S* values over the region between the pumping and observation wells. In contrast to the *S* estimate, the transmissivity, *T*, estimate is a weighted average of all *T* values in the entire domain with relatively high weights near the pumping well and the observation well. In concordance with the finding by *Oliver* [1993], *Wu et al.* [2005] concluded that the *T* estimate can be influenced by any large-sized or strong anomaly within the cone of depression. Thus interpretation of the meaning of *T* estimates can be highly uncertain. As a result, previous assessments of transmissivity distributions of aquifers may be subject to serious doubt.

[3] Hydraulic tomography [*Gottlieb and Dietrich*, 1995; *Yeh and Liu*, 2000; *Liu et al.*, 2002; *Bohling et al.*, 2002; *Zhu and Yeh*, 2005], based on the computerized axial tomography (CAT) scan concept of medical sciences, is potentially a viable technology for characterizing detailed spatial distributions of the hydraulic properties. Hydraulic tomography, in a simple term, is a series of cross-well interference tests. More precisely, hydraulic tomography involves stressing an aquifer by pumping water from or injecting water into a well, and monitoring the aquifer's response at other wells. A set of stress and response data 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 induced by the stresses at different locations and thus 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 estimates approach reality.

[4] Interpretation of hydraulic tomography surveys however is a numerical challenge. The large number of well hydrographs generated during tomography often leads to information overload, substantial computational burdens, and numerical instabilities [*Hughson and Yeh*, 2000]. To overcome these difficulties, *Yeh and Liu* [2000] developed a sequential successive linear estimator (SSLE) approach. This approach eases the computational burdens by sequentially including information obtained from different pumping tests; it resolves the nonuniqueness issue by providing an unbiased mean of an abstracted stochastic parameter rather than the actual parameter. That is, it conceptualizes hydraulic parameter fields as spatial stochastic processes and seeks their conditional effective parameter distributions. The conditional effective parameters are conditioned on the data obtained from and governing physical principles of hydraulic tomography, as well as our prior knowledge of the geologic structure, and directly measured parameter values (such as from slug tests, or core samples). The SSLE approach in essence is the Bayesian formalism. Sand box experiments by *Liu et al.* [2002] and W. A. Illman et al. (Steady-state hydraulic tomography in a laboratory aquifer with deterministic heterogeneity: Multiscale validation of hydraulic conductivity tomograms, submitted to *Water Resources Research*, 2005) proved that the combination of hydraulic tomography and SSLE is a viable, 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.

[5] Several researchers have investigated THT. *Bohling et al.* [2002] exploited the steady shape flow regime of transient flow data to interpret tomographic surveys. 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. As in X-ray tomography, their approach relies on the assumption that the pressure pulse travels along a straight line or a curved 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 interpret pneumatic cross-borehole tests in fractured rocks.

[6] Recently, *Zhu and Yeh* [2005] extended the SSLE approach to THT to estimate both spatially varying hydraulic conductivity and specific storage fields in 3-D random media. They demonstrated that it is possible to obtain detailed hydraulic conductivity and specific storage fields of aquifers using few wells with THT surveys. However, as the size of the field site to be characterized increases and the demands of resolution increases, the computational burden increases significantly. A computationally efficient algorithm therefore must be developed for speedy analysis of the THT surveys. For basin-scale naturally recurrent tomographic surveys (such as river stage tomography, *Yeh et al.* [2004]), development of such a technology is imperative.

[7] In this study, inspired by the moment generating function approach by *Harvey and Gorelick* [1995], we develop the temporal moment generation equation for impulse pumping tests, similar to the recent work by *Li et al.* [2005]. While *Li et al.* [2005] focus on applying temporal moments to a single impulse pumping test, we apply the temporal moments to transient hydraulic tomography. Specifically, we incorporate the zeroth and first temporal moments of well hydrographs into the SSLE inverse approach [*Yeh and Liu*, 2000] for THT. In addition, we implement a loop iteration scheme [*Zhu and Yeh*, 2005] to avoid the effects of sequential addition of moment information. By directly comparing the estimation using the temporal moments with that using transient heads, we thereafter investigate the temporal moment approach in terms of computational efficiency and accuracy of estimation. Last, limitations of the moment approach are discussed.