## 1. Introduction

[2] Detailed models of subsurface flow properties are necessary for accurate predictions of solute and multi-phase flow and transport. Such models are increasing in number due to new technologies, such as tracer and pumping tests in networks of multilevel samplers [*Freyberg*, 1986; *Butler et al.*, 1999]. A related method, crosswell pressure testing or hydraulic tomography [*Hsieh et al.*, 1985; *Paillet*, 1993; *Cook*, 1995; *Masumoto et al.*, 1995; *Karasaki et al.*, 2000; *Yeh and Liu*, 2000; *Vesselinov et al.*, 2001; *Vasco and Karasaki*, 2001] has evolved over the past two decades and promises to provide higher resolution estimates of flow properties between wells. Additional methods, such as permanent downhole pressure [*Gringarten et al.*, 2003] and surface tilt monitoring [*Fabian and Kuempel*, 2003; *Vasco*, 2004] can also provide a wealth of observations for the characterization of subsurface flow properties.

[3] There are several important issues associated with the acquisition and interpretation of a large suite of pressure or head observations. For example, in order for a cross-well pressure test to be completed in a reasonable amount of time, it is not generally feasible to attain steady state conditions. Rather, a truncated test must be run [*Karasaki et al.*, 2000] for a specified period of time for each source-receiver pair. Thus, one may not invoke steady state conditions in the interpretation of most cross-well hydraulic tests [*Bohling et al.*, 2002]. There is also the burden associated with archiving and analyzing the recorded data for a large number of transient tests or for long observation intervals. Large amounts of data must be manipulated and stored, adding to the expense and effort required to use the observations for characterization. The data themselves may be quite noisy, particularly for early observation times when the signal is rather small. In addition, long period background variations may be superimposed upon the signal due to atmospheric pressure and temperature fluctuations [*Butler et al.*, 1999].

[4] The inversion of transient head data can be a computational challenge, requiring hours or days of computation, even on a sizable parallel computer [*Vesselinov et al.*, 2001]. This is particularly true when steady state conditions cannot be invoked and the full transient nature of the test must be considered. All inverse methods require repeated solution of the forward problem in order to estimate model parameters which are compatible with a set of observations. This is true for both stochastic methods, such as simulated annealing [*Mauldon et al.*, 1993; *Nakao et al.*, 2001] and Markov Chain Monte-Carlo, as well as for deterministic methods such as the Gauss-Newton technique [*Sun*, 1994]. Even the more efficient methods, such as the adjoint state approach [*Sun*, 1994], require repeated solution of the full transient flow problem.

[5] There have been some investigations aimed at developing more efficient inversion techniques which are applicable to the fully transient problem. *Vasco et al.* [2000] introduced an approach based upon a high-frequency asymptotic solution to the governing equation for time-varying head. Equations governing the arrival time and amplitude of a transient head variation allow for a form of diffusive traveltime tomography. The approach has been applied to field data by *Vasco et al.* [2000], *Brauchler et al.* [2003], *Vasco and Finsterle* [2004], and *He et al.* [2006]. To date, these applications have utilized head and pressure data associated with constant rate tests in which pumping was started and continued for some time. For such a test one must use the time derivative of the head variation to define an arrival time. Computation of the derivative can be difficult in the presence of early time variations due to regional background and well effects. An alternative approach, which is more general and does not require the use of the time derivative, is to deconvolve the source-time function associated with a truncated pumping test. *Bohling et al.* [2002] proposed a steady shape analysis for the efficient inversion of cross-well head and pressure data. The approach requires the attainment of steady shape, unsteady state conditions (steady shape), in which the drawdown continues to change while the hydraulic gradient does not. The methodology is motivated by an analytic solution for radial flow by *Butler* [1988] which indicates that the hydraulic gradients are equivalent for steady shape and steady state conditions. Thus, steady state methods may be used to interpret steady shape observations. For heterogeneous media the onset of steady shape conditions may be delayed by the presence of low-conductivity regions which fail to equilibrate with the surrounding medium [*Bohling et al.*, 2002].

[6] In this paper we consider an approach based upon a low-frequency asymptotic solution to the equation governing transient head variations. In many respects our approach is complementary to the high-frequency asymptotic methodology of *Vasco et al.* [2000]. The advantages of this approach is that by working in the frequency domain we can attain a significant reduction in the number of observations that must be considered. In addition, the forward and inverse modeling only require the solution of two problems which are equivalent to the steady state pressure equation. Thus, the level of computation is equivalent to that required for inverting steady state head observations. The method makes use of the entire transient head variation and is applicable to a truncated pumping tests. Finally, we present a semi-analytic expression relating perturbations in frequency domain data to perturbations in hydraulic conductivity. The expression forms the basis for a very efficient inversion algorithm. We demonstrate the utility of this approach by an application to a set of hydraulic tomographic data from the Raymond field site in California.