An asymptotic expression for transient head variations, valid at low-frequencies, forms the basis for an efficient scheme for estimating hydraulic conductivity. The critical computational step is equivalent to solving the governing equation for steady state head. Thus, model parameter sensitivities, relating changes in head to changes in hydraulic conductivity, of the fully transient problem can be computed with the equivalent of four steady state head computations. A comparison of model parameter sensitivities computed using the low-frequency asymptotic approach and sensitivities computed using a purely numerical approach indicates good agreement. An inversion of synthetic hydraulic tomography data indicates that it is possible to estimate overall permeability variations using the technique. In an actual application to truncated crosswell pressure tests from the Raymond field site, we image two high permeability fracture zones, in agreement with a conceptual model of the region. The location of the two fracture zones correlates with the position of transmissive fractures, as measured by borehole conductivity logs.
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 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].
 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.
 There have been some investigations aimed at developing more efficient inversion techniques which are applicable to the fully transient problem. Vasco et al.  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. , Brauchler et al. , Vasco and Finsterle , and He et al. . 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.  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  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].
 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. . 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.
 In this section we discuss the low-frequency asymptotic solution for head variations. The detailed derivation, which is a variation of methods used in modeling wavefields [Friedlander and Keller, 1955; Virieux et al., 1994; Keller and Lewis, 1995; Mendez-Delgado et al., 1999], is presented in Appendix A. Procedures to account for a time-varying source and the windowing of the observations are also included in this section. Accounting for source and truncation effects is important in any practical algorithm and necessary when treating field observations. We conclude this section by presenting a semi-analytic expression for model parameter sensitivities. This expression is the cornerstone of an efficient inverse modeling algorithm.
2.1. An Asymptotic Solution
 Because we are interested in long-period or low-frequency variations in head, it is natural to work in the frequency or Fourier domain. That is, we transform the head variation, H(x, t), a function of space (x) and time (t), into a function which depends upon frequency ω, (x, ω). This transformation is accomplished using the integral transform [Arsac, 1966],
known as the Fourier transform. From this point on we shall consider only quantities in the frequency domain. For example, we shall assume that our data has been transformed into the frequency domain using a fast Fourier transform [Bracewell, 1978, p. 370]. In the frequency domain the pressure equation is given by
 We are interested in the interpretation of the low-frequency end of the spectrum. For this reason we consider the following power series representation of the head
which is dominated by the first few terms of the sum when ω is small. The functional form is motivated by the solution of the diffusion equation in a homogeneous medium [Virieux et al., 1994] for an impulsive source. In a uniform medium the solution takes the form of a modified Bessel function of zeroth order K0(αr) where α is a constant which depends on the properties of the medium and r is the distance from the source [Bowman, 1958; Gradshteyn and Ryzhik, 1980]. The power series solution follows from an approximation to the modified Bessel function for ω small [Gradshteyn and Ryzhik, 1980, p. 959].
 The series (2) is characterized by the function σ(x) and the set of functions Hn (x). The quantity σ(x) is often referred to as the phase. For high-frequency head variations the phase may be interpreted as a traveltime for the propagation of the head disturbance [Virieux et al., 1994; Vasco et al., 2000]. However, this interpretation is not possible for the low-frequency components of head. The infinite set of functions Hn(x) represent successive corrections to the amplitude of the propagating head variation. To determine σ(x) and Hn(x), n = 0, 1,… we substitute the series (2) into the governing equation for head (1). The result is an infinite expression containing terms of various powers of . The details of the procedure are described in Appendix A. For low-frequency head variations, ω ≪ 1, the 1/ term will dominate and
is a good approximation to the head variation in the frequency-domain.
 As shown in Appendix A, the zeroth-order amplitude, H0(x), in equation (3) satisfies a linear differential equation which is equivalent to the equation for steady state head
where u = ∇ ln K (x). Thus, at low-frequencies the head amplitude depends upon the spatial gradient of the hydraulic conductivity and not upon the specific storage S(x). The dependence of low-frequency head variations on K(x) is in accordance with earlier results, based upon an integral formulation [Vasco and Karasaki, 2001].
 In Appendix A we also derive an equation for the diffusive phase σ(x)
where we have defined
Thus, having solved equation (4) to determine H0(x), we can compute the coefficients Ω(x) and (x). Then we solve equation (5) for the phase using a numerical method, such as finite differences. Note that equations (4) and (5) are independent of frequency and only need to be solved once for a given well configuration.
 The functional form of the low-frequency approximation (3) is similar to a high-frequency solution for pressure [Vasco et al., 2000]. However, there are significant differences between these two approximations. As is evident from equation (5), at low frequencies the function σ(x), depends upon the hydraulic conductivity and the steady state head distribution and does not depend upon the specific storage. This means that σ(x) is sensitive to factors influencing the steady state head, such as the boundary conditions. In the limit of high frequencies one can interpret σ(x) as a diffusive ‘traveltime’ from a source to an observation point [Vasco et al., 2000]. Such an interpretation is not possible for low-frequency head variations. At low frequencies the function σ(x) is governed by a linear second order differential equation. At high frequencies σ(x) is determined by the eikonal equation, a non-linear first order differential equation [Vasco et al., 2000]. Similarly, the zeroth-order amplitude H0 (x) is governed by a second order differential equation which is identical to the equation for steady state head. As is clear from equation (4), the zeroth-order amplitude only depends upon the gradient of the hydraulic conductivity and not upon σ(x). In contrast, at high frequencies the zeroth-order amplitude is determined by a transport equation [Vasco et al., 2000] which contains the phase and phase gradient in its coefficients.
2.2. Validity of the Approximation and Errors
 The low-frequency asymptotic approach is an approximation which is valid within a specified frequency range. In particular, the changes in head should vary over a time-interval which is longer than a second. Fortunately, the appropriateness of the approximation is easily evaluated by examining higher-order terms in the series given by equation (2). A rough estimate of the error in the asymptotic approximation is obtained by considering the relative size of the powers of ω in the sum [equation (2)]. A more accurate estimate is obtained by formally calculating the next higher-order term in the series, e.g. computing terms of order ω using equation (A10) and comparing them with the zeroth-order approximation [equation (3)]. The availability of a simple error estimate is one advantage of the asymptotic methodology [Dingle, 1973]. Because the validity of the approach depends upon the frequency we shall postpone a specific discussion until we treat particular sets of head variations as in the application to data from the Raymond field site. By way of a general comment, we note that for tests lasting several minutes, hours, or days the next highest order term will be at least an order of magnitude smaller than the zeroth-order approximation.
2.3. Source and Windowing Effects
 The zeroth-order solution provided by expression (3) exhibits a square-root singularity when ω is 0. However, the particular frequency-dependence of the solution is a consequence of the idealized impulsive source, which is implicit in the derivation of (3), and the rate of decay of the solution. Typically, the source, which is given by the time variation of injection, is not simply an impulse. For example, in the case of the Raymond field experiment, discussed below, the source is better described by a box-car or rectangular time variation. Thus, we must convolve the impulse response with the source-time function in the time-domain. Convolution in the time-domain corresponds to multiplication of the Fourier transforms in the frequency-domain [Bracewell, 1978]. The Fourier transform of a box-car is the ‘sinc’ function, sin ω/ω. The frequency dependence due to a rectangular source-time variation is represented by the function Ψ (ω) which is given by
More general rate variations are obtained by multiplying by the Fourier transform of the source-time function in the frequency-domain.
 In addition to the source-time variation we must also account for windowing the observations. That is, we always consider a finite observation interval which is delineated by a starting time and an ending time. The time interval may be determined by cost considerations, as it is not feasible to run experiments for long periods of time. The time interval may also be constrained by experimental considerations, such as imposed when conducting crosswell pressure testing. For example, at the Raymond field site each test was truncated in order to keep the data collection time within reasonable limits. This truncation or windowing of the observations is modeled by multiplication by a box-car function in the time-domain. In the frequency-domain the corresponding operation is convolution by the Fourier transform of the box-car, the ‘sinc’ function. Therefore, the complete response due to the source variation and windowing in time, is given by
For general source-time variations it is not possible to represent this function analytically and a numerical method must be employed. However, this is easily accomplished through the use of the Fourier transform and a frequency-domain convolution algorithm [Bracewell, 1978].
2.4. Model Parameter Sensitivities
 Model parameter sensitivities relate a perturbation in hydraulic conductivity at a point y in the Earth, to a change in the observationed head at a point x [Sun, 1994]. Model parameters sensitivities are a vital component of any iterative inversion scheme as described below. As shown in Appendix B, a perturbation in the observed head at a point x, δ(x, ω), is related to perturbations in hydraulic conductivity, δK(y), via an integral over the model volume V,
In these equations H0(y, x) signifies the zeroth-order head amplitude at the point x due to a source at y, and H0(xs, y) signifies the head amplitude recorded at y due to a source at xs. The term σ(y, x) signifies the diffusive phase at the point x due to a source at y. The sensitivity, the partial derivative of the head at the observation point x due to a perturbation of the hydraulic conductivity at y, is given by the multiplier of δK(y) in the integrand of (10)
The sensitivities are computed by solving equations (4) and (5) twice, once for a source at the point xs, and again for a source at the observation point x. Note that this approach is equivalent to an adjoint state method for the steady state problem, and the two sets of solutions correspond to forward- and back-propagation, respectively. However, we only have to solve the steady state equations rather than the full equations for transient flow.
 The semi-analytic expression for model parameter sensitivities (13) involves some degree of approximation. For example, we are neglecting terms of order and greater in the asymptotic expansion (2). In order to test the validity of the low-frequency approximation we compare it with model parameter sensitivities computed using a purely numerical approach. In the comparison an injector introduces a pulse of fluid, as illustrated in Figure 1. A receiver, designated by a star in Figure 1, records the resulting head change at a nearby observation point. The aquifer consists of a single uniform layer with a permeability of 100 md. The resulting pulse at the observation point is shown in Figure 2, along with the real and imaginary components of its Fourier transform. Note that the frequencies considered are in the 0.1 milli-Hz range. Thus, the neglected terms, of order-ω and higher, in the asymptotic series [equation (2)] are 10,000 times smaller than the zeroth-order terms we retain.
 We compute the reference model parameter sensitivities using a straight-forward, but computationally intensive, numerical perturbation approach. A reservoir simulator is used to compute the head at the observation well due to the pulse. Then the permeability in a grid block is perturbed and the head is recomputed. The resulting perturbed and unperturbed head variations are Fourier transformed and differenced. The change in the Fourier transform at the observation well due to the perturbation is divided by the permeability deviation. The result is a numerical estimate of the model parameter sensitivity to a change in permeability for the block in question. The single layer model is sub-divided into a 21 (lateral) by 21 (vertical) grid of cells for the numerical modeling. Thus, a total of 442 transient head calculations were required to compute the sensitivities using the perturbation technique. The numerical computation took roughly 37 minutes of CPU time on a workstation. In Figure 3 we display the model parameter sensitivities for three frequencies (−0.07, 0.0, and 0.07 mHZ). Note that the sensitivities are similar in pattern but different in magnitude at the three frequencies. The sensitivities, which are largest near the injection and observation wells, agree with previous estimates [Oliver, 1993; Vasco et al., 2000].
 For comparison, we compute the sensitivities using the semi-analytic formula (13) where Π(xs, x, ω) and Σ(xs, x) are given by (11) and (12) respectively. Equation (11) implies that the peak sensitivities will be found at the injection locations where the gradient of the head variations is greatest. Note that, because we include a source term at the observation location, the gradient is also large at the observation well. The contributions from the scalar product of the amplitude gradients and the sum of the phase terms are shown in Figure 4. The total contribution, which is the product of the contributions in Figure 4, is shown in Figure 5. The entire procedure required less than a CPU second to complete. Note the agreement with the numerical sensitivities (Figure 3), with the peak sensitivities found near the injection and observation wells. Also, the sensitivity pattern is not a strong function of frequency. The sensitivity magnitude does depend on frequency, as in the numerical computations (Figure 3), and as suggested by the semi-analytic expression (13). Two steady state head computations were required to get the amplitude terms. An additional two solutions of the equation for phase are needed for the sensitivities, equivalent to two more solutions of the steady state equation for head.
 For single well experiments, such as a flow test, the computations are reduced by half. That is, only a single amplitude computation is necessary because the source and observation points coincide. Similarly, a single phase calculation is required for a single well test. The situation is illustrated in Figure 6, where we compare numeric and asymptotic sensitivities for a flow test. In this figure we only display the sensitivities corresponding to ω = 0. The sensitivities, which are given in Pascals per milli-darcy, are similar in magnitude and pattern, centered about the location of the borehole.
 We begin this section with a numerical test case which is based upon an actual field experiment conducted at the Raymond field site in California. This test provides an indication of how our approach will work, given the geometry of a cross well experiment. In particular, we can check our result against the actual model used to generate the synthetic head values. In the Raymond field test which follows, we have no reference with which to compare our results and we must resort to other means to validate the results.
3.1. Numerical Illustration
 The synthetic head variations were generated using a cross well geometry. The overall experimental setup is indicated in Figure 7 for the two cross well tests. In the first numerical test, 13 pumping intervals were located in the left-most well, indicated by the stars in Figure 7. Three widely spaced observation intervals are indicated by the unfilled circles in the well at the right. The permeability variation is indicated by the gray scale plot in the region between the wells. A high permeability channel is located in the upper-middle portion of the cross well region, indicated by the dark band in Figure 7. For the second test a total of 14 pumping tests were simulated in the right-most well with three observation points in the well on the left (the open circles in Figure 7). A truncated pumping test was modeled, in which fluid was injected for 600 seconds at each location at a constant rate. The head variation at each observation point is Fourier transformed. For this illustration we only consider the real component of the Fourier transform for ω = 0.
 We adopt a linearized least squares algorithm to invert the synthetic head values. That is, the inverse problem is linearized about some initial or background model, based upon a Taylors series expansion [Parker, 1994]. The non-linear nature of the inverse problem follows from the governing equation (1). The equation contains both the unknown field (x, ω) and the unknown hydraulic conductivity K(x) as products, providing a non-linear constraint equation, as noted by Vasco [1999, 2000]. The background model consists of a uniform half-space with a permeability of 100 md. A numerical simulator provides estimates of head variations at the observation points for each test. The estimated head variations at each receiver point are Fourier transformed and residuals, δ(x, ω), are computed by subtracting the estimated values from the reference values. In the linearized approximation, the head residuals are related to updates in the background permeability by equation (10). We first discretize the problem by subdividing the region between the wells into a 15 (lateral) by 60 (vertical) grid, a total of 900 cells. We solve for the perturbation of the average permeability within each grid block or cell. The constraint provided by an observation at point xi is given by a discrete version of equation (10)
where δKj represents the perturbation of the average permeability in grid block j. For a set of pumping tests recorded by an array of receivers we will have a set of M equations, where M is the total number of receivers. If we define the vector of residuals by δ, the matrix of coefficients by M, and the vector of permeabilities by δK the collection of equations may be written in matrix-vector form
 As indicated in the Methodology section, the coefficients contained in M require two solutions of the equation for static head for sources situated at both the pumping location and the receiver location. This can be accomplished for all source-receiver pairs by simply running a forward problem for each source point and for each receiver point and then combining the results as needed. Thus, the number of static problems will be Ns + Nr where Ns is the number of distinct source locations and Nr is the number of distinct receiver locations. The results are then combined, as indicated by equation (11). In the synthetic tests under consideration there are a total of 27 distinct source locations and 6 distinct receiver positions, a total of 33 simulations. We must solve the equation for static head 66 times in order to generate sensitivities for all 81 source-receiver pairs. A similar strategy may be followed with respect to the phase terms. However, for ω = 0 the phase factor drops out and we need only consider the amplitude term.
 Because the number of constraints provided by the 81 head residuals is less than the 900 unknown permeability values, the system of equations (15) is under-determined. Any solution of the equations will be unstable with respect to perturbations in the data vector δ or perturbations in the matrix coefficients M. Hence, small changes in δ and M may lead to large changes in our estimates of δK. We can stabilize the system and reduce the sensitivity to noise by adding regularization equations in addition to the data constraints [Parker, 1994]. In terms of a least-squares formulation, rather than minimize the sum of the squares of the residuals we minimize the penalized misfit function
where the vertical bars denote the L2 norm of a vector, δK0 is a prior model, D is a discrete approximation to the spatial gradient operator, and Wn and Wr are the weights given to the norm and roughness penalties, respectively. In minimizing (16) we seek a model that fits the data yet lies close to δK0 and is as smooth as possible. We solve the necessary equations for a minimum of P using the LSQR algorithm [Paige and Saunders, 1982], as described by Vasco et al. . The values of Wn and Wr were chosen by trial and error on the basis of several test inversions. The goal was to balance fitting the head values against obtaining a smooth model.
 The final result is shown in Figure 8 as a gray scale plot. The model contains the high permeability channel which is the main feature of the reference model. The anomaly appears somewhat smeared in the vertical direction, perhaps due to the roughness penalty. The final permeability model is compatible with the observations and results in predictions which agree with the synthetic head values (Figure 9). The discrepancies may be due to the presence of the roughness penalty. Because the inverse problem is ill-posed, the choice for Wn and Wr may depend on the level of noise in the reference values. If more noise is present, we cannot resolve details of the model nor can we recover the reference anomaly amplitudes as precisely. Note that we did not iterate the procedure, in order to account for the non-linearity of the inverse problem. Rather, we chose to simply take one step in the inversion of the head values. Additional iterations may further reduce the misfit by accounting for non-linearity.
3.2. Hydraulic Tomography at the Raymond Field Site, California
 The Raymond Field site was a facility for multi-disciplinary studies aimed at characterizing fractured rock [Karasaki et al., 2000]. It consisted of nine 75–90 m deep wells arranged in a rough V-shaped pattern (Figure 10a). Numerous geophysical and hydrologic tests were conducted in the wells in order to assess characterization technology. Early studies indicated that the wells were intersected by two westwardly dipping transmissive fracture zones at depths of 20–40 m and 60–85 m (Figure 10b). Some 130 injection tests were conducted during the lifetime of the facility, resulting in roughly 4100 cross-hole transient pressure measurements [Karasaki et al., 2000].
 Systematic injection tests were conducted in all nine wells [Cohen, 1993]. In this study we shall analyze observations from two sets of tests conducted in wells 0-0 and SE-1. In these tests a straddle packer string 6 m in length was used to isolate and inject water in a packed-off interval [Karasaki et al., 2000]. A pneumatically controlled downhole-valve was used to initiate and stop the injections. Pressure in the source water tank was kept constant using compressed air. Both the flow rate and the pressure in the injection interval were monitored for the duration of the experiment, typically 10 minutes. After the injection, the packer system was lowered in the well by approximately 6 m. There were a total of 13 injection events in well 0-0 and 14 injection events in well SE-1. During the injection event, pressure in 31 intervals within the other nine wells was monitored. The flow rate for 12 of the 13 injection events in well 0-0 are shown in Figure 11. Note the large variation in flow rate amplitude for each event, the transient nature of the injection, and the complicated flow rate associated with some events (for example event 10). The corresponding head changes for the three monitoring intervals in well SE-1 are displayed in Figure 12. Note the large variation in observed head changes between events and between monitoring intervals. Flow rates for injection events 1 to 12 in well SE-1 are shown in Figure 13. The resulting head changes in well 0-0 induced by the injections are shown in Figure 14.
 The head changes in Figures 12 and 14 form the basic set of observations we shall use to infer permeability between wells 0-0 and SE-1. The first task involves computing the Fourier transform of the head variations. Selected Fourier transforms of the head changes are shown in Figure 15 for observation intervals in well 0-0. As in the synthetic test, we shall only invert the real component at ω = 0. As indicated in Figure 15, the frequency band varies over a few milli-hertz. Thus, the order-ω and higher terms in the series [equation (2)] will be a 1,000 times smaller than the zeroth-order terms. In order to account for the complicated flow rate we multiply its Fourier transform by the theoretical response, given by equation (3). This is equivalent to convolving the flow rate with the theoretical impulse response.
 We followed the same procedure for the inversion of the field data as we did in the numerical illustration. In particular, the region between wells 0-0 and SE-1 is divided into a 15 (lateral) by 60 (vertical) grid of cells. We adopted a penalized least-squares formulation in which the function P in equation (16) is minimized. The weighting factors Wn and Wr were determined by a trial and error procedure. The reference model, a uniform background permeability of 1 md, was used to compute the sensitivities using equations (4), (10), and (11). Because the permeability of the fractures may be many orders of magnitude larger than the permeability of the host rock, we adopt a linearized iterative inversion scheme. That is, we iteratively update our permeability model in a series of linearized inversion steps. At each step we minimize the penalized sum of squares, equation (16), using the LSQR algorithm [Paige and Saunders, 1982] in order to find the model parameter perturbations δK. The values of Wn and Wr were determined by conducting a series of trial inversions. The sum of the squares of the residuals as a function of iteration is shown in Figure 16. The largest reduction in misfit is obtained in the first iteration, a reduction of over two orders of magnitude. Subsequent iterations do not reduce the squared error by much. The initial and final fits to the real components of the Fourier transform are shown in Figure 17. Initially, the Fourier components are much too large, suggesting the need for large changes in the starting permeability model. In the final model, we fit the extreme variations of the data though there is still considerable scatter in the residuals.
 The final permeability model, shown in Figure 18, contains two zones is which the permeability is one or more orders-of-magnitude larger than the background. The high permeability features roughly coincide with two fracture zones detected by other geophysical and hydrologic tests (Figure 10b). For example, we have plotted the results of a conductivity logging test in well 0-0 next to a seismic tomographic image (Figure 19). In the conductivity logging test, the fluid in the well is replaced by distilled water. Then a conductivity log is run repeatedly along the well. As the more saline formation fluid enters the well the conductivity within the well changes. Thus, high conductivity zones coincide with fluid bearing fractures in the surrounding rock. It is apparent that the high permeability zones in our inversion result correlate with the anomalies in the conductivity log. The seismic tomogram contains two low-velocity features which are thought to coincide with the two fracture zones shown in Figure 10b [Vasco et al., 1996].
 We have introduced a low-frequency asymptotic inverse modeling scheme. The approach has several advantages over current techniques for inverting head data. First, the method is very general and can be applied to a wide variety of pumping tests. For example, the technique works for an arbitrary flow rate and may be used to interpret the results of a truncated pump test. The methodology may even be used to invert results from single well tests, including tests in which fluid is pumped from an isolated interval in a well and the head is observed in various packed-off intervals within the same well. Second, the algorithms and software required for this approach are simple extensions of methods which are in common use today. The critical computation entails solving equations which are identical in form to the governing equation for static head. Hence, no special software needs to be developed in order to implement this approach. Third, the method is efficient, requiring essentially four solutions of the governing equation for static head in order to compute sensitivities necessary for the inverse modeling. By working in the frequency domain we can also dramatically reduce the volume of data we must model by considering a few frequencies instead of thousands or tens of thousands of data points. Fourth, the low-frequency data are directly sensitive to the gradient of the permeability and not influenced by the specific storage. In that sense, the low-frequency approach is complementary to the high-frequency methodology of Vasco et al.  which is sensitive to the ratio of the specific storage to the conductivity. Note that we can apply the high-frequency method to a truncated pumping test by deconvolving the source function. This suggests a two-step procedure in which we first invert the low-frequency amplitude data, using methods presented in this paper, and follow this with an inversion of the high-frequency arrival time data.
 As noted above, the asymptotic expression for head requires the equivalent of two static head solutions. The amount of computation is independent of the number of frequencies considered. In general, this level of computation is much less than that required for methods which rely on transient head solutions. For example, in our synthetic test the simulation of the entire suite of transient head variations required 44.9 CPU seconds. This contrasts with the 5.5 CPU seconds needed for two static head computations. The transient simulations for the Raymond field case, which are similar to the synthetic test, required 45.3 CPU seconds. The two static problems were solved in 5.6 seconds of computation. Thus, in this case the asymptotic approach is almost an order of magnitude faster than a method requiring the solution of the fully transient problem.
 In the future we would like to explore the potential of this approach for higher resolution imaging of flow properties. In particular, we will incorporate multiple frequencies and the imaginary component into the inversion. As shown here, at very low frequencies the sensitivities tend to be similar in pattern. However, this redundancy may be used to improve the signal to noise ratio. We will also combine this approach with the diffusive traveltime tomography of Vasco et al. . The method should work well in a three-dimensional setting such as provided by multi-level pressure data [Butler et al., 1999]. In addition, it is applicable to a wide range of experiments such as single well tests and downhole monitoring. In the future we shall explore these various applications. Finally, it should be possible to combine this inverse modeling method with several types of geophysical observations. For example, tilt meter observations [Vasco, 2004] or electrical methods can be sensitive to head variations and may provide a wealth of data.
Appendix A:: Asymptotic Solution for Low-Frequency Head Variations
First, we substitute the proposed solution (A1) into the governing equation (A2) and consider the resulting expression. We shall need to consider the spatial derivatives ∇ and ∇ · ∇ in the resulting equation. The gradient of (x, ω) is given by
and similarly for the Laplacian, ∇ · ∇(x, ω),
Substituting the expressions for (x, ω) and ∇ · ∇(x, ω) into the governing equation (A2) produces
The form of this equation suggests that w(x, ω) depends upon powers of ω. Thus, we represent w(x, ω) as a power series in ω
where Hn(x) are general functions of position. The form of the series is also motivated by the solution of the diffusion equation in a homogeneous medium [Virieux et al., 1994] which is a modified zeroth-order Bessel function, K0(αr). The constant α in the argument depends on the properties of the medium and r is the distance from the source [Bowman, 1958; Gradshteyn and Ryzhik, 1980]. A power series in ω follows from an approximation to the modified Bessel function for ω small [Gradshteyn and Ryzhik, 1980, p. 959].
 Note that for ω ≪ 1, the terms in the sum rapidly decrease in magnitude. Substituting the power series representation of w(x, ω) into equation (A3) results in an infinite series of terms of various orders in . Because we are interested in low-frequency head variations, ω small, we only need to consider the terms of lowest order in . In the two sub-sections that follow we consider terms of order −1, 0 ∼ 1, and , respectively.
A2. Terms of Order −1
 By considering terms of the lowest order in we obtain an equation for the zeroth-order amplitude, H0(x),
This expression is identical to the steady state equation governing the head distribution. Defining the vector u = ∇ ln K (x), we can re-write (A5) as
Thus, by solving a linear differential equation we can find zeroth-order amplitude H0 (x). Note that equation (A5) does not contain frequency and only needs to be solved once.
A3. Terms of Order 0 ∼ 1
 Collecting terms of order , the next highest order in ω, we arrive at the equation
where we have defined
Equation (A7) is a linear differential equation for σ(x) with coefficients which contain the zeroth-order amplitude H0(x).
A4. Terms of Order
 Considering terms of order produces a relationship between σ(x), H0(x), and H1(x)
This equation may be used to find the amplitude term H1(x), given H0(x) and σ(x). Note that the expression in brackets on the right-hand-side is the eikonal equation which is used to determine the diffusive ‘traveltime’ for high-frequency transient pressure tomography [Vasco et al., 2000]. Thus, deviations from the eikonal equation serve as source terms for the higher-order correction H1(x). For the purposes of our study we shall not make use of the -order term H1(x). We shall only consider the lowest-order representation for head
which is adequate for the interpretation of low-frequency head variations.
 The procedure for constructing the lowest-order solution for (x, ω) involves first solving for H0(x) using equation (A5). This may be accomplished using a numerical simulator because the equation is identical to the equation for steady state head. One then substitutes the solution H0(x) into (A8) and (A9) and finds σ(x) using equation (A7). Again, the equation may be solved numerically using a method such as finite differences.
Appendix B:: Sensitivity Calculations
 In this appendix we use a perturbation approach to derive model parameter sensitivities. Our derivation follows that of Vasco et al.  and we refer readers to that work for additional details. Here, we shall concentrate on the overall approach and the differences which result from our interest in the low-frequency component of head.
 Model parameter sensitivities relate a perturbation in a model parameter, in our case hydraulic conductivity at a point y in the Earth, to a change in the observations at a point x. We compute sensitivities by considering a slight change in hydraulic conductivity at y from a background value Kb(y)
and derive an expression for the change in the head observation at position x with respect to a background head value b(x, ω)
It is shown by Vasco et al.  that one can derive an equation for δ(x, ω) by substituting expressions for K(y) and (x, ω) into the equation for head (1) and considering terms of first order in the perturbations. The result is a differential equation identical in form to the equation for head, but with additional source terms. We solve the equation for δ(x, ω) using a Green's function solution, G(x, y, ω), which gives the head at point x due to a point source at y. In terms of the Green's function the expression for δH(x, ω) is the integral over the volume of interest V [Vasco et al., 2000],
 We use our asymptotic expressions for the Green's function and the head field. In particular, for low frequencies, we have the zeroth-order (3) representation of the Green's function, which we write in a slightly more explicit form
where Ψ(ω) contains frequency-dependent terms related to the time-variation of the source and the windowing of the observations [see equation (9)]. The zeroth-order amplitude function H0(y, x) represents the amplitude at x due to a source at y and similarly for the phase term σ(y, x). The zeroth-order representation of the head at point y due to a source at xs is given by
The spatial gradients of G(x, y, ω) and Hb(xs, y, ω) are
respectively. Substituting the expressions (B6) and (B7) into the integral (B3) produces
The sensitivity, the partial derivative of the head at the observation point x due to a perturbation of the hydraulic conductivity at y, is given by the integrand
 This work was supported by the Assistant Secretary, Office of Basic Energy Sciences, of the U.S. Department of Energy under contract DE-AC03-76SF00098. All computations were carried out at the Center for Computational Seismology, Berkeley Laboratory.