In this study the potential of an inversion approach based on hydraulic travel time and hydraulic attenuation tomography was assessed. Both hydraulic travel time and hydraulic attenuation tomography are based on the transformation of the transient groundwater flow equation into the eikonal equation using an asymptotic approach. The eikonal equation allows the calculation of pressure propagation and attenuation along trajectories, which is computationally efficient. The attenuation and travel time-based inversion approaches are naturally complementary: hydraulic travel times are determined by the hydraulic diffusivity, a combination of hydraulic conductivity and specific storage, whereas the attenuation is determined solely by specific storage. The potential of our hydraulic tomographical approach was investigated at a well-characterized sand and gravel aquifer located in the Leine River valley near Göttingen, Germany. The database for the hydraulic inversion consists of 392 cross-well slug interference tests performed between five wells, in which the positions of the sources (injection ports) and the receivers (observation ports), isolated with double packer systems, were varied between tests. The results have shown that the combination of hydraulic travel time and hydraulic attenuation tomography allows the reconstruction of the diffusivity and storage distribution in two and three dimensions with a resolution and accuracy superior to that possible with type curve analysis.
 Information about the spatial distribution of hydraulic properties between wells is generally gained by performing hydraulic tests. Traditionally, hydraulic tests are evaluated assuming a homogeneous parameter distribution and fitting an analytical solution to recorded pressure responses. The design, performance, and evaluation of hydraulic tests under various conditions are described in detail in compendia, such as Dawson and Istok , Kruseman and de Ridder , Batu , Butler , and Streltsova . These solutions provide hydraulic bulk properties averaged over a larger area or point information.
 However, the quality of subsurface transport predictions depends strongly on the scale of investigation and resolution. Neglecting relatively thin structures, such as sedimentary intrachannel deposits constituting high-permeability zones, does not allow sufficiently accurate predictions required in the planning of remediation strategies and geothermal installations. Dipole flow tests [Zlotnik and Zurbuchen, 1998; Peursem et al., 1999], multilevel slug tests [Butler et al., 1994; Brauchler et al., 2010], or borehole flowmeter tests [Molz and Young, 1993; Boman et al., 1997] allow the identification of such thin structures, which possibly dominate subsurface transport. The degree of lateral connectivity of such thin structures, which act as preferential flow paths, however, cannot be detected with these techniques since they provide only information in the vicinity of the well.
 Over the last decade and a half, several research groups started work on a new approach, i.e., hydraulic tomography. This approach has the potential of providing enough information to image the spatial continuity and interconnectivity of preferential flow paths between wells at sufficient resolution. Hydraulic tomography consists of a series of short term pumping or slug tests. Varying the location of the source stress (pumping or slug interval) and the receivers (pressure transducers) generates streamline patterns that are comparable to the crossed ray paths of a seismic tomography experiment [Butler et al., 1999].
 An alternative to the above mentioned inverse methods for the evaluation of a suite of hydraulic tomographic data is the travel time–based inversion scheme. This inversion scheme follows the procedure of seismic ray tomography and is based on the transformation of the transient groundwater flow equation into the eikonal equation using an asymptotic approach [Virieux et al., 1994]. The eikonal equation can be solved with ray tracing techniques or particle tracking methods, which allow the calculation of pressure propagation along trajectories [e.g., Vasco et al., 2000; Vasco and Karasaki, 2006; Kulkarni et al., 2001; Datta-Gupta et al., 2001; Brauchler et al., 2003, 2007, 2010; He et al., 2006]. The main feature of this procedure is a travel time integral relating the square root of the peak travel time, assuming a Dirac point source at the origin, to the inverse square root of the hydraulic diffusivity D [Vasco et al., 2000; Kulkarni et al., 2001; Datta-Gupta et al., 2001]. Brauchler et al.  have further proposed the travel time–based approach to invert all data of a transient pressure signal for a Dirac source. Their analysis also covers a Heaviside source at the origin.
 As the travel time of a hydraulic signal depends on the diffusivity of the investigated medium, it is difficult to separate it into its components hydraulic conductivity and specific storage. In order to overcome this problem we perform, in addition to the travel time inversion, an inversion that is based on the relationship between attenuation of a hydraulic signal traveling between source and receiver, and the specific storage of the investigated medium. Both inversion techniques are based on ray tracing, which allows the inversion of large data sets in a short time on a common PC. For the transformation of the diffusivity equation into the eikonal equation a hydraulic parameter distribution is assumed, which varies smoothly with respect to the spatial wavelength of the propagation and attenuation of the pressure pulse. However, Vasco et al.  and Brauchler et al.  have shown, that parameter variations of several orders of magnitude can be reconstructed. A prerequisite therefore is a regularly arranged distribution of test (sources) and observation intervals (receivers). The attenuation and travel time-based inversion approaches are naturally complementary: hydraulic travel times are determined by the hydraulic diffusivity, a combination of hydraulic conductivity, and specific storage, whereas the attenuation is determined solely by specific storage. Thus, combining these two approaches will allow the independent identification of the high-resolution spatial variability of hydraulic conductivity and specific storage. The potential of our hydraulic tomographic approach was investigated at a well-characterized sand and gravel aquifer located in the Leine River valley near Göttingen, Germany. Figure 1 illustrates the well-network of the test site. The database for the hydraulic inversion consists of 392 cross-well interference slug tests performed between five wells in which the positions of the sources (injection ports) and the receivers (observation ports), isolated with double packer systems, were varied between tests. It was possible to reconstruct the hydraulic properties of a three-dimensional area of 5 m × 5 m × 2 m by applying the hydraulic travel time and attenuation inversion technique to the 392 transient pressure response curves.
 In the first part of this section a brief review of the travel time inversion approach, in particular the inversion of additional travel time diagnostics besides the peak time, is given. Note a travel time diagnostic is defined as the time of occurrence of a certain feature of the transient pressure pulse. For example, the t-10% diagnostic is the time at which the pressure pulse rises to 10% of its ultimate peak value. In the second part of this section the derivation of the attenuation-based inversion approach is illustrated.
2.1. Travel Time–Based Inversion Approach
 The 3-D propagation of a pressure pulse in the subsurface can be described by a line integral relating the arrival time of a “hydraulic signal” to the reciprocal value of diffusivity [Vasco et al., 2000; Kulkarni et al., 2001],
where tpeak is the travel time of the peak of a signal from the point x1 (source) to the observation point x2 (receiver), and D is the diffusivity as a function of arc length along the propagation path (s). Diffusivity is defined as the quotient of hydraulic conductivity k over specific storage Ss. The inversion of additional travel time diagnostics besides the peak time was accomplished by introducing a transformation factor [Brauchler et al., 2003],
where is the travel time (arrival time) of a selected point of the signal and the related transformation factor. The subscript d stands for a Dirac source. Equation (2) allows to relate any recorded travel time with the diffusivity D by using the corresponding transformation factor. The transformation factor is defined as follows
where W stands for the Lambert's W function. The head ratio enables the comparison of the peak time with the respective travel time diagnostic as follows: where hd (r, t) is the hydraulic head depending on space and time. A detailed derivation of the transformation factor is given by Brauchler et al. .
2.2. Attenuation-Based Inversion Approach
 After the work by Häfner et al. , the solution of the diffusion equation in an infinite domain for a Dirac source is
where rc is the casing radius, H0 is the initial displacement, and hd (r, t) is the hydraulic head depending on space and time. Ss denotes the specific storage coefficient, and k denotes the hydraulic conductivity. The peak time of a pressure pulse can be determined from the first derivative of equation (4),
 The first derivative becomes 0, when . The amplitude of the signal can be determined by inserting tpeak into equation (5),
Equation (6) states that the decay of the amplitude (total attenuation) of an impulse source is only a function of the flow property specific storage Ss and the distance r apart from test specific parameters. The test specific parameter can be summarized introducing the parameter B,
where h(r, tpeak) is the peak response at position r.
 Based on equation (8) a trajectory describing the attenuation of a hydraulic pressure pulse in a media with a heterogeneous distribution of the parameter specific storage can be defined. Thereby, we follow the derivation of the travel time integral (equation (1)) proposed by Vasco et al.  introducing a curvilinear coordinate system. Constraints are provided by the requirements of the test setup that the pressure change starts at the source point x1 and ends at another specified point. Hence, specific storage is only changing along the defined trajectory (s). In our treatment, specific storage is adopted locally as homogeneous and thus the whole trajectory is a sum of a defined number of homogeneous sections. The number of homogeneous section is defined by the mesh during the inversion process. This point of view is in accordance with Huygens–Fresnel principle that states each point of an advancing wavefront is in fact the center of a fresh disturbance and the source of a new train of waves. The advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already traversed. In our treatment the propagation as a whole is described by the trajectory that is composed of a number of homogeneous sections, each reflecting a change in specific storage of the investigated medium.
Equation (9) relates the attenuation of a pressure signal with a Dirac source signal at the origin to changes in specific storage S(s) integrated along the trajectory defined by start and end points x1 and x2. Note, the proposed integrals (equations (2) and (9)) are a result of a high-frequency assumption, i.e., that the hydraulic parameters vary smoothly with respect to the spatial wavelength of the propagation and attenuation of the pressure pulse [Vasco et al., 2000].
3. Test Site Characterization
 The presented methodology was applied to hydraulic tomographic measurements performed at the research site “Stegemühle,” which is located in the Leine River valley near Göttingen, Germany. The test site consists of a network of 26 wells comprising 1″, 2″, and 6″ and also multitube wells (Figure 1). The 6″ wells were drilled with a top drive drilling rig, whereas all other wells were installed using direct-push technology [e.g., Dietrich and Leven, 2006].
 The shallow unconsolidated braided river sediments largely consist of 3.5 m silt and clay overlying 2.5 m sand and gravel. The bedrock consists of Triassic (Middle Keuper) mudstone. The geologic interpretation is based on an electrical conductivity log obtained from direct-push electrical probing, termed in the following DP-EC logs [e.g., Christy et al., 1994; Schulmeister et al., 2003], and a gamma-ray log performed at each well location. All recorded logs show the same basic pattern with variable thicknesses of the confining unit (silt and clay layers) and the aquifer material (sand and gravel). Figure 2 shows an example of the vertical profile of the unconsolidated sediments of well P0/M25 (Figure 1). Note the DP-EC log was recorded prior to well installation, which allows to determine the aquifer bottom by penetrating the first few centimeters of the bedrock. The tomographic measurements were performed in the vicinity of well P0/M25. Here, the aquifer is approximately 2 m thick with an average hydraulic conductivity of 5.0 × 10−4 m/s (determined by pumping tests). Four additional wells were installed at a distance of 2.5 m from the center well P0/M25. The spatial position of the five wells (in the following called five-point star) within the test site is shown in Figure 1. Brauchler et al.  summarize much of the subsurface geometry and hydraulic characterization at the “Stegemühle” site. Experience at the site has demonstrated that the aquifer behaves as a confined system.
 The database for the hydraulic inversion consists of cross-well interference slug tests, which were performed in a tomographic array between five wells arranged as a five-point star. The suite of tomographic cross-well slug interference tests were recorded by employing double packer systems with a screened interval of 0.25 m and an internal tube diameter (ID) of 0.031 m in the test and observation well. By varying the test and observation interval, the suite of cross-well interference slug tests produce a pattern of crossing trajectories between the test and observation wells, similar to the paths of a radar or seismic experiment. All cross-well interference slug tests were initiated pneumatically to avoid large fluctuations of the initial readings of the response signal [Butler, 1998]. The test well was equipped with two pressure transducers, one in the air column and another in the screened section of the double packer installed in the test well. The comparison of the pressure drop in the air and in the water column in the test well allows to identify nonlinear behavior of the change of the water column. These nonlinear effects can lead to a dependence of the pressure response on the initial displacement [McElwee and Zenner, 1998; Zurbuchen et al., 2002; Butler and Zhan, 2004; Chen and Wu, 2006]. The test design chosen allows to identify and to quantify such type of dependence.
 Two test series, whose test set-ups are shown in Figure 3 were performed. Note the black lines are not identical with the trajectories calculated during the inversion process, but the black lines illustrate the measured configurations as well as the spatial position of the test and observation intervals. (1) Test series 1 comprises four profiles. Each profile consists of seven injection and seven observation intervals. For all four profiles the center well of the five-point star configuration served as test well and the four surrounding wells served as observation wells (Figure 3a). For test series 1, the minimum number of three pressure transducers was used at the same time. That means, two pressure transducers were installed in the test well to measure the pressure drop in the air as well as in the water column of the test well and the third pressure transducer was installed in the screened section of the double packer system used in the observation well. They recorded at a temporal resolution of 7 Hz. Note the temporal resolution depends on limitations of the data loggers, i.e., the number of connected pressure transducers. Every test was conducted at least two times to assure repeatability and data quality. These data were used to reconstruct the hydraulic properties of the diagonals of the cube defined by the five wells. (2) Test series 2 was initiated in order to record the remaining four profiles between the four outer wells (Figure 3b). The recorded response curves of test series 1 have shown that a temporal resolution of 5 Hz is sufficient for the investigated media. This allowed us to increase the number of pressure transducers from three to five and to record the pressure response in three different wells, each equipped with a double-packer system, at the same time. This resulted in a decrease in the sampling rate to 5 Hz and increases the efficiency by a factor of three. The data of test series 1 and 2 were employed in order to show the potential of the proposed inversion approach by estimating the hydraulic properties of the cube, defined by the five-point star configuration, in three dimensions. For the reconstruction of the individual profiles the pressure interference responses of 49 source-receiver configurations were used (seven sources times seven observation points). The three-dimensional reconstruction is based on the data of eight profiles comprising 392 pressure interference responses.
Vasco et al.  proposed a transformation of the transient groundwater flow equation into the eikonal equation using an asymptotic approach [Virieux et al., 1994]. Hence, the eikonal equation can be solved with ray tracing techniques or particle tracking methods, which allow the calculation of pressure propagation along trajectories. This method was already applied successfully to the inversion of travel times by several researchers [Vasco et al., 2000; Vasco and Karasaki, 2006; Kulkarni et al., 2001; Datta-Gupta et al., 2001, Brauchler et al., 2003, 2007, 2010; He et al., 2006] in order to reconstruct the diffusivity distribution between two wells. The analogy between equations (1) and (9) allows calculating the specific storage distribution, in the same way as the diffusivity distribution, after the amplitude data have undergone proper data processing. Hence, within this study the same ray tracing technique was utilized to perform (1) the hydraulic travel time and (2) the hydraulic attenuation inversion using the commercial software package GeoTom 3-D [Jackson and Tweeton, 1996], which is based on the SIRT (simultaneous iterative reconstruction technique) algorithm [Gilbert, 1972]. The inversion of the travel times and attenuation were performed separately. The algorithm allows the calculation of curved ray paths and trajectories through the 3-D model grid, respectively, and is therefore well suited for applications in seismic and hydraulic tomography. The detailed derivation of the SIRT algorithm with regard to the travel time inversion of transient pressure responses is provided by Brauchler et al. . For the inversion a homogeneous starting model was used. The starting values for the velocity/attenuation fields are derived from the mean values of the measured source–receiver combinations. The model domain of the two-dimensional inversion consists of 6 × 7 cells and for the three-dimensional inversion the model domain consists of 6 × 10 × 10 voxels. The travel time and attenuation inversion of the three-dimensional data set took less than 1 min on a PC with a 3.2 GHz CPU. Note the three-dimensional data set allowed the reconstruction of the full three-dimensional hydraulic property field of the cube defined by the five-point star configuration.
5.1. Travel Time Inversion
 The presented results of the travel time inversion are based on the inversion of the travel time diagnostics t-10%, which is a compromise between obtaining high data quality (avoiding early time noise) and the findings of Brauchler et al.  and Cheng et al.  that early travel time diagnostics are more suited to resolve hydraulic heterogeneity. Note the calculated trajectories follow Fermat's principle, and thus, small travel time diagnostics follow the fastest pathway, which are usually identical to preferential flow paths, while large travel time diagnostics reflect the integral behavior of the investigated system. Wellbore storage effects as well as inertial effects, which are typical for cross-well slug interference tests performed in small-diameter wells installed in relatively high permeability material, cause a delay in the time at which the pressure change is observed at some distance from the test well [Prats and Scott, 1975]. For this purpose an analytical model method reported by Brauchler et al. [2007, 2010] was applied. The key element of this method is an analytical solution, which is utilized to generate a reference model for the quantification of the associated travel-time delays.
5.2. Attenuation Inversion
 For the attenuation inversion, which were performed independently from the travel-time inversion, the slug test interference data were processed following the procedure for the analysis of slug tests, performed in partially penetrating wells in formations of high hydraulic conductivity, presented by Butler et al. . Thereby, the deviation data recorded in the observation well are divided by the change in water table that initiated the test (H0) to obtain the normalized attenuation of the pressure response signal traveling between test and observation interval. The initial water level change was estimated using the pressure transducer reading in the air column above the well water level. After manipulation of the normalized attenuation according to equation (9), the spatial distribution of the specific storage can be reconstructed.
 For comparison with the proposed inversion techniques, the cross-well slug interference test data were analyzed by analytical solutions. The results were presented by Brauchler et al. . Since the data are utilized to validate and verify the inversion results, a short summary is given below.
6.1. Evaluation Based on Type Curve Analysis
 For the evaluation (type curve matching) of the pressure response curves the solution developed by Hyder et al.  was applied. If curve matching was not possible with this solution, we have applied the Butler and Zhan  solution that accounts additionally for inertial effects. Both solutions allow to evaluate water-level response at the test and observation wells in a homogeneous confined aquifer, with infinite lateral areal extent, for fully and partially penetrating wells; whereby the Butler and Zhan  solution allows, in addition, the evaluation of underdamped responses typical for high K-aquifer sections. Beyond this the Butler and Zhan  solution accounts for frictional losses in small-diameter wells and inertial effects in the test and observation wells. For the application of the solutions the following assumptions were made: the aquifer is confined, isotropic, of infinite extent, test and observation wells are partially penetrating, and there is no well skin.
 The pressure responses recorded in the observation intervals of test series 1 are evaluated and the derived hydraulic parameters are illustrated in Figure 4 by means of four profiles. The illustrated K and Ss estimates are obtained for the test configurations, where the double packer systems are positioned at the same depth, and the depth of the test intervals refers to the center of the double packer systems. The profiles illustrated in Figures 4b and 4c show an increasing of the K values and a decreasing of Ss values with depth. The K values decrease from ∼10−3 ms−1 close to the bottom to ∼10−4 ms−1 at the top of the aquifer. The Ss distribution shows an opposite behavior. The Ss values increase from ∼10−5 ms−1 close to the bottom of the aquifer to ∼10−3 ms−1 at the top of the aquifer. The profile P0/M25-P0/M22.5 shows no significant variation with depth (Figure 4a).
 An important point, to be taken into account in the evaluation of cross-well slug interference tests in relative high K-material, is a potential dependence of the pressure response on the initial displacement. Such a nonlinear behavior of the change of the water column were discussed, e.g., by McElwee and Zenner , Zurbuchen et al. , Butler and Zhan , and Chen and Wu . The influence of these nonlinear effects on the observation well responses were identified and assessed by repeating cross-well slug interference tests with different initial displacements. Comparison of the observation well responses showed that the travel time of the signals was not influenced by these nonlinear effects. However, the amplitude, and consequently the storage estimates derived from the analytical as well as the attenuation inversion, showed an influence of nonlinear effects. The investigation of selected test configurations recorded in the high-permeability zone close to the bottom of the aquifer has shown that the specific storage estimates vary within a factor of two [Brauchler et al., 2010]. This has to be taken into account by the interpretation of the specific storage tomograms.
 Four reconstructed diffusivity and specific storage profiles (in the following called diffusivity or specific storage tomograms), inverted using the procedure described above, are displayed in Figure 5. The diffusivity and specific storage tomograms show similar spatial distribution, but with a negative correlation. The correlation coefficient for collocated values of the diffusivity and specific storage tomograms varies between −0.6 and −0.8. All tomograms show, except for the left part of the diffusivity and specific storage tomograms P0/M25 – P0/M22.5, horizontally layered structures. That means that the highest diffusivity values and the lowest specific storage values are close to the bottom of the aquifer and decrease/increase to the top of the aquifer, respectively. This general pattern agrees with the parameter distribution derived from the type curve analysis illustrated in Figure 4.
 The left parts of the diffusivity and specific storage tomograms P0/M25 – P0/M22.5 display a homogeneous parameter distribution in the vicinity of well P0/M22.5 (Figure 5). This indicates that the zone characterized by high diffusivity and low specific storage values pinches out toward well P0/M22.5. The existence of the lateral change is supported by the parameter distribution derived from the type curve analysis. The determined values for hydraulic conductivity and specific storage between well P0/M25 and well P0/M22.5 show no significant variation with depth in contrast to the other three configurations (Figure 4). K estimates derived from multilevel single-well slug tests performed in different depths of each well of the five-point star configuration, illustrated by Brauchler et al. , also support the existence of the lateral change.
 The agreement between diffusivity and specific storage tomograms, and the results derived from type curve analysis, indicates the ability of the proposed inversion scheme to reconstruct lateral and vertical changes in hydraulic properties with spatial high resolution. However, in this regard, it has to be mentioned that the K and Ss estimates derived from type curve analysis, assuming a homogeneous parameter distribution, do not reflect a uniformly weighted average, but the weight depends on the test and observation interval and the heterogeneity of the subsurface [Wu et al., 2005]. The interpretation of the proposed tomograms is an important point because for every reconstruction usually an inverse problem has to be solved, likely leading to a mathematically nonunique solution [Tarantola, 2005]. Possible reasons for a mathematically nonunique solution, leading to consequential prediction errors are e.g., sparse data and/or spatially varying data density. Beyond this, insufficient measuring accuracy can lead to inaccurate estimates. Thus, reconstructed parameter distributions might be characterized by smearing effects and ambiguity. In order to overcome this problem a zonation strategy was applied. This approach is based on an assumption that the “true” parameter distributions are better described as discrete than as continuously varying. Based on the diffusivity and storage reconstruction k-means cluster analysis [McQueen, 1967] was utilized to generate zones of constant hydraulic properties. The cluster analysis was performed purely in variable space (diffusivity, specific storage) without using any spatial adjacency. Cluster analysis has proven to be a powerful tool to objectively identify major common trends and groupings in various combinations of tomographic data [Eppstein and Dougherty,1998; Tronicke et al., 2004; Paasche et al., 2006; Dietrich and Tronicke, 2009].
 In this study three zones of constant hydraulic properties were generated to interpret the tomograms. The number of zones was chosen in accordance with the number of significant subsurface features recognizable in the diffusivity and specific storage tomograms. In cases where the tomograms are ambiguous and no significant subsurface features can be identified, the optimum number of clusters can be specified by statistical criteria, e.g., the variance ratio criterion method introduced by Calinsky and Harabasz . Note that the absence of significant subsurface features indicates a homogeneous hydraulic parameter distribution and hence, type curve methods assuming a homogeneous parameter distribution of the subsurface might be the better choice for the estimation of hydraulic subsurface parameters. The final zonation is illustrated in Figure 5c and interpreted as follows:
 (1) Zone 1 represents the area close to the bottom of the aquifer characterized by high-diffusivity values and low specific storage values.
 (2) Zone 2 represents a transition zone between the high diffusivity/low specific storage values close to the bottom and the low diffusivity/high specific storage values close to the top of the aquifer as well as the left part of the tomograms P0/M25 – P0/M22.5.
 (3) Zone 3 represents the area close to top of the aquifer characterized by low diffusivity values and high specific storage values.
 For the three zones, the calculated hydraulic conductivity (based on the relationship K = D × Ss) values range from 10−3 m/s to 5 × 10−4 m/s and the specific storage values from 10−4 m−1 to 8 × 10−4 m-1. These values agree with the hydraulic property estimates derived from type curve analysis and demonstrate that the calculated hydraulic conductivity values based on the diffusivity and specific storage reconstructions are reliable. Naturally, the range of the hydraulic properties of the three zones is smaller than the range of the values derived from type curve analysis, since the zones represent values integrated over a larger area. However, the three zones represent the significant subsurface features, particularly the pinching out of the high hydraulic conductivity/low specific storage zone close to the bottom of the aquifer and their hydraulic properties. The final zonation can serve as a starting model for further investigations with the goal to resolve the multiscale heterogeneity.
 Data of eight profiles comprising 392 pressure interference responses were employed in order to show the potential of the proposed inversion approach by reconstructing the full three-dimensional hydraulic property field of the cube, defined by the five-point star configuration. For the three-dimensional reconstruction the pressure response data of all available source-receiver configurations are inverted simultaneously. The inversion for the travel time and amplitude inversion took less than 1 min on a PC with a 3.2 GHz CPU. Next to the calculation efficiency, the comparison between the two- and three-dimensional reconstructions, displayed in Figure 6, demonstrates the reliability of the inversion scheme. The three-dimensional reconstruction shows no increased trend to smearing effects and ambiguity, despite the imperfect source–receiver distribution, which is predetermined by the five-point star configuration.
 The tomograms based on the two-dimensional inversion are shown together in Figure 6a and 6b. It is apparent that the north-south and west-east sections do not match in detail in the vicinity of the center well P0/M25, in particular the diffusivity tomograms. The three-dimensional reconstruction illustrated in Figure 6c and 6d, however, can resolve the mismatch at the interface. A possible explanation for the differences between the two- and three-dimensional reconstructions at their interface in the vicinity of well P0/M25 could be horizontal anisotropy within sedimentary architectural elements because of complex sedimentation processes.
7. Summary and Conclusions
 The developed attenuation inversion scheme complements the travel time–based inversion scheme efficiently. Travel times are sensitive to the diffusivity distribution between pumping and observation interval, whereby the attenuation of hydraulic signal is sensitive to the storage distribution. The combination of both provides information on the spatial variation of hydraulic conductivity, specific storage, and diffusivity with a high resolution between the wells. The travel time and attenuation inversion were developed and tested using cross-well interference slug tests. The cross-well slug interference tests were performed in a shallow confined sand and gravel aquifer. In total, eight profiles were recorded between five wells arranged in a five-point star. Each profile consisted of 49 cross-well interference slug tests, in which the position of the sources and observation points, isolated by double packer systems, were varied between the tests.
 The results of the field study showed that travel-time and amplitude inversion have the ability to reconstruct lateral and vertical changes in hydraulic properties with spatial high resolution. Beyond this, the field study has shown that travel-time and amplitude inversion are very computationally efficient. The applied ray tracing technique enables to invert a huge amount of data in a short time on a PC. In this study, the inversion of the travel time and attenuation took less than 1 min on a PC with a 3.2 GHz CPU.
 A comparison between type curve analysis with tomographic reconstructions demonstrated that the combination of both inversion schemes allows the reconstruction of the diffusivity and storage distribution in two-and three-dimensions with a spatial resolution and accuracy, which goes beyond type curve analysis. In particular, it was possible to detect lateral changes in hydraulic parameters, which is not possible with conventional hydraulic testing methods. The fence diagram of the two-dimensional reconstructions indicates a mismatch at their interface, which, however, can be resolved by the three-dimensional reconstructions. A possible explanation for the differences between the two- and three-dimensional reconstructions at their interface could be horizontal anisotropy within sedimentary architectural elements because of complex sedimentation processes.
 An area that warrants further research is the coupling of ray tracing–based inversion schemes with inversion schemes based on the inversion of the whole transient pressure signals. The computational efforts could be considerably reduced by using tomograms, reconstructed by travel time– and attenuation-based inversion, as starting models for the computationally intensive full signal inversion.
 The advantages of performing tomographic measurements with cross-well interference slug tests comprise the opportunity to record a large number of pressure responses in a short period of time, and the low cost level in terms of equipment and manpower. A measurement program can be performed by a single person, or, at most, two individuals using a pressure transducer, data logger, and double packer systems. Additionally, no water needs to be handled, a very important advantage of slug tests at sites with potential groundwater contamination. The short range of a pressure pulse, however, limits the application of tomographic measurements with cross-well interference slug tests. With increasing distance between test and observation interval the measurement accuracy is decreasing. The lower signal-to-noise ratio of observed pressure responses can affect the quality of the inversion results [Illman et al., 2008].
 In cases where the range of cross-well slug interference tests is not sufficient, short-term pumping tests can be performed. Note the presented travel time and attenuation inversion approach can be transferred to the evaluation of pumping tests without any restrictions. In combination with direct push technology, hence, it is possible to perform and evaluate tomographic measurements in unconsolidated sediments, independent of existing well fields, at low costs in terms of time, effort, and finances. Thus, the inversion scheme presented could complement the spectrum of methods and technologies available for an adaptive site investigation approach.
 The investigations were conducted with the financial support of the German Research Foundation to the project “High resolution aquifer characterization based on direct-push technology: An integrated approach coupling hydraulic and seismic tomography” under grant BR3379/1-2. This manuscript greatly benefited from comments and suggestions by Geoffrey Bohling, Walter Illman, and one anonymous reviewer for Water Resources Research.