## 1. Introduction

[2] Hydraulic parameters estimated by conventional aquifer investigation methods like pumping or slug tests lead to integral information averaged over a large volume. This kind of information is insufficient to develop groundwater models requiring detailed information about the spatial distribution of the subsurface. To circumvent this problem tomographical methods are applied. The principle of tomography was developed for medical applications, e.g., to determine the spatial distribution of X-ray attenuation over cross sections of the head or body [*Mersereau and Oppenheim*, 1974; *Scudder*, 1978]. The application of geophysical methods in a tomographic array to create a three-dimensional image of the subsurface has been established since several years. It has to be noted, however, that geophysical methods like radar tomography [*Davis and Annan*, 1989; *Hubbard et al.*, 1997], seismic tomography [*Bois et al.*, 1972; *Gelbke*, 1988] or electrical tomography [*Yorkey et al.*, 1987; *Kohn and Vogelius*, 1984] yield a geophysical parameter distribution which does not have to be in accordance with hydraulic properties of the subsurface. Consequently, it is necessary to perform a complicated and ambiguous parameter transformation [*Dietrich et al.*, 1995, 1998, 1999].

[3] As opposed to geophysical methods, hydraulic tomography allows to directly determine hydraulic subsurface properties. The procedure of hydraulic tomography consists of a series of short-term tests conducted in a tomographical array. Up-to-date, several researchers have evaluated hydraulic tests by using drawdown as a function of time [*Bohling*, 1993; *Gottlieb and Dietrich*, 1995; *Butler et al.*, 1999; *Yeh and Liu*, 2000]. Another approach is based upon the inversion of travel times of pressure changes. This approach follows the procedure of seismic ray tomography. In seismic ray tomography the influence of the velocity field on the travel time between a source and a receiver are described as a line integral relating the travel time to the velocity field. According to that, *Vasco et al.* [2000] derived a line integral relating the arrival time of a “hydraulic signal” to the reciprocal value of diffusivity. The line integral relates the square root of the drawdown peak arrival time of a transient pressure curve obtained for a Dirac source at the origin directly to the square root of the reciprocal value of diffusivity. The derivation of the line integral is based on an asymptotic approach described in detail by *Datta-Gupta et al.* [2001], *Vasco et al.* [1999], *Vasco and Datta-Gupta* [1999], *Datta-Gupta and King* [1995], and *Virieux et al.* [1994]. Diffusivity is the quotient of hydraulic conductivity to storage and is a quantitative measure for the response of a formation on hydraulic stimulation.

[4] In this paper we propose a further development of the travel time line integral. This advanced approach relates any recorded travel time corresponding to the arrival of a certain percentage of the maximum signal (e.g., 1%, 10%, 20% of the maximum amplitude) to the peak time of a signal associated with a Dirac source. This relationship is derived for a Dirac source as well as for a Heaviside source. The advantage of simultaneously inverting several travel times is to exploit their different information content. As the signal propagation in a hydraulic tomographic experiment follows Fermat's principle, early arrivals characterizing the initial part of a signal, follow the fastest pathways between source and receiver. The fastest pathways are usually identical to preferential flow paths, and thus early travel times are dominated by preferential flow (if available). Whereas, late travel times, characterizing the final part of a signal, reflect integral behavior more or less, since the pressure difference between source and receiver stretches out over the complete system. Hence it can be assumed that the different inversion results provide improved knowledge about the properties of the overall system. Additionally, the comparison of the inversion results allows the identification of potential artifacts. For the inversion we have applied a least squares based inverse procedure which is established since several years in seismic tomography.

[5] Beyond this, the inversion results depend on the spatial position of the used grid because the hydraulic properties are averaged over one voxel. On account of this it could be that little inhomogeneities are not resolved because of the average effect within one voxel. Practically it is not possible to refine the grid as far as the averaging effects can be neglected because for each voxel, one model parameter has to be determined. Thus the number of performed short term tests is the upper limit for the model parameters we can safely invert [*Menke*, 1984]. To circumvent this problem without using any a priori information we use the method of staggered grids, which *Vesnaver and Böhm* [2000] proposed for the inversion of seismic tomographic data. Especially, for hydraulic travel time tomography the method of staggered grids is of great interest, because the time exposure for the data acquisition is much higher in comparison to geophysical methods. By the method of staggered grids, the initial grid is displaced in several ways by a known factor. For each grid we get a slightly different image of the parameter distribution because inside each voxel we get a different averaged value. Finally, we have determined the arithmetic mean of the parameter distribution of all grids by staggering them. This leads to a refining of the grid and to a better resolved image without conducting additionally measurements or using a priori information.

[6] The introduced procedure has been applied to data from a set of pneumatic short term tests conducted in an unsaturated fractured sandstone cylinder. We have performed pneumatic instead of hydraulic short term tests in order to reduce the duration of the experiments. The experimental setup was designed to produce a streamline pattern covering the complete cylinder, whereby the source can be considered as a point source. The produced streamline patterns can be compared to the ray patterns of geophysical cross-hole tomography experiments. These recorded data make it possible to reconstruct a three-dimensional image of the diffusivity distribution of the cylinder and to verify the results by geological surface mapping. In this paper we use the terminology hydraulic instead of pneumatic tomography because the practical application of this method is the high resolution characterization of aquifers.