## 1. Introduction

[2] Ground-penetrating radar (GPR) measurements are not directly related to soil hydraulic parameters in the vadose zone. However, they are highly sensitive to fluid distribution (and changes thereof) and are therefore potentially useful for inferring soil hydraulic parameters. The use of GPR methods for characterizing the distribution and movement of fluids in the subsurface is well established. However, only recently has the potential for using time-lapse GPR measurements to infer soil hydraulic properties—which can then be used to model flow and transport—been explored [*Binley et al.*, 2002; *Kowalsky*, 2003; *Kowalsky et al.*, 2004a; *Lambot et al.*, 2004; *Rucker and Ferré*, 2004]. The response of a hydrological system to external stimuli, such as the injection of water in the subsurface or ponding of water on the ground surface, depends primarily on the soil hydraulic functions and their variations in space (and on the initial and boundary conditions). Corresponding GPR measurements of the same system also depend on the soil hydraulic functions—although indirectly—because the soil hydraulic functions influence the water distribution, which in turn influences the GPR response.

[3] A review of GPR applications in hydrological investigations is given by *Annan* [2005]. The soil property that most directly affects the speed at which GPR waves travel in the subsurface is the dielectric permittivity. For simplicity we use the (relative) dielectric constant κ, defined as the dielectric permittivity of a material normalized by that of free space. For common earth materials and under favorable conditions (e.g., where highly conductive materials, such as clays, are sparse), the dielectric constant is related to the electromagnetic (EM) wave velocity (*V*) through

where *c* is the EM wave velocity in free space [*Davis and Annan*, 1989]. The presence of water affects the dielectric constant of soil mixtures [*Daniels*, 1996] in a manner that can be modeled with relationships that are purely empirical [*Topp et al.*, 1980; *Persson et al.*, 2002], semi-empirical [*Birchak et al.*, 1974; *Roth et al.*, 1990], or theoretical [*de Loor*, 1964; *Dobson et al.*, 1985; *Friedman*, 1998; *Sihvola*, 1999]. Hereafter, we refer to the functions that relate water content (or water saturation) to the dielectric constant of soil mixtures as petrophysical functions, and to the parameters that describe such functions as petrophysical parameters.

[4] Because of their high sensitivity to the pore water distribution, cross-borehole ground-penetrating radar (GPR) measurements are used increasingly for imaging transient flow in the vadose zone [e.g., *Alumbaugh et al.*, 2002; *Binley et al.*, 2001; *Kowalsky et al.*, 2004b]. Tomographic inversion techniques [*Peterson et al.*, 1985] are typically applied to cross-borehole GPR data sets [*Eppstein and Dougherty*, 1998; *Peterson*, 2001] to obtain spatial distributions of EM velocity (tomograms), which can be converted to water saturation using a petrophysical function.

[5] While tomography is especially useful for gaining a qualitative understanding of flow processes in the subsurface (e.g., to help identify preferential flow paths), there are some well known limitations, such as the occurrence of artifacts, like smoothing and smearing, that can be introduced through the tomographic inversion procedure [*Stewart*, 1991; *Peterson*, 2001]. For demonstration, Figure 1 depicts the traditional application of cross-borehole tomography for the case of a simulated water injection in the vadose zone. An injected water plume corresponds to the region with decreased EM velocity (Figure 1a). Simulation of a dense set of GPR measurements (Figure 1b), followed by tomographic inversion, results in a tomogram (Figure 1c) that is similar to the true velocity model but is distorted. An implicit assumption in such a procedure is that the water distribution, which determines the EM velocity distribution, remains constant during the survey; in reality it may change appreciably, especially in dynamic flow situations.

[6] Such limitations make the direct use of tomograms potentially problematic for hydrological applications [*Day-Lewis and Lane*, 2004; *Moysey et al.*, 2005]. A more fundamental limitation is that traditional cross-borehole tomography cannot in general be used to obtain quantitative estimates of vadose zone flow parameters, like the permeability and the soil hydraulic parameters of the capillary pressure and relative permeability functions, except in some limited cases [e.g., *Hubbard et al.*, 1997].

[7] As opposed to the typical use of cross-borehole data described above and depicted in Figure 1, an alternative application involves their direct integration in hydrological inversion schemes in a coupled fashion (i.e., where the geophysical and hydrological data are simulated simultaneously and are explicitly linked to hydrological parameters). Few such applications have been reported in the literature, especially for vadose zone applications. *Hyndman et al.* [1994] and *Hyndman and Gorelick* [1996] developed an inversion algorithm for estimating lithological zones and the hydrological parameters of the zones in fully saturated aquifers using seismic and tracer data. *Rucker and Ferré* [2004] used an analytical GPR ray-tracing model and an analytical unsaturated flow model to invert for the average hydraulic conductivity value with synthetic GPR cross-borehole travel time measurements collected through time at a single depth. They also demonstrated that two additional hydraulic parameters can be estimated if pressure head measurements are included in the inversion.

[8] Here we describe an approach for estimating soil hydraulic parameter distributions such as in the vadose zone through the coupled numerical simulation (and inversion) of multiple-offset cross-borehole GPR and hydrological data collected during transient flow experiments. Our approach uses GPR travel times directly and does not aim to obtain geophysical images (tomograms), avoiding some difficulties of cross-borehole tomography and allowing for a far less dense set of GPR data to be collected (thus allowing data sets to be collected in less time). While the methodology can be applied to any type of hydrological data, we focus on neutron probe measurements, which provide near-borehole estimates of water content. Coupling between the hydrological and GPR simulators links the simulated water saturation distributions and the generated porosity distributions to the simulated GPR data (e.g., travel times), thus indirectly linking the geophysical data to the soil hydraulic parameters. Joint inversion proceeds by perturbing the soil hydraulic parameters—which alters the simulated hydrological and geophysical data—until the simulated and measured hydrological and geophysical data are in good agreement. A flow chart depicting the joint inversion procedure, which we implemented in iTOUGH2 [*Finsterle*, 1999], is given in Figure 2, and details of the inversion methodology are discussed below.

[9] The methodology we employ is an extension of the work by *Kowalsky et al.* [2004a], which involved the joint use of geophysical and hydrological data within a maximum a posteriori (MAP) inversion framework [*McLaughlin and Townley*, 1996] that employed concepts from the pilot point method [*RamaRao et al.*, 1995; *Gomez-Hernandez et al.*, 1997]. The approach of *Kowalsky et al.* [2004a] allowed for estimation of unknown log-permeability values, at so-called pilot point locations, and other hydrological parameters, resulting in hydrological models that honored geophysical and hydrological data and that contained permeability distributions with specified patterns of spatial correlation and that honored available log-permeability point measurements. The method was shown to be useful for accurately predicting flow phenomena and quantifying parameter uncertainty. However, the forward model used to simulate GPR data was limited to a simple data acquisition configuration, disallowing the use of multiple-offset GPR measurements (discussed below) and generalized three-dimensional models. In addition, the petrophysical function was assumed to be known and error free, despite the fact that inaccuracies easily enter into the field-scale petrophysical function when it is derived using non-site-specific data or laboratory-scale measurements [e.g., *Moysey and Knight*, 2004; *Lesmes and Friedman*, 2005].

[10] At present the aforementioned method of *Kowalsky et al.* [2004a] is extended to allow for (1) inclusion of GPR measurements (travel times) collected using any transmitter and receiver geometry within a possibly three-dimensional model, and (2) estimation of petrophysical parameters. (We have also extended the method to allow for possible estimation of spatial correlation parameters, but this possibility is not currently explored herein.) These extensions permit investigations under more realistic conditions (e.g., where there is uncertainty in the petrophysical function) and increase the flexibility of GPR data acquisition configurations that may be considered, which allows for soil hydraulic parameter estimates with increased resolution and accuracy. Following a description of the methodology, given in section 2, synthetic examples and an application using field data are presented in sections 3 and 4, respectively.