2.1. Hydrological Measurements
 The hydrological process considered in this study is variably saturated flow in the vadose zone. Variably saturated flow of incompressible water in non-deformable porous media can be modeled with the Richards' equation:
where K and Pc, both functions of water saturation Sw, are the unsaturated hydraulic conductivity and the capillary pressure, respectively, ρw is the water density, g is the gravitational constant, ϕ is the porosity, and z is the vertically oriented unit vector (positive upward). The hydraulic conductivity is defined as:
where krel is the dimensionless relative permeability (the only component of K that is a function of water saturation), μw is the dynamic viscosity of water, and k is the absolute permeability, which is a scalar k for the case of isotropic media, and which has horizontal and vertical components kh and kv, respectively, for the case of anisotropic media. For this study, we model the relative permeability and capillary pressure with the functions given by van Genuchten  as:
where S is the normalized water saturation, and Swres and Swsat are the soil-specific residual and maximum water saturation values, respectively, and m (dimensionless) and α (Pa−1) are soil-specific parameters. Hysteresis of the relative permeability function can affect the redistribution of water following infiltration [Philip, 1991], but is not considered in this study.
 To simulate flow in the vadose zone using equations (2)–(6), the soil hydraulic parameters ϕ, α, m, Swres, Swsat, and k must be specified. Although a number of studies suggest that spatial variability of these parameters can be significant, data describing such variability for modeling applications are limited [Jury et al., 1987; Russo and Bouton, 1992]. The absolute permeability is commonly the parameter to which flow in the vadose zone is most sensitive, arguably making its characterization of primary importance. In the present work, all parameters are considered spatially uniform, except for k, which can be treated as a space random function (SRF).
 The joint inversion approach described below aims to estimate soil hydraulic parameters using a combination of hydrological measurements (e.g., water content values inferred from neutron probe logging) and ground-penetrating radar measurements (e.g., cross-borehole travel times), which are highly sensitive to the time- and space-varying distribution of Sw, which is in turn affected by the soil hydraulic parameters. The framework used for the coupled simulation (and inversion) of hydrological and geophysical data (discussed next) is iTOUGH2 [Finsterle, 1999], a code that provides parameter estimation capabilities for the TOUGH2 flow simulator [Pruess et al., 1999].
2.2. Ground-Penetrating Radar Measurements
2.2.1. Petrophysical Function Relating Water Saturation and Porosity to Dielectric Constant
 Application of GPR measurements in the subsurface requires a petrophysical function that relates the soil water saturation and porosity to the dielectric constant (e.g., reviews are given by Huisman et al.  and Lesmes and Friedman ). One of the most commonly used models, by Topp et al. , is given as a third-order polynomial:
where θ is the water content (the product of water saturation and porosity), and where the coefficients were determined through laboratory measurements on several inorganic soils. However, the dielectric constant of soils is sensitive to additional soil properties, such as the mineral composition of the solid soil particles [Roth et al., 1990], organic matter and bulk density [Jacobsen and Schjonning, 1993], temperature [Roth et al., 1990; Or and Wraith, 1999], and grain geometry and cementation [Lesmes and Friedman, 2005], all of which suggest the need for alternative petrophysical relationships that allow for site-specific variations.
 Alternatively, there are theoretically based models [Lesmes and Friedman, 2005], such as volumetric mixing formulae, which account for the volume fraction and geometrical arrangement of materials with known or measurable dielectric constants. An expression used for two-phase mixtures [Birchak et al., 1974] and extended to three-phase mixtures of air, water, and solids [Alharthi and Lange, 1987; Roth et al., 1990] is given by
where ϕ is the porosity, κs is the dielectric constant for the solid components, κw and κa are the known dielectric constants for water and air, respectively, and n is a parameter related to the geometric arrangement of materials relative to the applied electric field [Ansoult et al., 1984]. The value of n is commonly assumed to be 0.5, which is expected in isotropic media [Birchak et al., 1974], but measured values for sediments have been observed to range between 0.4 and 0.65 [Chan and Knight, 1999].
 The petrophysical model of equation (8) can be adjusted to site-specific conditions, given estimates of κs and ϕ. The measurement of porosity from cores can be problematic, since the in-situ packing of unconsolidated or semi-consolidated materials is difficult to preserve through the coring process. Values of κs are sometimes assumed (e.g., a “representative” value is taken from the literature) or are determined with cores in the laboratory using time-domain reflectometry (TDR) methods [Topp et al., 1980; Martinez and Byrnes, 2001]. However, errors can be unintentionally introduced from several sources during this process. As petrophysical functions are frequency dependent [Robinson et al., 2003, 2005], relationships derived in the laboratory from TDR measurements, for example, are not necessarily appropriate for application at the field scale [Huisman et al., 2003; Moysey and Knight, 2004], where the frequencies employed are typically lower than in the laboratory. Petrophysical functions may also be derived in the field by correlating dielectric constant estimates, derived from cross-borehole GPR, with estimates of water content inferred from co-located neutron probe (NP) data [Hubbard et al., 1997]. However, tomogram artifacts (such as those resulting from smoothing, noisy data, or smearing due to limited ray coverage [Stewart, 1991]) and errors in NP data [Yao et al., 2004; Fares et al., 2004] can introduce errors into the estimated petrophysical function [Huisman et al., 2003; Day-Lewis and Lane, 2004; Moysey et al., 2005; N. Linde et al., Inversion of tracer test data using tomographic constraints, submitted to Water Resources Research, 2005].
 As will be described in section 2.3, our inversion approach allows for the estimation of petrophysical parameters in equation (8)—at present, we consider the estimation of κs—thus helping to overcome potential errors introduced by scale discrepancy and measurement error. It should be noted that we currently assume the error in the petrophysical function is contained entirely in the parameters, not in the petrophysical model itself; the flexible form of equation (8) ensures that it can be applied in most real-world situations.
2.2.2. Simulation of GPR Data
 Numerous techniques are available for simulating GPR data, ranging from ray-based methods [Cai and McMechan, 1995; Sethian and Popovici, 1999], to pseudo-spectral methods [Casper and Kung, 1996], to time-domain finite-difference full-waveform methods [Kunz and Luebbers, 1993; Bergmann et al., 1998; Kowalsky et al., 2001]. Ray-based methods are the simplest and most computationally efficient for the simulation of GPR travel times; they are based on a high frequency approximation that calculates the arrival time of the first break of the transmitted wave (i.e., the time at which the wave amplitude departs from zero) and ignores the remainder of the waveform [Bregman et al., 1989]. While a full-waveform finite-difference method is used in the synthetic examples of this study to simulate GPR data, the straight-ray method is used for inversion; we examine whether the straight-ray approximation, chosen for computational efficiency, significantly impacts the estimated parameters and predicted system behavior. As will be discussed in the synthetic examples given below, significant errors can arise in travel times simulated using the straight-ray method, depending on the corresponding water distribution, leading in some cases to a systematic error (over-prediction) in simulated travel times that must be accounted for in the inversion procedure.
 The travel time T for an EM wave traveling between the transmitting and receiving antennas in a domain characterized by discrete grid blocks can be approximated by defining a straight ray between the antennas and summing the travel times through each grid block that the ray travels:
where Li is the length of the travel path (linear line segment) in block i, N is the number of blocks through which the ray passes, and Vi is the EM velocity in block i. For the present work, the petrophysical function is modeled using the volumetric mixing formula of equation (8), which, through combination with equations (1) and (9), allows for the travel time T to be calculated as follows:
where Sw,i and ϕi are the water saturation and porosity in grid block i, respectively. Since porosity is presently assumed to be constant ϕi = ϕ.
 While more sophisticated methods (such as curved-ray methods) are reasonable alternatives for use in joint inversion, they would be computationally demanding, especially for three-dimensional models. Note that our methodology involves large numbers of parameter perturbations, each of which requires an entire flow simulation and a full set of simulated GPR surveys. Furthermore, curved-ray paths would need to be calculated for all antenna combinations that are used for each survey and for each parameter perturbation. Application of curved-ray methods to models with irregular grids would pose additional difficulties.
 The straight-ray travel time approximation (see equation (10)) allows for most calculations to be done only once and before the inversion begins (e.g., the list of values Li for each combination of transmitting and receiving antennas). Additionally, irregular and three-dimensional models are easily and efficiently simulated. The appropriateness of the straight-ray approximation is further discussed in section 3.1.1.
 In this work, the simulation of cross-borehole GPR travel times in a domain undergoing transient fluid flow was made possible through solution of equation (10) within iTOUGH2 [Finsterle, 1999]. Each travel time is thus a function of the distributions of water saturation and porosity, the variable dielectric parameters (n, κs), and remaining known parameters.
2.3. Joint Inversion Methodology
 Here we extend the method developed by Kowalsky et al. [2004a] for estimating flow parameter distributions in the vadose zone using hydrological and geophysical data collected during transient flow experiments. The approach used a maximum a posteriori (MAP) inversion framework [McLaughlin and Townley, 1996; Rubin, 2003] and employed concepts from the pilot point method [RamaRao et al., 1995; Gomez-Hernandez et al., 1997]. Application of pilot point methods involves the generation of spatially correlated parameter fields, which are subsequently perturbed by changing the values at select conditioning points (referred to as pilot points) to minimize the misfit between measured and simulated data (which could include both hydrological and geophysical data).
 At present, the permeability distribution is anisotropic and treated as a lognormal SRF. The spatially varying component of permeability is introduced through the permeability modifier ξ(x), defined through the following relationships:
where kv and kh are the mean values of vertical and horizontal permeability, respectively, and ξ(x) is an SRF with known patterns of spatial correlation (i.e., known semivariograms). Given the frequent availability of semivariograms derived from well data and investigations of nearby outcrops or sites with similar geology, it is considered reasonable to assume that the spatial correlation patterns are known. However, we note again that the methodology does not prevent the spatial correlation parameters from being treated as unknowns. The mean of ξ(x) is zero and its variance is equivalent, as is the semivariogram, to that of the log distributions of kh(x) and kv(x).
 The permeability modifier field is parameterized using pilot points, giving a vector of unknowns (ξpp) at the pilot point locations. Through sequential simulation [Deutsch and Journel, 1992], a permeability modifier field conditional to ξpp is generated. During the inversion procedure, the vector ξpp is repeatedly perturbed as ξ(x) is updated through sequential simulation, until the log-permeability field (calculated with equations (11) and (12)) is found that provides (1) an optimal match to the observed hydrological and geophysical data, and (2) minimal deviation of the unknowns from prior estimates, if available. The remaining unknown model parameters are simultaneously estimated with ξpp.
 In previous work, the petrophysical function was assumed to be known and error free. In addition, the technique used for simulating ground-penetrating radar measurements was limited to one simple data acquisition configuration, the zero-offset profile (ZOP), in which the transmitting and receiving antennae are kept at equal depths during each measurement. This acquisition geometry yields a single depth profile of GPR travel times. While ZOP surveys are useful for gaining depth-averaged information, they cannot be used to resolve lateral variations in material properties.
 Presently our methodology incorporates equation (10) for the calculation of GPR travel times for any transmitter and receiver combination (i.e., multiple-offset profile surveys). We accommodate two- and three-dimensional models with grid cells of any shape (regularly-spaced grids are not required). Additionally, we expand the parameters that can be considered as unknowns to include the vector of soil hydraulic parameters, such as ah = [ϕ, α, m, Swres, Swsat, kh, kv, …] from equations (2)–(6); the vector of petrophysical parameters aκ = [κs, n] from equation (8); and, while not explored in this study, semivariogram parameters, including, for example, the range parameter occurring in most semivariogram models [Deutsch and Journel, 1992]. These extensions permit investigations under more realistic conditions, such as where there is uncertainty in the petrophysical function as well as in the spatial correlation function. Moreover, as opposed to the ZOP surveys previously considered, the wealth of data available in multiple-offset profile surveys improves the accuracy and resolution of estimated soil hydraulic parameters.
 The general goal of the inverse problem in this study is the estimation of vectors a = [ah, aκ] and ξpp given the following measurements:
 1. GPR travel time measurements (T), given as zGPR = T(xTxt, xRxt) + eGPR, for survey times t (length n) and taken for the transmitting and receiving antenna positions xTxt and xRxt, such that
where Mi,j is the number of receiving antenna positions for transmitting antenna at survey time ti. The length of the vectors xTxt and xRxt depends on the number of transmitting and receiving antenna combinations used for each survey time, and is given by Mi,j. The measurement error vector associated with measurement of zGPR is of the same length and is given by eGPR. This general notation allows for data to be collected for varying subsets of receiving and transmitting antenna positions at different times.
 2. Hydrological measurements, given here as local water content measurements (e.g., inferred from NP data) zH = ϕ · + eH, taken at borehole positions xH (length MH) and at survey times tH (length nH), where eH is the measurement error associated with measurement of zH (length MH × nH). As noted previously, the porosity is assumed to be spatially uniform in the examples given below, although the methodology does not require it.
 Accurate characterization of eGPR is significant but challenging, since its magnitude and distribution depend on antenna separation distance and vertical offset. We hypothesize that accounting for spatial variations in eGPR is of secondary importance for the current application due to spatial averaging effects and data redundancy (overlapping of ray paths traveling between different antenna positions). For simplicity we model the distribution of eGPR as independent of time and antenna position and leave related investigations for future research.
 In order to test the approach with minimal data requirements, we do not work with point measurements of permeability in this study, although they can easily be included if they are available [Kowalsky et al., 2004a]. As mentioned above, the geostatistical information describing the log-permeability SRF is assumed known from outcrop studies or previously characterized sites with similar geology [cf. Rubin, 2003] (section 2.3).
 Assuming that (a) the measurement error vectors eGPR and eH are characterized by known normal distributions, (b) the log-permeability field is uncorrelated with other soil-hydraulic parameters, and (c) the prior information of the parameters is normally distributed, then the objective function (OF) that is minimized during inversion can be written as:
where pp and are the prior means of ξpp and a, respectively, and and Ca are the corresponding covariance matrices. The variances of form the diagonal terms in Ca (the remaining terms are zero). For the case in which log-permeability point measurements are available, pp and its variance values (which are used as the diagonal terms in are calculated through kriging [Deutsch and Journel, 1992]. Since we currently assume that no log-permeability point measurements are available, the prior values of pp equal zero (i.e., each pilot point is penalized equally for deviating from the mean log-permeability value, which is one of the unknowns in aH); in addition, the diagonal values of contain the known log-permeability variance, ensuring that the log-permeability values at the pilot point locations stay reasonably close to the mean value k.
 For models with spatially uniform soil properties (as in the example given in section 3.1), only one inversion realization is performed, giving MAP estimates that are equivalent to the weighed least squares solution. In models with heterogeneous permeability (as in the example given in section 3.2 and the application to field data given in section 4), multiple inversions are performed, each giving one realization of the MAP solution, and each obtained using a different initial log-permeability field (i.e., a seed number that is unique to each inversion realization is used for sequential simulation [Deutsch and Journel, 1992]).
 In the following examples, the Levenberg-Marquardt algorithm [Levenberg, 1944; Marquardt, 1963] was used to minimize the objective function.