## 1. Introduction

[2] Quantitative subsurface hydrologic analysis is based on the conceptualization, development, and testing of hydrologic models [*Neuman et al.*, 2003]. Model conceptualization is the process of observing a system and proposing a simplified representation of the system that incorporates the features deemed important to the processes under observation (e.g., water flow or solute transport). Model development translates the proposed conceptual model (or models) to a mathematical or numerical model(s) that can be used to test a hydrologic hypothesis. Model testing is an analytical process, wherein the predictions of the model(s) are compared quantitatively with the available data. The goals of model testing are to find the values of the adjustable parameters in a model that result in the best fit of the model predictions to the data (calibration) [e.g., *Kim et al.*, 1999; *Poeter and Hill*, 1999, *Vrugt et al.*, 2009b]; to quantify the goodness of fit and assess parameter nonuniqueness for the calibrated model [e.g., *Vrugt et al.*, 2003; *Mugunthan and Shoemaker*, 2006]; and, increasingly, to compare the goodness of fit of multiple models to the data [*Neuman*, 2003; *Ye et al.*, 2004; *Troldborg et al.*, 2007, *Vrugt and Robinson*, 2007]. The results of model testing can be used to revise model conceptualization and/or development as well as to guide the design of future data collection activities. In short, the three phases of model-based hydrologic analysis can be described as (1) hypothesizing which subsurface structures and processes are significant, (2) translating this hypothesis into mathematical expressions and parameterizations, and (3) testing the hydrologic models against observations.

[3] To test multiple conceptual or mathematical models effectively, the inverse approach used for parameter calibration must be efficient in extracting relevant information from the observed data. The efficiency of parameter calibration depends on both the inverse algorithm selected and the way in which the inverse problem is posed. Inverse analysis is common to most scientific disciplines. Hydrologic science has adopted and developed many inverse approaches. But, until recently, there have been few studies regarding the formulation of efficient inversion strategies that incorporate multiple measurement types, including indirect measurements. The development of such inversion strategies for hydrologic problems is critical as the use of geophysical methods becomes increasingly common for monitoring subsurface flow and transport.

[4] Geophysics is a mature discipline that has made fundamental contributions to a range of scientific disciplines [e.g., *National Academy of Sciences*, 2000]. Many of these contributions stem from the ability of geophysical methods to provide unparalleled views into the earth. As a result, geophysics is a cornerstone of oil and mineral exploration and production. Increasingly, geophysical imaging of the subsurface is also being used to conceptualize and develop hydrologic models through mapping subsurface structures and improving estimates of the spatial distribution of hydrologic properties [e.g., *Hubbard and Rubin*, 2000; *Vereecken et al.*, 2004]. Advanced joint inversion methods have been developed to use multiple geophysical methods to reduce the nonuniqueness of the estimated structural models [e.g., *Vozoff and Jupp*, 1975; *Gallardo and Meju*, 2003, 2004; *Linde et al.*, 2006a]. In addition, geostatistical methods have been developed to estimate hydrologic property distributions based on statistical correlations present in geophysical images [e.g., *Cassiani et al.*, 1998; *Hubbard et al.*, 1999; *Yeh et al.*, 2002; *Chen et al.*, 2004]. Finally, hydrologic structure and parameter distributions have been estimated simultaneously using geophysical and hydrologic data [e.g., *Hyndman et al.*, 1994; *Hyndman and Gorelick*, 1996; *Chen et al.*, 2006, *Linde et al.*, 2006b]. In general, there is wide and growing recognition of the value of geophysics for subsurface imaging to aid in the conceptualization of hydrologic models.

[5] The use of geophysical data for testing hydrologic models also has a long history. For example, many studies have used electrical [e.g., *Daily et al.*, 1992; *Park*, 1998; *Kemna et al.*, 2002; *Sandberg et al.*, 2002; *French and Binley*, 2004; *Halihan et al.*, 2005; *Vanderborght et al.*, 2005; *Cassiani et al.*, 2006; *Chambers et al.*, 2006] and/or electromagnetic [e.g., *Binley et al.*, 2001; *Day-Lewis et al.*, 2003; *Cassiani et al.*, 2004; *Lambot et al.*, 2004; *Turesson*, 2006; *Deiana et al.*, 2008; *Looms et al.*, 2008b] methods to monitor changes in water content or solute concentration with time. Despite the differences in hydrologic targets and geophysical methods used in these studies, they have all followed the sequential approach to using the geophysical data to test a hydrologic model outlined in Figure 1a, which we refer to as uncoupled hydrogeophysical inversion. Uncoupled hydrogeophysical inversion follows three independent steps.

[6] 1. Geophysical survey data are inverted to estimate the spatial distribution of a geophysical property throughout the subsurface region of interest (Figure 1a, geophysical inversion).

[7] 2. A petrophysical relation [e.g., *Archie*, 1942; *Topp et al.*, 1980; *Binley et al.*, 2005] is used to convert the geophysical property map[s] to hydrologic state distributions at each measurement time (Figure 1a, dashed line linking geophysical inversion to hydrologic inversion).

[8] 3. The inferred hydrologic states are used either independently or together with directly measured hydrologic states to test hydrologic models (Figure 1a, hydrologic inversion).

[9] The gray process boxes in Figure 1a are present to illustrate how multiple data streams could be included in the uncoupled analysis. In the examples presented here, only one data stream (electrical conductivity data) is used. This approach, using a single data stream, is illustrated by the black process boxes.

[10] Uncoupled inversion propagates measurement errors and uncertainties related to parameter resolution and uniqueness that arise during the independent inversion of the geophysical data to the hydrologic analysis through the petrophysical relation. A particular issue stems from the fact that geophysical imaging typically requires a large number of parameters to retain the flexibility necessary to capture arbitrary, complex distributions of properties in the subsurface (e.g., the electrical conductivity distribution associated with a contaminant plume). As a result, geophysical inverse procedures commonly require the use of prior information (e.g., a smoothness constraint) to stabilize underconstrained problems [e.g., *Menke*, 1984]. It has been recognized that this regularization may not reflect the hydrologic conditions and can limit the value of hydraulic property estimates derived from geophysical observations [*Day-Lewis et al.*, 2005; *Chen et al.*, 2006; *Slater*, 2007].

[11] There have been several alternatives proposed to improve the uncoupled inversion approach. The petrophysical conversion can be improved using apparent calibration relationships that vary with location to compensate for the impacts of the spatially variable sensitivity of measurement methods and associated inversion artifacts [e.g., *Moysey et al.*, 2005; *Singha and Moysey*, 2006]. In addition, temporal relaxation techniques can be used in the geophysical inversion to interpret multiple time slices simultaneously, thereby effectively reducing the number of free parameters to be estimated [*Day-Lewis et al.*, 2002]. Finally, some joint inversion approaches allow for simultaneous determination of geophysical property distributions and petrophysical relations [e.g., *Hyndman et al.*, 1994; *Hyndman and Gorelick*, 1996; *Chen et al.*, 2006; *Linde et al.*, 2006b]. However, all of these proposed advances still rely on an independent geophysical inversion step to infer hydrologic states.

[12] We examine an alternative approach to uncoupled hydrogeophysical inversion for model testing. The approach, hereafter referred to as coupled hydrogeophysical inversion [*Ferré et al.*, 2009], is based on the premise that the goal of model testing is to determine the degree of consistency between a proposed hydrologic model and associated observations, thereby assessing the likely validity of a proposed hydrologic model. From this basis, it seems most reasonable that the geophysical data should be interpreted in the context of the proposed hydrologic model. This differs from the joint inversion approaches outlined previously. In the joint approaches, a relationship between hydrologic and geophysical (or between two geophysical) properties is assumed, but the hydrologic model is not used to guide the geophysical interpretation [*Ferré et al.*, 2009].

[13] The workflow of coupled inversion, shown in Figure 1b, is similar to that used by *Rucker and Ferré* [2004] and *Kowalsky et al.* [2005], which was summarized by *Ferré et al.* [2009]. It is typical of nonlinear optimization problems where an initial parameter set is proposed, used to simulate observed measurements, and then updated based on misfit between the simulated and observed data values. The distinguishing factor in the coupled inversion strategy is that, for any observed geophysical data set, we couple a hydrologic and geophysical model to represent the forward model in the optimization. In practice, this is achieved through a straightforward and flexible process where an initial set of hydraulic parameters is proposed and a forward hydrologic model (e.g., flow and reactive transport) is run using these parameter values. The model-predicted hydrologic states (e.g., water content) are converted to geophysical properties using petrophysical relations. The resulting geophysical property distributions are used to predict the response for each measurement method at each observation time and location using geophysical (forward) models. During inversion, the hydraulic properties and the petrophysical model parameter values are optimized to minimize the difference between predicted and measured geophysical observations. In Figure 1b, the gray process boxes illustrate how the coupled inversion approach could use multiple data streams. In the examples presented here, only one data stream (electrical conductivity) is used as illustrated by the black process boxes.

[14] The coupled and uncoupled approaches to inversion have several distinct, yet important differences that can impact both the computational effort and the uniqueness of the interpretations. Coupled inversion does not require a geophysical imaging step, thereby avoiding geophysical resolution problems related to the estimation of a large number of poorly constrained parameters. This alleviates the need to construct point-specific apparent calibration relationships to account for the effects of the spatially variable measurement sensitivity [*Moysey et al.*, 2006]. This is especially important when the spatial sensitivity of the geophysical method depends on the spatial distribution of the hydrologic state of interest [*Klenke and Flint*, 1991; *Ferré et al.*, 1996; *Furman et al.*, 2003]. Another advantage is that explicit assumptions regarding the spatial continuity of geophysical properties (e.g., smoothness) are no longer required to stabilize the geophysical component of the inverse problem because the hydrologic model defines the spatial arrangement of geophysical properties using physically based predictions of the hydrologic properties. Temporal regularization methods are also no longer required because the temporal dynamics are also enforced by the physics underlying the hydrologic model. Furthermore, because the underlying hydraulic properties are estimated directly, there is no need to consider joint inverse techniques explicitly. Rather, all of the data are considered in a single inversion and any of the correlations among measurement types that form the basis of joint inversion techniques (e.g., cross gradients in work by *Gallardo and Meju* [2004]) arise naturally by coupling the process model (i.e., the hydrologic model) and the geophysical models. In summary, because coupled inversion interprets the geophysical data in the context of the proposed hydrologic model, it provides a better test of the consistency of the proposed hydrologic model with the geophysical observations.

[15] Despite the potential advantages of coupled inversion, relatively few investigators have used this method to constrain hydrologic models with geophysical data [*Rucker and Ferré*, 2004; *Kowalsky et al.*, 2004, , 2005, 2009; *Sicilia and Moysey*, 2007; *Finsterle and Kowalsky*, 2008; *Looms et al.*, 2008a; *Lehikoinen et al.*, 2009]. None of these investigations has directly compared coupled and uncoupled approaches for model testing.

[16] In this study, we demonstrate the advantages and limitations of coupled hydrogeophysical inversion using an illustrative example of infiltration into the unsaturated zone monitored by electrical conductivity surveys. We use a synthetic example with known hydrologic and petrophysical properties to allow for a quantitative comparison of the accuracy of uncoupled and coupled hydrogeophysical inversion approaches. This relatively simple example allows us to isolate the effects of coupled and uncoupled inversion from complications arising from soil and petrophysical parameter heterogeneity, boundary condition uncertainty, and model structural error. In particular, we examine the impacts of structural errors in the model on hydrogeophysical inversion by comparing inversions performed with two different data sets. The first data set is based on the analytical infiltration model of *Philip* [1957], which assumes a homogeneous soil. The second data set is generated using a numerical model for unsaturated flow in a heterogeneous medium (HYDRUS 1D) [*Simunek et al.*, 1999]. The homogeneous model is used for the inversion of both data sets, thereby introducing model structural errors for the case where the subsurface is actually heterogeneous. This latter analysis is intended to represent the common practice of applying hydrologic models and parameter distributions that are considerably simpler than reality to make numerical inversion tractable.