## 1. Introduction and Scope

[2] The predictive capability of unsaturated flow and transport models relies heavily on accurate estimates of the soil water retention and unsaturated soil hydraulic characteristics at the application scale of the model. To enable such accurate soil physical characterization, methodologies need to be developed that allow a rapid, reliable, and cost-effective estimation of the hydraulic properties of the considered soil domain, including its spatial variability. Most of the early work reporting on the estimation of hydraulic properties of unsaturated soils has focused on relatively small soil samples using static or steady state flow experiments. These static or steady state flow experiments have the advantage of being relatively simple to implement. However, these methods are typically time consuming and require restrictive initial and boundary conditions to satisfy the assumptions of the corresponding analytical solutions.

[3] Significant advances in computational capabilities have resulted in Inverse Modeling (IM) applications for the estimation of soil hydraulic properties from small soil cores [*Durner et al.*, 1997; *Hopmans et al.*, 2002a]. However, when using an IM approach, the soil hydraulic properties can no longer be estimated by direct inversion but are determined using an iterative solution, thereby placing a heavy demand on computational resources. In this iterative process, the soil water retention and unsaturated soil hydraulic conductivity characteristics are indirectly determined from repeated numerical simulations of the governing Richards' equation:

thereby minimizing the difference between the observed and model predicted flow variables such as water content and fluxes. Using one-, two-, or three-dimensional forms, this transient equation solves for soil water matric potential, water content and water flux density as a function of time and space. In equation (1), *C* denotes the soil water capacity (L^{−1}), *K* is the unsaturated hydraulic conductivity tensor (L T^{−1}), *h*_{m} is the soil water matric head (L), *z* (L) denotes the gravitational head to be included for the vertical flow component only, and *A* (L^{3} L^{−3} T^{−1}) is the volumetric sink term, representing sources and/or sinks of water. For isotropic soils, *K* simplifies to a scalar that is a function of both *h*_{m} and the spatial coordinates. Boundary conditions must be included to allow for specified soil water potentials or fluxes at all boundaries of the simulated unsaturated soil domain. Moreover, user-specified initial conditions and time-varying source/sink terms need be specified. Both, the soil water retention and unsaturated hydraulic conductivity functions (referred to as soil hydraulic functions) are highly nonlinear, with *h*_{m} and *K* varying many orders of magnitude over the water content range that significantly contributes to water flow.

[4] Research on the applicability and suitability of the inverse approach toward identification of the soil hydraulic properties has focused primarily on five issues, (1) the type of transient experiment and kind of prescribed initial and boundary conditions suited to yield a reliable characterization of the soil hydraulic properties [*Hopmans et al.*, 2002a; *van Dam et al.*, 1992, 1994; *Ciollaro and Romano*, 1995; *Santini et al.*, 1995; *Šimůnek and van Genuchten*, 1996, 1997; *Šimůnek et al.*, 1998a; *Romano and Santini*, 1999; *Durner et al.*, 1997; *Wildenschild et al.*, 2001], (2) the determination of the appropriate quantity and most informative kind of observational data [e.g., *Zachmann et al.*, 1981; *Kool et al.*, 1985; *Parker et al.*, 1985; *Kool and Parker*, 1988; *Valiantzas and Kerkides*, 1990; *Toorman et al.*, 1992; *Eching and Hopmans*, 1993; *Eching et al.*, 1994], (3) the selection of an appropriate model of the soil hydraulic properties [*Zachmann et al.*, 1982; *Russo*, 1988; *Zurmühl and Durner*, 1998], (4) the construction and weighting of multiple sources of information in an objective function [*van Dam et al.*, 1994; *Hollenbeck and Jensen*, 1998; *Vrugt and Bouten*, 2002], and (5) the adoption and development of Bayesian and multiple-criteria parameter estimation strategies that can be used to quantify the uncertainty (probabilistic and multiobjective) associated with the inversely estimated soil hydraulic properties [*Kool and Parker*, 1988; *Hollenbeck and Jensen*, 1998; *Vrugt and Bouten*, 2002; *Vrugt et al.*, 2003a]. With these developments, the capabilities and limitations of the inverse approach for the identification of soil hydraulic properties from laboratory soil cores may be considered reasonably well understood. Despite this progress made, still little is known about the suitability of the inverse approach for the identification of vadose zone properties at larger spatial scales.

[5] It has been a major challenge to integrate these small-scale measurements of soil hydraulic properties in hydrologic models that apply across a range of spatial and temporal scales [*Gelhar*, 1986; *Grayson and Blöschl*, 2001]. In most applications, prediction of soil-water dynamics at larger spatial scales uses soil hydraulic properties determined from laboratory core or small field plot measurements, and are included in hydrologic models with a grid or element size much larger than the core or field plot scale. Because of the high nonlinearity of the soil hydraulic functions, their application across spatial scales is inherently problematic. Specifically, the averaging of processes determined from discrete small-scale samples may not be representative of the key hydrologic processes of the larger spatial domain. In addition, the dominant hydrologic flow processes may vary between spatial scales, so that potentially different models need to be used to describe water flow at the soil pedon, field, or watershed scale.

[6] Typically, in hydrologic studies of large spatial dimensions, one may apply a deterministic approach, using a distributed physically based model with upscaled effective soil properties [*Blöschl et al.*, 1995] or use stochastic modeling. A stochastic model preserves the small-scale characteristics of the measurement, but provides estimates of effective properties at the larger spatial scale after accounting for spatial heterogeneity of hydraulic properties. Stochastic approaches to upscale soil hydrologic processes from the local to the field scale include analytical models, based primarily on perturbation approximations of Richards' equation [e.g., *Zhang*, 2002]. Alternatively, numerical stochastic models have used Monte Carlo (MC) simulations to derive effective field-scale hydraulic properties and to predict field-scale hydraulic behavior based on local-scale measurements [*Hopmans and Stricker*, 1989; *Harter and Yeh*, 1998; *Harter and Zhang*, 1999]. Any of these approaches can become computationally intensive, requiring a large number of model simulations.

[7] Instead of a formal upscaling technique that incorporates nonlinear effects on upscaled soil properties applicable across a range of spatial domains, we here propose using a deterministic inverse modeling approach [*Hopmans et al.*, 2002b]. To estimate effective vadose zone parameters, a parameter optimization technique will be applied that is consistent regarding the spatial and temporal scale of the measurement and model parameter support. However, unlike in small-scale experiments, boundary and initial conditions at the larger spatial scales are not as clearly defined because direct measurement techniques are mostly not available. Furthermore, the data available to characterize large-scale vadose processes are sparse, both in space and time. Both these factors lead to significant uncertainty when estimating effective large-scale parameters. Therefore the application of deterministic inverse modeling at the watershed scale must account for the uncertainty of the estimated large-scale parameters and the associated prediction uncertainty. This requires the use of statistically based parameter estimation algorithms. An additional key element of the proposed distributed approach is that the applied vadose zone model must be able to simulate the key hydrologic processes that dominate the larger spatial domain, by using appropriate effective hydrologic parameters. This requires the use of a process-based model that incorporates functional distributed vadose zone parameters that can account for the relevant observed hydrologic processes.

[8] Current computational capabilities have evolved to a point, where it is now possible to use multidimensional physically based watershed models to study spatial and temporal patterns of water flow in the vadose zone [*Beven*, 2001; *Madsen*, 2003; *Panday and Huyakorn*, 2004]. With the availability of powerful personal computers, efficient computational methods, and sophisticated GIS, remote sensing and advanced visualizations tools, the hydrologic community is beginning to take advantage of the potential and utility of these physically based numerical models. With few exceptions, these models are based on complex multidimensional governing equations. They have received limited attention, primarily because of their computational, distributed input, and parameter estimation requirements.

[9] Considerable progress has been made in the application of automated optimization algorithms to estimate hydrologic model parameters across a range of spatial scales. However, emphasis is mostly placed on the estimation of a single optimal set of model parameters, thereby effectively neglecting the influence of parameter uncertainty. Such uncertainties arise mainly from the inability of the calibration process to uniquely identify a single optimal parameter set, from measurement errors associated with system input and output and from model structure errors. The hydrologic community is increasingly aware that hydrologic model identification and evaluation procedures should explicitly include uncertainty estimates [*Kuczera and Parent*, 1998; *Bates and Campbell*, 2001; *Thyer et al.*, 2002; *Vrugt et al.*, 2003a, 2003b, 2003c, 2004]. To acknowledge the presence of parameter uncertainty and to develop a tool that can be used to estimate this uncertainty, *Vrugt et al.* [2003b], recently developed the Shuffled Complex Evolution Metropolis-University of Amsterdam/Arizona (SCEM-UA) global optimization algorithm. The SCEM-UA algorithm is a general purpose global optimization algorithm that provides an efficient estimate of the most likely parameter set and its underlying posterior probability distribution within a single optimization run. The algorithm is an extension of the SCE-UA population evolution method developed by *Duan et al.* [1992].

[10] The aim of the present paper is to explore the usefulness and applicability of this inverse method to estimate vadose zone parameters at the small catchment and watershed scale by using spatially distributed tile drainage data as calibration targets. We hypothesize that the proposed inverse modeling approach will significantly improve our understanding of unsaturated water flow at larger spatial and temporal scales. To test the proposed model calibration approach, we selected the 3880-ha Broadview Water District (BWD), located on the west side of the San Joaquin Valley of California. The BWD has been the subject of various investigations [*Vaughan and Corwin*, 1994; *Vaughan et al.*, 1995, 1999; *Bourgault et al.*, 1997; *Corwin et al.*, 1999]. These research efforts have resulted in a comprehensive measurement data set of spatially distributed, weekly tile drainage flows and groundwater table depths throughout the district. This data set provides a unique opportunity to study spatial and temporal patterns of soil water flow by inverse modeling. As in other watersheds, however, this data set includes considerable uncertainties, arising from unknown soil properties, limited information about the spatial variations in rainfall, crop transpiration and soil evaporation across BWD, and the unknown spatial distribution of groundwater flow. In this paper, we compare three different mathematical models (representing different levels of model complexity) and consider three spatial resolution scales (field, drainage unit, and water district scale) for their ability to minimize uncertainty in the calibration parameters while also minimizing model prediction errors.

[11] The remainder of this paper is organized by sections. Section 2 discusses the BWD including an overview of the measurements that are available for model calibration, presents a condensed description of the MODHMS and BUCKET hydrologic models, and describes the SCEM-UA algorithm, which is used to solve for the single criterion optimization problem. In section 3, we explore the usefulness and applicability of the combined MODHMS-inverse methodology for the identification of vadose zone properties across a range of spatial scales using varying model complexity and spatial resolution of the boundary conditions. Fully integrated three-dimensional solutions of the Richards' equation (1) with spatially distributed boundary conditions are compared with results from a simplified conceptual bucket model and with one-dimensional solutions of the unsaturated flow equation with upscaled, spatially averaged boundary conditions. Finally, a summary with conclusions is presented in section 4.