Inverse Estimation of the Unsaturated Soil Hydraulic Properties from Column Outflow Experiments Using Free-Form Parameterizations

is often still limited by the ill-posedness of the problem being considered. Ill-posedness in general is affected Inverse methods are increasingly used to estimate the hydraulic by nonoptimal experimental design, poor measurement properties of unsaturated soils. The method generally uses a weighted least-squares approach in which numerically simulated data are fitted quality, errors in the assumed process model, or unsuitto measured data. In this study we used inverse methods to estimate the able parameterizations of the hydraulic functions (Horunsaturated soil hydraulic properties from continuous and multistep nung, 1983, 1990; Kool et al., 1987; Yeh and Simunek, column outflow experiments. The method employs piecewise polyno2002; Vrugt et al., 2003). mial functions to obtain a free-form parameterization of the hydraulic Parameterizations of the soil hydraulic functions serve properties, rather than fixed functional forms typical of the van Gento constrain the inverse problem. The number of degrees uchten–Mualem and Brooks–Corey–Burdine models. For the polyof freedom determines the flexibility of the parameternomial functions we used quadratic B-splines and piecewise cubic ized hydraulic functions. Most commonly used are the Hermite interpolation. The method leads to local parameterizations functions of van Genuchten–Mualem (van Genuchten, that can also be hierarchic, depending on the invoked number of de1980) and Brooks–Corey–Burdine (Brooks and Corey, grees of freedom. Since a suitable number of degrees of freedom cannot be defined a priori, we embedded the estimation method into a 1966), which contain a limited number of adjustable multilevel procedure, which also included a stability analysis, based coefficients. Since the individual coefficients are related on singular value decomposition of the sensitivity matrix. The optimito properties of the pore-size distribution, they are genzation procedure was made more stable by imposing monotonicity erally referred to as “parameters” (e.g., parameter n in constraints on the hydraulic functions. Tests with synthetic and meathe van Genuchten equation refers to the width of the sured data from column outflow experiments show the validity and pore-size distribution). The low number of parameters robustness of the method. may be achieved in part by coupling the retention function with the conductivity function through common parameters (again, parameter n can be considered as K of the hydraulic properties of unsatuan example). Furthermore, in these low-parameterized rated soils (i.e., the retention function relating the functions, the degrees of freedom have a global characwater content with the pressure head , and the hyter in that parameter changes will affect the hydraulic draulic conductivity function K as a function of or ) functions over the entire pressure head domain ( ∞,0), is essential for most or all studies involving water flow even if only one degree of freedom is modified. and solute transport in the vadose zone. Most traditional The use of low-parameterized functions can cause methods to determine these properties require relatively problems in cases where the selected hydraulic functions restrictive initial and boundary conditions, and thus can are not flexible enough to represent the actual shape be time-consuming, laborious, and expensive. Moreover, of the hydraulic properties, such as for structured soils these methods generally aim at identifying either the or for soils having pore-size distributions that do not retention or conductivity properties. Because of major have an approximately lognormal shape. In such cases, advances in computational techniques and computer systematic errors in the fit of the retention function may power, estimation of the unsaturated soil hydraulic proppropagate into the estimated conductivity function in erties by means of inverse modeling has now become a disproportionate manner (Durner et al., 1999a). To an attractive alternative to traditional methods (Hopovercome this problem, piecewise continuous and commans et al., 2002). posite (bimodal) functions have been introduced (Othmer Popular laboratory approaches for the inverse estimaet al., 1991; Durner, 1992; Wilson et al., 1992; Ross and tion of the soil hydraulic properties have been one-step Smettem, 1993; Mohanty et al., 1997). Unfortunately, or multistep outflow methods (Kool et al., 1987; Durner when such functions are used in models that couple et al., 1999a). Whereas the general feasibility of outflow the conductivity function with the retention curve (e.g., experiments for identifying hydraulic functions has been Burdine, 1953; Mualem, 1976), the approach may lead to demonstrated in several studies (e.g., Hopmans and Siinstability in the conductivity prediction near saturation munek, 1999), the applicability and success of this method (Durner, 1994). Furthermore, the identifiability of individual parameters in these composite functions is generS. Bitterlich and P. Knabner, Institute of Applied Mathematics, Univ. ally much worse than for the low-parameterized funcof Erlangen-Nuremberg, Martensstr. 3, 91058 Erlangen, Germany; tions. Consequently, their use in inverse procedures for W. Durner and S.C. Iden, Institute of Geoecology, Braunschweig estimating the hydraulic properties has been limited to Technical Univ., Langer Kamp 19c, 38106 Braunschweig, Germany. feasibility studies so far (e.g., Zurmühl and Durner, Received 27 Nov. 2003. Original Research Paper. *Corresponding author ([email protected]). 1998). One major difference between the use of unimodal and bimodal descriptions is that some of the Published in Vadose Zone Journal 3:971–981 (2004). parameters in the bimodal approach no longer have an  Soil Science Society of America 677 S. Segoe Rd., Madison, WI 53711 USA effect on the global shape of the hydraulic functions.