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a Institute of Applied Mathematics, Univ. of Erlangen-Nuremberg, Martensstr. 3, 91058 Erlangen, Germany
b Institute of Geoecology, Braunschweig Technical Univ., Langer Kamp 19c, 38106 Braunschweig, Germany
* Corresponding author (w.durner{at}tu-bs.de)
Received 27 November 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORYAPPLICATIONRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYAPPLICATIONRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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with the pressure head 
, and the hydraulic conductivity function K as a function of or ) is essential for most or all studies involving water flow and solute transport in the vadose zone. Most traditional methods to determine these properties require relatively restrictive initial and boundary conditions, and thus can be time-consuming, laborious, and expensive. Moreover, these methods generally aim at identifying either the retention or conductivity properties. Because of major advances in computational techniques and computer power, estimation of the unsaturated soil hydraulic properties by means of inverse modeling has now become an attractive alternative to traditional methods (Hopmans et al., 2002). Popular laboratory approaches for the inverse estimation of the soil hydraulic properties have been one-step or multistep outflow methods (Kool et al., 1987; Durner et al., 1999a). Whereas the general feasibility of outflow experiments for identifying hydraulic functions has been demonstrated in several studies (e.g., Hopmans and Simunek, 1999), the applicability and success of this method is often still limited by the ill-posedness of the problem being considered. Ill-posedness in general is affected by nonoptimal experimental design, poor measurement quality, errors in the assumed process model, or unsuitable parameterizations of the hydraulic functions (Hornung, 1983, 1990; Kool et al., 1987; Yeh and Simunek, 2002; Vrugt et al., 2003).
Parameterizations of the soil hydraulic functions serve to constrain the inverse problem. The number of degrees of freedom determines the flexibility of the parameterized hydraulic functions. Most commonly used are the functions of van Genuchten–Mualem (van Genuchten, 1980) and Brooks–Corey–Burdine (Brooks and Corey, 1966), which contain a limited number of adjustable coefficients. Since the individual coefficients are related to properties of the pore-size distribution, they are generally referred to as "parameters" (e.g., parameter n in the van Genuchten equation refers to the width of the pore-size distribution). The low number of parameters may be achieved in part by coupling the retention function with the conductivity function through common parameters (again, parameter n can be considered as an example). Furthermore, in these low-parameterized functions, the degrees of freedom have a global character in that parameter changes will affect the hydraulic functions over the entire pressure head domain (–
,0), even if only one degree of freedom is modified.
The use of low-parameterized functions can cause problems in cases where the selected hydraulic functions are not flexible enough to represent the actual shape of the hydraulic properties, such as for structured soils or for soils having pore-size distributions that do not have an approximately lognormal shape. In such cases, systematic errors in the fit of the retention function may propagate into the estimated conductivity function in a disproportionate manner (Durner et al., 1999a). To overcome this problem, piecewise continuous and composite (bimodal) functions have been introduced (Othmer et al., 1991; Durner, 1992; Wilson et al., 1992; Ross and Smettem, 1993; Mohanty et al., 1997). Unfortunately, when such functions are used in models that couple the conductivity function with the retention curve (e.g., Burdine, 1953; Mualem, 1976), the approach may lead to instability in the conductivity prediction near saturation (Durner, 1994). Furthermore, the identifiability of individual parameters in these composite functions is generally much worse than for the low-parameterized functions. Consequently, their use in inverse procedures for estimating the hydraulic properties has been limited to feasibility studies so far (e.g., Zurmühl and Durner, 1998). One major difference between the use of unimodal and bimodal descriptions is that some of the parameters in the bimodal approach no longer have an effect on the global shape of the hydraulic functions.
This study focuses on local rather than global parameterization (i.e., perturbations of single parameters affect just segments of the hydraulic functions). We investigate the use of free-form parameterizations of the hydraulic properties using piecewise polynomial functions in which each degree of freedom influences the hydraulic functions only locally within a small range of water potentials. The term free-form is used to indicate that no a priori assumption is made about the specific shape of the hydraulic functions and the number of parameters (degrees of freedoms). Splines and other interpolation methods have been used previously to describe the soil hydraulic properties (Erh, 1972; Kastanek and Nielsen, 2001; Prunty and Casey, 2002). However, to our knowledge, our study is the first where this type of description is used for inverse modeling of transient flow experiments.
The free-form parameterization inverse approach will be used here to analyze several transient outflow experiments in which an initially saturated soil column is drained by decreasing the pressure head at the lower outflow boundary. Water flow across the upper boundary is forced to be zero. The suitability of outflow experiments to identify hydraulic parameters has been investigated since the early 1980s (e.g., see review by Kool et al., 1987). The current standard in applying the technique was described by Hopmans et al. (2002), and Durner et al. (1999a) published a review on problems and current limitations of the methodology.
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THEORY |
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 | [1] |
(L) is the pressure head, () represents the water retention curve, and K() (L T–1) is the hydraulic conductivity function. Inherent to this description is the assumption that water flow takes place as isothermal one-phase flow in a rigid porous medium. Equation [1] is subject to the following initial and boundary conditions:
 | [2] |
Free-Form Parameterizations
DuChateau (1997) showed that values of the pressure head during an outflow experiment satisfy the inequality
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LB(t) is continuously differentiable with 'LB < 0 for 0 < t < T and LB(0) = L. Consequently, an outflow experiment does not provide information for estimating the hydraulic functions outside of the pressure interval [, 0], where = LB(T) – L. Furthermore, the hydraulic functions are assumed to be constant when the soil is saturated ( > 0), such that it is sufficient to describe the hydraulic functions on the interval [, 0]. For physical reasons the hydraulic functions are additionally assumed to be nonnegative and monotonously increasing. The problem of monotonicity will be discussed in more detail below.
For the interval [, 0] we use a free-form cascaded parameterization. For this purpose we partition the interval [, 0] into:
 | [4] |
For simplicity we will use equidistant partitioning with h = j – j–1 for j = 2, ..., r. We can now proceed in several slightly different ways. Below we describe two possible free-form parameterizations: quadratic B-spline parameterization and piecewise cubic Hermite interpolation.
Quadratic B-Spline Parameterization
Quadratic B-splines consistent with the partitioning of Eq. [4] are defined as (e.g., de Boor, 1978; Prenter, 1975)
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1, 2 and r, r+1 are defined within the boundaries at and 0, respectively. Outside the interval [, 0] the functions are kept constant. These quadratic B-splines serve as shape functions such that the hydraulic functions are defined by
 | [6] |
= r + 1 is the number of degrees of freedom for each function () and K() assuming partitioning (Eq. [4]) with r nodes, and pj (j = 1, ..., 2) denotes the degrees of freedom of the parameterization (which must be determined). Because of the local support of the shape functions, each degree of freedom pj or pj+ influences or K, respectively, only locally on the subinterval (j–2, j+1). It is possible to choose several alternative piecewise polynomial shape functions instead of quadratic B-splines. However, linear B-splines are unsuitable as shape functions because they lead to a nondifferentiable function () with a corresponding discontinuous water capacity function.
Since parameterization (Eq. [6]) does not automatically ensure monotonicity in the soil hydraulic functions, we require the pi to satisfy additional constraints of the form
 | [7] |
= 
1 < ... < 
= 0 is a second partitioning of the interval [, 0] with a finer subdivision. For the parameterization using quadratic B-splines we need at least one additional subdivision of partitioning (Eq. [4]). Even then, however, monotonicity cannot be guaranteed completely. In some cases, numerical tests showed minor violations of the monotonicity requirement of the hydraulic functions. Having nonmonotonous hydraulic functions is physically unrealistic and could hamper convergence of numerical solutions of the Richards equation. For these reasons we also considered piecewise cubic Hermite interpolation since in this method monotonicity in the degrees of freedom implies monotonicity in the parameterized hydraulic functions.
Piecewise Cubic Hermite Interpolation
We again use Eq. [4] to partition the interval [, 0]. The degrees of freedom are now defined as the values of the hydraulic functions at the nodes of the partitioned interval; that is,
 | [8] |
On each subinterval (j, j+1) the functions () and K() are determined by cubic Hermite interpolation between pj and pj+1 or pr+j and pr+j+1, respectively (Prenter, 1975). Hermite interpolation requires derivatives of the and K functions at each node in addition to the values of and K. From Fritsch and Carlson (1980) we know that the derivatives can be computed from the values pj, j = l, ..., 2r, such that the resulting interpolates are monotonously increasing if the degrees of freedom satisfy
 | [9] |
) is the number of degrees of freedom for each unknown function.
Solution of the Forward Problem
The Richards equation was numerically solved using the hybrid mixed finite element method of Schneid and Knabner (1997). This method uses the Darcian flow rate
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Treatment of the Inverse Problem
In the following, the relative permeability and capillary pressure functions are both partitioned into the same number of intervals (alternative approaches are possible). Consequently, let p represent the vector of the 2 adjustable degrees of freedom of the invoked parameterization. The inverse problem is solved by means of a least-squares approach in which the objective function
 | [11] |
According to DuChateau (1997), measurements of the pressure head at the upper boundary of the soil column (z = 0) and the cumulative outflow rate at the lower boundary (z = L) permit a unique solution of the inverse problem. We hence use these two data types in our objective function. Since only the derivative of the water content appears in Eq. [1], the saturated water content, s, and the residual water content, r, cannot be determined simultaneously by inverse modeling of the outflow data. For this reason we fixed the value of (0) = s in the minimization. To improve estimation of the hydraulic conductivity near saturation, the saturated conductivity K(0) = Ks was also fixed. Minimization of the objective function O(p), with the monotonicity restrictions, was accomplished using a sequential quadratic programming method. Details of this procedure are given by Schittkowski (1988).
Sensitivity Analysis and Multilevel Procedure
Inverse modeling results using free-form parameterization depend very much on the number of degrees of freedom in each of the hydraulic functions. The number of degrees of freedom determines the flexibility of the parameterized hydraulic functions. This flexibility and the effects of measurement errors in the data both increase with increasing . Hence, an appropriate value of cannot be established a priori but must be determined during the solution of the inverse problem. The aim is to achieve a good balance between having flexibility in the hydraulic functions and limiting the effect of errors in the data.
We performed a linear sensitivity analysis similar to that used by Chavent et al. (1994). Assume that the minimum of Eq. [11] for data set Ym occurs at p, and the minimum for the data Ym + 
Ym at p + p, where Ym denotes the measurement errors and p the associated parameter deviations. A first-order expansion leads to the approximate problem of minimizing
 | [12] |
The minimum of Eq. [12] in the least squares sense occurs at
 | [13] |
This equation shows that the measurement errors and parameter changes are coupled via the sensitivity matrix [
Ys(p)]/p containing the derivatives of the simulated data with respect to the parameters. We assume here that 2 < N and that the sensitivity matrix has a full rank 2.
Singular value decomposition (Golub and van Loan, 1996) of the sensitivity matrix leads to a relationship between the identification error and the measurement error as follows
 | [14] |
Ym and the sensitivity matrix are appropriately modified to incorporate the weights in W. Furthermore, 
j (j = l, ..., 2) are the singular values of the sensitivity matrix, whereas {uj} and {vj} form orthogonal bases of the range of the sensitivity matrix and the range of the transposed sensitivity matrix, respectively. Since the singular values determine the influence of data errors on the identification errors, we define two characteristic numbers, the spectral condition number of the sensitivity matrix.
 | [15] |
 | [16] |
The minimization approach for the objective function and the sensitivity analysis were embedded into a multilevel algorithm. Minimization starts with the least possible number of degrees of freedom. The number of degrees of freedom is subsequently increased step by step until a certain stopping criterion is satisfied, or the maximum number of degrees of freedom is achieved. At each level the initial parameter values for the minimization are set by interpolating the minimization result of the previous level to the new parameterization. By making use of the sensitivity analysis, a stopping criterion is formulated that terminates the multilevel identification process if the characteristic numbers exceed given tolerance parameters:
 | [17] |
indicates the presumed measurement error of the adopted experimental setup. Suitable tolerance parameters can be found from numerical experiments by comparing the development of singular values with the fitting errors. Most problems will show a slow increase in the characteristic numbers. When the characteristic numbers increase dramatically, the multilevel estimation process should be terminated. We refer the reader to Bitterlich and Knabner (2001)(2002a, 2002b, 2003) and Igler and Knabner (2000) for additional details about the identification method. |
APPLICATION |
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The synthetic data were generated by simulating an outflow experiment for a soil column of Guelph loam (drying curve) with a length of 15 cm. The hydraulic functions used for this simulation were given by the van Genuchten–Mualem model with s = 0.52, r = 0.218, 
= 0.0115 cm–1, n = 2.03, m = l – l/n, Ks = 1.3167 cm h–1, and 
= 0.5 (van Genuchten, 1980). The pressure applied at the lower boundary was decreased continuously from the initial equilibrium condition with LB(0) = +15 cm to LB(40) = –200 cm after 40 h. The simulated cumulative outflow and the pressure head at the upper boundary were used in the optimization, first without noise and then with 5% Gaussian noise added.
The measured data were obtained by means of both a continuous and a multistep outflow experiment on an undisturbed core taken from a field site near Bayreuth, Germany. The sample (diameter 10 cm, vertical length 15 cm) was taken at the 10- to 25-cm depth from the Ap (0–35 cm) horizon of a Stagnic Cambisol. The soil is a very dense loamy sand (78% sand, 18% silt, 4% clay, 1% organic matter content) with a volumetric saturated water content of 0.31 and a saturated hydraulic conductivity of 100 cm d–1. The samples were mounted on a sintered glass plate (height 0.7 cm, air entry = –150 cm, Ks = 120 cm d–1) and slowly saturated from the bottom with de-aired water. The saturated conductivity was measured by establishing a constant-gradient flux from the bottom to the top of the column. The outflow experiments started by applying a continuously or stepwise increasing suction to the lower boundary. A fully automated computer controlled device was used to control the experiment and monitor outflow at the bottom, while a tensiometer at a depth of 3.5 cm from the top of the soil core provided pressure head data. A detailed description of the laboratory device is given by Zurmühl (1998).
Two types of outflow experiments were performed sequentially on the same sample. In the "Continuous" experiment, the pressure at the bottom of the column was decreased continuously (and linearly in time) from +15.2 cm at t = 0 to –60 cm at t = 127 h. In the "Multistep" experiment the pressure at the bottom was decreased stepwise from +15.2 cm to –60 cm at t = 100 h. The sample was resaturated between the two experiments as described above.
As mentioned, values of s and Ks were fixed during the optimization. Since the objective function consisted of two types of data with different units and magnitudes, we weighted the data with the inverse of the squares of their mean values (Kool et al., 1987). Also, the multilevel algorithm requires the parameterization to be refined by equidistant bisectioning of the pressure interval [, 0]. This means that we used a sequence of equidistant partitions with r = 2, 3, 5, 9, 17, and 33 nodes in the multilevel algorithm.
To compare the free-form optimization method with a more classic approach, additional optimizations were performed with the standard van Genuchten–Mualem functions. In this approach the van Genuchten coefficients , n, and r were optimized using the Levenberg–Marquardt method. A modified version of the HYDRUS-1D code (Simunek et al., 1998) was used to solve the direct problem. Both s and Ks were again fixed at their measured values as in the free-form parameterization method, the tortuosity parameter was assumed to be 0.5, and m was constrained by the relation m = 1 – l/n. The objective function was taken to be the same as for the free-form method.
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RESULTS |
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= –200 cm. This illustrates the poor sensitivity of the experiment near the lower limit of the experimental pressure head range. Identification here is accurate and robust only up to the lowest observed pressure at about –175 cm (Fig. 1, second row). By comparison, identification of the hydraulic functions in the wet part of the curves was very stable since the curves here were fixed at their saturated values.
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For the case without added noise (results not further shown here), we obtained almost identical results for r = 2, 3, and 5, with nearly perfect fits of the data, while values of the objective function were essentially zero. Optimization of the data without noise produced about the same low fitting errors in the hydraulic conductivity as optimization of the data with 5% noise. Not fixing Ks produced much larger errors in both cases.
Cubic Hermitian parameterization was found to produce very similar results as for the B-splines (not further shown here) for both the noise-free and noisy data. Initially, for low r values (r < 5), values of the objective function and the fitting error were greater than those for quadratic parameterization because of less flexibility (one less degree of freedom). For the higher parameterization levels, we obtained almost identical results for the objective functions and hydraulic properties. Hence, except for the monotonicity issues discussed above, no reasons exist for preferring one method over the other. Rather, one may want to decide a posteriori whether one of the two methods is superior for a specific case.
Fit to Experimental Data
The free-form parameterization approach appears particularly suited for the hydraulic functions of porous media having bimodal pore-size distributions, as are commonly found for fine-textured and aggregated soils, or fractured rock (Othmer et al., 1991; Coppola, 2000; Tuller and Or, 2002; Peters and Klavetter, 1988). While the Bayreuth soil used in this study had a relatively unimodal pore-size distribution, attempts to describe its flow behavior with the classic van Genuchten–Mualem model were not very successful (Durner et al., 1999b). Neither did the results improve by increasing the number of degrees of freedom in the classic functions (e.g., by allowing the saturated conductivity, Ks, and tortuosity coefficient, , to be adjustable parameters). Moreover, the classic functions obtained for this soil using inverse modeling differed for different experimental procedures (continuous vs. multistep outflow) and for different methods of weighting the data in the objective function (outflow vs. outflow and pressure head data). These difficulties led us to believe that for this problem either the measurements were incorrect, the Richards equation was invalid, or the invoked analytical van Genuchten–Mualem functions for the hydraulic properties were too restrictive.
We now discuss the two consecutive outflow experiments (continuous and multistep) performed on the same sample. We first discuss the continuous outflow experiment and compare results obtained with the free-form model with those using the classic van Genuchten–Mualem approach. Figure 3 shows the measured cumulative outflow (top row) and pressure head (second row) data. Also shown are the simulated data using the free-form model with r = 2 (left column), 9 (central column), and 33 (right column), as well as amplifications of the fitting errors (black area plots). The third and fourth rows of Fig. 3 depict the corresponding hydraulic functions. We selected r = 9 for the central row based on an analysis of the error functional and the condition number for increasing levels of parameterization (Fig. 4) . The fitting error decreased significantly until r = 9, whereas the condition number increased continually with increasing parameterization level. From this we conclude that r = 9 provides an appropriate degree of flexibility in fitting the data. To compare this fit with the optimal fit that can be obtained by using the classic van Genuchten parameterization, we include the latter one in the central column of Fig. 3, together with an enlargement of the deviations (gray area).
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Inspection of the hydraulic functions (Fig. 3, central column) shows that the estimated free-form retention function with r = 9 and the van Genuchten–Mualem retention function are almost identical. This suggests that the differences in simulated flow behavior were caused primarily or only by differences in the estimated hydraulic conductivity functions.
Figures 3 and 4 showed the results for the continuous outflow experiment. Similar results for the multistep outflow experiment are given in Fig. 5 and 6 . Analysis of the error functional and condition number (Fig. 6) shows that r = 9 again provides an appropriate degree of flexibility since the condition number increased substantially whereas the fitting error had reached its minimum. Similarly as for the continuous outflow experiment, the classic van Genuchten–Mualem model fitted the first part of the experimental data very well (Fig. 5), whereas the second part again showed systematic deviations. The free-form model nearly perfectly reproduced both the outflow and the pressure head data for the entire experiment, except for some dynamic effects near the air entry pressure (to be discussed below). Even the amplified fitting errors did not show significant differences between the measurements and the simulations.
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Comparison of Fig. 3 and 5 shows that the free-form retention functions identified from the continuous and multistep outflow experiments are very similar. Some differences are apparent between the conductivity functions, especially near saturation where the sensitivity is too low. We conclude from the almost perfect fit of the observed outflow and pressure head data, and from the similarity of the resulting hydraulic functions for the two outflow experiments, that the outflow processes were described well with the governing Richards equation. The only notable deviation during multistep outflow occurred at water contents close to the air entry pressure. Probably, the air-phase permeability in this pressure range was not high enough to allow for an unrestricted flow of air. This dynamic equilibrium phenomenon is often observed during multistep outflow experiments (e.g., Wildenschild et al., 1999) and may require application of a two-phase (air, water) flow model (Luckner et al., 1989; Schultze et al., 1999). Since this effect is generally restricted to only a relatively narrow pressure range, and its magnitude is often small, it should be negligible in most practical applications.
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DISCUSSION |
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The proposed method also avoids potential problems related to the shape of the hydraulic conductivity function near saturation. Inverse modeling with low-parameterized global hydraulic functions is unable to address the frequently observed inconsistency between the optimized and measured saturated hydraulic conductivity, especially for structured media (Mohanty et al., 1997; Schaap and Leij, 2000). This has also lead to the recommendation not to use Ks as a matching factor in hydraulic conductivity prediction models, but to prefer some measured value away from saturation (van Genuchten and Nielsen, 1985; Hopmans et al., 2002). Free-form estimation of the conductivity function requires fixing the conductivity at saturation because of the extremely low sensitivity of the function on outflow information near saturation. However, the resulting interpolation between the estimated unsaturated conductivity and the measured value at saturation will always be consistent with respect to the observed flow process.
Potential drawbacks of the proposed free-form parameterization method are (i) difficulties to implement free-form descriptions of the hydraulic properties in numerical models for variably saturated flow, (ii) the empirical (nonphysical) nature of the free-form parameterization coefficients, (iii) problems of extrapolating the resulting hydraulic functions beyond the moisture range covered by the experiment, and (iv) increased risk of obtaining hydraulic functions that are physically meaningless when the experimental data are incomplete or erroneous.
The first drawback should be relatively unimportant since many numerical models now read hydraulic properties from tables and/or operate internally with interpolations between tabulated values (e.g., Simunek et al., 1998). Since Hermitian and B-spline interpolations lead to smooth functions with monotonous and continuous derivatives for both the retention and conductivity function, their use in simulation models should not pose undue problems.
The second drawback (the empirical nature of the free-form parameters) could actually be an advantage in that it immediately reflects the nonphysical nature of most or all currently available analytical hydraulic property descriptions. Assigning physical meaning to soil hydraulic parameters (e.g., the tortuosity coefficient and other parameters in the van Genuchten–Mualem functions) can be misleading (Schaap and Leij, 2000) since this ignores the fact that the functions are essentially empirical expressions. This is further reflected by the debate on the physical meaning of the residual water content (e.g., Nimmo, 1991), and the saturated water content and saturated hydraulic conductivity (van Genuchten and Nielsen, 1985). This is even more the case for descriptions of the hydraulic parameters of soils having bimodal pore-size distributions. Such descriptions were initially presented within a purely empirical context to improve the ability to fit observed data (Durner, 1994), but were later redefined in terms of parameters reflecting the statistical properties of textural and structural pore systems (Coppola, 2000; Miyamoto et al., 2003).
A seemingly more serious problem of the proposed free-form method is the extrapolation of the functional relationships to moisture conditions outside the range of calibration. Since the free-form parameterization assumes local estimation, any extrapolation must be done by users with or without additional information of soil texture and related data. While in some cases such extrapolation may be justified (e.g., with certain pore-size distribution models and/or using independent estimates of the residual water content using pedotransfer function [Schaap et al., 1998]), measurements in the wet range generally provide little information about the shape of the hydraulic functions in the dry range. Overall, the free-form functions reflect the real information content that can be gained from the calibration experiment.
The fourth potential drawback addresses the independent parameterization of the retention and the conductivity functions. Decoupling enhances the risk of obtaining physically unrealistic results (e.g., a constant water content across a certain range of pressure heads and a simultaneously decreasing hydraulic conductivity function), especially when the measurements are incomplete or erroneous. In the classic approaches, the conductivity function is often coupled with the retention function by means of a statistical pore-size distribution model. One major reason for sticking to this coupling in the inverse analysis is to limit the number of unknown coefficients, thus avoiding nonuniqueness problems. However, our study indicates that the optimal number of parameters will depend on both the design of the selected flow experiment and the quality of the measured data. Coupling the retention and conductivity functions using shared parameters is not always necessary and may actually reduce flexibility in the hydraulic description. Still, we recognize that coupling serves as a stabilizing feature in hydraulic parameter estimation analyses since it forces the constitutive functions into shapes that are physically consistent. It may also allow identification of the hydraulic functions from outflow data alone in cases where the decoupled case could fail. Since coupling the conductivity curve with free-form parameterized retention curve using the model of Mualem (1976), or any other predictive pore-size distribution model poses no inherent problems, we will address this approach in a future study.
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CONCLUSIONS |
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