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U.S. Salinity Lab., USDA-ARS, 450 W. Big Springs Road, Riverside, CA 92507 USA
mschaap{at}ussl.ars.usda.gov
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTheoryMaterials and methodsNOTESResults and discussionSummary and conclusionREFERENCES |
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). We used a data set of 235 soil samples with retention and unsaturated hydraulic conductivity data to test and improve predictions with the MVG equation. The standard practice of using
and
resulted in a root mean square error for log(K) (RMSEK) of 1.31. Optimization of the matching point (Ko) and L to the hydraulic conductivity data yielded a RMSEK of 0.41. The fitted Ko were, on average, about one order of magnitude smaller than measured Ks. Furthermore, L was predominantly negative, casting doubt that the MVG can be interpreted in a physical way. Spearman rank correlations showed that both Ko and L were related to van Genuchten water retention parameters and neural network analyses confirmed that Ko and L could indeed be predicted in this way. The corresponding RMSEK was 0.84, which was half an order of magnitude better than the traditional MVG model. Bulk density and textural parameters were poor predictors while addition of Ks improved the RMSEK only marginally. Bootstrap analysis showed that the uncertainty in predicted unsaturated hydraulic conductivity was about one order of magnitude near saturation and larger at lower water contents.
Abbreviations: MVG, Mualem–van Genuchten model
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsNOTESResults and discussionSummary and conclusionREFERENCES |
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As an alternative to measurements, one can use estimation methods that utilize physical or empirical relations between hydraulic properties and other soil variables. The advantage of such methods, also called pedotransfer functions, is that the input variables can be measured more easily—and, hence, are more widely available—than hydraulic properties. For the prediction of water retention and saturated hydraulic conductivity, this approach has led to a number of pedotransfer functions that use soil texture, bulk density, or other soil variables as input (e.g., Rawls and Brakensiek, 1985; Ahuja et al., 1989; Vereecken et al., 1989; Schaap et al., 1998). Far fewer alternatives exist for unsaturated hydraulic conductivity. Although some pedotransfer functions are available (Saxton et al., 1986; Schuh and Bauder, 1986; Vereecken et al., 1990), pore-size distribution models by Burdine (1953) and Mualem (1976), among others, are more popular.
Generally speaking, the Burdine and Mualem models infer the pore-size distribution of a soil from its water retention characteristic. By making assumptions about continuity and connectivity of pores, integral expressions can be derived that describe unsaturated conductivity in terms of water content or pressure head. A general expression can be given as (after Hoffmann-Riem et al., 1999):
 | (1) |
= 2.
van Genuchten (1980) defined the following water retention:
 | (2) |
is the volumetric water content (cm3cm-3). The parameters r and s are residual and saturated water contents respectively (cm3cm-3), 
(>0, in cm-1) is related to the inverse of the air entry pressure, and n (>1) is a measure of the pore-size distribution (cf., van Genuchten, 1980; van Genuchten and Nielsen, 1985). Combination of Eq. [1] and [2] and for the Mualem parameters yields the following closed-form expression for K(Se):
 | (3) |
Equation [3] is frequently used to estimate unsaturated hydraulic conductivity using Eq. [2] (e.g., Powers et al., 1998; Vanderborght et al., 1998; Jones and Or, 1999; Wildenschild and Jensen, 1999), which requires that Ko and L must also be specified. Commonly, the saturated hydraulic conductivity, Ks, is used for Ko since it can be measured in a simple experiment. However, van Genuchten and Nielsen (1985) and Luckner et al. (1989) argued that Ks may not be an especially suitable matching point because Ks is sensitive to macropore flow, whereas unsaturated flow occurs in the soil matrix. They postulated that a matching point at slightly unsaturated conditions would yield better results. The term SLe in Eq. [3] is an empirical correction factor that was introduced by Burdine (1953) and Fatt and Dykstra (1951) to account for pore tortuosity (see Hoffmann-Riem et al., 1999 for a review). Mualem (1976) noted that L may be positive or negative. However, if SLe is to be interpreted in terms of pore continuity and tortuosity, SLe should always be smaller than 1 and hence L > 0 since 0 
Se 1. Mualem (1976) found that
was an optimal value for a data set of 45 disturbed and undisturbed samples. For a subset of Mualem's data, Yates et al. (1992) found that L varied between -3.31 and values much greater than 100. For a data set of 75 samples, Schuh and Cline (1990) reported that L varied between -8.73 and 14.80 with increased variability for smaller mean particle diameters. The geometric mean of L was 0.63 with a 95% confidence interval between -0.88 and 2.44. Using Eq. [2] with an exponent 1 instead of 1 - 1/n, Vereecken (1995) determined that L had a positive correlation with the sand fraction and n parameter, and a negative correlation with Ks. Wösten et al. (1995) found that L was related to the organic matter content. Despite these insights,
and
are the most common default values in predictive uses of Eq. [3].
The international database UNSODA has recently become available (Leij et al., 1996) and allows a more detailed investigation of pore-size models than was previously possible. Using 200 soils from the UNSODA database, Kosugi (1999) evaluated the standard Burdine
and Mualem
versions of Eq. [1] in conjunction with a lognormal function for water retention (Kosugi, 1994). The measured conductivity closest to saturation was used as a matching point. Kosugi (1999) found that usage of the standard parameters led to considerable errors in the description of unsaturated hydraulic conductivity. Much better results were obtained if L, ß or were optimized. Kosugi (1999) developed two models to predict relative unsaturated hydraulic conductivity from water retention parameters. The first model used optimal values for the entire data set
whereas the second model used
and utilized a power function between ß and a distribution parameter in the retention equation. Compared with the standard Burdine and Mualem models, the variance between measured and predicted relative hydraulic conductivities decreased by about 90%. Although this is a considerable improvement in the ability to estimate unsaturated hydraulic conductivity, Kosugi's models estimate relative hydraulic conductivity data. Predictive uses of Kosugi's models therefore need additional measurements or estimates of a matching point.
The objective of this paper is to develop predictive expressions for Ko and L for the Mualem–van Genuchten model (Eq. [3]) and compare their effectiveness and reliability with those of the standard model of Mualem (1976), where
and
. To this end, we will fit both Ko and L to the unsaturated conductivity data of 235 samples from the UNSODA database. We set ß and to 1 and 2, respectively. In this way, we can keep Eq. [3] mathematically simple and avoid strong correlations among fitted L, ß or , which would complicate any further modeling efforts. We will investigate the correlation structure among Ko, L, and potential predictors such as retention parameters, Ks, soil texture and bulk density and we will develop predictive models for Ko and L using a combined bootstrap-neural network approach (Schaap et al., 1999). Neural networks were used because of their ability to find patterns in complex data. The bootstrap was used to quantify the uncertainty in predicted Ko and L, and therefore uncertainty in K(Se), due to variability and ambiguity in the data set. Such information is useful for interpreting the reliability of predictions and essential when predicted Ko and L are used in stochastic analyses of water flow in unsaturated soils.
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Theory |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsNOTESResults and discussionSummary and conclusionREFERENCES |
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 | (4) |
This formulation makes it easier to plot unsaturated hydraulic conductivity and, more importantly, it minimizes a bias towards high conductivities when Eq. [4] is fitted to experimental data. Furthermore, Eq. [4] makes it easier to understand the effects of Ko and L on the shape of K(Se).
The effect of Ko on the shape of K(Se) can be easily understood as a scaling of K(Se). The roles of the second and third terms on the right hand side of Eq. [4] are more complex. Figures 1a through c show the sensitivity of the second term (Fig. 1a) and the relative conductivity
to variations in L for
(Fig. 1b) and
(Fig. 1c). The second term in Eq. [4] diverges at low Se for different L and has a positive slope for positive L (Fig. 1a). The third term always increases with Se but is invariant for L and is therefore represented by the curves at L = 0 in Fig. 1b and 1c. This term is dependent on the n parameter of Eq. [2] and, indirectly, also on because Se must be computed with Eq. [2]. The combination of the second and third terms in Fig. 1b and c demonstrate that higher values of L lead to a stronger reduction in K(Se) at low Se and vice versa. Although the contribution of the second term is the same for both values of n, there seems to be a difference in sensitivity of log10(Kr) to L for
and
. This effect is apparent because it results from the range that we used to plot log10(Kr), which was chosen to contain most of the measurements. In practice, however, only the hydraulic conductivity of soils with high n (predominantly sandy soils) can be measured over the full Se scale. Differences in sensitivity to L may therefore occur for different n.
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The sharp increase in hydraulic conductivity near saturation in Figure 1c is caused by the nonzero slope of Eq. [2] near
for n < 2. We refer to Vogel and Cislerova (1988) for a more detailed discussion of using Eq. [2] and [3] near saturation for low n.
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Materials and methods |
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- h pairs and at least five K - h pairs. Samples with chaotic data or with limited retention or conductivity ranges were omitted. Almost all samples were from temperate zones in the northern hemisphere (Belgium, 47; France, 1; Germany, 54; India, 1; Japan, 9; the Netherlands, 13; Switzerland, 54; United Kingdom, 11; USA, 22; former USSR, 23). Figure 2
provides the textural distribution of the samples and their classification in four textural groups: Sands (100), Loams (41), Silts (58) and Clays (36).
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Fitting Retention and Conductivity Parameters to Hydraulic Data
Equations [2] and [4] were fitted to measured data with the Simplex or Amoeba algorithm (Nelder and Mead, 1965; Press et al., 1988). Because we used parameters from Eq. [2] to predict unsaturated hydraulic conductivity with Eq. [4], we did not carry out simultaneous optimizations of Eq. [2] and [4] (cf., Yates et al., 1992). Instead, we first optimized Eq. [2] and then used the n parameters to compute Se and optimize Ko and L in Eq. [4].
Some samples had insufficient retention points to describe the entire retention curve. To avoid fitted parameters with unreasonable values (e.g., s > 1), we imposed the following constraints during the optimization: 0.0 < r < 0.3 cm3 cm-3; 0.6 
< s < cm3 cm-3 (where is the total porosity); 0.0001 < < 1.000 cm-1; 1.0001 < n < 10. The objective function that was minimized is given by:
 | (5) |
i and i' are the measured and predicted water contents respectively, Nw is the number of measured water retention points for each sample and p is the parameter vector {r, s, , n}.
The parameters Ko and L in Eq. [4] were subject to the constraints 0.001 (cm day-1) < Ko Ks and -2 - 2/(n - 1) < L < 100. The lower constraint for Ko was never violated; the upper constraint was necessary for 66 samples, mainly because of missing conductivity data near the wet end which would otherwise have led to Ko > > Ks. The following objective function was minimized:
 | (6) |
. As mentioned earlier, logarithmic values of Ki were used in Eq. [6] to avoid bias towards high conductivities in the wet range.
The goodness of fit of Eq. [2] and [4] was quantified with the root mean square error:
 | (7) |
Prediction of Ko and L
The results from the previous section will show how well Eq. [4] describes existing unsaturated hydraulic conductivity data. Our goal, however, is to evaluate how well we can predict unsaturated hydraulic conductivity when these data are not available. To this end, we need to estimate Ko and L from potential predictors like sand and clay percentages, bulk density, r, s, , n, and/or Ks. We computed Spearman rank correlations (e.g., Press et al., 1988) to investigate the correlation structure among fitted Ko, L, and these predictors. Spearman rank correlations were chosen over linear correlations because the variables were not always normally distributed.
Subsequently, we developed neural network models to estimate Ko and L from the potential predictors. Neural networks have previously been used to predict water retention and saturated hydraulic conductivity and, for the sake of brevity, we refer to Pachepsky et al. (1996), Schaap et al. (1998), and Schaap and Leij (1998) for details. The primary reason to use neural networks is that they can discover and implement complex nonlinear relations among variables (Hecht-Nielsen, 1991; Haykin, 1994). Similar to Schaap and Leij (1998), we used feed-forward backpropagation networks with one hidden layer, containing six nodes, and sigmoidal transfer functions. Instead of minimizing the variance of fitted and predicted Ko and L during the neural network optimization (cf., Schaap and Bouten, 1996), we used Eq. [6] as an objective function. Although computationally more intensive, this choice was motivated by the difference in sensitivity of Eq. [4] to different values of n and L, as discussed earlier. The change in objective function led to RMSEK values that were about 10% lower.
The neural network analysis was combined with the bootstrap method (Efron and Tibshirani, 1993) to study the uncertainty in predicted Ko and L. As was shown by Schaap and Leij (1998) for water retention and Ks, uncertainty and ambiguity in a data set used for model calibration leads to uncertainty in predicted hydraulic parameters. The theory behind the bootstrap method assumes that multiple alternative realizations of a population can be simulated from a single data set. By calibrating a (neural network) model on each of these alternative data sets, slightly different models result—leading to uncertainty in the prediction. The alternative data sets have the same size as the original data set and are generated by random sampling with replacement. Therefore, in a data set of N samples, each sample has a chance of 1 -[(N - 1)/N]N to be selected once or multiple times. Because some samples are selected more than once, each alternative data set contains about 63% of the original data. Neural networks were calibrated on each of the alternative data sets and validated with the 37% of the samples that were not selected. We will present only the validation results in this study.
The bootstrap method was combined with the TRAINLM routine of the neural network toolbox (Demuth and Beale, 1992) of MATLAB1 (version 5.0, MathWorks Inc., Natick, MA). The neural network code was modified to avoid local minima in the objective function. We used 100 bootstrap data sets resulting in 100 neural network models. In effect, each prediction yielded 100 values for Ko and L, which were subsequently summarized with means and standard deviations.
As error criteria, we will present coefficients of determination (R2) between predicted and fitted Ko and L. Further, we will compute RMSEK values (Eq. [7]) of our model predictions. Although it is not necessary to correct model predictions for the number of model parameters (np), we will still carry out these corrections to be able to compare the RMSEK values of the predictions with those of the optimizations.
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Results and discussion |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsNOTESResults and discussionSummary and conclusionREFERENCES |
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) were higher for the Loams than for the Sands while we also found that the Loams had lower log10(n) values than the Clays.
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in Eq. [4] leads to an overestimation of unsaturated hydraulic conductivity. At the same time, however, we have to realize that using Ko as a matching point will lead to a hydraulic conductivity that is too low at or near saturation since at
the hydraulic conductivity should be equal to Ks. This indicates that Eq. [4] cannot describe the shape of the measured conductivity characteristics near saturation. With the present data set it is impossible to use a bi-modal pore structure (e.g., Durner, 1994) to account for effects of macro-porosity on retention and conductivity curves. The results in this paper thus only pertain to conditions away from saturation (suctions stronger than a few centimeters).
We found negative L for all textural groups with the lowest values for the Loams and Clays. Although not all individual samples had a negative L, the fact that SeL > 1 for negative L makes it impossible to interpret SeL in Eq. [4] as a simple reduction factor that accounts for pore tortuosity and connectivity (Mualem and Dagan, 1978). Because Ko is constant for all Se of a sample, the negative L seem to compensate the third term in Eq. [4]. This term is equivalent to the curve for
in (cf. Fig. 1a–1c) and it could be argued that it drops off too quickly with decreasing Se because of the factor 2, which follows from in Eq. [1]. However, using a different water retention model, Kosugi (1999) reported negative L values even after setting
. Apparently, negative L are found irrespective of the chosen water retention model and the ß and configuration of Eq. [1]. This therefore seems to point to a failure in the pore-tortuosity or pore-interaction concepts in models defined by Eq [1]. Likewise, Hoffmann-Riem et al. (1999) concluded that models based on Eq. [1] should not be interpreted as being physically based. In the remainder of the study we will therefore treat Ko and L as empirical parameters.
Although we found negative L values for many samples, the results in Table 1 by themselves do not prove that using a negative L leads to substantially lower RMSEK (Eq. [7]) compared with a fixed value
(Mualem, 1976). We therefore varied L between -3 and 3 in steps of 0.5 while we fitted Ko and computed the average RMSEK for each soil textural group and the entire dataset (Fig. 3)
. The Sands have a clear minimum RMSEK of 0.60 at
, which is substantially lower than at
where the RMSEK is 1.21. The other textural groups also have lower RMSEK values at negative L but with smaller decreases relative to
. The more gradual slopes of the RMSEK curves for these groups are caused by their lower n values and limited Se range of the measurements. These factors cause that the shape of Eq. [4] is less sensitive to variations in L (cf. Table 1, Fig 1b and 1c). For all samples (All), the optimum value is also
. This value does not agree with the value of -3.09 in Table 1 because in that case both Ko and L were optimized and the reported L is an average for all 235 samples. Because all RMSEK curves are near-optimal at L = -1, it makes sense to use this value for all samples. It should be noted, however, that the optimal fixed value of L might depend on ratio of coarse textured soils (with L
-1) and fine textured soils (L < -1) in the data set. Furthermore, by fixing L and only optimizing Ko we must make a concession to the goodness of fit. Average RMSEK for all samples at
is 0.75 (Fig. 3), whereas it is 0.41 when both Ko and L were optimized (Table 1).
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r, s, , n, and Ks) and Ko and L for all 235 samples (All). Significance levels at
, 0.01, and 0.001 indicate where the Null Hypothesis (no correlation present) should be rejected. We also show truncated matrices for the four textural groups with the correlations between Ko, L, and the potential predictors. The correlation structures of the omitted parts are largely similar to the matrix of all 235 samples but they are less significant due to a reduced number of samples. Correlations for the case where we fitted only Ko and assumed
are not shown because these were largely similar to the results in Table 2.
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s can be understood because the total porosity is determined by the bulk and particle densities while s should be smaller or equal to the porosity. The autocorrelation of r is not equal to 1 because r had to be constrained to 0.0 for more than 100 samples when we fitted Eq. [2] to the retention data. Therefore, these samples obtain the same rank, resulting in a reduction in the Spearman correlation.
Table 2 shows that Ko has a significant positive correlation with Ks. This is partly caused by the constraint Ko Ks that we imposed during the optimization of Eq. [4] in which we forced
if the optimal value of Ko was greater than Ks. However, Table 2 shows that the correlation is not near 1; in fact, it is quite poor for the Silt group (0.28). Table 1 already showed that fitted Ko values were considerably lower, while standard deviations were higher, than measured Ks. This indicates that Ko cannot be calculated from Ks with a simple reduction factor. The Ko parameter shows a positive correlation with for all textural groups which is easily explained by the fact that larger indicate larger average pore sizes. Negative correlation coefficients are found for bulk density, except in the case of the Sand group. Lower bulk densities increase the pore space and therefore, potentially, increase the conductive path for water flow. Similarly, negative correlations are also found for r; in this case, lower r values effectively increase hydraulically active pore space. Conversely, s has a positive correlation coefficient with Ko for the Loams and the Silts because higher s values increase the amount of potentially mobile water. Note that, if significant, the correlation coefficients between r, s and bulk density, and Ko are higher for the individual textural groups than for all 235 samples. This finding points to possibly complex relations between Ko and its predictors. Correlations of sand and clay percentages with Ko are more ambiguous than correlations with retention parameters and usually small or insignificant.
In the case of L, we find positive correlation coefficients with n for all textural groups as was also reported by Vereecken (1995) for a combination of Mualem's (1976) model with a different retention function. This correlation implies that when n decreases, L decreases, causing the hydraulic conductivity to decrease less with Se (cf. Fig. 1a–c). Because L is predominantly negative and seems to compensate the third term in Eq. [4], it is difficult to interpret physically this correlation. The parameter exhibits a weaker correlation with L, except for the sands where no significant correlation was found. The negative correlation means that smaller values lead to larger values of L and a stronger reduction in hydraulic conductivity. Because small values of both and n are normally associated with finer textures, it seems that and n have opposite effects on L. A significant correlation between L and bulk density, sand, or clay is absent for most or all textural groups.
Prediction of Ko and L with Neural Networks
Three models for predicting Ko and L were tested. Model A reflects the traditional Mualem–van Genuchten model (van Genuchten, 1980) with
and
. In the case of Model B, we constructed neural network models that predicted only Ko while assuming that
. Model C predicts both Ko and L. Based on the results in Table 2, we used four different sets of predictors for Models B and C: (i) sand and clay percentages and bulk density, (ii) retention parameters {r, s, , n}, (ii) retention parameters and Ks, and (iv) Sets 1 and 3 combined. Models B and C were indexed according to these four sets of predictors (i.e., B1...B4, C1...C4).
Table 3 shows that the RMSEK of Model A is more than one order of magnitude, with a very high value for the Clays (1.70). Models B1 and C1 yielded somewhat lower RMSEK values by using sand, silt, clay, and clay (SSCBD) as predictors. Models B2 and C2 clearly show that water retention parameters make more effective predictors, as was already inferred from the correlation matrices in Table 2. Results for Models B3 and C3 show that adding Ks to the retention parameters increased the coefficients of determination of Ko. However, comparison with Models B2 and C2 shows that RMSEK were only marginally reduced, if at all. Models C2, C3, and C4 had lower RMSEK than the B2, B3, and B4 models, especially for the Loams and Clays. These findings suggest that both Ko and L should be predicted and should not be fixed to predetermined values.
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and n are also needed to compute Se with Eq. [2]. It is interesting to note that Model C2 uses less data than Model A (Ks is not required), yet it has a RMSRK that is about half an order of magnitude lower. Comparison of our results with those of the AB and FN models reported in Table 3 of Kosugi (1999) shows that Model C2 performs better. Assuming that Kosugi (1999) also used an average of 18.5 K(Se) observations per sample (his and our data sets were derived from UNSODA and partially overlap), the converted RMSEK results for Model AB and FN are 1.11 and 0.94 respectively, whereas Model C2 had an error of 0.84. Still, the RMSEK of Models C2, C3, and C4 are significantly higher (approximately 0.8) than the RMSEK of the direct fit to the measured K - h data (0.41, Table 1). The results for direct fit are essentially averages for individual samples and are insensitive to variations among samples due to systematic differences in, for example, measurement methods (cf. Stolte et al., 1994). The neural network models, however, attempt to implement relations that are valid for all the samples in the calibration data set. Consequently, the RMSEK of the model predictions are sensitive to systematic and random differences that exist from sample to sample.
Finally, Fig. 4
shows the predictions of Models A, B2, and C2 for the average hydraulic parameters of the sands and clays in Table 1. The inserts in Fig. 4 provide the values of the estimated Ko and L parameters as well as their uncertainty, which was generated with the bootstrap method. Figure 4 also shows the 90% confidence interval for Model C2 as shaded areas. These intervals were generated by evaluating Eq. [4] at many values of Se for each of the Ko and L pairs following from the 100 bootstrap–neural network analyses that were performed to create C2. Especially for the sand, Model A predicts systematically higher K(Se) values at Se > 0.15 for the sand and Se > 0.45 for the clay than Model B2 or C2. Below these values, Model A predicts smaller K(Se) than B2 or C2. The overprediction of Model A follows from the use of
. The underprediction at small Se is caused by setting L to 0.5 which causes a steeper drop in K(Se) than Models B2 and C2, which have smaller L values (cf., Fig. 1a–c). Predictions by Models B2 and C2 are largely similar. The confidence intervals of C2 show that even though Model C2 has a near-optimal minimum RMSEK, the uncertainty in predicted K(Se) can still be large. For example, the confidence intervals increase from less than one order of magnitude near saturation to about two orders of magnitude at low Se. We note that this uncertainty does not directly relate to variability found in the field but rather reflects variability and ambiguity in the data set that was used. Schaap and Leij (1998) showed that using larger data sets generally leads to smaller confidence intervals.
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Model C2 is implemented in Rosetta, a Windows 95/98 program that permits the prediction of retention, saturated, and unsaturated hydraulic parameters with pedotransfer functions. Rosetta is available at: http://www.ussl.ars.usda.gov/models/rosetta/rosetta.htm; verified January 31, 2000.
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Summary and conclusion |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsNOTESResults and discussionSummary and conclusionREFERENCES |
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and
, leads to relatively poor predictions of unsaturated hydraulic conductivity
. The unsaturated hydraulic conductivity is overpredicted in the wet range and underpredicted in dry range. When we fitted Eq. [4] to hydraulic conductivity data, we found that Ko was, on average, almost one order of magnitude smaller than Ks while L was often negative, with smaller values for finer textured soils. The difference between Ko and Ks can be attributed to effects of macroporosity which does affect Ks but which hardly has an effect on unsaturated hydraulic conductivity. The negative values for L make a physical interpretation of the Mualem (1976) model difficult because they imply that the tortuosity decreases and/or the connectivity increases with decreasing water contents. In the light of Ko < Ks, the negative values can be understood as correction factors that cause a more gradual drop in conductivity than positive values for L. The results in the present study and those of Kosugi (1999) indicate that neither Ko nor L can be interpreted as physically meaningful parameters and should be considered only as empirical shape factors of the Mualem–van Genuchten model. This, ultimately, suggests that the models proposed by Mualem (1976) or Burdine (1953) give a too-simplified conceptualization of the hydraulic conductivity of a porous medium. Still, as was shown by Kosugi (1999) and the current study, these pore-size distribution models can be used to successfully predict unsaturated hydraulic conductivity from water retention parameters.
Using Spearman rank correlation matrices, we showed that Ko and L have low correlations with sand and clay percentages and bulk density. Water retention parameters were better and more consistent predictors of Ko and L. A neural network model that predicted both Ko and L from retention parameters (r, s, , n) yielded a RMSEK of 0.84, which was a substantial improvement over the traditional Mualem–van Genuchten model. Uncertainty analysis with the bootstrap method suggested that the data set created a large degree of uncertainty in the prediction of unsaturated hydraulic conductivity. Near saturation, the uncertainty was less than one order of magnitude, at lower water contents the uncertainty grew to two orders of magnitude. The uncertainty intervals can probably be reduced when the models would be based on more unsaturated hydraulic conductivity characteristics. Effects of macroporosity are not included in the model; such effects need to be accounted for when our models are to be used near saturation.
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ACKNOWLEDGMENTS |
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NOTES |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsNOTESResults and discussionSummary and conclusionREFERENCES |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsNOTESResults and discussionSummary and conclusionREFERENCES |
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