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Soil Physics

Soil Water Retention

II. Derivation and Application of Shape Index

^a Laboratoire d'Etude des Transferts en Hydrologie et Environnement, LTHE (UMR 5564, CNRS, INPG, UJF, IRD), BP 53X, 38041, Grenoble, Cedex 9, France
^b Instituto Mexicano de Tecnología del Agua (IMTA), Paseo Cuauhnáhuac 8532, Col. Progresso 62550 Jiutepec, Morelos, Mexico
^c CSIRO Land and Water, Indooroopilly, Qld. 4067, Australia

^* Corresponding author (haverkamp{at}hydrowide.com)

	ABSTRACT

TOP ABSTRACT INTRODUCTION PARAMETRIC SHAPE INDEX CONVERSION EQUATIONS CONVERSION OF SHAPE PARAMETERS CONSTRAINED OPTIMIZATION CONCLUSIONS REFERENCES

 
The concept of a shape index was introduced in Part I to characterize the water retention curve. A potential problem with the parameterization of retention data is the interdependency of parameter values and their conversion to other parametric models. In this Part II the shape index is used to derive formulae for the conversion between: (i) hydraulic parameters of the Brooks-Corey (BC) and van Genuchten (vG) equations, (ii) parameter sets with and without the constraint that the residual water content _r = 0, and (iii) vG-shape parameters with different constraints. The dependency of hydraulic parameters on the optimization strategy and the applicability of conversion equations were investigated with 660 retention curves from the GRIZZLY database. The BC-shape parameter and the vG-shape factor mn are poorly correlated for larger and mn (overall r² = 0.96 for _r = 0). Instead of the commonly used equality = mn, conversion based on the shape indices P_BC and P_vG is more accurate (r² = 0.99). The shape index is also convenient to accurately predict shape parameters for different constraining scenarios involving _r and m = 1 – k/n. The values for _r and the shape parameters and mn are not unique. They are positively correlated to maintain the same soil water capacity for a particular soil. For the optimization of parameters, we recommend constraining _r. Accurate optimization of retention data, in particular the shape parameters, is possible by constraining _r to a non-zero value inferred from the shape index.

Abbreviations: BC, Brooks-Corey Model • RMSE, root mean square error • vG, van Genuchten Model

	INTRODUCTION

TOP ABSTRACT INTRODUCTION PARAMETRIC SHAPE INDEX CONVERSION EQUATIONS CONVERSION OF SHAPE PARAMETERS CONSTRAINED OPTIMIZATION CONCLUSIONS REFERENCES

 
IN HAVERKAMP et al. (2005) we illustrated how different parameterizations may complicate the analysis of soil water retention data. In particular, the conversion of retention parameters between different models was prone to error. We introduced the shape index P to characterize the retention behavior of a particular soil with a single number. The value for P is directly linked to the mass fractal dimension, the upper limit for the shape index being 3 for a soil exhibiting fractal behavior. Values for P computed from water retention points were shown as a function of soil texture. In this Part II, we are interested to use the shape index to provide a benchmark for obtaining consistent and comparable hydraulic parameters.

We will again consider the parametric models by Brooks and Corey (1964) and van Genuchten (1980). These are respectively defined as:

[1]

[2a,b]

where is the volumetric water content (L³ L^–3), _r and _s are the water content scaling parameters. The variable h is the pressure head (L), expressed in centimeters of water and taken as the absolute value of the matric head, is scaled by h_BC or h_vG (Haverkamp et al., 2002). Finally, , m, and n are shape parameters and k is a user index to constrain the vG-shape parameters (which are denoted by m_k and n_k when constrained). The product of m and n (or of m_k and n_k) will be referred to as the shape factor mn (or mn_k). As was shown in Part I (Haverkamp et al., 2005), the common approach for conversion between vG and BC equations by assuming that = mn involves considerable error, especially for coarse-textured soils.

The shape index is defined by:

[3]

where P is an integral measure of the slope of the log-transformed retention curve. In addition to obtaining a value for P directly from experimental data (experimental P), we also want to be able to calculate a value by substituting parameters of any water retention model into an expression obtained by evaluating Eq. [3] for that model (parametric P). Conversion between parameters obtained from different parameterizations might then be possible based on the premise that the shape index is constant. The description of the slope, and hence the shape index, will typically vary depending on the optimization, but this effect is considered unrelated to the conversion itself. Conversions are important for comparing and applying data from different studies with varying parameterizations, and for constraining so as to allow a smaller number of parameters. Of particular interest are constraints on _r and m_k.

The objectives of this paper are: (i) to derive expressions for the shape index in terms of parameters in the vG and BC equations, (ii) to formulate conversions for shape parameters from different optimization scenarios, and (iii) to apply the shape index to improve conversion and optimization of hydraulic parameters.

	PARAMETRIC SHAPE INDEX

TOP ABSTRACT INTRODUCTION PARAMETRIC SHAPE INDEX CONVERSION EQUATIONS CONVERSION OF SHAPE PARAMETERS CONSTRAINED OPTIMIZATION CONCLUSIONS REFERENCES

 
Figure 1 illustrates the organization of the material for different scenarios and includes pertinent equation numbers and parameters considered here. Additional conversions are possible and the shape index concept is not restricted to the BC and vG equations. Based on the formal definition of the shape index, we can obtain a value for the index using experimental retention data (as was already done in Part I [Haverkamp et al., 2005]) or according to formulae based on a parametric model for the retention curve (as will be done here). Different formulae are used for _r = 0 (upper dashed box in Fig. 1) and _r > 0 (lower dashed residual mode box). Equations for the conversion of shape parameters of the BC and vG equations will be derived. For the vG equation, the shape index concept will be reviewed both with unconstrained m and n, and with the constraint m_k = 1 – k/n_k (the dotted user mode box). Transformations between different k can be readily done for _r = 0.

View larger version (36K): [in this window] [in a new window]   Fig. 1. Outline of presentation of shape index with equation numbers for parameter conversion.

Nonresidual Mode
Eliminating _r as an optimization variable by setting _r = 0 is attractive in view of possible convergence problems during optimization, the lack of a conceptual underpinning of _r, and the poor predictability of _r (Leij et al., 2002). Conversions for the nonresidual mode are illustrated in the upper part of Fig. 1.

Substitution of the BC equation with _r = 0 in Eq. [3] reveals that the index is simply equal to the shape parameter:

[4]

where the subscript "0" indicates the nonresidual mode.

The shape index for the vG equation with or without the user index is given by:

[5]

Note that the shape parameters in the user mode, m_0,k and n_0,k, and their product mn_0,k will normally not be equal to their unconstrained values m₀, n₀, and mn₀. If the retention curve exhibits fractal behavior, we may require that P_vG < 3. In the user mode, this implies the following constraint on the shape factor for a particular value of the user index k:

[6]

This condition is met for any k if mn_0,k < 3. Typically, values for the user index k do not exceed 2 (i.e., the value for the conductivity model by Burdine, 1953) in which case mn_0,k may be as much as 5.16 without violating the condition. Such high values for mn_0,k were rarely encountered for the UNSODA and GRIZZLY databases (Fig. 3 in Haverkamp et al., 2005).

Figure 2 provides a qualitative illustration of P_vG as a function of sand and clay percentages for a subsample of the GRIZZLY database (every thirteenth sample out of a total of 660). The retention data were optimized assuming _r = 0 using the "Burdine" constraint (i.e., k = 2). The shape index, P_vG, was computed from the optimized mn_0,k using Eq. [5]. The insert shows the corresponding retention curves for P_vG equal to 0.05, 0.25, and 1.25. Despite the different sample populations and the methods to estimate the shape index, the behavior of P with texture shown in Fig. 2 is quite similar to that for experimental P shown in Fig. 9 of Haverkamp et al. (2005). This gives credence to the concept of a shape index to quantify retention with a single number.

View larger version (25K): [in this window] [in a new window]   Fig. 2. Shape index PvG computed for GRIZZLY according to Eq. [5] with k = 2, r = 0 as a function of clay and sand percentage.

[7]

Because we postulate that P_BC is independent of the parameterization, we must have ₀ (Eq. [4]). The difference in and ₀ should compensate for the effect of _r.

The shape index is more complicated for the vG equation. Insertion of Eq. [2a] into Eq. [3] and evaluation of the integral for unconstrained m and n yields

[8]

where ₂F₁() is the hypergeometric function (Oberhettinger, 1964). The index for the constrained case is identical except for the substitutions n_k = k/(1 – m) and m_k = m. A first-order approximation is:

[9a,b]

This expression is very similar to Eq. [7] for P_BC except for the (un)constrained shape parameters. A slightly more precise second-order approximation is:

[10a,b]

The correction factors ß₁ and ß₂ account for the effect of a nonzero _r. Note that we have given equations both without and with a user index. A comparison between Eq. [5] and [9a] or [10a] shows that mn_k > mn_0,k for a given k (Fig. 5 of Haverkamp et al., 2005). As k, the value of P_vG approaches mn_0,k and the vG and BC equations become equivalent (Fig. 6 in Haverkamp et al., 2005).

The parametric shape indices P_vG and P_BC are plotted versus the experimental shape index P for UNSODA in Fig. 3 . The parameters in the BC and vG equations were optimized using _r and, for Eq. [2], with both m and n among the fitting parameters. We used all 220 samples that yielded a positive _r for both the vG and BC equations. P_BC was predicted with Eq. [7], the exact expression P_vG with Eq. [8], and the first- and second-order approximations (P_vG1 and P_vG2) with Eq. [9a] and [10a]. The hypergeometric function in Eq. [8] was evaluated with the Mathematica software (Wolfram Research, Inc., Champaign, IL). The coefficients of correlation between all shape indices, which are provided in the insert, are fairly high. The parametric shape index for the BC equation exhibits a lower correlation with the "experimental" P (r² = 0.921) than the indices for the vG equation because the latter fits the data better. The exact expression P_vG has a negligibly better correlation with P than the first- and second-order approximations (P_vG1 and P_vG2). The highest correlation (r² = 0.999) is found between P_vG1 and P_vG2 due to their similar formulation. The simpler first-order approximation appears more than adequate to estimate P_vG.

View larger version (26K): [in this window] [in a new window]   Fig. 3. Parametric shape index PBC according to Eq. [7] and PvG according to Eq. [9a], [10a], and [8], that is, the first- and second-order approximation and the exact expression, computed from optimized r, s, , m, and n as a function of the experimental shape index, P, using samples from UNSODA with a positive r. Coefficients for the correlation between all shape indices are provided in the insert.

View larger version (13K): [in this window] [in a new window]   Fig. 4. Effect of r as an optimization parameter for k = 2 on 247 samples of GRIZZLY database: (a) mn2 as a function of mn0,2 and (b) PvG computed from optimization results with r > 0 versus results for r fixed at 0.

	CONVERSION EQUATIONS

TOP ABSTRACT INTRODUCTION PARAMETRIC SHAPE INDEX CONVERSION EQUATIONS CONVERSION OF SHAPE PARAMETERS CONSTRAINED OPTIMIZATION CONCLUSIONS REFERENCES

Brooks-Corey and van Genuchten Equations
Conversion between the shape parameters in the BC and vG equations is easily done for _r = 0. We equate Eq. [4] and Eq. [5], with the user index, to obtain:

[11]

[12]

For m and n optimized independently, we can still use Eq. [5] to compute ₀. The reverse, that is, computing both m₀ and n₀ from a single ₀, is not possible without additional assumptions. For the residual mode (_r > 0), we can still use the above equations by first converting and mn_k to ₀ and mn_0,k.

User Mode
For the vG equation, we also like to be flexible regarding the user index. Conversion of the shape parameters is again done for _r = 0. We consider conversions to, from and within the user mode, respectively.

We may determine m_0,k—and hence n_0,k because of the constraint posed by Eq. [2b]—for an arbitrary k from unconstrained m₀ and n₀ by rewriting Eq. [5] as:

[13a,b]

where the shape factor mn₀ denotes the product of unconstrained m₀ and n₀.

Conversion from the user mode, that is, predicting the unconstrained shape parameters m₀ and n₀ from n_0,k, is possible if we are making the additional assumption that mn₀ mn_0,k. This assumption is reasonable for small mn_0,k (e.g., mn₀ < 0.8; see Haverkamp et al., 2005, Fig. 6). In this case:

[14]

Conversion within the user mode between constrained shape parameters obtained for different user indices, say k₁ and k₂, may be done according to (Eq. [5]):

[15]

Instead of using these conversions, we can also estimate shape parameters for the desired constraining scenario from the shape index—computed from experimental data or from a different parameterization—according to Eq. [5].

Residual Mode
We know that changes in _r imply a commensurate change in , m, and n because the shape index presumably remains constant and because the other water content scale factor, _s, is invariant (Fig. 4, in Haverkamp et al., 2005).

For conversion to and from the residual mode, we equate the shape indices for _r = 0 and _r > 0. The conversion of the BC-shape parameter is governed by:

[16]

This follows directly from Eq. [4] and [7]. Conversion of vG-shape parameters is governed by (Eq. [5] and [9a] or [10a]):

[17a,b]

where ß denotes either ß₁ or ß₂ according to Eq. [9b] or [10b]. Note that if _r = 0, we have ß = 1 and a trivial conversion since there is no residual mode. The parameter ß accounts for the dependency of the shape parameters on _r/_s. The behavior of the individual shape parameter m_k in the residual (and user mode) is related to that in the nonresidual mode according to:

[18]

This expression follows from Eq. [2b] and [17b]. If the shape parameters are not constrained by a user index, we may invoke the assumption mn₀ ~ mn_0,k for small mn₀ (Eq. [14]) to obtain from Eq. [17a]:

[19]

	CONVERSION OF SHAPE PARAMETERS

TOP ABSTRACT INTRODUCTION PARAMETRIC SHAPE INDEX CONVERSION EQUATIONS CONVERSION OF SHAPE PARAMETERS CONSTRAINED OPTIMIZATION CONCLUSIONS REFERENCES

 
Brooks-Corey and van Genuchten Equations
As was discussed in Part I (Haverkamp et al., 2005), the retention functions according to Eq. [1] and [2] yield equivalent results for large h/h_vG or n, provided that h_BC = h_vG and equals either mn_k or mn. Figure 5a shows ₀ as a function of mn_0,2—the Burdine constraint with user index k = 2—for 655 samples of GRIZZLY. Beyond the threshold value of, say, mn_0,2 = 0.8, the commonly assumed identity = mn is no longer satisfied. This threshold increases with k (Fig. 6 of Haverkamp et al., 2005). Equation [11] is convenient to assess the equivalence between vG and BC equations. The equality ₀ = mn_0,k only holds if mn_0,k << k (i.e., mn_0,k0 and/or k), while Eq. [12] similarly implies that mn_0,k = ₀ only if ₀<<k (i.e., ₀0). The condition ₀ = mn_0,k therefore constitutes a limiting case for large k.

View larger version (11K): [in this window] [in a new window]   Fig. 5. Parameter optimization to retention data from GRIZZLY with r = 0 using the BC and vG equations (k = 2): (a) 0 as a function of mn0,2 and (b) PBC as a function of PvG.

To further assess the utility of the shape index for the conversion of ₀ to mn_0,2, we also compared the RMSE of optimized water contents for unconstrained optimization of _s, mn_0,2, and h_vG with the RMSE for constrained optimization using P_BC = ₀. In the latter case, only _s and h_vG are optimized while mn_0,2 is obtained from ₀ according to Eq. [12]. The close correspondence between RMSE-values for the "constrained" and the "unconstrained" optimizations suggests again that the shape index is useful to convert BC and vG parameters (Fig. 6) .

View larger version (17K): [in this window] [in a new window]   Fig. 6. Optimizing vG equation to water content of 655 soils from GRIZZLY: RMSE if shape factor mn2 is independently predicted from BC equation with PBC = PvG versus RMSE if mn2 is included in the optimization.

View larger version (12K): [in this window] [in a new window]   Fig. 7. Shape parameter 0 estimated from parameters of the vG equation, as a function of optimized 0 for 660 samples from GRIZZLY: (a) estimation from shape index, PvG, and (b) estimation with = mn1.

[20]

which gives in combination with Eq. [2b]:

[21]

Figure 8a depicts mn_0,k optimized for k = 0.5, 1, 2, 3, 5, 7, and 10 as a function of mn_0,2 (i.e., the Burdine constraint). The results indicate that the vG shape factor mn_0,k depends on the value of the user index; mn_0,k becomes larger than mn_0,2 for k < 2 and smaller for k > 2. The correspondence between mn_0,k and mn_0,2 becomes worse for higher mn_0,k. By no means can mn_0,k be considered an intrinsic soil characteristic because it depends on the user index.

View larger version (15K): [in this window] [in a new window]   Fig. 8. The effect of the user index, k, for optimizations of GRIZZLY data with r = 0: (a) shape parameters mn0,k for user index k = 0.5, 1, 3, 5, 7, and 10 as a function of mn0,2, and (b) shape index, PvG, for user index k = 0.5, 1, 3, 5, 7, and 10, as a function of PvG for k = 2.

The relationship between shape index and shape factor is further explored in Fig. 9 , which shows P_vG as a function of mn_0,k for different k according to Eq. [20]. As previously noted, k is often chosen independently based on the hydraulic conductivity model. The increase in P_vG does not keep pace with that of mn_0,k, especially for lower k. Since P_vG is constant for a particular soil, there is an appropriate mn_0,k for each k. If k, we already noticed that P_vG tends to mn_0,k (Eq. [5b]) and hence m_0,k0 (Eq. [21]). The shape index can be used to convert mn_0,k-values for different user indices k (Fig. 1).

View larger version (20K): [in this window] [in a new window]   Fig. 9. The water retention shape index PvG as a function of mn0,k according to Eq. [5] for user index k = 0, 1, 2, 10 for the vG equation and k for the BC equation.

Dependency of and mn_k on _r
The theoretical behavior of /₀ as a function of _r/_s, defined by Eq. [16], is shown in Fig. 10a (left-hand axis). The ratio /₀ increases rapidly with _r. For example, when _r/_s = 0.20, increases by 67% compared with ₀ for _r/_s = 0. We postulated in Haverkamp et al. (2005) that to maintain the soil water capacity for a particular soil—equivalent to the shape index—an increase in the shape factor is compensated by a larger _r. Note that coarse-textured soils have larger and smaller _r than fine-textured soils. To put the deviation of from ₀ in perspective, we have also included the distribution of _r/_s among GRIZZLY samples with a nonzero _r in Fig. 10a. Although the majority of samples have a zero or small value for _r/_s, for a sizeable number of samples _r/_s > 0.05 and the difference between ₀ and will be at least 25%. It is clear that the use of -values obtained for different optimizations of _r can easily lead to invalid comparisons.

View larger version (34K): [in this window] [in a new window]   Fig. 10. The effect of variations in r on shape parameters: (a) /0 and mn2/mn0,2 using a first- and second-order approximation (left-hand axis) and the distribution of samples with a nonzero r (right-hand axis) from GRIZZLY as a function of r/s and (b) mn2/mn0,2 using different m0,2 (left-hand axis) and the distribution of samples with a nonzero r (right-hand axis) from UNSODA.

Predictability of _r and mn_k
Here we consider how well _r and mn_k can be predicted from one another assuming that _s is invariant with residual mode. The first-order approximation ß₁ was defined by Eq. [9b] and may also be derived by rewriting Eq. [17b]:

[22]

Note that m_0,k and m_k follow directly from mn_0,k and mn_k because of the user mode constraint.

First, consider the hypothetical example that the shape factors mn_0,k and mn_k are known (i.e., nonresidual and residual modes), but that _r is not known. The shape factors are substituted in Eq. [22], which is then solved for _r/_s. The resulting _r/_s predicted in this way are plotted as a function of the optimized _r/_s for both k = 1 and 2 in Fig. 11a . Obviously, only soils with _r > 0 are considered because _r = 0 yields the trivial result ß₁ = 1. Even though the prediction of _r/_s is good for small _r/_s, the scatter around the 1:1-line increases considerably for larger _r/_s. This suggests that _r is difficult to predict and that the shape index concept is not suitable to quantify _r—a scale parameter.

View larger version (22K): [in this window] [in a new window]   Fig. 11. Predictability of residual water content and shape factor with the first-order approximation of the shape index using Eq. [22]: (a) predicted versus optimized r/s and (b) predicted versus optimized mnk.

	CONSTRAINED OPTIMIZATION

TOP ABSTRACT INTRODUCTION PARAMETRIC SHAPE INDEX CONVERSION EQUATIONS CONVERSION OF SHAPE PARAMETERS CONSTRAINED OPTIMIZATION CONCLUSIONS REFERENCES

 
It might be desirable to constrain _r to a nonzero value during parameter optimization. We can use the shape index to do this using the following procedure. First, optimize {_s, h_vG, mn_0,k} with _r = 0 to estimate P_vG from Eq. [5]. To obtain the correction factor ß₁, we need to repeat the optimization in the residual mode to obtain {_r, _s, h_vG, mn_k}. The shape factor from this second optimization is then used to compute ß₁ with Eq. [9a] using the independent value for P_vG from the first optimization. A non-zero value for _r that can be used as constraint follows from Eq. [22] using the value for _s from the second optimization. This value for _r is then used as a fixed parameter in a third optimization to yield where the asterisk denotes a non-zero constraint on _r.

As an example, consider the silt loam used in Table 1. Table 2 contains the values for the retention parameters, the standard error (SE), and 95% confidence limits (CL), the sum of squared differences between observed and optimized water contents (SSQ), and values for the shape index P_vG. We used the Burdine constraint (k = 2). The first optimization with _r = 0 yields the poorest fit with SSQ equal to 0.00262 cm⁶ cm^–6. From Eq. [5] we obtain P_vG = 0.158. The second optimization without a constraint on _r yields the closest description of the retention data (SSQ = 0.00151 cm⁶ cm^–6), but there is a more than six-fold increase in the SE for n₂ compared with the first optimization. Also notice the greater uncertainty associated with optimizing _r than _s. Using the optimized mn₂ from the second and P_vG from the first optimization, we obtained ß = 0.506 with Eq. [9]. By solving Eq. [22] with _s = 0.427 cm³ cm^–3 using Mathematica, we found _r = 0.119 cm³ cm^–3. The latter is then used as constraint for the third optimization. There is a marginally larger SSQ for the third than for the second, unconstrained optimization. More importantly, the SE is smaller and the confidence interval narrower for _s, h_vG, and especially n₂. The strong dependency of the shape factor on _r was already shown in Fig. 10. Incidentally, the shape indices for optimizations 2 and 3 are nearly identical. The value for P_vG is larger for Optimization Model 1 where the fit was not as good.

View this table: [in this window] [in a new window]   Table 2. Sequential optimization to constrain r to a positive value for San Fernando silt loam.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION PARAMETRIC SHAPE INDEX CONVERSION EQUATIONS CONVERSION OF SHAPE PARAMETERS CONSTRAINED OPTIMIZATION CONCLUSIONS REFERENCES

Figure 2 illustrates the dependency of the parametric shape index on texture, which is similar to that shown in Haverkamp et al. (2005) for the experimental shape index. The shape index was fairly constant with parameterization (Table 1, Fig. 3), the variation is mainly due to the different description of the retention curve and its slope. Unlike the shape factor mn, the shape index is barely affected by the magnitude of _r (Fig. 4). The parametric shape index hence appears a useful tool to quantify retention data with a single number and to allow for comparison and conversion of parameters from different optimizations. The conversion of shape parameters involved: (i) the BC and vG equations, (ii) parameter sets with and without _r equal to zero, and (iii) vG parameters for different user indices. The equations used in the conversions are also shown in Fig. 1.

Figure 5a showed that the popular assumption mn for conversions between BC and vG parameters was inaccurate even if _r = 0. Albeit mathematically convenient, this conversion rests on the specious hypothesis that and mn_k do not depend on _r and k. Conversions between and mn_k are better done with Eq. [11] and [12] using equivalence of the shape index (Fig. 5b). There is virtually no difference in RMSE for the fit of water content if mn_k is included in the optimization or independently predicted from BC parameters using the shape index (Fig. 6). Predictions of the BC shape parameter are also best made using the shape index rather than the vG shape factor (Fig. 7).

The choice of a particular conductivity model in which retention data are to be used often determines the value for the user index k. Unlike the shape factor, the shape index is virtually independent of k (Fig. 8). Shape factors for different k can be readily obtained from the shape index. The theoretical change in mn with P_vG is illustrated in Fig. 9 for different k.

The scale factor _r determines the range of water contents included in parameterization of retention data. The value for _r might affect optimization of retention parameters such as , m, and n. Figure 10 illustrates that for both the BC and vG equations, the value for the shape parameters strongly depends on _r. The theoretical results suggest that the shape parameters increase with _r to presumably maintain the same slope (shape index) for a particular soil. Optimization of data in the GRIZZLY and UNSODA databases revealed that _r is larger than zero for an appreciable number of samples. The erratic behavior of _r implies the difficulty of predicting this parameter (Fig. 11). Hence we recommend quantifying hydraulic parameters by constraining _r. Commonly, _r has been set to zero. Constraining _r to a non-zero value might be more accurate. We did this in a sequential optimization procedure that yielded parameters—particularly the shape factors—with smaller standard errors than if _r were set to zero or not constrained (Table 2).

An extension of the current work on shape parameters, would be a further analysis of the water content and pressure head scale parameters—for example, we did not discuss conversion between h_vG and h_BC—and parameters for the hydraulic conductivity. Additional indices pertaining to the cumulative pore-size and/or particle-size distribution functions would be useful for this purpose.

	ACKNOWLEDGMENTS

 
This study was partially supported by the European Union DGXII-Environment Program through funding of the project AgriBMPWater (EVK1-1999-00117).

Received for publication July 2, 2004.

	REFERENCES

TOP ABSTRACT INTRODUCTION PARAMETRIC SHAPE INDEX CONVERSION EQUATIONS CONVERSION OF SHAPE PARAMETERS CONSTRAINED OPTIMIZATION CONCLUSIONS REFERENCES

 

• Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Hydrol. Paper 3. Colorado State University, Fort Collins.
• Burdine, N.T. 1953. Relative permeability calculations from pore-size distribution data. Petr. Trans. Am. Inst. Mining Metall. Eng. 198:71–78.
• Haverkamp, R., C. Zammit, F. Bouraoui, K. Rajkai, J.L. Arrúe, and N. Heckmann. 1998. GRIZZLY, Grenoble Catalogue of Soils: Survey of soil field data and description of particle-size, soil water retention and hydraulic conductivity functions. Laboratoire d'Etude des Transferts en Hydrologie et Environnement (LTHE), Grenoble Cedex 9, France.
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