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Estimation of water retention curve of granular soils from particle-size distribution — a Bayesian probabilistic approach
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C.F. Chiu, W.M. Yan, and Ka-Veng Yuen
Abstract: This study proposes two empirical relationships to estimate the parameters of van Genuchten’s formula for modeling the water retention curve from the particle-size distribution. The relationships are determined by the Bayesian probabilistic method for selecting the most plausible class of models based on a database of 90 soil samples. The highest plausibility model among the selected relationships shows that the parameter a can be expressed as a first-order function of the particle size at 50% passing (d50) and parameter n is expressed as a third-order polynomial of the reciprocal of the standard deviation of geometric mean particle size (sg). The predictability of proposed relationships for other soils outside the calibrated database is also presented. It is found that the model prediction is highly consistent with the measurements for sands. However it only matches well with the measurements in the low suction regime for soils with at least 20% of fines content. Key words: Bayesian analysis, particle-size distribution, saturation, soil suction, unsaturated soils. Résumé : Cette étude propose deux relations empiriques pour estimer les paramètres de la formule de van Genuchten servant à modéliser la courbe de rétention d’eau à partir de la distribution granulométrique. Les relations sont déterminées à l’aide de la méthode probabiliste bayésienne pour sélectionner la classe de modèles la plus plausible à partir d’une base de données de 90 échantillons de sol. Le modèle étant le plus plausible parmi les relations sélectionnées suggère que le paramètre a peut être exprimé comme une fonction de premier ordre de la taille des particules à 50 % passant (d50) et que le paramètre n peut être exprimé comme une fonction polynomiale de troisième ordre de la réciproque de l’écart-type de la moyenne géométrique de la taille des particules (sg). La prédictibilité des relations proposées pour d’autres sols en dehors de la base de données calibrée est aussi présentée. On observe que la prédiction du modèle est très consistante avec les mesures pour les sables. Cependant, les prédictions correspondent bien aux mesures seulement pour des sols en faible succion ayant un contenu en particules fines d’au moins 20 %. Mots‐clés : analyse bayésienne, distribution granulométrique, saturation, succion du sol, sols non saturés. [Traduit par la Rédaction]
Introduction “Water retention curve” (WRC) refers to the relationship between water content and soil suction. It is an essential input function for modeling the retention and transport of water and contaminants in the vadose zone of soils. However, the measurement of WRC is time-consuming, involving high labor costs. Thus, many empirical methods have been developed to correlate the WRC with some commonly available soil index properties, e.g., particle-size distribution, organic content, plasticity index, and bulk density. These empirical methods can be classified into two categories: (i) statistical approach and (ii) physicoempirical modeling approach. In the statistical approach, regression techniques are used such that the WRC is correlated to the soil index properties. These regression formulas are usually referred to as pedotransfer functions, PTFs (Bouma 1989). The PTF can estimate either the
water contents at pre-defined suction values (Gupta and Larson 1979; Rawls et al. 1982) or the fitting parameters of a closed-form expression of WRC (Saxton et al. 1986; Vereecken et al. 1989; Schaap and Bouten 1996; Scheinost et al. 1997; Zapata et al. 2000). It is generally recognized that the shape of the WRC depends on the pore-size distribution (Simms and Yanful 2002, 2005). In the physicoempirical modeling approach, a physics-based conceptual model is used to model the pore-size distribution based on the particle-size distribution (PSD), from which a relationship between water content and associated soil suction, i.e., WRC, can be derived (Arya and Paris 1981; Haverkamp and Parlange 1986; Tyler and Wheatcraft 1989; Fredlund et al. 2002; Aubertin et al. 2003). It should be noted that the estimation methods based on PSD cannot address the effects of density, stress history, fabric, and hydraulic hysteresis on the shape of the WRC. Furthermore, it is more difficult to pre-
Received 12 September 2011. Accepted 13 June 2012. Published at www.nrcresearchpress.com/cgj on XX August 2012. C.F. Chiu. Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Xikang Road, Nanjing 210098, China. W.M. Yan. Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China. K.-V. Yuen. Department of Civil and Environmental Engineering, University of Macau, Taipa, Macau, China. Corresponding author: C.F. Chiu (e-mail:
[email protected]). Can. Geotech. J. 49: 1024–1035 (2012)
doi:10.1139/T2012-062
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Chiu et al.
dict the WRC of fine-grained soils using these methods due to their complex fabric. The main criticism of the empirical PTF is its predictability with respect to other soils outside the calibrated database. A more complex function can always fit the data better than a simpler one because adding more adjustable parameters can potentially reduce the error of fitting. If the error of fitting is the only measure used to select the optimal function, overfitting may occur and the function will be too dependent on the errors or noise of the data. As a result, the prediction by the overfitted function for other soils outside the calibrated database is not reliable. The Bayesian probabilistic approach has been widely used in geotechnical engineering; for instance, modeling uncertainty in soil and rock properties (Miranda et al. 2009; Yan et al. 2009; Ching et al. 2010; Wang et al. 2010; Chiu et al. 2012), characterizating model uncertainty (Zhang et al. 2009a, 2009b); and assessing slope reliability (Cheung and Tang 2005). Furthermore, Bayesian inference can also address overfitting of data by penalizing the model complexity. It has been applied recently for model class selection in different civil engineering problems where a large number of models are under consideration for the data (Beck and Yuen 2004; Hoi et al. 2009; Yan et al. 2009; Yuen 2010b). One of the merits of the Bayesian approach is that in contrast to a classical statistical approach, no adhoc penalty term, e.g., the Akaike’s information criterion (Akaike 1974), is used. Instead, a model class is penalized in a natural measure of complexity by the Bayesian procedures. In this paper, the potential physical correlations between the WRC and PSD are first addressed, from which key variables derived from the PSD are identified to estimate the WRC. Then a brief review of the Bayesian model class selection is presented. Based on the framework, two PTFs for estimating the WRC from PSD are proposed using the data from sand and silt loam. Finally, the predictability of the proposed PTFs for some sandy and silty soils outside the calibrated database is discussed.
Relationships between WRC and PSD Figure 1 depicts a typical unimodal WRC measured during the drying process. It should be noted that the wetting curve is different from the drying curve due to the hydraulic hysteresis. This paper considers only the drying curve. The key features of WRC are air-entry value (AEV), saturated volumetric water content (qs), and residual volumetric water content (qr). AEV is the soil suction where air commences to enter the largest pore of saturated soil. qr is the water content where a substantial change in suction is required to remove additional water in the soil. Both of them can be estimated by the empirical procedures suggested by Fredlund and Xing (1994). Table 1 summarizes some common PTFs found in the literature that estimate the WRC. The PTFs can be classified into two main groups: (i) point estimation (Gupta and Larson 1979; Rawls et al. 1982) where the water contents at pre-defined suction values are the outputs of estimation, and (ii) parameter estimation of a closed-form expression of WRC (Saxton et al. 1986; Vereecken et al. 1989; Schaap and Bouten 1996; Scheinost et al. 1997; Zapata et al. 2000). The predicted WRC formulas include Campbell (1974), van Gen-
1025 Fig. 1. Typical unimodal water retention curve.
uchten (1980) and Fredlund and Xing (1994). It is seen from Table 1 that the input parameters of PTF are basic soil properties, such as soil texture, PSD, plasticity index, organic matter content, and bulk density. Many past studies have attempted to correlate the WRC and the PSD. Based on the assumption of uniform-size spherical particles and cylindrical pores, Arya and Paris (1981) postulated that the pore radius is related to the particle radius. Then the estimated pore-size distribution is used to derive the WRC according to the capillary model (Brutsaert 1966). An empirical scaling parameter is required to scale the pore radius of uniform-size spherical particles to that of a corresponding natural soil with nonspherical particles. Arya et al. (1982) suggested that the average value of the scaling parameter ranges from 1.1 for fine-grained soils to 2.5 for coarsegrained soils. However, the effect of density or packing arrangement of particles was not considered in the Arya and Paris (1981) model. Later Fredlund et al. (2002) proposed a different physicoempirical model that includes the effect of density or porosity on estimating the WRC from the PSD. In this model the PSD is first divided into numerous individual fractions of uniformly sized particles. The WRC of each individual fraction is then estimated and their sum is the WRC of soil. Unlike the Arya and Paris (1981) model, the pore volume of the individual fraction is calculated by a packing factor that accounts for the packing arrangement of particles. Fredlund et al. (2002) assumed a constant packing factor for all particle sizes. However, they also suggested that it could be a function of the particle sizes. It should be noted that the sum of the estimated pore volumes for the individual fractions may be different from the actual pore volume of soil. If the actual pore volume is smaller, the contribution of the voids greater than the actual porosity on the WRC is ignored in Fredlund et al. (2002). Thus, it is required to estimate the optimal value of the packing factor by fitting the predicted WRC with the measured values. The simplest approach to represent the PSD is the soil texture class, i.e., sand, silt, and clay contents (Gupta and Larson 1979; Rawls et al. 1982; Saxton et al. 1986; Vereecken et al. 1989). On the other hand, Scheinost et al. (1997) and Minasny et al. (1999) found that the geometric mean particle size (dg) and its standard deviation (sg) are useful parameters to represent the PSD and these parameters are given by Published by NRC Research Press
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Table 1. Existing PTFs for WRC in literature. Reference Gupta and Larson (1979) Rawls et al. (1982) Saxton et al. (1986)
PTF q ¼ a1 S þ a2 SL þ a3 CL þ a4 OM þ a5 BD q ¼ a0 þ a1 S þ a2 SL þ a3 CL þ a4 OM þ a5 BD þ a6 q33 þ a7 q1500 for 1500 kPa > j > 10 kPa ðCampbell 1974Þ j ¼ AqB A ¼ expða0 þ a1 CL þ a2 S2 þ a3 S2 CLÞð100Þ B ¼ a4 þ a5 CL2 þ a6 S2 þ a7 S2 CL q ¼ qr þ ðqs qr Þ½1 þ ðajÞn 1
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Vereecken et al. (1989)
ðvan Genuchten 1980Þ
lnðaÞ ¼ a0 þ a1 S þ a2 CL þ a3 C þ a4 BD lnðnÞ ¼ a0 þ a1 S þ a2 CL þ a3 S2 q ¼ qr þ ðqs qr Þ½1 þ ðajÞn ð1=nÞ1
Schaap and Bouten (1996)
ðvan Genuchten 1980Þ
a ¼ a0 þ a1 dp lnðnÞ ¼ n0 þ n1 lnðnp Þ þ n2 BD q ¼ qr þ ðqs qr Þ½1 þ ðajÞn 1
Scheinost et al. (1997)
a ¼ a0 þ a1 dg n ¼ n0 þ n1 s 1 g qS ln ½1 þ ðj=jr Þ C ¼1 c b fln½e þ ðj=aÞ g ln ½1 þ ð106 =jr Þ ðFredlund and Xing 1994Þ
Zapata et al. (2000)
q¼C
PI > 0
a ¼ a0 þ a1 mPI þ a2 ðmPIÞa3
c ¼ c0 þ c1 ðmPIÞc2 PI ¼ 0 a1 a ¼ a0 d60
b ¼ b0 þ b1 ðmPIÞb2 c
jr ¼ h1 eh2 ðmPIÞ a
b ¼ b0
c ¼ c0 þ c1 ln d60
jr 1 ¼ h0 þ d60 a
Note: q, volumetric water content; S, mass fraction of sand (%); SL, mass fraction of silt (%); CL, mass fraction of clay (%); OM, organic matter content (%); BD, bulk density (kg/m3); q33, volumetric water content corresponding to suction of 33 kPa; q1500, volumetric water content corresponding to suction of 1500 kPa; j, suction (kPa); qs, saturated volumetric water content; qr, residual volumetric water content; dp, fitting parameter of eq. [2] (mm); np, fitting parameter of eq. [2]; dg, geometric mean particle size (eq. [1a], mm); sg, standard deviation of dg (eq. [1b], mm); jr, suction corresponding to qr (kPa); C, carbon content (%); PI, plasticity index; m, mass fraction of particle finer than 75 mm; d60, particle size corresponding to 60% of particles passing (mm).
" ½1a
dg ¼ exp
t X
# fi lnðmi Þ
i¼1
½1b
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# t t X 2 X fi ln 2 ðmi Þ fi lnðmi Þ s g ¼ exp i¼1
i¼1
where t is the number of particle fractions, fi is the amount of the ith fraction, and mi is the logarithmic mean of the ith fraction. Alternatively, the PSD can be fitted by a closedform expression, e.g., Schaap and Bouten (1996) used the following logistic expression to fit the PSD: np ð1=np Þ1 dp ½2 FðdÞ ¼ 1 þ d where d is the particle size, and dp and np are fitting parameters that represent the average particle size and uniformity of the PSD, respectively. It turns out that these parameters can be classified into two groups: (i) one controls the average particle size, e.g., dg and dp; (ii) the other one governs the spread of size distribution, e.g., sg and np. By the same to-
ken, the average particle size may be represented by a combination of selected critical points on the PSD, e.g., 10% (d10), 30% (d30), and 50% (d50) quartiles of PSD. Furthermore, the coefficient of uniformity (CU = d60 / d10), which is a shape parameter for PSD, may be an alternative choice to represent the spread of size distribution, where d60 is the 60% quartile of the PSD.
Empirical relationships of WRC Various empirical relationships have been proposed to describe the WRC (Gardner 1958; Brooks and Corey 1964; Campbell 1974; van Genuchten 1980; Fredlund and Xing 1994). The number of fitting parameters ranges between two and three with qs and qr considered as known values. The current study does not intend to discuss the pros and cons of each formula. In fact, the same Bayesian analyzing procedures can be applied to different formulas to shed light on the correlation between PSD and the WRC. Chiu et al. (2012) studied the reliability of the WRC formula proposed by van Genuchten (1980). It is concluded that the two-parameter (a and n as shown in eq. [3]) version is adequate to characterize WRC from a fitting perspective. The use of an additional Published by NRC Research Press
Chiu et al.
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fitting coefficient (denoted by m in their paper) provides negligible contribution for sandy soils. Furthermore, statistical distribution of the fitting parameters is given and most importantly the confidence intervals of WRC for some soil types, including sand, sandy loam, and silt loam, are presented. In this study, the same formula by van Genuchten (1980) is employed due to its fitting capability yet simplicity
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½3
qw ¼ qr þ ðqs qr ÞS
and
S¼
1 1 þ ðajÞn
where qw is the volumetric water content, S is the normalized water content bounded by 0 and 1; j is the matric suction, and a and n are two nonnegative curve-fitting parameters. It should be noted that a controls the AEV of the curve while n characterizes the gradient of desorption (i.e., change in water content with respect to suction). Equation [3] assumes a symmetrical and log-normal pore-size distribution (from the first derivative of eq. [3]). Thus, a is related to the modal pore size and n is inversely proportional to the standard deviation of the modal pore size.
Bayesian approach for model class selection Bayesian probabilistic model class selection has been successfully applied to different civil engineering problems (Beck and Yuen 2004; Yan et al. 2009; Yuen 2010a, 2010b). This section briefly reviews its key components and analyzing procedures. Consider a dataset D containing N sets of measuring input and output records. A number of predictive model classes Mj for j = 1, 2, …, NM are under consideration and the prime goal is to select the most plausible model class among them. The plausibility of a class of models given data D can be characterized by using Bayes’ theorem ½4
pðDjMj ; UÞPðMj jUÞ ; PðMj jD; UÞ ¼ pðDjUÞ NM P
½6
y ¼ 4T x ¼ 41 x1 þ 42 x2 þ ::: þ 4Nx xNx
where y is the target quantity; 4 is the column vector for the uncertain coefficients; and x is a column vector that represents the independent measurements (in this study they are quantities representing the information of PSD). Due to modeling error and measurement noise (collectively known as prediction error), the relationship between the measurement ey and the model output in eq. [6] can be expressed as follows: ½7
e y ¼ 4T x þ 3
where 3 is the modeling error, adequately modeled by Gaussian random variables with zero mean and variance s 23 . The variable s 23 controls the magnitude of the fitting-error variance and it is an unknown parameter to be identified using the measurements. Therefore, the uncertain model parameter vector x consists of 4 and s 23 : x ¼ ½4T ; s 23 T . The updated PDF of the model parameter x given the data D and model class Mj can be expressed as (Beck and Katafygiotis 1998; Yuen and Kuok 2011) ½8
pðxjD; Mj Þ ¼ c0 pðxjMj Þ
N ð2pÞN=2 s N exp J ð4jD; M Þ g j 3 2s 23
j ¼ 1; 2; :::NM
where pðDjUÞ ¼
likelihood function that measures the fitting of data. A better fit to the data implies a higher value of the likelihood function. The other factor in eq. [5] — ½pðbxjMj Þ½ð2pÞNj =2 jHj ðbxÞj1=2 — is the Ockham factor, which can be interpreted as a penalty against parameterization (Gull 1988; MacKay 1992; Yuen and Mu 2011). It penalizes the model classes with parameters that are highly sensitive to the measurement noise. Consider the following multivariate linear model system:
pðDjMj ; UÞPðMj jUÞ
j¼1
by the theorem of total probability, U represents the user’s judgment on the initial plausibility of the model classes, and P(Mj|U) is the prior probability on the model class Mj. The probability density function p(D|Mj,U) is called the evidence for the model class Mj provided by the data D. Equation [4] implies that the most plausible model class is the one that maximizes [p(D|Mj,U)][P(Mj|U)] with respect to j. Beck and Yuen (2004), based on the work by Papadimitriou et al. (1997), proposed the evidence for the model class Mj provided by the data D which can be written as h i ½5 pðDjMj Þ pðDjbx; Mj ÞpðbxjMj Þ ½ð2pÞNj =2 jHj ðbxÞj1=2 ; j ¼ 1; 2; :::; NM where Nj is the number of uncertain parameters for the model class Mj and bx is the most probable value of the model parameter vector x. This vector can be obtained by maximizing the updated PDF, which is proportional to [p(D|x, Mj)][p(x| Mj)]. The matrix Hj ðbxÞ is the Hessian matrix of lnf½pðDjx; Mj Þ½pðxjMj Þg evaluated at bx. The factor pðDjbx; Mj Þ in eq. [5] is called the
where c0 is a normalizing constant such that the volume under pðxjD; Mj Þ is unity; N is the number of records in D; and pðxjMj Þ is the prior PDF of the x model parameters expressing the user’s judgment about the relative plausibility of the values of the model parameters before the data is used. The goodness-of-fit function Jg ð4jD; Mj Þ indicates the level of data fitting and is given by ½9
Jg ð4jD; Mj Þ ¼
N 1X ½y xðnÞ ; 4; Mj e y ðnÞ 2 N n¼1
where yðxðnÞ ; 4; Mj Þ is the model-predicted output of the nth record based on the model class Mj with coefficients 4, and eyðnÞ is the corresponding measured value. The most probable b , is obtained by maximizing the model coefficient vector, 4 updated PDF pðxjD; Mj Þ. It is equivalent to minimizing the goodness-of-fit function Jg ð4jD; Mj Þ over all parameters in 4 b can be obif a uniform prior pð4jMj Þ is used. Moreover, 4 tained by solving the following simultaneous linear algebraic equation ½10
@Jg ð4jD; Mj Þ ¼0 @4k
for k ¼ 1; 2; :::; Nj
where Nj is the number of parameters in model class Mj. It turns out that Published by NRC Research Press
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2 ½11
6 6 6 b¼6 4 6 4
b11
b12
b1Nx
b21 .. .
b22 .. .
.. .
b2Nx .. .
bNx 1
bNx 2
bNx Nx
31 7 7 7 7 7 5
2 XN
x e y n¼1 1ðnÞ ðnÞ
3
6 7 6 XN 7 6 7 e x y 2ðnÞ ðnÞ 6 7 n¼1 6 7 6 7 .. 6 7 . 6X 7 4 5 N e x y n¼1 Nx ðnÞ ðnÞ
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where the coefficients bkl are given as follows: ½12
bkl ¼
N X
xkðnÞ xlðnÞ
n¼1
Moreover, it can also been shown that the most probable value of the prediction-error variance is given by ½13
b 23 ¼ min ½Jg ð4jD; Mj Þ ¼ Jg ðb 4 jD; Mj Þ s
The updated PDF pðxjD; Mj Þ can be well approximated by a Gaussian distribution centered at its optimal point for large N. Therefore, the uncertainty of the parameter estimation can be represented by its covariance matrix given by Sx ¼ Hj ðbxÞ1 where Hj ðbxÞ is given by (Yuen 2010a) 2 3 b11 b12 b1Nx 0 6 7 6 b21 b22 b2Nx 0 7 6 7 16 . .. .. .. .. 7 .. ½14 Hj ðbxÞ ¼ 2 6 . . . . 7 7 b3 6 s 6 7 6 bN 1 bN 2 bN N 0 7 x x x 4 x 5 0 0 0 2N The Bayesian procedures for model class selection are summarized as follows. For a model class Mj, the most probb is obtained by eqs. [11] and able model coefficient vector 4 [12]. The most probable value of the prediction-error variance (b s 23 ) can be calculated by eq. [13]. Then, the Hessian matrix can be easily formed by eq. [14]. With the most probable values of the model parameters and Hessian matrix, the evidence of model class Mj is computed using eq. [5]. The model classes are ranked according to eq. [4] and the selected model class is the one with the highest value of plausibility. In this study, a uniform prior plausibility is chosen for PðMj jUÞ over the model classes Mj, j = 1, …, NM. Details of the computational procedures are presented in Yan et al. (2009).
Soil database The Bayesian framework for model class selection is used to analyze the WRCs reported in the soil database UNSODA (Leij et al. 1996). The database contains 780 soil samples. The soils in the database are grouped in accordance with the USDA-SCS classification scheme (Soil Survey Division Staff 1993). A total of 90 soil samples were used in the study. Figure 2 shows the texture plot of the soil samples. Among them, 77, 10, 2, and 1 samples are classified as sand, silt loam, sandy loam, and loamy sand, respectively. As the input parameters of the proposed PTFs are derived from the information of PSD in this study, only soil samples with more than three independent measurements of the PSD have been
chosen. In addition, the upper limit of coefficient of uniformity and coefficient of curvature are 70 and 5, respectively. Soil samples beyond these ranges have not been considered.
Computational procedures In this study, model class selection is performed for the empirical multivariate linear correlation of the model parameters a and n of eq. [3]. The effective degree of saturation, S, is computed using S ¼ ðqw qr Þ=ðqs qr Þ. For a particular soil sample, the saturated volumetric water content, qs, was either identified from the records or estimated from the one at the minimum suction (ranging from 0 to 2.4 kPa). On the other hand, the residual water content qr was estimated either using the method suggested by Fredlund and Xing (1994) or from the one at the maximum suction in the case where no inflection point was identified in the high-suction range. Equation [3] assumes a symmetrical and log-normal poresize distribution (from the first derivative of eq. [3]). Thus, a is related to the modal pore size and n is inversely related to the standard deviation of the modal pore size. The pore size could be assumed proportional to the particle size. Existing PTFs (Schaap and Bouten 1996; Scheinost et al. 1997) have used the average particle size and spread of the PSD to correlate a and n, respectively. It is proposed that the average particle size can be represented by the linear combination of d10, d30, d50, dg, sg, and d10 ln(CU). By the same token, the spread of PSD is inversely proportional to some forms of sg and CU. Consider the following forms of eq. [6]: ½15a
a ¼ 4Ta xa
½15b
n ¼ 4Tn xn
where 4a and 4n are the column vectors for the parameters and xa and xn are the column vectors for the independent measurements. Based on the aforementioned discussion of the relationships between the WRC and PSD, the following variables have been selected for the vectors xa and xn: ½16a
xa ¼ ½1; d10 ; d30 ; d50 ; dg ; s g ; d10 lnCU ÞT "
½16b
xn ¼ 1;
2 2 1 1 ð1=CU Þ 1 2 1 ; ;e ; ; ; eð1=CU Þ ; s g CU sg CU # 3 3 3 T 1 1 ð1=CU Þ ; ; e sg CU
Table 2 tabulates the ranges of PSD’s parameters appeared in eqs. [16a] and [16b]. These ranges were selected with reference to the PSD of typical sand and silt loam. It is seen that they are not identical, but vary substantially. In other words, one may have very different prior distributions pðxjMj Þ for different parameters x. Note that the plausibility for each model class is influenced heavily by pðxjMj Þ. In this study, therefore, xa and xn are transformed to new variables Xak and Xnk , each with the same bounds, 0 to 1, to neglect the adverse effect of different prior distributions for different xk. We have ½17a
Xka ¼ mak xk þ cak
for k ¼ 2 to 7 Published by NRC Research Press
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Fig. 2. Texture plot of soil samples.
Table 2. Ranges of model parameters. d10 (mm) (0.0005, 0.5)
½17b
Xkn ¼ mnk xk þ cnk
d30 (mm) (0.004, 0.75)
d50 (mm) (0.009, 1)
for k ¼ 2 to 10
where mak , mnk , cak , and cnk are constants adopted for the transformation. Equations [16a] and [16b] are rewritten and now become ½18a
xa ¼ ½1; X2a ; X3a ; X4a ; X5a ; X6a ; X7a T
½18b
n T xn ¼ ½1; X2n ; X3n ; X4n ; X5n ; X6n ; X7n ; X8n ; X9n ; X10
The corresponding column vectors of parameters are ½19a
4a ¼ ½4a1 ; 4a2 ; 4a3 ; 4a4 ; 4a5 ; 4a6 ; 4a7 T
½19b
4n ¼ ½4n1 ; 4n2 ; 4n3 ; 4n4 ; 4n5 ; 4n6 ; 4n7 ; 4n8 ; 4n9 ; 4n10 T
The model class candidates are constructed with different linear combinations of the transformed variables in eqs. [18a] and [18b]. In this study, 17 and nine model class candidates are considered for eqs. [15a] and [15b], respectively. The prior plausibilityPðMj jUÞ on the model class Mj in eq. [4] is treated as an equal prior. In eq. [15a], PðMj jUÞ ¼ 1=17 for j = 1, 2, …, 17. On the other hand, PðMj jUÞ ¼ 1=9 for j = 1, 2 …, 9 in eq. [15b]. The prior distributions pðxjMj Þ for the parameters xk (k = 1, 2, …, Nj) are assumed to be an in-
dg (mm) (0.01, 1)
sg (mm) (0.001, 0.01)
Cu (1, 70)
dependent uniform distribution. As the ranges of transformed variables in the vectors Xak and Xnk are identical, the same pðxjMj Þ can be assumed for each parameter xk. In this study, a uniform prior is adopted and the parameters 4a and 4n are bounded by (–10, 10) and (–100, 100), respectively. For each model class Mj, the optimal values of the paramb are computed using eqs. [11] and [12]. The correeters 4 sponding most probable value of the prediction-error variance b b 23 is computed using eq. [13]. Tables 3 and 4 summarize 4 s b 23 for the parameters a and n, respectively. and s The Hessian matrix is obtained from eq. [14]. Then, the evidence of the model class pðDjMj Þ is obtained by substituting the prior PDF and Hessian matrix into eq. [5]. After computing the evidences of all model classes, the plausibility of a model class PðMj jDÞ is evaluated by eq. [4] and the most plausible model class is the one with the highest plausibility.
Discussion of results Optimal PTFs A total of 17 and nine model class candidates were evaluated for the parameters a and n, respectively. Figures 3 and 4 summarize the computational results for the logarithm of the “Ockham factor”, logarithm of “likelihood”, and plausibility of model class PðMj jDÞ for the parameters a and n, rePublished by NRC Research Press
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Table 3. Optimal parameters and prediction-error variance of the model classes for the parameter a. Model class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
b 41 0.0584 0.0120 0.0110 0.0015 0.0040 0.0079 0.0019 0.0046 –0.0492 0.0051 –0.0862 0.0080 0.0066 0.0035 –0.0225 –0.0768 –0.0619
b 42 1.1451 — — — — 0.1208 –0.0768 –0.0478 — — — 0.1315 –0.0594 –0.1152 — — —
b 43 — 0.9964 — — — — — — — — — –0.0436 0.6628 0.4283 0.2678 0.5904 0.4767
b 44 — — 0.9508 — — 0.8869 — — — 0.9468 — 0.9208 — — 0.6999 — —
b 45 — — — 1.2383 — — 1.2980 — — — 1.2486 — 0.4732 — — 0.5253 —
b 46 — — — — — — — — 0.2023 0.0637 0.3217 — — — 0.1227 0.3030 0.2424
b 47 — — — — 5.2583 — — 5.4096 5.2401 — — — — 3.4906 — — 2.8636
b 23 (× 10–3) s 11.86 4.40 3.28 4.76 3.86 3.21 4.74 3.85 3.50 3.25 3.85 3.21 4.26 3.46 3.19 3.47 3.01
Table 4. Optimal parameters and prediction-error variance of the model classes for the parameter n. Model class 1 2 3 4 5 6 7 8 9
b 41 0.6735 1.1291 1.2776 2.0670 0.8381 0.9172 –0.0269 0.8335 0.8280
b 42 6.8711 — — –4.9940 — — 24.7076 — —
b 43 — 4.2454 — — 7.0011 — — 7.0878 —
b 44 — — 5.2715 — — 9.7981 — — 11.9550
b 45 — — — 22.1583 — — –103.183 — —
Fig. 3. Model class results for parameter a: (a) logarithm of “Ockham factor”, (b) logarithm of “likelihood”, and (c) plausibility of model class.
b 46 — — — — –4.8422 — — –5.1944 —
b 47 — — — — — –10.2429 — — –21.3833
b 48 — — — — — — 158.829 — —
b 49 — — — — — — — 0.3832 —
b 410 — — — — — — — — 15.0617
b 23 s 1.0835 1.0772 1.0941 1.0385 1.0612 1.0623 1.0142 1.0612 1.0608
Fig. 4. Model class results for parameter n: (a) logarithm of “Ockham factor”, (b) logarithm of “likelihood”, and (c) plausibility of model class.
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spectively. Figure 3 shows that model class 3 of the parameter a (a ¼ 4a1 þ 4a4 X4a or a ¼ 4a1 þ 4a4 (1.0091d50 – 0.0091) has the highest plausibility among 17 model class candidates, PðM3 jDÞ ¼ 0:93. The second highest, PðM6 jDÞ ¼ 0:03, is model class 6 (a ¼ 4a1 þ 4a2 X2a þ 4a4 X4a or a ¼ 4a1 + 4a2 ð2:002d10 0:001Þ þ 4a4 ð1:0091d50 0:0091Þ). It should be noted that the prediction-error variance s 23 of model class 6 is smaller than that of model class 3 (see Table 3). However, the plausibility of model class 6 is lower than that of model class 3 because model class 6 is penalized more heavily due to the additional variable d10. Thus, the Bayesian approach takes into account both the goodness of fit for the data (s 23 ) and the penalty for the parameterization (Ockham factor) to select the most suitable model class. Figure 4 shows that the three highest plausibility model classes of the parameter n among nine candidates are model class 7 (n ¼ 4n1 þ 4n2 X2n þ 4n5 X5n þ 4n8 X8n ), model class 1 (n = 4n1 þ 4n2 X2n ), and model class 4 (n ¼ 4n1 þ 4n2 X2n þ 4n5 X5n ). The corresponding plausibility is PðM7 jDÞ ¼ 0:49, PðM1 jDÞ ¼ 0.14, and PðM4 jDÞ ¼ 0:13, respectively. Model class 7 has the highest plausibility and also the smallest s 23 . In this case the penalty of the additional variables in model class 7 does not play a significant role in the selection of optimal model class because its s 23 is substantially smaller than those of model classes 1 and 4. Figures 5a and 5b depict the measured and predicted parameters a and n, respectively. Compared with the measured values, the predicted values of a are more consistent than those of n. To illustrate the reliability of the proposed empirical relationships, the WRCs of four different soil classes from the calibrated database have been predicted. The PSDs of some soil samples in the calibrated database are shown in Fig. 6. Figures 7a to 7d depict the corresponding predicted WRCs and the measured data. It is shown that the predictions are consistent with the measured data for three soil classes except for soil samples ID 1281 (a silt loam). The prediction underestimates a of the silt loam.
Fig. 5. Measured and predicted values of parameters (a) a and (b) n.
Fig. 6. Particle-size distribution of soil samples in the calibrated database.
Comparison with existing PTFs Based on Bayesian model class selection results, eqs. [20a] and [20b] are selected as the optimal model of the most plausible model class among the 17 and nine model class candidates, respectively ½20a
a ¼ 0:00236 þ 0:959d50
½20b
n ¼ 2:2 107 =s 3g 1:9 104 =s 2g þ 0:0594=s g 4:258
Parameter a can be expressed as a first-order function of d50 while n is expressed as a third-order polynomial of the reciprocal of sg. Hence, a single variable of PSD is sufficient to estimate the model parameters of the van Genuchten’s (1980) formula. Compared to other existing PTFs, Vereecken et al. (1989) used four soil variables in their PTF. PTFs proposed by Schaap and Bouten (1996) and Scheinost et al. 1997 are similar to eqs. [20a] and [20b]. For the estimation of the parameter a, existing PTFs used other average particle size parameters in the correlations, such as dg and dp in eqs. [1a] and [2], respectively. The merit of variable d50 over
dg and dp is that it can be obtained directly from the PSD without additional computational effort. Despite a better goodness of fit to the measured data that can be achieved by using a linear combination of d30, sg, and d10 ln(Cu) to estimate the parameter a (see model class 17 in Table 3), the Bayesian method does not select this model class because of the penalty of the additional variables. Furthermore, Scheinost et al. (1997) proposed a linear relationship between the reciprocal of sg and n. This study has shown that a better fit to the data will be achieved without overfitting if a third-order polynomial of the reciprocal of sg is used. Similar to other estimation methods based on PSD, eqs. [20a] and [20b] cannot address the effects of density, stress history, fabric, and hydraulic hysteresis on the shape of the WRC. Published by NRC Research Press
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Fig. 7. Predicted and measured WRC for soil samples (a) sand ID-1466, (b) loamy sand ID-1465, (c) sandy loam ID-1120, and (d) silt loam ID-1281.
Predictions of WRC for sandy and silty soils Five sandy and silty soils taken from the other data sets have been used to validate the proposed empirical relationships for the parameters a and n. The PSDs of the soils are depicted in Fig. 8. Among the five soils, there are three sands, one loamy sand, and one silt loam. The measured WRCs and model predictions are shown in Figs. 9a to 9e. The predictions by two existing PTFs (Vereecken et al. 1989; Scheinost et al. 1997) are also depicted in the figures for comparison. It is seen from the figures that eqs. [20a] and [20b] give better estimation than the two existing PTFs. It appears that the model prediction for the three sand samples matches very well with measured WRCs. This is consistent with other estimation methods based on PSD that give reliable predictions for sands. For loamy sand and silt loam (fines content > 20%), the predicted WRC is consistent with the measured data in the low suction regime (i.e., a better estimation of AEV or higher level of confidence for parameter a), but it underestimates the normalized water content in the high suction regime (i.e., a worse estimation of gradient of desorption or lower level confidence for parameter n). It seems that one of the main limitations of the proposed PTFs is that the simple variables derived from the PSD (e.g., d50 and sg) may not be able to address the issue of complex fabric such as interparticle pores within clay aggregates and bimodal pore-size distribution commonly found in the fine-grained soils.
Fig. 8. Particle-size distribution of soil samples outside the calibrated database.
Conclusions Two empirical pedotransfer functions (PTFs) for estimating the model parameters of van Genuchten’s (1980) formula of the water retention curve from the particle-size distribution (PSD) are proposed. The PTFs are determined by the Bayesian probabilistic method for selecting the most plausible Published by NRC Research Press
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Fig. 9. Predicted and measured WRC for soil samples (a) Ottawa sand, (b) sand No. 350, (c) sand No. 10720, (d) loamy sand No. 10741, and (e) silt loam.
class of models. The criteria for selecting the optimal PTF are based on the error of fitting and model complexity. The Bayesian method is then applied to analyze the measured data of 90 soil samples and the most plausible classes of PTF for model parameters a and n are identified. The results show that a single variable derived from the PSD is sufficient to estimate a and n for sandy and silty soils. Parameter a is a first-order function of d50. On the other hand, parameter n is expressed as a third-order polynomial of the reciprocal of sg.
The predictability of proposed PTFs for other soils outside the calibrated database is also presented. It is found that the proposed PTFs give better estimation of the WRC than two existing PTFs. The model prediction is highly consistent with the measurements for sands. On the other hand, the predicted WRC matches well only with the measured data for soils with at least 20% of fines content in the low suction regime. However, it underestimates the normalized water content in the high suction regime. The discrepancy may be Published by NRC Research Press
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attributed to the limitation of the PSD-based estimation methods to model the WRC of coarse-grained soil with a considerable amount of fines as complex fabric cannot be addressed in these methods.
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Acknowledgements The authors would like to acknowledge support from the Seed Funding Program for Basic Research provided by the University of Hong Kong (Project Code 200908159002) and the National Natural Science Foundation of China under Grant No. 50878076.
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