3.3 Probabilistic Framework
3.3.1 Evaluation of various constraints among the SWCC model
SWCC expresses the relationship between two primary state variables, water content (w/θw/S) and suction (ψ). There are various empirical or semi-empirical models reported in the literature (Brooks and Corey, 1964; van Genuchten, 1980; Fredlund and Xing, 1994) for quantifying a con- tinuous SWCC using the measured suction and water content data. Among them all, the one proposed by van Genuchten (1980) remains the most popular and is widely used in geotechnical and geoenviromental literature (Carsel and Parrish, 1988; Phoon et al., 2010; Chiu et al., 2012, e.g.,). The vG model in terms of degree of saturation is given as:
Se= S−Sr
1−Sr = 1
[1 + (αψ)n]m (3.1)
Where Se= effective degree of saturation, Sr= residual degree of saturation,ψ= suction head, α= parameter related to inverse of air entry value, n= parameter related to rate of desaturation and m is a symmetry parameter very often represented in following forms m = 1−1/n , m = 1−2/n , m = 1 after Mualem (1976), Burdine (1953) and Gardner (1958) respectively. Along with independent m, a total of 4 cases are possible. The constraints mentioned above regarding m are often applied to reduce the number of unknown parameters. Another common constraint very often applied is Sr=0. However, these constraints arises mainly from the lack of knowledge and confidence regarding the definition and inference of residual degree of saturation (Likos and Yao (2014)). Combining the constraintSr=0 and Sr retained as a fitting parameter, with 4 cases already mentioned, a total of 8 constraints are possible. In this study hereafter these 8 constraints will be referred to as 8 cases and are illustrated in Table 3.2. These 8 cases were fitted to the SWCC(S,ψ) data using the nlinfit subroutine in Matlab. This subroutine utilizes the Levenberg- Marquardt algorithm for optimization. More details about the algorithm can be found elsewhere (Seber and Wild, 2003, e.g.,).
As already discussed, in most of the studies, the constraints are selected arbitrarily without any proper statistical evaluation for the particular class of soil being considered. However, it is essential to evaluate the impact of various constraints on the SWCC parameters due to following reasons (1) since the measurement of SWCC of bentonite is way too complicated and time-consuming, a variety of pedotransfer (PTF) approaches are common to estimate the SWCC from more easily measured soil properties. The approach used is to develop correlations among measured soil properties and parameters of vG model. However, often the SWCC constraints applied to obtain the parameters are not specified or are unknown (Likos and Yao, 2014). (2) Each of the constraints although applied arbitrarily, imposes some restriction on the curve fitting capability of the model. While some constraints can be suitable for sandy soil, it may not be suitable for a clayey material like bentonite. (3) The degree of correlation among the SWCC parameters, which is an important aspect for reliability studies can vary widely with the choice of constraints (ref. Table 3.3). (4)
Key properties being estimated such as AEV, which is closely related to the inverse ofαvalue can be varying in the order of MPa due to the choice of constraints. An ideal analysis would entail the evaluation of various available SWCC equations other than vG, but for the sake of brevity, only vG equation has been evaluated in this study. The aim of this study is not only to evaluate and compare various constraints but thereafter also develop a joint probability distribution of the parameters using the best fit constraint from the aforementioned study.
There are various approaches available in the literature for performance evaluation or model se- lection such as Akaike Information Criterion (AIC) (Akaike, 1974), Bayesian Information Criterion (BIC) (Schwarz et al., 1978), and Deviance Information Criterion (DIC) (Van Der Linde, 2005). In this study, AIC and BIC are used. They are defined as follows:
BIC =N ln[
PN
j=1(yobsj −yf itj (b))2
N ] +kln(N) (3.2)
AIC =N ln[
PN
j=1(yjobs−yjf it(b))2
N ] + 2k (3.3)
Where N= sample size i.e. no of data points, k= no of free parameters in the model, Yjobs is observed S (measured S in database) and yjf it (b) is the S obtained using Eq. 3.1 respectively at ψj (measuredψ in database), PN
j=1(yjobs−yjf it(b))2 = residual sum of squares (RSS) and b is the parameters vector containing the k parameters that need to be estimated. The best fit model is selected on the basis of minimum BIC and AIC criterion. It can be noted that the AIC and BIC not only take into account the minimization of the residuals (RSS) but also assign a penalty term for the extra number of parameters. This is beneficial since a model with a large number of parameters is more likely to accurately fit the data (e.g., in terms of RSS). But this is not always desirable as it can lead to overfitting and multiple local optima during the optimization (curve fitting) process.
For models with a similar degree of fit in terms of RSS the penalty termskln(N) and 2k in Eqs.
3.2 and 3.3 ensure a larger BIC and AIC value for the model with a larger number of parameters.
Since the criterion for selection is minimum BIC and AIC, the model with extra parameters but similar RSS (compared to alternate models) gets eliminated. The procedure adopted in this study for performance evaluation is summarized using a flowchart in Fig. 3.1. For further comparison of goodness of fit for various models, RMSE andR2 given in Eq. 3.4 and Eq. 3.5 respectively are also evaluated.
RM SE = s
PN
j=1(yobsj −yjf it(b))2
N (3.4)
R2 = 1−
PN
j=1(yjobs−yjf it(b))2 PN
j=1(yobsj −(N1 PN
k=1ykobs)2 (3.5)
Compile 𝑺 − 𝝍 database for the bentonite
Fit vG equation with various constraints 𝒋 = 𝟏: 𝟖 mentioned in
Table 2 to SWCC data (𝒊 = 𝟏: 𝟔𝟎
Compute the 𝑹𝑴𝑺𝑬, 𝑹𝟐, 𝑨𝑰𝑪 and 𝑩𝑰𝑪 for each of the fit (i.e.
𝑺𝒊𝒛𝒆 𝒊 × 𝑺𝒊𝒛𝒆 𝒋 = 𝟒𝟖𝟎)
For each of the model 𝒋 = 𝟏: 𝟖 , average the above values for
whole database = ∑𝑽𝒂𝒍𝒖𝒆
𝟔𝟎 Where 𝑽𝒂𝒍𝒖𝒆 = [ 𝑹𝑴𝑺𝑬, 𝑹𝟐, 𝑨𝑰𝑪, 𝑩𝑰𝑪]
Select the model 𝒋 with minimum average 𝑨𝑰𝑪 and 𝑩𝑰𝑪
Start
Stop
Figure 3.1: Procedure used for identification of best fit van Genuchten (1980) model among various constraints mentioned in Table 3.2.