Chain Monte Carlo (MCMC) samples of (a) mean µα (b) mean µn (c) standard deviation σα (d) standard deviation σn and (e) correlation coefficientρ for fly ash. 109 6.13 Empirical and calculated PDF and CDF for detrended data ln(Kd) in fig.
General
The reliability-based analysis can be achieved using various frameworks such as Monte Carlo simulation (Ang and Tang, 2007). The output of the governing equations can be linked with performance criteria for the structure (e.g.
Objectives
Scope and organization of the thesis
The multivariate probabilistic characterization task in all the chapters mentioned above is achieved using copula theory. This chapter presents the literature background that leads to the motivation for the objectives of this study.
Probabilistic characterization of SWCC
UNSODA (Nemes et al., 2001)) of SWCC for a particular soil texture (class) and then construct the joint probability distribution of the SWCC parameters. Very few studies (e.g. Zhang et al., 2018) have used the Bayesian approach to model multivariate non-normal distribution.
Probabilistic estimation of CEC and SSA
However, construction of non-normal joint distribution using conventional approaches such as translational methods (e.g. Phoon et al., 2010) are difficult to integrate into the Bayesian approach. The measurement of CEC and SSA is not easy, but rather a challenging and time-consuming task (Khorshidi et al., 2016; Khorshidi and Lu, 2017).
Stochastic seepage analysis with consideration to uncertainties in hydraulic parameters 12
The Vine-copula approach overcomes this limitation (Bedford and Cooke, 2001; Aas et al., 2009) and is therefore used in this study. However, even traditional copula approaches have some serious limitations regarding generalization in n dimensions (Aas et al., 2009), which is essential for random fields.
Database
Probabilistic Framework
Evaluation of various constraints among the SWCC model
It can be observed that AIC and BIC not only consider residual minimization (RSS), but also assign a penalty for the additional number of parameters. This is useful because a model with a large number of parameters is more likely to fit the data accurately (eg in terms of RSS).
Copula approach
It should be noted that modeling the dependence for a bivariate case is relatively simple, as the copula is characterized by only one parameter in the dependence matrix, θ. Φ−1(un)) (3.9) where C(ui. .un;θ) is the copula function with parameter matrix θ, Φθ(.) represents the multivariate standard normal function with linear correlation matrixθ, and Φ−1 gives the quantile or inverse CDF of one-dimensional standard normal distribution.
Results and Discussion
Effects of constraints on vG parameters for modelling SWCC of bentonites . 27
For the most popular case 4, it can be observed that the correlation coefficient of (α,n) is -0.66. Although it can be noted that the correlation between the parameters (eg α,n) for different cases differs significantly.
Construction of a multivariate probabilistic model for bentonite SWCC
3.6, and it can be noted that the empirical and the observed (using Lognormal) CDF are well comparable. It can be observed that the prescribed copulas together with lognormal marginals can adequately represent the measured vG parameters.
Practical implications of this study
It can be observed that at 1 MPa suction the frequency is highest for Se close to 1. It can be easily observed that there is considerable wide spread in the confidence intervals.
Summary
2012) reported the confidence intervals of SWCC for sand, sandy loam and silty loam using the Bayesian approach. The above confidence intervals are valuable for assessing uncertainty limits related to water flow and seepage modeling in the engineering projects/applications using SWCC of bentonites.
General
Copula based Bayesian approach
This aspect can be avoided by using a general class of acceptance-rejection sampling algorithm called Markov Chain Monte Carlo (MCMC) (Gelman et al., 2013, p. It can be observed that even if the target distribution in Equation 4.13 was computed including z term K−1 would be eliminated during the calculation in step 5.
Database used in this study
Regarding the coefficient of variation (COV) of the parameters, it can be noted that the COV for parameter α is highest for bentonite database at 205% and lowest for fly ash at 46. However, it should be noted again that this implication only applies to the constraint ofm= 1−1 /n.
Discussion
Probabilistic characterization of SWCC
It can be easily seen that given only 2 observed data points (N = 2) previously in Table 4.2 and the proposed approach, µα, σα,µn, σn and τα,n for the MCMC samples seem to approach the actual statistic population. High-density (inner) as well as low-density (outer) regions for MCMC samples are also representative of population density.
Influence of number of available data points
This is also in line with the philosophy of the Bayesian approach (e.g. Wang and Cao, 2013), where the reduction in uncertainty is greater at low data points, and with the increase in the number of available data points, estimates from both Bayesian approaches (MCMC samples) and conventional approaches (based on available data) converge. It is also important to note that the trend of decreasing dispersion with an increase in the number of data points could not be observed in a strict sense.
Influence of prior width
These observations are expected and in line with the philosophy of the Bayesian approach (Wang and Cao, 2013) and (K'ery and Schaub, 2011, for example P36,Tab 2, Lines 5-8), where the effect of previous experiences should gradually decrease as the number of data points increases. If the prior is not well defined, the effect of prior may not decrease with an increase in the number of data points.
Comparison with conventional approach
In the conventional approach (Prakash et al., 2018b), only the available data will be used to estimate the parameters of the joint distribution, i.e. 4.11, it can be easily observed that at a lower number of data points (e.g. N = 2, N = 5) the spread in the parameter estimate using the proposed approach is significantly lower than the conventional approaches.
Practical Application: Site specific unsaturated RBD
It can be noted in Table 4.4 that the distribution of SWCC parameters is classified as unknown. It can be noted that the fraction of FS < 1 becomes lower with the increase of ψ range.
Summary
Examples of the implementation of the global multivariate distribution for probabilistic estimation of CEC and SSA are also shown. This means that the uncertainty associated with the estimation of CEC and SSA is strictly defined.
Database: CLAY/C-S/5/278
The proposed approach is also crucial for extending reliability-based design and analysis approaches in projects where CEC and SSA are key control parameters. Therefore, the highest CEC and SSA values for montmorillonite reported in the literature (Low, 1980; Lu and Mitchell, 2019) and also in the database, i.e.
Multivariate distribution
According to the Sklar theorem (1959) any multivariate distribution can be expressed in terms of its marginals and a copula. Detailed algorithms for the efficient implementation of the above procedure in an arbitrary dimension n can be found in Aas et al.
Results and Discussion
However, for LL and PI, it can be observed that there is a large discrepancy between the measured and simulated values. It can be observed that for all ten measured and simulated pairs the τ values are in close agreement.
Probabilistic estimation of CEC and SSA: Implementation of the multivariate joint
All expressions for the conditional density of CEC and SSA can be obtained analytically and are summarized in Table 5.7. It can be easily concluded that the mode of the PDFs (dots) for CEC and SSA are consistent with the actual measured data.
Some example geotechnical applications
Frost Heave susceptibility
This section repeats the same problem shown in Konrad (1999) for the Saints-Martyrs-Canadiens site. In Konrad (1999) SSA was estimated deterministically, and in this study SSA will be estimated probabilistically from LL (ref. Table 5.7).
Swelling pressure- dry density relationship
From Eqs, the theoretical p−ρrelation can be established corresponding to some SSA and CEC values. However, the same was not demonstrated due to space constraints in this study and also as the main objective of this study was the probabilistic estimation of SSA and CEC.
Summary
This implies that the uncertainty associated with the estimation of CEC and SSA was rigorously characterized in the form of PDFs and therefore the proposed approach is superior to the popular transformation approach which can only be used to obtain a point estimate. Cholesky decomposition or Karhunen-Lo`eve (K-L) expansion) which can only handle the Gaussian spatial structure and interdependence in random fields of hydraulic parameters.
Random field: Vine copula approach
Univariate random field: Modelling spatial dependence
This implies that for C-vines, the first pair of trees can be uniquely determined from the spatial dependence model fitted to the autodependence data. The copula families among the selected candidates can be selected corresponding to the minimum AIC value.
Multivariate random fields: Modelling cross dependence
Borden aquifer hydraulic conductivity data: Implementation Example
It can be noted that for both locations in most cases the non-Gaussian copula is the best fit. 6.8 it can be observed that, except for core 4 and 9, for the rest of the cases non-Gaussian spatial dependence structures are identified as the best fit.
Stochastic seepage and slope stability analysis
Governing equations
Se(h, x, t){1−[1−Se(h, x, t)1/m(x)(x, t)]m(x)}2 (6.19) where Se is the effective degree of saturation , θwr and θws are residual and saturated volumetric water content, respectively, α and nare the unsaturated soil parameters related to air entry value and desaturation respectively, and m = 1 −1/n. To demonstrate the application, an example of an infinite unsaturated gradient is presented in this section.
Stochastic seepage and slope stability analysis under steady seepage: Impact
To investigate the impact of the spatial dependence structure ofk, the above exercise was performed using all other three copulas, and the resulting quantiles are shown in Figure. Nevertheless, this case study has shown for the first time that the potential impact of the spatial dependence structure of hydraulic parameters e.g.
Stochastic seepage and slope stability analysis under transient seepage: Im-
Apparently, it can also be noted that the effect of cross-dependence the structure in fig. The choice of cross-dependence structure was not found to be significant in this study.
Summary
It has been shown that the choice of the spatial dependence structure (copula) can significantly affect the pressure head quantiles, the safety factor and finally the probability of failure (Pf) of the slope. Apparently, it was also found that the choice of the spatial dependence structure (rarely considered) is more crucial than the cross-dependence structure (the main application area of copulas in geotechnical engineering) in the context of stochastic seepage and slope stability analysis.
Conclusions
It was found that the uncertainty (in terms of COV) in the estimates of CEC and SSA decreases as the values of LL, PI and CF increase. It was shown that a non-Gaussian dependence structure can be observed for most of Borden's aquifer data.
Contributions
Also, 8 out of 10 pairs of copulas required for joint density construction were found to be non-Gaussian, thereby indicating strong evidence of non-Gaussian dependence structure among the five parameters in the CLAY/C-S/5/278 database. In the context of stochastic seepage and slope stability analysis, this study presented a grapevine head-based multivariate random field framework to handle the non-Gaussian (along with Gaussian) spatial and cross-dependence structure under the random fields of hydraulic conductivity and of Genuchten (1980) soil-water characteristic curve parameters (α and n).
Limitations
Future scope
2011), ‘Quantitative estimation of clay mineralogy in fine-grained soils’, Journal of Geotechnical and Geoenvironmental engineering Reliability analysis of the soil-water characteristics curve and its application to slope stability analysis’, Engineering Geology 135, 83–91. 2003), “Identification of Statistically Homogeneous Soil Layers Using Modified Bartlett Statistics,” Journal of Geotechnical and Geoenvironmental Engineering.
Acceptable failure probability P f after Salgado and Kim (2014)
Details of the compiled database for bentonite SWCC
Comparison of van Genuchten (1980) parameter statistics obtained under various
Measured rank correlation coefficient Kendall’s τ among van Genuchten (1980) pa-
Performance evaluation of various constraints in van Genuchten (1980) model
Measured and simulated Kendall’s rank correlation for van Genuchten (1980) pa-
Identification of best fit marginals for curve fitting parameters in van Genuchten
Lognormal parameters for α,n,m
The goodness of fit for Gaussian and student t-Copula
Prior utilized in this study for mean µ, standard deviation σ and rank correlation
Slope parameters used for the application example in this study
Details of the compiled database CLAY/C-S/5/278
Some statistics for the five parameters in the CLAY/C-S/5/278 database
Marginal selection for the parameters in the CLAY/C-S/5/278 database
Measured and simulated univariate statistics for the parameters in the CLAY/C-
Comparison of measured and simulated univariate statistics for LL and PI using
Copula selection for the CLAY/C-S/5/278 database corresponding to the D-vine
Expressions for conditional/posterior probability density functions (PDFs) of CEC
Parameters for the application example in this study
