Reliability of an engineered slope considering the Regression Kriging (RK)-based conditional random field

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ABSTRACT

Many attempts have been made to apply random field theory to the slope reliability analysis in recent decades. However, there are only a few studies that consider real landslide cases by incorporating actual soil data in the probabilistic slope stability analysis with spatially variable soils. In this paper, an engineered slope located in Hong Kong was investigated using the probabilistic approach considering the Regression Kriging (RK)-based conditional random field. The slope had been assessed and considered to be safe by classical deterministic slope stability analyses but failed eventually. In this study, both deterministic slope stability analyses and probabilistic slope stability analyses were conducted, and the comparison was made between the probabilistic approach adopting RK-based conditional random field and that adopting Ordinary Kriging (OK)-based approach. The results show that the deterministic factor of safety (FS) for a slope may not be an adequate indicator of the safety margin. In particular, a slope with a higher deterministic FS may not always represent a lower probability of failure under the framework of probabilistic assessment, where the spatial variability of soil properties is explicitly considered. Besides, the critical portion of the slope could not be found using the OK-based approach that considers a constant trend structure.

KEYWORDS Slope reliability analysis; engineered slope; conditional random field; Regression Kriging; restricted maximum likelihood CONTACT Lei Huang

Received 14 February 2020

[email protected]

on a fixed critical surface were compared with those considering randomised search of slip surface. It was found that the probability of failure (Pf) of a slope would be underestimated when considering the fixed critical surface. In these previous studies, unconditional random fields were simulated where the site investigation data was unavailable. In order to take the site investigation data of different sample points into consideration, some researchers adopted conditional random fields to study the reliability of slopes (Gholampour and Johari, 2019a; Gholampour and Johari, 2019b; Jin and Sitar, 2013; Johari and Gholampour, 2018; Li et al., 2016; Liu et al., 2017a; Lo and Leung, 2017). In Liu et al. (2017a), the reliability after conditioning of a slope under drained conditions was investigated in two dimensions. Three-dimensional probabilistic slope stability analysis was implemented by Li et al. (2016) through conditional random field simulation. In Johari and Gholampour (2018), unsaturated soil characteristics were considered in probabilistic slope stability analysis by the conditional random field simulation method. When constructing conditional random field, Ordinary Kriging (OK) is often adopted and merged with traditional random field generators. The OK assumes an unknown constant trend. However, due to the deposition and complexity of natural soils, a constant trend may not be an optimal trend structure of soil properties in most cases (Asaoka and A-Grivas, 1982; Carsel and Parrish, 1988; Fredlund et al., 1978; Zhang, 2005). Thus, Regression Kriging (RK) would be a better choice if the trend structure can be obtained.

Besides, considering the sparse sampling data in most 1. Introduction

Landslides occasionally occur on engineered slopes that were assessed to be “safe” based on conventional deterministic slope stability analysis techniques. In Hong Kong, this kind of landslides can be observed almost every year (see Table 1). In the classical deterministic slope stability analysis, the soil properties of each soil layer are assumed to be homogeneous. This assumption ignores the local weak portion of a slope which might be a potential failure zone since the spatial variability of soil properties is neglected. In order to incorporate the soil spatial variability into slope stability analyses, classical slope stability techniques (e.g. finite element method and limit equilibrium method) have been combined with random field theory to develop advanced probabilistic analysis tools, such as random finite element method (RFEM) (Griffiths and Fenton, 2004; Griffiths et al., 2009; Hicks and Spencer, 2010; Hicks et al., 2014; Huang et al., 2010;

Jha and Ching, 2013; Jiang et al., 2014; Li et al., 2016; Liu et al., 2020) and random limit equilibrium method (RLEM) (Cho, 2010; Jiang et al., 2015; Jiang and Huang, 2016; Li et al., 2013; Li et al., 2015; Liu et al., 2018; Wang et al., 2011). In Griffiths and Fenton (2004), the reliability of a slope was investigated by a 2D probabilistic slope stability analysis, considering various autocorrelation distances and coefficients of variation of soil properties. Later on, the slope reliability in three dimensions was investigated by Griffiths et al. (2009) considering soil spatial variability.

In Cho (2010), simulation results by RLEM based

Reliability of an engineered slope considering the Regression Kriging (RK)-based conditional random field

Lei Huang^*, Andy Yat Fai Leung, Wenfei Liu and Qiujing Pan

Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, People’s Republic of China

*The first author who was at the age of 35 or below on the closing date of submission for the HKIE Outstanding Paper Award for Young Engineers / Researchers 2020.

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L HUANG ET AL.

184 HKIE Transactions | Volume 27, Number 4, pp.183-194

engineering practices, the performance of RK is usually better than OK (Batista et al., 2017; Baxter and Oliver, 2005).

Table 1. Number of landslide incidents involving slopes processed under the conventional slope safety system from 2008 to 2014 in Hong Kong.

Year No. of landslide incidents

2008 29

2009 12

2010 8

2011 7

2012 17

2013 14

2014 8

In this study, the conditional random field was considered in the slope reliability analysis to assess the safety of an engineered slope, and the comparative study was conducted between the approach using RK-based conditional random field and the approach using OK-based conditional random field. This slope had been assessed using the deterministic limit equilibrium method and accepted as “safe” by local authorities in 2001, but it eventually failed in 2010. The integrated framework proposed by Liu et al. (2017b), which combined the restricted maximum likelihood (REML) method with residual diagnostics and an optimal detrending process, was adopted to obtain the autocorrelation distance and the optimal trend used in the RK interpolation. For random field generations, the best estimates of soil properties were determined by Kriging interpolation techniques, and then adopted in the random field models. The Box-Cox transformation (1964) was used to help ensure the stationarity assumptions before generating the random fields. In the current work, the factor of safety (FS) was recalculated using the non-circular limit equilibrium method, while the probability of failure was computed by random limit equilibrium method. Also, the failure mechanisms obtained based on the deterministic analysis and the probabilistic analysis were investigated and presented.

2. Description of the case

2.1. Site description and background

The engineered slope investigated in this study comprises two parts, and is located in the South East of Sai Wan Terrace, Quarry Bay, Hong Kong (Figure 1(a)).

In the current study, the upper part of the engineered slope is named “Section A”, and the lower part is named

“Section B”. Section A is a soil cut slope with a width of approximately 90 m measured along its toe and a maximum height of about 15 m at its central portion. Section B is also

a soil cut slope located between the government office and Section A. The width and maximum height of Section B are about 63 m and 14 m, respectively. The average gradients of the two sections are both approximately 45°.

7

(a) 109

110

111

112 113 (b)

114 、

Figure 1. (a) The study area (Google map) and(b) section 1-1.

115 116 117

118

7

(a) 109

110

111

112 113 (b)

114 、

Figure 1. (a) The study area (Google map) and(b) section 1-1.

115 116 117

118

(a)

(b)

Figure 1. (a) The study area (Google map) and (b) section 1-1.

The stability assessment of the engineered slope was completed in 2001 with Section A and Section B analysed separately. An unsupported cross-section 1-1 (Figures 1(a) and 1(b)) , which involves the two sections, was identified as critical due to its height and gradient, and was therefore investigated at that time. The results of the slope stability analysis revealed that the minimum FS was 1.336 for the entire cross-section with a corresponding critical surface located within Section B, while the minimum FS of Section A was 1.429. Therefore, the reports concluded that based on the recommended FS against high loss of life and property in Hong Kong which is 1.4, the safety margin of Section A was adequate. Although the FS of Section B was less than the above threshold, no upgrading works were conducted since the FS was greater than 1.2 which corresponded to a recommended value against low loss of life and property and the corresponding site had undergone extensive geotechnical investigation and stability assessment.

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HKIE Transactions | Volume 27, Number 4, pp.183-194 185 However, a shallow landslide occurred on the

unsupported part (corresponding to cross-section 1-1) of Section A on 10 June 2010 due to rainfall (Leung et al., 2014; Li et al., 2014) with a failure volume of approximately 40 m³, whereas no failure record for Section B has been reported so far. This means the deterministic analyses failed to identify the higher failure probability associated with Section A, which was determined to have a higher FS.

2.2. Soil sampling

Soil samples from 12 boreholes (Figure 2) were considered in this study. A total of 33 sampled points for soil cohesion and friction angle were collected from an area of 100 m × 90 m based on the geotechnical investigation reports, and the number of samples retrieved from each borehole is shown in Figure 2. All these sampled points are located in the completely decomposed granite (CDG) layer with a lowest level of 23.5 mPD and a highest level of 43.08 mPD (mPD = metre above Principal Datum of Hong Kong, which is 1.230 m below the mean sea level).

9

sampled points for soil cohesion and friction angle were collected from an area of 100 142

m×90 m based on the geotechnical investigation reports, and the number of samples 143

retrieved from each borehole is shown in Figure 2. All these sampled points are 144

located in the completely decomposed granite (CDG) layer with a lowest level of 23.5 145

mPD and a highest level of 43.08 mPD (mPD = metre above Principal Datum of 146

Hong Kong, which is 1.230 m below the mean sea level).

147 148

149

150

151

Figure 2. Distribution of the boreholes.

152 153

3. Modelling spatial variability of soil properties

154 155

Figure 2. Distribution of the boreholes.

3. Modelling spatial variability of soil properties 3.1. Spatial random variables and random field

Soil properties can be modelled as spatially correlated random variables based on random field theory. The random field model can be given by

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) ( 2 ij j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³²₂

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

, (1)

In Equation (1),

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) ( 2 ij j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³²₂

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

and

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) ( 2 ij j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³₂²

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

denote the soil property and the expected value at a specified location, respectively.

σ is the standard deviation of the Gaussian random field and

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) ( 2 ij j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³₂²

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

is the jth element in the vector of the ith realisation of

the residual part of a random field.

3.2. Spatial autocorrelation and matrix decomposition method

In this study, a two-dimensional slope reliability analysis was carried out. However, soil samples and spatial autocorrelation of the soil properties are in three dimensions in reality. Therefore, the following theoretical three- dimensional autocorrelation function was used:

10

3.1.Spatial random variables and random field 156

Soil properties can be modelled as spatially correlated random variables based on 157

random field theory. The random field model can be given by 158

) ( 2 ij j

j µ σ ε

z = + (1)

159

In Equation (1), z and _j µ_jdenote the soil property and the expected value at a 160

specified location, respectively. σ is the standard deviation of the Gaussian random 161

field andε is the jth element in the vector of the ith realisation of the residual part of ⁽ⁱ_j⁾ 162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method 165

In this study, a two-dimensional slope reliability analysis was carried out. However, 166

soil samples and spatial autocorrelation of the soil properties are in three dimensions 167

in reality. Therefore, the following theoretical three-dimensional autocorrelation 168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³₂²

v

h θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2), τ₁and τ₂are the lag distances in the two horizontal directions, and 171

τ3is the lag distance in the vertical direction. θ_hand θ_vrepresent the autocorrelation 172

distances in the horizontal and vertical directions, respectively, which are determined 173

by REML method discussed in Section 5.

174

, (2)

In Equation (2),

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) 2 i(

j j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³²₂

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

and

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) 2 i(

j j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³₂²

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

are the lag distances in the two horizontal directions, and

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) ( 2 ij j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³₂²

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

is the lag distance in the vertical direction.

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) ( 2 ij j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³₂²

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

and

10

3.1.Spatial random variables and random field

156

Soil properties can be modelled as spatially correlated random variables based on

157

random field theory. The random field model can be given by

158

) ( 2 ij j

j

µ σ ε

z = + (1)

159

In Equation (1), z and

_j

µ denote the soil property and the expected value at a

_j

160

specified location, respectively. σ is the standard deviation of the Gaussian random

161

field and ε is the jth element in the vector of the ith realisation of the residual part of

⁽ⁱ_j⁾

162

a random field.

163

164

3.2.Spatial autocorrelation and matrix decomposition method

165

In this study, a two-dimensional slope reliability analysis was carried out. However,

166

soil samples and spatial autocorrelation of the soil properties are in three dimensions

167

in reality. Therefore, the following theoretical three-dimensional autocorrelation

168

function was used:

169

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ + +

−

=exp ( )

) , ,

( ₁ ₂ ₃ ¹² ₂²² ³₂²

v

h

θ

τ θ

τ τ τ

τ τ

ρ (2)

170

In Equation (2),

τ₁

and

τ₂

are the lag distances in the two horizontal directions, and

171

τ3

is the lag distance in the vertical direction.

θ_h

and

θ_v

represent the autocorrelation

172

distances in the horizontal and vertical directions, respectively, which are determined

173

by REML method discussed in Section 5.

174

represent the autocorrelation distances in the horizontal and vertical directions, respectively, which are determined by REML method discussed in Section 5.

Once the autocorrelation structure is determined, the spatial autocorrelation matrix R can be obtained, and the random field can be generated using Cholesky decomposition:

11

175

Once the autocorrelation structure is determined, the spatial autocorrelation 176

matrixRcan be obtained, and the random field can be generated using Cholesky 177

decomposition:

178

LLT

R = (3) 179

i Lsi

ε =⁽⁾ (4) 180

In Equations (3) and (4), L is the Cholesky factor of R, and s_iis anne × 1 vector of 181

independent standard Gaussian random variables. ε denotes the ith realisation of ⁽ⁱ⁾ 182

183

184

3.3. Cross-correlation of soil parameters 185

Soil parameters are usually found to be cross-correlated with each other. For cohesion 186

and friction angle, a negative correlation is often observed. According to the literature, 187

in general, the coefficient of the cross-correlation between cohesion and friction angle 188

ρc,ϕ is in the range of -0.7 <ρc,ϕ< -0.24( Lump, 1970; Rackwitz, 2000;Yucemen et al., 189

1973). The cross-correlation should be considered when generating a random field of 190

a c - ϕ soil (Fenton and Griffiths, 2003). In the current work, ρc,ϕ = -0.5 was adopted 191

as a base set, while ρc,ϕ = {-0.7,-0.3,-0.2} was also considered.

192

193

3.4. Conditioned random field 194

, (3)

11

175

decomposition:

178

LLT

R = (3) 179

i Lsi

ε =⁽⁾ (4) 180

183

184

192

193

, (4)

In Equations (3) and (4), L is the Cholesky factor of R, and

11

175

decomposition:

178

LLT

R = (3) 179

i Lsi

ε =⁽⁾ (4) 180

183

184

192

193

is an ne × 1 vector of independent standard Gaussian random variables.

11

175

decomposition:

178

LLT

R = (3) 179

i Lsi

ε =⁽⁾ (4) 180

183

184

192

193

denotes the ith realisation of the residual part of a random field.

3.3. Cross-correlation of soil parameters

Soil parameters are usually found to be cross- correlated with each other. For cohesion and friction angle, a negative correlation is often observed. According to the literature, in general, the coefficient of the cross-correlation between cohesion and friction angle ρc,ϕ is in the range of -0.7 < ρc,ϕ < -0.24 (Lump, 1970; Rackwitz, 2000; Yucemen et al., 1973). The cross-correlation should be considered when generating a random field of a c - ϕ soil (Fenton and Griffiths, 2003). In the current work, ρc,ϕ = -0.5 was adopted as a base set, while ρc,ϕ = {-0.7,-0.3,-0.2} was also considered.

3.4. Conditioned random field

Site investigation data provide information for characterising the spatial variability of soil properties.

Kriging interpolation approaches are usually adopted to make predictions for unsampled points. However,