# Cross-Correlation Analysis of the Stability of Heterogeneous Slopes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{f}increases when the correlation between c and tanφ increases. Griffiths et al. [8] and Jiang et al. [9] obtained similar conclusions by studying two-dimensional heterogeneous slopes using the random finite element method. The correlation between c and tanφ significantly influences the probability of slope failure. However, the influence of the correlation between c and tanφ on slope stability considering parameter statistical anisotropy has not been comprehensively studied. Likewise, studies have compared the effect of the conditional random field with the unconditional random field on slope stability analysis using MCS with the finite element method. They concluded that conditional random field simulations could address the reduction in uncertainty due to conditioning with sampled parameters in evaluating slope stability [28,29,30,31]. Nonetheless, these studies have not addressed the vital issue of reducing uncertainty by selecting sampling locations. This issue (where to sample and how many sampling locations) to reduce the uncertainty is critical since only a limited number of samples is available in field situations.

## 2. Methodology

#### 2.1. Random Field Modeling of Heterogeneity

_{ij}is the autocorrelation coefficient between the parameter at location (x

_{i}, y

_{i}) and location (x

_{j}, y

_{j}), and λ

_{x}and λ

_{y}are the horizontal and vertical correlation scales, respectively. Many other forms of the autocorrelation function are available. They all are ensemble statistics (i.e., general knowledge). We, therefore, chose the most simplistic one. This study considers the correlation scales of c and tanφ as the same.

_{c}= 15 kN/m

^{2}, σ

_{c}= 7.5 kN/m

^{2}), but with different correlation scales. These fields are called statistically isotropic when the λ

_{x}is the same as the λ

_{y}(Figure 1a,b), and statistically anisotropic when the λ

_{x}is different from the λ

_{y}(Figure 1c–f). In Figure 1c,d, the correlation scales are 1 m in the vertical direction and 5 m and 10 m in the horizontal direction, respectively. As the λ

_{x}becomes large, the strong and weak zones extend greatly in the horizontal direction, and the slope shows an apparent horizontal layered structure. Figure 1e,f display the cases where λ

_{x}= 1 m, λ

_{y}= 5 m and λ

_{x}= 1 m, λ

_{y}=10 m, respectively. The slope exhibits a vertically layered structure as the λ

_{y}becomes greater than the λ

_{x}. In this study, we consider different cases as follows. For statistical isotropy: λ

_{x}= λ

_{y}= 1 m; for statistical horizontal anisotropy: λ

_{y}= 1 m, λ

_{x}= 5 m, 10 m, 20 m, 40 m, 80 m, respectively; for statistical vertical anisotropy: λ

_{x}= 1 m, λ

_{y}= 5 m, 10 m, 20 m, 40 m, respectively.

#### 2.2. Slope Stability Analysis

_{x}= 0 m), the bottom boundaries are zero horizontal and vertical displacements (u

_{x}= 0 m, u

_{y}= 0 m), and the slope surface is free displacement. The slope is assumed to be subjected to gravity loads only and consists of elastic–perfectly plastic soils following the Mohr–Coulomb failure criterion. Specifically, the loading stress at each element is the total weight of the element above, and the shear strength of each element follows the Mohr–Coulomb failure criterion. Other complex or advanced constitutive models [35,36,37] or numerical simulation methods [38] could be used with corresponding randomized input parameters. Table 1 lists the statistics of soil mechanical parameters, except for their correlation scales. Subsequently, the corresponding FS for the entire slope was evaluated based on the finite element strength reduction method (SRM) [39,40]. The program for calculating FS in this study is mainly based on the program p64 [40], and the main difference lies in the automatic MCS and the search for the critical strength reduction factor. The SRM has been widely used due to its practicality and reliability. The factor of safety (FS) is defined as the proportion by which c and tanφ must be reduced in order to cause slope failure. The strength reduction based on the Mohr–Coulomb criterion is shown in the following equation [39,40]:

_{trial}and tanφ

_{trial}are the trial shear strength parameters, which decrease with an increase in SRF. Several gradually increasing values of the SRF are tested, and the updated c

_{trial}and tanφ

_{trial}are used for elastoplastic analysis. When the algorithm does not converge for 1000 iterations, the slope is considered as having failed in this study. The smallest value of SRF causing failure is then interpreted as the factor of safety FS. For example, the FS of this slope using the mean values (Table 1) as input parameters, considering the slope as homogeneous, is 1.141.

_{f}and the reliability index β according to the following formula [16]:

_{FS<1}is the number of realizations whose FS value is less than 1 (i.e., the slope fails). A small value of N

_{FS}

_{<1}implies that the probability of failure of the slope is small. In Equation (5), μ

_{FS}is the mean value and σ

_{FS}is the standard deviation of FS values of N realizations of MCS. (μ

_{FS}− 1) represents the slope stability and σ

_{FS}represents the uncertainty in the evaluating FS. Therefore, the larger β is, the more reliable the estimated FS is and the smaller the probability of slope failure.

#### 2.3. Cross-Correlation Analysis

_{FSc}(${x}_{i}$) and ρ

_{FS}

_{tanφ}(${x}_{i}$) are the cross-correlations between the cohesion and the internal friction angle at the location ${x}_{i}$ and FS, respectively. FS

_{k}is the FS at the k

_{th}realization; c(${x}_{i}$,k) is the cohesion at the location ${x}_{i}$ and k

_{th}realization; tanφ(${x}_{i}$,k) is the tangent of the internal friction angle at the location ${x}_{i}$ and k

_{th}realization. Lastly, σ

_{c}(${x}_{i}$) is the standard deviation of the cohesion; σ

_{tanφ}(${x}_{i}$) is the standard deviation of tanφ.

## 3. Results of Cross-Correlation Analysis for Statistical Isotropy

_{FSc}is positive at the toe and the top of the slope (Figure 3a), suggesting a large c value at these areas leads to the greater FS value of the slope. On the other hand, the interior of the slope areas has a positive correlation between tanφ and FS (Figure 3b). The remaining areas have correlation values close to zero, meaning that shear strength in these areas (i.e., most of the slope surface and the back of the slope.) has little effect on the slope stability.

_{c}, Figure 4a) and internal friction angle (the sensitivity of FS to tanφ, J

_{tanφ}, Figure 4b) display similar patterns to those in Figure 3a,b, derived from MCS. The sensitivity analysis of FS to the parameter takes the following steps. First, the parameter at each location is set as the mean value, and the FS of the homogeneous slope is evaluated by the finite element strength-reduction method. Then, we applied a perturbation of the parameter at a spatial location

**x**

_{i}, keeping the parameter at the other locations as the mean value. We subsequently evaluated the FS corresponding to this perturbation. The ratio of the change of FS to the perturbation is the sensitivity of FS to the parameter at this position. After calculating the sensitivity at each location, we derived a sensitivity map. The map shows that the sensitivity of FS to c is greater at the foot and top of the slope compared to other regions, and the sensitivity of FS to tanφ is more significant in the interior of the slope. Notice that the sensitivity analysis, based on the perturbation method, aims at the change of FS per change in the given mean value of the parameter, ignoring the variability (variance) and spatial structure (correlation scale) of the parameters [42]. Specifically, the cross-correlation analysis considers many possible slopes with heterogeneous parameter fields with the same mean parameter value but different perturbations and spatial structure patterns. It then summarizes the results statistically. Consequently, the sensitivity analysis results (Figure 4) differ from the cross-correlation analysis (Figure 3), and the cross-correlation analysis is most appropriate for cases where spatial parameter values are unknown (i.e., realistic field situations).

## 4. Results of Cross-Correlation Analysis for Statistical Anisotropy

#### 4.1. Statistical Horizontal Anisotropy (Horizontal Correlation Scale > Vertical Correlation Scale)

_{x}values (Figure 5) while λ

_{y}= 1 m. Figure 5a,c show the cross-correlation maps between FS and c, for λ

_{x}of c equal to 5, 10, and 20 m, respectively. The cross-correlation maps between FS and tanφ, for λ

_{x}of tanφ equal to 5, 10, and 20 m are presented in Figure 5d,f, respectively. We observe that the positive areas of ρ

_{FSc}and ρ

_{FS}

_{tanφ}expand as λ

_{x}increases, but the areas are confined to the areas at the slope toe. Comparing the results to the cross-correlation map of the statistically isotropic parameters (Figure 3), we notice that ρ

_{FSc}develops from the toe. In contrast, ρ

_{FS}

_{tanφ}develops from the inside of the slope.

_{f}), reliability β index, the mean of FS (μ

_{FS}), and the standard deviation of FS (σ

_{FS}) as a function of the normalized horizontal correlation scale, respectively. The normalized horizontal correlation scale is λ

_{x}/(H/tanα), the ratio of the horizontal correlation scale to the horizontal projection of the slope length, H/tanα. Notice that α is the slope inclination angle. Since the correlation between c and tanφ (φ is the friction angle) is generally unclear [8,9,16], this study also examines the effect of perfectly positively, zero, and negatively correlated c and tanφ perturbation relationships and they are indicated by the red, green, and blue lines in these figures, respectively.

_{FS}) in Figure 6d rapidly increase as λ

_{x}tanα/H approaches one and stabilize afterward. On the other hand, the reliability (β) (Figure 6b) and the mean of FS (μ

_{FS}) (Figure 6c) decrease exponentially. These results stem from the fact that for horizontally layered slopes, the layer with the lowest parameters (such as the weak interlayer or the stratum with highly developed joints and fractures) controls the stability of the slope. A longer correlation scale means that the layer with the weakest strength covers most of the slope, and the slope is less stable. On the other hand, from the physical meaning of the correlation between c and tanφ, the higher the correlation is, the lower c is at a location, and the lower the tanφ at the same location. Therefore, the slope is less stable (i.e., the red line is higher than the green line in Figure 6a).

_{x}tanα/H increases, the p

_{f}value increases slightly but remains very low (about 0.05), while the reliability β index increases and remains high at about 2.4. The value of μ

_{FS}increases from 1.075 to 1.085 and remains constant over the rest of λ

_{x}(tanα/H). The value of σ

_{FS}decreases first at λ

_{x}(tanα/H) = 0.5 and remains almost constant at a small value. The trends are distinctly different from when ρ

_{c}

_{tanφ}is 1 or 0. The negative correlation means that a large c is at one location, and a small tanφ is at the same location or vice versa. As a result, the slope stability no longer decreases significantly or even increases slightly.

_{x}can result in an overestimated slope stability when evaluating the stability of horizontally layered slopes—the importance of identifying the spatial structure of the slope is clear.

#### 4.2. Statistical Vertical Anisotropy (Horizontal Correlation Scale < Vertical Correlation Scale)

_{y}values when the slope has vertically stratified formations. Figure 7a–c illustrate the cross-correlation maps between FS and c, and λ

_{y}of c with 5, 10, and 20 m, respectively, while their horizontal correlation scales are 1 m. These figures indicate a distinctly positive correlation between FS and c at the toe of the slope. Furthermore, as λ

_{y}increases, the high correlation area becomes more concentrated and vertical, and the cross-correlation value weakens slightly.

_{y}of tanφ equal to 5, 10, and 20 m, respectively. We observe that FS and tanφ are positively correlated at the area x = 20 to 25 m and y = 0 to 5 m, and the cross-correlation decreases slightly as λ

_{y}increases.

_{f}, β, μ

_{FS}, and σ

_{FS}in the slopes with longer vertical correlation scales than the horizontal one as a function of λ

_{y}/H (the vertical correlation scale normalized by the height of the slope, H) are displayed in Figure 8a–c, and d, respectively. First, we notice that the value of probability failure (p

_{f}) in this case is much less than that in the horizontal layering slope (i.e., Figure 8a vs. Figure 6a), regardless of the effects of various factors as in Figure 6. In other words, vertical stratification (the orientation of the large-scale structures) plays a more dominant role than the others do in slope stability.

_{c}

_{tanφ}is 1 (the red line), with the increase in λ

_{y}, p

_{f}and σ

_{FS}decrease, and μ

_{FS}and β increase, indicative of the fact that as λ

_{y}increases, the stability of the slope increases and the uncertainty of the evaluation decreases. This result stems from the fact that when a slope is vertically layered, the high-strength layer controls the stability, similar to anti-slip piles. The longer λ

_{y}means that the layer with high strength is extensive, and when ρ

_{c}

_{tanφ}is 1, the c is large and so is φ large, and the anti-slip pile can be effective, leading to high slope stability.

_{c}

_{tanφ}is 0 or −1 (the green and blue line in Figure 8), the increase in λ

_{y}leads to increases in p

_{f}and σ

_{FS,}and it decreases μ

_{FS}and β values, suggesting that increasing λ

_{y}worsens the stability of the slope and increases the uncertainty of the evaluation. These trends are the opposite of when ρ

_{c}

_{tanφ}is 1, likely because c and tanφ are uncorrelated or perfectly negatively correlated, and the effect of anti-slip piles weakens.

#### 4.3. Effects of the Number of Realizations in MCS

_{f}and μ

_{FS}at λ

_{x}= 10 m and 20 m, respectively, and λ

_{y}= 1 m, in these conditions, σ

_{FS}are the maximums. As shown in the figure, the mean values of p

_{f}and μ

_{FS}fluctuate widely within 150 realizations but stabilize after more than 300 realizations, certifying the adequacies of the number of realizations used in the MCS and the results’ representativeness.

## 5. Effects of Conditional Random Fields

_{y}of the random fields in the two cases were identical, except that λ

_{x}was different.

_{FS}of sampling Scheme 3 is the closest to the FS of the reference since the sampling area is primarily in the high correlation region. Sampling Scheme 1, which samples a smaller portion of the highly correlated region, yields μFS value that is the second closest to the reference value. Since sampling Schemes 2 and 4 cover the minimal correlation between the parameters and FS, they yield a μ

_{FS}that differs significantly from the reference value and is close to the μ

_{FS}of the unconditional random simulation, indicative of their ineffectiveness for defining the actual factor of safety.

## 6. Conclusions

_{FSc}and ρ

_{FS}

_{tanφ}are different: the former distributes at the toe and top of the slope, and the latter in the interior of the slope.

_{x}is, the longer the extension of the area in the horizontal direction is. In addition, when c and tanφ are positively correlated or uncorrelated, the larger λ

_{x}is, and the less stable the slope is. When c and tanφ are perfectly negatively correlated, the effect of λ

_{x}on the stability of the slope decreases.

_{y}, the correlation in the high correlation region slightly decreases, and the region’s distribution becomes more vertical than others. Moreover, when c and tanφ are positively correlated, the larger λ

_{y}is, and the more stable the slope is. When c and tanφ are negatively correlated or uncorrelated, the larger λ

_{y}is, and the less stable the slope is.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Numerically generated realizations of c in different correlation scales: (

**a**,

**b**) λ

_{x}= λ

_{y}= 1 m, with different random seed, (

**c**) λ

_{x}= 5 m, λ

_{y}= 1 m, (

**d**) λ

_{x}= 10 m, λ

_{y}= 1 m, (

**e**) λ

_{x}= 1 m, λ

_{y}= 5 m, and (

**f**) λ

_{x}= 1 m, λ

_{y}= 10 m.

**Figure 3.**(

**a**) Cross-correlation map of FS and c, (

**b**) Cross-correlation map of FS and tanφ for statistical isotropy.

**Figure 5.**Cross-correlation maps of FS and c (

**a**–

**c**) and tanφ (

**d**–

**f**) under three different horizontal statistical anisotropic correlation scales.

**Figure 6.**(

**a**–

**d**) are pf, β, μ

_{FS}and σ

_{FS}of FS obtained by MCS for different λ

_{x}and ρ

_{c}

_{tanφ}values.

**Figure 7.**Cross-correlation maps of FS and c (

**a**–

**c**) and tanφ (

**d**–

**f**) under three different vertical statistical anisotropic correlation scales.

**Figure 8.**(

**a**–

**d**) are pf, β, μ

_{FS}, and σ

_{FS}of FS obtained by MCS for different λ

_{y}and ρ

_{c}

_{tanφ}values.

**Figure 9.**Pf and μ

_{FS}, when λ

_{y}= 1 m and λ

_{x}= 10 m and 20 m, respectively, as functions of the number of realizations.

Parameters | Values |
---|---|

Mean of cohesion, μ_{c} | 15 kN/m^{2} |

Coefficient of variation of cohesion, COV_{c} | 0.5 |

Mean of friction angle, μ_{φ} | 10° |

Coefficient of variation of friction angle, COV_{φ} | 0.5 |

Dilation angle, ψ | 0° |

Young’s modulus, E | 1 × 10^{5} kPa |

Poisson’s ratio, υ | 0.3 |

Unit weight, γ | 20 kN/m^{3} |

Scheme 1 | Scheme 2 | Scheme 3 | Scheme 4 | |
---|---|---|---|---|

μ_{FS} of Case 1 | 1.033 | 1.041 | 1.031 | 1.040 |

σ_{FS} of Case 1 | 0.066 | 0.072 | 0.058 | 0.072 |

μ_{FS} of Case 2 | 0.935 | 0.939 | 0.924 | 0.935 |

σ_{FS} of Case 2 | 0.094 | 0.105 | 0.082 | 0.112 |

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## Share and Cite

**MDPI and ACS Style**

Li, Y.; Zhang, F.; Yeh, T.-C.J.; Hou, X.; Dong, M.
Cross-Correlation Analysis of the Stability of Heterogeneous Slopes. *Water* **2023**, *15*, 1050.
https://doi.org/10.3390/w15061050

**AMA Style**

Li Y, Zhang F, Yeh T-CJ, Hou X, Dong M.
Cross-Correlation Analysis of the Stability of Heterogeneous Slopes. *Water*. 2023; 15(6):1050.
https://doi.org/10.3390/w15061050

**Chicago/Turabian Style**

Li, Yukun, Faming Zhang, Tian-Chyi Jim Yeh, Xiaolan Hou, and Menglong Dong.
2023. "Cross-Correlation Analysis of the Stability of Heterogeneous Slopes" *Water* 15, no. 6: 1050.
https://doi.org/10.3390/w15061050