Rock slope stability analysis and charts based on hybrid online sequential extreme learning machine model

Abstract

The stability of rock slopes is a difficult problem in the field of geotechnical and geological engineering. Less than 20% of all landslides are predictable each year, so a simple, fast, reliable and low-cost method to predict the stability of slopes is urgently needed. This study investigates a new regularized online sequential extreme learning machine, incorporated with the variable forgetting factor (FOS-ELM), based on intelligence computation to predict the factor of safety of a rock slope (F). The Bayesian information criterion (BIC) is applied to establish seven input combinations based on the parameters of the Hoek-Brown criterion and geometrical and mechanical parameters of the slope, such as the geological strength index (GSI), disturbance factor (D), rock material constant (m_i), uniaxial compressive strength (σ_ci), unit weight of the rock mass (γ), slope height (H) and slope angle (β). Seven models are established and evaluated to determine the optimal input combination. Various statistical indicators are calculated for the prediction accuracy examination. Compared to the classical extreme learning machine (ELM) model predictions of F, the results of the applied FOS-ELM model demonstrate a better prediction accuracy and are more effective when accounting for an increase in data. The FOS-ELM model with all seven input parameters is used to establish stability charts with the influence coefficient of slope angle change (η_β), disturbance change (η_D) and slope height change (η_H). Using stability charts with a combination of η_β, η_D and η_H can be used to quickly and preliminarily analyze rock stability as a guide for engineering practitioners in rock slope design.

Introduction

A landslide is a type of mass and energy wasting process that acts on natural and artificial slopes. Landslides cause more financial loss and casualties than has been generally recognized, generating an annual loss of property higher than those caused by other natural hazards, such as earthquakes, floods and windstorms (Guzzetti et al. 1999). This loss is more obvious in developing countries; approximately 0.5% of the worldwide gross national product (GNP) has been lost due to landslides (Chung and Fabbri 2003), while the majority of the recorded landslides worldwide are in developing countries; for example, landslides are a serious geologic hazard that cannot be ignored in China (Fig.1). Over vast areas and in complex geographical environments, for slope stability evaluation, geological engineers need to carry out a series of large-scale surveys and evaluations, and rock slope stability evaluation and landslide prevention will consume considerable time and money. The stability of rock slopes is a classical and difficult problem in the field of geotechnical and geological engineering (Li et al. 2011). According to the official website of the China Geological Survey, less than 20% of all landslides are predictable each year, so a simple, fast, reliable and low-cost method to judge the stability of slopes is urgently needed. In particular, very large, fast-moving rock landslides are probably the most destructive and hazardous mass movements (lower part of Fig.1). The stability assessment of rock slopes has become a research topic for many researchers (Li et al. 2018; Saade et al. 2016; Johari and Mehrabani Lari 2017). However, there are still many problems in the stability analysis of rock slopes for engineering practitioners. For example, the existing software that is suitable for stability analysis of rock slope used by engineering practitioners is not very accurate (Li et al. 2008), convenient and fast, and additional resources capable of providing information useful for decision-making are required (Li et al. 2016).

Fig. 1

Distribution map of China’s geological disaster prone areas

With the rapid development of computer and intelligent computing technology, it has been widely used in various fields due to its advantages of high efficiency, fast independent learning and accurate prediction (Hou et al. 2018); such approaches use gray models, artificial neural networks (ANNs) and fuzzy methods, etc. The application of intelligence models presented in the form of artificial intelligence (AI) models has been introduced to analyze the stability of slopes. There are a large number of landslides that need to be prevented and treated every year worldwide. These engineering and research examples provide valuable experience and research data. Therefore, many researchers use these valuable data to evaluate and predict slope stability (Erzin and Cetin 2013; Liu et al. 2014; Wang et al. 2010; Fallah-Zazuli et al. 2019; Shafizadeh-Moghadam et al. 2019) and invert unknown slope parameters (Li et al. 2016) by artificial intelligence calculations to improve the accuracy and efficiency of slope stability evaluations. Slope stability analysis provides a comprehensive evaluation of slope behaviors and often considers the factor of safety (F); thus the calculation of F is an important content of slope stability analysis. An AI model relies on gaining a range of slope parameters about the stability behavior of slopes and inputs those slope parameters to output the most influential factors that play a significant role in the slope stability or F. In the work of Deng et al. (Deng and Lee 2001) and Li et al. (Li et al. 2016), the advantage of using a trained ANN in slope stability and displacement analysis is evident.

In the analysis of slope stability, Mohr-Coulomb (M-C) linear failure criterion has been widely used for a long time, including in rock engineering modelling and design (Zhao 2000), because it uses clear concept of parameters to reflect the strength of rock and soil, such as cohesion c and internal friction angle φ. However, the M-C criterion could not reflect the nonlinear failure characteristics of rocks (Eberhardt 2012) because of the discontinuities and in-homogeneity of rock masses including joints, fractures, faults and bedding planes, and could not explain the influence of low stress zone, tensile stress zone and minimum principal stress on the strength. The Hoek-Brown (H-B) failure criterion (Hoek and Brown 2018), published in1980, can reflect the inherent non-linear failure characteristics of rock and rock mass, and make up for the deficiency of M-C criterion, and can be applied to description of anisotropic rock mass. In the almost 40 years since its introduction, the H-B failure criterion has become one of the most widely used failure criteria in the field of engineering applications, and researchers in the past decades have devoted great efforts to compensate for the shortcomings of the original criterion (Hoek and Brown 2018; Sonmez and Ulusay 1999; Hoek et al. 2002; Wenkai et al. 2018). The generalized H-B failure criterion and the associated geological strength index (GSI) have gained a great deal of attention for estimating the strength of heavily jointed rock masses and analyzing slope stability (Hoek and Brown 2018). To more effectively use the H-B criterion in various geotechnical software, following E. Hoek et al. (Hoek et al. 2002; Clausen and Damkilde 2008; Priest 2005), the nonlinear H-B criterion is mathematically modified or optimally linearized. The equivalent cohesion and friction angle of the rock mass are solved directly, and the stability of the rock slope is analyzed by using the M-C criterion, which can meet some characteristics of the linear strength criterion, i.e., the linearization of the nonlinear strength criterion.

As mentioned previously, the rock masses constituting the slope are heterogeneous and discontinuous. The generalized H-B failure criterion and limit equilibrium method can be effectively applied to the stability analysis of rock slopes (Li et al. 2008). At the same time, the existing intelligent calculation methods used to solve the problems of slope stability evaluation, prediction and parameter inversion have their unique advantages, as Li et al. noted (Li et al. 2016), and the application of the extreme learning neural network method to the stability analysis of rock slopes has obtained relatively ideal results. In this study, a regularized online sequential extreme learning machine with variable forgetting factor (FOS-ELM) as a new novel case of the ELM model is developed to evaluate and predict slope stability. While several AI models need internal parameters to be adjusted for more accurate results, the FOS-ELM algorithm needs no a priori knowledge of the control variables, resulting in a small influence on the optimization problem. The main objective of this study was to determine the feasibility and accuracy of the newly proposed hybrid FOS-ELM for evaluating and predicting the stability of rock slopes. In particular, the following contributions have been made in this paper:

• (1) BIC is used as a variable selection algorithm to optimize the parameters of the H-B criterion and slope geometry reflect the key factors of slope stability evaluation. Although, a few papers have used this method to select variables, this case declares its successful candidate for slope stability prediction.

• (2) To propose a novel hybrid AI model to evaluate and predict the stability of rock slope. If more slope engineering data can be collected, the proposed model can continuously increase parameters of the rock slope, and it is easy and fast to update the output layer parameters in the modeling dynamically without retraining. However, it is essential to modify almost all the parameters for the ANN and other algorithms mentioned above.

• (3) To add regularization and forgetting factors in the conventional ELM, the proposed model develops a new way to address the increase in the amount of data without retraining. The established method has better prediction accuracy and generalization ability than other parameter estimation proposals mentioned in the literature, because the regularization term can prevent abnormal point and overfitting, and the forgetting factor can reduce the negative influence of the old data.

• (4) To verify the accuracy of the established FOS-ELM model in evaluating and predicting the stability of a rock slope, a conventional ELM model (Li et al. 2016) is established.

• (5) In view of the minor problems existing in the current study of rock slopes stability charts (Li et al. 2008), slope angle influence coefficient (η_β), disturbance influence coefficient (η_D) and slope height influence coefficient (η_H) are proposed.

Previous studies

The generalized H-B failure criterion

The main factor controlling the stability of a rock slope is the presence of structural planes with different mechanical properties and scales in the rock mass. Hoek et al. proposed an empirical model of the strength of a jointed rock mass. The H-B criterion has nonlinear characteristics and can better reflect the strength characteristics of a jointed rock mass; thus, this criterion has been widely used. The development of the H-B failure criterion has mainly gone through four stages (Hoek and Brown 2018; Hoek et al. 2002). In 1980, Hoek and Brown (Hoek and Brown 2018; Hoek and Brown 1980) established the strength criterion for intact rock and applied it to estimate the strength parameters of jointed hard rock masses. In 1988, Hoek (Hoek and Brown 1988) applied the RMR classification index to establish empirical formulas for determining the rock mass material constants m and s and established the narrow H-B criterion. Hoek et al. (Hoek et al. 1992; Hoek 1994) revised the empirical formula in 1992 and revised it again in 1994. The formula for calculating the material constants mb, s and ɑ by using the GSI index was put forward. In 2002, Hoek (Hoek et al. 2002) introduced a disturbance factor D to revise the constants mb, s and ɑ. This version of the criterion is called the generalized H-B criterion and expressed as:

$$ {\sigma}_1={\sigma}_3+{\sigma}_{c\mathrm{i}}{\left({\mathrm{m}}_{\mathrm{b}}\frac{\sigma_3}{\sigma_{\mathrm{ci}}}+\mathrm{s}\right)}^{\alpha } $$

(1)

where σ₁ and σ₃ are the major and minor principal stresses, σ_ci is the uniaxial compressive strength. m_b, s and ɑ are the rock mass material constants, m_b and s are different empirical parameters of rock mass, s is used to quantify the degree of fragmentation of rock mass, given by

$$ {\mathrm{m}}_{\mathrm{b}}={\mathrm{m}}_{\mathrm{i}}\exp \left[\left( GSI-100\right)/\left(28-14D\right)\right] $$

(2)

$$ \mathrm{s}=\exp \left[\left( GSI-100\right)/\left(9-3D\right)\right] $$

(3)

$$ \alpha =1/2+1/6\left({e}^{- GSI/15}-{e}^{-20/3}\right) $$

(4)

These equations show that m_b and s are all dependent on GSI and D and that ɑ is determined by GSI. GSI was introduced to estimate rock mass strength in different geological conditions and ranges from approximately 10 (extremely fractured rock mass) to 100 (intact rock mass). The magnitude of D (0 ≤ D ≤ 1) is dependent on engineering experience; the D value of an undisturbed situ rock mass is 0, and for a disturbed rock mass, it is 1. m_i is a rock material constant related to rock type, which can be fitted by rock triaxial testing data or determined by considering the rock type, and the value of m_i ranges between 1 and 35.

At the same time, it should be emphasized that the H-B failure criterion has its limitations, because the H-B cning stresses within the range defined by and the transition from shear to ductile failure (Hoek and Brown 2018). The GSI system also has its applicability and limitations in the use of the geological strength index. A very detailed discussion can be found in the studies of Marinos (Marinos et al. 2005) and Sonmez (Sonmez and Ulusay 1999); it is only applicable to rock masses of Group I and Group III in Fig. 2, that is, complete rock masses and rock masses with at least 4 groups of joints (Li et al. 2008). In this study, the rock slopes collected are all heavily jointed rock masses, belonging to Group III.riterion is an empirical criterion. In the latest version, Hoek and Brown indicate that the H-B failure criterion is only applicable under confi.

Fig. 2

Applicability of H-B criterion for slope stability problems

Limit equilibrium method based on the nonlinear H-B criterion

The limit equilibrium method (LEM) is the earliest and most developed quantitative analysis method in slope stability analysis. Due to its advantages of a simple calculation and consideration of various shapes and sections, the LEM is one of the most widely used methods in slope stability analysis (Li et al. 2016). The most common limit equilibrium techniques are slice methods (Liu et al. 2015), the wedge stability analysis method, the slice method of the American Federation of Military Engineers, the residual thrust method, and the Sarma method. In slope stability analysis, many geotechnical software programs take the M-C strength theory as the yield criterion. The problem can be solved by analyzing the static equilibrium of the rock and soil when they are destroyed, and the LEM is generally based on the M-C failure criterion. Therefore, to analyze the stability of the rock slope, the values of the cohesion c and friction angle φ of the equivalent M-C parameters must be evaluated based on the strength parameters of the rock mass. The equivalent M-C parameters can be expressed as:

$$ \mathrm{c}=\frac{\sigma_{\mathrm{ci}}\left[\left(1+2\alpha \right)s+\left(1-\alpha \right){m}_b{\sigma}_{3n}\right]{\left(s+{m}_b{\sigma}_{3n}\right)}^{\alpha -1}}{\left(1+\alpha \right)\left(2+\alpha \right)\sqrt{1+\left[6{m}_b\alpha {\left(s+m{\sigma}_{3n}\right)}^{\alpha -1}\right]/\left(1+\alpha \right)\left(2+\alpha \right)}} $$

(5)

$$ \phi =\arcsin \left[\frac{6\alpha {m}_b{\left(s+{m}_b{\sigma}_{3n}\right)}^{\alpha -1}}{2\left(1+\alpha \right)\left(2+\alpha \right)+6\alpha {m}_b{\left(s+{m}_b{\sigma}_{3n}\right)}^{\alpha -1}}\right] $$

(6)

where σ_3n = σ_3max/σ_ci, and the value of σ3max must be determined in different problems. For slope stability analysis, Li et al. (Li et al. 2008) proposed new estimates for the determination of σ3max for gentle and steep slopes, as follows:

$$ {\upsigma}_{3\max }=0.41{\upsigma}_{\mathrm{cm}}{\left[\frac{\upsigma_{\mathrm{cm}}}{\upgamma \mathrm{H}}\right]}^{-1.23}\kern0.5em \left(\mathrm{gentle}\ \mathrm{slope},\upbeta <45{}^{\circ}\right) $$

(7)

$$ {\sigma}_{3\max }=0.2{\sigma}_{\mathrm{cm}}{\left[\frac{\sigma_{\mathrm{cm}}}{\gamma H}\right]}^{-1.07}\kern0.5em \left(\mathrm{steep}\ \mathrm{slope},\upbeta \ge 45{}^{\circ}\right) $$

(8)

$$ {\sigma}_{\mathrm{cm}}={\sigma}_{\mathrm{ci}}\frac{\left[{m}_b+4s-a\left({m}_b-8s\right)\right]{\left({m}_b/4+s\right)}^{\alpha -1}}{2\left(1+\alpha \right)\left(2+\alpha \right)} $$

(9)

where H is the slope height, γ is the material unit weight of the rock, and σ_cm is the compressive strength of the rock mass.

As previously mentioned, the M-C failure criterion is a linear equation, and the H-B failure criterion and equivalent M-C failure envelope do not exactly match. The equivalent values of c and φ could affect the accuracy of the results. When compared to the limit analysis results using the H-B criterion, the limit equilibrium results are at most 64% higher than the limit analysis results, and more details can be found in Li et al. (Li et al. 2008). Meanwhile, the limit equilibrium software Slide for slope stability analysis and Bishop’s simplified method based on the H-B criterion have been used in comparison with the upper and lower limit analysis. The results indicate that the solution using Slide software based on the H-B criterion is closer to the real solution of slope stability (Li et al. 2016). In this paper, the GEOSLOPE software based on the H-B criterion is used to calculate the stability factor of the slope which is collected worldwide.

The computational intelligence approaches used in slope stability analysis

Slope stability evaluation and prediction is a hot topic and important issue and thus continues to be a focus of scholars and experts. In recent years, the development of intelligent computing technology has provided new ideas for slope stability prediction and analysis (Zhang et al. 2014; Yun et al. 2018); for example, the ANNs and the fuzzy methods have been proposed for slope stability evaluation and prediction. These methods evaluate slope stability based on slope material properties, slope geometry and water conditions, which are gathered from the worldwide databases or simulation results. Many slope stability analyses in recent years have been based on various parameters used by different types of ANNs. An ANN-based response surface was used by Cho. (Cho 2009) to approximate the limit state function, thereby reducing the number of stability analysis calculations. In a study by Deng and Lee (Deng and Lee 2001), an ANN was trained to obtain a more reliable prediction of slope displacement. The ELM is introduced to evaluate and predict the stability of slopes by means of prediction of the factor of safety (F), and the ELM has been proven to be a promising technology with which to evaluate soil slope stability (Liu et al. 2014). In a recent study, ELM was first applied to stability analysis and parameter inversion of rock slopes based on the finite element upper and lower bound limit analysis method by Li et al. (Li et al. 2016).

However, almost all of the methods mentioned above have certain limitations despite their achievements. For example, ANN has difficulties in determining network structure, a slow training speed and local minimum solution problems, although great improvements have been made through iterative efforts (Sinha et al. 2010). The study of Li et al. (Li et al. 2016) pointed out the problems of overfitting and convergence to a local minimum. With more input parameters, conventional ANN (or ELM) approaches need to be retrained to inprove their prediction results, which certainly will take time.

Therefore, a fast, effective and accurate method of evaluation and prediction of slope stability needs to be further studied. The newly developed extreme learning machine (ELM) model (Huang et al. 2017) overcomes the shortcomings of traditional neural networks. To obtain a fast and effective method for evaluating and predicting slope stability, a regularized online sequential extreme learning machine with a variable forgetting factor (FOS-ELM) as a new novel case of the ELM model based on the H-B criterion is developed and used in this study.

Problem definition

Case study description

The current study was established by collecting parameters of 1235 homogeneous rock slopes around the world, most of which were gathered from China’s slope engineering database. To date, there is no research using the FOS-ELM model based on the latest version of the H-B criterion to evaluate and predict the stability of slopes. The parameters of the H-B criterion and geometric parameters of the slope,including the geological strength index (GSI), disturbance factor (D), rock material constant (m_i), unconfined compressive strength (σ_ci), unit weight of rock mass (γ), slope height (H) and slope angle (β), were chosen as the training inputs. The factors of slope safety (F) were the desired training output of the FOS-ELM model. Notably, the factors of slope safety were calculated in GEOSLOPE softwareby using LEM based on the H-B yield criterion.

Stability charts and the influencing factors

In the analysis of slope stability, slope stability charts were widely accepted and used as a design tool by geotechnical engineers as a part of an effective method for predicting the F of slopes. Since Taylor (Taylor 1937) first proposed the stability chart method to calculate the safety factor of a simple homogeneous clay slope in 1937, such as Gao et al. (Gao et al. 2013) and Steward et al. (Steward et al. 2010) have proposed a series of soil slope stability charts. For the stability charts of rock slopes, Zanbak (Zanbak 1983) proposed stability diagrams of rock slopes that are prone to collapse disasters, Siad (Siad 2003) put forward stability diagrams of rock slopes under earthquake action based on the upper bound solution. However, these charts are proposed based on the M-C criterion and need to provide shear strength parameters c and φ. In recent research, stability charts for rock slopes based on the H-B failure criterion can be found (Li et al. 2008). Qian Z.G. et al. (Qian et al. 2017) provided stability charts for disturbed rock slopes based on the finite element limit analysis method, and Xu. (Xu and Yang 2018) proposed seismic stability charts of a 3D rock slope on the basis of the strength reduction technique. In their studies, the stability number N_r is given, and the relation between N_r and F is used to solve the factor of safety from slope stability charts (Li et al. 2008; Li et al. 2016). The stability number Nr is defined as:

$$ {N}_{\mathrm{r}}=\frac{SR}{F} $$

(10)

where SR = σ_ci/γH, F is safety factor of slope.

The abovementioned slope stability charts were established based on the relationship between the H-B parameters and N_r. According to Eq.(10), if the corresponding values of GSI and m_i are equal and β is the same, N_r should be equal, then, it can be assumed that N_r1 = N_r2, and H₁F₁ = H₂F₂ can be deduced, where N_r1 and N_r2 represent the slope stability numbers of slopes of different heights (H₁ and H_2, respectively). Similarly, F₁ and F₂ represent the slope stability safety factor. Under the different conditions of H, the F of the slope is hyperbolic with the slope height H. However, a large number of studies show that the relationship between the slope stability safety factor F and slope height H is not simply hyperbolic (Gao et al. 2015; Chen et al. 2016; Sun et al. 1997). Therefore, the slope stability charts based on dimensional Nr will inevitably produce some errors. In addition, in the process of using the aforementioned stability charts, it is necessary to calculate F via the linearization technique, so some errors will inevitably occur, such as those related to the use of β and D. In engineering practice, the number of stability charts is high, which is not very convenient for geological engineers and requires considerable time. In the current study, in view of shortcomings described above, the influence coefficients of slope angle change (η_β), disturbance change (η_D) and slope height change (η_H) are proposed and can be expressed as:

$$ {\eta}_{\beta }=\frac{F_{\beta }}{F_{45{}^{\circ}}} $$

(11)

$$ {\eta}_D=\frac{F_D}{F_0} $$

(12)

$$ {\eta}_H=\frac{F_H}{F_{100}} $$

(13)

where F_β and F_45°are the safety factors of slopes with the slope angles of β and 45°, respectively; similarly, F_D and F₀ are the safety factors of slopes with the disturbance factors of D and 0, and F_H and F₁₀₀ are the safety factors of slopes with the slope heights of H and 100.

Furthermore, by taking β = 45°, D = 0 and H = 100 as constants, using GSI, m_i and SR as variables, and ensuring the prediction accuracy of the FOS-ELM model, the distribution map of the safety factor of rock slope stability is predicted. Once the relationships between η_β and β, η_D and D, and η_H and H are fitted by observed or predicted results, we can use these fitting relations and the corresponding stability charts to quickly determine the safety factor of any homogeneous rock slope and preliminarily evaluate the stability of the slope.

Methodology

Limit equilibrium method based on nonlinear H-B criterion

As mentioned earlier, the LEM based on the nonlinear H-B criterion is highly accurate. In the current research, GEOSLOPE software, which is widely used in geotechnical engineering, is used to calculate the F of slopes based on Bishop’s simplified method and the nonlinear H-B failure criterion. The generalized H-B failure criterion has been used with Bishop’s simplified method in GEOSLOPE software. When the H-B failure criterion is selected, the software will generate a series of instantaneous equivalent M-C parameters based on the normal stress at the bottom of each individual slice. Therefore, the equivalent M-C parameters (c and φ) will change with a given slip surface. A more accurate representation of the curved nature of the H-B yield criterion for a rock slope is obtained by calculating equivalent M-C parameters in this way. A.J.Li et al. (Li et al. 2008) proved the accuracy of this method by calculating the F of a slope in Slide software.

Regularized online sequential extreme learning machine with variable forgetting factor.

The ELM model was established based on the single-hidden-layer with random weight configuration, which was founded by Prof Huang et al.. Then the ELM model was extended a generalized single-layer feed-forward networks (SLFNs), in which the hidden layer neurons need not be similar (Huang et al. 2017). The single-hidden-layer feed-forward neural network of ELM is shown in Fig. 3. The core of ELM algorithm is to solve the least squares solution of loss function, and l2 regularization constraint is introduced into loss function of the ELM to improve the generalization ability, more details can be founded in a paper by Hou et al.(Hou et al. 2018).

Fig. 3

The network of classical extreme learning machine(ELM)

Unfortunately, to increase a large amount of new slope data, traditional artificial intelligence algorithms, such as ANNs, support vector machines, limit learning models, need to use both new and old data and then retrain them (Crisosto et al. 2018). Therefore, the traditional artificial intelligence algorithm has difficulty solving big-data, non-stationary, number-varying, and time-varying prediction problems. In this study, an online sequence extreme learning machine model (OS-ELM) is established to effectively solve the nonstationary and variable data flow problems.The learning of OS-ELM includes an initial ELM batch learning process and a continuous block-by-block learning process(Hou et al. 2018).

However, when the autocorrelation matrix of the hidden layer output matrix is singular or ill-conditioned in the learning process, the generalization ability of OS-ELM deteriorates sharply. To overcome the shortcomings of OS-ELM, Huynh (Bou-Rabee et al. 2017) combined Tikhonov regularization with OS-ELM and proposed a regularized OS-ELM to improve the stability and generalization of the algorithm. To solve the problem of time-varying and quantity-varying in the process of sample addition online learning, the forgetting factor (Paleologu et al. 2008) is introduced into OS-ELM in this study.

The R-ELM-FF algorithm is theoretically equivalent to minimizing the following least squares error functions based on FF and l2 regularization:

$$ {J}_{FR}\left({\beta}_k\right)=\sum \limits_{i=1}^k{\lambda}^{k-i}{\left|{t}_i-{h}_i{\beta}_k\right|}^2+{\delta \lambda}^k{\left\Vert {\beta}_k\right\Vert}^2 $$

(14)

in the formula, λ is the FF parameter of the weights of new and old samples, and δ is the regularization parameter to increase the stability and generalization ability of the algorithm.

Based on the recursive formula(Kumar et al. 2002), the recursive least squares method is used to solve the eq. (22). β_k can be deduced as:

$$ {\displaystyle \begin{array}{l}{P}_k=\frac{P_{k-1}}{\lambda }-\frac{P_{k-1}{h}_k^T{h}_k{P}_{k-1}}{\lambda \left(\lambda +{h}_k{P}_{k-1}{h}_k^T\right)}\\ {}{\beta}_k={\beta}_{k-1}+{P}_k{h}_k^T\left({t}_k-{h}_k{\beta}_{k-1}\right)\end{array}} $$

(15)

However, with the increase of k, δλ^k‖β_k‖² decreases exponentially to zero, which will lead to the gradual failure of the regularization function.Therefore, a new constant coefficient regularization term δ‖β_k‖² is introduced as substitute for exponential regularization term δλ^k‖β_k‖² in FOS-ELM cost function. Which can be expressed as:

$$ {J}_{FR}\left({\beta}_k\right)=\sum \limits_{i=1}^k{\lambda}^{k-i}{\left|{t}_i-{h}_i{\beta}_k\right|}^2+\delta {\left\Vert {\beta}_k\right\Vert}^2 $$

(16)

The FOS-ELM operates as follows

Let the data sample arrive in the form of data stream, the excitation function is G(w, b, x), the number of hidden neurons is n, the regularization parameter is δ, and the forgetting factor is λ.

The first step: Initialization. Set up the initial training subsetΩ_k ‐ 1 = {(x_j, t_j)|x_j ∈ Rⁿ, |t_j ∈ R^m, j = 1, …, k − 1} and perform the following actions:

• 1) Random generation of hidden layer neuron parameters

$$ \left({\mathbf{w}}_i,{b}_i\right),i=1,2,\cdots, n $$

• 2) Using formula (17) to calculate hidden layer output matrix

$$ {H}_{k-1}={\left[{\mathrm{h}}_1^T\ {\mathrm{h}}_2^T\cdots {\mathrm{h}}_{k-1}^T\right]}^T $$

3) Calculating the initial output weight matrix

$$ {\beta}_{k-1}={P}_{k-1}{H}_{k-1}^T{T}_{k-1} $$

$$ {P}_{k-1}={\left({H}_{k-1}^T{H}_{k-1}+\lambda I\right)}^{-1},{T}_{k-1}={\left[{t}_1\ {t}_2\cdots {t}_{k-1}\right]}^T $$

(and)

The second step: Online learning and prediction. The following operations are performed on the new sample (x_k, t_k):

• 1) Calculate the new sample inputx_k:h_k = [G(a₁, b₁, x_k)⋯G(a_n, b_n, x_k)]for the hidden layer output vector

2) Compute the network output, i.e. the predicted value of t_k:${\hat{t}}_k={h}_k{\beta}_{k-1}$.

3) Adjust the output weight according to the truth value t_k

$$ {\displaystyle \begin{array}{c}{P}_k^{\ast }=\frac{1}{\lambda }{P}_{k-1}-\frac{\delta \left(1-\lambda \right)}{\lambda^2}{P}_{k-1}\left(I-\frac{\delta \left(1-\lambda \right)}{\lambda }{P}_{k-1}\right){P}_{k-1}\\ {}{P}_k={P}_k^{\ast }-\frac{P_k^{\ast }{h}_k^T{h}_k{P}_k^{\ast }}{1+{h}_k{P}_k^{\ast }{h}_k^T}\\ {}{\beta}_k={\beta}_{k-1}+{P}_k{h}_k^T\left({t}_k-{h}_k{\beta}_{k-1}\right)-\delta \left(1-\lambda \right){P}_k{\beta}_{k-1}\end{array}} $$

4) Back to the second step.

To assess the developed FOS-ELM and the classical ELM model for F in rock slope prediction, multiple statistical indicators were used. These performance indicators can be described as follows.

• I. The Correlation coefficient (r) is represented as:

$$ \mathrm{r}=\left(\frac{\sum \limits_{\mathrm{k}=1}^n\left({F}_{re}-\overline {F_{re}}\right)\left({F}_{pr}-\overline {F_{pr}}\right)}{\sqrt{\sum \limits_{\mathrm{k}=1}^n{\left({F}_{re}-\overline {F_{re}}\right)}^2}\sqrt{\sum \limits_{\mathrm{k}=1}^n{\left({F}_{pr}-\overline {F_{pr}}\right)}^2}}\right) $$

(17)

• II. The root mean square error (RMSE) is given as:

$$ RMSE=\sqrt{\frac{1}{\mathrm{n}}\sum \limits_{k=1}^n{\left({F}_{pr}-{F}_{re}\right)}^2} $$

(18)

• III. The mean absolute error (MAE) can be shown as:

$$ MAE=\frac{1}{\mathrm{n}}\sum \limits_{k=1}^n\left|\left({F}_{pr}-{F}_{re}\right)\right| $$

(19)

where F_pr and F_re represent the predicted F and the real F of the rock slope, respectively, $\overline {F_{\mathrm{pr}}}$ and $\overline {F_{\mathrm{re}}}$are average values, and n indicates the number of testing data points.

Procedures of the methodology

Here we choose the optimal input combination (GSI, m_i, σ_ci, D, γ, β, H). Eventually, the proposed FOS-ELM is applied to establish the F prediction model, which is illustrated in Fig. 4.

Fig. 4

Main frame of the proposed FOS-ELM

Variable selection

For slope stability prediction, variable selection is critical. Once the variables that are not related to the phase response are selected, the understanding of the relationship between the variables will be disturbed, and geological work in the future may also be needed on the slope. Otherwise, studies have shown that the selection of variables with little correlation will lead to a significant decrease in forecasting accuracy. Thus, if some slope parameters are difficult to obtain in in geological investigations, it is very important to use other parameters to preliminarily predict slope stability. Therefore, it is of practical significance to optimize parameters and output F through the FOS-ELM model to determine the correlation and error.

The basic idea of BIC (Schwarz 1978) assumes that there is a uniform distribution on the candidate models; then, the sample distribution is used to find the posterior distribution on the models, and the model with the maximum posterior probability is ultimately selected. This is equivalent to thinking that the subset of variables that minimize the BIC value is optimal. Therefore, BIC can be used to optimize the combination of slope parameters and determine the errors under different combinations to make a reliable prediction of slope stability. The BIC value is expressed as:

$$ BIC=-2\ln L+k\log n $$

(28)

where n is the sample size, k is the number of estimated parameters in the model, and L is the maximized value of the likelihood function for the estimated model.

Results and discussion of the FOS-ELM model

In this study, as mentioned in Section 3.1, GSI, mi, σ_ci, D, γ, H and β were chosen as the inputs, and F was chosen as the output of the proposed FOS-ELM model. Thus, the model had seven inputs and one output. For the training, 1085 train data points were randomly selected, and the remaining data were used as validation data. The results are presented to assess the proposed FOS-ELM model for rock slope factor of safety prediction. The proposed FOS-ELM model is appraised in comparison with the ELM model, using statistical metrics and error distributions between the predicted F (F_pr) and real F (F_re) over the testing phase. F_re was obtained in GEOSLOPE software by using the LEM based on the H-B criterion. BIC was used for variable selection and generating the best combination of input parameters.

If the values of F_pr and F_re can be expressed by the linear fitting equation y = kx, when the correlation coefficient r² and the slope k are closer to 1, the prediction performance of the model is better. Therefore, we conducted linear fitting with y = kx in Origin software. Figure 5 indicates the goodness-of-fit and correlation coefficient r² of F_re and F_pr through scatter plots. The proposed FOS-ELM method outperforms the ELM in terms of r² and k (ELM:r² = 0.9673, k = 1.0132; FOS-ELM: r² = 0.9913, k = 1.0059) for Model 7 (M₇) using all inputs. Similarly, the proposed FOS-ELM model is more accurate than the ELM approach for Model 6 (M₆) with the input combination (GSI, σ_ci, m_i, D, H and β) according to r² (ELM: r² = 0.7068; FOS-ELM: r² = 0.9045) and k (ELM: k = 0.9811; FOS-ELM: k = 0.9909), followed by Models 5 (M₅), 4 (M₄), 3 (M₃), 2 (M₂), and 1 (M₁). On the basis of attaining r² and k values close to 1, the proposed FOS-ELM model shows better accuracy by incorporating all inputs(see M₇)to increase r² and k. And M₅ is second to the M₇ model of the FOS-ELM models, which indicates that the number of input parameters will affect the predicted results.

Fig. 5

Scatter-plot variance between F_re and F_pr over the testing modeling phase, FOS-ELM vs. ELM models and for all investigated input combination panels (a) Model 1 (M₁), (b) Model 2 (M₂), (c) Model 3 (M₃), (d) Model 4 (M₄), (e) Model 5 (M₅), (f) Model 6 (M₆) and (g) Model 7(M₇)

The ELM model was used to train and predict the stability number N_r which required a few hours to obtain all the results; this work included the training and validation data from the study of A.J.Li (Li et al. 2016). However, intelligent calculations based on big data are one of the directions of prediction pplications in the future. With the increase in the amount of engineering slope data available, we need to constantly increase the slope databases to improve the model prediction ability. The ELM model needs to be retrained to complete the prediction process, so we can take advantage of the FOS-ELM model which does not need to be retrained. The training performance using the FOS-ELM model with data increased to 500 datapoints can be seen in Fig.6. Therefore, the ELM will require a longer calculation time than the FOS-ELM model, and it has been proven that the FOS-ELM model mentioned is more accurate, Fig.7 shows the same finding, implying that the proposed FOS-ELM model is convenient and efficient because a rock assessment can be performed more accurately and quickly with more data. Most notably, The hyper parameters (the hidden neurons number L, forgetting factor λ, regularization parameter δ are 115, 0.975 and 10⁻¹¹, respectively) of FOS-ELM were tuned by a gridsearch strategy. Additionally, the sigmoid function was used as the activation function for the hidden layer.

Fig. 6

Training performance use the FOS-ELM model with 500 datapoints

Fig. 7

Performance of 500 datapoints trained by (a) ELM and (b) FOS-ELM with a150-validation datapoint set

The BIC values of all input variable combination were calculated in Matlab, the variable combination with smallest BIC value is the optimal combination of variables. For different numbers of input variables, seven different models (M₁, M₂, M₃, M₄, M₅, M₆, M₇) are established by optimal input combination, including the model with all input variables. In Table 1, based on the correlation coefficient (r), root mean square error (RMSE) and mean absolute error (MAE), the accuracy of the FOS-ELM models were assessed in comparison to the ELM model to each input combination. The proposed FOS-ELM model with all input parameters (M₇) attained the highest correlation coefficient (r) and the smallest value of RSME and MAE (r = 0.994, RSME = 0.069, MAE = 0.060) compared to the ELM model (r = 0.987, RSME = 0.100, MAE = 0.090). Moreover, for M₆ these metrics were the FOS-ELM model (r = 0.957, RSME = 0.181, MAE = 0.093), and the ELM model (r = 0.884, RSME = 0.291, MAE = 0.158). Similarly, the FOS-ELM model is better with (M₁, M₃, M₄, M₅) in terms of attaining largest magnitudes of correlation coefficient (r) and smallest value of RMSE and MAE, except M₂ model with data increased to 1085. It indicates that the proposed FOS-ELM model can be a better data-intelligent tool for rock slope stability prediction than the ELM.

Table 1 The seven input combination models were constructed based on the variability influence of each slope parameter (several statistical metrics were computed as determination factors

Supplementary stability charts and application

Supplementary stability charts

The F predicted by FOS-ELM for the most accurate M₇ model (FOS-ELM-M₇) with all input combinations demonstrates that this model can provide a direct and accurate estimate of the F of a rock slope. All the collected slope data were trained by the FOS-ELM-M₇ model, and the model was used to predict F under the conditions of 10 ≤ GSI ≤ 100, 5 ≤ m_i ≤ 35, SR ≥ 0.2, β = 45, H = 100 and D = 0. Then the slope stability charts under the conditions of β = 45, H = 100 and D = 0 were obtained, as shown in Fig.8. To predict the stability of rock slopes with different β, H and D values, it is necessary to obtain the slope angle influence coefficient (η_β), slope height influence coefficient (η_H) and disturbance influence coefficient (η_D). The η_β, η_H and η_D are defined in Eq.(11),(12) and (13) respectively.

Fig.8

F curve for stability analysis of rock slope with FOS-ELM-M₇ model (β = 45°, H = 100, D = 0)

N_r was proposed by A.J.Li et al. (Li et al. 2008; Li et al. 2016; Qian et al. 2017) via Eq.(10). Under certain β values, different D, GSI and mi have little effect on the value of Nr. However, under certain conditions of D, GSI and m_i, the N_r value is sensitive to β. Under the same conditions of D, GSI and m_i, σ_ci and γ are generally constants, and the relationship between F and the slope angle of slope with H is obvious, more details can be found in (Qian et al. 2017; Deng et al. 2016). Therefore, the influence of D, GSI, m_i, σci and γ can be neglected when establishing the relationship between F and β, and the relationship between η_β and β can be established by neglecting the influence of D, GSI, m_i, σ_ci and γ. Then η_β can be calculated by selecting a large number of slope data with β of 20 ~ 80° arbitrarily, and η_β under different β can be obtained by FOS-ELM-M₇. Then the relation curve between η_β and β can be obtained by fitting analysis in Origin software, as shown in Fig. 9.

Fig. 9

The fitting relationship between η_β and β with the M₇ model under different values of GSI

According to a study by Li et al. (2008), Nr is sensitive to changes in GSI. The influence of GSI on the relationship between H and F cannot be ignored. Therefore the relation curve between η_H and H (20 ≤ H ≤ 200) can be obtained by considering the value of GSI. Similarly, η_H and H can be calculated by predicting the results of the FOS-ELM-M₇ model. In Fig. 10, ${\eta}_H={A}_0+{A}_1{\mathrm{e}}^{\hbox{-} \mathrm{H}/{\mathrm{t}}_1}$ is used to fit the results, whereA₀,A₁and t₁ are fitting constants. The correlation r² is approximately 0.99, and the fitting curves of GSI = 30, 40, 50 and 60 are similar.

Fig. 10

The fitting relationship between η_H and H with the M₇ model under different values of GSI

In engineering practice, there are many disturbance factors affecting the slope rock mass, and it is difficult to accurately determine the value of D. E.Hoek and E.T.Brown (Hoek and Brown 2018) provided guidelines for estimating D value for different slope situations. The specific empirical values of D are 0, 0.7 and 1.0, and the determination of the D value in slope stability analysis is subjective. Therefore, to quantify the influence of the D value on the F of slope stability, the relation between η_D and D is established under different values of GSI (10 ≤ GSI ≤ 100), SR(1, 5, 15 and 30), m_i (5, 15, 25 and 35) by predicting the results of FOS-ELM-M₇ model in Fig. 11.

Fig. 11

The relationship between η_D and D with FOS-ELM-M₇ model

Application of charts

The stability charts illustrated in Fig. 8 provide a efficient way to determine F for rock slopes with slope angle β = 45°, slope height H = 100 and disturbance factor D = 0. Stability charts combining with Fig.9,10 and 11, the F of any rock slopes can be solved. A.J.Li et al. (Li et al. 2008) provided a slope to inversely calculate F by using charts of stability number N_r. It has the following parameters: GSI = 30, σ_ci = 20 MPa, m_i = 8, β = 60°, H = 20 m, D = 0, γ = 23kN/m³. In this study, the F of this case with above parameters can be obtained as follows. First, from values of σ_ci, γ, H = 100 m, we can calculate SR = σ_ci/γH ≈ 8.70. In Fig. 8, GSI = 30, SR = 8.70, β = 45°, D = 0, the F ≈ 6.15 and 3.33 are obtained with m_i = 5 and 10 respectively, the F₀ = 5.42 can be obtained by using linear interpolation when m_i = 8. η_β = 0.765 (β = 60°) and η_H = 2.09 (H = 20 m) can be obtained in Figs. 9 and 10 respectively. And η_D = 1, when D = 0 in Fig. 11. The F for this case can be calculated F=F₀η_βη_Hη_D = 8.665, the result is close to F = 8.7 determined by Li et al. (Li et al. 2008).

One of the rock slopes at Baskoyak barite open pit mine was investigated by H.Sonmez and R.Ulusay (Sonmez and Ulusay 1999). It is known that the slope angle is β = 34°, height of slope is H = 20 m, intact uniaxial compressive strength σ_ci = 5.2 MPa, geological strength index is GSI = 16, intact rock yield parameter is m_i = 7, disturbance factor is D = 0.7 and unit weight of the rock mass is γ = 22.2kN/m³. Hence, we can obtain SR = σ_ci/γH = 5200/(22.2 × 100) ≈ 2.34 by assuming H = 100. Then according to Fig. 8(a), for m_i = 5, we can easily obtain F = 0.17 and 0.53 with GSI = 10 and 20, and the F = 0.386 can be obtained by using linear interpolation when GSI = 16. Similarly, in Fig. 8(b), for m_i = 10, the F = 0.78 can be obtained. F₀ = 0.55 can be obtained by using linear interpolation with m_i = 7. That is, we calculate the value of F ≈ 0.55 with GSI = 16, m_i = 7, σ_ci = 5.2 MPa, β = 45°, H = 100 m, D = 0, and γ = 22.2 kN/m³. η_D = 0.51 with D = 0.7 can be obtained from Fig. 11(a) and (b). η_H = 1.57 with GSI = 16 is calculated from Fig.10 by using linear interpolation. η_β = 1.47 is quickly calculated from the fitting formula of Fig.9. Finally, we can calculate F=F₀η_βη_Hη_D = 0.55 × 1.47 × 1.57 × 0.51 = 0.65. The calculated results are consistent with the unstable slope results determined by H.Sonmez and R.Ulusay (Sonmez and Ulusay 1999).

Conclusions

In this study, a new ELM-based model, a regularized online sequential extreme learning machine with a variable forgetting factor, was proposed and first applied to predict slope stability and establish stability charts. Using the geological strength index (GSI), the disturbance factor (D), the rock material constant (m_i), the uniaxial compressive strength (σ_ci), the unit weight of the rock mass (γ), the slope height (H) and the slope angle (β) as seven different input variables of this study, we investigated the correlation between different variables and the factor of safety and the influence of different input combinations on the prediction accuracy of the model to select the optimal model input combination with different numbers of input variables by BIC. The statistical performance evaluation indicates that the prediction accuracy of the factor of safety can achieve excellent prediction accuracy with seven variable combinations. By comparing with ELM, the results clearly demonstrate the effectiveness and outstanding performance of the proposed FOS-ELM model. Compared with approaches used in the mentioned literature, the proposed model offers a new way to address an increase in data without retraining. The proposed model tries to update the output weight of the network and “fix” the network model with increasing data to achieve higher prediction accuracy. The stability charts were established by the model FOS-ELM-M₇. The relationships between the proposed influence coefficient of slope angle change(η_β), disturbance change (η_D), slope height change(η_H) and slope angle(β), disturbance factor (D) and slope height (H) were established on the basis of the results of FOS-ELM-M₇. Stability charts with a combination of η_β, η_D and η_H can be used to quickly and preliminarily analyze rock stability quickly as a guide for engineering practitioners in rock slope design.