Slope stability analysis based on big data and convolutional neural network

Abstract

The Limit Equilibrium Method (LEM) is commonly used in traditional slope stability analyses, but it is time-consuming and complicated. Due to its complexity and nonlinearity involved in the evaluation process, it cannot provide a quick stability estimation when facing a large number of slopes. In this case, the convolutional neural network (CNN) provides a better alternative. A CNN model can process data quickly and complete a large amount of data analysis in a specific situation, while it needs a large number of training samples. It is difficult to get enough slope data samples in practical engineering. This study proposes a slope database generation method based on the LEM. Samples were amplified from 40 typical slopes, and a sample database consisting of 20000 slope samples was established. The sample database for slopes covered a wide range of slope geometries and soil layers’ physical and mechanical properties. The CNN trained with this sample database was then applied to the stability prediction of 15 real slopes to test the accuracy of the CNN model. The results show that the slope stability prediction method based on the CNN does not need complex calculation but only needs to provide the slope coordinate information and physical and mechanical parameters of the soil layers, and it can quickly obtain the safety factor and stability state of the slopes. Moreover, the prediction accuracy of the CNN trained by the sample database for slope stability analysis reaches more than 99%, and the comparisons with the BP neural network show that the CNN has significant superiority in slope stability evaluation. Therefore, the CNN can predict the safety factor of real slopes. In particular, the combination of typical actual slopes and generated slope data provides enough training and testing samples for the CNN, which improves the prediction speed and practicability of the CNN-based evaluation method in engineering practice.

Graphical abstract

Keywords

slope stability / limit equilibrium method / convolutional neural network / database for slopes / big data

1 Introduction

Slopes are often used as the load carrier in engineering construction. If slope design or management is improper, it may impact engineering construction, induce slope instability, and even cause geological disasters [1,2]. Slope stability often becomes a key engineering geological problem in mountain engineering construction, which controls engineering projects’ feasibility, safety, and economy. Traditional slope stability analysis methods mainly included qualitative evaluation methods and limit equilibrium methods (LEM) according to the geological conditions [3,4]. The LEM is based on the principle of force balance and evaluates the slope stability through the relationship between the shear strength and sliding force on slope sliding body [5]. Commonly used LEMs include the Swedish arc method [6], simplified Bishop method [7], Spencer method [8], Sarma method [9], etc. Zhou and Cheng [10] established a strict LEM considering interstrip force to automatically search the sliding surface of a three-dimensional landslide, and determine its safety factor and its stability. Zheng et al. [11] combined the LEM with the genetic algorithm (LEM-GA) to predict rock slopes’ safety factor and failure surface under bending and toppling. In addition, some optimization calculation methods involving structural stability [12] and material fracture analysis [13] also provide new ideas for slope stability analysis. Zhang et al. [14] used a numerical method to optimize the discontinuous layout and analyzed the stability of the shotcrete supported crown, which achieved good results. Rabczuk and Belytschko [15] proposed a method of modeling discrete cracks in the meshless method, and he proposes that the crack can be arbitrarily oriented, but its growth is represented discretely by the activation of crack surfaces at individual particles.

For the evaluation of slope stability, it is necessary to establish an appropriate geotechnical model as the research object and calculate the safety factor (F_{\mathrm{S}}) representing slope stability. Slope stability evaluation involves many variables and the calculation of F_{\mathrm{S}} needs geometric data, geological material parameters (internal friction angle, pore water pressure information, and other physical data). These are very complex calculations, which are time-consuming and laborious. Therefore, it is not feasible to use a LEM when a large number of slopes need to be analyzed in a limited time. Due to the complexity of the slope system itself, it is necessary to adopt an effective prediction method to evaluate slope stability more efficiently. Therefore, quickly carrying on the preliminary analysis of the stability of a large number of slopes is particularly important.

Due to the obvious advantages of the artificial neural network in solving nonlinear problems [16,17] and its successful performance in modeling nonlinear multivariate problems, it has been used to evaluate slope stability. Kaunda et al. [18] applied an artificial neural network (ANN) to deal with the key factors affecting slope stability, predicted the position of the sliding surface of South Park slope in Miami through slope displacement, and further analyzed the factors influencing slope stability. Verma et al. [19] proposed a method using a neural network and fuzzy clustering system to effectively analyze the failure mechanism of a landslide and estimate the possible occurrence date of a landslide. The artificial neural network has a strong ability of feature extraction and learning, and it can extract features that cannot be described manually and complete classification and recognition. However, the traditional neural network is time-consuming and prone to overfitting [20], limiting its application in practical engineering. In recent years, the convolutional neural network (CNN) has been successfully applied in pavement crack recognition [21] and image processing [22]. However, although the CNN has good performance, it needs a large number of training samples [23]. When the samples are too few, they may not cover the slopes we need. Consequently, the neural network model cannot be properly trained and is easy to be over-fitted. A small number of training samples would lead to poor generalization ability of the model. One way to prevent over-fitting is to amplify training data sets. Teng et al. [24] used a finite element method to simulate the damage scenarios of a structure so that the number of training samples of the network was amplified and solved the problem that the data of damage conditions was difficult to measure in the process of structural health monitoring. To solve the problem of insufficient learning samples in deep learning, Yang et al. [25] used Generative Adversarial Network (GAN) to generate more new learning samples and applied deep learning in the field of pesticide residue detection. In the traditional CNN, the input is a two-dimensional matrix such as images, and the internal structure of the network, such as convolution kernel and feature graph are also two-dimensional. This results in a relatively long training process and high hardware requirements. Since the coordinate data of slope profiles and soil physical and mechanical parameters are a 1-D data, the 1-D CNN is used to input slope data, which is beneficial for a neural network to learn the relationship between slope stability and input parameter characteristics, thus enhancing the prediction speed of the neural network. The studies of slope stability mainly focus on the slope instability mechanism and reinforcement methods or use the traditional BP neural network to predict the slope stability, but few studies on the generation method of slope samples. Many samples can greatly increase the prediction speed and accuracy of neural networks, thus improving the valuable reference for slope engineering construction.

This paper proposes a method of generating a slope database using the LEM. From the perspective of numerical analyses, it provides a new framework for sample amplification and proposes a slope stability analysis method based on big data and a CNN.

2 Method

In this paper, the slope safety factor was obtained by a numerical analysis method, and the geometries, coordinates, and soil physical parameters of the slopes were used as the CNN inputs to predict the stability of the slope (Fig.1). The CNN can automatically extract slope stability characteristics from these data, which requires no complex calculation like traditional slope stability analysis methods. The methods are summarized as follows:

1) collect actual slope data and randomly modify slope coordinates and physical parameters (cohesion c, internal friction angle \phi, natural bulk density \gamma, and saturated bulk density {\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}), obtain a large number of new slope samples;

2) using the LEM principle, the slope safety factor is calculated in batches, and the slope database is established;

3) use the generated database to train the CNN;

4) test the trained CNN on actual slopes.

This paper took natural slopes as the research object and collected 55 typical slopes through field investigation and from Refs. [26–35]. There were 30 unstable slopes and 25 stable slopes. Among the 55 slopes, 40 were taken as basic samples for amplification to generate a sample database. The remaining 15 slopes were used as the testing samples to validate the performance of the CNN model. The 40 actual slopes were used as initial models, and 500 new samples were generated by sample amplification from each slope. The total number of slope samples generated was 40 × 500 = 20000. The slope stability analysis program was used to calculate and output the safety factors of the 20000 slopes.

2.1 Slope database

2.1.1 Actual sample preparation

In this paper, 40 actual slopes were used for sample amplification to enrich the types of the slope database. Four factors affecting the stability of natural slopes were selected as the input parameters of the neural network: cohesion c, internal friction angle \phi, natural bulk density \gamma, saturated bulk density {\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}. A slope consisting of five soil layers was taken as an example of extracted data (Fig.2). Some typical slope data are shown in Fig.3 and Tab.1.

2.1.2 Slope stability analysis method

In this paper, a new slope stability analysis program was written in Matlab language based on LEM (referring to the existing slope stability analysis program STAB). The operation of the program includes four steps:

1) reading the digital information of the slope profiles;

2) generating the sliding surface and soil strips;

3) calculating the weight of soil strip;

4) calculating the slope safety factor.

The specific calculation steps are as follows: firstly, read the profile data of the slopes (including the coordinates of each point constituting the slope and the physical and mechanical parameters of each soil layer: c, \phi ,\gamma, {\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}). Then, generate the sliding surface. If it was a homogeneous soil slope, a circular sliding surface was generated by reading the given slope profile information (Fig.4), and then the line segments where the upper and lower intersections of the sliding surface and the slope were located were found. The coordinates of the upper and lower intersection points were calculated, and the soil was divided into strips. Otherwise, an arbitrary sliding surface can be generated; that is, a number of smooth curves can be connected and approximated by a spline function to simulate an arbitrary sliding surface. Then, calculate the weight of soil by judging whether the slope line intersects with the center line of divided soil; the difference of the Y coordinate value of the adjacent two intersection points is calculated to obtain the height of each soil H₁, H₂, and H₃.

According to the soil layer number, the corresponding physical and mechanical parameters were found (Fig.5). Then, the soil strip’s weight was calculated. Finally, a safety factor was calculated according to the simplified method proposed by Chen and Morgenstern [36,37], which was used as the initial value of iteration (Eq. (1)). Then, the critical sliding surface corresponding to the minimum safety factor was found by searching and comparing all possible sliding surfaces according to the enumeration method. The final safety factor of the slope was calculated by the Spencer method as expressed in Eq. (1):

(1)F=\frac{{\int }_a^{b}A\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left[-\left({\displaystyle \frac{{\mathrm{t}\mathrm{a}\mathrm{n}\phi }^{\prime }}{F}}\alpha +{\displaystyle \frac{K_i}{F}}\right)\right]\mathrm{d}x}{{\int }_a^{b}B\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left[-\left({\displaystyle \frac{{\mathrm{t}\mathrm{a}\mathrm{n}\phi }^{\prime }}{F}}\alpha +{\displaystyle \frac{K_i}{F}}\right)\right]\mathrm{d}x} ,

where, with the slope foot being the origin, the positive direction of X axis is slope length direction, and the positive direction of Y axis is slope height direction. (X and Y here are different from x, which represents the width of a single soil strip). a and b were the x coordinates of the left and right endpoints of the sliding body, respectively, K_i was the coefficient considering the influence of the base dip angle and the limit internal friction angle, K_i={\sum _{i=1}^s\left[\mathrm{t}\mathrm{a}\mathrm{n}{\varphi }_i^{\prime }\cdot {\alpha }_i\right]}_i^r, and A and B were expressions related to weight, respectively, which can be specified as:

(2)\begin{array}{rl}A=& [\frac{\mathrm{d}W}{\mathrm{d}x}\left(\mathrm{c}\mathrm{o}\mathrm{s}\alpha -r_{\mathrm{u}}\mathrm{s}\mathrm{e}\mathrm{c}\alpha \right)\mathrm{t}\mathrm{a}\mathrm{n}{\varphi }^{\prime }+c^{\prime }\mathrm{s}\mathrm{e}\mathrm{c}\alpha \\ & -\eta \frac{\mathrm{d}W}{\mathrm{d}x}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{t}\mathrm{a}\mathrm{n}{\varphi }^{\prime }+q\mathrm{c}\mathrm{o}\mathrm{s}\alpha \mathrm{t}\mathrm{a}\mathrm{n}{\varphi }^{\prime }],\end{array}

(3)B=\left[\left(\frac{\mathrm{d}W}{\mathrm{d}x}+q\right)\mathrm{s}\mathrm{i}\mathrm{n}\alpha +\eta \frac{\mathrm{d}W}{\mathrm{d}x}\mathrm{c}\mathrm{o}\mathrm{s}\alpha \right],

where \alpha was the dip angle of soil strip, r_{\mathrm{u}} was the water pressure coefficient of void, and {\varphi }_i^{\prime } was the ultimate internal friction angle of the sliding surface i, s was the number of soil blocks divided.

The specific calculation steps are shown in Fig.6.

2.1.3 Generation method of sample database

The generation method of slope sample database is shown in Fig.7. Firstly, the actual slope samples were selected and input into the numerical analysis program, as shown in Fig.1, according to the data types introduced in Subsection 2.1. The new and different slope samples were obtained by randomly modifying the coordinates of the slope and the physical and mechanical parameters of the soil, and then the safety factor was automatically calculated.

Through sample amplification, the slope sample database eventually contains 20000 samples, covering the slope’s complete geometric and physical data. Geometric data refers to the coordinate information of the key points of each soil layer of the slope, which was arranged in a sequence according to the coordinates. Physical data refers to the physical and mechanical parameters of each soil layer: cohesion c, internal friction angle \varphi, natural bulk density \gamma, saturated bulk density {\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}. The database contains 7624 unstable slopes, 5846 metastable slopes, 2532 basically stable slopes, and 3998 stable slopes. Some typical sample data in the database are shown in Fig.8 and Fig.9. The variation rules of soil physical and mechanical parameters are shown in Tab.2, where R value represents the random variable between 0 and 20, which can better reflect the random variations of parameters.

The nine slopes in Fig.9(a)−Fig.9(i) were generated samples based on the original slope in Fig.8. By comparing Fig.8 and Fig.9, it can be seen intuitively that there were great differences in slope shapes and soil layer shapes between the generated and original samples. At the same time, the range of physical parameters of soil layers was controlled within a reasonable random value R to satisfy the richness of the physical and mechanical parameters of slopes. Therefore, the slope formed a new slope completely different from the original sample due to its change in shape and physical and mechanical parameters. The next step was to train the neural network using a sample database consisting of 20000 slopes and apply it to real slopes to test the accuracy of the trained CNN model.

2.2 Slope stability analysis based on CNN

2.2.1 Training samples

According to Subsubsection 2.1.3, 20000 slope samples were generated by numerical analyses. A 1D-CNN model captured key information about slopes and realized feature extraction of slope signals. After reasonably analyzing the influence of data arrangement on slope feature extraction in the neural network, after repeated attempts, the arrangement of input data is finally determined as follows.

Taking a set of slope data as an example, one-dimensional data were distributed in a row, and columns 1 to 4 were the data of the first soil layer of the slope, respectively: C,\phi ,\gamma ,{\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}. Columns 5 to 44 were the coordinates of the key points of the line segments of the first soil layer. When the key points of the line segments were less than 20, the left vacancies were filled with zeros, which does not affect the overall CNN characteristics and ensures the uniformity of the data format to facilitate the subsequent CNN training. Similarly, columns 45 to 88 were the geometric and physical data of the second layer, while the data of the third and fourth layers were at columns 89 to 132 and 133 to 176, respectively. The data of a slope are shown in Tab.3. Other slope data were also arranged in this manner.

2.2.2 CNN structure

In this paper, the CNN was designed in Matlab, and the internal parameters of the CNN were adjusted to achieve ideal detection results. After repeated attempts, it was found that some features would be lost after convolution and pooling of each set of data, resulting in inaccurate prediction results. To ensure the integrity of slope feature learning, pooling was not used, and the padding of the matrix was added. The entire model parameters are shown in Tab.4. The CNN consists of four convolutional layers, a fully connected layer, and an output layer [26]. The first layer has 16 convolutional kernels with the size and stride of 1 × 3 and 1, respectively; the second layer has 32 convolution kernels with the size of 1 × 3 and stride of 1; the third layer has 64 convolution kernels with the size of 1 × 3 and stride of 1; the fourth layer has 128 convolution kernels with the size of 1 × 3 and stride of 1). To avoid gradient disappeared, the activation function, Rectified Linear Unit (ReLU), was used. To get a better training model, 4 convolution layers were used. After ReLU activation, the model can better obtain relevant features and fit the training data. After the convolution and padding operations, the feature images of each convolution layer were multiplied. To improve the learning ability of the network, a variety of convolution kernel is added. After running the scenario, 128 feature images were finally obtained. Then, the 128 feature maps were expanded to obtain one-dimensional data, which was input to the fully connected layer, and finally a predicted value of slope safety factor was obtained. The CNN parameters are shown in Tab.1. The CNN structure used for classification is shown in Fig.10. The input parameters of CNN include the coordinates of key points (KP) of each slope and the physical and mechanical parameters of each soil layer (C,\phi ,\gamma ,{\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}), and the output is the slope safety factor and the stable state of the slope.

2.2.3 Analysis of slope stability

Based on the established slope database, the CNN, which was designed in Subsubsection 2.2.2, was trained and tested. There were 20000 sets of slope samples in the database, among which 18000 sets were used for training, and the remaining 2000 were used for testing. The trained CNN was used to predict the stability of another 15 real slopes to test the generalization ability of the CNN and verify its applicability by the database. According to the technical code for building slope engineering (GB50330-2013), the stability of a slope is classified into unstable, metastable, basically stable, and stable (Tab.5).

Before the slope stability evaluation, the data in Subsubsection 2.1.3 were normalized. Normalization is widely used in data processing because it keeps data within a specific range and makes data from different sources comparable. In this paper, the slope geometric dimensions and coordinates of key points and the physical and mechanical parameters of soil were normalized by Eq. (4).

(4)y=\frac{x-a}{b-a},

where x and y were values before and after normalization, respectively, and b and a were the maximum and minimum values of the sample data, respectively. After normalization, the data were input into the CNN for slope stability analysis. Accuracy was adopted to evaluate the prediction effect of the CNN, and the calculation formula is as follows:

(5)a=\frac{TP+TN}{TP+TN+FP+FN},

where TP was actually a positive sample and the prediction was also a positive sample; TN was actual negative sample and the prediction was also negative sample; FN was actual positive sample and was predicted to be negative sample; FP was actual negative sample and was predicted to be positive sample. The loss function calculated the deviation between the CNN output result and the label result. This paper adopted the mean square error loss function, which minimizes the distance between each training point and the optimal fitting line. The calculation formula was:

(6)J_{\mathrm{M}\mathrm{S}\mathrm{E}}=\frac{1}{M}\sum _{i=1}^M(y_i-y_i^{\prime }{)}^2,

where J_{\mathrm{M}\mathrm{S}\mathrm{E}} was the mean square error loss function value, y_i was the true value, y_i^{\prime } was the predicted value, and M was the number of samples.

At the same time, in order to make an effective comparison with the prediction effect of the CNN, a BP neural network was used in this paper to predict the same data set. Mean Squared Error (MSE) was used to evaluate the network performance of the BP neural network after training, which was defined as follows:

(7)MSE=\frac{1}{N}\sum _N^1(y -y_i{)}^2,

where y was the true value, y_i was the predicted value, and N was the number of samples.

3 Results

3.1 Slope stability analysis based on database

The slope sample database described in Subsubsection 2.1.3 is used to train and validate the CNN model. Fig.11 and Fig.12 and Tab.6 show the training process and validation results of the generated slope database. It can be seen from Fig.11 that there is a small oscillation in the first 10 rounds, and the accuracy in the training set reaches 100%, while the accuracy of the validation set also reached 99.4%. Fig.12 shows that the accuracy and loss function value have roughly opposite trends. The loss function value decreases on the training set until it approaches zero. After several training rounds, the loss function value drops to a very low level, and after about 15 rounds of training, the loss function value approaches zero. In the testing set, the loss function also decreases overall, except for some slight oscillations in the first 15 rounds. After 15 rounds, the loss function value tends to be stable, reaching about 0.04. In summary, the 1D-CNN was effectively trained on the training set with an accuracy rate of more than 99.4%, and no overfitting occurred. The confusion matrix of the CNN for the prediction results of the validation set is shown in Fig.13 (the number 438, 244, 542, 764 on the diagonal are the number of slopes that the CNN correctly predict the actual stability); however, one of the actually stable slopes is predicted to be basically stable, 4 basically stable slopes are predicted to be metastable, etc. Part of the prediction results are shown in Tab.6. According to statistics, the maximum absolute and relative errors of the CNN predictions are 2.52% and 1.44%, respectively, for the 2000 validation samples.

In this study, it can be seen from the training results of the BP neural network (Fig.14) that MSE gradually decreases to a small value on the training set and validation set. After 42 iterations, the error converges, and the MSE value for the training set at this time is 0.002, while the MSE for the validation set and testing set is 0.003. The training effect at this time was reasonable.

The fitting effect of the BP neural network is shown in Fig.15. For the training set, validation set, and testing set, the regression value R is above 98%, and the overall regression value reaches 98.5%, indicating a high correlation between the output and target values. The confusion matrix predicted by the BP neural network is shown in Fig.16 (the number 437, 209, 498, 749 on the diagonal are the number of slopes that the BP correctly predict the actual stability). However, 15 basically stable slopes and 11 metastable slopes are predicted to be stable; 23 basically stable slopes and 16 unstable slopes are predicted to be metastable; 1 basically stable slope and 3 metastable slopes are predicted to be unstable, etc. Finally, the results show that the prediction accuracy of the BP neural network for 2000 slopes samples, which are not used in training, reaches 94.6%.

Obviously, with a large number of slope samples in the database, the prediction accuracy of the CNN and BP neural network is 99.4% and 94.6%, respectively. According to the prediction effect of the two networks, the CNN performs better.

3.2 Slope stability analysis based on actual slope data

The trained CNN is used to predict the remaining 15 actual slopes to prove its generalization ability and accuracy. After the normalized parameters of 15 samples are input to the trained 1-D CNN model, the slope stability can be obtained immediately. The confusion matrix of the CNN prediction results is shown in Fig.17 (the diagonal numbers 3, 3, 1, 7 are the number of slopes that are predicted correctly). It can be seen that, among the 15 slopes, the stability of 14 slopes is correctly predicted. One basically stable slope is wrongly predicted to be metastable as the basically stable and metastable states are close to each other. The prediction accuracy for the slope stability of 15 samples is 93.3%.

Fig.18 and Tab.7 show the prediction results and prediction errors of the 15 slopes. Thus, the maximum absolute error of the CNN model is 5.42%, and the maximum relative error is 3.23%. The output results are basically consistent with the actual results, indicating that the 1D-CNN model has a strong generalization ability.

4 Discussion and conclusions

In this paper, the applicability of the CNN in slope stability prediction is studied. The LEM is used to build a slope sample database, and the prediction ability of CNN and BP neural networks is compared. The following observations can be obtained from the above studies.

1) New slope samples can be obtained by automatically modifying real slopes’ coordinates and physical and mechanical parameters in an appropriate range. In the process of sample generation with numerical simulations, two kinds of sliding modes, namely circular sliding surface and arbitrary sliding surface, are fully considered, making the numerical calculation results more reasonable.

2) A 1-D CNN prediction model is applied to slope stability analysis. The input of the network is slope coordinates and soil physical parameters. Data features are automatically extracted by the 1-D CNN, which effectively improves the classification and prediction ability of the network. After training, the prediction accuracy of the CNN is above 99.0%, which is higher than that of the traditional BP neural network (93.6%).

From the above discussion, the following conclusions can be drawn.

1) The slope sample database established based on the LEM solves the problem of insufficient samples in deep learning and augments the sample quantity and improves accuracy.

2) The CNN can be applied to slope stability prediction. Ideal and rapid slope prediction results can be achieved by inputting the coordinate information and physical and mechanical parameters of slopes. The prediction accuracy of the CNN is better than that of the BP neural network.

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