Generating Stochastic Structural Planes Using Statistical Models and Generative Deep Learning Models: A Comparative Investigation

Abstract

Structural planes are one of the key factors controlling the stability of rock masses. A comprehensive understanding of the spatial distribution characteristics of structural planes is essential for accurately identifying key blocks, analyzing rock mass stability, and addressing various rock engineering challenges. This study compares the effectiveness of four stochastic structural plane generation methods—the Monte Carlo method, the Copula-based method, generative adversarial networks (GAN), and denoised diffusion models (DDPM)—in generating stochastic structural planes and capturing potential correlations between structural plane parameters. The Monte Carlo method employs the mean and variance of three parameters of the measured factual structural planes to generate data that follow the same distributions. The other three methods take the entire set of measured factual structural planes as the overall input to generate structural planes that exhibit the same probability distributions. Five sets of structural planes on four rock slopes in Norway are examined as an example. The validation and analysis were performed using histogram comparison, data feature comparison, scatter plot comparison, and linear regression analysis. The results show that the Monte Carlo method fails to capture the potential correlation between the dip direction and dip angle despite the best fit to the measured factual structural planes. The Copula-based method performs better with smaller datasets, and GAN and DDPM are better at capturing the correlation of measured factual structural planes in the case of large datasets. Therefore, in the case of a limited number of measured structural planes, it is advisable to employ the Copula-based method. In scenarios where the dataset is extensive, the deep generative model is recommended due to its ability to capture complex data structures. The results of this study can be utilized as a valuable point of reference for the accurate generation of stochastic structural planes within rock masses.

1. Introduction

The structural planes are one of the key factors controlling the stability of the rock mass. Structural planes, which include numerous cracks, faults, and other discontinuities, impart discontinuous, heterogeneous, and anisotropic properties to the rock masses and have an important influence on the geomechanical and hydrogeological behaviors of the rock mass [1,2,3]. Structural planes cut the rock mass into blocks of various shapes. In the natural state, these spatial blocks remain in static equilibrium. When subjected to external forces, some blocks on the critical surface may lose their initial mechanical equilibrium, slipping along the structural planes, destabilizing them, and causing progressive failure. These blocks are referred to as “key blocks” [4]. Accurately identifying key blocks is crucial for ensuring the stability of the rock mass. The distribution of key blocks is primarily determined by the stochastic arrangement of structural planes. Therefore, a comprehensive understanding of the spatial distribution characteristics of structural planes is essential for accurately identifying key blocks, analyzing rock mass stability, and addressing various rock engineering challenges [5].

Large-scale structural planes such as faults, which exhibit a limited degree of development in a given area, can be measured using traditional deterministic methods [6]. There exists a class of structural planes within the rock mass that are numerous, widely distributed, and small in size that cannot be accurately measured and located using conventional engineering geological methods. These structural planes exhibit spatial distributions characterized by grouping and adherence to specific patterns. Therefore, based on data observed from surface outcrops, the internal distribution of structural planes within the rock mass can be inferred statistically using probabilistic statistical theory.

The Monte Carlo method is the most widely used method for generating stochastic structural planes within rock masses [7,8,9,10]. This method operates by generating stochastic numbers that adhere to the same probabilistic distribution as the variable, based on the known probability distribution of that variable. The accuracy of the Monte Carlo method has been validated in various case studies. For example, Shapiro and Delport [11] employed the Monte Carlo method to simulate jointed rocks. Wang et al. [12] conducted a Monte Carlo simulation of stochastic joints using the Dabeigou Tunnel on the Duolun second-class highway in Duolun Town, Nei Mongol, China, as a case study, implementing the identification of key blocks based on limit equilibrium theory. Li et al. [12] derived the characteristics of joints in the study area from five geological outcrops and subsequently constructed a network model of joints using the Monte Carlo method. Wang et al. [13] enhanced the Monte Carlo simulation by integrating measured data from over 100,000 discontinuities across 35 hydropower projects, providing an in-depth analysis of the statistical characteristics of rock geometric parameters of rock discontinuities.

However, the Monte Carlo method, which independently generates stochastic numbers based on the statistical properties of the geometric parameters of structural planes, has inherent limitations. It disregards the correlations between structural plane parameters that have been mentioned in the literature [9,14,15,16,17]. Neglecting the correlation between the parameters affects the accuracy of the generated results. Therefore, there is a necessity to investigate alternative stochastic structural plane generation methods for generating structural planes that can consider the correlation among parameters.

There are two types of dominant methods for characterizing the interrelationships between random variables. The first method utilizes the Copula-based method to construct joint distribution models that accurately match parameter correlations. The second method employs deep generative models to automatically capture parameter correlations during the generation process.

The Copula-based method can be employed to establish the correlation structure between variables by defining marginal univariate distributions and selecting an appropriate Copula function to construct multivariate distributions for variables with distinct marginal distributions [18,19,20,21]. The deep generative model, constructed using deep neural networks that emulate the structure and function of the human brain, processes data through multiple layers of nonlinear transformations and weight adjustments. In these networks, each layer of neurons receives inputs from the previous layer and applies a series of nonlinear mappings [22]. Comprising thousands of neurons, deep neural networks do not directly calculate linear relationships between data. Instead, they map data into a transformed space through successive layers of nonlinear mappings, facilitating the identification of patterns and regularities within the data and enabling the modeling of complex distributions and nonlinear relationships.

Both deep generative models and Copula-based method fitting models have been employed for generating stochastic structural planes. For example, Meng et al. [23] introduced a method for stochastic structural plane generation based on GAN, which takes into account parameter correlations. Similarly, Han et al. [24] analyzed the relationship between the length and aperture of structural planes using the Copula-based method, demonstrating its effectiveness in describing orientation data and illustrating the relationship between trace length and aperture diameter.

Although much research work has been conducted to generate stochastic structural planes, the identification of the most precise method that is consistently best with the measured factual structural planes remains ambiguous. Many studies substantiate their conclusions based on individual cases or operations, thereby undermining the reliability of their results. The Monte Carlo method fails to capture parameter correlations. The Copula-based method relies on the correlation between two variables, and weak correlations can lead to inaccurate computation of correlation coefficients, resulting in unclear dependency structures between variables. Deep generative models depend heavily on the quality and diversity of training data, with the training process often being unstable and prone to issues such as mode collapse and overfitting, potentially leading to a lack of diversity in the generated samples. Each of these four methods has its strengths and weaknesses. A comprehensive evaluation of stochastic structural plane generation methods is currently lacking. Therefore, it is essential to examine the accuracy and applicability of these methods through experiments involving multiple instances of rock slopes.

This paper presents a comprehensive comparison of four prominent stochastic structural plane generation methods: statistical models (including the Monte Carlo method and the Copula-based method) and deep generative models (including DDPM, and GAN). The performance of these methods is compared using five different datasets from four rock slopes: the Valle study area, Tunhovd study area, Straumklumpen study area, and Oernlia study area in Norway. Each method underwent multiple samplings, and the most favorable outcomes were chosen to evaluate the generative accuracy of the models and the captured parameter correlations both by quantitative calculations and quantitative analyses.

2. Methods

We employed two statistical probabilistic models and two deep generative models for generating stochastic structural planes. The chosen methods include the conventional and extensively applied Monte Carlo method; the Copula-based method, recognized for its effectiveness in capturing random variable dependencies; the GAN, the most prominent model of deep generative models; and the recently developed and esteemed DDPM. This section provides a detailed introduction to these four methods, as well as the methods employed for result validation.

2.1. The Employed Statistical Models

2.1.1. The Monte Carlo Method

The Monte Carlo method is a prominent and extensively utilized method for generating stochastic structural planes [7,16,17,25,26,27,28,29,30,31,32]. In the process of stochastic structural plane generation, the parameters of the structural planes, including dip direction, dip angle, and trace length, are considered stochastic variables characterized by specific probability distribution functions and cumulative distribution functions. The fundamental concept entails defining the probability density function for each geometric parameter of the structural planes through statistical evaluation of empirical data. Subsequently, the structural planes are randomly sampled based on the established probability density functions to produce simulated stochastic values that closely resemble the factual probability distribution functions.

To ensure the appropriate selection of the prior distribution, the probability distribution of each parameter is initially assessed through Q-Q plots or P-P plots. Subsequently, a Python code is utilized to produce random numbers that adhere to the specific probability distributions, considering the data attributes such as mean and variance. The sampling process is iterated until the most precise set of outcomes is achieved.

2.1.2. The Copula-Based Method

The Copula theory, initially introduced by Sklar in 1959 [33], has been utilized in various fields such as finance, civil engineering, medicine, etc. [34,35,36,37]. The Copula-based method serves to link or “couples” a multivariate distribution function with its one-dimensional marginal distribution function, where its density indicates the degree of the correlation between these marginal distributions [38]. The fundamental equation for the Copula-based method is presented in Equation (1).

$$F\left(x_1,x_2,\cdots ,x_n\right)=C\left(F_1\left(x_1\right),F_2\left(x_2\right),\cdots ,F_n\left(x_n\right);\theta \right)$$

(1)

where $F_1\left(x_1\right),F_2\left(x_2\right),\dots ,F_n\left(x_n\right)$ are the continuous marginal distribution functions of the random variables $x_1,x_2,\dots ,x_n$, respectively; C is a Copula-based method; and $\theta$ is the Copula parameter. $F(x_1,x_2,\dots ,x_n)$ represents the joint distribution function of the variables $x_1,x_2,\dots ,x_n$. Equation (1) illustrates that the construction of the joint probability distribution of random variables involves two main components: the marginal distribution function and the corresponding Copula-based method.

In this study, the Gaussian Copula-based method, which is widely utilized, is applied. The structural form of the Gaussian Copula-based method is presented as follows (Equation (2)):

$${\int }_{-\infty }^{{\mathsf{\Phi }}^{-1}\left(u\right)}{\int }_{-\infty }^{{\mathsf{\Phi }}^{-1}\left(v\right)}{\displaystyle \frac{1}{2\pi \sqrt{1-{\alpha }^2}}}exp\left[-{\displaystyle \frac{s^2-2\alpha st+t^2}{2\left(1-{\alpha }^2\right)}}\right]$$

(2)

where u and v represent binary variables, and $\alpha$ is the parameter of the Copula-based method.

2.2. The Employed Generative Deep Learning Models

Deep learning includes a range of neural network models, including deep generative networks (DGMs) [39], convolutional neural networks (CNNs) [40], long short-term memory networks (LSTMs) [41], and deep belief networks (DBNs) [42,43]. In recent years, generative deep learning emerged as a crucial branch within the field of deep learning, which employs deep neural networks to acquire data representations and facilitate the learning process based on the objective function [22]. This approach is widely employed in image generation [44], audio generation [45], text generation [46], and other areas [47,48,49].

Deep generative models encompass various categories such as autoregressive models (e.g., pixel CNN), flow models, DDPM, latent variable models (e.g., GAN, VAE), and energy-based models [50]. These models found extensive applications in engineering geology. For example, Li et al. [51] employed Wasserstein GAN with gradient penalty for digital rock reconstruction. Al-Najjar et al. [52] proposed an integrated approach for equalizing landslide data and spatial prediction using GAN. Esmaeili [53] applied DDPM to merge carbonate and sandstone data for creating digital rock images, thereby overcoming data limitations in digital rock analysis.

2.2.1. GAN

In 2014, Goodfellow introduced the GAN at the NIPS conference, which had a notable impact on the field of deep learning [54]. GAN is based on the concept of Nash equilibrium, where a specific set of strategies signifies a state where no participant can gain an advantage by independently altering their strategy. The key advancement of GAN is its fusion of generative and discriminative models, enabling them to engage in a competitive process, consequently enhancing their performance.

The schematic diagram of GAN is illustrated in Figure 1, which demonstrates the components of GAN and the process of enhancing performance during the training process. As depicted in the figure, GAN consists of two main components: the generator (G) and the discriminator (D). The generator is tasked with generating data, while the discriminator assesses the authenticity of the data. The training process follows a left-to-right flow.

First, a random noise vector z from the latent space, typically following a specific prior distribution (generally Gaussian), is input into the generator. Subsequently, the generator generates new samples $G\left(z\right)$ that resemble real sample data based on this noise vector. At the same time, the real sample is denoted as X. These generated samples $G\left(z\right)$ are presented to the discriminator alongside real samples X. The task of the discriminator is to distinguish between real and synthetic data. Functioning similarly to a binary classification model, the discriminator outputs a probability value. A discriminator output of 1 indicates that the input sample is authentic, while an output of 0 indicates that the sample is counterfeit, generated by the generator. The proximity of the output probability value to 1 reflects the degree to which the generated sample resembles a real sample.

In the training process, the generator and discriminator engage in a continuous cycle of confrontation and competition. The discriminator aims to accurately distinguish between real samples and generated samples. In contrast, the generator aims to produce data that closely resemble real samples, making it challenging for the discriminator to differentiate between the two. The loss function for the training process is formally defined as shown in Equation (3)).

$$\underset{G}{min}\underset{D}{max}V(D,G)=\underset{G}{min}\underset{D}{max}{\mathbb{E}}_{x\sim p_{\mathrm{data}}\left(x\right)}\left[\log D\left(x\right)\right]+{\mathbb{E}}_{z\sim p_z\left(z\right)}\left[\log (1-D\left(G\left(z\right)\right))\right]$$

(3)

In practice, the generator and discriminator are trained alternately, with the discriminator being trained first, followed by the generator, in a continuous iterative process aimed at enhancing both networks. Initially, the parameters of the generator remain constant while the parameters of the discriminator are optimized. The objective is to maximize the following function (Equation (4)):

$$\underset{D}{max}{\mathbb{E}}_{x\sim q\left(x\right)}\left[\log D\left(x\right)\right]+{\mathbb{E}}_{z\sim p\left(z\right)}\left[\log (1-D\left(G\left(z\right)\right))\right]$$

(4)

In the training process, the parameters of the discriminator are set to fixed values to facilitate the training of the generator. The primary objective for the generator is to enhance the discriminative probability $D\left(G\right(z\left)\right)$ of the generated samples, which is essentially achieved by minimizing $\log (1-D(G\left(z\right)\left)\right)$. As the training advances, the model converges, leading to the establishment of a Nash equilibrium between the two networks. The discriminator is unable to differentiate between the real and synthetic data generated by the generator, resulting in a predicted probability value of 1/2. This particular state indicates the successful completion of the training procedure.

2.2.2. DDPM

DDPM represents an emerging class of deep generative models. The foundational concepts were initially introduced by Sohl-Dickstein in 2015 [55], establishing the basis for diffusion modeling, and further improved by Jonathan Ho in 2020 [56]. Inspired by nonequilibrium thermodynamics, DDPM integrates diffusion modeling with deep learning techniques.

The principle of DDPM is illustrated in Figure 2. The figure demonstrates the workflow of DDPM, where data is iteratively modified by adding and removing noise. Initially, the initial data $X_0$, depicted on the left side of the Figure 1, resembling the clear image displayed above. The diffusion steps of the Markov chain are executed, progressively introducing stochastic noise to the data. This process generates a series of states ($X_1$, $X_{t-1}$, $X_t$). It is observable that as noise is gradually incorporated, the image associated with each state progressively blurs, ultimately resulting in pure noise. This process obeys the known conditional probability distribution. Then the reverse diffusion process is acquired, and the model systematically eliminates the noise, starting from the noisy data, generating a series of intermediate states, and ultimately restoring the sample data. The conditional probability distribution of this reverse process is unknown and necessitates approximation by training the neural network model.

• Forward diffusion process

The forward process involves the gradual addition of noise, which can be viewed as a Markov process. Noise is incrementally added to the sample $X_0$ over specified time steps, with each step corresponding to a different noise level. This ultimately transforms the original distribution of the sample data into a simple standard Gaussian distribution. The noise addition process can be described as a conditional normal distribution, as defined by the following equation (Equation (5)):

$$q\left(x_t\mid x_{t-1}\right)=N\left(x_t;\sqrt{1-{\beta }_t}{x}_{t-1},{\beta }_{t}I\right)$$

(5)

where q represents the conditional probability distribution that characterizes the noise addition process from the state $x_{t-1}$ at the previous time step to the state $x_t$ at the current time step. The parameter ${\beta }_t$ is a predefined noise parameter, and I denotes the identity matrix. Given the state $x_{t-1}$ at the previous time step, the sample $x_t$ at the current time step follows a normal distribution with mean and variance ${\beta }_t$. The parameter ${\beta }_t$ increases with t, and as t approaches infinity, the sample at this moment converges to an approximately Gaussian distribution.

• Reverse diffusion process

The inverse process, which is the reverse of the forward diffusion process, can also be considered as a Markovian process. Initially, $X_T$ is sampled from the noise distribution, typically a standard Gaussian distribution. The noise prediction network, denoted as ${ϵ}_t$, generates noise that is gradually removed in successive steps. At each step, a small amount of Gaussian noise is eliminated, transforming $X_T$ into $X_{T-1}$, then $X_{T-2}$, and so on. This process continues until the data closely approximate the true distribution $X_0$. The result is a set of samples that are close to the real data distribution, thus achieving the objective of data generation. This process is governed by the following formula (Equation (6)):

$$\mathrm{q}\left({\mathrm{x}}_{\mathrm{t}-1}\mid {\mathrm{x}}_{\mathrm{t}},{\mathrm{x}}_0\right)=\mathrm{N}\left({\mathrm{x}}_{\mathrm{t}-1};{\tilde{\mu }}_{\theta }\left({\mathrm{x}}_{\mathrm{t}},{\mathrm{x}}_0\right),{\tilde{\beta }}_{\mathrm{t}}\mathrm{I}\right)$$

(6)

where ${\tilde{\mu }}_{\theta }(x_t,x_0)$ and ${\tilde{\beta }}_{\theta }$ represent the predicted mean and constant variance of the generated sample distribution in the reverse diffusion process, respectively. By applying the conditional probability formula $p_{\theta }\left(x_{t-1}\mid x_t\right)$ and ${\tilde{\beta }}_t={\displaystyle \frac{1-{\overline{\alpha }}_{t-1}}{1-{\overline{\alpha }}_t}}·{\beta }_t$ to Equation (7), and interpreting the forward diffusion process as the reverse diffusion process, the following can be deduced:

$${\tilde{\mu }}_t={\displaystyle \frac{1}{\sqrt{{\alpha }_t}}}\left(x_t-{\displaystyle \frac{1-{\alpha }_t}{\sqrt{1-{\overline{\alpha }}_t}}}{\epsilon }_t\right)$$

(7)

• Loss function

Samples are taken from the structural plane data at a randomly chosen time t. A random noise is sampled from the Gaussian distribution, transforming $x_0$ into $x_t$. The noise prediction network then predicts the noise based on $x_0$ and $x_t$. The difference between the predicted noise and the true noise constitutes the loss function, and the objective of the training process is to minimize the error generated by the denoising process. The loss function is formulated as Equation (8), where $ϵ$ represents the true noise distribution at time step t. The network parameters are continuously optimized through gradient-based optimization, ensuring that the predicted noise distribution gradually approximates the real noise distribution in the same state. Upon completion of the training, new data can be gradually sampled from the denoising function to generate new data.

$$Loss\left(\theta \right):=\mathbb{E}\left[{∥\epsilon -{\epsilon }_{\theta }\left(\sqrt{{\alpha }_t}{x}_0+\sqrt{1-{\overline{\alpha }}_t}\epsilon ,t\right)∥}^2\right]$$

(8)

2.3. Metrics for Comparative Evaluation

After generating data using the four specified methods, a comparative analysis was conducted to evaluate their performance.

Table 1. The metrics for comparative evaluation.

Metrics		Principles of the Metrics
Qualitative analysis	Comparison of histograms of probability distributions	The histogram illustrates the distribution of values of a variable across a specified range. Each bar in the graph corresponds to the frequency of values within that range, presenting a visual depiction of the data distribution through a series of vertical bars or lines of different heights. A histogram serves as a precise graphical portrayal of the numerical data distribution.
	Scatter plot Comparison	A scatter plot represents the distribution of data points on the Cartesian coordinate system in regression analysis. It provides a visual assessment of the potential correlation strength between two variables, aiding in the selection of a suitable function to model the data points.
Quantitative calculation	Comparison of mean value and variance	Mean and variance are important statistics that describe the characteristics of the data and can provide a visual representation of the overall characteristics and distribution of the dataset.
	Pearson correlation coefficient	The Pearson correlation coefficient is the most commonly used method for conducting correlation analysis. It serves the purpose of the strength of correlation between two variables. A larger absolute value of the correlation coefficient indicates a stronger correlation, while a value closer to zero suggests a weaker correlation.

lists the metrics employed in the comparative evaluation, along with the characteristics of each metric. The analysis is divided into two primary categories: quantitative calculations and quantitative analyses. The qualitative analysis involves comparing the histograms of the probability distribution of each parameter of the generated structural planes with those of the measured factual structural planes to assess accuracy. It also involved the utilization of scatter plots to visualize the strength of the potential correlation between the two variables and patterns in the data distribution. Quantitative calculations involve calculating the mean and variance of each parameter of the generated structural planes, as well as the mean and variance for each parameter of the measured factual structural planes. The Pearson correlation coefficient was also calculated using linear regression analysis to assess the strength of the correlation between two parameters. A larger absolute value of the correlation coefficient indicates a stronger correlation, while a value closer to zero suggests a weaker correlation.

3. Results and Analysis

3.1. Experimental Data

The objective of our research is to investigate the ability of different methods to generate stochastic structural planes in terms of accuracy and capturing parameter correlations. Therefore, the selected structural plane groups must exhibit distinct parameter probability distribution patterns and correlations between the dip direction and dip angle. To ensure the comprehensiveness of the experiments, the chosen structural plane groups also needed to vary in data size. To achieve this, we conducted a detailed analysis of each group of structural planes, including the verification of probability distributions using histograms and Q-Q plots, and determining correlations through linear regression analysis. Ultimately, we selected five sets of structural planes that varied in terms of data size and correlation patterns as case studies to evaluate the efficacy of different methods.

The data utilized for the experiments were sourced from structural planes datasets provided by Darvell in the publicly accessible database [57]. The five sets of structural planes were distributed from four different study areas, including four rock slopes in Valle, Tunhovd, Straumklumpen, and Oernlia, Norway.

Figure 3 illustrates the locations of the four study areas. Dataset 1 pertains to the Valle study area, situated adjacent to the small town of Valle in the Setesdalen valley within Agder County. Both Dataset 2 and Dataset 4 are derived from the identical slope of the Tunhovd study area, located on the eastern bank of Tunhovdfjorden. The study areas from which Dataset 3 and Dataset 5 are derived are both located on the shores of Lake Straumklumpen. Dataset 3 encompasses a broader study area that encompasses nearly the entire southern inclines of Straumklumpen, while Dataset 5 is confined to a smaller study area on the fringes of Lake Straumklumpen.

Table 2. The probability distributions and data characteristics of geometric parameters of five datasets of structural planes.

Groups	Location	Number	Distribution Type			Correlation	Pearson Correlation Coefficient
			Dip Direction	Dip Angle	Trace Length
Dataset 1	Valle study area	40	Normal	Normal	Log-normal	Negative	−0.374
Dataset 2	Tunhovd study area	157	Normal	Normal	Log-normal	Positive	0.361
Dataset 3	Straumklumpen study area	257	Normal	Normal	Log-normal	Negative	−0.342
Dataset 4	Tunhovd study area	325	Normal	Normal	Log-normal	Negative	−0.252
Dataset 5	Oernlia study area	766	Normal	Normal	Log-normal	Negative	−0.685

presents the probability distributions of the geometric parameters and data characteristics of the five structural plane datasets, illustrating distinct variations among the datasets. The datasets are organized based on the volume of data, ranging from the smallest to the largest. Dataset 2 exhibits a positive correlation between the dip direction and dip angle, while the other four datasets display a negative correlation between these two parameters.

3.1.1. Dataset 1: Structural Plane Set of Valle Study Area

Dataset 1 comprises the structural planes in the Valle study area. The Valle study area is located next to the small town of Valle in the Setesdalen valley in the county of Agder. It is situated on the Nomelandsfjellet hillside at an altitude of 315 ∼ 665 m.a.s. The rocky side slopes face SE and are approximately 300 m high. The rocks consist of granite [57]. This set of structural planes is characterized by a limited dataset, comprising only 40 pieces of structural planes. The probability distribution and probabilistic verification of the structural planes in this study area are illustrated in Figure 4. Both the dip direction and dip angle exhibit normal distributions, while the trace length follows a log-normal distribution pattern. The linear regression analysis depicted in Figure 5 reveals a negative correlation between the dip direction and dip angle, as indicated by a Pearson correlation coefficient of −0.374.

3.1.2. Dataset 2: Structural Plane Set of Tunhovd Study Area

Dataset 2 and Dataset 4 comprise the structural planes in the Tunhovd study area. The Tunhovd study area is situated on the eastern shore of Tunhovdfjorden, near the southern end of the lake in Viken County. It is positioned along the Tunhovdvegen road at an elevation of 760 ∼ 890 m above sea level, extending parallel to the road for a distance of 360 m. The rocky slopes, which face south-southwest, are approximately 100 m high. Most of the slopes have an angle of 80 degrees and are composed of quartzite [57].

Dataset 2 includes 157 pieces of structural planes. The probability distribution and probabilistic verification of the first set of structural planes are illustrated in Figure 6. Both the dip direction and dip angle exhibit normal distributions, while the trace length follows a log-normal distribution pattern. The linear regression analysis depicted in Figure 7 reveals a positive correlation between the dip direction and dip angle, as indicated by a Pearson correlation coefficient of 0.361.

3.1.3. Dataset 3: Structural Plane Set of Straumklumpen Study Area

Dataset 3 comprises the structural planes in the Straumklumpen study area. The Straumklumpen study area is the largest, encompassing nearly the entire southern hillside of Straumklumpen at an altitude of 400 ∼ 700 m.a.s, with a length of 1340 m. It features gently sloping rocky terrain with well-developed talus. The rocky side slopes, oriented to the southwest, have a measured slope profile of approximately 540 m. The bedrock comprises granite and granitic gneiss [57]. This dataset includes 257 pieces of structural planes. The probability distribution and probabilistic verification of the structural planes in this study area are illustrated in Figure 8. Both the dip direction and dip angle exhibit normal distributions, while the trace length follows a log-normal distribution pattern. The linear regression analysis depicted in Figure 9 reveals a negative correlation between the dip direction and dip angle, as indicated by a Pearson correlation coefficient of −0.252.

3.1.4. Dataset 4: Structural Plane Set of Tunhovd Study Area

Dataset 2 and Dataset 4 comprise the structural planes in the Tunhovd study area. Dataset 4 includes 325 pieces of structural planes. The probability distribution and probabilistic verification of the second set of structural planes in this study area are illustrated in Figure 10. Both the dip direction and dip angle exhibit normal distributions, while the trace length follows a log-normal distribution pattern. The linear regression analysis depicted in Figure 11 reveals a negative correlation between the dip direction and dip angle, as indicated by a Pearson correlation coefficient of −0.342.

3.1.5. Dataset 5: Structural Plane Set of Tunhovd Study Area

Dataset 5 comprises the structural planes in the Oernlia study area. The Oernlia study area is situated at an altitude of 280 ∼ 425 m above sea level and extends for 230 m. The rocky side slopes, which are west-directed, have a measured slope section height of approximately 125 m and an angle of approximately 50 degrees. The rock in this area primarily consists of granitic gneiss [57]. This set of structural planes is the most numerous, containing 766 pieces of structural planes. The probability distribution and probabilistic verification of the structural planes in this study area are illustrated in Figure 12. Both the dip direction and dip angle exhibit normal distributions, while the trace length follows a log-normal distribution pattern. The linear regression analysis depicted in Figure 13 reveals a negative correlation between the dip direction and dip angle, as indicated by a Pearson correlation coefficient of −0.685.

3.2. Experimental Environments

The experimental tests were carried out on a Windows 10 system with 16 GB of memory, an AMD Ryzen 7 5800H processor with Radeon Graphics 3.20 GHz, and an NVIDIA GeForce RTX 3060 graphics card. The programming language is Python 3.8.8, and the deep learning framework is PyTorch1.9.0.

3.3. Experimental Results

This section presents results generated using each of the four methods for five different datasets of structural planes.

3.3.1. Generated Results of Dataset 1

The first group of structural planes is the least numerous, containing only 40 pieces of structural planes.

Table 3. Means and variances of the parameters of the structural planes generated by the four methods compared with those of the measured factual structural planes.

	Dip Direction (°)			Dip Angle (°)			Trace Length (m)
	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation
Measured factual	Normal	74.80	6.04	Normal	46.72	4.19	Log-normal	1.88	2.24
Monte Carlo method	Normal	73.30	6.56	Normal	46.73	3.69	Log-normal	1.75	1.61
Copula-based method	Normal	75.23	6.32	Normal	46.40	4.59	Log-normal	1.79	1.63
GAN	Normal	77.22	5.34	Normal	46.88	4.65	Log-normal	1.98	1.90
DDPM	Normal	75.73	8.94	Normal	45.97	4.67	Log-normal	4.01	3.00

illustrates the means and variances for each parameter of the structural planes generated by the four methods with those of the measured factual structural planes. Figure 14 presents the results of the histogram comparison between the dip direction of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 15 illustrates the results of the histogram comparison between the dip angle of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 16 displays the results of the histogram comparison between the trace length of the structural planes generated by the four methods and those of the measured factual structural planes. Finally, Figure 17 displays the linear regression analysis between the dip direction and dip angle of the structural planes generated by the four methods.

An analysis of Table 3 reveals that the mean and variance of the dip direction are 74.8 and 6.04, respectively, with parameter ranges generated by the four methods spanning 73.3–77.22 and 5.34–8.94. The mean and variance of the dip angle are 46.72 and 4.19, respectively, with parameter ranges generated by the four methods spanning 45.97–46.88 and 3.69–4.67. The mean and variance of the strike are 1.88 and 2.24, respectively, with parameter ranges generated by the four methods spanning 1.75–4.04 and 1.61–3.00. Significant errors were observed in the parameters generated by the GAN and DDPM, particularly with DDPM showing substantial discrepancies in the generated trace length.

The Pearson correlation coefficient between the dip direction and dip angle of the measured structural planes indicates a negative correlation of −0.374. In contrast, the Pearson correlation coefficient between the dip direction and dip angle of the structural planes generated by the Monte Carlo method is 0.164, suggesting a positive correlation. This suggests that the correlation has not been effectively captured.

The Pearson correlation coefficient between the dip direction and dip angle of the structural planes generated by the Copula-based method is −0.326, which is close to the Pearson correlation coefficient between the dip direction and dip angle of the measured structural planes. The Pearson correlation coefficient between the dip direction and dip angle of the structural planes generated by the DDPM is −0.313. However, the accuracy of the generation process is inferior to that of the Copula-based method, as evidenced by the trend in the data distribution observed in the scatter plots. The distribution trend of data in the scatter plots of the structural planes generated by the GAN is close to that of the measured structural planes. However, the Pearson correlation coefficient between the dip direction and dip angle is −0.694, indicating a deviation from the corresponding Pearson correlation coefficient between the dip direction and dip angle in the measured factual structural planes.

3.3.2. Generated Results of Dataset 2

This dataset includes 157 pieces of structural planes.

Table 4. Means and variances of the parameters of the structural planes generated by the four methods compared with those of the measured factual structural planes.

	Dip Direction (°)			Dip Angle (°)			Trace Length (m)
	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation
Measured factual	Normal	262.29	10.56	Normal	50.29	5.15	Log-normal	1.77	1.18
Monte Carlo method	Normal	261.83	9.93	Normal	50.08	5.45	Log-normal	1.75	1.61
Copula-based method	Normal	263.72	10.69	Normal	50.49	4.99	Log-normal	1.68	1.10
GAN	Normal	258.88	12.01	Normal	46.99	5.14	Log-normal	1.13	0.57
DDPM	Normal	261.41	12.07	Normal	49.49	5.59	Log-normal	1.78	1.17

illustrates a comparative analysis of the mean and variance of each parameter generated by the four methods with the measured factual structural planes. Figure 18 illustrates the results of the histogram comparison between the dip direction of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 19 illustrates the results of the histogram comparison between the dip angle of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 20 illustrates the results of the histogram comparison between the trace length of the structural planes generated by the four methods and those of the measured factual structural planes. Finally, Figure 21 displays the linear regression analysis between dip direction and dip angle of the structural planes generated by the four methods.

An analysis of Table 4 reveals that the mean and variance of the dip direction are 262.29 and 10.56, respectively, with parameter ranges generated by the four methods spanning 258.88–263.72 and 9.93–12.07. The mean and variance of the dip angle are 50.29 and 5.15, respectively, with parameter ranges generated by the four methods spanning 46.99–50.49 and 4.99–5.59. The mean and variance of the trace length are 1.77 and 1.18, respectively, with parameter ranges generated by the four methods spanning 1.13–1.78 and 0.57–1.61. Significant errors were observed in the parameters generated by the GAN and DDPM, particularly with the GAN showing substantial discrepancies in the generated trace length.

The Pearson correlation coefficient between the dip direction and dip angle of the measured structural planes is 0.361, and the Pearson correlation coefficient between the dip direction and dip angle of the structural planes generated by the Monte Carlo method is 0.09, which indicates that the correlation has not been captured effectively. The Pearson correlation coefficients between the dip direction and dip angle of the structural planes generated by the Copula-based method, the GAN, and the DDPM are 0.326, 0.351, and 0.271, respectively. These values closely approximate the Pearson correlation coefficient between the dip direction and dip angle of the measured factual structural planes. However, upon examining the scatter plots illustrating data distribution trends, it is evident none of the data distribution trends in the dip direction and dip angle of the structural planes generated by these three methods are particularly close to the data distribution trends of the measured factual structural planes.

3.3.3. Generated Results of Dataset 3

This dataset includes 257 pieces of structural planes.

Table 5. Means and variances of the parameters of the structural planes generated by the four methods compared with those of the measured factual structural planes.

	Dip Direction (°)			Dip Angle (°)			Trace Length (m)
	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation
Measured factual	Normal	217.12	7.84	Normal	38.13	4.58	Log-normal	6.17	3.24
Monte Carlo method	Normal	217.23	8.27	Normal	38.05	4.78	Log-normal	5.88	2.64
Copula-based method	Normal	218.22	7.85	Normal	37.45	4.52	Log-normal	5.94	2.72
GAN	Normal	216.87	5.62	Normal	38.06	4.27	Log-normal	6.21	3.14
DDPM	Normal	217.54	8.49	Normal	37.70	5.52	Log-normal	6.39	3.80

illustrates a comparative analysis of the mean and variance of each parameter generated by the four methods with the measured factual structural planes. Figure 22 presents the results of the histogram comparison between the dip direction of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 23 presents the results of the histogram comparison between the dip angle of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 24 presents the results of the histogram comparison between the trace length of the structural planes generated by the four methods and those of the measured factual structural planes. Finally, Figure 25 displays the linear regression analysis between dip direction and dip angle of the structural planes generated by the four methods.

An analysis of Table 5 reveals that the mean and variance of the dip direction are 217.12 and 7.84, respectively, with parameter ranges generated by the four methods spanning 216.87–218.22 and 5.62–8.49. The mean and variance of the dip angle are 38.13 and 4.58, respectively, with parameter ranges generated by the four methods spanning 37.45–38.06 and 4.27–5.52. The mean and variance of the trace length are 6.17 and 3.24, respectively, with parameter ranges generated by the four methods spanning 5.88–6.39 and 2.64–3.8. In conjunction with the probability distribution diagrams, the probability distributions of the parameters of the structural planes generated by the four methods are close to the probability distributions of the parameters of the measured factual structural planes.

The measured factual structural planes show a negative correlation between the dip direction and dip angle with a Pearson correlation coefficient of −0.252. The Pearson correlation coefficient between the dip direction and dip angle of the structural planes generated by the Monte Carlo method is 0.102, which indicates that the correlation has not been captured effectively. The Pearson correlation coefficients between the dip direction and dip angle of the structural planes generated by the Copula-based method, the GAN, and the DDPM are −0.351, −0.239, and −0.225, respectively. These values closely approximate the Pearson correlation coefficient between the dip direction and dip angle of the measured factual structural planes. Combined with the data distribution trends observed in the scatter plots, it can be seen that the correlation distribution trends generated by the Copula-based method and DDPM are closer to those of the measured factual structural planes.

3.3.4. Generated Results of Dataset 4

The fourth set of structural planes is the second set of structural planes in the Tunhovd study area. This dataset includes 325 pieces of structural planes.

Table 6. Means and variances of the parameters of the structural planes generated by the four methods compared with those of the measured factual structural planes.

	Dip Direction (°)			Dip Angle (°)			Trace Length (m)
	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation
Measured factual	Normal	80.50	6.35	Normal	54.63	4.04	Log-normal	2.25	1.41
Monte Carlo method	Normal	80.1	6.34	Normal	54.39	4.20	Log-normal	2.31	1.43
Copula-based method	Normal	76.69	6.49	Normal	54.72	4.13	Log-normal	2.24	1.14
GAN	Normal	80.73	5.70	Normal	54.70	3.06	Log-normal	2.22	1.03
DDPM	Normal	81.63	6.12	Normal	54.71	3.83	Log-normal	2.35	1.53

illustrates a comparative analysis of the mean and variance of each parameter generated by the four methods with the measured factual structural planes. Figure 26 presents the results of the histogram comparison between the dip direction of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 27 presents the results of the histogram comparison between the dip angle of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 28 presents the results of the histogram comparison between the trace length of the structural planes generated by the four methods and those of the measured factual structural planes. Finally, Figure 29 presents the linear regression analysis between dip direction and dip angle of the structural planes generated by the four methods.

An analysis of Table 6 reveals that the mean and variance of the dip direction are 80.5 and 6.35, respectively, with parameter ranges generated by the four methods spanning 76.69–81.63 and 5.7–6.49. The mean and variance of the dip angle are 54.63 and 4.04, respectively, with parameter ranges generated by the four methods spanning 54.39–54.72 and 3.06–4.20. The mean and variance of the trace length are 2.25 and 1.41, respectively, with parameter ranges generated by the four methods spanning 2.22–2.35 and 1.03–1.53. The data generated by the GAN exhibit some level of error. The probability distributions of the parameters of the structural planes generated by the other three methods are close to the probability distributions of the parameters of the measured factual structural planes.

The Pearson correlation coefficient between the dip direction and dip angle of the measured structural planes exhibits a negative correlation with a Pearson correlation coefficient of −0.342. The Pearson correlation coefficient between the dip direction and dip angle of the structural planes generated by the Monte Carlo method is −0.048, which indicates that the correlation has not been captured effectively. The Pearson correlation coefficients between the dip direction and dip angle of the structural planes generated by the Copula-based method, the GAN, and the DDPM are −0.383, −0.265, and −0.321, respectively. These values closely approximate the Pearson correlation coefficient between the dip direction and dip angle of the measured factual structural planes. Combined with the data distribution trends observed in the scatter plots, it can be seen that the correlation distribution trends generated by the Copula-based method and DDPM are more consistent with the correlation of the measured factual structural planes.

3.3.5. Generated Results of Dataset 5

This dataset is the largest, containing 766 pieces of structural planes.

Table 7. Means and variances of the parameters of the structural planes generated by the four methods compared with those of the measured factual structural planes.

	Dip Direction (°)			Dip Angle (°)			Trace Length (m)
	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation	Distribution Type	Mean Value	Standard Deviation
Measured factual	Normal	258.78	12.46	Normal	36.03	9.02	Log-normal	3.95	2.80
Monte Carlo method	Normal	258.78	12.44	Normal	36.02	9.01	Log-normal	3.85	1.67
Copula-based method	Normal	259.30	12.14	Normal	35.74	8.96	Log-normal	4.06	1.80
GAN	Normal	259.77	11.23	Normal	35.38	8.14	Log-normal	3.88	2.77
DDPM	Normal	258.59	11.94	Normal	35.53	8.41	Log-normal	4.57	2.92

illustrates a comparison of the mean and variance of each parameter generated by the four methods against the measured factual structural planes. Figure 30 presents the results of the histogram comparison between the dip direction of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 31 presents the results of the histogram comparison between the dip angle of the structural planes generated by the four methods and those of the measured factual structural planes. Figure 32 presents the results of the histogram comparison between the trace length of the structural planes generated by the four methods and those of the measured factual structural planes. Finally, Figure 33 presents the linear regression analysis between dip direction and dip angle of the structural planes generated by the four methods.

An analysis of Table 7 reveals that the mean and variance of the dip direction are 258.78 and 12.46, respectively, with parameter ranges generated by the four methods spanning 258.59–259.77 and 11.23–12.44. The mean and variance of the dip angle are 36.03 and 9.02, respectively, with parameter ranges generated by the four methods spanning 35.38–36.02 and 8.14–9.01. The mean and variance of the trace length are 3.95 and 2.80, respectively, with parameter ranges generated by the four methods spanning 3.85–4.57 and 1.67–2.92. The probability distributions of the parameters of the structural planes generated by the four methods are close to the probability distributions of the parameters of the measured factual structural planes.

The Pearson correlation coefficient between the dip direction and dip angle of the measured structural planes is −0.685, indicating a strong negative correlation. The Pearson correlation coefficient between the dip direction and dip angle of the structural planes generated by the Monte Carlo method is −0.059, which indicates that the correlation has not been captured effectively. The Pearson correlation coefficients between the dip direction and dip angle of the structural planes generated by the Copula-based method, the GAN, and the DDPM are −0.664, −0.610, and −0.646, respectively. These values closely approximate the Pearson correlation coefficient between the dip direction and dip angle of the measured factual structural planes. Combined with the data distribution trends observed in the scatter plots, it can be seen that the correlation distribution trends generated by the GAN and DDPM are more consistent with the correlation of the measured factual structural planes.

3.4. Comparative Analysis

The five sets of structural planes utilized in the experiment were arranged in ascending order according to the data size, exhibiting both positive and negative correlations between the dip direction and dip angle. The dip direction and dip angle of structural planes in Dataset 5 exhibited the most significant correlation, whereas the dip direction and dip angle of structural planes in other sets also displayed correlation but with noticeable dispersion. The validation of the results involved employing probability distribution histograms to compare the distribution of individual parameters of the generated structural planes by the four methods with that of the measured factual structural planes. Additionally, scatter plots were utilized to observe trends in the distribution of the dip direction and dip angle of the generated structural planes. Moreover, quantitative calculations of the mean and variance of individual parameters of the generated structural planes were performed for each dataset to ensure accuracy.

In the case of the Monte Carlo method, the accuracy of the probability distribution for each parameter of the generated structural planes is independent of the data quantity and remains consistently high. However, the absolute values of the Pearson correlation coefficients between the dip direction and dip angle of the generated structural planes range between 0 and 0.1 regardless of the inherent strength of the correlations of the measured factual structural planes, indicating that the Monte Carlo method fails to capture these correlations effectively.

For the Copula-based method, the accuracy of the probability distribution for each parameter of the generated structural planes is independent from data quantity and remains consistently high. In terms of capturing correlations, this method can effectively capture correlations between the dip direction and dip angle of the measured factual structural planes when the dataset is limited. However, the precision of these captured correlations may be compromised due to the small dataset and the dispersed nature of the data. When the dataset is sufficiently large, the accuracy and precision of the captured correlations improve significantly.

For deep generative models, data quantity significantly impacts their performance. When the dataset is small, there is a notable gap in the accuracy of the probability distribution of each parameter of the generated structural planes and the captured correlations compared to the Monte Carlo method and Copula-based method. As the data volume increases, both the accuracy of the probability distribution of each parameter of the generated structural planes and the captured correlations become more accurate. When the dataset is sufficiently large, deep generative models can fully exploit their capabilities to capture complex hidden patterns among the data. Among the two generative models examined, DDPM outperforms GAN. In the presence of noise, DDPM is more capable than GAN at capturing the original features and validity of the sample data. Consequently, the data generated by DDPM is more consistent with the sample data and exhibits higher generalization ability. In summary, the Monte Carlo method performs poorly compared to the other three methods, failing to capture the correlation between dip direction and dip angle. The Copula-based method performs better than the GAN and DDPM methods when the dataset is small. However, as the amount of data increases, GAN and DDPM show superior performance in capturing the correlation between inclination and dip angle and learning the probability distribution of real structural planes.

4. Discussion

This study compares the effectiveness of four stochastic structural plane generation methods—the Monte Carlo method, the Copula-based method, GAN, and DDPM—in generating stochastic structural planes and capturing potential correlations between structural plane parameters.

The structural plane sets utilized in the experiments were obtained from different research areas, each exhibiting significant variations in data volume and characteristics. The analysis of the results from these distinct research areas indicates that the Monte Carlo method consistently plays a crucial role in accurately learning the probability distributions of structural plane parameters regardless of the variations in the characteristics of the research areas. However, it fails to capture the correlation between the dip direction and dip angle. In the Valle research area, the limited data volume and scattered correlation patterns resulted in the suboptimal performance of the other three methods on this dataset. At the gradual increase in the data volume of datasets in the other research areas, the performance of these methods improves correspondingly. The Oernlia research area, characterized by the largest data volume and the most pronounced correlation between the dip direction and dip angle, demonstrated that the other three methods excel in both learning the probability distribution of parameters and capturing the correlations.

Upon further analysis of the performance of the four methods, it is evident that compared to the other three methods, the Monte Carlo method exhibits limitations in its inability to capture the correlation between the dip direction and dip angle. The Copula-based method performs better compared to the GAN and DDPM methods when the dataset is small. However, as the amount of data increases, GAN and DDPM exhibit superior performance in capturing the correlation between the dip direction and dip angle and in learning the probability distribution of the measured factual structural planes. Next, we will provide a detailed discussion.

4.1. Performance Analysis of the Statistical Methods

The Monte Carlo method performs poorly compared to the other three methods, failing to capture the correlation between dip direction and dip angle. The traditional Monte Carlo method is a well-established approach for generating stochastic structural planes. It operates under the assumption that the data adhere to a specific prior distribution. After determining the distribution of the function, the corresponding formula parameters are calculated from the sample data. This method is not influenced by the quantity of measured factual structural planes but is contingent upon the clarity of the probability distribution for each parameter of the measured factual structural planes. The precision of the generated results does not improve with an increase in the data size. Its inability to capture the correlation between parameters is also due to the principle that it samples independently based on the probability distribution of each parameter and that the individual parameters are unconnected during the generation process.

The Copula-based method successfully captures the correlation between the dip direction and dip angle. The initial four datasets exhibit small sizes and relatively weak correlations and the correlations captured by the Copula-based method do not precisely match those of the measured factual structural planes. Given that the Copula-based method is dependent on the correlation between two variables, a weak correlation can lead to imprecise calculations of correlation coefficients between the variables, resulting in an unclear dependence structure between them. In this case, the Copula-based method may rely more on the characteristics of the marginal distribution than on the correlation between the two variables. As the dataset size increases and the correlation strengthens, the Copula-based method generates data with increasing accuracy.

4.2. Performance Analysis of the Deep Generative Models

The correlations captured by the deep generative model exhibit discrepancies with the correlations observed in the measured factual structural planes when the dataset is limited in size. This difference can be attributed to the substantial influence that the quantity of training samples exerts on the performance of deep learning models.

Deep learning models utilize deep neural networks to learn data representations and complete the learning process according to an objective function. The effectiveness of a deep learning model is highly dependent on the quality of the data, as ‘invalid inputs’ often lead to ‘invalid outputs’ [58]. A neural network consists of a series of neurons that act as basic units. Each neuron is a function that performs a nonlinear mapping of inputs, incorporating trainable parameters such as weights and biases. Deep neural networks contain multiple hidden layers with various numbers of neurons in each layer, enabling more complex learning. However, this complexity necessitates a large amount of data to train these parameters effectively. With a small number of training samples, the training data may not be representative, preventing the model from adequately learning and generalizing. This can result in phenomena such as overfitting, which can limit the performance of the neural network and reduce model accuracy. As the amount of data increases, neural networks perform better.

The two generative models generally exhibit similar performance patterns in generating stochastic structural planes. When the dataset is small, there is a notable gap in the accuracy of the probability distribution of each parameter of the generated structural planes and the captured correlations compared to the Monte Carlo method and the Copula-based method. When the amount of data is large enough, the performance of the deep generative model is fully exploited to capture the hidden complex patterns among the parameters. Among the two generative models, DDPM performs better compared to GAN.

GAN is dominant in the field of generation and is the most widely used generative model. GAN can generate high-quality data, but the high quality is achieved through sophisticated parameter selection. GAN is extremely parameter-sensitive during training and is prone to pattern crashes. In addition, the limited amount of structural plane data available can lead to overfitting in GAN, thereby constraining their efficiency. In contrast, DDPM employs a gradual denoising generation process, which mitigates the risk of pattern collapse and enhances stability and controllability. DDPM exhibits reduced sensitivity to parameters during training and offers a simpler training process compared to GAN. Therefore, DDPM is more suitable for generating stochastic structural planes than GAN.

4.3. Comparative Analysis of Statistical Methods and Deep Generative Models

The Monte Carlo method fails to capture the correlation between the dip direction and dip angle. Both the Copula-based method and deep generative models successfully capture the correlation between the dip direction and dip angle. Although the Copula model has a certain amount of error when the amount of data is small, the correlations captured by the Copula-based method are more consistent with the correlations of the measured factual structure planes compared to the correlations captured by the deep generative model. When the dataset size is considerable, both the Copula-based method and the deep generative model demonstrate strong performance, accurately generating probability distributions for the parameters of the structural planes. However, the deep generative model excels in capturing the complex patterns inherent in the structural planes more effectively than the Copula-based method and achieves a correlation between the dip direction and dip angle that is closer to that of the measured structural planes.

The core concept of the Copula-based method is to couple the marginal distributions of multiple variables with a Copula-based method. This involves transforming marginal variables into uniformly distributed variables and then defining the correlation as a joint distribution over this uniform distribution, with the required parameters being the correlation coefficients between the variables. While this method simplifies the procedure, it also imposes certain constraints. In the Oernlia study area, the volume of the structural plane dataset is larger compared to other study areas. The data did not exhibit a typical univariate linear correlation, instead demonstrating a more complex correlation pattern. The results generated by the two deep generative models performed better than those generated by the Copula-based method. The two deep generative models effectively captured the underlying distributional properties of the data through a series of non-linear transformations, enabling them to generate new samples that adhere to the same distributional patterns. This capability allowed the models to more accurately capture the correlation trends between the dip direction and dip angle of the measured structural planes. Consequently, the deep generative models performed better than the Copula-based method in handling complex distribution patterns.

4.4. Future Work

In the future, we plan to integrate the Copula-based method with the deep generative model to capitalize on the respective advantages of both methods. This combination is intended to achieve more accurate data generation and better capture the correlation between structural plane parameters, irrespective of the data size. We also consider further applying these methods to the field of geohazard analysis. For example, employing deep generative models to interpolate short-term missing data in landslides can address issues of data sparsity and missing data in datasets, thereby improving the accuracy of landslide hazard predictions [22]. In addition, these methods can be applied to the field of Earth digital twins, coupling the uncertainty in the dynamics of the Earth system itself, combining observational data, predicting future conditions, and continuously performing high-precision, real-time, highly detailed, and more realistic Earth system simulations.

5. Conclusions

Structural planes are one of the key factors controlling the stability of rock masses. A comprehensive understanding of the spatial distribution characteristics of structural planes is essential for accurately identifying key blocks, analyzing rock mass stability, and addressing various rock engineering challenges. This study compares the effectiveness of four stochastic structural plane generation methods—the Monte Carlo method, the Copula-based method, GAN, and DDPM—in generating stochastic structural planes and capturing potential correlations between structural plane parameters. Five different sets of structural planes on four slopes, Valle, Tunhovd, Straumklumpen, and Oernlia in Norway, are examined as an example. The validation and analysis were performed using probability distribution histograms, data feature comparison, scatter plots, and linear regression analysis. The results indicate the following:

(1) The probability distributions of structural plane parameters generated by the Monte Carlo method are consistent with that of the measured factual structural planes but fail to capture the correlation between the dip direction and dip angle.

(2) When the dataset size is limited and the correlation is weak, the correlation between the dip direction and dip angle generated by the Copula-based method cannot fully fit that of the measured factual structural planes, but it exhibits a closer fit to the measured factual structural planes compared to those generated by the deep generative model.

(3) Both the Copula-based method and deep generative model can accurately generate parameter probability distributions when the dataset size is considerable, but the deep generative model is superior in capturing the complex patterns inherent in the structural plane.

In summary, in the case of a limited number of measured structural planes, the Copula-based method is recommended for stochastic structural plane generation. In contrast, in the case of a sufficiently large number of datasets, the deep generative model is recommended due to its ability in capturing complex data structures. In addition, DDPM is more recommended than GAN, and its training process is relatively stable, less prone to pattern collapse, and performs better in generation. The results of this study can serve as a valuable reference for the accurate generation of stochastic structural planes within rock masses.