Vector Decomposition of Elastic Seismic Wavefields Using Self-Attention Deep Convolutional Generative Adversarial Networks

Abstract

Vector decomposition of P- and S-wave modes from elastic seismic wavefields is a key step in elastic reverse-time migration (ERTM) to effectively improve the multi-wave imaging accuracy. Most previously developed methods based on the apparent velocities or the polarization characteristics of different wave modes are unable to accurately achieve the vector decomposition of P- and S-wave modes. To effectively overcome the shortcomings of conventional methods, we develop a vector decomposition method of P- and S-wave modes using self-attention deep convolutional generative adversarial networks (SADCGANs) to effectively separate the horizontal and vertical components of P- and S-wave modes from elastic seismic wavefields and accurately preserve their amplitude and phase characteristics for isotropic elastic media. For an elastic model, we use many time slices for a given source position to train the neural network, and use other time slices not in this training dataset to test the neural network. Numerical examples of different models demonstrate the effectiveness and feasibility of our developed method and indicate that it provides an effective intelligent data-driven vector decomposition method of P- and S-wave modes.

1. Introduction

Elastic seismic wavefields contain different wave modes, such as converted, P-, and S-waves, and therefore ERTM can provide multi-wave imaging results including PP, PS, SP and SS depth images [1,2], which is helpful to effectively reduce the ambiguity of multicomponent seismic data interpretation. Because the cross-correlation imaging calculation of coupled P- and S-wave modes will bring about some crosstalk artifacts in imaging results [3,4], ERTM usually requires one to accurately separate the different wave modes from the coupled elastic seismic wavefields before utilizing the cross-correlation imaging conditions [5,6].

There are three frequently used methods to achieve the separation and decomposition of P- and S-wave modes from the coupled elastic seismic wavefields. The first approach is the Helmholtz decomposition algorithm which can be theoretically regarded as applying the divergence and curl operators to, respectively, obtain the decoupled P- and S-wave modes from the coupled elastic seismic wavefields [7,8], but it has two serious deficiencies which will affect the multi-wave imaging accuracy of ERTM: one is that it will seriously damage the amplitude and phase characteristics of the decoupled P- and S-wave modes [9,10], and the other is that the corresponding PS and SP images will suffer from polarity reversals [11]. The second approach is the vector decomposition algorithm based on the decoupled elastic wave equations which can effectively simulate the decoupled vector P- and S-wave modes for a given elastic model at different times by discretely solving these decoupled wave equations [12,13,14,15,16,17], but it can only obtain the good vector decomposition results of different wave modes when this elastic model is sufficiently smoothed, because adopting the true elastic models to model the decomposed vector P- and S-wave modes will give rise to obvious artifacts at the strong reflectors when using these decoupled elastic wave equations [3,17,18]. The third approach is the vector decomposition algorithm in the wavenumber domain which can effectively preserve all the components of the decoupled vector P- and S-wave modes and has its advantages in terms of amplitude and phase accuracy [19,20], but its computational efficiency is poor because of the extra forward and inverse Fourier transforms. Although these above conventional approaches can achieve the separation and decomposition of P- and S-wave modes to a certain extent, they are usually dependent on accurate elastic model parameters and certain prior conditions and therefore have obvious limitations in practical application.

In recent years, the application of deep learning methods [21] in the fields of geophysics and applied geophysics has received great attention and promising achievements, such as detecting faults [22,23], classifying facies [24,25], attenuating noise [26,27], picking first arrivals [28,29], building velocity models [30,31] and reconstructing seismic data [32,33]. Inspired by deep learning methods, some scholars have proposed many effective separation and decomposition methods of P- and S-wave modes from the coupled elastic seismic wavefields based on different neural networks, such as multi-task learning [34], convolutional neural networks (CNNs) [35,36], generative adversarial networks (GANs) [37,38] and deep convolutional neural networks (DCNNs) [39], and these methods are intelligent data-driven algorithms for the separation and decomposition of P- and S-wave modes which are not dependent on elastic model parameters and certain prior conditions. However, these above methods mainly use the corresponding neural networks to separate two decoupled scalar P- and S-wave modes from the coupled elastic seismic wavefields, and therefore cannot obtain all the horizontal and vertical components of the decomposed vector P- and S-wave modes. It is worth noting that using deep learning methods to effectively achieve the vector decomposition of coupled elastic seismic wavefields has several key factors, such as establishing an appropriate neural network and generating the corresponding training dataset.

In this paper, we develop a vector decomposition method of P- and S-wave modes using SADCGANs to accurately separate all the horizontal and vertical components of P- and S-wave modes from the coupled elastic seismic wavefields, and this intelligent data-driven method can accurately preserve the amplitude and phase characteristics of the decomposed vector P- and S-wave modes. This developed method emphasizes establishing an appropriate neural network, generating an effective training dataset and training and testing this neural network. Firstly, we establish an effective framework based on SADCGANs for different wave mode decompositions. Secondly, we generate a large number of training datasets for different elastic models using the elastic wave equations, including many time slices for a given source position, and then we feed them to the neural network to train this neural network. After training, we use all remaining time slices for this source position and all time slices for other source positions to test this neural network. Numerical examples of different models demonstrate the effectiveness and feasibility of our developed method.

2. Methods

2.1. Wave Mode Decomposition Theory

For two-dimensional (2D) isotropic elastic media, the coupled elastic seismic wavefields ($u_x,u_z$) can be decomposed into the horizontal and vertical components of P- and S-wave modes ($u_{xP},u_{zP},u_{xS},u_{zS}$), which can be expressed as

$$u_x=u_{xP}+u_{xS},u_z=u_{zP}+u_{zS}$$

(1)

The essence of vector decomposition of P- and S-wave modes is to redistribute the energy from the coupled elastic seismic wavefields based on the waveform differences of different wave modes, which can be theoretically regarded as a point-to-point prediction problem. To effectively solve this nonlinear prediction problem, deep convolutional generative adversarial networks (DCGANs) [40] can be used to develop the intelligent data-driven vector decomposition method of P- and S-wave modes. In other words, the vector decomposition method of different wave modes based on DCGANs belongs to an intelligent image processing algorithm and therefore is not dependent on elastic model parameters and certain prior conditions compared with the conventional algorithms. According to Equation (1), the goal of vector decomposition is to accurately separate four decoupled wavefield components from two coupled wavefield components using DCGANs, and therefore its network structure should ideally have two channels for inputting the horizontal and vertical components of coupled elastic seismic wavefields ($u_x,u_z$) and four channels for outputting the horizontal and vertical components of the decoupled vector P- and S-wave modes ($u_{xP},u_{zP},u_{xS},u_{zS}$). However, the network structure is relatively complex according to this above strategy which may lead to an unstable training process, and therefore we use the workflow shown in Figure 1 for the vector decomposition of P- and S-wave modes to obtain better training results. From Figure 1, the network structure of DCGANs has only two channels to output the horizontal and vertical components of the decoupled vector P-wave mode ($u_{xP},u_{zP}$) after predicting the wave mode of each point for input samples, which can be expressed as

$$\left(u_{xP},u_{zP}\right)=f\left[\left(u_x,u_z\right);\theta \right]$$

(2)

where $f(·)$ is the DCGANs, and $\theta$ represents its network hyperparameters. By reducing the output channels of the neural network, we can simplify the network structure as much as possible and further stabilize the training process. The horizontal and vertical components of the decoupled vector S-wave mode can finally be obtained through the strategy of wavefield subtraction, which can be written as

$$u_{xS}=u_x-u_{xP},u_{zS}=u_z-u_{zP}$$

(3)

2.2. Network Structure of SADCGANs

In addition to the input and output channels, the network structure of DCGANs has two different models (Figure 1): a generator (G) and a discriminator (D). The generator is mainly used to generate the approximately real samples to deceive the discriminator as much as possible, and the discriminator is mainly used to accurately classify whether the input samples come from the real samples or the generated samples [41]. In other words, there is a competitive relationship between the generator and the discriminator, and their competition until the latter cannot distinguish between the generated samples and the corresponding real samples [42]. In our work, the generator is used to generate the horizontal and vertical components of the decoupled vector P-wave mode ($u_{xP},u_{zP}$) by inputting the coupled elastic seismic wavefields ($u_x,u_z$), and the discriminator is used to determine whether two generated vector P-wave modes ($u_{xP},u_{zP}$) are real. Due to the limitation of the size of the receptive field of a fixed-size convolution kernel, the convolution kernel, which can be regarded as a small patch, usually only acquires the local features of input samples; in other words, the global features of input samples will be lost for the conventional DCGAN-based methods. To effectively solve this problem, self-attention [43] is incorporated into the conventional DCGANs in our work and therefore our developed method can be called a self-attention deep convolutional generative adversarial network (SADCGAN). By introducing the self-attention modules into the generator and the discriminator, SADCGANs can simultaneously acquire the local and global features of input samples, which can greatly improve the training process of the neural network and further generate more accurate wavefield decomposition results.

The working principle of the self-attention module is shown in Figure 2. Firstly, the feature maps $x$ are transformed into the feature spaces $f\left(x\right)$, $g\left(x\right)$ and $h\left(x\right)$ using three 1 × 1 convolutions, and they can be written as

$$f\left(x\right)=W_{f}x,g\left(x\right)=W_{g}x,h\left(x\right)=W_{h}x$$

(4)

where $W_f$, $W_g$ and $W_h$ are the weight matrix of the convolutional layer.

Secondly, the attention feature map ${\beta }_{j,i}$ can be obtained by multiplying the transpose of $f\left(x\right)$ by $g\left(x\right)$ and then normalizing the corresponding result using the softmax function, which can be expressed as

$${\beta }_{j,i}=\frac{\exp \left(s_{ij}\right)}{{\displaystyle \sum _{i=1}^N\exp \left(s_{ij}\right)}},s_{ij}=f{\left(x_i\right)}^{T}g\left(x_j\right)$$

(5)

Thirdly, the global feature map $o_j$ can be obtained by multiplying the attention feature map ${\beta }_{j,i}$ by $h\left(x\right)$ and then dealing with the corresponding result using a 1 × 1 convolution, which can be expressed as

$$o_j=v\left({\displaystyle \sum _{i=1}^N{\beta }_{j,i}h\left(x_i\right)}\right),v\left(x_i\right)=W_v{x}_i$$

(6)

where $W_v$ is also the weight matrix of the convolutional layer.

Finally, the final output $y_i$ of the self-attention module can be obtained, and the corresponding expression can be described as

$$y_i=\gamma o_i+x_i$$

(7)

where $\gamma$ is a transition parameter with an initial value of 0, and it can effectively achieve the feature acquisition of local and global space positions.

The network structures of the generator and discriminator for SADCGANs, which are used in our work, are shown in Figure 3. According to Figure 3, the generator has 24 transposed convolutional layers to generate two samples of the decoupled vector P-wave mode ($u_{xP},u_{zP}$) from two samples of coupled elastic seismic wavefields ($u_x,u_z$), and uses the ReLU activation function and the spectral normalization for all layers except its output layer which uses the Tanh activation function to effectively improve the training dynamics of its neural network. In addition, two self-attention modules are, respectively, added to the first two layers of the generator to effectively extract the global features of the input samples. The discriminator has 8 convolutional layers to accurately distinguish between the generated samples and the corresponding real samples, and uses the LeakyReLU activation function and spectral normalization for all layers except its output layer to effectively improve the training dynamics of its neural network. Additionally, two self-attention modules are, respectively, added to the sixth and seventh layers of the discriminator to effectively extract the global features of the input samples.

Stability is a key consideration factor in the training process of neural networks. In the practical training process, the loss function of conventional DCGANs is often unable to stabilize the training process. To effectively overcome this problem and further obtain better training results, the loss function based on Wasserstein distance [44] which is used in our work can be expressed as

$$L=\underset{\tilde{x}\sim p_{\tilde{x}}}{E}\left[D\left(\tilde{x}\right)\right]-\underset{x\sim p_x}{E}\left[D\left(x\right)\right]+\lambda \underset{\hat{x}\sim p_{\hat{x}}}{E}\left[{\left({\Vert {\nabla }_{\hat{x}}D\left(\hat{x}\right)\Vert }_2-1\right)}^2\right]$$

(8)

where ($x,\tilde{x},\hat{x}$) are, respectively, the real, generated and interpolated samples; ($p_x,p_{\tilde{x}},p_{\hat{x}}$) are their corresponding distributions; and $\lambda$ is the penalty coefficient. Additionally, the interpolated samples meet the following condition $\hat{x}=\epsilon x+\left(1-\epsilon \tilde{x}\right)$, and $\epsilon$ is a coefficient between 0 and 1. The improved loss function shown in Equation (8) can be called the Wasserstein GAN with gradient penalty (WGAN-GP) [44]. Compared with the conventional loss functions, WGAN-GP can more effectively stabilize the training process of SADCGANs and generate higher-quality samples.

2.3. Training Dataset and Hyperparameters

Using the SADCGANs to achieve the vector decomposition of P- and S-wave modes requires preparing a set of training data for training the neural network. In our work, we first simulate the coupled elastic seismic wavefields at different times for a given source position through forward modeling based on the elastic wave equations for different isotropic elastic models, and then randomly select some of these coupled elastic seismic wavefields. Next, we use the horizontal and vertical components of the decoupled vector P- and S-wave modes for these selected elastic seismic wavefields separated by the vector decomposition algorithm in the wavenumber domain as training samples, and further use those unselected elastic seismic wavefields and the elastic seismic wavefields for other source positions as testing samples. During training, the coupled elastic seismic wavefields ($u_x,u_z$) are fed as input samples to the generator, which outputs the generated horizontal and vertical components of the decomposed vector P-wave mode ($u_{xP},u_{zP}$). Next, the discriminator compares two generated components with two real components for vector P-wave mode, and it outputs a value between 0 and 1 that indicates the probability of generated samples being real. Once the discriminator determines that the generated samples are real, the horizontal and vertical components of the decomposed vector P-wave mode ($u_{xP},u_{zP}$) can be outputted, and the horizontal and vertical components of the decomposed vector S-wave mode ($u_{xS},u_{zS}$) can be obtained using Equation (3).

Our work’s goal is to effectively and accurately separate the horizontal and vertical components of the decoupled vector P- and S-wave modes from the coupled elastic seismic wavefields. To better train the neural network, the appropriate hyperparameters are critical considerations. In our work, we carry out many experiments to select the optimal hyperparameters, including training epochs and learning rate. After repeated experiments, the Adam optimizer with 400 training epochs and a learning rate of 0.0002 is used to train the neural network, and the learning rate remains equal for the first 200 epochs and then gradually decreases to zero over the next 200 epochs.

3. Numerical Experiments

In this section, we verify and test the effectiveness, feasibility and adaptability of our developed method using a simple isotropic elastic model (Section 3.1) and the Hess isotropic elastic model (Section 3.2). During the experiments of different isotropic elastic models, we compare our developed method with two conventional methods, including the Helmholtz decomposition algorithm and the vector decomposition algorithm in the wavenumber domain.

3.1. Simple Isotropic Elastic Model

We first use a simple isotropic elastic model, which is shown in Figure 4, to demonstrate the effectiveness and feasibility of our developed method, and the density of this model is a constant in our work. During the training process of this model, we first define an explosive source located at ($2.56 \mathrm{km},0.02 \mathrm{km}$) of this model which can be mainly represented by P-wave incident energy and simulate the coupled elastic seismic wavefields at different times by discretely solving the elastic wave equations, and then we separate the vector P- and S-wave modes from these coupled wavefields using the vector decomposition algorithm in the wavenumber domain. Finally, we randomly select fifty pairs of time slices from these coupled elastic wave modes and the corresponding decoupled vector P-wave modes for training our neural network, and all remaining time slices for this given source position are used for testing our neural network. Additionally, we randomly crop 100 patches with a size of 256 × 256 for each training sample, and therefore we have a total of 5000 training samples in the training dataset, and the batch size is defined as 32.

The decomposition results of P- and S-wave modes from the coupled elastic seismic wavefields at $t=0.9 \mathrm{s}$, which are not included in training dataset, using different methods are shown in Figure 5. The horizontal and vertical components of the coupled elastic seismic wavefields are shown in Figure 5a,b, the decomposed P- and S-wave modes using the Helmholtz decomposition algorithm are shown in Figure 5c,d, the horizontal and vertical components of the decomposed vector P- and S-wave modes using the wavenumber domain decomposition algorithm are shown in Figure 5e–h, and the corresponding horizontal and vertical components of the decomposed vector P- and S-wave modes using our developed method are shown in Figure 5i–l. The corresponding wavenumber spectrums of these time slices (Figure 5a–l) are, respectively, shown in Figure 6a–l. From Figure 5 and Figure 6, the Helmholtz decomposition algorithm can generate the decomposed P- and S-wave modes from the coupled elastic seismic wavefields, but it cannot generate all components of the decomposed vector P- and S-wave modes and will damage their amplitude and phase characteristics, which will seriously affect the imaging quality of different wave modes from ERTM. The wavenumber domain decomposition algorithm can effectively generate the horizontal and vertical components of the decomposed vector P- and S-wave modes from the coupled elastic seismic wavefields and accurately preserve the amplitude and phase characteristic of different wave modes, but there is still some weak residual energy of other wave modes in these decomposed wave modes. Our developed method can accurately generate the horizontal and vertical components of the decomposed vector P- and S-wave modes from the coupled elastic seismic wavefields which are similar to the decomposition results separated by the vector decomposition algorithm in the wavenumber domain, which verifies the correctness and effectiveness of the intelligent data-driven vector decomposition method using SADCGANs. To compare the performance of different methods more intuitively, we further extract and compare the amplitude of the decomposed P- and S-wave modes (Figure 5c–l) at $\mathrm{depth}=1.28 \mathrm{km}$ which are shown in Figure 7a–d. From Figure 7, we can draw the same conclusion: our developed method can effectively achieve the vector decomposition of P- and S-wave modes from the coupled elastic seismic wavefields and accurately preserve the amplitude and phase characteristic of different wave modes which are similar to the wavenumber domain decomposition algorithm, whereas the Helmholtz decomposition algorithm cannot achieve this goal.

To test the adaptability of our neural network, the time slices for another source position located at ($2.56 \mathrm{km},1.28 \mathrm{km}$) of this simple isotropic elastic model are also used to test our neural network. It is worth noting that all the time slices for this source position simulated by the elastic wave equations are not included in the training dataset. The horizontal and vertical components of the coupled elastic seismic wavefields at $t=0.4 \mathrm{s}$ are shown in Figure 8a,b, and then we separate the P- and S-wave modes from the coupled wavefields using different methods. The decomposed P- and S-wave modes using the Helmholtz decomposition algorithm are shown in Figure 8c,d, the horizontal and vertical components of the decomposed vector P- and S-wave modes using the wavenumber domain decomposition algorithm are shown in Figure 8e–h, and the corresponding horizontal and vertical components of the decomposed vector P- and S-wave modes using our developed method are shown in Figure 8i–l. The corresponding wavenumber spectrums of these time slices (Figure 8a–l) are, respectively, shown in Figure 9a–l. From Figure 8 and Figure 9, even if the source position changes, our developed method can still effectively and accurately generate the horizontal and vertical components of the decomposed vector P- and S-wave modes from the coupled elastic seismic wavefields which are similar to those of the vector decomposition algorithm in the wavenumber domain, which reveals that our developed method also has good adaptability when the source position changes. To compare the performance of different methods more clearly, the amplitude comparisons of the decomposed P- and S-wave modes (Figure 8c–l) at $\mathrm{depth}=2.0 \mathrm{km}$ are shown in Figure 10a–d. From Figure 10, we can also find that our developed method can effectively separate the vector P- and S-wave modes from the coupled elastic seismic wavefields for different source positions and accurately preserve their amplitude and phase characteristic when using the time slices for a given source position as a training dataset.

3.2. Hess Isotropic Elastic Model

We then use the Hess isotropic elastic model, which is shown in Figure 11, to demonstrate the effectiveness and feasibility of our developed method in the complex elastic model with an anomalous body, and the density of this model is also a constant in our work. The experimental process for this Hess isotropic elastic model is similar to the previous model. To obtain an effective training dataset, we first simulate the coupled elastic seismic wavefields at different times by discretely solving the elastic wave equations, and this source wavelet is an explosive source which is located at ($3.5 \mathrm{km},2.2 \mathrm{km}$) of this model and can be mainly represented by the P-wave incident energy. We then use the vector decomposition algorithm in the wavenumber domain to generate the decomposed vector P- and S-wave modes from these coupled wavefields. Finally, we also randomly select fifty pairs of time slices from these coupled elastic wave modes and the corresponding decomposed vector P-wave modes for training our neural network. Additionally, we also randomly crop 100 patches with a size of 256 × 256 for each training sample, and therefore we have a total of 5000 training samples in the training dataset, and the batch size is also defined as 32. After training, all remaining time slices for this given source position are used for testing our neural network.

Figure 12a,b show the horizontal and vertical components of the coupled elastic seismic wavefields at $t=0.6 \mathrm{s}$ for this Hess isotropic elastic model, and then we use different methods to separate the decomposed P- and S-wave modes from the above coupled wavefields and further compare their decomposition results. Figure 12c,d show the decomposed P- and S-wave modes using the Helmholtz decomposition algorithm, Figure 12e–h show the horizontal and vertical components of the decomposed vector P- and S-wave modes using the wavenumber domain decomposition algorithm, and Figure 12i–l show the corresponding horizontal and vertical components of the decomposed vector P- and S-wave modes using our developed method. Figure 13a–l show the corresponding wavenumber spectrums of these time slices shown in Figure 12a–l. According to Figure 12 and Figure 13, similar to the previous elastic-model testing process, our developed method can also effectively separate the horizontal and vertical components of the decomposed vector P- and S-wave modes from the coupled elastic seismic wavefields when using the time slices which have the same source position as the training samples as the testing samples for complex isotropic elastic models, and these decomposition results of different wave modes are close to that of the vector decomposition algorithm in the wavenumber domain. However, although the Helmholtz decomposition algorithm can generate the decomposed P- and S-wave modes, it will damage their amplitude and phase characteristics. Figure 14 show the amplitude comparisons of the decomposed P- and S-wave modes shown in Figure 12c–l at $\mathrm{depth}=2.5 \mathrm{km}$. As shown in Figure 14, similar to the vector decomposition algorithm in the wavenumber domain, our developed method can also accurately preserve the amplitude and phase characteristics of the decomposed vector P- and S-wave modes when it is applied to complex elastic models, whereas the Helmholtz decomposition algorithm cannot achieve this goal.

Next, the time slices for this Hess isotropic elastic model with another source position located at ($3.0 \mathrm{km},0.2 \mathrm{km}$) are further used to test the adaptability of our neural network to variations in source position. It is noteworthy that all the time slices for this source position are excluded from the training dataset and are obtained using the elastic wave equations. Figure 15a,b show the horizontal and vertical components of the coupled elastic seismic wavefields at $t=0.8 \mathrm{s}$ for this Hess isotropic elastic model with the above source position, and then we also use different methods to separate the decomposed P- and S-wave modes from these above coupled wavefields and further compare their decomposition results. Figure 15c,d show the decomposed P- and S-wave modes using the Helmholtz decomposition algorithm, Figure 15e–h show the horizontal and vertical components of the decomposed vector P- and S-wave modes using the wavenumber domain decomposition algorithm, and Figure 15i–l show the corresponding horizontal and vertical components of the decomposed vector P- and S-wave modes using our developed method. Figure 16a–l show the corresponding wavenumber spectrums of these time slices shown in Figure 15a–l. According to Figure 15 and Figure 16, for complex isotropic elastic models, our developed method can still effectively and accurately generate the horizontal and vertical components of the decomposed vector P- and S-wave modes from the coupled elastic seismic wavefields even if the source position changes, which further reveals that our developed method also has good adaptability for complex isotropic elastic models and variant source positions. Figure 17 show the amplitude comparisons of the decomposed P- and S-wave modes shown in Figure 15c–l at $\mathrm{depth}=1.5 \mathrm{km}$. According to Figure 17, our developed method can effectively generate the horizontal and vertical components of the decomposed vector P- and S-wave modes form the coupled elastic seismic wavefields and accurately preserve their amplitude and phase characteristic when using the time slices which have different source positions from the training samples as the testing samples for complex isotropic elastic models.

4. Discussion

To evaluate the performance of our developed method based on SADCGANs, we use two popular performance metrics to effectively quantify the neural network output. The first performance metric is the structural similarity index (SSIM) [38,45] which provides the perceptual difference between two samples and has proven to be highly effective for measuring the signal fidelity. The expression of SSIM can be written as

$$SSIM=\frac{\left(2{\mu }_x{\mu }_y+C_1\right)\left(2{\sigma }_{xy}+C_2\right)}{\left({\mu }_x^2+{\mu }_y^2+C_1\right)\left({\mu }_x^2+{\mu }_y^2+C_2\right)}$$

(9)

and

$$C_1={\left(K_{1}L\right)}^2, C_2={\left(K_{2}L\right)}^2$$

(10)

where ${\mu }_x$ is the mean value of sample $x$, ${\mu }_y$ is the mean value of sample $y$, ${\sigma }_{xy}$ is the covariance of samples $x$ and $y$, ${\mu }_x^2$ is the variance of sample $x$, ${\mu }_y^2$ is the variance of sample $y$, $C_1$ and $C_2$ are the stability factors, $L$ is the dynamic range of the pixel values, and $K_1<1$ and $K_2<1$ are small constants.

The second performance metric is the coefficient of determination ($R^2$) [38] which is a statistical measure of closeness between two samples, and its expression can be described as

$$R^2=1-\frac{{\displaystyle \sum _{i=1}^N{\left(z^i-{\tilde{z}}^i\right)}^2}}{{\displaystyle \sum _{i=1}^N{\left(z^i-{\overline{z}}^i\right)}^2}}$$

(11)

where $z^i$ and ${\tilde{z}}^i$ are the original samples and the generated samples using the developed SADCGANs, ${\overline{z}}^i$ is the mean value of the original samples, and $N$ is the number of samples.

Table 1. Structural similarity index (SSIM) and determination coefficient ($R^2$) for different models with different source positions.

Models	Simple Model ($2.56 \mathbf{km},0.02 \mathbf{km}$)	Simple Model ($2.56 \mathbf{km},1.28 \mathbf{km}$)	Hess Model ($3.5 \mathbf{km},2.2 \mathbf{km}$)	Hess Model ($3.0 \mathbf{km},0.2 \mathbf{km}$)
$u_{xP}$(SSIM)	0.999	0.985	0.997	0.978
$u_{xP}$$(R^2$)	0.993	0.964	0.990	0.955
$u_{zP}$(SSIM)	0.998	0.981	0.992	0.976
$u_{zP}$$(R^2$)	0.990	0.963	0.984	0.953
$u_{xS}$(SSIM)	0.989	0.978	0.983	0.959
$u_{xS}$$(R^2$)	0.976	0.953	0.964	0.940
$u_{zS}$(SSIM)	0.983	0.970	0.976	0.954
$u_{zS}$$(R^2$)	0.965	0.949	0.952	0.933

lists the SSIM and $R^2$ values for different models with different source positions mentioned in the previous section. According to Table 1, we can find that the decomposed P- and S-wave modes using the developed algorithm based on SADCGANs and the wavenumber domain decomposition algorithm are close to each other even if the source position changes, which effectively proves that our developed vector decomposition method based on SADCGANs has good application performance.

Deep learning methods often require a large number of sample labels to train the neural network. For our vector decomposition method of the P- and S-wave modes, it requires a large number of decomposed vector P-wave modes as sample labels to train the network structure of SADCGANs. In our work, we use the vector decomposition algorithm in the wavenumber domain to effectively separate the decomposed vector P-wave modes from the time slices of simulated elastic seismic wavefields at different times as training sample labels. After training, the decomposed vector P-wave modes can be separated from the coupled elastic seismic wavefields which are not included in training sample labels using the trained neural network, and the decomposed vector S-wave modes can be obtained using the strategy of wavefield subtraction shown in Equation (3). Therefore, the decomposition results of our developed method based on SADCGANs will be as close as possible to that of the vector decomposition algorithm in the wavenumber domain, because the training results of the neural network can only be as close to the sample labels as possible. There is still some weak residual energy of S-wave modes in these decomposed vector P-wave modes as sample labels for our neural network, and therefore each wave mode generated by our developed method contains other weak wave modes which are similar to those of the wavenumber domain vector decomposition algorithm. In other words, the accuracy of sample labels largely determines the accuracy of our developed method. Thus, studying how to effectively improve the accuracy of the decomposed vector P-wave modes as training sample labels as much as possible is worth considering.

Our developed method is currently applied to the elastic wavefield extrapolation and decomposition process of ERTM to obtain the multi-wave imaging results, and therefore its application object is the time slices of coupled elastic seismic wavefields at different times. In our work, we mainly use the time slices at some certain times for two theoretical elastic models as training sample labels, and therefore the training data have the following main limitations. Firstly, the training results of the neural network theoretically improve as the number of sample labels gradually increases, but we can only select the simulated wavefields at some certain times as training data for a given elastic model, and therefore this number needs to be carefully considered. Secondly, due to only testing two models, the training dataset we used in our work lacks diversity, which may result in our developed method not achieving good vector decomposition results for more complex elastic models, and this problem can be effectively solved by introducing various theoretical elastic models. Thirdly, various theoretical models are only the approximation of the real geological models, and therefore the simulated wavefields with fixed simulation parameters as training data are different from the real elastic seismic wavefields in subsurface media, such as time and space grid sizes, source and receiver positions, etc., which may lead to the fact that our developed method cannot effectively and accurately separate the horizontal and vertical components of the vector P- and S-wave modes from the real elastic seismic wavefields. Fourthly, at present, there is no effective method to completely separate the vector P- and S-wave modes from the coupled elastic seismic wavefields, and therefore the training dataset which we use in our work is not absolutely perfect. In future work, our developed method will be applied to multicomponent seismic data to effectively and accurately separate the seismic data of different wave modes, so that it can promote the development and practical application of multicomponent seismic exploration technology. Compared with the simulated seismic data, the wavefield characteristics of real multi-component seismic data are more complex, and therefore how to generate the corresponding training sample labels is also a key consideration.

Because the elastic wavefield extrapolation of ERTM itself requires a huge computing time, using deep learning methods to achieve the vector decomposition of coupled elastic seismic wavefields will require more computing time, and therefore one practical limitation of our developed method is its computational efficiency. In other words, among the three decomposition algorithms used in our numerical experiments, the Helmholtz decomposition algorithm has the highest computational efficiency, the vector decomposition algorithm in the wavenumber domain requires a longer computing time than the previous algorithm, and our developed algorithm based on SADCGANs requires the longest computing time. The biggest concern of using SADCGANs is the computational cost of the training process. Once trained, the computational cost for the prediction process of the neural network is comparatively inexpensive. During our numerical experiments, the central processing unit (CPU) which is used for elastic wavefield decomposition is the Intel Xeon Gold 5218 16-core 2.30 GHz processor, and the graphics processing unit (GPU) which is used for elastic wavefield decomposition is the 32G NVIDIA Tesla V100 processor. To obtain the decomposed wave modes from the coupled elastic seismic wavefields, the computing time of the Helmholtz decomposition algorithm is so small that it can be ignored, the vector decomposition algorithm in the wavenumber domain usually takes approximately 12 to 16 s, and our developed method based on SADCGANs with 5000 training samples requires approximately 48 h for the training process and 10 to 15 s for the subsequent prediction process. Based on these statistical time data, the significant computational cost of our developed vector decomposition algorithm based on SADCGANs is mainly caused by the training process. With the rapid development of advanced hardware, such as CPUs, GPUs and computer clusters, the computational cost will not be a problem for our developed method based on SADCGANs in the future.

With the rapid development of deep learning methods, more and more neural networks are widely used in seismic data processing, such as DCNN, DenceNet, ResNet and GANs, etc. To achieve the vector decomposition of coupled elastic seismic wavefields, we use SADCGANs which are an improvement of conventional GANs in this paper. Compared with other generative models, GANs are a type of generative model with several advantages. Firstly, GANs can be well generalized without requiring a large amount of annotated training data [38]. Secondly, Markov chains may have a lower efficiency in high-dimensional spaces because their convergence rate may be slower, and therefore GANs have the advantage of not requiring Markov chains. Most of the other neural networks either use stochastic approximations to select the training samples to minimize the objective function or use Markov chains to repeatedly extract samples from the distribution and update them to ultimately converge to the true model [38,46]. Thirdly, GANs can use multimodal outputs where a single input may correspond to multiple correct answers. Compared with conventional machine learning models, this significant advantage can minimize the mean squared error (MSE) between the predicted and the desired outputs and thus lead to the blurring of the detailed features because of the averaging effect [38,46]. Finally, the GAN framework is a promising alternative to the neural networks operating on MSE because it provides a rich internal representation of the structural information of the datasets without any blurring or averaging effect [38,47]. However, the conventional GANs also have some disadvantages, the most serious problem of which is the difficulty of training. To effectively alleviate this problem, we use the more effective SADCGANs instead of the conventional GANs to stabilize the entire training process and obtain better wavefield decomposition results. At present, we only study the developed vector decomposition method based on SADCGANs, and in future work, we will test other neural networks used in elastic wavefield decomposition and conduct a comprehensive comparative analysis of their performance, such as DCNN, DenceNet and ResNet, etc.

5. Conclusions

In this paper, we develop a vector decomposition method of the P- and S-wave modes from the coupled elastic seismic wavefields using SADCGANs for isotropic elastic media. To stabilize the training process and achieve more accurate wavefield decomposition results, we introduce the self-attention modules into the generator and the discriminator to simultaneously extract the local and global features of input samples. To demonstrate the effectiveness, feasibility and adaptability of our developed method, it is applied to a simple isotropic elastic model and the Hess isotropic elastic model. According to the numerical experimental results of different models, we can draw the following conclusions. Our developed method can effectively generate the horizontal and vertical components of the decomposed vector P- and S-wave modes from the coupled elastic seismic wavefields and accurately preserve their amplitude and phase characteristics for different elastic models even if the source position changes. The accuracy of training sample labels largely determines the accuracy of our developed method, and its vector decomposition results of different wave modes can only be as close as possible to the training sample labels. Our developed method is an intelligent data-driven algorithm which is not dependent on elastic model parameters and certain prior conditions compared with the conventional algorithms. Our developed method can be well applied to the elastic wavefield extrapolation process of ERTM to effectively achieve the vector decomposition of the P- and S-wave modes to further improve the multi-wave imaging accuracy.