Crossline Reconstruction of 3D Seismic Data Using 3D cWGAN: A Comparative Study on Sleipner Seismic Survey Data

Abstract

In seismic data acquisition, data loss can occur, particularly with the use of streamer systems in marine seismic exploration. These systems often cause spatial aliasing problems by having close inline intervals and wide crossline intervals to maximize the exploration range. To improve the resolution of seismic data in the crossline direction, various machine learning techniques have been employed for crossline data reconstruction. In this study, we introduce a 3D cWGAN (conditional Wasserstein generative adversarial network) for interpolating 3D seismic data. We evaluate the model’s performance by comparing it with 2D cWGAN and 3D U-Net. In this study, two interpolation strategies are employed to reconstruct missing data in the crossline direction. The first strategy uses a 2D network, which trains a model using inline data and applies the trained model to the crossline direction via 2D cWGAN. The second strategy employs a 3D network, which uses the 3D volume of the seismic data directly via 3D cWGAN and 3D U-Net. We demonstrate the effectiveness of the proposed method using the Sleipner CO2 4D seismic survey dataset. Our results show that the 3D cWGAN is more efficient in enhancing resolution and computation compared to the 2D cWGAN or 3D U-Net.

1. Introduction

In the acquisition of seismic data, there is a risk of losing some of the data due to economic and environmental constraints, mechanical issues, and other such factors. In particular, the use of conventional streamer systems in marine seismic exploration, in particular, often leads to spatial aliasing problems because these systems tend to have close inline intervals and wide crossline intervals in order to maximize the system’s survey area. This loss of data can have a negative effect on the subsequent processing and interpretation of data, and it may be necessary to conduct additional surveys in order to restore the data, which incurs additional time and financial costs [1]. To improve the resolution of 3D seismic data, trace interpolation in the crossline direction is required. With the advancement of machine learning, there has been an increase in research into various machine learning-based interpolation methods.

Chai et al. (2020) proposed a convolutional neural network model using U-Net architecture for the interpolation of 2D seismic data. The proposed model outperformed the FGFT (fast-generalized Fourier transform) interpolation method, and the synthetic and field data had higher signal-to-noise ratios [2,3]. Chang et al. (2019) proposed a TF-CGAN (conditional generative adversarial networks in time and frequency domain) model using a cGAN model. The TF-CGAN model uses frequency domain information from Fourier transform in its generator and discriminator to increase the prediction accuracy of the model [4]. Li et al. (2022) proposed a CA-Unet (coordinate attention Unet) model based on a U-Net with a coordinate attention block. The CA-Unet model uses a coordinate attention block to optimize the use of feature maps and achieve high-accuracy predictions [5]. Park et al. (2022) proposed a CF-Unet (coarse-refine Unet) model using a coarse-refine U-Net with frequency wavenumber (FK) loss. The CF-Unet model solves the over-smoothing problem and accelerates network convergence with FK loss while mitigating the high-frequency loss problem [6]. Kaur et al. (2019) proposed a model consisting of two generators and one discriminator using the Cycle GAN structure. The proposed model showed an improved prediction speed and accuracy compared to the RNA method [7] in frequency spatial prediction filter interpolation [8].

Kong et al. (2022) proposed a multi-resolution U-Net model that utilizes the correlation of 3D data for interpolating 3D seismic data. The model outperformed the traditional U-Net model as it was able to accurately predict weak events and utilize both horizontal and vertical correlations. The model also reduced computation costs by pre-setting initial CNN weights [9]. Qian et al. (2022) proposed a deep tensor autoencoder (DTAE) model that is able to learn data-based, nonlinear, and high-dimensional mapping. The DTAE model accurately captures relationships between traces even when using data associated with complex geologic structures, resulting in accurate predictions [10]. Jin et al. (2023) proposed a DS-U-Net model based on 3D U-Net, which uses depthwise separable convolution instead of traditional convolution. Depthwise separable convolution reduces the computational cost and the number of parameters by dividing traditional convolution into two parts: depthwise convolution and pointwise convolution [11].

Compared to other machine learning algorithms, convolutional neural network (CNN) based and generative adversarial network (GAN) based algorithms are widely used in the field of seismic data interpolation. Yu and Yoon (2022) conducted an evaluation and comparison of the U-Net and conditional Wasserstein generative adversarial network (cWGAN) to determine an interpolation method that is able to accurately reconstruct 2D seismic data. After thorough experiments, they concluded that cWGAN achieved higher accuracy and excellent performance compared to U-Net [12].

Based on this, in this paper, we propose to extend the 2D cWGAN to a 3D model and apply it to the interpolation of 3D seismic data. We also propose to evaluate the performance of the developed model by comparing the results of the 2D cWGAN and 3D U-Net for the interpolation of 3D seismic data.

In this paper, we first introduce cWGAN and describe the study strategy and the seismic data used. We then present the structure and performance evaluation indicators of the 2D cWGAN, 3D cWGAN and 3D U-Net models. Finally, we compare the interpolation results obtained from each model.

2. Approach

2.1. cWGAN

GAN is a type of generative model that was proposed in 2014 by Goodfellow [13]. GAN is composed of a generator and a discriminator; both networks compete with each other in an adversarial relationship while learning and generating data. The latent variable used to generate data passes through the generator to generate fake data and is then input into the discriminator along with real data. The discriminator performs the task of distinguishing between fake and real data and obtains discriminator loss and generator loss during the discrimination process. Through the process of minimizing discriminator loss and generator loss, the abilities of the discriminator and generator are improved. The final goal is for the generator’s ability to improve to the point where the discriminator can no longer distinguish between fake and real data.

CGAN (conditional GAN) [14] was developed to address the issue of controlling the noise vector that was unable to be adjusted in traditional GANs. By adding conditions to traditional GANs, it became possible to control the data generation process. It also uses a generator based on pix2pix, making it a supervised learning technique [15].

WGAN (Wasserstein GAN) [16] was developed to solve the issues of mode collapse and vanishing gradients that occur in traditional GANs. When using the Wasserstein loss function and gradient penalty regularization, WGAN can further improve the stability and performance of the training process.

cWGAN is a combination of CGAN and WGAN. cWGAN utilizes Wasserstein loss and applies supervised learning to control the noise vector in the same way as CGAN. As a result, cWGAN can effectively address issues, such as mode collapse, gradient vanishing, and controlling data generation by being able to control the noise vector at the same time [17,18].

The structure of the cWGAN used in this study is depicted in Figure 1. Figure 1a illustrates the structure of the 2D cWGAN, where two-dimensional input data are used, and Figure 1b illustrates the structure of the 3D cWGAN, where three-dimensional input data are used. In both models, the input data x, which are a patch of seismic data with missing traces, are fed into the generator. The predicted result G(x), which is the interpolated data, and the input data x are simultaneously fed into the discriminator to undergo discrimination, and its output is D(x, G(x)). Additionally, the input data x and the corresponding target data y are used for training the discriminator, and its output is D(x, y). Once the model is trained, only the generator G is used to conduct the interpolation.

2.2. Generator

The structure of the generator of both the 2D cWGAN and 3D cWGAN used in this study is illustrated in Figure 2. The generator for the 2D cWGAN is a 2D U-Net that employs 2D convolution, as shown in Figure 2a. The input size of the model is 80 × 80 × 1 for the corrupted data used as input and conditioning data, and the model is designed to output data of the same size. The model consists of contraction and expansion stages in an autoencoder format. The contraction stage employs ReLU (rectified linear unit) activation functions and consists of 3 × 3 convolutions and 4 max poolings. The expansion stage also uses ReLU as the activation function and is made up of 3 × 3 convolutions and 2 × 2 upsamplings. During the contraction stage, the input is downsampled, which reduces its spatial resolution and results in the loss of important details. Similarly, during the expansion stage, the upsampled feature maps must be merged with the feature maps from the contracting stage to recover the lost spatial information. To address these issues, skip connections are added. Detailed information about the model is provided in

Table 1. Structure of 2D generator.

Layer (Type)	Output Size
input_1 (Input layer)	(80, 80, 1)
conv2d (Conv2D)	(80, 80, 64)
conv2d_1 (Conv2D)	(80, 80, 64)
max_pooling2d (MaxPooling2D)	(40, 40, 64)
conv2d_2 (Conv2D)	(40, 40, 128)
conv2d_3 (Conv2D)	(40, 40, 128)
max_pooling2d_1 (MaxPooling2D)	(20, 20, 128)
conv2d_4 (Conv2D)	(20, 20, 256)
conv2d_5 (Conv2D)	(20, 20, 256)
max_pooling2d_2 (MaxPooling2D)	(10, 10, 256)
conv2d_6 (Conv2D)	(10, 10, 512)
conv2d_7 (Conv2D)	(10, 10, 512)
max_pooling2d_3 (MaxPooling2D)	(5, 5, 512)
conv2d_8 (Conv2D)	(5, 5, 1024)
conv2d_9 (Conv2D)	(5, 5, 1024)
up_sampling2d (Upsampling2D)	(10, 10, 1024)
conv2d_10 (Conv2D)	(10, 10, 512)
concatenate (Concatenate)	(10, 10, 1024)
conv2d_11 (Conv2D)	(10, 10, 512)
conv2d_12 (Conv2D)	(10, 10, 512)
up_sampling2d_1 (Upsampling2D)	(20, 20, 512)
conv2d_13 (Conv2D)	(20, 20, 256)
concatenate_1 (Concatenate)	(20, 20, 512)
conv2d_14 (Conv2D)	(20, 20, 256)
conv2d_15 (Conv2D)	(20, 20, 256)
up_sampling2d_2 (Upsampling2D)	(40, 40, 256)
conv2d_16 (Conv2D)	(40, 40, 128)
concatenate_2 (Concatenate)	(40, 40, 256)
conv2d_17 (Conv2D)	(40, 40, 128)
conv2d_18 (Conv2D)	(40, 40, 128)
up_sampling2d_3 (Upsampling2D)	(80, 80, 128)
conv2d_19 (Conv2D)	(80, 80, 64)
concatenate_3 (Concatenate)	(80, 80, 128)
conv2d_20 (Conv2D)	(80, 80, 64)
conv2d_21 (Conv2D)	(80, 80, 64)
conv2d_22 (Conv2D)	(80, 80, 2)
conv2d_23 (Conv2D)	(80, 80, 1)

.

The 3D cWGAN generator uses a 3D U-Net model with 3D convolution, as shown in Figure 2b. The model is configured to take in input data of a size of 80 × 80 × 80 × 1 and output results of the same size. It is also composed of contraction and expansion stages, similar to the 2D cWGAN, with the contraction stage using ReLU as the activation function, which is composed of 3 × 3 × 3 convolutions and 4 max pooling operations. The expansion stage also uses ReLU as the activation function, which is composed of 3 × 3 × 3 convolutions and 2 × 2 × 2 upsamplings. The accuracy of the prediction is improved by adding a skip connection, which is the same as with the 2D cWGAN. Detailed information on the model is shown in

Table 2. Structure of 3D generator.

Layer (Type)	Output Size
input_1 (Input layer)	(80, 80, 80, 1)
conv3d (Conv3D)	(80, 80, 80, 64)
conv3d_1 (Conv3D)	(80, 80, 80, 64)
max_pooling3d (MaxPooling3D)	(40, 40, 40, 64)
conv3d_2 (Conv3D)	(40, 40, 40, 128)
conv3d_3 (Conv3D)	(40, 40, 40, 128)
max_pooling3d_1 (MaxPooling3D)	(20, 20, 20, 128)
conv3d_4 (Conv3D)	(20, 20, 20, 256)
conv3d_5 (Conv3D)	(20, 20, 20, 256)
max_pooling3d_2 (MaxPooling3D)	(10, 10, 10, 256)
conv3d_6 (Conv3D)	(10, 10, 10, 512)
conv3d_7 (Conv3D)	(10, 10, 10, 512)
max_pooling3d_3 (MaxPooling3D)	(5, 5, 5, 512)
conv3d_8 (Conv3D)	(5, 5, 5, 1024)
conv3d_9 (Conv3D)	(5, 5, 5, 1024)
up_sampling3d (Upsampling3D)	(10, 10, 10, 1024)
conv3d_10 (Conv3D)	(10, 10, 10, 512)
concatenate (Concatenate)	(10, 10, 10, 1024)
conv3d_11 (Conv3D)	(10, 10, 10, 512)
conv3d_12 (Conv3D)	(10, 10, 10, 512)
up_sampling3d_1 (Upsampling3D)	(20, 20, 20, 512)
conv3d_13 (Conv3D)	(20, 20, 20, 256)
concatenate_1 (Concatenate)	(20, 20, 20, 512)
conv3d_14 (Conv3D)	(20, 20, 20, 256)
conv3d_15 (Conv3D)	(20, 20, 20, 256)
up_sampling3d_2 (Upsampling3D)	(40, 40, 40, 256)
conv3d_16 (Conv3D)	(40, 40, 40, 128)
concatenate_2 (Concatenate)	(40, 40, 40, 256)
conv3d_17 (Conv3D)	(40, 40, 40, 128)
conv3d_18 (Conv3D)	(40, 40, 40, 128)
up_sampling3d_3 (Upsampling3D)	(80, 80, 80, 128)
conv3d_19 (Conv3D)	(80, 80, 80, 64)
concatenate_3 (Concatenate)	(80, 80, 80, 128)
conv3d_20 (Conv3D)	(80, 80, 80, 64)
conv3d_21 (Conv3D)	(80, 80, 80, 64)
conv3d_22 (Conv3D)	(80, 80, 80, 2)
conv3d_23 (Conv3D)	(80, 80, 80, 1)

. In this study, the model prediction was performed using only the generator; this model is referred to as the 3D U-Net.

2.3. Discriminator

The discriminator of 2D cWGAN is based on PatchGAN [19]. PatchGAN is a modification of the traditional GAN architecture that uses a discriminator network to classify input images as either real or fake. Instead of classifying the entire image as real or fake, the PatchGAN discriminator classifies local patches of the image and outputs a score for each patch, indicating whether it is real or fake. The scores are averaged to produce a single value, which represents the overall score for the entire image. This enables the discriminator to capture more fine-grained details of the image and provide more accurate feedback to the generator network [20]. The PatchGAN-based discriminator used for the 2D cWGAN in this study consists of a series of 2D convolutional layers followed by fully connected layers. The convolutional layers extract features from the input patch, and the fully connected layers transform the extracted features into a probability score. The activation function used is ReLU, which is the same as that used for the generator. The discriminator of the 3D cWGAN is also based on PatchGAN, which applies 3D convolution and is composed of an encoder only, which is the same as the discriminator of the 2D cWGAN and uses ReLU as the activation function.

Table 3. Structure of 2D discriminator.

Layer (Type)	Output Size
input_1 (Input layer)	(80, 80, 2)
conv2d (Conv2D)	(80, 80, 64)
conv2d_1 (Conv2D)	(80, 80, 64)
max_pooling2d (MaxPooling2D)	(40, 40, 64)
conv2d_2 (Conv2D)	(40, 40, 128)
conv2d_3 (Conv2D)	(40, 40, 128)
max_pooling2d_1 (MaxPooling2D)	(20, 20, 128)
conv2d_4 (Conv2D)	(20, 20, 256)
conv2d_5 (Conv2D)	(20, 20, 256)
max_pooling2d_2 (MaxPooling2D)	(10, 10, 256)
conv2d_6 (Conv2D)	(10, 10, 512)
conv2d_7 (Conv2D)	(10, 10, 512)
max_pooling2d_3 (MaxPooling2D)	(5, 5, 512)
conv2d_8 (Conv2D)	(5, 5, 1024)
conv2d_9 (Conv2D)	(5, 5, 1024)
max_pooling2d_4 (MaxPooling2D)	(2, 2, 1024)
flatten (Flatten)	(4096)
dense (Dense)	(1)

and

Table 4. Structure of 3D discriminator.

Layer (Type)	Output Size
input_1 (Input layer)	(80, 80, 80, 2)
conv3d (Conv3D)	(80, 80, 80, 64)
conv3d_1 (Conv3D)	(80, 80, 80, 64)
max_pooling3d (MaxPooling3D)	(40, 40, 40, 64)
conv3d_2 (Conv3D)	(40, 40, 40, 128)
conv3d_3 (Conv3D)	(40, 40, 40, 128)
max_pooling3d_1 (MaxPooling3D)	(20, 20, 20, 128)
conv3d_4 (Conv3D)	(20, 20, 20, 256)
conv3d_5 (Conv3D)	(20, 20, 20, 256)
max_pooling3d_2 (MaxPooling3D)	(10, 10, 10, 256)
conv3d_6 (Conv3D)	(10, 10, 10, 512)
conv3d_7 (Conv3D)	(10, 10, 10, 512)
max_pooling3d_3 (MaxPooling3D)	(5, 5, 5, 512)
conv3d_8 (Conv3D)	(5, 5, 5, 1024)
conv3d_9 (Conv3D)	(5, 5, 5, 1024)
max_pooling3d_4 (MaxPooling3D)	(2, 2, 2, 1024)
flatten (Flatten)	(8192)
dense (Dense)	(1)

represent the structure of the 2D cWGAN and 3D cWGAN discriminators, respectively.

2.4. Wasserstein Loss

In this study, we trained 2D and 3D cWGAN models with Wasserstein loss. Wasserstein loss, represented by W, calculates the Wasserstein distance, proposed by Dobrushin in 1970 [21] and is defined as

$$\mathrm{W}\left(P_r,P_g\right)=in{f}_{\gamma \in \pi (P_r,P_g)}{E}_{\left(x,y\right)~r}\left[d\left(x\left(w\right),y\left(w\right)\right)\right],$$

(1)

where w represents a sample from population $\mathsf{\Omega }$ and $\mathsf{\pi }\left(P_r,P_g\right)$ represents the set of all joint probability distributions $\mathsf{\gamma }(x,y)$, with $P_r=\int \gamma \left(x,y\right)dy$ and $P_g=\int \gamma \left(x,y\right)dx$. In other words, $\mathrm{W}(P_r,P_g)$ is the estimated value that minimizes the expected value of $d(x,y)$ among $\mathsf{\pi }(P_r,P_g)$. As the sampling is repeated, $\mathsf{\gamma }(x,y)$ converges to the optimal connection, and the distribution of $d(x,y)$ varies depending on the connection between x and y. In this way, the Wasserstein distance has a probability distribution with the minimum expected value of $d(x,y)$ among many $\mathsf{\gamma }(x,y)$ [21].

The KL (Kullback–Leibler) divergence or JS (Jensen–Shannon) divergence is discontinuous when $P_r$ and $P_g$ do not overlap with each other, whereas the Wasserstein distance is continuous, which is an advantage in determining convergence because it is flexible. If $d(x,y)$ is specified as the Euclidean distance $‖x-y‖$, and the probability measures of $P_r$ and $P_g$ are defined as $\mathsf{\mu }$ and $\mathsf{\nu }$, respectively, it can be derived according to the Kantorovich–Rubinstein duality as

$$\mathrm{W}\left(P_r,P_g\right){=}_{{‖f‖}_L\le 1}\left\{E_{y~\mu }\left[f\left(x\right)\right]-E_{x~\nu }\left[f\left(x\right)\right]\right\}.$$

(2)

${‖f‖}_L\le 1$ means that the function $f$ is 1-Lipschitz continuous, and a 1-Lipschitz continuous function is a function that satisfies the following equation for all x, y within its domain [22,23].

$$\left|f\left(y\right)-f\left(x\right)\right|\le ‖y-x‖.$$

(3)

2.5. Interpolation Strategy

In this study, two interpolation strategies were applied in order to perform the data interpolation of 3D seismic data in the crossline direction. The first is an interpolation strategy using a 2D network, and the generated data and data extraction direction are shown in Figure 3. First, to train the model, data with a regular 50% missing trace were generated from the inline direction and 80 × 80-sized patches were extracted, and this was used as the training data (Figure 3a). Next, to test the model, data with a regular 50% missing trace were generated from the crossline direction and 80 × 80-sized patches were extracted, and this was used as the test data (Figure 3b). Finally, to predict data in the inline direction, the predicted data in the crossline direction, marked with a blue box in Figure 3c, were combined with the overall predicted data in the crossline direction, and the lost inline data prediction was performed (Figure 3c). In this study, a 2D cWGAN was used as the 2D network.

The second strategy is an interpolation strategy using a 3D network, and the generated data and data extraction direction using this strategy are shown in Figure 4. To model training and testing, regularly 50% of the traces were decimated from the crossline direction, and an 80 × 80 × 80-sized cube was extracted to generate the data. In total, 70% of the extracted data were used as the training data (Figure 4a), and 30% were used as the test data (Figure 4b). In this strategy, crossline reconstruction was performed using the 3D U-Net and 3D cWGAN.

3. Experiment

The present study uses the publicly available Sleipner CO₂ 4D seismic survey dataset for the interpolation of 3D seismic survey data, which targets a reservoir layer located near the North Sea in Norway [24] (Figure 5). The data were obtained from repeated surveys of a 4 km × 7 km area that includes the reservoir layer, with the reference data collected in 1994 and the monitoring data collected in 2001, 2004, 2006, 2008, and 2010. The seismic data up to and including the 2001 survey were processed by WesternGeco, and the seismic data from 1994, 2001, 2004, and 2006 were fully reprocessed by PGS. Time-lapse processing in 2008 and 2010 was carried out in a similar procedure. The 2010 data set includes time-lapse and seismic imaging data with four offset stacks featuring full, near, mid and far folds. This study used the seismic image processed 2010 data with full fold, which consists of 1121 crosslines, 216 inlines, and 1001 samples, with a total of 242,136 traces [25,26].

Table 5. Acquisition parameter of Sleipner 4D seismic data 2010.

Parameter	Data of 2010
Shot interval (m)	12.5
Shoot direction	N-S
Group interval (m)	12.5
Sample interval (ms)	2
Record length (ms)	4608
$\mathrm{Amount}\mathrm{of}\mathrm{injected}{\mathrm{C}\mathrm{O}}_2$ (kt)	12,080

provides further detailed information on the data.

To train each model, the two strategies described were applied to generate train data. For the 2D cWGAN, regularly 50% of the traces were missed in the inline direction, and 80 × 80 sized patches were extracted and applied to train the model. For the 3D cWGAN, data were generated by regularly missing 50% of the traces in the crossline direction, and 80 × 80 × 80 sized cubes were extracted; 70% of these were used for model training. For the 3D U-Net that only used the generator of 3D cWGAN, the data were generated in the same way as those generated with the 3D cWGAN, with 80 × 80 × 80 sized cubes being extracted; 70% of these were used for the training data, 20% for the validation data, and 10% for the testing data. We used the RMSprop optimizer with a learning rate of 5 × 10⁻⁵ and 1 × 10⁻⁵ for 2D and 3D cWGAN, respectively, and the Adam optimizer with a learning rate of 1 × 10⁻⁴ for 3D U-Net. The batch sizes are 4, 4, and 32 for 3D U-Net, 3D cWGAN and 2D cWGAN, respectively. All network training was carried out on TensorFlow 2 framework, and two GPUs (GeForce RTX 2090 Ti, NVIDIA, Santa Clara, CA, USA) were used to accelerate computation.

Six different test datasets were generated for testing the trained model, and these are shown in Figure 6. For testing, data with misses of 50%, 67%, and 75% in a regular/irregular pattern were generated from the same dataset in the crossline direction and were used. The same testing data were applied to the 2D cWGAN, 3D cWGAN, and 3D U-Net models.

The test data use data that were missed at rates of 50%, 67%, and 75% in a regular/irregular manner in the crossline direction, and the same test data were applied to the 2D cWGAN, 3D cWGAN, and 3D U-Net models.

4. Results

The six test results of the 2D cWGAN, 3D cWGAN, and 3D U-Net models used in this study are shown in Figure 7, Figure 8 and Figure 9. Figure 7 and Figure 8 are the test results in the crossline direction, and Figure 9 is the test results in the inline direction for data with regularly 75% missing traces.

By comparing the prediction results in the crossline direction of each model through Figure 7 and Figure 8, it can be seen that the difference between the label data and the prediction result of the 3D cWGAN is smaller compared to that of the 2D cWGAN and 3D U-Net and that it shows a smoother prediction result compared to the prediction results of the two models. On the other hand, in the case of the 3D U-Net, as the missing rate of the data increases, it is possible to see that most of the data are not predicted.

In the case of Figure 9a, which represents the input data and label data in the inline direction, it can be confirmed that there are no data in the inline direction due to the missing data in the crossline direction in the input data. By comparing the prediction results in the inline direction in Figure 9b, it can be seen that the difference between the label data and the prediction result is smaller in the 3D cWGAN than in the 2D cWGAN and 3D U-Net.

Additional accuracy verification of the interpolation result was performed by converting it into an f-k spectrum for comparison. The f-k spectrum is a frequency-wavenumber domain where the frequency (f) and wavenumber (k) are used as independent variables, which are the results of a two-dimensional Fourier transform of seismic exploration records or sections. It is mainly used in the analysis of seismic direction and apparent velocity and in the design of velocity filters in seismic data analysis. Data that exceeds the Nyquist frequency ($f_N$) determined by discrete space sampling can spread around and be mixed with signals, and the Nyquist wavenumber ($K_N$) proceeds from right ${+K}_N$ to left ${-K}_N$. The aliasing observed in the f-k spectrum is due to the interval of the receiver or missing data, resulting in spatial aliasing [27].

The results of the interpolation in the crossline direction shown in Figure 7c, where 75% of the trace is regularly missed, are presented in Figure 10 for the f-k spectrum of the data. Figure 10a shows the input data and label data for the corresponding data, and Figure 10b shows the interpolation results of each model and the f-k spectrum of the residuals corresponding to each result. When comparing the interpolation results shown in Figure 10b, the 3D U-Net showed large residuals compared to other models, while the 3D cWGAN showed relatively small residuals and highly accurate prediction results. In addition, similar f-k spectra were observed for other cases. Overall, our results confirmed that the 3D cWGAN approach achieves higher accuracy in crossline interpolation compared to other models.

The f-k spectrum corresponding to the inline direction interpolation results shown in Figure 9 is shown in Figure 11. Figure 11a shows the label data in the inline direction, and Figure 11b shows the interpolation results and corresponding residuals for each model in the inline direction. When comparing the interpolation results of each model in Figure 11b, the 3D U-Net showed relatively large residuals and poor performance, while the 3D cWGAN showed relatively small residuals and accurate prediction results compared to other models. Through analysis of the f-k spectrum, we confirmed that the 3D cWGAN outperformed other methods.

To quantify the performance of the trained models, we use three metrics: MSE (mean squared error), PSNR (peak signal-to-noise ratio) and SSIM (structural similarity index).

MSE is widely utilized as a full-reference metric for evaluating image quality, with lower values indicating better performance. MSE is calculated by following Equation (4).

$$\mathrm{M}\mathrm{S}\mathrm{E}=\frac{1}{N}{\sum }_{i=0}^n\left[G\right(x_i)-y_i{]}^2,$$

(4)

where $G\left(x_i\right)$ represents the predicted result; $y_i$ represents the actual value; N represents the amount of data.

PSNR is a metric commonly used to measure the quality of image representation by calculating the ratio between the maximum possible signal power and the power of the distorting noise. PSNR is calculated by following Equation (5).

$$\mathrm{P}\mathrm{S}\mathrm{N}\mathrm{R}=10{log}_{10}\left({peakval}^2\right)/MSE,$$

(5)

where $peakval$ (peak value) is the maximal in the image data. A larger PSNR indicates better quality of the image.

SSIM is a perceptual image quality assessment metric used to measure the quality of an image. This metric compares the contrast, brightness, structure, and other aspects of an image to measure the structural similarity between the original and altered images. SSIM is calculated by following Equation (6).

$$\mathrm{S}\mathrm{S}\mathrm{I}\mathrm{M}\left(x,y\right)=\frac{\left(2{\mu }_x{\mu }_y+C_1\right)\left(2{\sigma }_{xy}+C_2\right)}{{(\mu }_x^2+{\mu }_y^2+C_1){(\sigma }_x^2+{\sigma }_y^2+C_2)},$$

(6)

where x and y represent the two images to be compared; ${\mu }_x,{\mu }_y,{\sigma }_x^2,{\sigma }_y^2$, and ${\sigma }_{xy}$ are the mean, variance, and covariance of images x and y, respectively; $C_1$ and $C_2$ are small constant values. The $\mathrm{S}\mathrm{S}\mathrm{I}\mathrm{M}\left(x,y\right)$ value ranges between 0 to 1, with values closer to 1 indicating greater similarity between $x$ and $y$.

We obtain the average values of these metrics by applying the trained model to the test dataset (Figure 12). The standard deviation of each is shown as error bars on the graph, and the x-axis represents the amount of missing data as a percentage. Figure 12a–c represents the graph of the comparison of MSE, PSNR, and SSIM, respectively, and those values for all the test cases are presented in

Table 6. Performance comparison of 3D cWGAN, 2D cWGAN, and 3D U-Net interpolation methods on the Sleipner Seismic dataset using MSE, PSNR, and SSIM metrics. Bold: Best values.

Metric	Model	Crossline						Inline
		Regular			Irregular			Regular
		50%	67%	75%	50%	67%	75%	75%
MSE	2D cWGAN	0.00061	0.0015	0.0016	0.00096	0.0016	0.0016	0.00064
	3D cWGAN	0.00010	0.00045	0.00064	0.00031	0.00061	0.00077	0.00052
	3D U-Net	0.00023	0.0049	0.012	0.0086	0.014	0.016	0.012
PSNR	2D cWGAN	33.56	28.93	29.03	30.97	29.02	28.41	28.46
	3D cWGAN	39.34	35.91	34.89	32.70	31.19	32.14	35.45
	3D U-Net	40.02	28.84	22.89	25.69	22.33	22.51	24.69
SSIM	2D cWGAN	0.975	0.952	0.948	0.965	0.949	0.942	0.907
	3D cWGAN	0.994	0.983	0.980	0.977	0.978	0.975	0.969
	3D U-Net	0.994	0.862	0.650	0.828	0.640	0.517	0.594

.

When 50% of the traces are regularly missing, as shown in Figure 12, the results indicate that both the 3D U-Net and 3D cWGAN exhibit similar accuracy, with the 3D U-Net having a higher MSE (indicating worse performance) and PSNR (indicating better performance), but equal SSIM compared to the 3D cWGAN. The 2D cWGAN model performs worse than other models in this case. These findings suggest that both the 3D U-Net and 3D cWGAN models can be effective in cases where 50% of traces are regularly missing.

However, in all other cases except regularly missing 50%, the 3D cWGAN consistently outperformed the other two models. The 3D U-Net performs worse as the missing rate increases and is even worse than the 2D cWGAN. Therefore, our findings suggest that the 3D cWGAN is a promising approach to use for interpolating 3D seismic data since it can achieve highly accurate predictions in all our test results.

5. Discussion and Conclusions

In this study, we aimed to interpolate 3D seismic data using three deep learning models, namely, 2D cWGAN, 3D cWGAN, and 3D U-Net, and evaluate their prediction accuracy by comparing the interpolation results. To accomplish this goal, we used the Sleipner CO2 4D seismic survey dataset and assessed the performance of each model using MSE, PSNR, and SSIM metrics.

Our analysis revealed that the 3D U-Net exhibited a high accuracy only when the training and test data were generated using similar patterns of missing traces, such as regularly 50% missing traces in both sets. However, its performance significantly decreased when applied to other types of missing data. In contrast, the 3D cWGAN demonstrated high-accuracy predictions not only for the test data generated with similar patterns to the training data but also for other types of missing data. Moreover, when comparing the prediction results in the inline direction, the 3D cWGAN showed smaller residuals compared to the other two models. Therefore, among the three models, the 3D cWGAN performed data prediction in the inline and crossline directions with relatively high accuracy.

Although the 3D cWGAN exhibited the best performance in this study, it is important to note that there may be other deep learning models or optimization techniques that could further improve the accuracy of the interpolation of the data. Nevertheless, our findings suggest that the 3D cWGAN is a promising approach to use for interpolating 3D seismic data as it can achieve highly accurate predictions even when dealing with data patterns that are not similar to the training data. This could have significant implications for the field of geophysics and seismic exploration, as the accurate interpolation of 3D seismic data is critical for identifying and characterizing subsurface geological structures.

In conclusion, our study demonstrates that deep learning models, such as the 3D cWGAN, have the potential to significantly improve the accuracy of interpolating 3D seismic data. Our results also highlight the need to conduct further research on the development of advanced deep-learning models and optimization techniques that can further enhance the performance of these models. Ultimately, our findings could help pave the way for a more accurate and efficient interpretation of subsurface geological structures, which could have important implications for a wide range of industries, including oil and gas exploration, mining, geothermal energy production, and CO₂ injection monitoring.