Seismic data reconstruction is a crucial step in seismic data processing, which faces numerous challenges in accurately capturing the complex patterns and structures within the data. Deep learning is revolutionizing the processing of seismic exploration data, enabling more precise imaging of subsurface structures and improving the detection of potential oil and gas reservoirs. We propose a modified least-squares generative adversarial network incorporating the inverse interpolation (LSGAN-II) method for seismic data reconstruction. Our approach integrates inverse interpolation algorithms into a generative adversarial network (GAN) framework to enhance both the accuracy and efficiency of seismic data reconstruction. Initially, a GAN is employed to predict local event slopes in seismic data, where a least-squares loss function is utilized to improve the learning capacity and convergence speed of the deep learning network. Subsequently, the predicted local event slopes are used as regularization operators in inverse interpolation, effectively enhancing the precision of the reconstructed seismic data. A dual-discriminator-based LSGAN-II is developed to implement the seismic data reconstruction process. This network combines the benefits of both GAN and inverse interpolation techniques, utilizing dual discriminator loss functions and a least-squares error loss function to optimize the prediction of local event slopes and the reconstruction of seismic data. The testing of the LSGAN-II method is conducted using two synthetic datasets and one actual dataset, demonstrating its effectiveness and practicality in seismic data reconstruction.

Simultaneous source technology can enhance the efficiency of seismic data acquisition, however, blending noise significantly hinders subsequent seismic data processing. To deal with blending noise, an improved iterative deblending algorithm is proposed to separate simultaneous source data. Considering the spatial incoherence between useful signal and blending noise in certain sorting domains, including common receiver domain and common offset domain, blended data was transformed from the time-space (t−x) domain to the frequency-wavenumber (f−k) domain to extract useful signal. However, the sparse representation capability of two-dimensional (2D) fast Fourier transform (FFT) is limited, particularly, for the case that weak coherent signal is contaminated by strong incoherent noise. To relieve this issue, a filter named multistage median bilateral filtering (BFMLM) was integrated into the iterative deblending framework to enhance the sparse representation of 2D FFT by attenuating blending noise. This improvement can maximize the extraction of useful signal and significantly reduce noise leakage, thereby achieving better deblended data. Two sets of synthetic data and two sets of field data were applied to evaluate the deblending performance of the methods based on 2D FFT, curvelet transform (CT) and the proposed method. With BFMLM, the deblending quality of the proposed method is better than that of FFT method. Comparing the deblending method by CT, the proposed method can obtain the similar deblending performance with less computational cost.

Deep learning methods have gained widespread popularity for effectively suppressing random noise in seismic data. The recent progress in techniques based on supervised learning for attenuating seismic random noise underscores their potential, particularly when an abundant set of training examples is accessible. Unfortunately, collecting an adequate amount of representative training samples is not always feasible. DIP aims to capture a lot of low-level statistical information by using rich implicit prior knowledge inherent in the structure of the generation network itself. Therefore, it is not essential to provide a training database or uncontaminated data as a truth label, whereas only requires a noisy seismic image. In order to boost the performance, we add an explicit prior, weighted total variation, to the standard DIP, which leverages sparsity-promoting priors and restricts the solutions of DIP to satisfy a prior inherent in the seismic data. The proposed method is tested on synthetic seismic data with random noise that follows different distributions, then is applied to field pre- and post-stack seismic data. Furthermore, a comparison is drawn between the new method and the traditional DIP-based denoising method in terms of signal to noise ratio and local similarity. Our method shows more promising results because prior information from both the structure of the network and the seismic data is considered in the denoising processing.

The geostatistics stimulation method, as an important tool in subsurface modeling, is crucial for hydrocarbon reservoir characterization. Using geostatistical methods to reproduce complex heterogeneous structures is still challenging because of nonstationarity and computational consumption. We develop a stabilized stochastic simulation method by introducing the generative adversarial network based on a single image (SinGAN). It can preserve multiscale features contained in an individual training image by using a multistage training framework. In order to stabilize the training of the discriminator, the spectral normalization is integrated. We also introduce spatial attention and channel attention mechanism into the network to focus on the most significant features in each training stage, so that these features can be reproduced in the realizations. An adaptive strategy is adopted to automatically choose training stages, which balances the diversity and quality of simulation results and decreases the man-made factor on SinGAN. Several experiments are tested on synthetic and actual training images, respectively. We evaluate the simulation results from many perspectives, including variability, connectivity, probability density distribution, and time-consuming. The successful application of the new method on both categorical and continuous variables indicates that it has a strong ability to reproduce complex subsurface models, even for nonstationary geologic phenomena.

Simultaneous source acquisition is becoming more promising than the traditional seismic acquisition by firing multiple sources with a short interval time, which improves acquisition efficiency and enhances data quality. However, the blended interference severely obscures the coherent signal, challenging the conventional seismic data processing methods. Recently, convolution neural network (CNN) has been successfully implemented to address the blended interference. Different from CNN, the self-attention mechanism based Transformer neural network is good at capturing the global features. In this letter, we propose a Deblending Transformer (DT) based on Transformer module to separate the simultaneous source data. The DT architecture mainly includes linear embedding operation, patch partition based Transformer block and output projection layer. The patch partition algorithm is embedded into the multi-head self-attention module, which extracts the vertical, horizontal and local information. In addition, with the help of linear embedding operation and output projection algorithm, the DT can easily extract the global features from the input. Experiments on synthetic and field data demonstrate that the proposed method has better deblending performance than the U-net based and curvelet based methods.

Seismic data interpolation and denoising play vital roles in obtaining complete and clean data in seismic data processing. Seismic data usually misses along various spatial axes and always mix with random noise. In order to obtain complete and clean seismic data, reconstruction technology can interpolate missing data and attenuate random noise. Nonsubsampled contourlet transform is an effective transform to obtain multi-scale and multi-direction sparse domain data for compression sensing interpolation and denoising. However, conventional iterative shrinkage/thresholding cannot handle ill-posed and ill-conditioned equations for solving linear inverse problem. We present a two-step iterative log thresholding method to overcome ill-posed and ill-conditioned problems and improve the convergence rate and solution accuracy, which can interpolate and denoise seismic data simultaneously in the nonsubsampled contourlet transform framework. First, we use nonsubsampled contourlet transform to convert the seismic missing data with random noise to sparse domain. Then, we apply two-step iterative log thresholding algorithm to interpolate and denoise data in sparse domain. The result of each iteration is based on the results of the previous two iterations, which can accelerate convergence rate. In addition, log thresholding can further improve convergence rate and solution accuracy. Finally, we use inverse nonsubsampled contourlet transform to obtain the interpolated and denoised seismic data. The new method can reconstruct the irregularly missing data and attenuate random noise to obtain complete and clean seismic data with high accuracy, which is crucial for seismic imaging and inversion. We demonstrate the applicability and effectiveness of this simultaneous interpolation and denoising technique with successful applications to both synthetic and field data examples.

Unlike the conventional seismic acquisition, simultaneous-source acquisition allows the overlap in the record, which can enhance acquisition efficiency and improve image quality. To further explore the advantage of simultaneous-source technology, a double blending survey is proposed, which contains the self interference and cross interference. In the designed survey, the self and cross interference are controlled by two factors (inline shot interval and source number). When the inline shot interval is larger than the efficient record length (ERL), the blending scheme is degraded to the cross simultaneous-source survey. When the source number is equal to one, the blending scheme is degraded to the self simultaneous-source survey. To suppress the intense hybrid interference, the mixed regularization term integrating sparse constraint and rank-reduction constraint is applied to provide the stronger regularization ability. Compared with the individual penalty term, the mixed constraint can further suppress hybrid interference and obtain better deblending performance. Deblended results on synthetic and field data examples confirm the performance of mixed constraint and demonstrate that the designed hybrid simultaneous-source survey can further enhance the efficiency of acquisition.

Acquisition technology of multisource data has outstanding advantages in enhancing collection efficiency and reducing cost. However, traditional seismic data processes cannot be applied to multisource blended data. Therefore, deblending technology of multisource data is the key to the research. In this letter, we propose a deblending method of multisource seismic data based on a periodically varying cosine code (PVCC). First, we design a PVCC to blend the seismic data. Next, the blending model is transformed into the minimum problem of the objective function. Then, the blended data are decomposed in the curvelet domain. Finally, the main source data are separated based on sparse inversion. Furthermore, we use the edge processing to eliminate the boundary effect in the processed seismic data. The examples of synthetic data and field data are adopted to demonstrate that the proposed method has great potential in the deblending of multisource data. In addition, the edge processing can effectively suppress the boundary effect.

A sparse dictionary is more adaptive than a sparse fixed-basis transform since it can learn the features directly from the input data in a data-driven way. However, learning a sparse dictionary is time-consuming because a large number of iterations are required in order to obtain the dictionary atoms that best represent the features of input data. The computational cost becomes unaffordable when it comes to high-dimensional problems, e.g., 3-D or even 5-D applications. We propose an efficient high-dimensional dictionary learning (DL) method by avoiding the singular value decomposition (SVD) calculation in each dictionary update step that is required by the classic$K$-singular value decomposition (KSVD) algorithm. Besides, due to the special structure of the sparse coefficient matrix, it requires a much less expensive sparse coding process. The overall computational efficiency of the new DL method is much higher, while the results are still comparable or event better than those from the traditional KSVD method. We apply the proposed method to both 3-D and 5-D seismic data reconstructions and demonstrate successful and efficient performance.

Simultaneous-source acquisition, breaking the limit of conventional seismic acquisition, is a rapidly evolving research field, due to its advantage in reducing survey time and improving data quality. The benefits of simultaneous-source acquisition are compromised by the intense blending interference. Separating a blended record into a group of individual records, known as “deblending” is one of the most popular solution to the problem. However, the blended records are often corrupted by random noise, which causes difficulties in separation. In an iterative deblending algorithm, the incoherent interference can be simulated and subtracted from the blended record. When the random noise is strong, it is difficult to simulate the incoherent interference. In this paper, we propose a hybrid-sparsity constraint model that applies the dictionary learning into the deblending framework that is based on the sparsity-promoting transform to deal with extremely noisy simultaneous source data. The dictionary learning with fine-tuned adaptation can learn the incoherent interference into atoms and reject random noise. Then, the sparse transform-based framework is implemented to iteratively separate the signal and interference. We use two synthetic examples to demonstrate the advantage of the proposed method in extremely noisy situations. Two field examples further confirm the superior deblending performance of the proposed method for the noisy simultaneous-source data over the curvelet transform-based and rank reduction-based methods.

Seismic data interpolation and reconstruction play an important role in seismic data processing. Seismic data are often inadequately sampled along various spatial axes. We have developed a new approach to interpolate aliased multidimensional seismic data based on the multidimensional adaptive prediction-error filter in frequency domain. First, we estimate the adaptive prediction-error filter coefficients, then interpolate missing traces using the estimated coefficients. Shaping regularization is used to control the smoothness of frequency-domain multidimensional adaptive prediction-error filter coefficients. Instead of estimating prediction-error filter coefficients only along one direction space, we estimate the prediction-error filter coefficients using more information along different direction spaces. So, multidimensional adaptive prediction-error filter using regularized nonstationary autoregression can adaptively estimate seismic events whose slopes vary in multidimensional space. The frequency-domain multidimensional interpolation method can input data at temporal frequency, which can save computer memory and time. For multidimensional seismic data, which miss different number of traces regularly in different axes, the proposed method can be used to interpolate missing traces to obtain more accurate results. The proposed method improves the calculation efficiency by applying shaping regularization and implementation in the frequency domain. The applicability and effectiveness of the proposed method are examined by synthetic and field data examples.

We propose a novel approach for removing noise from multiple reflections based on an adaptive randomized-order empirical mode decomposition (EMD) framework. We first flatten the primary reflections in common midpoint gather using the automatically picked normal moveout velocities that correspond to the primary reflections and then randomly permutate all the traces. Next, we remove the spatially distributed random spikes that correspond to the multiple reflections using the EMD-based smoothing approach that is implemented in the f-x domain. The trace randomization approach can make the spatially coherent multiple reflections random along the space direction and can decrease the coherency of near-offset multiple reflections. The EMD-based smoothing method is superior to median filter and prediction error filter in that it can help preserve the flattened signals better, without the need of exact flattening, and can preserve the amplitude variation much better. In addition, EMD is a fully adaptive algorithm and the parameterization for EMD-based smoothing can be very convenient.

Observed seismic data are mostly irregularly sampled and seismic data interpolation is an essential procedure to provide accurate complete data for seismic data analysis, such as amplitude-versus-offset analysis, multiple suppression, and wave-equation migration. The well-known minimum weighted norm interpolation (MWNI) method could achieve a relatively good result. However, the algorithm needs many iterations and thus the total calculation is expensive. In this letter, we propose a fast interpolation algorithm. Instead of wavenumber spectrum, singular spectrum can give a more accurate description of the sampled data. We use shaping regularization to control the smoothness of the singular values matrix. Compared with the conventional MWNI method, we test the proposed method on both synthetic and field data sets. The results confirm that our proposed method is more effective.

The proximal splitting algorithm, which reduces complex convex optimization problems into a series of smaller subproblems and spreads the projection operator onto a convex set into the proximity operator of a convex function, has recently been introduced in the area of signal processing. Following the splitting framework, we propose a novel three-operator proximal splitting (TOPS) algorithm for 3-D seismic data reconstruction with both singular value decomposition (SVD)-based low-rank constraint and curvelet-domain sparsity constraint. Compared with the well-known forward-backward splitting (FBS) method, our proposed TOPS algorithm can be flexibly employed to recover a signal satisfying double convex constraints simultaneously, such as low-rank constraint and sparsity constraint used in this letter. We have used both synthetic and field data examples to demonstrate the superior performance of the TOPS method over traditional SVD-based low-rank method and curvelet-domain sparsity method based on the FBS framework.

Optimal stacking of multiple data sets plays a significant role in many scientific domains. The quality of stacking will affect the signal-to-noise ratio and amplitude fidelity of the stacked image. In seismic data processing, the similarity-weighted stacking makes use of the local similarity between each trace and a reference trace as the weight to stack the flattened prestack seismic data after normal moveout correction. The traditional reference trace is an approximated zero-offset trace that is calculated from a direct arithmetic mean of the data matrix along the spatial direction. However, in the case that the data matrix contains abnormal misaligned trace, erratic, and non-Gaussian random noise, the accuracy of the approximated zero-offset trace would be greatly affected, and thereby further influence the quality of stacking. We propose a novel weighted stacking method that is based on principal component analysis. The principal components of the data matrix, namely, the useful signals, are extracted based on a low-rank decomposition method by solving an optimization problem with a low-rank constraint. The optimization problem is solved via a common singular value decomposition algorithm. The low-rank decomposition of the data matrix will alleviate the influence of abnormal trace, erratic, and non-Gaussian random noise, and thus will be more robust than the traditional alternatives. We use both synthetic and field data examples to show the successful performance of the proposed approach.

The low-rank approximation method is one of the most effective approaches recently proposed for attenuating random noise in seismic data. However, the low-rank approximation approach assumes that the seismic data has low rank for its f - x domain Hankel matrix. This assumption is seldom satisfied for the complicated seismic data. Besides, the low-rank approximation approach is usually implemented in local windows in order to satisfy the principal assumption required by the algorithm itself. When implemented in local windows, the rank is even more difficult to choose because the seismic data is highly nonstationary in both time and spatial dimensions and the optimal rank for different local windows is not consistent with each other. In order to preserve enough useful energy, one needs to set a relatively large rank when implementing the low-rank approximation method, which makes the traditional method incapable of attenuating enough noise. Considering such difficulties described above, we propose an empirical low-rank approximation approach. We adaptively decompose the input data into several components that have truly low ranks via empirical mode decomposition. An interpretation of the proposed empirical low-rank approximation method is that we empirically decompose a multi-dip seismic image that is not of low rank into multiple single-dip seismic images that are low-rank individually. We use both synthetic and field data examples to demonstrate the superior performance of the proposed approach over traditional alternatives.

The seislet transform can be used to interpolate regularly undersampled seismic data if an accurate local slope map can be obtained. The dealiasing capability of such method highly depends on the accuracy of the estimated local slope, which can be achieved by using the low-frequency components of the aliased seismic data in an iterative manner. Previous approaches to solving this problem have been limited to the unstable estimation of local slope via a large number of iterations. Here, we propose a new way to obtain the slope estimation. We first estimate the NMO velocity and then use a velocity-slope transformation to get the optimal local slope. The new method allows us to avoid the iterative slope estimation and can obtain an accurate slope field in one step. The one-step slope estimation can significantly accelerate the iterative seislet domain thresholding process and can also stabilize the iterative inversion. Both synthetic and field data examples are used to demonstrate the performance by using the proposed approach compared with alternative approaches.

Seismic data interpolation or reconstruction plays an important role in seismic data processing. Many processing steps, such as high resolution processing, wave-equation migration, amplitude-versus-offset and amplitude-versus-azimuth analysis, require regularly sampled data. The reconstruction can be posed as an inverse problem, which is known to be ill posed and requires constraints to obtain unique and stable solutions. In this letter, we propose an iterative scheme to interpolate the big gaps with a slope constraint. In the first iteration, the smooth radius must be large to estimate the smooth dip from the decimated data, and a large scaling parameter can guarantee the stability of the inversion. In the later iterations, the smooth radius will be shortened in order to get a more accurate dip estimation from the updated result. When the dip estimation is accurate, a small scaling parameter can not only guarantee the convergence of the inversion but also obtain a result with high signal-to-noise ratio. We compare the proposed method with the well-known projection-onto-convex-sets method on synthetic and field data examples. The interpolation results illustrate the advantage of the proposed method in interpolating the big gaps.