We have explored an unsupervised deep learning (DL)-based approach for the efficient and effective reconstruction of noisy and incomplete (N-I) seismic data. This method does not require clean and complete (C-C) seismic data as labeled data. In each iteration, the seismic data input to the network is first subjected to patching techniques, dividing the 2-D or 3-D data into many 1-D signals. Then, to boost the efficiency, a reconstruction error-based patch selection (REBPS) is employed to choose patches that contain more complex structures, which are then fed into the network. The network adopts an encoder and corresponding decoder architecture to compress and reconstruct data features, attenuating noise within the seismic data and performing an initial reconstruction of the missing parts. To improve the reconstruction accuracy, we employ the projection onto convex sets (POCS) algorithm, ultimately obtaining reconstructed data from one iteration. In this process, the output results of each POCS iteration serve as the input for the next round. Through experimental verification using both synthetic and field seismic data, the results show that our proposed method surpasses other comparative methods in the quality of seismic data reconstruction.

Microseismic data are typically characterized by a low signal-to-noise ratio (SNR), which makes their processing challenging. Therefore, increasing the SNR value of microseismic data by attenuating noise is a critical step in data processing. Traditional denoising methods mostly rely on prior knowledge and have low efficiency in large-scale data scenarios. Recently, intelligent denoising methods have become a focal point in the field of microseismic denoising. However, the existing deep learning methods commonly focus on either the time or frequency domain, thus failing to adequately leverage the time-frequency information in the process of microseismic signal separation from noise. To address this shortcoming, this study proposes a dual cross-attention fusion network named DCAF-Net, which is designed by combining the time and wavelet domains. The proposed DCAF-Net replaces the pooling layers used for downsampling with a discrete wavelet transform, capturing frequency information from the decomposition coefficients and enhancing a microseismic signal’s ability to represent details across multiple scales. In addition, a spatial-transformer attention block based on the Transformer encoding structure is designed to improve the feature extraction capacity of microseismic signals prior to the skip connections. Further, an adaptive attention fusion block is introduced to the proposed network to enhance the fusion of microseismic data at different scales, thus effectively achieving signal-to-noise separation. The proposed model is verified and compared with the representative methods, such as bandpass filtering, wavelet transform, the DnCNN model, and the U-Net model. The results indicate that at a low SNR value, the proposed DCAF-Net model can improve the SNR by an average of 52.46% compared to the U-Net model. Meanwhile, the proposed approach is compared with two representative frequency-domain-based deep learning denoising methods, namely the DeepDenoiser and DeepSeg methods. The results of extensive experiments conducted on both synthetic and field microseismic data demonstrate that the proposed DCAF-Net model can enhance the amplitude preservation and noise suppression capabilities of microseismic signals compared to the existing models, ensuring optimal performance.

Full waveform inversion (FWI) plays an important role in velocity modeling due to its high-resolution advantages. However, its highly non-linear characteristic leads to numerous local minimums, which is known as the cycle-skipping problem. Therefore, effectively addressing the cycle-skipping issue is crucial to the success of FWI. Well-log data contain rich information about subsurface medium parameters, providing inherent advantages for velocity modeling. Traditional well-log data interpolation methods to build velocity models have limited accuracy and poor adaptability to complex geological structures. We propose a well interpolation algorithm based on a generative diffusion model (GDM) to create initial models for FWI, trying to address the cycle-skipping problem. By integrating well-log data, migration images to encode physics-based geological priors in the velocity-model building process, our approach can provide much more detailed information of the faults and stratigraphic features. Numerical experiments demonstrate that the method produces accurate and reliable initial models. Compared to the conventional purely data-driven methods, our approach significantly enhances FWI performance and effectively mitigates the cycle-skipping issues.

3-D salt segmentation is important for many research topics spanning from exploration geophysics to structural geology. In seismic exploration, 3-D salt segmentation is directly related to the velocity modeling building that affects many processing steps, such as seismic migration and full waveform inversion. Manually picking the salt boundary becomes prohibitively time-consuming when the data size is too large. Here, we develop a highly generalized fully convolutional DenseNet for automatic salt segmentation. A squeeze-and-excitation network is used as a self-attention mechanism for guiding the proposed network to extract the most significant information related to the salt signals and discard the others. The proposed framework is a supervised technique and shows robust performance when applied to a new dataset using transfer learning and a small amount of training data. We test the robustness of the proposed framework on the Kaggle TGS salt segmentation dataset. To demonstrate the generalization ability of the framework, we further apply the trained model to an independent dataset synthesized from the 3-D SEAM model. We apply transfer learning to finely tune the trained model from the TGS dataset using only a small percentage of data from the 3-D SEAM dataset and obtain satisfactory results.

Accurate estimation of reservoir parameters (e.g., permeability and porosity) helps to understand the movement of underground fluids. However, reservoir parameters are usually expensive and time-consuming to obtain through petrophysical experiments of core samples, which makes a fast and reliable prediction method highly demanded. In this article, we propose a deep learning model that combines the 1-D convo- lutional layer and the bidirectional long short-term memory network to predict reservoir permeability and porosity. The mapping relationship between logging data and reservoir parameters is established by training a network with a combination of nonlinear and linear modules. Optimization algorithms, such as layer normalization, recurrent dropout, and early stopping, can help obtain a more accurate training model. Besides, the self-attention mechanism enables the network to better allocate weights to improve the prediction accuracy. The testing results of the well-trained network in blind wells of three different regions show that our proposed method is accurate and robust in the reservoir parameters prediction task.

Diffractions in the seismic data are associated with the small-scale subsurface structures, thus their separation and imaging are helpful in characterizing the underground discontinuities with a high resolution that cannot be reached by traditional reflection imaging methods. Traditional seismic slope-based diffraction separation methods are strongly affected by the accuracy and stability of the slope estimation methods, e.g., the plane-wave destruction (PWD) method. When the local seismic slope is not properly estimated, the separated seismic diffraction waves suffer from the mixture between the reflection and diffraction energy due to their coupling in the slope map. We propose an automatic local rank-reduction (LRR) method to separate 3D diffraction waves from zero-offset seismic data, based on which we conduct 3D migration to output the diffraction images. Due to the difficulty in choosing the rank in each local 3D window, we apply an adaptive strategy to obtain the optimal rank. The proposed LRR method with adaptively selected ranks (LRRA) is applied to several 3D synthetic and field data examples and demonstrated to perform better than the traditional PWD, the LRR, and the global rank-reduction (GRR) methods.

The footprint noise is a type of coherent noise that arises from the acquisition geometry. The footprint noise is commonly seen in 3-D seismic volumes and greatly affects the amplitude-based processing and interpretation steps in the seismic exploration workflow. Thus, removal of the footprint noise is necessary for warranting a reliable interpretation output of seismic data processing. However, because footprint noise is usually weak and also spatially coherent, it is inevitable to cause damages to useful signals during its removal steps. Here, we propose a dictionary learning (DL)-based method to effectively remove the footprint noise. We design an algorithm framework to effectively learn the dictionary atoms of the signal waveforms and separate the features of the footprint noise from the learned atoms. Considering the special features of the footprint noise in the dictionary atoms, we propose a statistics-guided way to separate the dictionary atoms into footprint-affected and footprint-free atoms. Then, the footprint-affected atoms are processed via a 2-D median filtering step. The combination between the untouched footprint-free atoms and filtered footprint-affected atoms result in a better dictionary of the signal waveforms and the footprint atoms. We use residual DL to encode the input data by a linear combination of signal atoms and footprint atoms. Removal of footprint atoms and their corresponding sparse coefficients leads to a successful footprint removal. We use both 3-D synthetic and field data examples to demonstrate the effectiveness of the proposed method.

The omnidirectional plane-wave destruction (OPWD) algorithm can estimate the large dip by using circle-interpolating plane-wave destruction (PWD) filter but at the expense of causing potential instabilities due to the small values of the denominator in the regularized division problem. To mitigate the instability, one needs to use a relatively larger smoothing radius for a stronger regularization of the element-wise division, which however significantly decreases the resolution of dip estimation. We propose a new OPWD method without compromising the dip resolution for the large dip calculation by applying a nonstationary smoothness constraint to the model. We use a larger smoothing radius for areas that tend to cause instabilities and vice versa. The nonstationary smoothing is carried out in a simple and efficient recursion way. The synthetic and real seismic data examples demonstrate the performance of the proposed algorithm.

Due to its time-variant nature, the computationally expensive hyperbolic Radon transform (RT) is not easy to be accelerated, e.g., based on the convolution theorem in the frequency domain. However, the hyperbolic RT better matches the trajectories of reflection events in prestack gathers than other time-invariant RTs, e.g., linear or parabolic RTs. Hence, despite its large computational cost, the time-domain hyperbolic RT is still preferred in many seismic processing applications. We propose a fast high-resolution hyperbolic RT (HRHRT) with a fast butterfly algorithm. The forward and adjoint RTs can be greatly accelerated based on a fast butterfly algorithm by reformulating the time-space domain Radon operator as a frequency-domain Fourier integral operator (FIO). The fast butterfly algorithm solves the FIO problem by a blockwise low-rank approximation scheme. The single-step hyperbolic RT can be much faster (e.g., hundreds of times faster for a large problem) than the traditional implementation, resulting in a significant computational boost when the transform is taken in an iterative fashion to estimate the high-resolution Radon coefficients. We demonstrate the similar performance and the much different computational efficiencies between the proposed fast HRHRT and the traditional method over several different problems, i.e., random noise suppression, big-gap seismic reconstruction, and multiples attenuation.

The simultaneous-source technique improves the acquisition efficiency in marine seismic exploration. A high-quality deblending procedure is an important step when implementing the simultaneous-source technique. Deblending is often implemented in a domain (other than the common-source domain) where the blending noise is incoherent. In this article, we present a denoising method based on a deep convolutional neural network (CNN) for deblending. The denoising generalization ability of CNN-based denoising on real seismic data is limited because the training dataset, especially the labeled data, is often difficult to acquire. To make the CNN applicable to real data, we generate the training dataset directly from the real common-shot blended record itself, which has the same dynamic seismic wavefield characteristics as the real data. The input dataset for the CNN is acquired by adding A to B. A is the random time-delay encoding data of each trace of the common-shot blended data, while B is the data of each trace of the common-shot blended data; then, the blended data naturally become labeled data. The CNN model obtained through deep learning is used to remove the blending noise to complete the separation of blended data. Numerical tests using synthetic data and real data show that our method can provide high-precision separation results.

The local signal-to-noise orthogonalization algorithm has been widely used in the community of seismic processing and imaging. It helps orthogonalize the signal-and-noise components in an elegant way so that the noise does not contain the signal leakage in seismic denoising. The traditional local signal-to-noise orthogonalization is based on solving a highly underdetermined, ill-posed inverse problem with local smoothness constraint. Due to the inversion nature, the local orthogonalization method requires a large number of iterations and thus is computationally demanding in large-scale applications. Here, we proposed a much accelerated signal-and-noise orthogonalization method, where we design an efficient way for calculating the orthogonalization weight. When new samples are involved in the calculation, we calculate the orthogonalization weight of the new samples by connecting them with the calculated weights of the previous samples. The orthogonalization weight needs to be smoothed and scaled after all samples have been processed to make the resulted orthogonalization weight smooth across the seismic data and match the amplitude level of the initially suppressed noise. In this way, we avoid iterations when calculating the orthogonalization weight. We apply the proposed method to several synthetic and field data examples, have a benchmark comparison with state-of-the-art algorithms, and demonstrate its much accelerated efficiency compared with the traditional local signal-and-noise orthogonalization.

We propose to use unsupervised deep learning (DL) and attention networks to mute the unwanted components of the single-channel earthquake data. The proposed algorithm is an unsupervised technique that does not require any prior information about the input data, i.e., no need for the labeled data. The imaginary and real parts of the short-time frequency transform (STFT) are divided into several overlapped patches to be the input of the proposed DL network, while the output target is the absolute value of the STFT. The proposed DL network utilizes a customized loss function to reconstruct the signal mask, where the STFT components related to the seismic noise are muted. An adaptive thresholding technique is utilized to obtain the binary mask, which is multiplied by the real and imaginary parts of the input seismic data. The binary mask has zero values for the samples corresponding to the unwanted components and ones for the seismic signal components. Then, inverse STFT is used to reconstruct the denoised signal. The proposed algorithm is evaluated using samples from the STanford EArthquake Dataset (STEAD) and the results are compared to the benchmark denoising method, i.e., DeepDenoiser. As a result, the proposed algorithm shows a robust denoising performance and outperforms the DeepDenoiser method by 1.95 dB in terms of signal-to-noise ratio.

Effective random noise attenuation is critical for subsequent processing of seismic data, such as velocity analysis, migration, and inversion. Thus, the removal of seismic random noise with an uncertainty level is meaningful. Attenuating 3-D random noise in a supervised way based on deep learning (DL) is challenging because clean labels are difficult to obtain. Therefore, it is necessary to develop an adaptive unsupervised-based method for random noise attenuation. In this article, we propose a deep-denoising unsupervised learning (DDUL) network to attenuate random noise in 2-D/3-D seismic data. A patching technique is used to split 2-D/3-D seismic data into several patches to be fed into the network, which helps to expand the number of samples for training. We use the fully symmetrical structure of the autoencoder to construct the network. In each corresponding encoder and decoder layer, skip connections are added to enhance the learning of seismic data features. We construct three blocks to extract waveform features in seismic data, i.e., encoder, decoder, and skip blocks. Among them, the skip is connected between the encoder and decoder blocks of each hidden layer. The use of multiple blocks not only improves the network’s ability to extract seismic data features but also solves the problem of excessive training parameters caused by hidden layer stacking. Five 2-D/3-D synthetic and field seismic datasets are used to test the denoising performance of our proposed method. The denoising results demonstrate that our proposed method has good signal-preserving and noise attenuation capabilities in real-world applications.

Deblending plays an important role in preparing high-quality seismic data from modern blended simultaneous-source seismic data. The conventional methods become problematic when the random ambient noise becomes extremely strong and the inversion iteratively fits the random noise instead of the signal and blending interference. In this article, an improved method is proposed to separate blended seismic data. We treat the deblending problem as a regularization problem under the frame of compressed sensing (CS), and propose a new nonlinear Bregman iterative shaping (BIS) algorithm to solve the minimization problem. Based on the principle of CS, we transform the simultaneous-source data to the seislet-domain for inversion and separation, and verify the effectiveness and superiority of the proposed method by two theoretical model data and an actual marine streamer blended data. Compared with the traditional iterative shaping (TIS) algorithm, BIS adds data residuals to the observed data (the blended data) in each iteration, so that the error between the calculated data after each iteration and the main source signal in the known blended data is the smallest, and helps to recover the weaker information in the seismic data and obtain higher precision separation results. BIS uses a fixed soft threshold, which speeds up the convergence speed. In addition, BIS can effectively improve the separation accuracy when the given sparse domain threshold is not suitable.

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.

It is widely known that strong noise can decrease the quality of seismic data. However, the anelastic attenuation could be more important to account for the weak amplitude and low quality of seismic data. Here, we develop an inversion framework to simultaneously compensate for the attenuation of seismic data and remove noise, thereby enhancing the quality of seismic data. Instead of directly applying a compensation operator to the input seismic data, we formulate an inverse problem that connects the sparse reflectivity model and the raw seismic data via the convolution and attenuation functions. The random noise is assumed to be the unpredicted part of the forward modeling process. We use the L2-norm regularization for the data misfit and impose a sparsity constraint onto the reflectivity series, e.g., using the L1-norm constraint. We use an iterative preconditioned conjugate gradient method to solve the L1-norm constrained least-squares optimization problem and obtain the reflectivity series. The denoised and compensated data are obtained by applying the convolution operator to the reflectivity. We use several synthetic and field seismic data to illustrate the effectiveness of the presented method.

Because a high signal-to-noise ratio (SNR) is beneficial to the subsequent processing procedures, the noise attenuation is important. We propose an adaptive random noise attenuation framework based on convolutional neural networks (CNNs). The framework transforms the target function from effective signal learning to noise learning through residual learning, so as to improve the training efficiency. After sufficient training, the network transfers the learned seismic data features using a large synthetic data set to the testing of complex field data with unknown noise levels and, thus, attenuates the noise in an unsupervised way. Unsupervised noise reduction requires certain representativeness of the training data and a sufficient amount of training data sets. In the network architecture, we introduce residual learning and batch normalization (BN) to reduce the training parameters of the network, thereby shortening the time for feature learning. The activation function with leakage correction function can effectively retain negative information, and its combination with the double convolutional residual block can enhance the generalization ability and feature extraction performance of the network. In the test of synthetic data and complex field data with unknown noise levels, by comparing the noise reduction results of some classic denoising algorithms, the adaptive CNN proposed in this article can more effectively attenuate the noise and reconstruct the seismic waveform.

Inspired by image-to-image translation, we applied deep learning (DL) to regularly missing data reconstruction, aimed at translating incomplete data into their corresponding complete data. With this purpose in mind, we first construct a network architecture based on an end-to-end U-Net convolutional network, which is a generic DL solution for various tasks. We then meticulously prepare the training data with both synthetic and field seismic data. This article is implemented in Python based on Keras (a high-level DL library). We described the network architecture, the training data, and the training settings in detail. For training the network, we employed a mean-squared-error loss function and an Adam optimization algorithm. Next, we tested the trained network with several typical data sets, achieving good performances (even in the presence of big gaps) and validating the feasibility, effectiveness, and generalization capability of the assessed framework. The feature maps for a sample going through the well-trained network are uncovered. Compared with the f-x prediction interpolation method, DL performs better and is capable of avoiding several assumptions (e.g., linearity, sparsity, etc.) associated with conventional interpolation methods. We demonstrated the influences of the network depth, the kernel size of the convolution window, and the pooling function on the DL results. We applied the trained network to dense data reconstruction successfully. The proposed method can overcome noise to some extent. We finally discussed some practical aspects and extensions of the evaluated framework.

The seismic data usually contains strong random noise, which impedes the effective usage of the seismic signals for imaging and inversion. We propose an effective seismic denoising method based on a least-squares decomposition model. We assume that each trace in the multi-channel seismic data can be decomposed into several smoothly variable components. Since the decomposition is basically an inverse problem, we apply the temporal smoothness to constrain the inversion and control the stability. Considering the spatial coherency in a multi-channel seismic data, we also apply the spatial smoothness constraint to the decomposition. The space constraint is applied along the structural direction of the seismic events to preserve the dipping energy. The structural constraint is equivalent to applying a structure-oriented smoothing that requires the estimation of the local slope from the input noisy data. We validate the effectiveness of the proposed algorithm via several synthetic and real seismic data. The proposed method outperforms the state-of-the-art single-channel and multi-channel algorithms even in the case of strong random noise.

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.

Multichannel singular spectrum analysis (MSSA) is an effective tool for processing multidimensional time-series such as the reconstruction of high-dimensional seismic data. Low-rank estimation is a key stage in MSSA algorithm that can affect the recovery process. Truncated singular value decomposition (TSVD) and singular value thresholding (SVT) are two conventional options for rank reduction, which, however, do not result in satisfactory outcomes, especially in low signal-to-noise-ratio cases. In this letter, we propose to leverage the optimal low-rank estimator that emerges from random matrix theory known as OptShrink. The OptShrink can obtain more robust low-rank estimation in comparison with TSVD and SVT. In addition, we propose to constrain the singular values using a damping factor. The proposed damped OptShrink method is applied on real and synthetic 3-D seismic data. The comprehensive experiments and discussion verify the superior reconstruction ability of the proposed method in comparison with MSSA.

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.