Understanding bird migration across full annual cycles is critical for effective conservation. The recent BirdFlow framework allows researchers to model population-scale migration without physical tracking tags by inferring likely flight paths from weekly population snapshots (such as eBird abundance maps). However, the original framework is Markovian, meaning it assumes a bird’s next move depends only on its current location, completely ignoring its history. Because of this short-term focus, the model cannot capture essential year-long biological behaviors, most notably annual site fidelity, where birds return to the exact same breeding or wintering grounds year after year.

The growth of seismic data recorded by dense arrays has stimulated the development of new array-based receiver function (RF) imaging techniques. This study examines the feasibility and performance of the least-squares migration (LSM) method, a state-of-the-art technique used in exploration seismology, to lithospheric imaging using teleseismic RFs. Taking advantage of a pair of forward (demigration) and adjoint (migration) operators, the LSM casts migration as a regularized least-squares optimization problem. We employ the split-step Fourier method to design the two operators and conduct wavefield propagation in heterogeneous media. Synthetic tests with a two-layered crustal model and varying ratio of missing traces demonstrate that LSM is capable of suppressing imaging artifacts and improving imaging resolution compared to conventional migration. Real data application is conducted using teleseismic data recorded by the Himalayan-Tibetan continental lithosphere during mountain building (Hi-CLIMB) array deployed on the Tibetan Plateau. Considering the irregular and noisy recordings from field acquisition, we adopt signal processing algorithms, including the Radon transform and singular spectrum analysis (SSA) filter, to regularize the wavefields and precondition the RFs. The proposed workflow resolves fine-scale crustal structures that are consistent with earlier studies. Overall, our study offers a new high-resolution RF imaging tool and inspires the future development of advanced array processing workflows.

Distributed acoustic sensing (DAS) is emerging in seismic monitoring due to its ultra-dense spatial sampling, durability to harsh environments, and sensitivity to weak ground vibration. Compared with traditional nodal geophones that are normally sparsely distributed, DAS offers unprecedented detectability for small-magnitude earthquake events, very subtle reservoir dynamics, and other weak signals among various applications. The appealing detectability of weak signals is compromised by the terabyte-scale daily continuous record that causes prohibitive storage problems. The current solution is to save only the segmented data of interest, e.g., a certain length around a target event. Here, we tackle the urgent storage problem of DAS monitoring by designing a deep-learning (DL) based compression algorithm. The compression algorithm can be split into two major components. The first part is the encoder based on the vision transformer architecture, where the input multi-channel DAS dataset goes through an encoding process to output the key features from the input. The second part is the decoder, where the features are optimally combined to reconstruct the data of the original scale. The optimal network parameters are obtained via an unsupervised training process, aiming at minimizing the difference between the reconstructed and input data. In the proposed DL-based compression algorithm, only the decoder’s weight parameters and extracted features from the input data through the encoder are saved on the disk, which is sufficient to reconstruct a high-fidelity dataset. The proposed compression algorithm can reach around 50 times the compression rate for a gigabyte-scale DAS dataset without unsatisfactory reconstruction performance.

Distributed acoustic sensing (DAS) has emerged as a widely used technology in various applications, including borehole microseismic monitoring, active source exploration, and ambient noise tomography. Compared with conventional geophones, the fiber optic cable has unique characteristics that allow it to withstand high-temperature and high-pressure environments. However, due to its high sensitivity, the obtained seismic records are often corrupted with unavoidable background noise, which introduces more uncertainty in the subsequent seismic data processing and interpretation. Thus, the development of robust denoising techniques for DAS data is crucial to minimize the impact of noise and enhance the reliability of seismic data processing and interpretation. In this work, we propose a ground-truth-free method for strong background noise suppression in DAS vertical seismic profiling (DAS-VSP) data. Compared to existing deep learning (DL) methods, the proposed approach demonstrates promising generalizability in handling field examples across different surveys. The proposed method consists of four stages: training set extension with a patching scheme, feature selection with a kurtosis-based method, denoising with a deep image prior (DIP)-based unsupervised neural network, and an unpatching approach for denoised data reconstruction. Numerical experiments conducted on synthetic data and several profiles from the Utah FORGE project and the Groß Schönebeck site demonstrate that the proposed method can effectively suppress most of the background noise while preserving hidden signals. Furthermore, the unsupervised learning (USL) approach is unconditionally generalizable when applied to vastly different field data because it does not require pre-labeled datasets for training. The codes related to this article are fully open-source viahttps://github.com/cuiyang512/Unsupervised-DAS-Denoising.

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.

The popularity of blended acquisition is surging in the field of seismic exploration because of its higher efficiency and lower cost. As a compromise, a sophisticated deblending framework should be applied to remove interference noise. However, blended sources are usually fired at off-the-grid (OTG) samples, and the recorded data are incomplete because of some inevitable barriers and instrument errors, increasing the challenges to apply classic deblending methods to OTG data. Typically, the binning process and data reconstruction will be introduced for OTG incomplete data before subsequent deblending. Nevertheless, the binning process may cause amplitude and phase distortion, degrading the deblending accuracy. To overcome this problem, we propose a deep learning (DL)-based method without a binning process for OTG deblending and reconstruction, namely, OTGDR, avoiding the errors related to preprocessing routines. The proposed OTGDR framework contains two components: deep image prior (DIP)-inspired coherency-enhancing network and bilinear operator-guided projection onto convex set (POCS) iteration. The DIP incorporates several fully connected (FC) layers, attention mechanism, and skip connection to extract useful features selectively for superior performance, and the following POCS aims to remove the blending and ambient noise iteratively for enhanced signal-to-noise ratio (SNR). Moreover, the proposed OTGDR is completely data-driven and does not require labels for training, which increases its generality and enables it to adapt to different datasets. In our experiments, we compare the proposed OTGDR with classic deblending methods, and the results demonstrate that OTGDR shows superior performance on OTG denoising and reconstruction in terms of fidelity and SNR.

Seismic data is commonly contaminated by different types of noise with varying amplitudes, increasing the challenges of retrieving effective signals from strong background noise. Such issues become even worse when attenuation effects are considered because the amplitude of seismic waves will dissipate after propagation, making seismic signals easier to be inundated by noise, especially at deeper positions. To implement data enhancement and attenuation compensation without noise amplification, we propose a robust framework in a blind manner to reconstruct attenuation-compensated seismic data with a higher signal-to-noise ratio (SNR), namely, structural-oriented blind Q-compensated denoising (SBQD) method. Unlike classic Q-compensated denoising methods, the proposed SBQD does not require wavelet as a prior and can iteratively estimate wavelet and reflectivity series simultaneously, and the final stationary seismic data can be obtained by convolving the estimated wavelet with the reflectivity series. Moreover, guided by local adaptive structural regularization, the proposed SBQD can provide superior results with higher accuracy and fidelity and remove those noise-related artifacts during denoising and compensation. Compared with conventional two-step methods (e.g., denoising and attenuation compensation), the proposed SBQD can implement denoising and compensation simultaneously, which avoids introducing compensation-related errors (e.g., noise amplification) and effectively preserves useful signals. Synthetic and field examples are used to validate the robustness of the proposed SBQD method on noise removal and attenuation compensation for seismic data.

Natural fractures play a crucial role in the storage and seepage of oil shale. However, identifying fractures using conventional logging techniques presents challenges due to complex response characteristics and severe data imbalance. Here, we propose a highly accurate integrated ensemble learning method, called BSI-extreme gradient boosting (XGBoost), for identifying the natural fracture development, which combines several steps including isolation forests (iForests), synthetic minority oversampling techniques (SMOTEs), and XGBoost, and incorporates rock brittleness as a controlling factor in the model construction process. The proposed model effectively addresses several challenges encountered in fracture identification, including complex logging response characteristics, low precision and recall of fractured labels, and excessive sensitivity of ensemble learning to noise. To do so, the relationship between fracture density and brittle mineral content is analyzed through core analysis and X-ray diffraction (XRD). Then, conventional logging and rock brittleness are used as features for training the model. Herein, by screening the outliers of iForest, SMOTE oversampling, and feature selection, optimal hyperparameters of the model are obtained through the grid search method. The results demonstrated that using BSI-XGBoost, the testing set achieved an accuracy of 92.45%. Comparatively, this accuracy is 4.86% higher than the original XGBoost model and 3.73% higher than the B-XGBoost model, which incorporated brittleness curves but did not include oversampling and outlier removal. Collectively, this workflow provided an effective method for intelligent identification of fractures in oil shale with high accuracy based on easily accessible conventional logging curves.

The inherent non-uniqueness problem challenges acoustic impedance inversion, and thus it is meaningful to explore the possible solutions via advanced strategies, e.g., incorporating uncertainty estimation. At present, several generative adversarial network (GAN)-based inversion methods have been shown to offer advantages in terms of inversion accuracy. However, most of them have primarily focused on deterministic predictions, limiting their ability to explore the range of the solution space. Furthermore, the scarcity of labeled data pairs in field data tasks can reduce inversion accuracy. To address these shortcomings, we introduce a Bayesian GAN (BGAN) based onBayes by Backprop, and integrate it into a closed-loop framework. Synthetic data experiments demonstrate that the closed-loop BGAN performs better than cycle-consistent GAN (cycle-GAN) with insufficiently labeled data pairs. Moreover, unlike the cycle-GAN, the closed-loop BGAN possesses the capability of assessing prediction uncertainties. Compared with the Bayesian linearized inversion (BLI) and Monte Calor (MC) dropout methods, the closed-loop BGAN is more accurate and robust in the inversion of noisy seismic data with lower uncertainty. Therefore, the closed-loop BGAN can achieve high accuracy inversion while estimating potential solutions more reasonably. The field data example also demonstrates that compared with BLI and MC dropout, the closed-loop BGAN can obtain more reasonable inversion results with more reliable uncertainty estimation.

The focal mechanism provides seismological constraints on the geological faults that generate the earthquakes and thus is important for regional seismotectonic research. Focal mechanism calculation based on the P-wave first-motion-polarity is a widely used method, particularly helpful for small to moderate-size earthquakes. However, determining the P-wave first-motion polarity can be challenging and subjective for smaller earthquakes. Here, we propose a deep-learning method (EQpolarity) for determining the P-wave first-motion polarity using the vertical-component seismic waveforms. The proposed deep-learning method was trained using a large-scale dataset from South California and then adapted to the Texas earthquake data via a transfer learning method. The original and secondary models obtained 95.43% and 98.82% accuracy on the Texas database, respectively, indicating the effectiveness of transfer learning. We further apply the deep learning method to thousands of events on the TexNet catalog to determine the focal mechanisms. Most of the focal mechanism solutions align well with the strikes, dips, and rakes of the known faults that were explored previously using full-waveform-based methods. The generation of the large focal mechanism database offers significant insights into the seismotectonic status of West Texas. The open-source package of EQpolarity can be accessed at https://github.com/chenyk1990/eqpolarity.

Seismic data acquired from the field inevitably suffer from noise pollution, which covers the useful signals and affects the reliability of subsequent seismic data processing and interpretation. Many two-dimensional (2D) multi-channel seismic denoising methods depend on the assumption that the receiver array is spatial coherent in field microseismic data acquisition, which limits their performance when dealing with field data. However, the single-channel methods are more flexible when faced with real microseismic data because they do not require any assumptions regarding spatial coherency. Therefore, we propose a one-dimensional (1D) dictionary learning (DL) framework based on variational sparse representation to suppress background noise in seismic data. Compared with the 2D multi-channel denoising method, the proposed method takes into account the waveform characteristics and requires no spatial coherency of single-channel seismic data, thus achieving better denoising performance. Additionally, the 1D dictionary learning method requires fewer training samples than the 2D method to reach a promising result, thereby causing less consumption time. Numerical results show that compared with the bandpass (BP) filtering, structure-oriented filtering (SOF), and K-SVD methods, the proposed DL framework can significantly improve the signal-to-noise ratio (SNR) of seismic data and protect effective signals better without causing extra computation. Furthermore, we discuss how to further suppress the residual horizontal noise and erratic noise based on the proposed method.

High signal-to-noise ratio (SNR) seismic data are crucial for oil and gas exploration, particularly for advanced high-precision migration processes and sophisticated reservoir inversion techniques. The erratic and random noise present in nature increases the difficulty of oil and gas exploration by reducing seismic signal quality. Deep learning (DL) networks facilitate rapid extraction of signal characteristics and effective noise reduction. The unsupervised DL U-Net network reconstructs signals through encoding and decoding. However, it overlooks the morphology and continuity of seismic data reflection waves, potentially missing detailed information in the reflected waveform. To address this, we introduce a waveform multibranch attention network (WMANet) for unsupervised learning (UL), designed to attenuate erratic and random noise using the U-Net architecture. WMANet uses the waveform attention mechanism to enhance the weight of waveform information in the network. In addition, the proposed loss function combining Welsch loss and local similarity enhances the erratic noise removal performance while effectively reducing the signal leakage phenomenon. Specifically, during the encoding, waveform information is captured using a fully connected layer capable of isolating critical data points. In the decoding stage, we integrate the waveform attention module with the fully connected module. This strategy gradually restores local information of the waveform while improving the efficiency of interbranch information utilization. We have evaluated our proposed method on both the synthetic and field datasets. The results demonstrate superior signal preservation and noise reduction compared with traditional denoising techniques.

Multiple attenuation plays an important role in marine seismic data processing, and a slew of methods have been developed for multiple attenuation. However, due to acquisition limitation, the performance of such methods will degrade when it comes to incomplete seismic data. Here, we proposed a novel method to implement data reconstruction and multiple suppression simultaneously. Compared with the two-step methods (e.g., interpolation and multiple attenuation), the proposed method can restore and highlight primary reflections directly without data reconstruction in advance, which avoids introducing interpolation-related errors and simplifies the data processing routines. Based on the incomplete data, we first sort seismic data into common midpoint (CMP) gathers and use normal moveout (NMO) to flatten the primary reflections, and the multiples still retain parabolas. Then, we reassigned the seismic traces randomly to decrease the coherency of the multiples from near-offset to far-offset and applied the damped rank reduction (DRR) method to the disordered traces to reconstruct seismic data and remove the unexpected multiples simultaneously. Compared with conventional two-step methods, the proposed one-step method can suppress multiples and benefit useful signal preservation more effectively. Moreover, in addition to multiples, the proposed method can remove random ambient noise and provide superior results with an improved signal-to-noise ratio (SNR). Synthetic and field examples are used to validate the validity of the proposed method on data reconstruction and multiple attenuation for incomplete seismic data.

Quality factor Q is an important parameter that accounts for the amplitude dissipation and phase distortion of seismic waves propagating in the Earth’s interior. Q estimation with improved accuracy benefits nonstationary seismic inversion, seismic imaging, fluid identification, and so on. Usually, logarithmic spectral ratio (LSR) is widely used to estimate Q based on vertical seismic profile (VSP) and poststack data. However, LSR is very sensitive to noise, and the effect of normal moveout (NMO) distorts the spectrum of the stacked seismic data, leading to an inferior Q estimation result. To weaken the effect of NMO and enhance the accuracy of Q estimation, we expand an improved LSR method in the zero-offset traveltime-local slope (e.g.,$t_{0}-p$) domain and propose a robust prestack Q estimation method based on common midpoint (CMP) gathers. The proposed method incorporates warped mapping (WM) and shaping regularization to stabilize it during Q estimation in the case of low signal-to-noise ratio (SNR). Additionally, we incorporate nonzero-offset information for Q estimation, which weakens the strong dependence on zero-offset information during prestack Q estimation. Compared with the single-gather prestack Q estimation methods (SPQEM), we make the most of the spatial coherence between the adjacent CMP to eliminate the unexpected noise-related outliers during spectral division for improved robustness and accuracy. Numerical examples are used to validate the superior performance of the proposed method, even in the presence of strong ambient noise.

Distributed acoustic sensing (DAS) is a new technology for recording vibration signals using optical fibers, and is advantageous over traditional seismic geophones given high spatial sampling density, real-time monitoring capabilities and relatively low cost for large-scale data acquisition. In recent years, progress in applications of the DAS technique has been achieved in near-surface imaging, earthquake detection, and urban traffic monitoring. In this study, we propose to apply a machine learning (ML) method to recognize and extract vehicle signals from DAS data acquired in a typical urban environment in Hangzhou, China. To design an efficientML framework, we apply a series of processing steps to eliminate noise and strengthen the vehicle signal, which is crucial for preparing high-quality labels. Initially, a total of 190 features (62 1-D features and 128 2-D features) are extracted from raw data, which are filtered down to 31 through univariate feature selection, random forest, and similarity analyses. These selected features are classified into (traffic) signal or (non-traffic) noise using the classic ML method of support vector machine (SVM). The resulting model enables robustly extracting vehicle signals with only a small (e.g., 10) training dataset and achieve an overall accuracy of about 80% on the test data. We further demonstrate the application of city traffic monitoring by considering the slope and coherence of the extracted vehicle signals. The preservation of car signals leads to a more accurate estimate in vehicle speed and volume. This study highlights the potential of real-time monitoring of speed and volume of traffic flow using existing city infrastructure and sheds light on the promising applications of the DAS technique in developing smart cities.

Predicting the peak ground acceleration from the first few seconds after the P-wave arrival time is crucial in estimating the ground motion intensity of the earthquake. The early estimation of peak ground acceleration supports the earthquake early warning system to generate the warning. Here, we propose to use the vision transformer to predict the peak ground acceleration using 4-sec three-channel single-station seismograms, i.e., 1s prior to the P-wave arrival and 3s subsequent to the arrival. The vision transformer can significantly extract remarkable information from the data resulting in superior prediction performance. The core layer of the vision transformer is the multi-head attention network which highlights the significant features of the input data. We train and evaluate the proposed algorithm using the Italian earthquake waveform data, where the proposed algorithm shows a promising result. The proposed vision transformer network utilizes an augmentation strategy to improve the learning ability of the model. Our proposed method is compared to the benchmark deep learning methods and empirical ground-motion models and outperforms all of them. The proposed algorithm can also predict the peak ground acceleration accurately using only 2-sec data after the P-wave arrival time. The proposed vision transformer architecture can also be integrated into a peak ground acceleration classification framework. Finally, the proposed algorithm is tested using real-time data and shows accurate results, indicating its applicability in real-time monitoring.

Earthquake data recorded in Texas are dramatically different from other places because of the various types of noise caused by oil and gas production or anthropogenic activities. This causes a relatively lower signal-to-noise ratio (SNR) and a strong challenge to leverage a globally trained deep learning model for earthquake detection. To combat the challenging data characteristics when monitoring seismicity using deep learning, we propose to apply transfer learning to a globally optimal phase-picking model using regional earthquake data compiled from Texas. Specifically, we first train an advanced deep learning model based on the compact convolutional transformer (EQCCT) using a global earthquake dataset. Then, we construct individual datasets from each of the main basins in Texas and apply transfer learning to each basin-scale database, intending to obtain optimal picking performance in each basin. As a result, the precision, recall, and$F1$-score significantly increased from the original EQCCT model to the fine-tuned model in the Delaware and Midland basins. The standard deviations of the picking errors of both P- and S-wave phases accordingly decrease significantly. The greatly improved EQCCT models help detect more P- and S-wave arrivals, facilitating a more successful association and location. Transfer learning models using Texas data and Texas basin-based transfer learning models with detailed documentation can be downloaded fromhttps://github.com/omarmohamed15/Picking-Texas/tree/main.

Coherent and incoherent noise in seismic data inevitably reduces the quality of subsequent processing, e.g., migration and inversion. Different from random noise, erratic noise follows the non-Gaussian distribution and has high amplitude, which is a challenge to the conventional denoising frameworks based on deep learning (DL). In this study, we propose an unsupervised learning framework with a multi-branch attention mechanism (MANet) to attenuate the erratic and random noise in 2-D and 3-D seismic data. MANet can adaptively attenuate noise in multi-dimensional seismic data without the need to manually generate labels to train the network. MANet integrates global features of waveforms extracted from multiple branches in a weighted way to enhance attention to significant features, thus obtaining a global and comprehensive representation of weights. To enhance the migration ability of shallow-level to deep-level features, we add some skip connections in the corresponding encoder and decoder. We use a robust mean-Huber loss function that is less sensitive to outliers to improve the denoising performance of erratic noise. We apply the proposed network for both 2-D and 3-D synthetic and field data. The denoising results demonstrate that the proposed method has better signal preservation and noise attenuation abilities compared with the conventional denoising methods and the state-of-the-art unsupervised learning framework. We improve the interpretability of the network by visualizing the weight matrices and different encoders. Besides, the visualization schemes proposed in this paper can be applied to more research, such as geological event interpretation, geological resource detection, and surface morphology analysis.

Salt body segmentation is a critical part of structural interpretation and oil and gas exploration for subsalt reservoirs. Existing automatic salt body segmentation techniques mostly use supervised learning strategies. It is challenging to generate a large number of labels by manual labeling, especially for 3-D salt bodies. Here, we propose a self-supervised learning (SSL) framework called Salt3DNet, for 3-D salt body segmentation. This framework is divided into two stages: pretraining and fine-tuning of downstream tasks. In the pretraining stage, we use the Barlow twins (BTs) method to pretrain the encoder and reduce redundancy in a contrastive learning manner to learn high-level data representations. In the fine-tuning stage, we construct two encoders to reconstruct 3-D seismic data and segment salt bodies in a multitask collaborative learning way. The encoder and decoder are composed of the 3-D fully convolutional DenseNet and soft attention mechanism, where the latter represents the selective kernel block (SKB) with multiple kernels of different sizes. Salt3DNet calculates the correlation matrix of features from different perspectives in the pretraining stage and makes it close to the identity matrix to obtain a more prosperous feature representation. Then, Salt3DNet uses a limited number of labeled samples for training. According to the evaluation metrics, the proposed network has demonstrated promising salt segmentation performance in 3-D SEG advanced modeling (SEAM) synthetic data and$F3$block real seismic data. In addition, the proposed network is demonstrated to have higher prediction accuracy than state-of-the-art salt segmentation frameworks through ablation experiments.

We present a novel deep-learning method using the ConvMixer network for automatic earthquake location. The proposed ConvMixer network utilizes three-component waveform recordings of single stations for estimating the hypocenter location. The ConvMixer network is a patch-based architecture that combines depthwise and pointwise convolutions to extract the global and local information of the earthquake waveforms. We train and test the proposed method using the Italian seismic dataset (INSTANCE). The ConvMixer network estimates the earthquake hypocenter locations with high accuracy, reaching a mean absolute error (MAE) of 2.71 km for the epicenter distance, and 1.15 km for the depth. In addition, we use the global STanford EArthquake Dataset (STEAD) to further evaluate the performance of the ConvMixer. As a result, the ConvMixer network achieves MAEs of 2.27 km and 1.19 km for the distance and the depth, respectively. The proposed ConvMixer network is compared to the benchmark methods, i.e., ResNet, AlexNet, MobileNet, and Xception, and outperforms all of them.

Passive seismic source location imaging is important to various scientific and engineering research topics spanning from unconventional reservoir development in exploration seismology to seismic hazard prevention in the earthquake seismology community. The emerging machine-learning (ML) techniques enable the location of passive seismic sources with unprecedented efficiency and accuracy. Most of the state-of-the-art ML methods are based on waveforms, as required by the most popular convolutional neural network (CNN) architecture, which is prone to the sensitivity of velocity models. Here, we present a traveltime-based ML method, RFloc3D, to locate passive seismic sources from manually or automatically picked P- and S-wave arrivals. The proposed method is similar to traditional traveltime-based location methods, where the inverse mapping from arrival times to the passive source location is obtained by inverting a nonlinear inverse problem, but differs in leveraging the random forest (RF) method to learn the inverse mapping relation from numerous eikonal-based forward simulations. Details and analyses of the proposed RFloc3D method are illustrated based on a microseismic monitoring setup. Numerical and real data examples show that the proposed method is capable of real-time location. The inclusion of S-wave arrivals, most importantly, the differential time between P- and S-wave arrivals, helps significantly to reduce the depth error (e.g., decreasing the mean absolute error (MAE) to a half) of the located sources.

Signal attenuation estimation is a critical task in signal processing and is essential for analyzing media characteristics and compensating for energy loss. Current centroid frequency shift-based methods for estimating attenuation are mostly based on the assumptions of full-band analysis and frequency-dependent or independent quality factor (Q). In this paper, we propose a novel frequency-independent centroid frequency shift (FiCFS) method for signal attenuation estimation with higher adaptability. It is based on arbitrary frequency bands instead of the full-band spectrum defined by the conventional centroid frequency shift (CFS) method, and accordingly, the derivation is performed by incomplete gamma functions instead of ordinary gamma functions. Through rigorous mathematical derivations, the first moment (centroid frequency) and the second moment (variance) are proved to be frequency insensitive for arbitrary frequency bands, and then the arbitrary frequency band-based CFS method, i.e., the FiCFS method, is derived with the frequency-weighted exponential spectrum assumption. The matching ability of the frequency-weighted exponential spectrum to other signal spectra is verified, demonstrating the method’s adaptability to most attenuated signals. Experimental results using synthetic and field data sets demonstrate that the proposed method is adaptive, noise-immune, and reliable.

We propose to implement a compact convolutional transformer (CCT) for picking the earthquake phase arrivals (EQCCT). The proposed method consists of two branches, with each of them responsible for picking the arrival times of the P- or S-wave phases. We use the STEAD dataset to train and validate the proposed EQCCT algorithm. We split the STEAD dataset into 85% for training, 5% for validation, and 10% for testing To facilitate the training process, we implement several data augmentation strategies to the training set by adding Gaussian noise, randomly shifting the waveforms, adding a second earthquake to the input window, and dropping one or two channels from the seismogram in the STEAD dataset. As a result, the EQCCT model outperforms both EQTransformer and PhaseNet, the two most popular deep-learning-based phase-picking methods. Considering the true positive criterion as the picked phases arriving within 0.5 s of the reference times, the EQCCT achieves the lowest mean absolute error (MAE) compared to the EQTransformer and PhaseNet methods for the STEAD, Japanese, Instance and Texas datasets. Our EQCCT network also demonstrates superior performance in other metrics such as precision, recall, and F1 score. We apply the pre-trained model to three independent datasets (not included in the training set), i.e., the Japanese, Texas, and Instance datasets, and achieve higher picking accuracy than the EQTransformer and the PhaseNet in terms of various statistical metrics, demonstrating a stronger robustness and generalization ability of the EQCCT. The real-time application of EQCCT in the Texas Seismological Network (TexNet) further demonstrates its production-ready performance in terms of detection and phase-picking accuracy.

Noise suppression for seismic data can meliorate the quality of many subsequent geophysical tasks. In this work, we propose a novel self-supervised learning method, the deep nonlocal regularizer (DNLR), for 3D seismic denoising. Our DNLR fully exploits the nonlocal self-similarity of seismic data under a self-supervised learning framework for noise attenuation. It can be flexibly combined with different hand-crafted regularizers, e.g., total variation, nuclear norm, and correlated total variation, by performing the regularizer on nonlocal self-similar patches, which more effectively characterizes the intrinsic structures underlying seismic data. Our DNLR can be easily plugged into existing self-supervised denoising methods, e.g., deep image prior and Self2Self, and consistently improve their performance. To make the optimization model tractable, an algorithm based on the alternating direction multiplier method is introduced to solve the DNLR-based seismic denoising problem. Extensive seismic denoising experiments on synthetic and field data validate the superior performances of our DNLR as compared with state-of-the-art model-based and deep learning seismic denoising methods. Code is available at https://github.com/XuZitai/DNLR.

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.

We develop a random forest (RF) model for rapid earthquake location with an aim to assist earthquake early warning (EEW) systems in fast decision making. This system exploits P-wave arrival times at the first five stations recording an earthquake and computes their respective arrival time differences relative to a reference station (i.e., the first recording station). These differential P-wave arrival times and station locations are classified in the RF model to estimate the epicentral location. We train and test the proposed algorithm with an earthquake catalog from Japan. The RF model predicts the earthquake locations with high accuracy, achieving a mean absolute error (MAE) of 2.88 km. As importantly, the proposed RF model can learn from a limited amount of data (i.e., 10% of the dataset) and much fewer (i.e., three) recording stations and still achieve satisfactory results (MAE < 5 km). The algorithm is accurate, generalizable, and rapidly responding, thereby offering a powerful new tool for fast and reliable source-location prediction in EEW.

Discrimination between earthquakes and quarry blasts is crucial for precise seismic analysis, e.g., seismic hazard mitigation, earthquake cataloging, etc. However, the discrimination process is challenging due to the similarity of waveforms between the local earthquakes and quarry blasts. We propose to use the scalogram and the capsule neural network to distinguish between earthquakes and quarry blasts. First, we obtain the scalogram for 60s 3-channel waveforms, where we extract 10s before and 50s after the first arrival time of the seismic event. Secondly, we utilize the capsule neural network to extract the important information from the input scalogram which leads to robust classification performance. The proposed capsule neural network consists of the convolutional layer, primary capsule layer, and digit caps layer. The convolutional layer extracts the important information from the input data, and the primary capsule layer extracts the spatial relationship between different feature maps. Thirdly, we use the dynamic routing process to connect the primary capsule to the digit caps layer. We train and test the proposed capsule network using a small and unbalanced dataset which is recorded by the Egyptian Seismic Network (ENSN) in the Red Sea and the surrounded area in Egypt. Accordingly, the proposed method achieves a test accuracy of 96.08%. The proposed method is compared to the benchmark methods, i.e., convolutional neural network (CNN), AlexNet, VGG, and ResNet networks, and demonstrated to outperform all of the competing methods. Finally, we apply the proposed method to classify real-time seismic events and obtain promising results.

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.

We propose a novel deep learning method using an attention-based fully convolutional dense network (FCDNet) for automatic earthquake detection. The FCDNet consists of encoder-decoder parts with skip connections, where each encode-decoder block contains a block of densely connected layers to enhance the feature learning capability. The spatial attention mechanism is added within the FCDNet to assign greater attention to useful features and hence improve the accuracy of earthquake detection. The time-frequency representations of three-component seismograms produced by the Stockwell transform are used for better extracting the hidden data features. The attention-based FCDNet extracts the time-frequency features needed for distinguishing the seismic signal from the background noise. We evaluate the performance of the proposed method using a Mediterranean dataset. The attention-based FCDNet is trained using 90% of the Mediterranean dataset and tested using the remaining 10%. Accordingly, the training and testing accuracies are 97.71% and 97.02%, respectively. The intersection over union (IoU), precision, recall, and F1-score of the attention-based FCDNet are 93.80%, 99.72%, 99.55%, and 99.64%, respectively. Moreover, to evaluate the generalization ability of the trained model, we utilize 100,000 seismic waveforms recorded in different seismic regions from the global STanford EArthquake Dataset (STEAD) dataset for testing, which shows robust performance. We also apply the attention-based FCDNet to the Japanese seismic data and compare the performance to the CRED and SCALODEEP methods. The attention-based FCDNet outperforms the benchmark methods and achieves a higher detection accuracy of 99.46%. The attention-based FCDNet is additionally evaluated using one-day continuous seismic data recording a seismic swarm that occurred in the Helike region. As a result, the attention-based FCDNet recognizes 135 earthquakes and raises 15 false alarms with a detection accuracy of 90.06%.

Predictive filtering methods are widely used in industry to remove random noise, owing to their stability and efficiency. Traditional predictive filtering methods use a fixed autoregressive order (filtering factor) in the spatial direction. However, denoising spatially varying seismic data have thus far been ineffective. In this study, the effect of nonstationary seismic data variations is overcome using local windowing or nonstationary denoising methods. We propose a nonstationary predictive filtering method in which an autoregressive model is constructed using spatially varying filtering factors. First, we evaluate the structural complexity of the global data based on a local window selection using plane-wave destruction. Then, adaptive filtering factors are proposed depending on the structural complexity of the seismic data. Finally, the autoregressive model containing the adaptive filtering factors is solved. We use three quality measures to evaluate the denoising performance: the signal-to-noise ratio ($S/N$), the peak signal-to-noise ratio (PSNR), and local similarity (LS). The proposed method acts on both synthetic and field data, including both pre- and post-stack seismic data. Compared with prior nonstationary filtering methods, experimental results demonstrate that structural complexity-guided (CG) predictive filtering enables efficient random noise attenuation and reduces signal leakage.

Random noise attenuation of seismic data is a fundamental problem in seismic data processing. It is not only an important problem itself but also is a crucial step for the subsequent tasks, e.g., migration and inversion. We propose a local orthogonalization constrained unsupervised deep learning denoiser (LOUD) to suppress seismic random noise based on a new loss function that specifically adapts to seismic data. Through unsupervised learning, we eliminate the common need in supervised learning approaches of collecting or generating sizable clean and noisy image pairs, which is challenging and expensive, especially for seismic data. We utilize a deep convolutional autoencoder to reconstruct the clean seismic image and leverage the local signal-and-noise orthogonalization as a constraint to guarantee that the removed noise component is orthogonal to the recovered signal. Experimental results on both synthetic and field datasets exhibit the effectiveness of our proposed method over traditional denoising methods.

Predictive filtering is one of the most widely used denoising algorithms in the seismic data processing community because of its high efficiency and stability in different situations. The traditional predictive filtering, however, is not able to deal with structurally complex data set unless applied in local windows. We develop a novel noncausal predictive filtering method that is free of the windowing step but is able to denoise complicated data set. We extend the stationary predictive filtering method to its nonstationary version, where the predictive filter coefficients vary across the frequency-space domain. The nonstationary predictive filtering (NPF) model requires solving a highly underdetermined inverse problem using an iterative shaping regularization method. The traditional shaping regularization method solves an inverse problem by applying a constant smoothing operator and thus does not consider the heterogeneity of the filter coefficients in the frequency–space domain. We propose to apply a nonstationary smoothing operator to constrain the model in the shaping regularization framework. The smoothing radius in the nonstationary smoothing operator is chosen based ona prioriinformation of the model, e.g., the nonstationarity of the data in the frequency–space domain. The proposed NPF method offers the flexibility in controlling the smoothness and sharpness of the calculated filter coefficients in both frequency and space dimensions. Several synthetic data sets and complicated real data examples are used to demonstrate the advantages of the new method.

Prestack seismic inversion has emerged as a powerful technique for reconstructing parameters attribute to the subsurface properties and building the geophysical parameter models. However, the inversion algorithms always suffer from spatial blur and low resolution. Total variation (TV) regularization preserves the spatial variation boundary of data by highlighting the sparsity of the first-order difference, which is regarded as an important technical means for image restoration. However, when the data do not change along the spatial grid direction, TV regularization is prone to a staircase effect. In this article, a directional TV (DTV) method is proposed to conduct the prestack amplitude variation with offset/angle (AVO/AVA) inversion. The method consists of three essential steps: estimating the seismic slope attribute from the seismic data, introducing seismic slope attribute to the TV regularization to establish the objective function, and optimizing the objective function by the split-Bregman algorithm. Finally, the conventional and proposed methods are applied to the synthetic and the real seismic data. The comparison of different methods demonstrates that the proposed method is applicable to reveal the detailed subsurface models, alleviate the staircase effect or artifact substantially, and further upgrade the quality of prestack inversion 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.

In this work, a Robust Worst-Case (RWC) estimation is presented for recovering sparse reflectivity series from uniformly quantized seismic signals. First, theOrthogonal Matching Pursuit(OMP) algorithm is applied on a quantized seismic trace. A set of conservative estimates of the reflectivity impulses and a dimension-reduced system are accordingly obtained. Second, the error induced by the quantization is modeled as a systemic uncertainty in a multiplication way. This modeling imposes a greater model uncertainty on the estimated reflectivity impulses with small values and vice versa. Finally, a RWC deconvolution scheme is designed for the dimension-reduced system. Among those roughly estimated impulses from the OMP algorithm, the ones statistically less affected by the quantization process are assigned with higher confidence weights and the ones statistically with higher magnitudes of quantization error are assigned with lower confidence weights. By rescaling the quantization error, the proposed scheme significantly increases the robustness of the solution to the quantization error and hence better improves the visual saliency of seismic signals than that of the OMP algorithm. This scheme is tested on both synthetic and real seismic data and the performance is evaluated by compared to that of the OMP algorithm. The results shows that the new scheme significantly outperforms the OMP algorithm by observing the following two aspects: first, the falsely or overly estimated impulses are significantly suppressed in the RWC estimation compared with that of the OMP algorithm, and second, the RWC estimation exhibits enhanced robustness to the change of the quantization interval.

High-amplitude noise could interfere with useful seismic signals, affecting our ability in processing and interpreting seismic data. Thus, attenuating seismic noise is an important task in seismic processing. Total variation (TV) has played an important role in many steps of seismic data processing but always neglected the seismic structural information. Directional TV (DTV), however, considers the structural direction of seismic events but tends to cause the staircasing effect on seismic records based on the first-order formulation. Here, we develop a high-order DTV (HDTV) method for seismic denoising. It considers the local structural direction of the seismic data and calculates the higher order derivatives of seismic images to avoid the staircasing effect. We design several synthetic models that are contaminated by various types of random noise to test the denoising ability. The denoising performance of our new method is compared with the first-order DTV, conventional high-order TV, and TV regularization methods from two aspects, i.e., the signal-to-noise ratio and the effective signal leakage degree. Then, the advantages of the proposed method are further validated via several field seismic data sets.

Lithofacies classification is an indispensable procedure in well logging and seismic data interpretation. We propose a novel deep classified autoencoder learning approach to identify lithofacies for high-dimensional data and complex problems. Deep autoencoder (DAE) is an unsupervised learning method via layerwise pretraining multiple autoencoders. It can learn deep data features automatically and reconstruct the original data with a small error. Introducing sparse constraint (i.e., sparse autoencoder) potentiates the learning ability of autoencoder. On this foundation, additional regularization terms constructed by labeled samples are considered in the new DAE approach in order to boost the performance. The new method can adaptively preserve the most significant input features and remove insensitive properties to decrease computational complexity. At the same time, we embed the class information into the loss function of autoencoder to measure intraclass similarity and improve the classification accuracy. Several experiments on well data and seismic data show that the proposed method achieves promising results. Compared with the traditional deep autoencoder (DAE), the proposed method is more competitive in terms of classification accuracy and robustness.

Seismic multicomponent, here refers to P-P wave (PP) and P- shear wave (SV), joint inversion is an important approach to improve the accuracy of S-wave velocity and density prediction. Compared with P-P wave data, converted wave data are more sensitive to these parameters. Thus, introducing these data to the seismic inversion could improve the inversion accuracy of S-wave velocity and density. Due to the travel-time gap between the multicomponent data, multicomponent inversion usually needs to be prepared for registration processing in advance. However, registration methods often suffer from matching errors, which ultimately affect the quality of the inversion results. Based on the optimal transmission idea, a registration-free multicomponent joint amplitude variation with offset/angle (AVO/AVA) inversion algorithm is developed in this article. The proposed method adopts the Earth mover’s distance between the synthetic P-SV wave and the observed P-P wave by calculating the optimal transport path. Besides${\ell _{1-2}}$norm is exploited as the penalty norm, where the regularized misfit function is minimized by the alternating direction method of multipliers (ADMM) algorithm. The synthetic data test and real seismic data application show that the proposed method can retrieve the S-wave velocity and density information better than the conventional method. The proposed method can not only effectively avoid the cumbersome registration steps, but also minimize the registration error and avoid the subsequent error accumulation.

Recently, studies on multidimensional seismic data interpolation through rank-constrained matrix or tensor completion have led to many effective methods, with satisfactory results. Despite the success of the rank-constrained matrix completion methods, e.g., damped rank reduction (DRR), and the rank-constrained tensor completion methods, e.g., high-order orthogonal iteration (HOOI), strong noise and highly decimated traces could still make the reconstruction results not acceptable. In this article, we find that implementing only one rank constraint to solve the multidimensional seismic data recovery problem is not sufficient. Therefore, we consider a hybrid method to reconstruct the noisy and incomplete traces based on a new mixed rank-constrained (MRC) algorithm. The proposed MRC algorithm aims to take advantage of the merits of both the rank-constrained matrix and tensor completion models to restore the missing data. We first apply the unfolding and folding operator to the 4-D spatial hypercube data. Then, for each iteration, we connect the DRR and the HOOI approaches in the same framework to solve the proposed MRC model. The proposed MRC model aims to provide an enhanced level of rank constraint to improve the signal-to-noise ratio (SNR) of the recovered data. Synthetic and field 5-D seismic data are used to compare the performance of the new method with the HOOI and DRR methods. The comparison via visual inspection and numerical analysis reveals the better performance of the proposed MRC algorithm.

We develop a capsule neural network (CapsPhase) for seismic data classification and picking. CapsPhase consists of several layers, e.g., convolutional, primary capsule, and digit capsule layer. The convolutional layer extracts the significant features from the seismic data, while the primary capsule combines the extracted features into several vector representations named capsules. Afterward, the primary capsule is connected to the digit capsule layer using a dynamic routing strategy to obtain the vector representation of each output class, i.e.,$P$-wave,$S$-wave, and noise class. CapsPhase is trained using 90% of the Southern California seismic dataset, which contains 4.5 million 4 s-three-component seismograms, and is validated and tested using the remaining 10%. Accordingly, the training accuracy reaches 98.70%, while the validation accuracy is 98.67% and the testing accuracy is 98.66%. Furthermore, the CapsPhase is tested using 300 000 earthquake waveforms recorded worldwide from the STanford EArthquake Dataset (STEAD). Accordingly, the precision, recall, and F1-score of the$P$-picks corresponding to the CapsPhase reach 94.50%, 99.86%, and 97.10%, respectively, whereas the precision, recall, and F1-score of the$S$-picks corresponding to the CapsPhase are 88.05%, 99.87%, and 93.60%, respectively. In addition, CapsPhase is evaluated using the Japanese seismic data and is compared to benchmark methods, e.g., short-time average/long-time average (STA/LTA), generalized phase detection (GPD), and CapsNet methods. As a result, CapsPhase reaches F1-scores of 99.10% and 98.64% for$P$-wave and$S$-wave arrival times, respectively, and outperforms the benchmark methods. The results show that the CapsPhase has the ability to pick the arrival times accurately despite the existence of strong background noise, e.g., the signal-to-noise-ratio (SNR) can be as low as −4.97 dB. Besides, the CapsPhase detects the arrival time when the earthquake has a small local magnitude, e.g., as low as$0.1~M_{L}$. In addition, we find that the proposed algorithm has the ability to train using a small dataset, which is valuable for regions that have limited training data.

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.

We develop a deep learning framework based on deep image prior (DIP) and attention networks for 3-D seismic data enhancement. First, the 3-D noisy data are divided into several overlapped patches. Second, the DIP network has a U-NET architecture, where the input patches are encoded to extract the significant latent features, while the decoder tries to reconstruct the input patches using these extracted features. Besides, the attention network is used to scale the extracted features from the encoder and the decoder. Third, the attention network output of the encoder is concatenated with that of the decoder to obtain high-order features and guide the network to extract the most significant information related to the seismic signals and discard the others. Finally, the 3-D seismic data are reconstructed using the output patches obtained by the DIP network. The proposed algorithm is an iterative and unsupervised approach, which does not require labeled data. We evaluate the proposed algorithm using several synthetic and field data examples. As a result, the proposed algorithm shows the ability to enhance the 3-D seismic data by attenuating the random noise and preserving the 3-D seismic signal with minimal signal leakage. Moreover, the proposed algorithm shows good denoising performance when tested using various types of events, e.g., linear, hyperbolic, low and high dominant frequencies, and weak amplitude. In addition, the proposed method outperforms the predictive filtering (PF) and damped rank-reduction (DRR) methods. To further understand the principle of the proposed method inside the DIP network, we analyze the weighting matrices in the encoder and decoder parts in detail. We attribute the denoising ability of the DIP network to the improvement of the extracted basis features from the encoder to the decoder layers through a deep network.

The unsplit complex frequency-shifted perfectly matched layer (CFS-PML) has been widely used in the first-order wave equation in velocity and stress while rarely formulated for the wave equation recast as a second-order system in displacement. Among different variants of PML, the unsplit CFS-PML for the second-order wave equation enjoys better absorbing performance and numerical stability, compared to the traditional PML, due to the presence of the general form of CFS stretching factors, as well as higher computational efficiency over the split PMLs since it avoids wave equation order reduction and splitting the state variables into multiple directional components. This study aims to develop an unsplit CFS-PML scheme for the second-order wave equation and devote specific attention to fractional viscoacoustic simulation where fractional time derivatives are involved and hard to be reformulated into a first-order form. In the complex space, PML is typically regarded as an analytical continuation of the real coordinates; thus, we define an explicit coordinate stretching operator acting on the Laplacian operator. This stretching operator consists of several convolution terms; each of them can be efficiently resolved by a recursive convolution updating strategy. Viscoacoustic simulations on homogeneous Pierre Shale, Marmousi model, and 3-D SEG/EAGE overthrust model verify the feasibility and absorbing the performance of our proposed scheme.

5D data is the original recorded form in the 3D seismic acquisition, which includes sufficient information from all five dimensions. However, environmental and economic logistic difficulties often severely impact the data acquisition geometry, leading to raw data with missing traces and strong contaminating random noise. This deficiency often causes troubles in subsequent processing. Thus, an efficient interpolation and denoising method is required to recover useful signals. Unfortunately, practical applications of many existing reconstruction algorithms are limited by their intensive computational cost when applied to the 5D data. Additionally, the stability of these algorithms is also challenged by complex geological structures, which often degrades the reconstruction performance. To seek solutions to the aforementioned problems, we design a simple and effective framework for fast reconstruction and denoising of under-sampled 5D seismic data via a two-step process. First, we prepare the initial model from the original recordings by constructing a 3D gather at each common offset point. This step effectively interpolates the missing traces in 3D common offset gathers by exploiting the data coherency in the adjacent areas (i.e., nearby mid-points). In the second step, the processed 5D data is reorganized into 3D common mid-point gathers, with each of them further sorted into a 2D section according to absolute offset values. Then a conventional 2D processing algorithm (e.g., F-X prediction, wavelet thresholding, or multichannel singular spectrum analysis) is invoked to filter the obtained 2D section. The proposed workflow has a low overall computational cost and preserves signal fidelity. We use this framework to simultaneously denoise and interpolate the low-quality and extremely sparse seismic data. The synthetic and field examples both demonstrate the superb performance of the proposed framework in comparison with conventional methods.

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.

Coherent seismic noise is usually difficult to attenuate due to the similar morphological patterns between noise and useful signals. To attenuate coherent noise, special preknowledge should be utilized in a state-of-the-art approach, which causes significant inconvenience. Here, we develop an automatic method to attenuate coherent noise based on the adaptive dictionary learning algorithm. The adaptive dictionary algorithm can learn the features of both signals and coherent noise and leave obvious morphological differences in the dictionary atoms. These differences in the dictionary atoms can be transformed into statistical differences, which can be measured and then used to distinguish between signal and noise atoms. We evaluate several statistical metrics in characterizing the dictionary atoms and their feasibilities in distinguishing between signal and noise atoms. We find that the kurtosis metric can best represent the differences between signal and noise atoms, and then we design a kurtosis-based filter to reject those high-kurtosis atoms and their corresponding coefficient vectors for suppressing the coherent noise. Synthetic and real data examples demonstrate the performance of the proposed algorithm.

Simultaneous-source acquisition saves enormous acquisition costs and greatly enhances the quality of seismic data. However, it brings challenges to subsequent data processing due to the intense interferences of neighbor sources. Deblending is one of the methods to solve the problem of crosstalk noise. The deblending applied in the shot domain does not require the random scheduling that is used in the conventional simultaneous-source acquisition, so it has more flexibility. However, because the records of different sources show continuous traces in common shot gathers, they have the same characteristics and are difficult to separate directly. In this article, we use the least-squares Gaussian beam transform (LSGBT) to separate the useful seismic signals from crosstalk noise in the distance-separated simultaneous-sourcing (DSSS) survey and propose a novel deblending framework based on the LSGBT in the shot domain. Unlike most state-of-the-art Gaussian beam approaches that construct Gaussian beams in the frequency domain, the LSGBT constructs time-domain Gaussian beams, which can be considered as functions of amplitude, position, dip field, and so on. The essence of the proposed algorithm is that the blended data can be decomposed into the Gaussian beams that represent different dip-angle components. Thus, the single-source seismic records can be reconstructed in terms of a carefully selected combination of dip-oriented Gaussian beams. Two synthetic examples and one field data example show that the iterative deblending scheme based on LSGBT obtains better performance than the conventional frequency-wavenumber-based methods.

S-wave velocity and mass density, which are two essential parameters in distinguishing lithology and hydrocarbon detection, are very difficult to retrieve even when long-offset gather data are used. P-P and P-SV waves joint inversion has been verified as an effective tool to accurately extract such fluid related parameters. However, registering the travel-time of P-P and P-SV waves on the identical coordinate is a significant but knotty problem for the joint inversion. Therefore, an improved strategy of prestack seismic joint inversion is proposed for accurately transforming the recorded data to elastic-parameter-based interpretive information. First, to overcome the weaknesses of artificial strenching and the local cross correlation-based method, a nonstrenching and globally optimal registration algorithm, i.e., dynamic time warping (DTW), is exploited to match the P-P and P-SV waves. Then, a new method has been developed for the joint inversion method by combining the logarithmic absolute-criterion-based misfit function withl1-2norm-based penalty, which can significantly improve the vertical resolution and stability of inversion results. The numerical examples demonstrate that the DTW algorithm can obtain better registered results than the conventional method. Moreover, the proposed joint inversion method performs better than the conventional prestack inversion in both resolution and accuracy, especially for the S-wave velocity.

Time-lapse (4-D) seismic exploration is one of the essential means for accurately reconstructing the underground geological model, enhancing oil/gas recovery (EOR), and predicting remaining oil distribution. Under the assumption that the rock skeleton is almost unchanged in the process of development, inversion of time-lapse seismic difference data can be used to characterize the dynamic reservoir parameter. However, the inaccuracy of the forward operator, lack of constraints, and inappropriate regularization weight may lead to ill-posedness. Thus, the ill-posedness is still a key factor affecting the accuracy and stability of time-lapse seismic inversion. In this article, an improved difference-and-joint inversion strategy is proposed using the modified linear approximation as the forward operator. First, based on the modified approximation as the forward operator, time-lapse seismic difference inversion is exploited to obtain the variations of elastic parameter reflectance caused by the production-induced model perturbation. Then, we innovatively proposed to take the production-induced model perturbation as constraints, thereby achieving more accurate geological models for multiperiod seismic data joint inversion. Moreover, the L-curve method is exploited in this article to acquire the optimal regularization weight adaptively in each iteration step. Finally, the difference inversion results are used as constraints to precisely invert the geological model by combining multiperiod seismic data.

Reconstructing an accurate and high-resolution subsurface model is attractive in the fields of both geology and seismology. However, due to the band-limited characteristics of seismic data, the inversion greatly depends on the reliability of the initial model. A fairly acceptable initial model could lay a good foundation for seismic inversion. In this article, we first introduce a well-log interpolation method with the local slope as a constraint for building a high-fidelity starting model in prestack amplitude versus offset/angle (AVO/AVA) inversion. First, we briefly review the basic theory of general seismic inversion. Then, instead of using the conventional preconditioned least-squares method, we introduce shaping regularization theory into the geological structure-guided well-log interpolation to accelerate the convergence. We use the plane-wave destruction (PWD) algorithm to extract the slope attribute from seismic data, images, or velocity models. The slope is used as the constraint to solve the inverse problem based on the shaping regularization method. Numerical examples demonstrate that the proposed initial model building method performs better than the conventional ones. It greatly improves the accuracy of inversion results. Furthermore, we apply the proposed model building method to the inverse problems of AVO/AVA inversion and reservoir parameter estimation of several field data sets for the first time, which demonstrate encouraging performance.

Dynamic changes of reservoir parameters within a reservoir are usually estimated by a history matching process based on the production data. However, it is difficult to accurately acquire the spatial distribution using production data as the constraints. We formulate a nonlinear inversion method to dynamically characterize the reservoir parameter variations from time-lapse prestack seismic data. Besides, we introduce the modified Hertz-Mindlin (H-M) model to simulate the changes in the elastic behavior of a turbidite sandstone during production by establishing a link between reservoir parameters and elastic parameters. Then, the exact Zoeppritz equation is exploited as a forward engine to convert these parameters into synthetic time-lapse seismic data. Numerical examples verify that the porosity parameter has a higher reflection sensitivity than the other two parameters, which is followed by the effective pressure. The water saturation parameter, however, is the least sensitive. Following a Bayesian approach, the baseline and monitor seismic data are used in the seismic inversion to establish the regularized augmented function. Combining the modified H-M rock physics model with the exact Zoeppritz equation as a forward operator, a regularized function is minimized with the Gauss-Newton optimization method to determine a model update. We further apply the proposed inversion algorithm to a real seismic data set, including baseline and monitor seismic data. The results demonstrate that the proposed inversion method can not only yield an accurate description of subsurface static reservoir properties but also improve the accuracy of dynamic reservoir parameter characterization.

Patchy saturation of gas and brine within the porous rock can induce significant attenuation and velocity dispersion effects, which in turn have a profound impact on seismic data. Thus, there should be a close relationship between dispersion and reservoir parameters, such as saturation, porosity, and permeability. Hence, using seismic data to determine dispersion information has been the subject of intensive research in recent years. It can help quantitatively predicting gas saturation and monitoring CO2sequestration, a key strategy for mitigation of global warming. Here, a frequency-dependent prestack waveform inversion workflow was proposed to extract the wave-induced fluid flow (WIFF) effect from seismic data (WIFF prestack waveform inversion strategy). The proposed approach consists of three essential parts, comprising spectral decomposition,$Q$-compensated prestack waveform inversion, and frequency-dependent prestack waveform inversion. First,$\ell _{1}$norm constrained inverse spectral decomposition is exploited to provide time-frequency amplitude and phase spectra with high resolution and reliable accuracy. Then, constant-$Q$compensated prestack waveform inversion is used to generate relatively precise$P$- and$S$-wave velocities, density, and Fréchet derivative for the subsequent dispersion inversion. Finally, frequency-divided seismic data and the estimated parameters and Fréchet derivative are used to invert the$P$-wave velocity dispersion. A comparison between the inversion results and band-limited rock physics analysis results shows that the proposed method can provide a quantitative inversion of the velocity dispersion to some extent. The proposed strategy provides a possibility for quantifying dispersion and further estimating gas saturation.

Seismic data are usually contaminated by various noises. Noise suppression plays an important role in seismic processing. In this article, we propose a new denoising method based on the nonlocal weighted robust principal component analysis (RPCA). First, seismic data are divided into many patches and grouped based on the nonlocal similarity. For each group, then, we establish a similar block matrix and set up the objective function of the RPCA. Next, we introduce the iterative log-thresholding algorithm into the augmented Lagrangian method to solve the problem. Furthermore, varying weights are specified to different singular values when minimizing the objective function. Finally, aggregating all recovered matrices can obtain the denoised seismic data. The proposed method considers the nonlocal similarity and adaptively sets weights with local noise variance. It performs well also owing to the superiority of the iterative log-thresholding method. The presented method is assessed using a synthetic seismic section with several crossover events. We also apply this novel approach to a real seismic data, which shows good results. Comparison with other approaches reveals the effectiveness of the proposed approach.

Earthquake detection is an essential step in observational earthquake seismology. We propose to utilize a capsule neural network (CapsNet) to automatically identify and detect earthquakes. CapsNet is the new generation of deep learning architecture. It has the capability of learning with a great generalization performance from a small dataset. We train the CapsNet using 50% of the Southern California seismic data (2.25 million 4-s-three-component seismic windows) and use 222 395 waveforms from different seismic areas to evaluate the CpasNet performance, e.g., western United States, Europe, and Japan. As a result, the CapsNet misses 367 events and detects 217 305 events with an accuracy of 97.71%. Among these picked events, 210 498 events have an arrival time error below 0.2 s (96.86%) and 197968 waveforms with an arrival time error below 0.1 s (91.11%). The CapsNet precision, recall, and F1-score are 97.78%, 99.83%, and 98.79%, respectively. In addition, the CapsNet is tested using 100 000 60-s-three-component seismic noise waveforms. CapsNet shows a low false alarms rate of 1384, which gives the CapsNet an accuracy of 98.61%. In addition, CapsNet is tested using continuous seismic data associated with the 24-hours microearthquakes swarm that occurred in the Arkansas area. Accordingly, the CapsNet detects 221 earthquakes and releases 37 false alarms with a detection accuracy of 85.65%. CapsNet detects many microearthquakes with a small magnitude, as low as -1.3 Ml, and detects earthquakes that have a low signal-to-noise ratio (SNR), e.g., as low as -8.07 dB. The results of the CapsNet are compared to the benchmark methods, e.g., short-time average/long-time average (STA/LTA) and GPD methods. The CapsNet shows the highest picking accuracy and outperforms the benchmark methods.

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.

The plane-wave destruction (PWD) method has been a widely used local slope estimation method in the seismic community. It is based on the discretization of the plane-wave partial differential equation (PDE) and the linearization of the PDE with respect to the local slope. Solving the linearized inverse problem for the slope perturbation is equivalent to solving a smoothness constrained optimization based on a shaping regularization method. The smoothness constraint in the shaping regularization is important in that it not only controls the stability and smoothness of the solution, i.e., slope perturbation, but also affects the accuracy and resolution of the solution. The traditional PWD algorithm is not easy to compromise between the smoothness and resolution of the estimated local slope because it uses a stationary triangle smoothing operator as the shaping operator. Here, we propose to improve the robustness of the PWD algorithm by introducing a nonstationary triangle smoothing operator into the shaping regularization framework in order to adaptively constrain the solution according to the local signal reliability. The smoothing is weak in areas with a higher probability of signals and is strong in areas with a higher likelihood of noise. The smoothing radius can be adaptively estimated based on an optimization model, which is solved by a line-search method. The proposed new slope estimation method is referred to as a nonstationary method compared with the traditional stationary one. The effectiveness and benefits of the new slope estimation method are validated via several synthetic and field data examples.

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.

We propose a new method to tackle the mapping challenge from time-series data to spatial image in the field of seismic exploration, i.e., reconstructing the velocity model directly from seismic data by deep neural networks (DNNs). The conventional way of addressing this ill-posed inversion problem is through iterative algorithms, which suffer from poor nonlinear mapping and strong nonuniqueness. Other attempts may either import human intervention errors or underuse seismic data. The challenge for DNNs mainly lies in the weak spatial correspondence, the uncertain reflection-reception relationship between seismic data and velocity model, as well as the time-varying property of seismic data. To tackle these challenges, we propose end-to-end seismic inversion networks (SeisInvNets) with novel components to make the best use of all seismic data. Specifically, we start with every seismic trace and enhance it with its neighborhood information, its observation setup, and the global context of its corresponding seismic profile. From the enhanced seismic traces, the spatially aligned feature maps can be learned and further concatenated to reconstruct a velocity model. In general, we let every seismic trace contribute to the reconstruction of the whole velocity model by finding spatial correspondence. The proposed SeisInvNet consistently produces improvements over the baselines and achieves promising performance on our synthesized and proposed SeisInv data set according to various evaluation metrics. The inversion results are more consistent with the target from the aspects of velocity values, subsurface structures, and geological interfaces. Moreover, the mechanism and the generalization of the proposed method are discussed and verified. Nevertheless, the generalization of deep-learning-based inversion methods on real data is still challenging and considering physics may be one potential solution.

Facies identification is a powerful means to predict reservoirs. We achieve facies identification using a relevance vector machine (RVM) and develop a facies discriminant method based on a multikernel RVM (MKRVM). An RVM has the same functional form as a support vector machine (SVM) that is widely used in geophysics and shows a promising performance in disposing of small-samples, nonlinear and high-dimensional problems. The RVM inherits these superiorities, and its training is implemented under the Bayesian framework. Thus, it can provide probability information about the classified facies, which is critical to evaluate uncertainty of the result. Besides, the penalty parameter of the RVM does not depend on human experience. Compared with single-kernel learning, multikernel learning (MKL) is more flexible. After mapping the original data into a combined space by MKL, the features can be more accurately expressed in the new space, thereby improving the classification accuracy. Therefore, we introduce the RVM into facies classification and extend it to the MKRVM-based facies identification. The proposed method has advantageous properties such as strong generalization ability and high accuracy. First, we apply the approach to well log facies classification with different input features. Then, it is applied to seismic lithofacies identification with inverted elastic attributes to predict the target reservoirs. All the examples verify the effect and potential of the new method.

Incoherent noise is one of the most common noise widely distributed in seismic data. To improve the interpretation accuracy of the underground structure, incoherent noise needs to be adequately suppressed before the final imaging. We propose a novel method for suppressing seismic incoherent noise based on the robust low-rank approximation. After the Hankelization, seismic data will show strong low-rank features. Our goal is to obtain the stable and accurate low-rank approximation of the Hankel matrix and then reconstruct the denoised data. We construct a mixed model of the nuclear norm and the$l_{1}$norm to express the low-rank approximation of the Hankel matrix constructed in the frequency domain. Essentially, the adopted model is an optimization for the subspace similar to the online subspace tracking method, thus avoiding the time-consuming singular value decomposition (SVD). We introduce the orthonormal subspace learning to convert the nuclear norm to the$l_{1}$norm to optimize the orthonormal subspace and the corresponding coefficient. Finally, two optimization strategies—the alternating direction method and the block coordinate descent method—are applied to obtain the optimized orthonormal subspace and the corresponding coefficient for representing the low-rank approximation of the Hankel matrix. We perform incoherent noise attenuation tests on synthetic and real seismic data. Compared with other denoising methods, the proposed method produces small signal errors while effectively suppressing the seismic incoherent noise and has a high computational efficiency.

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 seismic noise attenuation approach based on least-squares Gaussian beam transform (LSGBT). Gaussian beam transform uses time-domain Gaussian beam (TGB), which can be characterized by a particular location, arrival time, amplitude, slope, curvature, and width. We implement the local attributes such as beam center, spacing, and width to perform Gaussian beam decomposition. In this approach, we first introduce the plane-wave decomposition (PWD) theory to implement TGB decomposition of noisy seismic data and then apply data reconstruction. Unlike most state-of-the-art algorithms, random noise is attenuated in the process of Gaussian beam reconstruction. In the reconstruction records, the useful events are well preserved simultaneously removing random noise. Comparisons of experimental results on field data using traditional$f$-$x$deconvolution (FX Decon) and median filter (MF) methods are also provided, which suggest that our method achieves better denoising performance than FX Decon and MF methods. Taking into account that signal loss is sometimes unavoidable in almost all existing denoising methods. In addition to the signal-to-noise ratio (SNR) measurement, we also use local similarity as an efficient tool to evaluate denoising performance. A group of synthetic and field examples demonstrates the effectiveness of the proposed approach.

Noise attenuation plays an important role in seismic data processing. We propose a novel denoising method for seismic data based on unsupervised sparse feature learning. Our goal is to obtain the identifiable feature of the noisy seismic data and then to represent the effective signals. By preprocessing the raw data and training the autoencoder neural network with sparse constraint, the sparse feature of the seismic data can be learned and stored in the neural network. We use the adaptive moment estimation as a backpropagation algorithm to minimize the cost function with a sparse penalty term and combine the dropout technique in the training process to improve the feature extraction and generalization capability of the neural network. Then, the test data set can be reconstructed by the most important sparse features. The final denoising result can be obtained by rearranging the output test data set. Compared with three commonly used state-of-the-art denoising methods, the proposed method performs well in applications to denoising for synthetic and real seismic data.

Least-squares reverse time migration (LSRTM) can provide higher quality images than conventional reverse time migration, which is helpful to image simultaneous-source data. However, it still faces the problems of the crosstalk noise, great computation time, and storage requirement. We propose a new LSRTM approach by using the excitation amplitude (EA) imaging condition to suppress the crosstalk noise. Since only the maximum amplitude or limited local maximum amplitudes at each imaging point and the corresponding travel time step(s) need to be saved, the great storage problem can be naturally solved. Consequently, the proposed algorithm can avoid the frequent memory transfer and is suitable for the graphics processing unit (GPU) parallelization. Besides, the shared memory with high bandwidth is used to optimize the GPU-based algorithm. In order to further improve the image quality of EA imaging condition, we adopt the shaping regularization as a constraint. The single-source tests with Marmousi and salt models show the feasibility of our algorithm to image the complex and subsalt structures, among which a wrong background velocity is used to test its sensitivity to the velocity error. The noise-free and noise-included simultaneous-source examples demonstrate the ability of EA imaging condition to suppress the crosstalk noise. During the implementation of the GPU parallelization, we find that the shared memory cannot always optimize the GPU parallel algorithm and just works well for the eighth- or higher order spatial finite difference scheme.

Recovering signal from under-sampled and erratic noise-corrupted seismic data is indeed a challenging task because of its difficulty in simultaneous modeling of erratic noise and missing signal. Assuming that the recorded data are the superposition of low-rank and sparse components, many related works have been reported using a hybrid rank-sparsity constraint. Those published works typically detect the rank and erratic noise using empirical and global thresholds, which often fail to well characterize the varying sparsity and easily cause biased estimation in case of their nonstationary distribution. We propose an iterative double Laplacian-scaled low-rank optimization to adaptively select the sparsity and rank regularizer parameters for robust signal recovery. Comparing with the published approaches with global threshold, Laplacian-scaled mixture, which is obtained by multiplying Laplacian variable with a Gamma variable, is used to locally model the sparsity of erratic noise and the low-rank feature of signal. Then, the expectation-maximization (EM) algorithm is used to transform the Laplacian-scaled mixture problem into a localized reweighted ℓ1minimization scheme. The weighted coefficient appearing in its EM solver provides a variable constraint to locally address the rank and erratic noise, and hence, their regularizer parameters can dynamically reflect the different importance of those coefficients. We tested the effectiveness of the proposed method using under-sampled synthetic and field data that are corrupted by erratic noise and used other state-ofthe-art methods as comparisons. The results showed that more exact estimations of the signal and erratic noise can be obtained using the proposed method.

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.

Simultaneous-source shooting can help reduce the acquisition time cost, but at the expense of introducing strong interference (blending noise) into the acquired seismic data. It has been demonstrated previously that the deblending problem can be considered as an inversion process. In this letter, we propose a new iterative approach to solve this inversion problem. In the proposed approach, a new coherency-promoting constraint, called structuring median filtering (SMF), is proposed and used to regularize the estimated model in each iteration. The SMF processes the signal by the interactions of the input signal and another given small section of signal, namely, the structuring element. The SMF is more robust than other coherency-promoting filtering such as the median filtering and mathematical morphological filtering. Numerical experiments demonstrate that the iterative deblending based on the SMF constraint obtains a better performance and a faster convergence than the low-rank and compressed sensing constraint-based deblending approaches.

Time-frequency analysis always plays a central role in the field of seismic processing due to the advantage in characterizing nonstationary signals. In this letter, we present a novel technique for seismic time-frequency analysis based on the high-order synchrosqueezing transform, which obtains more accurate instantaneous frequencies by using the higher order approximations for both amplitude and phase in order to achieve a highly energy-concentrated time-frequency representation. A synthetic example is employed to demonstrate the validity of the proposed method in sharpening time-frequency representation. Application on field data example further proves its potential in enhancing time-frequency resolution and delineating stratigraphic characteristics with higher precision and renders that this technique is promising for seismic data analysis.

Amplitude-versus-offset (AVO) inversion always plays an important role in reservoir fluid identification, which allows the estimation of various rock and fluid properties from prestack seismic data. In this paper, we propose a new method for discrimination of hydrocarbon accumulation that combines frequency-dependent AVO inversion scheme and variational mode decomposition (VMD). VMD is a recently developed algorithm for adaptive signal decomposition that is able to nonrecursively decompose a multicomponent signal into a number of quasi-orthogonal intrinsic mode functions and avoid mode mixing effectively. VMD is superior to other state-of-the-art approaches in obtaining high-resolution and high-fidelity local time-frequency depiction performance. Two synthetic signals are employed to illustrate that VMD achieves higher temporal and frequency resolution than the conventional continuous wavelet transform (CWT) decomposition. Other synthetic examples, elastic and dispersive, are utilized to demonstrate that the proposed method is more reliable for the detection of hydrocarbon saturation and a comparison is made with the CWT-based inverted results. Application on field data has further shown that the proposed approach has the potential in identifying the reservoir related to hydrocarbon.

Sparse domain thresholding filters operating in a sparse domain are highly effective in removing Gaussian random noise under Gaussian distribution assumption. Erratic noise, which designates non-Gaussian noise that consists of large isolated events with known or unknown distribution, also needs to be explicitly taken into account. However, conventional sparse domain thresholding filters based on the least-squares (LS) criterion are severely sensitive to data with high-amplitude and non-Gaussian noise, i.e., the erratic noise, which makes the suppression of this type of noise extremely challenging. In this paper, we present a robust sparsity-promoting denoising model, in which the LS criterion is replaced by the Huber criterion to weaken the effects of erratic noise. The random and erratic noise is distinguished by using a data-adaptive parameter in the presented method, where random noise is described by mean square, while the erratic noise is downweighted through a damped weight. Different from conventional sparse domain thresholding filters, definition of the misfit between noisy data and recovered signal via the Huber criterion results in a nonlinear optimization problem. With the help of theoretical pseudoseismic data, an iterative robust sparsity-promoting filter is proposed to transform the nonlinear optimization problem into a linear LS problem through an iterative procedure. The main advantage of this transformation is that the nonlinear denoising filter can be solved by conventional LS solvers. Tests with several data sets demonstrate that the proposed denoising filter can successfully attenuate the erratic noise without damaging useful signal when compared with conventional denoising approaches based on the LS criterion.

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.

Hydrocarbons can cause anomalies in the energy density of seismic signals when seismic waves pass through them. Teager-Kaiser energy (TKE) is an important attribute that can be utilized to depict the energy density of a seismic signal and the energy distribution of a seismic wavefield. In this letter, a novel spectral decomposition-based approach for hydrocarbon detection is proposed that applies the variational mode decomposition (VMD) associated with TKE to seismic data, which is called the VMDTKE algorithm. The proposed method not only possesses the better performance of TKE in focusing instantaneous energy, but also inherits the merit of high time-frequency resolution from VMD. The Marmousi2 example is used to demonstrate that the VMDTKE approach is capable of depicting the location and extent of strong anomalies which correlate to hydrocarbons more clearly. We compare the spectral decomposition results with that from the conventional VMD-based method. Application on field data further confirms the potential of the VMDTKE algorithm in delineating strong amplitude anomalies that are associated with hydrocarbon reservoirs.

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.

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.

Spikelike noise is a common type of random noise existing in many geoscience and remote sensing data sets. The attenuation of spike-like noise has become extremely important recently, because it is the main bottleneck when processing the simultaneous source data that are generated from the modern seismic acquisition. In this letter, we propose a novel low-rank decomposition algorithm that is effective in rejecting the spike-like noise in the seismic data set. The specialty of the low-rank decomposition algorithm is that it is applied along the morphological direction of the seismic data sets with a prior knowledge of the morphology of the seismic data, which we call local slope. The seismic data are of much lower rank along the morphological direction than along the space direction. The morphology of the seismic data (local slope) is obtained via a robust plane-wave destruction method. We use two simulated field data examples to illustrate the algorithm workflow and its effective performance.

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 traditional high-order staggered-grid finite-difference (SGFD) method has high-order accuracy in space, but only the second-order accuracy in time, which makes the traditional SGFD method suffer from a large temporal dispersion error during long-distance wave propagation. This paper develops temporal fourth- and sixth-order and spatial arbitrary evenorder SGFD schemes to model isotropic elastic wave propagation. The temporal high-order SGFD schemes have smaller temporal dispersion than the traditional temporal second-order scheme, and thus allow larger time steps to attain a similar accuracy. The developed temporal high-order SGFD schemes are applied to simulate a quasi-stress–velocity wave equation (QWE) that is derived in the framework of a$k$-space approach. A split QWE (SQWE) is further developed, and numerical simulation of SQWE results in separated P (compressional)-wave and S (shear)-wave. Theoretical computational cost analysis verifies that the numerical simulation of QWE using the temporal fourthand sixth-order SGFD schemes is more efficient than the numerical simulation of the traditional stress–velocity wave equation using the traditional temporal second-order SGFD scheme in 2-D. In 3-D, the temporal fourth-order SGFD scheme still runs faster than the traditional temporal second-order scheme; however, the temporal sixth-order scheme is more efficient only when a longer stencil length than 12 is adopted. Numerical examples confirm the correctness of the developed elastic wave modeling schemes.

A real-world signal is always corrupted with noise. The separation between a signal and noise is an indispensable step in a variety of signal-analysis applications across different scientific domains. In this paper, we propose a double least-squares projections (DLSPs) method to estimate a signal from the noisy data. The first least-squares projection is to find a signal-dimensional optimal approximation of the noisy data in the least-squares sense. In this step, a rough estimation of the signal is obtained. The second least-squares projection is to find an approximation of a signal in another crossed signal-dimensional space in the least-squares sense. In this step, a much improved signal estimation that is close to orthogonal to the separated noise subspace can be obtained. The DLSP implements projection operation twice to obtain an almost perfect estimation of a signal. The application of the DLSP method in seismic random noise attenuation and signal reconstruction demonstrates the successful performance in seismic data processing.

We propose a novel method for simultaneously removing random noise and increasing the resolution of seismic data. We formulate the problem of simultaneous denoising and increasing resolution as an inverse problem and solve it using the shaping regularization framework. The resolution of seismic data is increased by shaping the estimated seismic wavelet into a user-defined wavelet that is easier for interpretation. The random noise is removed by a simple FK thresholding step in the shaping regularization framework. There are two shaping processes here: 1) shaping the estimated wavelet to a more ideal wavelet; and 2) shaping the inverted model to a more admissible model during the iterations. We use a wedge synthetic model and land poststack seismic data to demonstrate the performance of the proposed approach. It is the first time that a simultaneous resolution enhancement and noise attenuation framework is proposed to process poststack seismic data.

The distance-separated simultaneous-sourcing (DSSS) technique can make the smallest interference between different sources. In a distance-separated simultaneous-source acquisition with two sources, we propose the use of a novel iterative seislet-frame thresholding approach to separate the blended data. Because the separation is implemented in common shot gathers, there is no need for the random scheduling that is used in conventional simultaneous-source acquisition, where random scheduling is applied to ensure the incoherent property of blending noise in common midpoint, common receiver, or common offset gathers. Thus, DSSS becomes more flexible. The separation is based on the assumption that the local dips of the data from different sources are different. We can use the plane-wave destruction algorithm to simultaneously estimate the conflicting dips and then use seislet frames with two corresponding local dips to sparsify each signal component. Then, the different signal components can be easily separated. Compared with the FK-based approach, the proposed seislet-frame-based approach has the potential to obtain better separated components with less artifacts because the seislet frames are local transforms while the Fourier transform is a global transform. Both simulated synthetic and field data examples show very successful performance of the proposed approach.

Time-frequency analysis is able to reveal the useful information hidden in the seismic data. The high resolution of the time-frequency representation is of great importance to depict geological structures. In this letter, we propose a novel seismic time-frequency analysis approach using the newly developed empirical wavelet transform (EWT). It is the first time that EWT is applied in analyzing multichannel seismic data for the purpose of seismic exploration. EWT is a fully adaptive signal-analysis approach, which is similar to the empirical mode decomposition but has a consolidated mathematical background. EWT first estimates the frequency components presented in the seismic signal, then computes the boundaries, and extracts oscillatory components based on the boundaries computed. Synthetic, 2-D, and 3-D real seismic data are used to comprehensively demonstrate the effectiveness of the proposed seismic time-frequency analysis approach. Results show that the EWT can provide a much higher resolution than the traditional continuous wavelet transform and offers the potential in precisely highlighting geological and stratigraphic information.

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.

Simultaneous sources acquisition attracts intensive attention from both academia and industry due to its greatly improved efficiency in acquiring high-density seismic data. Unfortunately, its merits are compromised by the strong interference noise between adjacent shots. In this letter, we propose a stepwise rank-increasing (RI) method to estimate the crosstalk noise in simultaneous sources acquisition. The proposed algorithm assumes that an ideal common offset gather (COG) can be represented via a low-rank matrix in the time-space domain. The coherent signals are estimated from low-rank decomposition and transformed to the crosstalk noise by employing a priori information about random dithering code, and then the blending noise is subtracted from the blended data. By increasing the rank of coherent signals step-by-step, the crosstalk noise can be gradually estimated with high accuracy. In this letter, singular value decomposition is utilized to increase the rank of COG data. Applications on synthetic and field data sets demonstrate the better performance of the proposed RI method not only by more effectively suppressing noise but also by accelerating the convergence rate.

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.

It has been previously shown that blended simultaneous-source data can be successfully separated using an iterative seislet thresholding algorithm. In this letter, I combine iterative seislet thresholding with a local orthogonalization technique via a shaping regularization framework. During the iterations, the deblended data and its blending noise section are not orthogonal to each other, indicating that the noise section contains significant coherent useful energy. Although the leakage of useful energy can be retrieved by updating the deblended data from the data misfit during many iterations, I propose to accelerate the retrieval of the leakage energy using iterative orthogonalization. It is the first time that multiple constraints are applied in an underdetermined deblending problem, and the new proposed framework can overcome the drawback of a low-dimensionality constraint in a traditional 2-D deblending problem. Simulated synthetic and field data examples show the superior performance of the proposed approach.

Bandpass filtering is a common way to estimate ground-roll noise on land seismic data, because of the relatively low-frequency content of ground roll. However, there is usually a frequency overlap between ground roll and the desired seismic reflections that prevents bandpass filtering alone from effectively removing ground roll without also harming the desired reflections. We apply a bandpass filter with a relatively high upper bound to provide an initial imperfect separation of ground roll and reflection signal. We then apply a technique called “local orthogonalization” to improve the separation. The procedure is easily implemented, since it involves only bandpass filtering and a regularized division of the initial signal and noise estimates. We demonstrate the effectiveness of the method on an open-source set of field data.

Interpolating regularly missing traces in seismic data is thought to be much harder than interpolating irregularly missing seismic traces, because many sparsity-based approaches cannot be used due to the strong aliasing noise in the sparse domain. We propose to use the seislet transform to perform a sparsity-based approach to interpolate highly undersampled seismic data based on the classic projection onto convex sets (POCS) framework. Many numerical tests show that the local slope is the main factor that will affect the sparsity and antialiasing ability of seislet transform. By low-pass filtering the undersampled seismic data with a very low bound frequency, we can get a precise dip estimation, which will make the seislet transform capable for interpolating the aliased seismic data. In order to prepare the optimum local slope during iterations, we update the slope field every several iterations. We also use a percentile thresholding approach to better control the reconstruction performance. Both synthetic and field examples show better performance using the proposed approach than the traditional prediction based and the F-K-based POCS approaches.