Seismic Data Reconstruction Research Articles

Fast iterative adaptive approach for 3D seismic data reconstruction

ABSTRACT In this paper, we introduce a fast iterative adaptive approach for the reconstruction of 3D seismic data with randomly missing traces. Our method starts by transforming 3D seismic volume to 2D harmonic signal for each frequency slice, and then the power spectrum of the frequency slice is iteratively estimated by a weighted least square fitting criterion. The missing data can be recovered with the obtained spectral estimate using a linear minimum mean-squared error estimator. However, estimation of the power spectrum depends on matrix-vector multiplications for each iteration that leads to high computation complexity when the data are large scale and high dimension. To solve the associated spectrum estimation problem above, a fast iterative adaptive scheme is adopted by utilising 2D fast discrete Fourier transform, which makes use of the block-Vandermonde structure and the 2D Fourier property of the steering matrix. Simulation experiments on synthetic and field data verify the effectiveness of the proposed algorithm. Compared with other reconstruction methods based on frequency slice, such as minimum-weighted norm interpolation approach, multi-channel singular spectrum analysis, and Curvelet transform method, the proposed method achieves better reconstruction performance and low computational complexity.

Robust damped rank-reduction method for simultaneous denoising and reconstruction of 5D seismic data

We have developed a new method for simultaneous denoising and reconstruction of 5D seismic data corrupted by random noise and missing traces. Several algorithms have been developed for seismic data restoration based on rank-reduction (RR) methods. More recently, a damping operator has been introduced into the conventional truncated singular-value decomposition (TSVD) formula to further remove residual noise, the presence of which disturbs the quality of the seismic results. Despite the success of the damped RR (DRR) method when the observed data have an extremely low signal-to-noise ratio (S/N), random noise is still a limiting factor for obtaining a perfect quality of the result. Therefore, how to accurately solve the simultaneous denoising and reconstruction problem with high fidelity is still challenging. We assume that introducing only the damping operator into the TSVD formula is not enough to remove the random noise and restore the useful signal well. By combining the soft thresholding (ST) operator and the moving-average (MA) filter, we have developed a new operator, which we call the STMA operator. Then, by introducing the STMA operator into the DRR framework, we have developed a new algorithm known as the robust DRR method, which aims at mixing the advantages of the STMA operator and the damping operator. The STMA operator is applied to the Hankel matrix after damped TSVD to better remove residual noise. Examples of our approach on synthetic and field 5D seismic data demonstrate the better performance in terms of the visual examination and numerical test compared with the DRR approach. Our method aims at producing an effective low-rank filter and, thus, can perfectly enhance the S/N of the simultaneously denoised and reconstructed results with higher accuracy.

Seismic data reconstruction method based on morphological component analysis

Seismic data reconstruction method based on morphological component analysis

Seismic data reconstruction method based on morphological component analysis

The real seismic data is usually under-sampled in space domain because of the physical or economic limitations. So the incomplete seismic data need to be reconstructed before the subsequent processing. A method based on morphological component analysis (MCA) is discussed in the paper, which uses the curvelet dictionary and local discrete cosine transform (LDCT) dictionary to reconstruct the smooth components and the singular components respectively. Block coordinate relaxation (BCR) algorithm is adopted to complete the sparse optimisation. The validity of the proposed method was tested by numerical experiments on synthetic and real data demonstrate. As the method based on curvelet combining with POCS is widely used in practice, we compare its reconstructed results with the MCA-based method. The numerical results validate that the proposed method has higher reconstruction performance.

Retraction notice: Multi-step damped multichannel singular spectrum analysis for simultaneous reconstruction and denoising of 3D seismic data

Retraction notice: Multi-step damped multichannel singular spectrum analysis for simultaneous reconstruction and denoising of 3D seismic data

Retraction notice: Adaptive rank-reduction method for seismic data reconstruction

Retraction notice: Adaptive rank-reduction method for seismic data reconstruction

Application of Envelope in Salt Structure Velocity Building: From Objective Function Construction to the Full-Band Seismic Data Reconstruction

For full-waveform inversion (FWI), building the salt structure velocity model is a challenging problem. In the absence of a high-fidelity initial model, low-frequency seismic data are indispensable for accurate reconstruction of the long-wavelength components of salt domes, which are often missing in field data. The envelope isolates the amplitude information from the seismic data and has a stronger linear relationship with the velocity than the seismic data. It is used to enhance the convexity of the objective function in an envelope inversion (EI). However, the properties of the envelope are not effectively utilized in EI due to the lack of their proper understanding. In order to solve the problems that EI has in building the salt structure velocity, such as the cycle-skipping problem, the gradient calculation problem caused by the waveform Frechet derivative, and the multisolution problem caused by the missing polarity information of the envelope, we propose a multiscale direct signed EI (MSDSEI) based on the direct envelope Frechet derivative and multiscale signed envelope. The development process from EI to MSDSEI also inspired us to understand the physical mechanism involved in the low-frequency components of the envelope: the signed envelope can be regarded as a low-pass filter of the reflection sequences in the subsurface. Thus, we use the smoothness and phase-independent properties of the envelope to propose an envelope-based full-band seismic data reconstruction method for multiscale FWI. Finally, the performance of various EI methods is verified on the Sigsbee2A model.

Five-dimensional seismic data reconstruction using the optimally damped rank-reduction method

SUMMARY It is difficult to separate additive random noise from spatially coherent signal using a rank-reduction (RR) method that is based on the truncated singular value decomposition (TSVD) operation. This problem is due to the mixture of the signal and the noise subspaces after the TSVD operation. This drawback can be partially conquered using a damped RR (DRR) method, where the singular values corresponding to effective signals are adjusted via a carefully designed damping operator. The damping operator works most powerfully in the case of a small rank and a small damping factor. However, for complicated seismic data, e.g. multichannel reflection seismic data containing highly curved events, the rank should be large enough to preserve the details in the data, which makes the DRR method less effective. In this paper, we develop an optimal damping strategy for adjusting the singular values when a large rank parameter is selected so that the estimated signal can best approximate the exact signal. We first weight the singular values using optimally calculated weights. The weights are theoretically derived by solving an optimization problem that minimizes the Frobenius-norm difference between the approximated and the exact signal components. The damping operator is then derived based on the initial weighting operator to further reduce the residual noise after the optimal weighting. The resulted optimally damped rank-reduction method is nearly an adaptive method, i.e. insensitive to the rank parameter. We demonstrate the performance of the proposed method on a group of synthetic and real 5-D seismic data.

Tubal-Sampling: Bridging Tensor and Matrix Completion in 3-D Seismic Data Reconstruction

The 3-D seismic data reconstruction can be understood as an underdetermined inverse problem, and thus, some additional constraints need to be provided to achieve reasonable results. A prevalent scheme in 3-D seismic data reconstruction is to compute the best low-rank approximation of a formulated Hankel matrix by rank-reduction methods with a rank constraint. However, the predefined Hankel structure is easily damaged by the low-rank approximation, which leads to harming its recovery performance. In this article, we present a structured tensor completion (STC) framework to simultaneously exploit both the Hankel structure and the low-tubal-rank constraint to further enhance the performance. Unfortunately, under the assumption of elementwise sampling used by existing methods, STC is intractable to be solved since Hankel constraints cannot be expressed as linear tensor equations. Instead, tubal sampling is proposed to describe the missing trace behavior more accurately and further build a bridge between tensor and matrix completion (MC) to overcome the solving issue in two aspects: through the bridge from tensor to MC, STC can be solved efficiently using MC from random samplings of each frontal slice in the Fourier domain. Through the bridge from matrix to tensor completion, various tensor models within the framework can be developed from noise-specific MC to meet the need for data reconstruction in changeable noise environments. Moreover, alternating-minimization and alternating-direction methods of multipliers are developed to solve the proposed STC. The superior performance of STC is demonstrated in both synthetic and field seismic data.

Seismic Data Reconstruction Using Deep Bidirectional Long Short-Term Memory With Skip Connections

Due to environmental and economic constraints on their acquisition, seismic data are always irregularly sampled and include bad or missing traces, which can cause problems for seismic data processing. Recently, many researchers have attempted to improve seismic data reconstruction using machine learning (ML) techniques, such as convolutional neural networks, which are inspired by computer vision and imaging processing. In this letter, we propose a novel approach for reconstructing missing traces in seismic data using ML techniques, especially recurrent neural network (RNN) algorithms. Instead of processing seismic data as an image, the proposed approach performs seismic trace interpolation using traces that are sequences of time-series data. More specifically, we adopt deep bidirectional long short-term memory (LSTM) for seismic trace interpolation and test models with and without skip connections. Field seismic data are used to demonstrate the effectiveness of the proposed approaches, and the deep bidirectional LSTM (DBiLSTM) with skip connections shows the best performance compared to cubic interpolation, minimum weighted norm interpolation (MWNI), and DBiLSTM without skip connection.

Robust dictionary learning for erratic noise-corrupted seismic data reconstruction

Dictionary learning methods adaptively train their bases from the given data in an iterative manner; hence, they can capture more detailed features and achieve sparser representation than a method that uses a fixed basis. However, there also exists a good chance of erratic noise corrupting the dictionary because of the insufficiency of the L1-norm regularization in distinguishing the signal and erratic noise, causing the inaccuracy of both dictionary learning and its application. A robust dictionary learning method is proposed to reconstruct the missing data even in the presence of strong erratic noise. Specifically, a projected operator, which is constructed with the robust Huber misfit, is used to damp erratic noise to a low-amplitude level. In the dictionary learning step, erratic noise is gradually reduced to an acceptable level with the help of this projected operator, from which the signal prototypes (or atoms) can be safely extracted in each iteration. Then, the learned dictionary is used to iteratively estimate the signal from the noisy and under-sampled data, performing noise attenuation as well as data interpolation at non-recording locations. We test the proposed dictionary learning method using 2D and 3D noisy seismic examples and compare it with other state-of-the-art methods. Numerical results demonstrate its superior effectiveness at recovering missing data and increasing signal-to-noise ratio in the presence of erratic noise.

A novel thresholding method for simultaneous seismic data reconstruction and denoising

Seismic trace missing together with random noise contamination may seriously influence the performance of some multichannel processing techniques. Simultaneous seismic data reconstruction and denoising is a popular antidote to these two issues since traces missing and random noise both generate random spectra in the frequency domain. However, aliasing in the FK spectrum and random noise residuals are often seen in the reconstructed data with some existing methods. In this paper, we developed a novel thresholding method based on the L1 norm regularization to improve simultaneous reconstruction and denoising quality. We proposed a modified thresholding method by inducing a weighting parameter to the iterative process. The threshold values are reduced exponentially during iteration to ensure stable outputs. Since the curvelet transform gives a very sparse expression of seismic data, we choose the curvelet transform as the sparse representation. Numerical tests demonstrate that the proposed method achieves both random noise elimination and missing trace reconstruction effectively.

Expression of Concern: Five-dimensional seismic data reconstruction using the optimally damped rank-reduction method

Expression of Concern: Five-dimensional seismic data reconstruction using the optimally damped rank-reduction method

2-D Seismic Data Reconstruction via Truncated Nuclear Norm Regularization

Rank-reduction (RR) methods have been widely applied for reconstructing seismic data. The popular convex relaxation formulation of RR is nuclear norm minimization (NNM), which is capitalized on its convexity. Consequently, global optimization can be effectively achieved with NNM method. However, NNM minimizes the summation of all singular values and is therefore not equivalent to the minimum of the rank function; in fact, the rank of the matrix is equal to the number of nonzero singular values. Thus, NNM is not a good approximation to the original RR problem, which affects the reconstruction performance. The truncated nuclear norm regularization (TNNR) method only minimizes the smallest $\min (m,n)-r$ singular values ( $m, n$ is the size of matrix and $r$ is the rank), yielding more accurate approximations to the rank function in real applications. In this article, we propose a new method for seismic reconstruction via TNNR based on the texture patch matrix, and further provide a strategy to estimate $r$ , which can efficiently reduce the cost of computation. Numerical experiments performed on both synthetic and real data indicate that the reconstruction quality based on our proposed algorithm is better than that based on the singular value thresholding algorithm and the low-rank factorization model (low-rank matrix fitting).

Editor's note: Multi-step damped multichannel singular spectrum analysis for simultaneous reconstruction and denoising of 3D seismic data

Editor's note: Multi-step damped multichannel singular spectrum analysis for simultaneous reconstruction and denoising of 3D seismic data

Editor's note: Adaptive rank-reduction method for seismic data reconstruction

Editor's note: Adaptive rank-reduction method for seismic data reconstruction

Two-dimensional seismic data reconstruction using patch tensor completion

Seismic data are often undersampled owing to physical or financial limitations. However, complete and regularly sampled data are becoming increasingly critical in seismic processing. In this paper, we present an efficient two-dimensional (2D) seismic data reconstruction method that works on texture-based patches. It performs completion on a patch tensor, which folds texture-based patches into a tensor. Reconstruction is performed by reducing the rank using tensor completion algorithms. This approach differs from past methods, which proceed by unfolding matrices into columns and then applying common matrix completion approaches to deal with 2D seismic data reconstruction. Here, we first re-arrange the seismic data matrix into a third-order patch tensor, by stacking texture-based patches that are divided from seismic data. Then, the seismic data reconstruction problem is formulated into a low-rank tensor completion problem. This formulation avoids destroying the spatial structure, and better extracts the underlying useful information. The proposed method is efficient and gives an improved performance compared with traditional approaches. The effectiveness of our patch tensor-based framework is validated using two classical tensor completion algorithms, low-rank tensor completion (LRTC), and the parallel matrix factorization algorithm (TMac), on both synthetic and field data experiments.

Fast and Robust Low-Rank Approximation for Five-Dimensional Seismic Data Reconstruction

Five-dimensional (5D) seismic data reconstruction becomes more appealing in recent years because it takes advantage of five physical dimensions of the seismic data and can reconstruct data with large gap. The low-rank approximation approach is one of the most effective methods for reconstructing 5D dataset. However, the main disadvantage of the low-rank approximation method is its low computational efficiency because of many singular value decompositions (SVD) of the block Hankel/Toeplitz matrix in the frequency domain. In this paper, we develop an SVD-free low-rank approximation method for efficient and effective reconstruction and denoising of the seismic data that contain four spatial dimensions. Our SVD-free rank constraint model is based on an alternating minimization strategy, which updates one variable each time while fixing the other two. For each update, we only need to solve a linear least-squares problem with much less expensive QR factorization. The SVD-based and SVD-free low-rank approximation methods in the singular spectrum analysis (SSA) framework are compared in detail, regarding the reconstruction performance and computational cost. The comparison shows that the SVD-free low-rank approximation method can obtain similar reconstruction performance as the SVD-based method but with a large computational speedup.

Based on Compressed Sensing of Orthogonal Matching Pursuit Algorithm Image Recovery

Compressive sensing theory mainly includes the sparsely of signal processing, the structure of the measurement matrix and reconstruction algorithm. Reconstruction algorithm is the core content of CS theory, that is, through the low dimensional sparse signal recovers the original signal accurately. This thesis based on the theory of CS to study further on seismic data reconstruction algorithm. We select orthogonal matching pursuit algorithm as a base reconstruction algorithm. Then do the specific research for the implementation principle, the structure of the algorithm of AOMP and make the signal simulation at the same time. In view of the OMP algorithm reconstruction speed is slow and the problems need to be a given number of iterations, which developed an improved scheme. We combine the optimized OMP algorithm of constraint the optimal matching of item selection strategy, the backwards gradient projection ideas of adaptive variance step gradient projection method and the original algorithm to improve it. Simulation experiments show that improved OMP algorithm is superior to traditional OMP algorithm of improvement in the reconstruction time and effect under the same condition. This paper introduces CS and most mature compressive sensing algorithm at present orthogonal matching pursuit algorithm. Through the program design realize basic orthogonal matching pursuit algorithms, and design realize basic orthogonal matching pursuit algorithm of one-dimensional, two-dimensional signal processing simulation.

Nonconvex Log-Sum Function-Based Majorization–Minimization Framework for Seismic Data Reconstruction

Because of the fact that complete seismic data can have a low rank in the frequency-space (f-x) domain, rank-reduction methods are classical techniques used for seismic data reconstruction. Models that employ nuclear-norm minimization signify convex relaxation methods in traditional rank minimization problems. However, the results obtained after solving the nuclear-norm minimization problem are usually suboptimal because the nuclear norm indicates a loose approximation of the rank function. To overcome the limitations of the nuclear norm, we propose a new method of seismic data reconstruction based on the log-sum function minimization, which is closer to the rank function than the convex nuclear norm. However, the problem based on the log-sum function is nonconvex. Consequently, a majorization–minimization framework has been adopted to solve the associated minimization problem. Numerical experiments performed using synthetic and real data demonstrate that the quality of reconstruction derived from our proposed algorithm is better than that of the singular value thresholding algorithm and the weighted nuclear norm method.