Benfeng Wang

An Improved Weighted Projection Onto Convex Sets Method for Seismic Data Interpolation and Denoising

Due to the environment effects, economy restrictions, and acquisition equipment limitations, observed seismic data always have several traces missing and contain some random noise, affecting the performance of surface-related multiple elimination (SRME), wave-equation-based imaging, and inversion. Projection onto convex sets (POCS) is an effective interpolation algorithm, while the performance is unsatisfactory in noisy situations. Weighted POCS (WPOCS) method can weaken the random noise effects to some extent, but the performance is still unsatisfactory. Thus, an improved WPOCS (IWPOCS) method is proposed in this paper, for seismic data interpolation and denoising simultaneously based on Curvelet transform. First, the POCS formula is derived from the iterative hard threshold (IHT) view. Then, its shortcoming is analyzed because there is an implicit assumption that the observed seismic data should have a high signal-to-noise ratio (SNR). Finally, a novel method named IWPOCS is proposed based on WPOCS method, which can achieve simultaneous interpolation and denoising. Among the above three methods, the IWPOCS method is the most effective to interpolate and denoise seismic data in terms of recovered SNR and visual view. Numerical experiments on the synthetic data and the real seismic data from the marine acquisition with towed streamers confirm the validity of the proposed IWPOCS method.

Accelerating seismic interpolation with a gradient projection method based on tight frame property of curvelet

Seismic interpolation, as an efficient strategy of providing reliable wavefields, belongs to large-scale computing problems. The rapid increase of data volume in high dimensional interpolation requires highly efficient methods to relieve computational burden. Most methods adopt the L1 norm as a sparsity constraint of solutions in some transformed domain; however, the L1 norm is non-differentiable and gradient-type methods cannot be applied directly. On the other hand, methods for unconstrained L1 norm optimisation always depend on the regularisation parameter which needs to be chosen carefully. In this paper, a fast gradient projection method for the smooth L1 problem is proposed based on the tight frame property of the curvelet transform that can overcome these shortcomings. Some smooth L1 norm functions are discussed and their properties are analysed, then the Huber function is chosen to replace the L1 norm. The novelty of the proposed method is that the tight frame property of the curvelet transform is utilised to improve the computational efficiency. Numerical experiments on synthetic and real data demonstrate the validity of the proposed method which can be used in large-scale computing. A gradient projection method for seismic interpolation based on the tight frame property of curvelet transform is proposed. Some smooth L1 norm functions were analysed, and the Huber function was chosen to replace the L1 norm. The tight frame property of the curvelet transform is utilised to improve the computational efficiency.

An Amplitude Preserving S-Transform for Seismic Data Attenuation Compensation

The S-transform (ST), as a time-frequency analysis tool, has been widely used, but the amplitude preserving property is a little poor near the boundary of the selected discrete signal. The reason lies that the summation of the product between the analytical window and the comprehensive window over the sliding step deviates from unity near the boundary in the discrete cases. In order to hold the amplitude preserving property for the discrete signal recovery analysis, an amplitude preserving S-transform (APST) is proposed based on a novel analytical window selection. First, lots of numerical tests are used to analyze the shortcomings of the ST near the boundary for the selected discrete signal and demonstrate the effectiveness and the validity of the proposed APST using the novel analytical window. After that, the proposed APST is used for seismic data attenuation compensation, during which the attenuation function is estimated based on the minimum phase assumption using a statistical variable-step hyperbolic smoothing method. Numerical examples on synthetic and field data demonstrate the validity of the proposed method using the seismogram and time-frequency spectrum comparisons. Besides, the proposed APST can be easily extended into a generalized ST which is more flexible compared with the ST, and it can also be used in seismology, remote sensing, and other related discrete signal analysis fields.

Renormalized nonlinear sensitivity kernel and inverse thin-slab propagator in T-matrix formalism for wave-equation tomography

We derived the renormalized nonlinear sensitivity operator and the related inverse thin-slab propagator (ITSP) for nonlinear tomographic waveform inversion based on the theory of nonlinear partial derivative operator and its De Wolf approximation. The inverse propagator is based on a renormalization procedure to the forward and inverse transition matrix scattering series. The ITSP eliminates the divergence of the inverse Born series for strong perturbations by stepwise partial summation (renormalization). Numerical tests showed that the inverse Born T-series starts to diverge at moderate perturbation (20% for the given model of Gaussian ball with a radius of 5 wavelength), while the ITSP has no divergence problem for any strong perturbations (up to 100% perturbation for test model). In addition, the ITSP is a non-iterative, marching algorithm with only one sweep, and therefore very efficient in comparison with the iterative inversion based on the inverse-Born scattering series. This convergence and efficiency improvement has potential applications to the iterative procedure of waveform inversion.

An Efficient POCS Interpolation Method in the Frequency-Space Domain

Sampling irregularity in observed seismic data may cause a significant complexity increase in subsequent processing. Seismic data interpolation helps in removing this sampling irregularity, for which purpose complex-valued curvelet transform is used, but it is time-consuming because of the huge size of observed data. In order to improve efficiency as well as keep interpolation accuracy, I first extract principal frequency components using forward Fourier transform. The size of the principal frequency-space domain data is at least halved compared with that of the original time-space domain data because the complex-valued components of the representation of a real-valued signal (i.e., a complex-valued signal with zero as its imaginary component) exhibit conjugate symmetry in the frequency domain. Then, the projection onto convex projection (POCS) method is used to interpolate frequency-space data based on complex-valued curvelet transform. Finally, interpolated seismic data in the time-space domain can be obtained using inverse Fourier transform. Synthetic data and field data examples show that the efficiency can be improved more than two times and the performance is slightly better in the frequency-space domain compared with the POCS method directly performed in the time-space domain, which demonstrates the validity of the proposed method.

Q estimation and its application in the 3D shallow weathering zone

AbstractAttenuation in the shallow weathering zone is relatively strong, causing severe energy loss during wave propagation. It is difficult to estimate accurate Q values in the shallow weathering zone, and the influence of shallow weathering zone is seldom considered into attenuation estimation and compensation in the deep part. We achieved Q value estimation where there exist microlog data in the shallow weathering zone using the generalized S transform (GST); then, we establish an empirical formula using the velocity and Q value estimated with microlog data; finally, the Q value in the 3D shallow weathering zone can be obtained using the established formula and the velocity information. During the first procedure, the GST is used to provide reasonable time-frequency resolution, and linear regression is used in the obtained logarithmic spectral ratio to get the estimated Q value. An empirical formula is established using the estimated Q value and the velocity where there exists microlog data in the seco...

A new central compact finite difference scheme with high spectral resolution for acoustic wave equation

Abstract Based on the existing cell-node and cell-centered compact finite difference schemes, we developed a new central compact scheme with a high spectral resolution for the acoustic wave equation. In the new scheme, both the function values on the cell-nodes and cell-centers are used to compute the second-order spatial derivatives on the cell-nodes. The cell-centered values are stored and updated as independent variables in the modeling. The spatial derivatives on the cell-centers are evaluated by half shifting the indices in the formula designed for the cell-nodes. Compared to the conventional compact interpolation scheme, the proposed approach can avoid introducing transfer errors. Either Taylor-series expansion-based or optimized least-squares-based methods are used to calculate the finite difference coefficients. Theoretical analysis and synthetic examples demonstrate that the optimized least-squares-based method can provide higher accuracy than the Taylor-series expansion-based method. This new scheme is not a simple combination of the cell-node and cell-centered compact schemes and outperforms them in three scenarios. Firstly, it can promise higher accuracy considering the same formal truncation errors and model parameters. Thus, it can maintain superior precision while using a shorter spatial finite difference stencil. Secondly, compared to the cell-node compact scheme with half of grid spacing, the new scheme can yield equally as accurate results with less time consuming, together with saving approximately 25% and 29% of memory in 2D and 3D modeling, respectively. Finally, for similar memory requirements, the new method can more efficiently provide solutions with higher accuracy. The synthetic examples on the 2D homogeneous and the 3D horizontally-layered models demonstrate the advantages of the proposed scheme. The numerical simulations with 2D Marmousi model further validate its accuracy, efficiency and flexibility in complex media.

Comparison of the projection onto convex sets and iterative hard thresholding methods for seismic data interpolation and denoising

Because of the environment limitations, irregularity appears in the observed seismic data. In addition, the observed seismic data contains random noise from the acquisition equipment and surrounding environment, which affects the performances of multi-channel techniques, such as surface related multiple elimination (SRME) and amplitude variation with offset (AVO) analysis. The projection onto convex sets (POCS) method, known as an efficient interpolation method, is suitable for high signal-to-noise ratio (SNR) situations; however, the existing random noise may affect its final performance. In our previously published paper, the POCS formula was deduced in the view of iterative hard thresholding (IHT) method using a projection operator. In this paper, more physical illustrations about its detailed deduction are provided to show the differences between IHT and POCS in noise-free and noisy situations with easy understanding for readers. Then, performances of the POCS and IHT methods are compared in both noise-free and noisy situations, in terms of seismograms, frequency wavenumber (FK) spectra and single traces. For noise-free data, both the POCS and IHT methods can achieve good interpolation results. For noisy data, the POCS method is unsuitable because of the observed noisy data insertion, while the IHT performance is satisfactory because it uses a thresholding operator to eliminate random noise. Numerical examples on noise-free datasets demonstrate the validities of the POCS and IHT methods for interpolation. Tests on noisy data contaminated with additive white Gaussian noise prove the ability and superiority of the IHT method with anti-noise property compared with the POCS method. The projection onto convex sets (POCS) method is deduced using the iterative hard thresholding (IHT) algorithm and a projection operator with more detailed physical illustrations. The interpolation performances on noise-free and noisy data are explained in detail and the reasons behind these performances are fully discussed, which provide clues to further improve interpolation accuracy.

An Events Rearrangement Strategy-Based Robust Principle Component Analysis

Random noise in seismic data can affect the performance of reservoir characterization and interpretation, which makes denoising become an essential procedure. This letter focuses on suppressing random noise in poststack seismic data while preserving the edges of desired signals. Due to the lateral continuity of seismic data, polynomial fitting (PF) method can be a good alternative in attenuating random noise. However, discontinuities exist widely in poststack seismic data, which might be damaged by the PF filter. By contrast, principle component analysis (PCA)-based filters have better performance in edge preserving, but there appear artifacts in the denoised results using the PCA-based filters. Thus, we propose an edge-preserving polynomial PCA filter which combines advantages of the PF and PCA methods by optimizing a PCA problem with a weighted polynomial constraint. The weight coefficient is determined adaptively according to the signal-to-noise ratio estimation and the energy proportion in the selected analysis window, which can help distinguish the horizontal continuous events and the edges effectively. To deal with the complicated slopes which make the local linear hypothesis invalid, we introduce a robust local slope estimation method and apply the slope estimation-based event tracing strategy to horizontally align the data set. Synthetic and field data examples show that the proposed method has a better performance in noise attenuation and edge preserving, compared with the edge-preserving PF method. In addition, the denoised results are free from artifacts.

Multi-Attribute Classification Based On Sparse Autoencoder - A Gas Chimney Detection Example

Multi-attribute classification technologies, including supervised and unsupervised methods, play an important role in seismic interpretation. Providing that enough labeled samples are available, supervised methods usually obtain some credible results. However, for seismic data, the unlabeled data is huge but the labeled data is usually limited. Therefore, the combination of supervised and unsupervised methods is a feasible idea. As a common unsupervised feature learning algorithm used in deep learning, sparse autoencoder can realize automatic feature extraction using the unlabeled data. In this abstract, we introduce the sparse autoencoder and design a semi-supervised learning framework for multi-attribute classification applications by combining unsupervised feature learning and supervised classification. In the proposed framework, the original data of both labeled and unlabeled samples are used to train a sparse autoencoder at first, and then encoded to get some features for better representation. After that, the features of these labeled samples are used to train a classifier. At last, we apply this classifier on these unlabeled samples to obtain the final classification results. To demonstrate the validity of the designed technology, we give a gas chimney detection example. Results show that the multi-attribute classification technology based on sparse autoencoder outperforms the traditional multilayer perceptron method.