Jingjie Cao

A review on restoration of seismic wavefields based on regularization and compressive sensing

Restoration of seismic data as an ill-posed inverse problem means to recover the complete wavefields from sub-sampled data. Since seismic data are typically sparse in the curvelet domain, this problem can be solved based on the compressive sensing theory. Meanwhile three major problems are modelling, sampling and solving methods. We first construct l 0 and l 1 minimization models and then develop fast projected gradient methods to solve the restoration problem. For seismic data interpolation/restoration, the regular sub-sampled data will generate coherence aliasing in the frequency domain, while the random sub-sampling cannot control the largest sampling gap. Therefore, we consider a new sampling technique in this article which is based on the controlled piecewise random sub-sampling scheme. Numerical simulations are made and compared with the iterative soft thresholding method and the spectral gradient-projection method. It reveals that the proposed algorithms have the advantages of high precision, robustness and fast calculation.

Regularizing active set method for nonnegatively constrained ill-posed multichannel image restoration problem

In this paper, we consider the nonnegatively constrained multichannel image deblurring problem and propose regularizing active set methods for numerical restoration. For image deblurring problems, it is reasonable to solve a regularizing model with nonnegativity constraints because of the physical meaning of the image. We consider a general regularizing l(p)-l(q) model with nonnegativity constraints. For p and q equaling 2, the model is in a convex quadratic form, therefore, the active set method is proposed since the nonnegativity constraints are imposed naturally. For p and q not equaling 2, we present an active set method with a feasible Newton-conjugate gradient solution technique. Numerical experiments are presented for ill-posed three-channel blurred image restoration problems.

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.

ON TIKHONOV REGULARIZATION AND COMPRESSIVE SENSING FOR SEISMIC SIGNAL PROCESSING

Using compressive sensing and sparse regularization, one can nearly completely reconstruct the input (sparse) signal using limited numbers of observations. At the same time, the reconstruction methods by compressing sensing and optimizing techniques overcome the obstacle of the number of sampling requirement of the Shannon/Nyquist sampling theorem. It is well known that seismic reflection signal may be sparse, sometimes and the number of sampling is insufficient for seismic surveys. So, the seismic signal reconstruction problem is ill-posed. Considering the ill-posed nature and the sparsity of seismic inverse problems, we study reconstruction of the wavefield and the reflection seismic signal by Tikhonov regularization and the compressive sensing. The l0, l1 and l2 regularization models are studied. Relationship between Tikhonov regularization and the compressive sensing is established. In particular, we introduce a general lp - lq (p, q ≥ 0) regularization model, which overcome the limitation on the assumption of convexity of the objective function. Interior point methods and projected gradient methods are studied. To show the potential for application of the regularized compressive sensing method, we perform both synthetic seismic signal and field data compression and restoration simulations using a proposed piecewise random sub-sampling. Numerical performance indicates that regularized compressive sensing is applicable for practical seismic imaging.

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.

3D seismic denoising based on a low-redundancy curvelet transform

Contamination of seismic signal with noise is one of the main challenges during seismic data processing. Several methods exist for eliminating different types of noises, but optimal random noise attenuation remains difficult. Based on multi-scale, multi-directional locality of curvelet transform, the curvelet thresholding method is a relatively new method for random noise elimination. However, the high redundancy of a 3D curvelet transform makes its computational time and memory for massive data processing costly. To improve the efficiency of the curvelet thresholding denoising, a low-redundancy curvelet transform was introduced. The redundancy of the low-redundancy curvelet transform is approximately one-quarter of the original transform and the tightness of the original transform is also kept, thus the low-redundancy curvelet transform calls for less memory and computational resource compared with the original one. Numerical results on 3D synthetic and field data demonstrate that the low-redundancy curvelet denoising consumes one-quarter of the CPU time compared with the original curvelet transform using iterative thresholding denoising when comparable results are obtained. Thus, the low-redundancy curvelet transform is a good candidate for massive seismic denoising.

Seismic data restoration with a fast L1 norm trust region method

Seismic data restoration is a major strategy to provide reliable wavefield when field data dissatisfy the Shannon sampling theorem. Recovery by sparsity-promoting inversion often get sparse solutions of seismic data in a transformed domains, however, most methods for sparsity-promoting inversion are line-searching methods which are efficient but are inclined to obtain local solutions. Using trust region method which can provide globally convergent solutions is a good choice to overcome this shortcoming. A trust region method for sparse inversion has been proposed, however, the efficiency should be improved to suitable for large-scale computation. In this paper, a new L1 norm trust region model is proposed for seismic data restoration and a robust gradient projection method for solving the sub-problem is utilized. Numerical results of synthetic and field data demonstrate that the proposed trust region method can get excellent computation speed and is a viable alternative for large-scale computation.

A new staggered grid finite difference scheme optimised in the space domain for the first order acoustic wave equation

Staggered grid finite difference (FD) methods are widely used to synthesise seismograms theoretically, and are also the basis of reverse time migration and full waveform inversion. Grid dispersion is one of the key problems for FD methods. It is desirable to have a FD scheme which can accelerate wave equation simulation while still preserving high accuracy. In this paper, we propose a totally new staggered grid FD scheme which uses different staggered grid FD operators for different first order spatial derivatives in the first order acoustic wave equation. We determine the FD coefficient in the space domain with the least-squares method. The dispersion analysis and numerical simulation demonstrated the effectiveness of the proposed method. In this paper, we propose a new finite difference (FD) scheme which uses different staggered grid FD operators for different first order spatial derivatives in the first order acoustic wave equation. The dispersion analysis and numerical simulation demonstrated the effectiveness of the proposed method.