Noncausal f–x–y regularized nonstationary prediction filtering for random noise attenuation on 3D seismic data

Seismic noise attenuation is very important for seismic data analysis and interpretation, especially for 3D seismic data. In this paper, we propose a novel method for 3D seismic random noise attenuation by applying noncausal regularized nonstationary autoregression (NRNA) in f–x–y domain. The proposed method, 3D NRNA (f–x–y domain) is the extended version of 2D NRNA (f–x domain). f–x–y NRNA can adaptively estimate seismic events of which slopes vary in 3D space. The key idea of this paper is to consider that the central trace can be predicted by all around this trace from all directions in 3D seismic cube, while the 2D f–x NRNA just considers that the middle trace can be predicted by adjacent traces along one space direction. 3D f–x–y NRNA uses more information from circumjacent traces than 2D f–x NRNA to estimate signals. Shaping regularization technology guarantees that the nonstationary autoregression problem can be realizable in mathematics with high computational efficiency. Synthetic and field data examples demonstrate that, compared with f–x NRNA method, f–x–y NRNA can be more effective in suppressing random noise and improve trace-by-trace consistency, which are useful in conjunction with interactive interpretation and auto-picking tools such as automatic event tracking.     

An algorithm for interpolation in the pyramid domain‡

With the pyramid transform, 2D dip spectra can be characterized by 1D prediction‐error filters (pefs) and 3D dip spectra by 2D pefs. These filters, contrary to pefs estimated in the frequency‐space domain (ω, x) , are frequency independent. Therefore, one pef can be used to interpolate all frequencies. Similarly, one pef can be computed from all frequencies, thus yielding robust estimation of the filter in the presence of noise. This transform takes data from the frequency‐space domain (ω, x) to data in a frequency‐velocity domain (ω, u=ω·x) using a simple mapping procedure that leaves locations in the pyramid domain empty. Missing data in (ω, x) ‐space create even more empty bins in (ω, u) ‐space. We propose a multi‐stage least‐squares approach where both unknown pefs and missing data are estimated. This approach is tested on synthetic and field data examples where aliasing and irregular spacing are present.     

Non‐convex compressed sensing with frequency mask for seismic data reconstruction and denoising

Compressed Sensing has recently proved itself as a successful tool to help address the challenges of acquisition and processing seismic data sets. Compressed sensing shows that the information contained in sparse signals can be recovered accurately from a small number of linear measurements using a sparsity‐promoting regularization. This paper investigates two aspects of compressed sensing in seismic exploration: (i) using a general non‐convex regularizer instead of the conventional one‐norm minimization for sparsity promotion and (ii) using a frequency mask to additionally subsample the acquired traces in the frequency‐space () domain. The proposed non‐convex regularizer has better sparse recovery performance compared with one‐norm minimization and the additional frequency mask allows us to incorporate a priori information about the events contained in the wavefields into the reconstruction. For example, (i) seismic data are band‐limited; therefore one can use only a partial set of frequency coefficients in the range of reflections band, where the signal‐to‐noise ratio is high and spatial aliasing is low, to reconstruct the original wavefield, and (ii) low‐frequency characteristics of the coherent ground rolls allow direct elimination of them during reconstruction by disregarding the corresponding frequency coefficients (usually bellow 10 Hz) via a frequency mask. The results of this paper show that some challenges of reconstruction and denoising in seismic exploration can be addressed under a unified formulation. It is illustrated numerically that the compressed sensing performance for seismic data interpolation is improved significantly when an additional coherent subsampling is performed in the domain compared with the domain case. Numerical experiments from both simulated and real field data are included to illustrate the effectiveness of the presented method.     

A FREQUENCY DOMAIN MAPPING APPROXIMATION OF MOVEOUT FILTERS WITH APPLICATIONS*

Wavenumber aliasing is the main limitation of conventional optimum least-squares linear moveout filters: it prevents adequate reject domain weighting for efficient coherent noise rejection. A general frequency domain multichannel filter design technique based on a one-to-one mapping method between two-dimensional (2D) space and one-dimensional (1D) space is presented. The 2D desired response is mapped to the 1D frequency axis after a suitable sorting of the coefficients. A min-max or Tchebycheff approximation to the desired response is obtained in the 1D frequency domain and mapped back to the 2D frequency domain. The algorithm is suitable for multiband 2D filter design. No aliasing damage is inherent in the linear moveout filters designed using this technique because the approximation is done in the frequency-wavenumber (f, k)-domain. Linear moveout filters designed by using the present coefficient mapping technique achieve better pass domain approximations than the corresponding conventional least-squares filters. Compatible reject domain approximations can be obtained from suitable mappings of the origin coefficient of the desired (f k)-response to the 1D frequency axis. The (fk)-responses of linear moveout filters designed by using the new technique show equi-ripple behavior. Synthetic and real data applications show that the present technique is superior to the optimum least-squares filters and straight stacking in recovering and enhancing the signal events with relatively high residual statics. Their outputs also show higher resolution than those of the optimum least-squares filters.     

A FILTER,DELAY AND SPREAD TECHNIQUE FOR 3D DMO1

The calculation of dip moveout involves spreading the amplitudes of each input trace along the source-receiver axis followed by stacking the results into a 3D zero-offset data cube. The offset-traveltime (x–t) domain integral implementation of the DMO operator is very efficient in terms of computation time but suffers from operator aliasing. The log-stretch approach, using a logarithmic transformation of the time axis to force the DMO operator to be time invariant, can avoid operator aliasing by direct implementation in the frequency-wavenumber (f–k) domain. An alternative technique for log-stretch DMO corrections using the anti-aliasing filters of the f–k approach in the x-log t domain will be presented. Conventionally, the 2D filter representing the DMO operator is designed and applied in the f–k domain. The new technique uses a 2D convolution filter acting in single input/multiple output trace mode. Each single input trace is passed through several 1D filters to create the overall DMO response of that trace. The resulting traces can be stacked directly in the 3D data cube. The single trace filters are the result of a filter design technique reducing the 2D problem to several ID problems. These filters can be decomposed into a pure time-delay and a low-pass filter, representing the kinematic and dynamic behaviour of the DMO operator. The low-pass filters avoid any incidental operator aliasing. Different types of low-pass filters can be used to achieve different amplitude-versus-offset characteristics of the DMO operator.     

A fast joint seismic data reconstruction by sparsity‐promoting inversion

Seismic field data are often irregularly or coarsely sampled in space due to acquisition limits. However, complete and regular data need to be acquired in most conventional seismic processing and imaging algorithms. We have developed a fast joint curvelet‐domain seismic data reconstruction method by sparsity‐promoting inversion based on compressive sensing. We have made an attempt to seek a sparse representation of incomplete seismic data by curvelet coefficients and solve sparsity‐promoting problems through an iterative thresholding process to reconstruct the missing data. In conventional iterative thresholding algorithms, the updated reconstruction result of each iteration is obtained by adding the gradient to the previous result and thresholding it. The algorithm is stable and accurate but always requires sufficient iterations. The linearised Bregman method can accelerate the convergence by replacing the previous result with that before thresholding, thus promoting the effective coefficients added to the result. The method is faster than conventional one, but it can cause artefacts near the missing traces while reconstructing small‐amplitude coefficients because some coefficients in the unthresholded results wrongly represent the residual of the data. The key process in the joint curvelet‐domain reconstruction method is that we use both the previous results of the conventional method and the linearised Bregman method to stabilise the reconstruction quality and accelerate the recovery for a while. The acceleration rate is controlled through weighting to adjust the contribution of the acceleration term and the stable term. A fierce acceleration could be performed for the recovery of comparatively small gaps, whereas a mild acceleration is more appropriate when the incomplete data has a large gap of high‐amplitude events. Finally, we carry out a fast and stable recovery using the trade‐off algorithm. Synthetic and field data tests verified that the joint curvelet‐domain reconstruction method can effectively and quickly reconstruct seismic data with missing traces.     

Seismic data analysis using local time‐frequency decomposition

Many natural phenomena, including geologic events and geophysical data, are fundamentally nonstationary ‐ exhibiting statistical variation that changes in space and time. Time‐frequency characterization is useful for analysing such data, seismic traces in particular. We present a novel time‐frequency decomposition, which aims at depicting the nonstationary character of seismic data. The proposed decomposition uses a Fourier basis to match the target signal using regularized least‐squares inversion. The decomposition is invertible, which makes it suitable for analysing nonstationary data. The proposed method can provide more flexible time‐frequency representation than the classical S transform. Results of applying the method to both synthetic and field data examples demonstrate that the local time‐frequency decomposition can characterize nonstationary variation of seismic data and be used in practical applications, such as seismic ground‐roll noise attenuation and multicomponent data registration.     

基于seislet变换的反假频迭代数据插值方法

许多地震资料处理方法需要完整的数据信息,但是受野外施工条件等因素的影响,观测系统很难记录完整的地震波场,如空间采样率不足和地震道缺失等现象,尤其是缺失的叠前地震数据时常产生空间假频现象,给后续处理流程中很多重要环节带来严重的影响.传统数据插值方法通常很难同时解决数据缺失和空间假频问题,因此开发有效的反空间假频数据插值方法具有重要的意义.本文通过同时改变时间和空间方向采样比例,利用预测误差滤波器的尺度缩放不变性,计算反空间假频地震倾角模式,构建可有效压缩含空间假频不完整地震数据的反假频seislet变换方法,通过压缩感知Bregman迭代算法,对缺失地震数据进行反假频插值.理论模型和实际数据的处理结果验证了基于seislet变换的迭代插值方法可以有效地恢复含有假频的缺失地震信息.     

Reconstruction of the near‐offset gap in marine seismic data using seismic interferometric interpolation

In conventional seismic exploration, especially in marine seismic exploration, shot gathers with missing near‐offset traces are common. Interferometric interpolation methods are one of a range of different methods that have been developed to solve this problem. Interferometric interpolation methods differ from conventional interpolation methods as they utilise information from multiples in the interpolation process. In this study, we apply both conventional interferometric interpolation (shot domain) and multi‐domain interferometric interpolation (shot and receiver domain) to a synthetic and a real‐towed marine dataset from the Baltic Sea with the primary aim of improving the image of the seabed by extrapolation of a near‐offset gap. We utilise a matching filter after interferometric interpolation to partially mitigate artefacts and coherent noise associated with the far‐field approximation and a limited recording aperture size. The results show that an improved image of the seabed is obtained after performing interferometric interpolation. In most cases, the results from multi‐domain interferometric interpolation are similar to those from conventional interferometric interpolation. However, when the source–receiver aperture is limited, the multi‐domain method performs better. A quantitative analysis for assessing the performance of interferometric interpolation shows that multi‐domain interferometric interpolation typically performs better than conventional interferometric interpolation. We also benchmark the interpolated results generated by interferometric interpolation against those obtained using sparse recovery interpolation.     

地震数据Bregman整形迭代插值方法

人工地震方法由于受到野外观测系统和经济因素等的限制,采集的数据在空间方向总是不规则分布.但是,许多地震数据处理技术的应用（如：多次波衰减,偏移和时移地震）都基于空间规则分布条件下的地震数据体.因此,数据插值技术是地震数据处理流程中关键环节之一.失败的插值方法往往会引入虚假信息,给后续处理环节带来严重的影响.迭代插值方法是目前广泛应用的地震数据重建思路,但是常规的迭代插值方法往往很难保证插值精度,并且迭代收敛速度较慢,尤其存在随机噪声的情况下,插值地震道与原始地震道之间存在较大的信噪比差异.因此开发快速的、有效的迭代数据插值方法具有重要的工业价值.本文将地震数据插值归纳为数学基追踪问题,在压缩感知理论框架下,提出新的非线性Bregman整形迭代算法来求解约束最小化问题,同时在迭代过程中提出两种匹配的迭代控制准则,通过有效的稀疏变换对缺失数据进行重建.通过理论模型和实际数据测试本文方法,并且与常规迭代插值算法进行比较,结果表明Bregman整形迭代插值方法能够更加有效地恢复含有随机噪声的缺失地震信息.     

Seismic data interpolation using generalised velocity‐dependent seislet transform

Data interpolation is an important step for seismic data analysis because many processing tasks, such as multiple attenuation and migration, are based on regularly sampled seismic data. Failed interpolations may introduce artifacts and eventually lead to inaccurate final processing results. In this paper, we generalised seismic data interpolation as a basis pursuit problem and proposed an iteration framework for recovering missing data. The method is based on non‐linear iteration and sparse transform. A modified Bregman iteration is used for solving the constrained minimisation problem based on compressed sensing. The new iterative strategy guarantees fast convergence by using a fixed threshold value. We also propose a generalised velocity‐dependent formulation of the seislet transform as an effective sparse transform, in which the non‐hyperbolic normal moveout equation serves as a bridge between local slope patterns and moveout parametres in the common‐midpoint domain. It can also be reduced to the traditional velocity‐dependent seislet if special heterogeneity parametre is selected. The generalised velocity‐dependent seislet transform predicts prestack reflection data in offset coordinates, which provides a high compression of reflection events. The method was applied to synthetic and field data examples, and the results show that the generalised velocity‐dependent seislet transform can reconstruct missing data with the help of the modified Bregman iteration even for non‐hyperbolic reflections under complex conditions, such as vertical transverse isotropic (VTI) media or aliasing.     

Recursive evaluation of interaction forces of unbounded soil in the time domain from dynamic-stiffness coefficients in the frequency domain

The interaction forces of the linear unbounded soil in a non-linear soil-structure-interaction analysis can be calculated recursively, starting directly from the dynamic-stiffness coefficients in the frequency domain. Two possibilities of choosing a recursive equation are discussed.

• (i) The recursive equation in the frequency domain. For each frequency, the interaction force at a specific time station is expressed as a function of the corresponding interaction force at the previous time station and of the displacements at the current time station and at the two most recent past time stations. This recursive evaluation of the convolution integral. which can be derived using the z-transformation, is rigorous. By using interpolation in the frequency domain, an approximate procedure results, which leads to a significant reduction in computational effort.
• (ii) The recursive equation in the time domain. By approximating the dynamic-stiffness coefficients as the ratios of two polynomials in frequency using a curve-fitting technique based on the least-squares method and by applying the partial-fraction expansion and using the z-transformation, the recursive coefficients can be determined explicitly. Alternatively, the ratio of two polynomials can also be transformed to an ordinary differential equation together with the initial conditions.

The recursive equations using interpolation in the frequency domain and based on a ratio of two polynomials lead to a reduction in the computational effort of one and up to three orders of magnitude, respectively.     

The design of visco‐acoustic weighted L1‐error frequency–space wavefield extrapolators

Seismic imaging is an important step for imaging the subsurface structures of the Earth. One of the attractive domains for seismic imaging is explicit frequency–space (f – x) prestack depth migration. So far, this domain focused on migrating seismic data in acoustic media, but very little work assumed visco‐acoustic media. In reality, seismic exploration data amplitudes suffer from attenuation. To tackle the problem of attenuation, new operators are required, which compensates for it. We propose the weighted L ₁ ‐error minimisation technique to design visco‐acoustic f – x wavefield extrapolators. The L ₁ ‐error wavenumber responses provide superior extrapolator designs as compared with the previously designed equiripple L ₄ ‐norm and L_∞‐norm extrapolation wavenumber responses. To verify the new compensating designs, prestack depth migration is performed on the challenging Marmousi model dataset. A reference migrated section is obtained using non‐compensating f – x extrapolators on an acoustic dataset. Then, both compensating and non‐compensating extrapolators are applied to a visco‐acoustic dataset, and both migrated sections are then compared. The final images show that the proposed weighted L ₁ ‐error method enhances the resolution and results in practically stable images.     

Prestack nonstationary deconvolution based on variable-step sampling in the radial trace domain

The conventional nonstationary convolutional model assumes that the seismic signal is recorded at normal incidence. Raw shot gathers are far from this assumption because of the effects of offsets. Because of such problems, we propose a novel prestack nonstationary deconvolution approach. We introduce the radial trace （RT） transform to the nonstationary deconvolution, we estimate the nonstationary deconvolution factor with hyperbolic smoothing based on variable-step sampling （VSS） in the RT domain, and we obtain the high-resolution prestack nonstationary deconvolution data. The RT transform maps the shot record from the offset and traveltime coordinates to those of apparent velocity and traveltime. The ray paths of the traces in the RT better satisfy the assumptions of the convolutional model. The proposed method combines the advantages of stationary deconvolution and inverse Q filtering, without prior information for Q. The nonstationary deconvolution in the RT domain is more suitable than that in the space-time （XT） domain for prestack data because it is the generalized extension of normal incidence. Tests with synthetic and real data demonstrate that the proposed method is more effective in compensating for large-offset and deep data.     

低信噪比地震资料FDOC-seislet变换阈值消噪方法研究

地震数据本质上是时变的,不仅有效同相轴表现出确定性信号的时变特征,而且复杂地表和构造条件以及深部探测环境总是引入时变的非平稳随机噪声.标准的频率-空间域预测滤波只适合压制平面波信号假设下的平稳随机噪声,而处理非平稳地震随机噪声时,需要将数据体分割为小窗口进行分析,但效果不够理想,而传统非预测类随机噪声压制方法往往适应性不高,因此开发能够保护地震信号时变特征的随机噪声压制方法具有重要的工业价值.压缩感知是近年出现的一个新的采样理论,通过开发信号的稀疏特性,已经在地震数据处理中的数据插值以及噪声压制中得到了应用.本文系统地分析了压缩感知理论框架下的地震随机噪声压制问题,建立了阈值消噪的数学反演目标函数;针对时变有效信息具有的可压缩性,利用有限差分算法求解炮检距连续方程,构建有限差分炮检距连续预测算子(FDOC),在seislet变换框架下,提出一种新的快速稀疏变换域———FDOC-seislet变换,实现地震数据的高度稀疏表征;结合非平稳随机噪声不可压缩的特征,提出了一种整形迭代消噪方法,该方法是一种广义的迭代收缩阈值(IST)算法,在无法计算稀疏变换伴随算子的条件下,仍然能够对强噪声环境中的时变有效信息进行有效恢复.通过对模型数据和实际数据的处理,验证了FDOC-seislet稀疏变换域随机噪声迭代压制方法能够在保护复杂构造地震波信息的前提下,有效地衰减原始数据中的强振幅随机噪声干扰.     

Three‐term inversion of prestack seismic data using a weighted l2, 1 mixed norm

We present a new inversion method to estimate, from prestack seismic data, blocky P‐ and S‐wave velocity and density images and the associated sparse reflectivity levels. The method uses the three‐term Aki and Richards approximation to linearise the seismic inversion problem. To this end, we adopt a weighted mixed l_{2, 1}‐norm that promotes structured forms of sparsity, thus leading to blocky solutions in time. In addition, our algorithm incorporates a covariance or scale matrix to simultaneously constrain P‐ and S‐wave velocities and density. This a priori information is obtained by nearby well‐log data. We also include a term containing a low‐frequency background model. The l_{2, 1} mixed norm leads to a convex objective function that can be minimised using proximal algorithms. In particular, we use the fast iterative shrinkage‐thresholding algorithm. A key advantage of this algorithm is that it only requires matrix–vector multiplications and no direct matrix inversion. The latter makes our algorithm numerically stable, easy to apply, and economical in terms of computational cost. Tests on synthetic and field data show that the proposed method, contrarily to conventional l₂‐ or l₁‐norm regularised solutions, is able to provide consistent blocky and/or sparse estimators of P‐ and S‐wave velocities and density from a noisy and limited number of observations.     

基于f-x域流式预测滤波器的地震随机噪声衰减方法

随机噪声的影响在地震勘探中是不可避免的,常规的随机噪声压制方法在处理中往往会破坏具有时空变化特征的非平稳有效地震信号,影响地震数据的准确成像.当前油气勘探的目标已经转变为“两宽一高”,随着数据量的增大,对去噪方法的处理效率也提出了更高的要求.因此,开发高效的非平稳地震数据随机噪声压制方法具有重要意义.预测滤波技术广泛用于地震随机噪声的衰减,本文基于流式处理框架提出一种新的f-x域流式预测滤波方法,通过在频率域建立预测自回归方程,运用直接复数矩阵逆运算代替迭代算法求解非平稳滤波器系数,实现时空变地震同相轴预测,提高自适应预测滤波的计算效率.通过与工业标准的FXDECON方法和f-x域正则化非平稳自回归(RNA)方法进行对比,理论模型和实际数据的测试结果表明,提出的f-x域流式预测滤波方法能更好地平衡时空变有效信号保护、随机噪声压制和高效计算三者之间的关系,获得合理的处理效果.     

A seismic interpolation and denoising method with curvelet transform matching filter

A new seismic interpolation and denoising method with a curvelet transform matching filter, employing the fast iterative shrinkage thresholding algorithm (FISTA), is proposed. The approach treats the matching filter, seismic interpolation, and denoising all as the same inverse problem using an inversion iteration algorithm. The curvelet transform has a high sparseness and is useful for separating signal from noise, meaning that it can accurately solve the matching problem using FISTA. When applying the new method to a synthetic noisy data sets and a data sets with missing traces, the optimum matching result is obtained, noise is greatly suppressed, missing seismic data are filled by interpolation, and the waveform is highly consistent. We then verified the method by applying it to real data, yielding satisfactory results. The results show that the method can reconstruct missing traces in the case of low SNR (signal-to-noise ratio). The above three problems can be simultaneously solved via FISTA algorithm, and it will not only increase the processing efficiency but also improve SNR of the seismic data.     

基于F-K偏移和反偏移的地震道插值方法研究

野外采集的地震数据经常存在空道或者坏道的情况,为了满足地震数据处理精度的要求,就必须首先进行插值.本文提出了一种新的地震数据插值方法：F-K偏移和反偏移插值法,该方法是通过F-K偏移和反偏移的串联使用来实现的.与其他的插值方法相比,这种插值方法的优点在于计算速度快,没有经过近似处理,精度高.F-K偏移/反偏移能够实现道插值的原理和Kirchhoff偏移/反偏移插值方法类似,也是由于数据是有限带宽的原因引起的.通过对模型的试算,可以看到F-K偏移和反偏移插值方法有着比较好的插值效果,是一种高效、准确、可行的方法.