基于Bregman迭代的复杂地震波场稀疏域插值方法

在地震勘探中,野外施工条件等因素使观测系统很难记录到完整的地震波场,因此,资料处理中的地震数据插值是一个重要的问题。尤其在复杂构造条件下,缺失的叠前地震数据给后续高精度处理带来严重的影响。压缩感知理论源于解决图像采集问题,主要包含信号的稀疏表征以及数学组合优化问题的求解,它为地震数据插值问题的求解提供了有效的解决方案。在应用压缩感知求解复杂地震波场的插值问题中,如何最佳化表征复杂地震波场以及快速准确的迭代算法是该理论应用的关键问题。Seislet变换是一个特殊针对地震波场表征的稀疏多尺度变换,该方法能有效地压缩地震波同相轴。同时,Bregman迭代算法在以稀疏表征为核心的压缩感知理论中,是一种有效的求解算法,通过选取适当的阈值参数,能够开发地震波动力学预测理论、图像处理变换方法和压缩感知反演算法相结合的地震数据插值方法。本文将地震数据插值问题纳入约束最优化问题,选取能够有效压缩复杂地震波场的OC-seislet稀疏变换,应用Bregman迭代方法求解压缩感知理论框架下的混合范数反问题,提出了Bregman迭代方法中固定阈值选取的H曲线方法,实现地震波场的快速、准确重建。理论模型和实际数据的处理结果验证了基于H曲线准则的Bregman迭代稀疏域插值方法可以有效地恢复复杂波场的缺失信息。     

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.     

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.     

Velocity model updating in prestack Kirchhoff time migration for PS converted waves: Part I – Theory

A velocity model updating approach is developed based on moveout analysis of the diffraction curve of PS converted waves in prestack Kirchhoff time migration. The diffraction curve can be expressed as a product of two factors: one factor depending on the PS converted‐wave velocity only, and the other factor depending on all parameters. The velocity‐dependent factor represents the hyperbolic behaviour of the moveout and the other is a scale factor that represents the non‐hyperbolic behaviour of the moveout. This non‐hyperbolic behaviour of the moveout can be corrected in prestack Kirchhoff time migration to form an inverse normal‐moveout common‐image‐point gather in which only the hyperbolic moveout is retained. This hyperbolic moveout is the moveout that would be obtained in an isotropic equivalent medium. A hyperbolic velocity is then estimated from this gather by applying hyperbolic moveout analysis. Theoretical analysis shows that for any given initial velocity, the estimated hyperbolic velocity converges by an iterative procedure to the optimal velocity if the velocity ratio is optimal or to a value closer to the optimal velocity if the velocity ratio is not optimal. The velocity ratio (V_P/V_S) has little effect on the estimation of the velocity. Applying this technique to a synthetic seismic data set confirms the theoretical findings. This work provides a practical method to obtain the velocity model for prestack Kirchhoff time migration.     

OPTIMUM HYPERBOLIC MOVEOUT FILTERS WITH APPLICATIONS TO SEISMIC DATA*

Optimum multichannel filters can be designed to process seismic events falling on hyperbolic moveout curves using the conventional least-squares method. Contrary to the linear moveout filters, autocorrelation and crosscorrelation functions inherent in the normal equations have to be computed numerically. However, computation times of filter coefficients are comparable to linear moveout operators. For a given source-receiver geometry and assuming straight ray-path, relative moveout of a seismic reflection event is dependent on the two way arrival time and rms velocity. Consequently, to avoid overlapping of pass and reject moveout windows, hyperbolic moveout filters have to be designed over time gates rather than for the whole record lengths. Hyperbolic and hyperbolic-linear moveout filters applied to synthetic and field seismic reflection traces show good signal-to-noise (S/N) ratio improvements. Results of some combined synthetic and field data examples are presented.     

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.     

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.     

基于压缩感知的Curvelet域联合迭代地震数据重建

由于野外采集环境的限制,常常无法采集得到完整规则的野外地震数据,为了后续地震处理、解释工作的顺利进行,地震数据重建工作被广泛的研究.自压缩感知理论的提出,相继出现了基于该理论的多种迭代阈值方法,如CRSI方法（Curvelet Recovery by Sparsity-promoting Inversion method）、Bregman迭代阈值算法（the linearized Bregman method）等.CSRI方法利用地震波形在Curvelet的稀疏特性,通过一种基于最速下降的迭代算法在Curvelet变换域恢复出高信噪比地震数据,该迭代算法稳定,收敛,但其收敛速度慢.Bregman迭代阈值法与CRSI最大区别在于每次迭代时把上一次恢复结果中的阈值前所有能量都保留到本次恢复结果中,从而加快了收敛速度,但随着迭代的进行重构数据中噪声干扰越来越严重,导致最终恢复出的数据信噪比低.综合两种经典方法的优缺点,本文构造了一种新的联合迭代算法框架,在每次迭代中将CRSI和Bregman的恢复量加权并同时加回本次迭代结果中,从而加快了迭代初期的收敛速度,又避免了迭代后期噪声干扰的影响.合成数据和实际数据试算结果表明,我们提出的新方法不仅迭代快速收敛稳定,且能得到高信噪比的重建结果.     

高阶seislet变换及其在随机噪声消除中的应用

Seislet变换是一种小波类数学变换方法,主要根据不同小波级数上地震同相轴的局部倾角的不同来分析数据.一般意义上,测线方向上的离散小波变换（DWT）是一种特殊的零局部地震倾角的seislet变换.早期的工作基于低阶版本的离散小波变换来构建seislet变换,在本文中,通过使用Cohen-Daubechies-Feauveau (CDF) 9/7双正交小波变换（常用于JPEG2000压缩标准）作为框架,扩展高阶seislet变换方法.通过分析理论模型和实际数据的处理结果,并对比傅里叶变换、离散小波变换和低阶seislet变换,高阶seislet变换可以为地震数据提供更好的压缩比.因此更加适用于地震数据去噪处理.     

Prestack Kirchhoff time migration of 3D coal seismic data from mining zones

Conventional seismic data processing methods based on post‐stack time migration have been playing an important role in coal exploration for decades. However, post‐stack time migration processing often results in low‐quality images in complex geological environments. In order to obtain high‐quality images, we present a strategy that applies the Kirchhoff prestack time migration (PSTM) method to coal seismic data. In this paper, we describe the implementation of Kirchhoff PSTM to a 3D coal seam. Meanwhile we derive the workflow of 3D Kirchhoff PSTM processing based on coal seismic data. The processing sequence of 3D Kirchhoff PSTM includes two major steps: 1) the estimation of the 3D root‐mean‐square (RMS) velocity field; 2) Kirchhoff prestack time migration processing. During the construction of a 3D velocity model, dip moveout velocity is served as an initial migration velocity field. We combine 3D Kirchhoff PSTM with the continuous adjustment of a 3D RMS velocity field by the criteria of flattened common reflection point gathers. In comparison with post‐stack time migration, the application of 3D Kirchhoff PSTM to coal seismic data produces better images of the coal seam reflections.     

基于Seislet-TV双正则化约束的地震随机噪声压制方法

人工地震数据总是受到随机噪声的干扰,地震数据时-空变的特性使得常规去噪方法处理效果并不理想,容易导致有效信号的损失.目前广泛应用的预测滤波类方法存在处理时变数据能力不足的问题.随着压缩感知理论的不断完善,稀疏变换阈值算法能够解决时变地震数据噪声压制问题,但是常规的稀疏变换方法,如傅里叶变换,小波变换等,并不是特殊针对地震数据设计的,很难提供地震数据最佳的压缩特征,同时,常规阈值算法容易导致去噪结果过于平滑.因此开发更加有效的时-空变地震数据信噪分离方法具有重要的工业价值.本文将地震数据信噪分离问题归纳为数学基追踪问题,在压缩感知理论框架下,利用特殊针对地震数据设计的VD-seislet稀疏变换方法,结合全变差(TV)算法,构建seislet-TV双正则化条件,并利用分裂Bregman迭代算法求解约束最优化问题,实现地震数据的有效信噪分离.通过理论模型和实际数据测试本文方法,并且与工业标准FXdecon方法进行比较,结果表明基于seislet-TV双正则化约束条件的迭代方法能够更加有效地保护时-空变地震信号,压制地震数据中的强随机噪声.     

Normal moveout velocity for pure‐mode and converted waves in layered orthorhombic medium

We study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi‐compressional, quasi‐shear, and converted waves in one‐dimensional multi‐layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface‐offset azimuth and applying our derived relationship between the surface‐offset azimuth and phase‐velocity azimuth, we obtain the normal moveout velocity versus the phase‐velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time‐domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local‐angle‐domain common image gathers. The angle and azimuth parameters of the local‐angle‐domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψ_phs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi‐layer structure is based on the fact that, for a one‐dimensional model assumption, the horizontal slowness and the azimuth of the phase velocity ψ_phs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface‐offset azimuth and phase‐velocity azimuth domains.     

Perturbation‐based moveout approximations in anisotropic media

The moveout approximations play an important role in seismic data processing. The standard hyperbolic moveout approximation is based on an elliptical background model with two velocities: vertical and normal moveout. We propose a new set of moveout approximations based on a perturbation series in terms of anellipticity parameters using the alternative elliptical background model defined by vertical and horizontal velocities. We start with a transversely isotropic medium with a vertical symmetry axis. Then, we extend this approach to a homogeneous orthorhombic medium. To define the perturbation coefficients for a new background, we solve the eikonal equation with horizontal velocities in transversely isotropic medium with a vertical symmetry axis and orthorhombic media. To stabilise the perturbation series and improve the accuracy, the Shanks transform is applied for all the cases. We select different parameterisations for both velocities and anellipticity parameters for an orthorhombic model. From the comparison in traveltime error, the new moveout approximations result in better accuracy comparing with the standard perturbation‐based methods and other approximations.     

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.     

Mapping of moveout attributes using local slopes

Decomposing seismic data in local slopes is the basic idea behind velocity‐independent imaging. Using accurate moveout approximations enables computing moveout attributes such as normal moveout velocity and nonhyperbolic parameters as functions of zero‐offset travel time. Mapping of moveout attributes is performed from the pre‐stack seismic data domain into the time‐migrated image domain. The different moveout attributes have different accuracy for a given moveout approximation that depends on the corresponding order of travel‐time derivative. The most accurate attribute is the zero‐offset travel time, and the nonhyperbolic parameter has the worst accuracy, regardless of the moveout approximation. Typically, the mapping of moveout attributes is performed using a point‐to‐point procedure, whereas the generalized moveout approximation requires two point‐to‐point mappings. Testing the attribute mapping on the different models shows that the accuracy of mapped attributes is model dependent, whereas the generalized moveout approximation gives practically exact results.     

Moveout approximation for horizontal transversely isotropic and vertical transversely isotropic layered medium. Part I: 1D ray propagation‡

Anisotropy in subsurface geological models is primarily caused by two factors: sedimentation in shale/sand layers and fractures. The sedimentation factor is mainly modelled by vertical transverse isotropy (VTI), whereas the fractures are modelled by a horizontal transversely isotropic medium (HTI). In this paper we study hyperbolic and non‐hyperbolic normal reflection moveout for a package of HTI/VTI layers, considering arbitrary azimuthal orientation of the symmetry axis at each HTI layer. We consider a local 1D medium, whose properties change vertically, with flat interfaces between the layers. In this case, the horizontal slowness is preserved; thus, the azimuth of the phase velocity is the same for all layers of the package. In general, however, the azimuth of the ray velocity differs from the azimuth of the phase velocity. The ray azimuth depends on the layer properties and may be different for each layer. In this case, the use of the Dix equation requires projection of the moveout velocity of each layer on the phase plane. We derive an accurate equation for hyperbolic and high‐order terms of the normal moveout, relating the traveltime to the surface offset, or alternatively, to the subsurface reflection angle. We relate the azimuth of the surface offset to its magnitude (or to the reflection angle), considering short and long offsets. We compare the derived approximations with analytical ray tracing.     

Non-hyperbolic moveout inversion of wide-azimuth P-wave data for orthorhombic media

The azimuthally varying non‐hyperbolic moveout of P‐waves in orthorhombic media can provide valuable information for characterization of fractured reservoirs and seismic processing. Here, we present a technique to invert long‐spread, wide‐azimuth P‐wave data for the orientation of the vertical symmetry planes and five key moveout parameters: the symmetry‐plane NMO velocities, V⁽¹⁾_nmo and V⁽²⁾_nmo , and the anellipticity parameters, η⁽¹⁾, η⁽²⁾ and η⁽³⁾ . The inversion algorithm is based on a coherence operator that computes the semblance for the full range of offsets and azimuths using a generalized version of the Alkhalifah–Tsvankin non‐hyperbolic moveout equation. The moveout equation provides a close approximation to the reflection traveltimes in layered anisotropic media with a uniform orientation of the vertical symmetry planes. Numerical tests on noise‐contaminated data for a single orthorhombic layer show that the best‐constrained parameters are the azimuth ? of one of the symmetry planes and the velocities V⁽¹⁾_nmo and V⁽²⁾_nmo , while the resolution in η⁽¹⁾ and η⁽²⁾ is somewhat compromised by the trade‐off between the quadratic and quartic moveout terms. The largest uncertainty is observed in the parameter η⁽³⁾ , which influences only long‐spread moveout in off‐symmetry directions. For stratified orthorhombic models with depth‐dependent symmetry‐plane azimuths, the moveout equation has to be modified by allowing the orientation of the effective NMO ellipse to differ from the principal azimuthal direction of the effective quartic moveout term. The algorithm was successfully tested on wide‐azimuth P‐wave reflections recorded at the Weyburn Field in Canada. Taking azimuthal anisotropy into account increased the semblance values for most long‐offset reflection events in the overburden, which indicates that fracturing is not limited to the reservoir level. The inverted symmetry‐plane directions are close to the azimuths of the off‐trend fracture sets determined from borehole data and shear‐wave splitting analysis. The effective moveout parameters estimated by our algorithm provide input for P‐wave time imaging and geometrical‐spreading correction in layered orthorhombic media.     

Fast hyperbolic deconvolutive Radon transform using generalized Fourier slice theorem

The hyperbolic Radon transform has a long history of applications in seismic data processing because of its ability to focus/sparsify the data in the transform domain. Recently, deconvolutive Radon transform has also been proposed with an improved time resolution which provides improved processing results. The basis functions of the (deconvolutive) Radon transform, however, are time-variant, making the classical Fourier based algorithms ineffective to carry out the required computations. A direct implementation of the associated summations in the time–space domain is also computationally expensive, thus limiting the application of the transform on large data sets. In this paper, we present a new method for fast computation of the hyperbolic (deconvolutive) Radon transform. The method is based on the recently proposed generalized Fourier slice theorem which establishes an analytic expression between the Fourier transforms associated with the data and Radon plane. This allows very fast computations of the forward and inverse transforms simply using fast Fourier transform and interpolation procedures. These canonical transforms are used within an efficient iterative method for sparse solution of (deconvolutive) Radon transform. Numerical examples from synthetic and field seismic data confirm high performance of the proposed fast algorithm for filling in the large gaps in seismic data, separating primaries from multiple reflections, and performing high-quality stretch-free stacking.     

基于波动方程预测和双曲Radon变换联合压制表面多次波

基于波动方程预测的表面多次波压制方法可处理复杂地下介质的地震资料,但计算成本较高.基于滤波的多次波压制方法计算效率较高,但其成功应用仅局限于一次波和多次波有明显时差差别的地震数据,对来自速度逆转等复杂介质数据则较难获得满意的压制效果.本文将波动方程预测的反馈迭代法和滤波法有效结合,采用GPU(图形处理器)和CPU协同并行加速计算粗略预测表面多次波,随后在双曲Radon域比较分析原始数据和预测的多次波,设计合理有效的Butterworth型自适应滤波器,滤出原始数据Radon域中的多次波能量,进行Radon反变换后,在时空域将多次波从原始数据中减去,得多次波压制结果.文中对理论模拟的单炮数据、复杂的SMAART模型以及实际地震数据进行了计算,结果表明,结合基于波动方程预测和双曲Radon变换的方法有效突破了两种方法各自的局限性,可高效高精度地压制复杂地下介质的表面多次波.     

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Compensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.