高分辨率全谱分解方法研究
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摘要
地震信号是典型的非平稳信号，需要借助时频分析技术刻画其频率随时间的变化特征，这在地震勘探中被称为地震谱分解技术。随着我国陆上油气田勘探程度的增加，勘探目标逐渐转向地层、岩性等复杂油气藏。在这种情况下，常规的谱分解方法已经越来越不能满足地震解释对精度和分辨率的要求，在薄层探测中甚至会得出错误的结论。为此，本文研究了两种聚焦性好、分辨率高的谱分解方法，分别为基于Gabor变换的时频谱重排方法（Reassignment）和基于稀疏反演的谱分解方法，并研究了与之相关的一些算法和技术。文中将能同时产生时频能量谱和有效时频相位谱的谱分解方法称为全谱分解方法，是相对于传统谱分解只利用时频能量信息，而忽略时频相位信息而提出的概念。本文研究力求在一定程度上提高解释能力，为生产和开发提供技术支持，从而降低储层勘探和开发的风险。
传统时频分析方法是研究高分辨谱分解方法的基础，本文首先回顾了几种常用的时频分析方法，包括短时窗Fourier变换法（STFT）、Gabor变换、连续小波变换（CWT）、S变换和Wigner-Ville分布（WVD）方法。重点从时频分辨率和相位特征两方面对其进行了对比分析。为提高计算效率，并给后续反演谱分解方法提供正反变换算子，本文利用基函数的概念，将几种线性时频分析方法正变换推导成互相关运算形式，逆变换推导成褶积运算形式，并在频率域实现。
本文研究的第一种高分辨率谱分解方法是基于Gabor变换的时频谱重排方法。系统阐述了时频谱重排方法的原理并推导了实现过程。将谱重排法用于地震数据谱分解，测试例子表明谱重排方法产生的时频谱聚焦性要比常规谱分解方法高很多，计算量仅是Gabor变换方法的2～3倍。但是，对于地震数据这种方法在频率方向聚焦效果受高斯窗参数的影响，高值时低频信号聚焦好，而高频信号在频率方向聚焦不足；低值时高频信号聚焦效果好而低频信号在频率方向聚焦不足。为了解决这个问题，本文提出使用时变窗参数方法，有效的改善了Gabor变换谱重排效果，使其在整个频带内产生聚焦性好、分辨率高的时频谱，本文将提出的这种方法称为变窗参数的Gabor谱重排方法。
在谱重排域，信号能量高度聚焦，随机噪声能量分散，信号和噪声水平有明显的区分。利用这一特点，本文提出了基于重排谱域阈值的地震数据随机噪声压制技术。利用此原理去噪的前提是存在一个信号可以从重排谱域反投影到时间域的逆过程。为此，本文提出了一种时频谱重排逆过程的实现方法。通过在正变换过程中记录一个加权因子和重排坐标函数，实现信号重构。拟合数据例子证明了谱重排逆过程的可行性和正确性。针对2D地震数据去噪，考虑地震记录的横向连续性和相关性，本文提出首先将2D地震数据转换到频率-空间域，再逐频率运算，在空间-波数域进行谱重排和阈值，称之为空间-波数域谱重排去噪方法。拟合例子和实际数据例子证明了谱重排在随机噪声去除上的有效性。
本文研究的另一种高分辨率、聚焦性好的谱分解方法是基于稀疏反演的谱分解方法。用非平稳地震褶积模型和基函数分解信号两种方式描述了反演谱分解问题的建立过程，前者被证明等价于CWT变换，是后者的一个特例。用线性反演理论系统的论述了反演谱分解的求解原理。为产生稀疏时频分布，本文使用了L1范数约束条件。快速迭代软阈值算法（FISTA）和In-Crowd算法都是用来求解L1范数正则化问题的快速算法，前者实现简单，适合复数运算，后者更重要的是体现了一种将大型矩阵优化问题降解为小的拥挤矩阵问题的思想。本文提出了将两种算法相结合的方法用于稀疏反演谱分解，拟合算例证明了组合方法比单独使用FISTA算法要快2～3倍。传统谱分解方法对单个地震道独立运算，忽略了地震数据的空间连续性，本文基于反演算法提出了一种考虑横向连续性的谱分解方法，将上一道的结果进行延时校正后作为当前道的初始解，热启动反演算法，速度较单道求解提高了5倍以上。将稀疏反演谱分解用于实际数据储层低频异常检测，用实例证明了稀疏反演谱的高分辨率特性及其在储层识别和层位识别上的优势。利用有效信号在反演时频谱域的稀疏性，本文将基于稀疏反演的谱分解技术用于地震随机噪声去除，拟合算例结果说明了这种方法具有很好的压制强随机噪声的能力。
将基于稀疏反演谱分解的时频相位谱概念引入到储层检测和层位识别中。由于分辨率的原因，常规谱分解方法得到的相位谱很难用于储层检测和层位识别，而由稀疏反演谱分解产生的时频相位谱，携带有地震记录的时频相位信息，具有与时频能量谱一样的高分辨率特性，并且可用于地层信息解释，因此，稀疏反演谱分解属于高分辨率全谱分解方法。实际数据例子说明，时频相位谱展示了时频能量谱无法展示的额外信息，作为时频能量谱的重要补充，在地震储层和层位识别上表现出了很强的优势和应用前景，有潜力成为地震解释的一把新的利刃。
两种高分辨率谱分解方法在文中的拟合和实际数据例子上都得到了很好的效果，但在实际生产中仍处于探索阶段，广泛推广前需要解决一些与之相关的问题。本文发现并解决其中两个问题。反演谱分解中使用的子波与地震记录真实子波存在误差时，会轻微影响谱分解的效果，为解决这一问题，本文提出了一种基于子波误差L2范数约束的子波校正技术，可以对误差较小的初始子波进行修正。高分辨率谱分解技术在得到高分辨率时频谱的同时，引起了分频数据体空间采样不足的问题，为此，本文基于压缩感知和离散余弦变换提出了谱数据的空间切片重构技术，实际数据例子证明，所提方法有效的解决了分频数据沿层切片和等时切片空间欠采样问题。高分辨率谱分解技术的发展也必将带动与之相关的新技术的发展。比如，薄层调谐效应在常规谱分解中会引起陷频效应，影响真实分频AVO曲线。本文提出了基于稀疏反演的分频AVO技术，有效的克服了调谐效应的影响。
Seismic signals are typically of non-stationary, which require time-frequencyrepresentations, known as spectral decomposition techniques in geophysics, todescribe its frequency characteristics varying with time. With the increased degree ofChina’s onshore oil and gas exploration, exploration targets are gradually shifting tothe complex oil and gas reservoirs such as stratigraphy or lithology reservoirs. In thiscase, common spectral decomposition methods have become increasingly unable tomeet the accuracy and resolution requirements of seismic interpretation. For thisreason, this paper is committed to research on spectral decomposition methods andtechniques to obtain high-resolution and concentrated time-frequency distributions.The research seeks to improve the interpretation power and provide technical supportfor the production and development, thereby reducing the risk of reservoir explorationand development.
Common time-frequency representations will play a basic role in the explorationof high-resolution methods. This paper first reviews several common time-frequencyrepresentations, including the STFT, Gabor transform, CWT, S transform andWigner-Ville distribution. These representations are compared particularly focus onthe resolution and phase characteristics. To provide forward and inverse operators forthe following-up high-resolution methods, utilizing the concept of basis function, theforward processes of linear time-frequency transformations are derived tocross-correlations and inverse transformations are derived to convolutions. Thecross-correlation and convolutions are adjoint operators and their frequencyimplementation increases the computational efficiency.
Two high-resolution spectral decomposition methods are developed in this paper,the first of which is the Gabor transform based reassignment method. The theory andderivation of reassignment are well stated, as well as some synthetic and real dataexamples. The test results show that, the reassigned spectrum are much moreconcentrated than the Gabor spectrum with only2~3times more computational cost.However, the frequency resolution of the reassigned spectrum is affected by theparameter sigma of Gaussian windows. Large sigma value leads to wider windowlength in the time domain, which is good for low frequency components and viceversa. To solve this problem, this paper propose to use a time varying sigma, since thefrequency components of seismic data are usually damped with time. The proposedapproach is called variable window parameter reassignment.
For seismic data contaminated by the random noise, thresholding in the time-frequency domain can help to attenuate the noise. However, it is hard toattenuate the noise without losing effective energy in the Gabor spectrum domain,since the signal energy spreads in the transformed domain, leading to a low disparityof the energy level with the noise. In the reassigned spectrum domain, the effectivesignal energy is strongly concentrated to points, while random noise spreads in thewhole time-frequency map with a much lower energy level. Consequently, a randomnoise attenuation method is proposed in this paper through thresholding in thereassigned Gabor domain. Time-frequency de-noising in the reassigned domain needsan inverse reassignment operator to go back to the time domain. Then, a weightingfactor is created during the forward reassignment process to save the information ofhow many components are transferred from a point to the desired point. In this way,after thresholding (or masking) the reassigned spectrum to attenuate the random noise,the Gabor spectrum can be reconstructed with this pre-computed weighting factor.Synthetic examples demonstrate the effectiveness of the proposed method. Thereassignment method is mainly demonstrated to work on1D signals in the timedomain. For2D seismic signals, events with varying dips in the time-space domainwill turn to be Chirp or chirp-like signals in the frequency-spacial domain. So, analternative choice is to perform the reassignment along the lateral spacial direction inthe space-wavenumber domain to preserve the lateral continuity of the seismic events,which is referred to as the space-wavenumber reassignment. The synthetic and realdata examples show that the reassignment method is a feasible option for spectraldecomposition and incoherent noise attenuation.
Another high-resolution decomposition method developed in this paper is thesparse inversion based method, also referred to as inverse spectral decompositionmethod. The inversion problem of spectral decomposition can be described throughtwo different approaches, one of which is using the concept of non-stationary seismicconvolutional model, expressing that the signal is composed from the sum of theconvolutions of wavelets of different frequencies and corresponding reflectivity. Thefrequency dependent reflectivity are referred to as the time-frequency distribution.Another approach states that the frequency dependent reflectivity are generated fromsignal decomposition by special basis functions. The first approach is shown to be thesame as the CWT, which is a special case of the second approach of statement. Thetheory of linear inversion is introduced to solve the inversion problems. To generatesparse spectra, L1norm regularization methods are used, such as the fast iterativesoft-thresholding algorithm (FISTA) and the In-Crowd algorithm. The FISTA ispopular due to its simply implement and the ability to deal with complex numbers.The In-Crowd algorithm is more like a flow or an idea rather than an algorithm. Thispaper integrate FISTA with In-Crowd algorithm, using FISTA to solve sub-problem ateach In-Crowd update. Synthetic examples show that the integrated method is nearly3times faster than FISTA used singly. Seismic data contains many traces with high spatial correlation. There is usually only a little time delay and some random noisedifference between two adjacent traces. Most of characteristic of the two traces aresimilar. Utilizing this property, the result of previous trace after a time-delaycorrection can be as a initial solution to warm start inversion of current trace. In thisway,6times of the computational cost are reduced. The sparse property in thespectrum is also used for the random noise attenuation, the robust ability of which isshown in a synthetic example.
The concept of time-frequency phase spectrum is introduced for the hydrocarbonreservoir detection and horizons identification. Due to the low-resolution, the phasespectra of common time-frequency distributions are difficult utilized for seismicinterpretation. While the phase spectrum from inversion method, carring theinformation of local phase, has the same high-resolution level with the time-frequencyenergy spectrum. The real data example show that horizons are more easily identifiedwith the help of the phase information than using the energy information only. As animportant complement to the time-frequency energy spectrum, the time-frequencyphase spectrum has the potential to become a new seismic interpretation tool.
The two high-resolution spectral decomposition methods both show good resultsin the synthetic and real examples. However, they are now still in the exploratorystage, so that before extensive applications, there are some related problems to solve.Two of them are investigated in this paper. The residual between the estimatedwavelet and the real wavelet will affect the result of the inversion spectrum. In orderto solve this problem, a wavelet correction technique is presented based on the L2norm constraint of the wavelet error. The sparse inversion spectral decompositionmethod pursues a high time-frequency resolution distribution and improves he abilityof interpretation of thin-layers. However, on the opposite side, the high resolutiondecomposition leads to under-sampled in the time slices and horizon slices of thespectra of3D seismic data. To overcome this drawback, we propose a discrete Cosinetransform based sparse recovery method which increases the visualization ofunder-sampled spectrum slices. A real data example is used to demonstrate theeffectiveness of the proposed method. The new techniques of high-resolution spectraldecomposition will promote the development of some related new techniques, one ofwhich is presented in this paper. Frequency-dependent AVO contains additionalattributes of the reservoir, which can enhance the accuracy of the seismicinterpretation. However, one difficult of extracting the additional attributes from thinlayers is that the tuning effects affect seismic amplitudes which can mask or at leastalter the features associated with permeability and fluid content. Therefore, this paperpresents one approach to remove the effects of thin-beds on frequency-dependentAVO analysis via spectral inversion.
传统时频分析方法是研究高分辨谱分解方法的基础，本文首先回顾了几种常用的时频分析方法，包括短时窗Fourier变换法（STFT）、Gabor变换、连续小波变换（CWT）、S变换和Wigner-Ville分布（WVD）方法。重点从时频分辨率和相位特征两方面对其进行了对比分析。为提高计算效率，并给后续反演谱分解方法提供正反变换算子，本文利用基函数的概念，将几种线性时频分析方法正变换推导成互相关运算形式，逆变换推导成褶积运算形式，并在频率域实现。
本文研究的第一种高分辨率谱分解方法是基于Gabor变换的时频谱重排方法。系统阐述了时频谱重排方法的原理并推导了实现过程。将谱重排法用于地震数据谱分解，测试例子表明谱重排方法产生的时频谱聚焦性要比常规谱分解方法高很多，计算量仅是Gabor变换方法的2～3倍。但是，对于地震数据这种方法在频率方向聚焦效果受高斯窗参数的影响，高值时低频信号聚焦好，而高频信号在频率方向聚焦不足；低值时高频信号聚焦效果好而低频信号在频率方向聚焦不足。为了解决这个问题，本文提出使用时变窗参数方法，有效的改善了Gabor变换谱重排效果，使其在整个频带内产生聚焦性好、分辨率高的时频谱，本文将提出的这种方法称为变窗参数的Gabor谱重排方法。
在谱重排域，信号能量高度聚焦，随机噪声能量分散，信号和噪声水平有明显的区分。利用这一特点，本文提出了基于重排谱域阈值的地震数据随机噪声压制技术。利用此原理去噪的前提是存在一个信号可以从重排谱域反投影到时间域的逆过程。为此，本文提出了一种时频谱重排逆过程的实现方法。通过在正变换过程中记录一个加权因子和重排坐标函数，实现信号重构。拟合数据例子证明了谱重排逆过程的可行性和正确性。针对2D地震数据去噪，考虑地震记录的横向连续性和相关性，本文提出首先将2D地震数据转换到频率-空间域，再逐频率运算，在空间-波数域进行谱重排和阈值，称之为空间-波数域谱重排去噪方法。拟合例子和实际数据例子证明了谱重排在随机噪声去除上的有效性。
本文研究的另一种高分辨率、聚焦性好的谱分解方法是基于稀疏反演的谱分解方法。用非平稳地震褶积模型和基函数分解信号两种方式描述了反演谱分解问题的建立过程，前者被证明等价于CWT变换，是后者的一个特例。用线性反演理论系统的论述了反演谱分解的求解原理。为产生稀疏时频分布，本文使用了L1范数约束条件。快速迭代软阈值算法（FISTA）和In-Crowd算法都是用来求解L1范数正则化问题的快速算法，前者实现简单，适合复数运算，后者更重要的是体现了一种将大型矩阵优化问题降解为小的拥挤矩阵问题的思想。本文提出了将两种算法相结合的方法用于稀疏反演谱分解，拟合算例证明了组合方法比单独使用FISTA算法要快2～3倍。传统谱分解方法对单个地震道独立运算，忽略了地震数据的空间连续性，本文基于反演算法提出了一种考虑横向连续性的谱分解方法，将上一道的结果进行延时校正后作为当前道的初始解，热启动反演算法，速度较单道求解提高了5倍以上。将稀疏反演谱分解用于实际数据储层低频异常检测，用实例证明了稀疏反演谱的高分辨率特性及其在储层识别和层位识别上的优势。利用有效信号在反演时频谱域的稀疏性，本文将基于稀疏反演的谱分解技术用于地震随机噪声去除，拟合算例结果说明了这种方法具有很好的压制强随机噪声的能力。
将基于稀疏反演谱分解的时频相位谱概念引入到储层检测和层位识别中。由于分辨率的原因，常规谱分解方法得到的相位谱很难用于储层检测和层位识别，而由稀疏反演谱分解产生的时频相位谱，携带有地震记录的时频相位信息，具有与时频能量谱一样的高分辨率特性，并且可用于地层信息解释，因此，稀疏反演谱分解属于高分辨率全谱分解方法。实际数据例子说明，时频相位谱展示了时频能量谱无法展示的额外信息，作为时频能量谱的重要补充，在地震储层和层位识别上表现出了很强的优势和应用前景，有潜力成为地震解释的一把新的利刃。
两种高分辨率谱分解方法在文中的拟合和实际数据例子上都得到了很好的效果，但在实际生产中仍处于探索阶段，广泛推广前需要解决一些与之相关的问题。本文发现并解决其中两个问题。反演谱分解中使用的子波与地震记录真实子波存在误差时，会轻微影响谱分解的效果，为解决这一问题，本文提出了一种基于子波误差L2范数约束的子波校正技术，可以对误差较小的初始子波进行修正。高分辨率谱分解技术在得到高分辨率时频谱的同时，引起了分频数据体空间采样不足的问题，为此，本文基于压缩感知和离散余弦变换提出了谱数据的空间切片重构技术，实际数据例子证明，所提方法有效的解决了分频数据沿层切片和等时切片空间欠采样问题。高分辨率谱分解技术的发展也必将带动与之相关的新技术的发展。比如，薄层调谐效应在常规谱分解中会引起陷频效应，影响真实分频AVO曲线。本文提出了基于稀疏反演的分频AVO技术，有效的克服了调谐效应的影响。
Seismic signals are typically of non-stationary, which require time-frequencyrepresentations, known as spectral decomposition techniques in geophysics, todescribe its frequency characteristics varying with time. With the increased degree ofChina’s onshore oil and gas exploration, exploration targets are gradually shifting tothe complex oil and gas reservoirs such as stratigraphy or lithology reservoirs. In thiscase, common spectral decomposition methods have become increasingly unable tomeet the accuracy and resolution requirements of seismic interpretation. For thisreason, this paper is committed to research on spectral decomposition methods andtechniques to obtain high-resolution and concentrated time-frequency distributions.The research seeks to improve the interpretation power and provide technical supportfor the production and development, thereby reducing the risk of reservoir explorationand development.
Common time-frequency representations will play a basic role in the explorationof high-resolution methods. This paper first reviews several common time-frequencyrepresentations, including the STFT, Gabor transform, CWT, S transform andWigner-Ville distribution. These representations are compared particularly focus onthe resolution and phase characteristics. To provide forward and inverse operators forthe following-up high-resolution methods, utilizing the concept of basis function, theforward processes of linear time-frequency transformations are derived tocross-correlations and inverse transformations are derived to convolutions. Thecross-correlation and convolutions are adjoint operators and their frequencyimplementation increases the computational efficiency.
Two high-resolution spectral decomposition methods are developed in this paper,the first of which is the Gabor transform based reassignment method. The theory andderivation of reassignment are well stated, as well as some synthetic and real dataexamples. The test results show that, the reassigned spectrum are much moreconcentrated than the Gabor spectrum with only2~3times more computational cost.However, the frequency resolution of the reassigned spectrum is affected by theparameter sigma of Gaussian windows. Large sigma value leads to wider windowlength in the time domain, which is good for low frequency components and viceversa. To solve this problem, this paper propose to use a time varying sigma, since thefrequency components of seismic data are usually damped with time. The proposedapproach is called variable window parameter reassignment.
For seismic data contaminated by the random noise, thresholding in the time-frequency domain can help to attenuate the noise. However, it is hard toattenuate the noise without losing effective energy in the Gabor spectrum domain,since the signal energy spreads in the transformed domain, leading to a low disparityof the energy level with the noise. In the reassigned spectrum domain, the effectivesignal energy is strongly concentrated to points, while random noise spreads in thewhole time-frequency map with a much lower energy level. Consequently, a randomnoise attenuation method is proposed in this paper through thresholding in thereassigned Gabor domain. Time-frequency de-noising in the reassigned domain needsan inverse reassignment operator to go back to the time domain. Then, a weightingfactor is created during the forward reassignment process to save the information ofhow many components are transferred from a point to the desired point. In this way,after thresholding (or masking) the reassigned spectrum to attenuate the random noise,the Gabor spectrum can be reconstructed with this pre-computed weighting factor.Synthetic examples demonstrate the effectiveness of the proposed method. Thereassignment method is mainly demonstrated to work on1D signals in the timedomain. For2D seismic signals, events with varying dips in the time-space domainwill turn to be Chirp or chirp-like signals in the frequency-spacial domain. So, analternative choice is to perform the reassignment along the lateral spacial direction inthe space-wavenumber domain to preserve the lateral continuity of the seismic events,which is referred to as the space-wavenumber reassignment. The synthetic and realdata examples show that the reassignment method is a feasible option for spectraldecomposition and incoherent noise attenuation.
Another high-resolution decomposition method developed in this paper is thesparse inversion based method, also referred to as inverse spectral decompositionmethod. The inversion problem of spectral decomposition can be described throughtwo different approaches, one of which is using the concept of non-stationary seismicconvolutional model, expressing that the signal is composed from the sum of theconvolutions of wavelets of different frequencies and corresponding reflectivity. Thefrequency dependent reflectivity are referred to as the time-frequency distribution.Another approach states that the frequency dependent reflectivity are generated fromsignal decomposition by special basis functions. The first approach is shown to be thesame as the CWT, which is a special case of the second approach of statement. Thetheory of linear inversion is introduced to solve the inversion problems. To generatesparse spectra, L1norm regularization methods are used, such as the fast iterativesoft-thresholding algorithm (FISTA) and the In-Crowd algorithm. The FISTA ispopular due to its simply implement and the ability to deal with complex numbers.The In-Crowd algorithm is more like a flow or an idea rather than an algorithm. Thispaper integrate FISTA with In-Crowd algorithm, using FISTA to solve sub-problem ateach In-Crowd update. Synthetic examples show that the integrated method is nearly3times faster than FISTA used singly. Seismic data contains many traces with high spatial correlation. There is usually only a little time delay and some random noisedifference between two adjacent traces. Most of characteristic of the two traces aresimilar. Utilizing this property, the result of previous trace after a time-delaycorrection can be as a initial solution to warm start inversion of current trace. In thisway,6times of the computational cost are reduced. The sparse property in thespectrum is also used for the random noise attenuation, the robust ability of which isshown in a synthetic example.
The concept of time-frequency phase spectrum is introduced for the hydrocarbonreservoir detection and horizons identification. Due to the low-resolution, the phasespectra of common time-frequency distributions are difficult utilized for seismicinterpretation. While the phase spectrum from inversion method, carring theinformation of local phase, has the same high-resolution level with the time-frequencyenergy spectrum. The real data example show that horizons are more easily identifiedwith the help of the phase information than using the energy information only. As animportant complement to the time-frequency energy spectrum, the time-frequencyphase spectrum has the potential to become a new seismic interpretation tool.
The two high-resolution spectral decomposition methods both show good resultsin the synthetic and real examples. However, they are now still in the exploratorystage, so that before extensive applications, there are some related problems to solve.Two of them are investigated in this paper. The residual between the estimatedwavelet and the real wavelet will affect the result of the inversion spectrum. In orderto solve this problem, a wavelet correction technique is presented based on the L2norm constraint of the wavelet error. The sparse inversion spectral decompositionmethod pursues a high time-frequency resolution distribution and improves he abilityof interpretation of thin-layers. However, on the opposite side, the high resolutiondecomposition leads to under-sampled in the time slices and horizon slices of thespectra of3D seismic data. To overcome this drawback, we propose a discrete Cosinetransform based sparse recovery method which increases the visualization ofunder-sampled spectrum slices. A real data example is used to demonstrate theeffectiveness of the proposed method. The new techniques of high-resolution spectraldecomposition will promote the development of some related new techniques, one ofwhich is presented in this paper. Frequency-dependent AVO contains additionalattributes of the reservoir, which can enhance the accuracy of the seismicinterpretation. However, one difficult of extracting the additional attributes from thinlayers is that the tuning effects affect seismic amplitudes which can mask or at leastalter the features associated with permeability and fluid content. Therefore, this paperpresents one approach to remove the effects of thin-beds on frequency-dependentAVO analysis via spectral inversion.
引文
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