基于第二代小波变换的各向异性参数反演
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摘要
经过长期的实际勘探和实验测试后发现地壳中大多数岩石对地震勘探是各向异性的,所谓各向异性是指当地震波在地球介质中传播时,地震波速度、偏振方向、振幅和衰减等物理性质都具有方向性特征,这一现象称为地震各向异性效应,相应的介质称为各向异性介质。一直以来把地球介质假设为各向同性的在油气田勘探中取得了瞩目的成绩,但是随着隐蔽油气藏勘探难度的日益加大,反演各向异性参数,精细描述油藏是目前油气勘探急需解决的问题。
各向异性参数反演的方法有很多种,其中利用波动方程的波形反演更多的利用了地震波的运动学和动力学的信息,能获得较好的效果。波形反演分为线性地震波形反演和非线性地震波形反演,在线性地震波形反演中,观测地震数据和速度的关系被近似线性化,非线性地震波形反演将反演问题转化为一个非线性问题,更接近于实际情况。由于目标函数中存在大量局部极小点,在处理合成数据和实际数据时,该方法效果很差。同时,利用波动方程的波形反演由于在反演过程中不断的重复正演算法,计算量较大。
小波分析被应用到波动方程反演问题中取得了较好的效果。小波分析是一种多分辨(多尺度)分析方法,将问题放在一系列嵌套空间中进行分析。由于较粗尺度下目标函数呈现较强的凸性及较少的局部极值,很有利于收敛至优化解。因此可先在粗尺度上迭代反演,得到一个较好的参数估计,再将这个估计作为较精细尺度的初值进行反演,直至原问题的全局最优解。另外,小波变换具有分解和恢复的快速算法,当计算量很大时,可以明显提高运算的效率。
本文利用有限元数值模拟的方法,实现了VTI介质的二维有限元正演模拟;利用正演模拟结果,将第二代小波变换和时间域叠前全波场各向异性参数反演方法相结合,实现了VTI介质各向异性参数的反演。
模拟算例表明将第二代小波变换应用于各向异性参数的反演方法提高了反演的效率和精度。并指出将最优估计理论应用于第二代小波变换将是本方法下一步的研究方向。
Most kinds of rocks are anisotropic to the seismic exploration, which has been discovered during long term of actual exploration and test. When seismic wave propagating in the mediums of earth, physical properties of it such as velocity,polarization direction,amplitude and attenuation all have directional properties, which is called seismic anisotropy and relevant mediums are called anisotropic mediums. The method to assume the medium of earth to be isotropic has benefited a lot to the field exploration all the times. However , as the exploration of subtle reservoir becoming more difficult ,to invert anisotropic parameters and describe reservoir delicatly has been the problem of subtle reservoir exploration activities that need to be solved now.
There are many methods to invert anisotropic parameters, among which waveform inversion based on wave equation uses more seismic wave information of kinematics and kinetics so it can get better effect. Waveform inversion includes lineal seismic waveform inversion and nonlinear seismic waveform inversion.In the lineal seismic waveform inversion the relationship between observed seismic data and velocity is linearized approximlately. And nonlinear seismic waveform inversion transforms inversion problem to nonlinear problem which is more close to reality. Because there are many local minimum points in objective function waveform inversion’s effectiveness is weak when processing anamorphic data and actual data. At the same time the calculated amount of waveform inversion using wave equation is big comparatively because it constantly repeats forward modeling algorithm in the inversion course .
The result is better when theory of wavelet analysis is used in the problem of wave equation inversion. Wavelet analysis is one kind of multiscale analysis method which analyses matters in a series of nesting air space. Objective function presents comparatively strong convexity and lesser local extreme values on wide scale which is favour of converging to optimized values . So firstly to get a better estimation of parameter using iterate inversion on wide scale , then using this estimation as initial value on mini scale till to get global optimum of original problem. Furthermore, wavelet analysis has fast algorithm of decomposition and reconstitution and this is can improve efficiency apparently when calculated amount is very big .
This paper realizes 2-D wavefield simulation of VTI medium by the finite element numerical simulation analysis. According to the results of forward modeling, combining the theory of second generation wavelet transforms with the method of the time domain pre-stack full-wave field parameters inversion, anisotropic parameters inversion of VTI medium are performed perfectly.
Numerical examples demonstrate that combining the theory of second generation wavelet transforms with the inversion method of anisotropic parameters is valuable for the improvement of efficiency and precision of inversion. The next step of the study in this field which is pointed out is to use the theory of second generation wavelet transforms with the idea of optimization.
各向异性参数反演的方法有很多种,其中利用波动方程的波形反演更多的利用了地震波的运动学和动力学的信息,能获得较好的效果。波形反演分为线性地震波形反演和非线性地震波形反演,在线性地震波形反演中,观测地震数据和速度的关系被近似线性化,非线性地震波形反演将反演问题转化为一个非线性问题,更接近于实际情况。由于目标函数中存在大量局部极小点,在处理合成数据和实际数据时,该方法效果很差。同时,利用波动方程的波形反演由于在反演过程中不断的重复正演算法,计算量较大。
小波分析被应用到波动方程反演问题中取得了较好的效果。小波分析是一种多分辨(多尺度)分析方法,将问题放在一系列嵌套空间中进行分析。由于较粗尺度下目标函数呈现较强的凸性及较少的局部极值,很有利于收敛至优化解。因此可先在粗尺度上迭代反演,得到一个较好的参数估计,再将这个估计作为较精细尺度的初值进行反演,直至原问题的全局最优解。另外,小波变换具有分解和恢复的快速算法,当计算量很大时,可以明显提高运算的效率。
本文利用有限元数值模拟的方法,实现了VTI介质的二维有限元正演模拟;利用正演模拟结果,将第二代小波变换和时间域叠前全波场各向异性参数反演方法相结合,实现了VTI介质各向异性参数的反演。
模拟算例表明将第二代小波变换应用于各向异性参数的反演方法提高了反演的效率和精度。并指出将最优估计理论应用于第二代小波变换将是本方法下一步的研究方向。
Most kinds of rocks are anisotropic to the seismic exploration, which has been discovered during long term of actual exploration and test. When seismic wave propagating in the mediums of earth, physical properties of it such as velocity,polarization direction,amplitude and attenuation all have directional properties, which is called seismic anisotropy and relevant mediums are called anisotropic mediums. The method to assume the medium of earth to be isotropic has benefited a lot to the field exploration all the times. However , as the exploration of subtle reservoir becoming more difficult ,to invert anisotropic parameters and describe reservoir delicatly has been the problem of subtle reservoir exploration activities that need to be solved now.
There are many methods to invert anisotropic parameters, among which waveform inversion based on wave equation uses more seismic wave information of kinematics and kinetics so it can get better effect. Waveform inversion includes lineal seismic waveform inversion and nonlinear seismic waveform inversion.In the lineal seismic waveform inversion the relationship between observed seismic data and velocity is linearized approximlately. And nonlinear seismic waveform inversion transforms inversion problem to nonlinear problem which is more close to reality. Because there are many local minimum points in objective function waveform inversion’s effectiveness is weak when processing anamorphic data and actual data. At the same time the calculated amount of waveform inversion using wave equation is big comparatively because it constantly repeats forward modeling algorithm in the inversion course .
The result is better when theory of wavelet analysis is used in the problem of wave equation inversion. Wavelet analysis is one kind of multiscale analysis method which analyses matters in a series of nesting air space. Objective function presents comparatively strong convexity and lesser local extreme values on wide scale which is favour of converging to optimized values . So firstly to get a better estimation of parameter using iterate inversion on wide scale , then using this estimation as initial value on mini scale till to get global optimum of original problem. Furthermore, wavelet analysis has fast algorithm of decomposition and reconstitution and this is can improve efficiency apparently when calculated amount is very big .
This paper realizes 2-D wavefield simulation of VTI medium by the finite element numerical simulation analysis. According to the results of forward modeling, combining the theory of second generation wavelet transforms with the method of the time domain pre-stack full-wave field parameters inversion, anisotropic parameters inversion of VTI medium are performed perfectly.
Numerical examples demonstrate that combining the theory of second generation wavelet transforms with the inversion method of anisotropic parameters is valuable for the improvement of efficiency and precision of inversion. The next step of the study in this field which is pointed out is to use the theory of second generation wavelet transforms with the idea of optimization.
引文
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[3] Christoffel, E. B. Uber die fortpflanzung yon St ssen durch elastische feste K rper[J]. Annali di Matematica, 1877; 8: 193-243
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