地震反射走时层析理论与应用研究
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摘要
速度建模是地震勘探中的一项重要技术。其中偏移速度分析和层析成像是两种很重要的手段。从理论上看，速度建模的精度应以层析成像为高，但层析成像在实践过程中受诸多因素的制约，其难点不在于理论方法上，而在于具体实现的技巧上。
本文对两种速度建模方法进行了系统的研究，并重点研究了利用反射地震资料进行走时层析成像速度建模的方法。考虑到层析成像在地震中应用的很多理论和实际上的困难，更由于反射层析中速度和反射深度的耦合性，本文确立了用偏移速度分析的结果为层析成像提供初始速度和反射深度的思想，使得层析成像能够具有一个好的起点。
本文对偏移速度分析的发展现状进行了简单的介绍，详细阐述了层析成像中存在的问题和国内外的发展现状，并针对这两种速度建模方法进行了详细的比较。本文对偏移速度分析的要求不高，偏移速度分析的结果只需给出常速模型即可，本文更对偏移速度分析提供的初始反射深度感兴趣。鉴于速度分析的迭代型和交互性，本文选取了速度较快的局部区域的Kirchhoff偏移方法和沿层进行偏移速度分析的策略。
关于地震层析成像，本文主要介绍了几个方面的内容：层析方程组的数值解法、正则化和归一化。层析方程组的求解，经过了矩阵直接求逆和迭代解两个阶段。矩阵直接求逆的解法，只对小型的矩阵适用。迭代解法主要分为ART类方法和投影类方法。ART类方法是一种行处理方法，是针对早期计算机能力的限制而采取的方法。投影类主要有Lanczos方法、共轭梯度法(CG)、最小二乘共轭梯度法(LSCG)以及最小二乘QR分解法(LSQR)。是目前最经常使用的一类方法。在地震层析中利用正则化对欠定分量和零空间分量进行约束是非常重要的，正则化对欠定分量和零空间分量的约束主要体现为最简单解的思想：包括最平坦解、最光滑解和最小长度解思想，最平坦解思想的出发点是使解的分量间只发生平缓的变化，最光滑解的思想是解的分量间的变化是光滑的，最小长度解思想是使解矢量相对于某个先验矢量发生最小的变化。本文将正则化的具体作用方式分为加法型和乘法型两种。加法型是指将正则化方程组补在原方程组的下面的处理方式，乘法型是指将正则化方程组与原方程组相乘的处理方式。在对正则化的作用范围和作用方式进行介绍后，本文又对正则化的矩阵表述形式、压缩存储进行了阐述，并针对简单模型对导数型正则化的效果进行了分析。归一化是地震层析成像中另一个非常重要的问题。反演计算中，数据d和模型参数m的物理量纲不同，而在多参数类型同时反演时，模型参数和数据中的元本身就带有不同的物理量纲。此时Frechet微商矩阵的列向量的模相差极大，导致问题的性状很坏。因此在进行数据反演计算之前，应该把所有的模型参数和数据进行归一化化为无量纲的量。
针对反射层析中速度和深度的耦合现象，本文综合多人的研究结果得出结论如下：速度深度的耦合性在本质上是由短偏移距引起的，大偏移距的存在能够缓解这种耦合。速度深度的耦合程度受几方面因素的影响：(1)偏移距与深度的比值，偏移距与深度的比值越大，速度深度的耦合性越弱；(2)拾取误差项，拾取误差越小，速度深度的耦合性越弱；(3)界面倾角，倾角越小，速度深度的耦合性越弱：(4)扰动波长与特征波长的比值，当扰动波长大于特征波长时慢度扰动占主要地位，当扰动波长小于特征波长时，深度扰动占主要地位：当扰动波长与特征波长相等时，速度和反射深度发生耦合。特征波长取决于异常体的厚度和异常体与反射面的距离。此外本文还分析了走时误差对反射层析中利用走时扰动分辨薄层内速度扰动的稳定性的影响，并得出结论：当层厚较薄时，由于走时信息中误差的影响，利用走时扰动分辨薄层内的速度扰动是非常不稳定的。层内速度越高、偏移距越小，不稳定性越强。本文针对反射层析的横向和垂向分辨能力也进行了分析，分析证明由于反射地震勘探中以近垂直方向射线为主，因而反射层析的横向分辨能力要优于垂向分辨能力。
本文针对矩形网格参数化的形式建立了层析反演体系，在正演模拟中选用了Langan射线追踪方法，这种方法主要有三个优点：(1)网格内速度的三参数(对二维情况)表述方式在本质上照顾到了射线理论对速度平滑性的要求，方法本身有一种内在的平滑效应；(2)Langan法射线追踪具有完美的解析表达形式，且能够保证精度和效率；(3)在模型矢量选为慢度矢量差，数据矢量选为走时差的情况下，射线在网格内的长度就是反映数据变化对参数变化灵敏度的Frechet微商。考虑到均一矩形网格参数化导致的巨大的矩阵维数给计算机存储量和计算量造成的压力，在对矩形网格参数化层析反演中，本文采用了行索引的压缩存储方法，并以压缩存储的方式补入了一阶导数型正则化和二阶导数型正则化矩阵，最后针对补入正则化矩阵的压缩矩阵采用了LSQR方法迭代求解，大大节省了存储量、计算量和计算时间。
尽管在目前的地震层析成像反演中，采用最多的是矩形网格参数化的方式，但矩形网格参数化方式在很多方面不容乐观：在矩形网格参数化形式下，对地下介质的精细描述必然导致巨大的矩阵维数(各种正则化约束加入后尤甚)，方程组的大型稀疏特性使得在求解过程中耗费了大量的计算机存储空间和计算时间。尽管本文针对方程组的稀疏特性设计了一种行索引的压缩存储方法以及针对压缩存储矩阵的迭代解法，大大节省了存储量、计算量和计算时间，但考虑到层析速度建模的迭代性和交互性，存储量和计算时间仍然是矩形网格参数化层析反演应用的一个限制因素。除此之外，矩形网格的均一剖分形式灵活性差，不论在构造简单区域和构造复杂区域都不能区别对待，增大了未知参数的个数，不仅在正演模拟时增加了计算量和计算时间，而且在层析反演时使问题的性态变得更坏，增加了层析反演的难度。矩形网格参数化在对界面的描述上也同样存在着缺陷，即速度模型和界面模型的不一致，由于矩形网格参数化只能用锯齿状的分布来表征界面上下的速度变化，在对界面的处理上增加了难度和引入了误差。针对矩形网格参数化方式在剖分上的不灵活给正反演方法带来的诸多不利因素，考虑到三角网格剖分的灵活性和本质上的正则化效应，本文进一步建立了可变三角网格参数化的层析反演体系，得到了很大的改善。三角网格参数化层析反演相对于矩形网格参数化层析反演无论在计算时间、计算量，还是在存储量上都占有非常大的优势，经验证两种参数化方式得到的层析结果具有可比性。
Velocity model building is an important technique in seismic exploration. Migrationvelocity analysis and tomography are two important manners among others. From thepoint-view of theory, the accuracy of tomography is higher, but tomography is restrictedby many factors in application, the difficulties of tomography don't lie with theories andmethods, but lie with the skills of implementation.
Considering the difficulties in the theories and applications of the tomography, andthe ambiguities between the velocity and the reflection depth, the idea of using theresults of migration velocity analysis as the initial model of the tomography isestablished, so that the tomography inversion can have a good start.
In this paper, the development status in quo of migration velocity analysis andtomography are introduced briefly, then the problems existed in tomography areexpounded, and the two velocity model building methods are compared. The migrationvelocity analysis is required less, the constant velocities in layers are fine, but thereflection depths of interfaces are necessary. Due to the iterative and alternatingcharacteristic of migration velocity analysis, Kirchhoff migration method suitable forlocal areas and the strategy of analyzing velocity along the reflector are adopted.
On tomography, Several aspects are introduced: the solver of tomography equationsystem, regularization and normalization. The solution process of tomography equationsystem has two stages: the direct solution and the iterative solution. The direct solutionis only suitable for matrix with small dimensions, and the iterative solution includesART methods and projection methods. ART method emerges in the early age oftomography development at that moment the memory and storage space of computer arelimit. Projection methods are the most used methods currently, mainly includes Lanczosmethod, conjugate gradient method, least square conjugate gradient and least square QRfactor decomposition method. It is very important using regularization to constrain theunderdetermined and the null space components in tomography inversion.Regularization usually has two manners: using the over-determined components toconstrain the under-determined and the null space components; and using the priorinformation to constrain the under-determined and the null space components. In thispaper, the fashion of regularization is classified into two classes: the addition style andthe multiplication style. The addition style usually pads the regularization equationsystem after the primal equation system; the multiplication style usually multiples theregularization equation system with the primal equation system. Except the action rangeand action style of regularization, the matrix expression and compression storage ofregularization are also introduced in this paper. In addition, the effects of variousregularization patterns are also analyzed. Normalization is another important problem inseismic tomography. In tomography, the data and the model have different physicaldimensions, and even elements of data and model parameter have different physical dimensions in multi-parameter tomography. The modules of different column vectors ofthe Frechet matrix are very different, which results in poor condition of tomography, sowe must normalize all the data and model parameters to nondimensional mount beforetomography.
About the ambiguity between velocity and reflection depth in reflectiontomography, research results of many geophysicists are integrated and the followingconclusions are obtained: In essence, the ambiguity between velocity and reflectiondepth is caused b y s mall offset, t he ambiguity can b e mitigated by t he existences o flarge offsets. The ambiguity between velocity and reflection depth is influenced by thefollowing factors:(1) the ratio between the offset and the depth, the larger the ratio, theweaker the ambiguity;(2) the pick error, the smaller the pick error, the weaker theambiguity;(3) the dip of the reflector, the smaller the reflection dip, the weaker theambiguity;(4) the ratio between perturbation wavelength and characteristic wavelength,when the ratio is greater than 1, velocity perturbation dominates, when the ratio issmaller than 1, the depth perturbation dominates, when the ratio is equal to 1, theambiguity occurs. The characteristic wavelength depends on the thickness of anomalyand the distance between the anomaly and the reflector.
In this paper, the tomography system based on the rectangle cell parameteration isestablished. In modeling based on rectangle cell, Langan ray tracing is adopted, whichhas three merits:(1) The three-parameters expression style of the velocity in one cell,considers the limits of ray theory——velocity must be smooth, thus the method has somesmooth effects in essence;(2)L angan ray tracing has perfect analytic expression, andaccuracy and efficiency can be guaranteed;(3) when the model vector is chosen asslowness vector difference, the data vector is chosen as the time difference, the length ofray in cell is the Frechet derivative which denotes the sensitivity of the data due to themodel perturbation. Considering the storage and computation pressure caused byrectangle cell parameteration, line-indexed compression storage is designed in this paper,and the various computation skills are also designed so that various projection iterationmethods suitable for primal tomography matrix are also suitable for compressed matrix.Various regularization matrixes are also stored and take part in tomography computationin compression style. The compression storage and solution of primal matrix andregularization matrix are very effective, large amount of computation time and storagecan be saved.
In current tomography, rectangle cell parameteration is the most adopted, butrectangle cell parameteration is not optimistic in many aspects: To finely expresssubsurface medium, the rectangle cell parameteration need large mount of cells, whichresults in huge matrix dimensions (even worse when considers regularization), thesparseness of matrix expenses large mount of computation time and storage space,although the line-indexed compression storage method and solution of compressedstorage equation system are designed so that large amount computation time and storagemount can be saved, the iterative and interactive characteristic of tomography still makes large amount of computation time and storage space as the limiting factors inrectangle cell tomography system. In addition, the uniform rectangle cell is not flexible,the same size of cell must be adopted both in complex structures areas and in simplestructures areas, thus the number of cell is very large, which results in morecomputation time, more computation mount and more storage space in modeling andresults in the worse condition and the larger difficulties in tomography. Moreover,rectangle cell parameteration has shortcomings in expressing the reflection interfaces,the velocity change above and under the interface can only expressed in saw-teethfashion, thus increase the difficulties and inducts the errors when treating interfaces.Aiming at the several adverse factors introduced by rectangle cell parameteration, thetomography system based on triangle cell parameteration is established. Triangle cellparameteration is not only flexible, but also has some regularization effects in essence.Compared with rectangle cell parameteration, the triangle cell parameteration has meritsin computation time, computation mount and storage mount. And the results obtainedby the two parameteration styles are comparative.
本文对两种速度建模方法进行了系统的研究，并重点研究了利用反射地震资料进行走时层析成像速度建模的方法。考虑到层析成像在地震中应用的很多理论和实际上的困难，更由于反射层析中速度和反射深度的耦合性，本文确立了用偏移速度分析的结果为层析成像提供初始速度和反射深度的思想，使得层析成像能够具有一个好的起点。
本文对偏移速度分析的发展现状进行了简单的介绍，详细阐述了层析成像中存在的问题和国内外的发展现状，并针对这两种速度建模方法进行了详细的比较。本文对偏移速度分析的要求不高，偏移速度分析的结果只需给出常速模型即可，本文更对偏移速度分析提供的初始反射深度感兴趣。鉴于速度分析的迭代型和交互性，本文选取了速度较快的局部区域的Kirchhoff偏移方法和沿层进行偏移速度分析的策略。
关于地震层析成像，本文主要介绍了几个方面的内容：层析方程组的数值解法、正则化和归一化。层析方程组的求解，经过了矩阵直接求逆和迭代解两个阶段。矩阵直接求逆的解法，只对小型的矩阵适用。迭代解法主要分为ART类方法和投影类方法。ART类方法是一种行处理方法，是针对早期计算机能力的限制而采取的方法。投影类主要有Lanczos方法、共轭梯度法(CG)、最小二乘共轭梯度法(LSCG)以及最小二乘QR分解法(LSQR)。是目前最经常使用的一类方法。在地震层析中利用正则化对欠定分量和零空间分量进行约束是非常重要的，正则化对欠定分量和零空间分量的约束主要体现为最简单解的思想：包括最平坦解、最光滑解和最小长度解思想，最平坦解思想的出发点是使解的分量间只发生平缓的变化，最光滑解的思想是解的分量间的变化是光滑的，最小长度解思想是使解矢量相对于某个先验矢量发生最小的变化。本文将正则化的具体作用方式分为加法型和乘法型两种。加法型是指将正则化方程组补在原方程组的下面的处理方式，乘法型是指将正则化方程组与原方程组相乘的处理方式。在对正则化的作用范围和作用方式进行介绍后，本文又对正则化的矩阵表述形式、压缩存储进行了阐述，并针对简单模型对导数型正则化的效果进行了分析。归一化是地震层析成像中另一个非常重要的问题。反演计算中，数据d和模型参数m的物理量纲不同，而在多参数类型同时反演时，模型参数和数据中的元本身就带有不同的物理量纲。此时Frechet微商矩阵的列向量的模相差极大，导致问题的性状很坏。因此在进行数据反演计算之前，应该把所有的模型参数和数据进行归一化化为无量纲的量。
针对反射层析中速度和深度的耦合现象，本文综合多人的研究结果得出结论如下：速度深度的耦合性在本质上是由短偏移距引起的，大偏移距的存在能够缓解这种耦合。速度深度的耦合程度受几方面因素的影响：(1)偏移距与深度的比值，偏移距与深度的比值越大，速度深度的耦合性越弱；(2)拾取误差项，拾取误差越小，速度深度的耦合性越弱；(3)界面倾角，倾角越小，速度深度的耦合性越弱：(4)扰动波长与特征波长的比值，当扰动波长大于特征波长时慢度扰动占主要地位，当扰动波长小于特征波长时，深度扰动占主要地位：当扰动波长与特征波长相等时，速度和反射深度发生耦合。特征波长取决于异常体的厚度和异常体与反射面的距离。此外本文还分析了走时误差对反射层析中利用走时扰动分辨薄层内速度扰动的稳定性的影响，并得出结论：当层厚较薄时，由于走时信息中误差的影响，利用走时扰动分辨薄层内的速度扰动是非常不稳定的。层内速度越高、偏移距越小，不稳定性越强。本文针对反射层析的横向和垂向分辨能力也进行了分析，分析证明由于反射地震勘探中以近垂直方向射线为主，因而反射层析的横向分辨能力要优于垂向分辨能力。
本文针对矩形网格参数化的形式建立了层析反演体系，在正演模拟中选用了Langan射线追踪方法，这种方法主要有三个优点：(1)网格内速度的三参数(对二维情况)表述方式在本质上照顾到了射线理论对速度平滑性的要求，方法本身有一种内在的平滑效应；(2)Langan法射线追踪具有完美的解析表达形式，且能够保证精度和效率；(3)在模型矢量选为慢度矢量差，数据矢量选为走时差的情况下，射线在网格内的长度就是反映数据变化对参数变化灵敏度的Frechet微商。考虑到均一矩形网格参数化导致的巨大的矩阵维数给计算机存储量和计算量造成的压力，在对矩形网格参数化层析反演中，本文采用了行索引的压缩存储方法，并以压缩存储的方式补入了一阶导数型正则化和二阶导数型正则化矩阵，最后针对补入正则化矩阵的压缩矩阵采用了LSQR方法迭代求解，大大节省了存储量、计算量和计算时间。
尽管在目前的地震层析成像反演中，采用最多的是矩形网格参数化的方式，但矩形网格参数化方式在很多方面不容乐观：在矩形网格参数化形式下，对地下介质的精细描述必然导致巨大的矩阵维数(各种正则化约束加入后尤甚)，方程组的大型稀疏特性使得在求解过程中耗费了大量的计算机存储空间和计算时间。尽管本文针对方程组的稀疏特性设计了一种行索引的压缩存储方法以及针对压缩存储矩阵的迭代解法，大大节省了存储量、计算量和计算时间，但考虑到层析速度建模的迭代性和交互性，存储量和计算时间仍然是矩形网格参数化层析反演应用的一个限制因素。除此之外，矩形网格的均一剖分形式灵活性差，不论在构造简单区域和构造复杂区域都不能区别对待，增大了未知参数的个数，不仅在正演模拟时增加了计算量和计算时间，而且在层析反演时使问题的性态变得更坏，增加了层析反演的难度。矩形网格参数化在对界面的描述上也同样存在着缺陷，即速度模型和界面模型的不一致，由于矩形网格参数化只能用锯齿状的分布来表征界面上下的速度变化，在对界面的处理上增加了难度和引入了误差。针对矩形网格参数化方式在剖分上的不灵活给正反演方法带来的诸多不利因素，考虑到三角网格剖分的灵活性和本质上的正则化效应，本文进一步建立了可变三角网格参数化的层析反演体系，得到了很大的改善。三角网格参数化层析反演相对于矩形网格参数化层析反演无论在计算时间、计算量，还是在存储量上都占有非常大的优势，经验证两种参数化方式得到的层析结果具有可比性。
Velocity model building is an important technique in seismic exploration. Migrationvelocity analysis and tomography are two important manners among others. From thepoint-view of theory, the accuracy of tomography is higher, but tomography is restrictedby many factors in application, the difficulties of tomography don't lie with theories andmethods, but lie with the skills of implementation.
Considering the difficulties in the theories and applications of the tomography, andthe ambiguities between the velocity and the reflection depth, the idea of using theresults of migration velocity analysis as the initial model of the tomography isestablished, so that the tomography inversion can have a good start.
In this paper, the development status in quo of migration velocity analysis andtomography are introduced briefly, then the problems existed in tomography areexpounded, and the two velocity model building methods are compared. The migrationvelocity analysis is required less, the constant velocities in layers are fine, but thereflection depths of interfaces are necessary. Due to the iterative and alternatingcharacteristic of migration velocity analysis, Kirchhoff migration method suitable forlocal areas and the strategy of analyzing velocity along the reflector are adopted.
On tomography, Several aspects are introduced: the solver of tomography equationsystem, regularization and normalization. The solution process of tomography equationsystem has two stages: the direct solution and the iterative solution. The direct solutionis only suitable for matrix with small dimensions, and the iterative solution includesART methods and projection methods. ART method emerges in the early age oftomography development at that moment the memory and storage space of computer arelimit. Projection methods are the most used methods currently, mainly includes Lanczosmethod, conjugate gradient method, least square conjugate gradient and least square QRfactor decomposition method. It is very important using regularization to constrain theunderdetermined and the null space components in tomography inversion.Regularization usually has two manners: using the over-determined components toconstrain the under-determined and the null space components; and using the priorinformation to constrain the under-determined and the null space components. In thispaper, the fashion of regularization is classified into two classes: the addition style andthe multiplication style. The addition style usually pads the regularization equationsystem after the primal equation system; the multiplication style usually multiples theregularization equation system with the primal equation system. Except the action rangeand action style of regularization, the matrix expression and compression storage ofregularization are also introduced in this paper. In addition, the effects of variousregularization patterns are also analyzed. Normalization is another important problem inseismic tomography. In tomography, the data and the model have different physicaldimensions, and even elements of data and model parameter have different physical dimensions in multi-parameter tomography. The modules of different column vectors ofthe Frechet matrix are very different, which results in poor condition of tomography, sowe must normalize all the data and model parameters to nondimensional mount beforetomography.
About the ambiguity between velocity and reflection depth in reflectiontomography, research results of many geophysicists are integrated and the followingconclusions are obtained: In essence, the ambiguity between velocity and reflectiondepth is caused b y s mall offset, t he ambiguity can b e mitigated by t he existences o flarge offsets. The ambiguity between velocity and reflection depth is influenced by thefollowing factors:(1) the ratio between the offset and the depth, the larger the ratio, theweaker the ambiguity;(2) the pick error, the smaller the pick error, the weaker theambiguity;(3) the dip of the reflector, the smaller the reflection dip, the weaker theambiguity;(4) the ratio between perturbation wavelength and characteristic wavelength,when the ratio is greater than 1, velocity perturbation dominates, when the ratio issmaller than 1, the depth perturbation dominates, when the ratio is equal to 1, theambiguity occurs. The characteristic wavelength depends on the thickness of anomalyand the distance between the anomaly and the reflector.
In this paper, the tomography system based on the rectangle cell parameteration isestablished. In modeling based on rectangle cell, Langan ray tracing is adopted, whichhas three merits:(1) The three-parameters expression style of the velocity in one cell,considers the limits of ray theory——velocity must be smooth, thus the method has somesmooth effects in essence;(2)L angan ray tracing has perfect analytic expression, andaccuracy and efficiency can be guaranteed;(3) when the model vector is chosen asslowness vector difference, the data vector is chosen as the time difference, the length ofray in cell is the Frechet derivative which denotes the sensitivity of the data due to themodel perturbation. Considering the storage and computation pressure caused byrectangle cell parameteration, line-indexed compression storage is designed in this paper,and the various computation skills are also designed so that various projection iterationmethods suitable for primal tomography matrix are also suitable for compressed matrix.Various regularization matrixes are also stored and take part in tomography computationin compression style. The compression storage and solution of primal matrix andregularization matrix are very effective, large amount of computation time and storagecan be saved.
In current tomography, rectangle cell parameteration is the most adopted, butrectangle cell parameteration is not optimistic in many aspects: To finely expresssubsurface medium, the rectangle cell parameteration need large mount of cells, whichresults in huge matrix dimensions (even worse when considers regularization), thesparseness of matrix expenses large mount of computation time and storage space,although the line-indexed compression storage method and solution of compressedstorage equation system are designed so that large amount computation time and storagemount can be saved, the iterative and interactive characteristic of tomography still makes large amount of computation time and storage space as the limiting factors inrectangle cell tomography system. In addition, the uniform rectangle cell is not flexible,the same size of cell must be adopted both in complex structures areas and in simplestructures areas, thus the number of cell is very large, which results in morecomputation time, more computation mount and more storage space in modeling andresults in the worse condition and the larger difficulties in tomography. Moreover,rectangle cell parameteration has shortcomings in expressing the reflection interfaces,the velocity change above and under the interface can only expressed in saw-teethfashion, thus increase the difficulties and inducts the errors when treating interfaces.Aiming at the several adverse factors introduced by rectangle cell parameteration, thetomography system based on triangle cell parameteration is established. Triangle cellparameteration is not only flexible, but also has some regularization effects in essence.Compared with rectangle cell parameteration, the triangle cell parameteration has meritsin computation time, computation mount and storage mount. And the results obtainedby the two parameteration styles are comparative.
引文
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