伪谱法在粘弹介质地震波场正演模拟中的应用
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摘要
粘弹性一词来源于模型理论，即这种性质可以用弹性元件和粘性元件串联或者并联组合而成的某种模型加以表示，如Maxwell模型、Kelvin-Voigt模型S.L.S模型Burgers模型等等。在不同场合、不同文献中我们还常常接触到一系列相关而不尽相同的名词，如蠕变、松弛、粘性、阻尼、内摩擦、粘滞性或弹性后效等等，这统称为介质的粘弹性性质或者流体性质。
在研究地球的各种长期变化(如地幔对流与板块运动、各种构造运动、地震孕育过程等)，以及精确研究瞬时过程时，我们都必须考虑地球介质的流变性质。地球自然振荡的最后消失就是地球介质是粘弹介质的最好证据。地震勘探中，地震波衰减受多种因素影响，虽然事实上现在没有一种机理可以描述所有环境条件下产生的损耗，但是很明显介质的粘滞性就是其中的一个主要原因。
其本构方程分为微分形式的本构方程和积分形式的本构方程，其中微分形式的本构方程和积分形式的本构方程是相互对应的，而且由微分形式的本构方程可以求出积分形式的本构方程。这两种表达形式各有优缺点。微分型是根据理想模型导出的，模型能直观地模拟和定性地解释介质的流变性质，模型中各流变常数的物理概念清楚。但是要较好地描述介质的流变性质，需要复杂的组合模型，导致求解复杂的本构方程。也正是由于这个原因，以往的粘弹性介质中地震波正演模拟只局限于二元的Maxwell模型、Kelvin-Voigt模型和S.L.S模型。另外，其蠕变和松弛仅以指数规律变化。积分型是以Boltzmann迭加原理为基础导出的，粘弹性体的本构方程统一为一个比较简单的积分方程。只要通过实验确定了介质的J(t)、E(t)就可求得本构方程，而且它们的表达式不仅是指数
摘要
函数，而且可以是任意的。其缺点是不像微分形式的本构方程那样直观
地与介质模型相对应。
本次研究力图从模型理论出发，由岩体力学中引入积分形式的本构
方程，然后在此基础上确立粘弹介质中地震波传播的波动方程;而且在
推导波动方程时本次研究采用对应原理，使其过程明显简洁明了，而且
具有一定的通用性。对于不同的粘弹性模型，只要求出松弛模量即可求
出相应的波动方程，使我们感到对更复杂的模型介质中地震波传播的研
究是有希望的。
正是由于上述工作，本次研究模拟了几种线性粘弹介质中地震波的
传播。通过一维粘弹波和弹性波的对比，可明显看出粘弹介质中波传播
存在很大的衰减。横波和纵波的模拟结果类似。虽然Burger体的参数直
接来自标准线性体的参数，可以看出在其中传播的地震波与标准线性体
中的波存在很大的差异。横波的波速非常小，而且由一维的对比结果看，
其衰减比标准线性体更加严重。这些都与理论分析的结果一致。
The word 'viscoelasticity' comes from model theory. In another word, we can represent that by some models, which are serially or in parallel made of the basic elements - spring and dashpot, such as, Maxwell model, Kelvin-Voigt model, standard linear solid model. In different occasion and in different books or magazines, we often find some words, such as, creep, relaxation, viscosity, damping, intrinsic friction, which all belong to the characters of viscoelasticity or fluid.
When we study the long-range movement (such as, convection of mantle, movement of plate, movement of formation, gestation of earth quake and so on) and accurately research instantaneous processes, we have to consider the viscoelasticity of the earth. On the other hand, the disappearance of surge of earth is the best evidence that proves that the medium in earth is viscoelastic medium. In the seism exploration, because of many factors, there is attenuation of wave. Although there is not one theory that can describe all factors, it is obvious that the viscoelasticity of medium is one of the important factors.
The constitutive relation of viscoelastic medium has two forms-differential operator form of stress strain constitutive relation and integral operator form of stress strain constitutive relation, and they are corresponding, and the later can be educed from the former. They have their own merit: differential form is directly educed from ideal model, so it can directly model and qualitatively explain viscoelasticity of medium and the parameters of the model have clear conception. However, it is essential for better describing viscoelasticity to use complicated model, which leads to perplexing
constitutive equation, and because of the difficulty researchers can only use some simple models, such as, Maxwell model, Kelvin-Voigt model, standard linear solid model to model wave propagation in viscoelastic medium. The integral form roots in principle of superposition and has a more simple uniform. Then we can get the constitutive equation if we have relaxation modulus E(t) and flexible modulus J(t).
In this research, on the base of the theory of model, we get the equation by integral form constitutive relation, and in this process we use corresponding principle. Then the process becomes compact and universal. So we have the hope that models wave propagation using more complicated models.
In this research, we model wave propagation in several kinds of model. According to 1-D modeling, the wave in viscoelastic medium has distinct attenuation by comparing with the wave in elastic medium. P-wave and S-wave all have these characteristic. Though the parameters of Burgers model near quantitatively these of S.L.S. Compared with modeling in S.L.S medium, the wave in burgers medium has more attenuation and the velocity of S-wave is very smaller. These phenomena all are consistent with what we theoretically analyze.
在研究地球的各种长期变化(如地幔对流与板块运动、各种构造运动、地震孕育过程等)，以及精确研究瞬时过程时，我们都必须考虑地球介质的流变性质。地球自然振荡的最后消失就是地球介质是粘弹介质的最好证据。地震勘探中，地震波衰减受多种因素影响，虽然事实上现在没有一种机理可以描述所有环境条件下产生的损耗，但是很明显介质的粘滞性就是其中的一个主要原因。
其本构方程分为微分形式的本构方程和积分形式的本构方程，其中微分形式的本构方程和积分形式的本构方程是相互对应的，而且由微分形式的本构方程可以求出积分形式的本构方程。这两种表达形式各有优缺点。微分型是根据理想模型导出的，模型能直观地模拟和定性地解释介质的流变性质，模型中各流变常数的物理概念清楚。但是要较好地描述介质的流变性质，需要复杂的组合模型，导致求解复杂的本构方程。也正是由于这个原因，以往的粘弹性介质中地震波正演模拟只局限于二元的Maxwell模型、Kelvin-Voigt模型和S.L.S模型。另外，其蠕变和松弛仅以指数规律变化。积分型是以Boltzmann迭加原理为基础导出的，粘弹性体的本构方程统一为一个比较简单的积分方程。只要通过实验确定了介质的J(t)、E(t)就可求得本构方程，而且它们的表达式不仅是指数
摘要
函数，而且可以是任意的。其缺点是不像微分形式的本构方程那样直观
地与介质模型相对应。
本次研究力图从模型理论出发，由岩体力学中引入积分形式的本构
方程，然后在此基础上确立粘弹介质中地震波传播的波动方程;而且在
推导波动方程时本次研究采用对应原理，使其过程明显简洁明了，而且
具有一定的通用性。对于不同的粘弹性模型，只要求出松弛模量即可求
出相应的波动方程，使我们感到对更复杂的模型介质中地震波传播的研
究是有希望的。
正是由于上述工作，本次研究模拟了几种线性粘弹介质中地震波的
传播。通过一维粘弹波和弹性波的对比，可明显看出粘弹介质中波传播
存在很大的衰减。横波和纵波的模拟结果类似。虽然Burger体的参数直
接来自标准线性体的参数，可以看出在其中传播的地震波与标准线性体
中的波存在很大的差异。横波的波速非常小，而且由一维的对比结果看，
其衰减比标准线性体更加严重。这些都与理论分析的结果一致。
The word 'viscoelasticity' comes from model theory. In another word, we can represent that by some models, which are serially or in parallel made of the basic elements - spring and dashpot, such as, Maxwell model, Kelvin-Voigt model, standard linear solid model. In different occasion and in different books or magazines, we often find some words, such as, creep, relaxation, viscosity, damping, intrinsic friction, which all belong to the characters of viscoelasticity or fluid.
When we study the long-range movement (such as, convection of mantle, movement of plate, movement of formation, gestation of earth quake and so on) and accurately research instantaneous processes, we have to consider the viscoelasticity of the earth. On the other hand, the disappearance of surge of earth is the best evidence that proves that the medium in earth is viscoelastic medium. In the seism exploration, because of many factors, there is attenuation of wave. Although there is not one theory that can describe all factors, it is obvious that the viscoelasticity of medium is one of the important factors.
The constitutive relation of viscoelastic medium has two forms-differential operator form of stress strain constitutive relation and integral operator form of stress strain constitutive relation, and they are corresponding, and the later can be educed from the former. They have their own merit: differential form is directly educed from ideal model, so it can directly model and qualitatively explain viscoelasticity of medium and the parameters of the model have clear conception. However, it is essential for better describing viscoelasticity to use complicated model, which leads to perplexing
constitutive equation, and because of the difficulty researchers can only use some simple models, such as, Maxwell model, Kelvin-Voigt model, standard linear solid model to model wave propagation in viscoelastic medium. The integral form roots in principle of superposition and has a more simple uniform. Then we can get the constitutive equation if we have relaxation modulus E(t) and flexible modulus J(t).
In this research, on the base of the theory of model, we get the equation by integral form constitutive relation, and in this process we use corresponding principle. Then the process becomes compact and universal. So we have the hope that models wave propagation using more complicated models.
In this research, we model wave propagation in several kinds of model. According to 1-D modeling, the wave in viscoelastic medium has distinct attenuation by comparing with the wave in elastic medium. P-wave and S-wave all have these characteristic. Though the parameters of Burgers model near quantitatively these of S.L.S. Compared with modeling in S.L.S medium, the wave in burgers medium has more attenuation and the velocity of S-wave is very smaller. These phenomena all are consistent with what we theoretically analyze.
引文
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