海底地震仪观测的粘弹性地层地震波场正演模拟
摘要
本文主要是通过波场数值正演模拟技术来研究粘弹性各向同性介质中地震波的传播问题。文中涉及粘弹性力学理论、地震波传播理论、有限差分理论、波场正演模拟技术等相关知识。所以首先介绍了粘弹性力学理论的基础知识，它是研究波在粘弹性介质中传播的理论基础，充分考虑了物质的粘弹性性质：蠕变、松弛、粘性、阻尼、内摩擦、滞弹性和弹性后效等等；其次根据粘弹性力学的基本理论介绍了粘弹性介质中地震波的传播问题，从而了解了地震波在介质中传播时能量变化同衰减系数和频率之间的关系，合理的解释了经典弹性波理论所不能解释的预期地震记录与实际接收的地震记录之间的差异；再次介绍了正演模拟用到的数值方法——有限差分方法的理论基础，包括常规有限差分法和多网格有限差分法。虽然数值模拟的方法有许多种，但是当精度要求一般时，有限差分技术是各种方法中实现最简单和最有效的方法，其中时间二阶和空间四阶的FD算法是一种好的选择(Carcione，2002)，尤其随着高阶交错网格有限差分法的发展与应用，我们可以根据需要对时间和空间做任意阶的有限差分，从而可以大大的提高计算精度、节约大量的计算时间；最后以前面各部分介绍的理论为基础，结合给出的地质模型进行波场正演数值模拟，并根据波场合成记录进行频谱分析，研究振幅谱随频率的变化、频率随偏移距的变化、频率随深度的变化。
地震模拟技术是模拟地震波在地下传播的技术，其目的就是在假定地下结构模型的情况下，预测在给定的检波点生成的地震记录，该技术作为一项重要的检验手段，一方面可用于检验解释结果和各种反演算法的正确性，另一方面可用于设计和评估野外观测系统。本文采用的是全波方程法模拟，即由方程的解中给出了全波场的值。根据论文前部分所介绍的理论基础结合前人的研究成果推导了适合粘弹性介质中地震波传播的波动方程。又根据需要进一步简化速度－应力公式，使其便于用交错网格有限差分法进行数值模拟，推导了交错网格中时间和空间上的高阶差分近似公
式，文中使用时间上二阶空间上四阶的交错网格有限差分数值方法。针对给出的地质模型(陆地地质模型和海洋地质模型)的特殊性，做了自由表面边界条件和流体－固体边界条件等特殊处理，成功的对两种地质模型进行了波场数值模拟。通过模拟得到了下面几点认识和结论：
粘弹性介质充分了考虑了实际介质对地震波的吸收衰减作用，合理的解释了经典弹性波理论不能解释的预期地震记录同实际接收的地震记录之间的差异，使理想介质理论研究向实际介质理论研究前进了一步。
交错网格有限差分法是一种高效的数值模拟方法，在提高计算速度和计算精度方面有着常规有限差分方法不可比拟的优势。随着高阶交错网格有限差分技术的应用，使得该方法的应用前景更为广阔。
正演模拟技术为我们直观的了解地震波在复杂介质中的传播规律提供了方便。我们可以通过模拟得到的地震波传播快照和地震波场合成记录来获取地震波在介质中传播的信息。从而为勘探地震数据反演提供了必要的依据。
从模拟得到的波场合成记录上可以清楚的看到粘弹性介质对地震波的吸收作用，但是对其具体影响，尚需通过AVO反射系数来分析，是需要以后进一步做的工作。
通过海洋介质模型模拟得到的波场合成记录可以看出，当海水层较深时地震勘探受海洋多次波影响较小；当海水层较浅时受海洋多次波影响较严重。所以海洋多次波消除技术对海洋地震勘探有着深远的影响。
振幅衰减特征：P波的衰减特征：零偏移距（即法向入射）时P波的振幅较强，随着偏移距的增大P波振幅减小。转换波的衰减特征：零偏移距时，没有转换波，随着偏移距的增大，出现转换波，当达到一定角度时，转换波的能量大于其衰减量，所以振幅增强，但随着偏移距的继续增大，其振幅减小。
频率的衰减特征：振幅谱随频率的增大而减小，但不是单调递减；频率随深度的增大而减小，因为地层对频率有吸收作用，并且对高频的吸收作用比对低频的吸收作用强。
In this paper, seismic wave propagation in viscoelastic isotropic media is mainly addressed by the technology of wave field numerical simulation. Many correlative techniques are involved, including seismic viscoelasticity, seismic wave theory, finite-difference and wave field modeling. The following sections character this paper: Firstly, the viscoelasticity is involved. Theoretical fundamental, studying wave propagation in viscoelastic media which sufficiently, takes into account the viscoelastic properties of media such as creep, relaxation, viscidity, damp, anelasticity and elastic drift. Secondly, the behavior of seismic wave in viscoelastic media is discussed, which clarified the relationship among the energy variation, attenuation coefficient and frequency of seismic wave, and rationally explained the difference between theoretical data sets and observed ones rather the classical elastic theory. Thirdly, the theoretical fundamental of finite-difference is introduced, including traditional FD and multi-grid FD. Although there are many approaches to numerical simulation, finite differences are simple to program and are efficient when compared to alternatives in cases where the accuracy requirements are fairly mild. In this sense, a good choice can be an FD algorithm that is second order in time and fourth order in space. Especially, with the development and application of high-order staggered FD, we can fulfill arbitrary order FD in both time and space, enhancing the accuracy of calculation and saving time greatly. Finally, the wave field is numerically simulated for the given geological models. Then, to analyze the characters of amplitude versus frequency, frequency variation with both offset and depth, spectral analysis of the synthetic seismogram is accomplished.
Seismic numerical modeling is a technique for simulating wave propagation in the earth. The objective is to predict the assumed structure of the subsurface. This technique is a valuable inversion algorithm. Another important application of seismic modeling is the evaluation and design of seismic surveys. The full-wave equation methods are adopted, namely the wave field is the solution of wave equation. Based on theory aforementioned and available publications, the wave equations are derived which suit to viscoelastic media. To perform staggered grid FD, velocity-stress formula is simplified. Furthermore, a general high-order FD expression is presented and a staggered grid FD of second order in time and fourth order in space is applied in this paper. After considering both free-surface boundary condition and fluid-solid boundary conditions, the numerical simulations of both continental and marine geological models are carried out. From the numerical simulation results, we can draw the following conclusions:
(1) The absorption and attenuation are adequately taken into account in the theory of viscoelasticity, which explain the difference between theoretical data sets and observed ones, rather the classical elastic theory.
(2) Staggered grid FD is an effective numerical simulation method and enjoys absolute advantage in improving the speed and accuracy of calculation. Accordingly the method of high-order staggered grid FD might have wilder perspective.
(3) The technique of forward modeling reveals the behaviors of seismic wave in the complex media. The characters of seismic wave could be obtained by synthetic snapshot and seismogram, which should be helpful for seismic inversion.
(4) The absorption of seismic wave in viscoelastic media could be clearly presented in the synthetic seismogram, however its concretive affection should be further studying by reflection coefficient.
(5) On the issue of synthetic seismogram for marine media model, the affection of multiple is not too much when the seawater is not very deep, whereas severe. As a result, suppression of multiples still remains a very important problem in seismic exploration, especially for the marine seismic survey.
(6) Attenuation of amplitude in viscoelastic media is
地震模拟技术是模拟地震波在地下传播的技术，其目的就是在假定地下结构模型的情况下，预测在给定的检波点生成的地震记录，该技术作为一项重要的检验手段，一方面可用于检验解释结果和各种反演算法的正确性，另一方面可用于设计和评估野外观测系统。本文采用的是全波方程法模拟，即由方程的解中给出了全波场的值。根据论文前部分所介绍的理论基础结合前人的研究成果推导了适合粘弹性介质中地震波传播的波动方程。又根据需要进一步简化速度－应力公式，使其便于用交错网格有限差分法进行数值模拟，推导了交错网格中时间和空间上的高阶差分近似公
式，文中使用时间上二阶空间上四阶的交错网格有限差分数值方法。针对给出的地质模型(陆地地质模型和海洋地质模型)的特殊性，做了自由表面边界条件和流体－固体边界条件等特殊处理，成功的对两种地质模型进行了波场数值模拟。通过模拟得到了下面几点认识和结论：
粘弹性介质充分了考虑了实际介质对地震波的吸收衰减作用，合理的解释了经典弹性波理论不能解释的预期地震记录同实际接收的地震记录之间的差异，使理想介质理论研究向实际介质理论研究前进了一步。
交错网格有限差分法是一种高效的数值模拟方法，在提高计算速度和计算精度方面有着常规有限差分方法不可比拟的优势。随着高阶交错网格有限差分技术的应用，使得该方法的应用前景更为广阔。
正演模拟技术为我们直观的了解地震波在复杂介质中的传播规律提供了方便。我们可以通过模拟得到的地震波传播快照和地震波场合成记录来获取地震波在介质中传播的信息。从而为勘探地震数据反演提供了必要的依据。
从模拟得到的波场合成记录上可以清楚的看到粘弹性介质对地震波的吸收作用，但是对其具体影响，尚需通过AVO反射系数来分析，是需要以后进一步做的工作。
通过海洋介质模型模拟得到的波场合成记录可以看出，当海水层较深时地震勘探受海洋多次波影响较小；当海水层较浅时受海洋多次波影响较严重。所以海洋多次波消除技术对海洋地震勘探有着深远的影响。
振幅衰减特征：P波的衰减特征：零偏移距（即法向入射）时P波的振幅较强，随着偏移距的增大P波振幅减小。转换波的衰减特征：零偏移距时，没有转换波，随着偏移距的增大，出现转换波，当达到一定角度时，转换波的能量大于其衰减量，所以振幅增强，但随着偏移距的继续增大，其振幅减小。
频率的衰减特征：振幅谱随频率的增大而减小，但不是单调递减；频率随深度的增大而减小，因为地层对频率有吸收作用，并且对高频的吸收作用比对低频的吸收作用强。
In this paper, seismic wave propagation in viscoelastic isotropic media is mainly addressed by the technology of wave field numerical simulation. Many correlative techniques are involved, including seismic viscoelasticity, seismic wave theory, finite-difference and wave field modeling. The following sections character this paper: Firstly, the viscoelasticity is involved. Theoretical fundamental, studying wave propagation in viscoelastic media which sufficiently, takes into account the viscoelastic properties of media such as creep, relaxation, viscidity, damp, anelasticity and elastic drift. Secondly, the behavior of seismic wave in viscoelastic media is discussed, which clarified the relationship among the energy variation, attenuation coefficient and frequency of seismic wave, and rationally explained the difference between theoretical data sets and observed ones rather the classical elastic theory. Thirdly, the theoretical fundamental of finite-difference is introduced, including traditional FD and multi-grid FD. Although there are many approaches to numerical simulation, finite differences are simple to program and are efficient when compared to alternatives in cases where the accuracy requirements are fairly mild. In this sense, a good choice can be an FD algorithm that is second order in time and fourth order in space. Especially, with the development and application of high-order staggered FD, we can fulfill arbitrary order FD in both time and space, enhancing the accuracy of calculation and saving time greatly. Finally, the wave field is numerically simulated for the given geological models. Then, to analyze the characters of amplitude versus frequency, frequency variation with both offset and depth, spectral analysis of the synthetic seismogram is accomplished.
Seismic numerical modeling is a technique for simulating wave propagation in the earth. The objective is to predict the assumed structure of the subsurface. This technique is a valuable inversion algorithm. Another important application of seismic modeling is the evaluation and design of seismic surveys. The full-wave equation methods are adopted, namely the wave field is the solution of wave equation. Based on theory aforementioned and available publications, the wave equations are derived which suit to viscoelastic media. To perform staggered grid FD, velocity-stress formula is simplified. Furthermore, a general high-order FD expression is presented and a staggered grid FD of second order in time and fourth order in space is applied in this paper. After considering both free-surface boundary condition and fluid-solid boundary conditions, the numerical simulations of both continental and marine geological models are carried out. From the numerical simulation results, we can draw the following conclusions:
(1) The absorption and attenuation are adequately taken into account in the theory of viscoelasticity, which explain the difference between theoretical data sets and observed ones, rather the classical elastic theory.
(2) Staggered grid FD is an effective numerical simulation method and enjoys absolute advantage in improving the speed and accuracy of calculation. Accordingly the method of high-order staggered grid FD might have wilder perspective.
(3) The technique of forward modeling reveals the behaviors of seismic wave in the complex media. The characters of seismic wave could be obtained by synthetic snapshot and seismogram, which should be helpful for seismic inversion.
(4) The absorption of seismic wave in viscoelastic media could be clearly presented in the synthetic seismogram, however its concretive affection should be further studying by reflection coefficient.
(5) On the issue of synthetic seismogram for marine media model, the affection of multiple is not too much when the seawater is not very deep, whereas severe. As a result, suppression of multiples still remains a very important problem in seismic exploration, especially for the marine seismic survey.
(6) Attenuation of amplitude in viscoelastic media is
引文
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侯安宁，何樵登，马在田，各向异性弹性波动交错网格高阶差分的误差研究，长春地质学院学报，1995，Vol.25(4)：P.446-451。
侯安宁，各向异性弹性波及其波动方程正反演研究，长春地质学院博士学位论文，1994。
周光泉，刘孝敏，粘弹性理论，中国科学技术大学出版社，1996。
张剑锋，李幼铭，水平成层介质中粘弹性波的计算，计算力学学报，1994，No.3，P.314-324。
埃蒙著，吕仲林译，波动方程与模型，石油物探译丛，1978：P.48-71。
孟凡顺等，复杂地质体粘滞弹性波正演模拟的有限差分法，青岛海洋大学学报，Vol.30(2)：P.315-320，2000。
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