基于FCT有限差分方法的瑞雷波数值模拟
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摘要
瑞雷波勘探作为一种新兴物探方法,近年来发展迅速,其经济、快捷、高效的特点逐渐为工程人员所认识,被广泛应用于工程地质勘察和工程无损检测等领域。目前,瑞雷波的理论研究主要集中在频散曲线的正反演方面。为更准确全面地了解瑞雷波的传播特性,一些学者在瑞雷波数值模拟方面做了有意义的研究。
基于弹性波方程的有限差分法是瑞雷波数值模拟的有力工具,但当单位波长内采样点数较少时(粗网格)会产生数值频散现象,这种严重的干扰,降低了波场模拟的精度和分辨率。为寻求快速、有效的波场模拟方法,本文将常用于流体动力学中的通量校正传输技术(FCT)与交错网格有限差分法相结合,对浅层各向同性弹性介质进行了包括瑞雷波和体波在内的全波场模拟。模拟结果表明,该方法有效地压制了粗网格条件下的数值频散现象,并保留了真实的波场特征,特别是层状介质中瑞雷波的频散特征得以凸显。同时,在同等精度条件下,该方法与细化网格方法相比,计算效率提高了1.3倍。
本文首先回顾了瑞雷波勘探、地震波有限差分模拟和FCT方法的研究现状；介绍了瑞雷波的基本理论；讨论了二维一阶应力—速度弹性波动方程的交错网格差分格式、边界条件、震源函数、以及差分格式的相容性、收敛性和稳定性,并给出了程序流程图；介绍了波的频散现象,推导了波动方程频散关系,区分了频散现象中的物理频散和数值频散,并说明了数值频散产生的原因以及消除方法；阐述了FCT方法的基本理论及计算步骤；最后选取11个算例进行了计算分析,给出了结论及进一步工作的方向。
As a new geophysical prospecting method, Rayleigh wave has been developing rapidly in recent years, its economic, fast and efficient features are gradually recognized by engineers, and it is widely used in engineering geological investigation and engineering non-destructive testing fields. At present, the Rayleigh wave of theoretical research focuses on the dispersion curves of forward and inversion terms. In order to have a more accurate and comprehensive understanding of the characteristics of Rayleigh wave propagation, some scholars have done a numerical significant research on simulation of Rayleigh wave.
Finite difference method based on elastic wave equation is a powerful tool for Rayleigh wave numerical simulation, but the phenomenon of the numerical dispersion appears when there are fewer samples in unit wavelength, which impairs the accuracy and resolution of wave-field simulation. To seek a fast and effective method for wave-field simulation, this dissertation has combined the flux-corrected transport method which is commonly used in fluid dynamics and the staggered-grid finite difference method. We have completed the simulation for the whole wave field, including Rayleigh surface wave and body wave by FCT finite difference method. The simulation results show that this method is effective to suppress the numerical frequency dispersion phenomenon by thick grids as well as to retain the real wave oscillation, especially to highlight the dispersion characteristics of Rayleigh surface wave in the layered mediums. The result points out that it improves the computing efficiency by 1.3 times compared with the thinning grids method with the same level of accuracy.
The dissertation firstly reviews the Rayleigh wave exploration, seismic wave simulation and FCT finite difference method of the status quo; and then introduces the basic theory of Rayleigh wave; thirdly, discusses the two-dimensional first-order stress-velocity elastic wave equation in staggered difference schemes, boundary conditions, source functions, differential format compatibility, and convergence and stability, and gives flow charts of the whole procedure; fourthly, describes the dispersion of wave phenomena, wave equation derived from dispersion relations, the distinction between dispersion phenomena in the physical dispersion and dispersion, and explains the causes of numerical dispersion and eliminating method; fifthly, describes the basic theory and the FCT method calculation steps; finally selects 11 examples to calculate and analyze, then gives conclusions on the study and directions for further work.
基于弹性波方程的有限差分法是瑞雷波数值模拟的有力工具,但当单位波长内采样点数较少时(粗网格)会产生数值频散现象,这种严重的干扰,降低了波场模拟的精度和分辨率。为寻求快速、有效的波场模拟方法,本文将常用于流体动力学中的通量校正传输技术(FCT)与交错网格有限差分法相结合,对浅层各向同性弹性介质进行了包括瑞雷波和体波在内的全波场模拟。模拟结果表明,该方法有效地压制了粗网格条件下的数值频散现象,并保留了真实的波场特征,特别是层状介质中瑞雷波的频散特征得以凸显。同时,在同等精度条件下,该方法与细化网格方法相比,计算效率提高了1.3倍。
本文首先回顾了瑞雷波勘探、地震波有限差分模拟和FCT方法的研究现状；介绍了瑞雷波的基本理论；讨论了二维一阶应力—速度弹性波动方程的交错网格差分格式、边界条件、震源函数、以及差分格式的相容性、收敛性和稳定性,并给出了程序流程图；介绍了波的频散现象,推导了波动方程频散关系,区分了频散现象中的物理频散和数值频散,并说明了数值频散产生的原因以及消除方法；阐述了FCT方法的基本理论及计算步骤；最后选取11个算例进行了计算分析,给出了结论及进一步工作的方向。
As a new geophysical prospecting method, Rayleigh wave has been developing rapidly in recent years, its economic, fast and efficient features are gradually recognized by engineers, and it is widely used in engineering geological investigation and engineering non-destructive testing fields. At present, the Rayleigh wave of theoretical research focuses on the dispersion curves of forward and inversion terms. In order to have a more accurate and comprehensive understanding of the characteristics of Rayleigh wave propagation, some scholars have done a numerical significant research on simulation of Rayleigh wave.
Finite difference method based on elastic wave equation is a powerful tool for Rayleigh wave numerical simulation, but the phenomenon of the numerical dispersion appears when there are fewer samples in unit wavelength, which impairs the accuracy and resolution of wave-field simulation. To seek a fast and effective method for wave-field simulation, this dissertation has combined the flux-corrected transport method which is commonly used in fluid dynamics and the staggered-grid finite difference method. We have completed the simulation for the whole wave field, including Rayleigh surface wave and body wave by FCT finite difference method. The simulation results show that this method is effective to suppress the numerical frequency dispersion phenomenon by thick grids as well as to retain the real wave oscillation, especially to highlight the dispersion characteristics of Rayleigh surface wave in the layered mediums. The result points out that it improves the computing efficiency by 1.3 times compared with the thinning grids method with the same level of accuracy.
The dissertation firstly reviews the Rayleigh wave exploration, seismic wave simulation and FCT finite difference method of the status quo; and then introduces the basic theory of Rayleigh wave; thirdly, discusses the two-dimensional first-order stress-velocity elastic wave equation in staggered difference schemes, boundary conditions, source functions, differential format compatibility, and convergence and stability, and gives flow charts of the whole procedure; fourthly, describes the dispersion of wave phenomena, wave equation derived from dispersion relations, the distinction between dispersion phenomena in the physical dispersion and dispersion, and explains the causes of numerical dispersion and eliminating method; fifthly, describes the basic theory and the FCT method calculation steps; finally selects 11 examples to calculate and analyze, then gives conclusions on the study and directions for further work.
引文
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