Elastic wave modelling in the Laplace domain is the foundation of Laplace-domain elastic wave full waveform inversion. In order to use Laplace-domain numerical modelling schemes efficiently, appropriate grid intervals must be chosen correctly. The determination of grid intervals is based on numerical dispersion analysis of Laplace-domain elastic wave equation. A new method of numerical dispersion analysis is developed for Laplace-domain 2-D elastic wave equation. The novelty of the method lies in two aspects: (1) Based on the concept of the pseudo-wavelength for both P- and S-waves, I introduce a general quantity which is a combination of the Laplace constant, velocity and grid interval. Therefore, the dispersion relations no longer directly depend on the concrete Laplace constant, velocity and grid interval. Just like the frequency-domain dispersion analysis, a general conclusion with regard to the general quantity can be made; (2) The ratio of numerical eigenvalue to analytical eigenvalue can be interpreted as the normalised attenuation propagation velocity. The number of grid points per S-wave pseudo-wavelength is determined by the accuracy of normalised P- and S-wave attenuation propagation velocities. Based on a commonly used finite-element scheme, the numbers of grid points per S-wave pseudo-wavelength are determined for different Poisson ratios and for different ratios of directional grid intervals. These results are important in applying the finite-element scheme to Laplace-domain elastic wave modelling and full waveform inversion. Comparisons with the analytical solution validate the criterion on the numbers of grid points per S-wave pseudo-wavelength.

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拉普拉斯域二维弹性波方程数值色散分析的一种新方法

拉普拉斯域弹性波建模是拉普拉斯域弹性波全波形反演的基础。为了有效地使用拉普拉斯域数值建模方案，必须正确选择适当的网格间隔。网格间隔的确定基于拉普拉斯域弹性波动方程的数值色散分析。为拉普拉斯域二维弹性波方程开发了一种新的数值色散分析方法。该方法的新颖之处在于两个方面： (1)基于P波和S波的伪波长概念，引入了拉普拉斯常数、速度和网格间隔的组合的一般量。因此，色散关系不再直接取决于具体的拉普拉斯常数、速度和网格间隔。就像频域色散分析一样，可以对一般量做出一般性结论；(2) 数值特征值与解析特征值的比值可以解释为归一化衰减传播速度。每个 S 波伪波长的网格点数由归一化 P 波和 S 波衰减传播速度的精度决定。基于常用的有限元方案，对于不同的泊松比和不同的定向网格间隔比率，确定每个 S 波伪波长的网格点数。这些结果对于将有限元方案应用于拉普拉斯域弹性波建模和全波形反演非常重要。