ABSTRACT We propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method.

中文翻译：

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使用李积公式和泰勒展开的一阶线性声/弹性波方程的时间演化

摘要 我们对一阶线性声波/弹性波方程提出了一种新的数值解。该数值解基于线性声波/弹性波方程的解析解并使用李乘积公式，其中解析解的时间演化算子写为指数矩阵的乘积，其中每个指数矩阵项近似为泰勒级数展开。最初，我们以数值方式检查所提出的方法，然后证明对指数函数恒等式应用泰勒展开比对指数函数本身应用更准确。公式化的数值解采用递归程序，并且还结合了分裂完美匹配层边界条件。因此，我们的方案可用于以比传统有限差分方案更大的时间步长以稳定的方式外推波场。通过对矩阵指数项使用 Chebyshev 展开方法的全声波方程的解进行比较，来检验这个新的数值解。此外，为了证明我们提出的解决方案的效率和适用性，给出了三个地质模型的地震建模结果，并将每个模型的处理时间与切比雪夫展开法的计算时间进行了比较。我们还展示了使用基于李乘积公式和泰勒级数展开的方案对垂直横向各向同性和倾斜横向各向同性介质中的一阶线性弹性波动方程进行地震建模的结果。最后，