Integral transformations, especially the inverse Laplace transform, are powerful techniques for resolving a wide range of geophysical and geodynamic simulation problems in viscoelastic materials. The exact location or distribution range of poles of the image function in a complex plane is usually necessary for applying numerical algorithms such as contour integration. Unfortunately, there are innumerable poles (such as those of post-seismic deformations) in a realistic Earth model with continuous stratification, finite compressibility, and self-gravitation. Here, an optimized method to effectively calculate the inverse Laplace transform is presented. First, the integral kernel is approximated as a rational function with two parameters (a and m). Thereafter, the residue theorem is analytically applied to the approximated integrand. Finally, a series formula of the inverse Laplace transform sampling of image functions along a contour line parallel to the image axis is obtained. The proposed approximate scheme of the inverse Laplace transform is discussed by some common geophysical signals and the optimized selection of two parameters (a = 6 and m = 4) is conducted after a detailed analysis. The proposed method is anticipated as being able to help performing certain theoretical studies related to geodynamic problems with viscoelastic deformations.

中文翻译：

---

优化的近似拉普拉斯逆变换，用于粘弹性地球模型中的地球变形计算

积分变换，尤其是拉普拉斯逆变换，是解决粘弹性材料中广泛的地球物理和地球动力学模拟问题的强大技术。图像函数的极点在复杂平面中的精确位置或分布范围通常对于应用数字算法（例如轮廓积分）是必需的。不幸的是，在现实的地球模型中，存在无数个极点（例如地震后变形的极点），具有连续的分层，有限的可压缩性和自重。这里，提出了一种有效计算拉普拉斯逆变换的优化方法。首先，积分核近似为具有两个参数（a和m）的有理函数。此后，将残差定理解析地应用于近似积分。最后，获得沿平行于图像轴的轮廓线的图像函数的逆拉普拉斯变换采样的序列公式。通过一些常见的地球物理信号讨论了提出的拉普拉斯逆变换的近似方案，并在详细分析后对两个参数（a = 6和m = 4）进行了优化选择。预期所提出的方法能够帮助进行与粘弹性变形的地球动力学问题有关的某些理论研究。通过一些常见的地球物理信号讨论了提出的拉普拉斯逆变换的近似方案，并在详细分析后对两个参数（a = 6和m = 4）进行了优化选择。预期所提出的方法能够帮助进行与具有粘弹性变形的地球动力学问题有关的某些理论研究。通过一些常见的地球物理信号讨论了提出的拉普拉斯逆变换的近似方案，并在详细分析后对两个参数（a = 6和m = 4）进行了优化选择。预期所提出的方法能够帮助进行与具有粘弹性变形的地球动力学问题有关的某些理论研究。