After the parabolic equation method was initially applied to scalar wave propagation problems in ocean acoustics and seismology, it took more than a decade before there was any substantial progress in extending this approach to problems involving solid layers. Some of the key steps in the development of the elastic parabolic equation are discussed here. The first breakthrough came in 1985 with the discovery that changing to an unconventional set of dependent variables makes it possible to factor the operator in the elastic wave equation into a product of outgoing and incoming operators. This innovation, which included an approach for handling fluid-solid interfaces, was utilized in the first successful implementations of the elastic parabolic equation less than five years later. A series of papers during that period addressed the issues of accuracy and stability, which require special attention relative to the scalar case. During the 1990s, the self-starter made it possible to handle all types of waves, rotated rational approximations of the operator square root made it possible to handle relatively thin solid layers, and there was some progress in the accurate treatment of sloping interfaces. During the next decade, an improved formulation and approach for handling interfaces facilitated the treatment of piecewise continuous depth dependence and sloping interfaces. During the last 10 years, the accuracy of the elastic parabolic equation was improved and tested for problems involving sloping interfaces and boundaries, and this approach was applied to Arctic acoustics and other problems involving thin layers. After decades of development, the elastic parabolic equation has become a useful tool for a wide range of problems in seismology, seismo-acoustics, and Arctic acoustics, but possible directions for further work are discussed.

中文翻译：

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地震学、地震声学和北极声学的抛物线方程技术

在抛物线方程方法最初应用于海洋声学和地震学中的标量波传播问题之后，在将这种方法扩展到涉及固体层的问题上，花了十多年的时间才取得实质性进展。这里讨论了弹性抛物线方程发展中的一些关键步骤。第一个突破出现在 1985 年，发现改变为一组非常规的因变量可以将弹性波方程中的算子分解为传出算子和传入算子的乘积。这项创新包括一种处理流固界面的方法，不到五年后首次成功实现了弹性抛物线方程。在此期间的一系列论文解决了准确性和稳定性问题，这需要特别注意标量情况。1990年代，自启动器使处理所有类型的波成为可能，算子平方根的旋转有理近似使得处理相对较薄的固体层成为可能，倾斜界面的精确处理也取得了一些进展。在接下来的十年中，处理界面的改进公式和方法促进了分段连续深度依赖性和倾斜界面的处理。在过去的 10 年里，弹性抛物方程的准确性得到了改进，并针对涉及倾斜界面和边界的问题进行了测试，这种方法被应用于北极声学和其他涉及薄层的问题。