Narrow Results

Several methods for handling sloping fluid–solid interfaces with the elastic parabolic equation are tested. A single-scattering approach that is modified for the fluid–solid case is accurate for some problems but breaks down when the contrast across the interface is sufficiently large and when there is a Scholte wave. An approximate condition for conserving energy breaks down when a Scholte wave propagates along a sloping interface but otherwise performs well for a large class of problems involving gradual slopes, a wide range of sediment parameters, and ice cover. An approach based on treating part of the fluid layer as a solid with low shear speed is developed and found to handle Scholte waves and a wide range of sediment parameters accurately, but this approach needs further development. The variable rotated parabolic equation is not effective for problems involving frequent or continuous changes in slope, but it provides a high level of accuracy for most of the test cases, which have regions of constant slope. Approaches based on a coordinate mapping and on using a film of solid material with low shear speed on the rises of the stair steps that approximate a sloping interface are also tested and found to produce accurate results for some cases.

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.