Modeling Rayleigh and Stoneley waves and other interface and boundary effects with the parabolic equation

Abstract

An improved approach for handling boundaries, interfaces, and continuous depth dependence with the elastic parabolic equation is derived and benchmarked. The approach is applied to model the propagation of Rayleigh and Stoneley waves. Depending on the choice of dependent variables, the operator in the elastic wave equation may not factor or the treatment of interfaces may be difficult. These problems are resolved by using a formulation in terms of the vertical displacement and the range derivative of the horizontal displacement. These quantities are continuous across horizontal interfaces, which permits the use of Galerkin's method to discretize in depth. This implementation extends the capability of the elastic parabolic equation to handle arbitrary depth dependence and should lead to improvements for range-dependent problems.

Elastic parabolic equation solutions for underwater acoustic problems using seismic sources

“…This approach is accurate and stable in part because interface conditions at the fluid-solid boundary are explicitly enforced and depth discretization is effectively handled using Galerkin's method. 15 For numerical implementations, Eq. (1) is written as @ @r…”

“…12 Current parabolic equation solutions for acoustic propagation in elastic sediments are based on the (u r , w) formulation of elasticity and are stable for a wide range of parameters. 15 Recently, this formulation has been used in rotated variable treatments for range-dependent underwater seismo-acoustic problems. 16 A single-scattering approximation in this formulation has been developed for purely elastic environments 17 and has recently been used to simulate propagation in complex multilayered range-dependent underwater acoustic environments, including beach and island topography.…”

A parabolic equation for under ice propagation

“…For Arctic environments, range dependence is considered in the form of an upward refracting sound speed profile, variable bathymetry, and variable ice thickness. The elastic parabolic equation is derived from the elastic equations of motion [3] in terms of the (u x , w) dependent variables, where u x is the horizontal derivative of the horizontal displacement and w is the vertical displacement [4,5]. In this form the equations are…”

Parabolic equation solution of seismo-acoustics problems involving variations in bathymetry and sediment thickness

“…In recent years, there has been a focus on improving accuracy for range-dependent seismo-acoustics problems. The introduction of the ͑u r , w͒ formulation, 6 where u r is the range derivative of the horizontal displacement and w is the vertical displacement, has led to progress in this area. An improved single-scattering solution in the ͑u r , w͒ formulation accurately handles range dependence within purely elastic media.…”