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.
The parabolic equation method can be used to model acoustic-wave propagation in elastic media. Current implementations do not accurately match the shear stress between two elastic layers, for which the expression involves a second-order depth derivative. This inaccuracy could be large in range-dependent problems. A reformulation of the elastic equations in terms of new variables permits all interface conditions to be handled accurately since no expressions contain second- or higher-order depth derivatives. However, the implementation in the new variables is found to be unstable because some evanescent modes are allowed to grow. Higher-order evanescent modes can be suppressed using a rational approximation to the depth operator with a polynomial in the denominator having a higher degree than the numerator. The lower-order evanescent modes can be controlled by imposing constraints on the choice of coefficients in the approximation. Examples are presented to show the accuracy and stability of the new parabolic equation in shallow-water environments. [Work supported by ONR.]
The parabolic equation method is used to model pulse propagation in an ocean overlying a lossy and dispersive elastic sediment. As for a fluid sediment [Kulkarni et al., J. Acoust. Soc. Am. 104, 1356–1362], loss and dispersion are included in the frequency domain propagation, and Fourier synthesis is used to recover the broadband signal. Loss is known to depend linearly on frequency in many materials. This is evidently because there are no relevant attenuation length scales in such materials. Lossy materials must also be dispersive in order to keep the waves causal, and the dispersive correction is a Hilbert transform of the attenuation as a consequence of the Kramers–Krönig relationship. In particular, the wave speed depends logarithmically on frequency when loss depends linearly on frequency. Examples are presented to illustrate the effects of dispersion on signals in shallow water environments. [Work supported by ONR.]
Parabolic equation techniques for solving seismic wave propagation problems are discussed. These techniques provide an excellent combination of accuracy and efficiency for problems involving laterally varying media. Due to recent improvements, the parabolic equation method is applicable to problems involving wide propagation angles, large variations in the properties of the medium, and all types of waves.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.