Analytic and numerical solutions to the seismic wave equation in continuous media

TL;DR

This paper develops analytical and numerical models for seismic wave propagation in continuous media, validating the numerical scheme with analytical solutions and demonstrating its efficiency for seismic forward modelling.

Contribution

It introduces a new analytical solution and a spectral numerical scheme with absorbing boundaries for seismic wave simulation in isotropic media.

Findings

Analytical model provides a validation benchmark for numerical methods.

Numerical scheme efficiently solves large systems on desktop hardware.

Visual comparison confirms accuracy of the numerical approach.

Abstract

This paper presents two approaches to mathematical modelling of a synthetic seismic pulse, and a comparison between them. First, a new analytical model is developed in two-dimensional Cartesian coordinates. Combined with an initial condition of sufficient symmetry, this provides a valuable check for the validity of the numerical method that follows. A particular initial condition is found which allows for a new closed-form solution. A numerical scheme is then presented which combines a spectral (Fourier) representation for displacement components and wave-speed parameters, a fourth order Runge-Kutta integration method, and an absorbing boundary layer. The resulting large system of differential equations is solved in parallel on suitable enhanced performance desktop hardware in a new software implementation. This provides an alternative approach to forward modelling of waves within isotropic media which is efficient, and tailored to rapid and flexible developments in modelling seismic structure, for example, shallow depth environmental applications. Visual comparisons of the analytic solution and the numerical scheme are presented.