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

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.

Large-Amplitude Elastic Free-Surface Waves: Geometric Nonlinearity and Peakons

“…In principle, the process for obtaining an exact solution to this linearized problem (2), with boundary conditions ( 12)-( 14) and subject to initial conditions of the sort described by ( 9), (10), should be reasonably straightforward. Similar to the method detailed in Walters et al [17], a Fourier Transform is taken in the spatial variable x, followed by a Laplace Transform in the time variable t. A fourth-order differential equation in the variable y is then solved, to give the complete solution in the Fourier-Laplace space. Such a process is used by Diaz & Ezziani [18], [19] in their investigations of a related elasticacoustic problem, for example.…”

“…From a physical point of view, this well-known difficulty with the linear two-dimensional elasticity equations is due in part to the fact that they contain two different wave speeds, corresponding to S and P waves, as discussed by Ockendon & Ockendon [16], and that energy can transfer back and forth between these shear and compressive transmission modes. When no free surface is present and the waves can simply move through an unbounded medium, closed-form mathematical solutions are occasionally possible, and some of these have been presented by Walters et al [17]. The analysis is very greatly complicated by the presence of a free surface, however, and can involve sophisticated manipulation of integrals in the complex Laplace-Transform space, containing awkward branch lines.…”

Time‐domain scaled boundary perfectly matched layer for elastic wave propagation

“…[66][67][68] Spatial discretization categories also differ, ranging from the temporally first-order system of velocity-stress 60,69 to the temporally second-order systems of displacement with auxiliary variables 63 and mixed displacement-stress. 70,71 In addition, PML formulations differ based on whether they are implemented in the final time-domain 15,72 or frequency-domain, 73 and whether they are based on finite difference methods, 53,74 finite element methods, [74][75][76][77] or spectral element methods. 52,78,79 Despite the widespread adoption of various PML formulations in solving the elastic wave problem in unbounded domains, there are still limitations.…”