A scheme to treat the singularity in global seismic wavefield simulation using pseudospectral method with staggered grids

Abstract

The pseudospectral method has been applied to the simulation of seismic wave propagation in 2-D global Earth model. When a whole Earth model is considered, the center of the Earth is included in the model and then singularity arises at the center of the Earth where r = 0 since the 1/r term appears in the wave equations. In this paper, we extended the global seismic wavefield simulation algorithm for regular grid mesh to staggered grid configuration and developed a scheme to solve the numerical problems associated with the above singularity for a 2-D global Earth model defined on staggered grid using pseudospectral method. This scheme uses a coordinate transformation at the center of the model, in which the field variables at the center are calculated in Cartesian coordinates from the values on the grids around the center. It allows wave propagation through the center and hence the wavefield at the center can be stably calculated. Validity and accuracy of the scheme was tested by compared with the discrete wavenumber method. This scheme could also be suitable for other numerical methods or models parameterized in cylindrical or spherical coordinates when singularity arises at the center of the model.

Quasi-Axisymmetric Finite-Difference Method for Realistic Modeling of Regional and Global Seismic Wavefield — Review and Application —

“…The extremely small lateral grid spacings near the center perturb the FDM stability criterion, and (2) the singularity of the elastodynamic equation at the center r = 0. To solve the first problem, Toyokuni & Takenaka (2011) applied the so-called multi-domain technique (e.g., Aoi & Fujiwara, 1999;Thomas et al, 2000;Wang & Takenaka, 2001;2010), in which several domains consisting of FD grids with different lateral grid spacings are connected in the r direction, with coarser lateral grids around the center. The second problem was solved using linear interpolation of wavefield variables in the r direction, giving values of particle velocity and stress at the center.…”

Global SH-wavefield calculation for a two-dimensional whole-Earth model with the parallel hybrid PSM/FDM algorithm

“…Furumura et al (1998) applied PSM to simulate global P-SV wave propagation for the 2-D whole-Earth model defined in cylindrical coordinates. Wang et al ( , 2010 extended this method to a whole 2-D Earth model including the center by solving numerical problems associated with the singularity at the center of the Earth. Wang and Takenaka (2011) presented a PSM scheme to simulate SH-wave propagation in a 2-D whole-Earth model with arbitrary lateral heterogeneities.…”

Three dimensional finite-differential simulation of the wave field taking sphericity of the Earth into account

“…Все численные методы моделирования распространения сейсмических волн в сферической Земле относящиеся к доменным по подходу можно условно разделить на два основных типа: осесимметричные и псевдоспектральные Wang, Takenaka, 2010]. Осесимметричные методы моделирования трехмерных сейсмических волновых полей основаны на аппроксимации структурной модели как вращательно-симметричной вдоль вертикальной оси источника колебаний и решении уравнения эластодинамики в цилиндрических или сферических координатах.…”