Spectral element modelling of three-dimensional wave propagation in a self-gravitating Earth with an arbitrarily stratified outer core

Abstract

S U M M A R YThis paper deals with the spectral element modelling of seismic wave propagation on a global scale. Two aspects relevant to low-frequency studies are particularly emphasized. First, the method is generalized beyond the Cowling approximation in order to fully account for the effects of self-gravitation. In particular, the perturbation of the gravity field outside the Earth is handled by a projection of the spectral element solution onto the basis of spherical harmonics. Second, we propose a new formulation inside the fluid which allows us to account for an arbitrary density stratification. It is based upon a decomposition of the displacement into two scalar potentials, and results in a fully explicit fluid-solid coupling strategy. The implementation of the method is carefully detailed and its accuracy is demonstrated through a series of benchmark tests.It has been recently established by several authors (Chaljub 2000; Komatitsch & Tromp 2002a,b;Chaljub et al. 2003) that the spectral element method (SEM) provides an efficient solution to the issue of computing synthetic seismograms in three-dimensional (3-D) models of the Earth. Whereas most of current spectral element studies aim to push calculations toward high frequencies, where the methods traditionally used on a global scale reach their limits, this paper focuses on some physical effects that are critical for the lower part of the seismic frequency band: (i) the full treatment of self-gravitation and (ii) the ability to take into account any density stratification in the fluid regions of the Earth.The first novelty of this paper stands in the incorporation of self-gravitation, the effect of which is important for seismic and gravimetric observations with periods larger than 100 s. All the previously mentioned studies based upon the SEM accounted for the effects of gravity within the Cowling approximation (Cowling 1941), i.e. by neglecting the perturbation of the gravity field by seismic waves. The main reason for making this assumption lies in the intrinsic difficulty of the problem. Considering the full effects of self-gravitation requires solving Poisson's equation for the perturbed gravitational potential which is defined over the whole space. Unlike spherical harmonics approaches, the use of a grid-based method such as the SEM does not provide a natural framework for the resolution of the exterior problem. Grid-based approximations in unbounded domains proceed first by restricting the computational domain, then by imposing an appropriate condition on the truncating boundary. Different methods arise depending on whether the artificial boundary condition (ABC) is local or not. Methods based upon a local ABC have the advantage of being computationally inexpensive and valid for arbitrary geometries. An example is the infinite-element method (e.g. Bettess 1992; Gerdes & Demkowicz 1996), in which the behaviour of the exterior solution is enforced in the radial direction. Methods of the second class, based upon a non-local ABC, are not as general sin...

Spectral-Element Simulations of Seismic Waves Generated by the 2009 L'Aquila Earthquake

“…If need be, the SEM can be implemented as a discontinuous Galerkin method (Kopriva et al, 2002;Kopriva, 2006;Acosta Minolia and Kopriva, 2011). The SEM has been used to study 2D and 3D local, regional, and global seismic-wave propagation problems (e.g., Cohen et al, 1993;Priolo et al, 1994;Faccioli et al, 1997;Komatitsch, 1997;Komatitsch and Vilotte, 1998;Komatitsch and Tromp, 2002a,b;Chaljub and Valette, 2004;Komatitsch et al, 2004Komatitsch et al, , 2005Liu et al, 2004;Stich and Morelli, 2007;Lee et al, 2008;Stich et al, 2009;Stupazzini et al, 2009;Tape et al, 2010;Zhu et al, 2012a,b). In particular, in the recent work of Smerzini and Villani (2012), the SEM implemented in the code GeoELSE (Stupazzini et al, 2009) has been used to simulate near-fault ground motions produced by the 2009 L'Aquila mainshock.…”

The spectral-element method in seismology

“…is the Brunt-Väisälä frequency [e.g., Valette, 1986;Dahlen and Tromp, 1998;Chaljub et al, 2003;Chaljub and Valette, 2004]. In previous work [Komatitsch and Tromp, 2002b] we made the assumption that the outer core was stably stratified and isentropic, which meant that N 2 = 0, and that perturbations in gravity could be ignored, i.e., = 0, based upon the Cowling approximation.…”

“…Such effects have been included in the SEM and lead to additional terms in the weak formulation (19). These terms were introduced in Chaljub [2000], Komatitsch and Tromp [2002b], Chaljub et al [2003] and Chaljub and Valette [2004], and the corresponding effects on seismic waves were carefully benchmarked. We summarize these results in the following sections for completeness.…”

“…All of these effects have been benchmarked in previous publications [Komatitsch and Vilotte, 1998;Komatitsch and Tromp, 1999;Komatitsch et al, 2000a, b;Komatitsch and Tromp, 2002a, b]. In this article, we use a simpler fluid-solid coupling method that does not require numerical iterations, based on the recent work of Chaljub and Valette [2004]. We illustrate how the method can be applied to high-resolution simulations of seismic wave propagation in the 3-D Earth on a very large parallel computer: the Earth Simulator at JAMSTEC in Japan.…”

“…The SEM has been used for two decades in computational fluid dynamics [Patera, 1984]. It has more recently been applied to problems related to two-dimensional (2-D) [Cohen et al, 1993;Priolo et al, 1994] and 3-D local or regional [Komatitsch, 1997;Faccioli et al, 1997;Komatitsch and Vilotte, 1998;Seriani, 1998;Komatitsch and Tromp, 1999;Komatitsch et al, 2004;Liu et al, 2004] and global [Chaljub, 2000;Komatitsch and Tromp, 2002a, b;Komatitsch et al, 2002;Chaljub et al, 2003;Chaljub and Valette, 2004] seismic wave propagation. We introduce the full complexity of the 3-D Earth, i.e., lateral variations in compressionalwave speed, shear-wave speed, and density in the mantle, a 3-D crustal model, anisotropy, ellipticity, surface topography and bathymetry, as well as the effects of the oceans, rotation, and self-gravitation.…”