01_introduction

Table of Contents

• Introduction
  • Citation
  • Support
  • References

Introduction

SPECFEM2D allows users to perform 2D and 2.5D (i.e., axisymmetric) simulations of acoustic, elastic, viscoelastic, and poroelastic seismic wave propagation as well as full waveform imaging (FWI) or adjoint tomography.

In fluids, SPECFEM2D uses the classical linearized Euler equation; thus if you have sharp local variations of density in the fluid (highly heterogeneous fluids in terms of density) or if density becomes extremely small in some regions of your model (e.g. for upper-atmosphere studies), before using the code please make sure the linearized Euler equation is a valid approximation in the case you want to study. For more details on that see e.g. (Jensen et al. 2011).

The 2D spectral-element solver accommodates regular and unstructured meshes, generated for example by Cubit, Gmsh or GiD. Even mesh creation packages that generate triangles, for instance Delaunay-Voronoi triangulation codes, can be used because each triangle can then easily be decomposed into three quadrangles by linking the barycenter to the center of each edge; while this approach does not generate quadrangles of optimal quality, it can ease mesh creation in some situations and it has been shown that the spectral-element method can very accurately handle distorted mesh elements.

Furthermore, the 2D spectral-element solver accommodates Convolution PML absorbing layers as well as higher-order time schemes (4th order Runge-Kutta and LDDRK4-6). Convolution or Auxiliary Differential Equation Perfectly Matched absorbing Layers (C-PML or ADE-PML) are described in (Martin, Komatitsch, and Ezziani 2008; Martin, Komatitsch, and Gedney 2008; Martin and Komatitsch 2009; Martin et al. 2010; Dimitri Komatitsch and Martin 2007).

The solver has adjoint capabilities and can calculate finite-frequency sensitivity kernels (Tromp, Komatitsch, and Liu 2008; Peter et al. 2011) for acoustic, (an)elastic, and poroelastic media. The package also considers 2D SH and P-SV wave propagation. Finally, the solver can run both in serial and in parallel. See the github repository SPECFEM2D for the source code.

The SEM is a continuous Galerkin technique (Tromp, Komatitsch, and Liu 2008; Peter et al. 2011), which can easily be made discontinuous (Bernardi, Maday, and Patera 1994; E. Chaljub 2000; Kopriva, Woodruff, and Hussaini 2002; E. Chaljub, Capdeville, and Vilotte 2003; Legay, Wang, and Belytschko 2005; Kopriva 2006; Wilcox et al. 2010; Acosta Minolia and Kopriva 2011); it is then close to a particular case of the discontinuous Galerkin technique (Reed and Hill 1973; Lesaint and Raviart 1974; Arnold 1982; Johnson and Pitkäranta 1986; Bourdel, Mazet, and Helluy 1991; Falk and Richter 1999; Hu, Hussaini, and Rasetarinera 1999; Cockburn, Karniadakis, and Shu 2000; Giraldo, Hesthaven, and Warburton 2002; Rivière and Wheeler 2003; Monk and Richter 2005; Grote, Schneebeli, and Schötzau 2006; Ainsworth, Monk, and Muniz 2006; Bernacki, Lanteri, and Piperno 2006; Dumbser and Käser 2006; J. D. De Basabe, Sen, and Wheeler 2008; Puente, Ampuero, and Käser 2009; Wilcox et al. 2010; Jonás D. De Basabe and Sen 2010; Étienne et al. 2010), with optimized efficiency because of its tensorized basis functions (Wilcox et al. 2010; Acosta Minolia and Kopriva 2011). In particular, it can accurately handle very distorted mesh elements (Oliveira and Seriani 2011).

It has very good accuracy and convergence properties (Maday and Patera 1989; G. Seriani and Priolo 1994; Deville, Fischer, and Mund 2002; Gary Cohen 2002; Jonás D. De Basabe and Sen 2007; G. Seriani and Oliveira 2008; Ainsworth and Wajid 2009, 2010; Melvin, Staniforth, and Thuburn 2012). The spectral element approach admits spectral rates of convergence and allows exploiting $hp$-convergence schemes. It is also very well suited to parallel implementation on very large supercomputers (Dimitri Komatitsch et al. 2003; Tsuboi et al. 2003; Dimitri Komatitsch, Labarta, and Michéa 2008; Carrington et al. 2008; D. Komatitsch, Vinnik, and Chevrot 2010) as well as on clusters of GPU accelerating graphics cards (Dimitri Komatitsch 2011; Michéa and Komatitsch 2010; Dimitri Komatitsch, Michéa, and Erlebacher 2009; Dimitri Komatitsch et al. 2010). Tensor products inside each element can be optimized to reach very high efficiency (Deville, Fischer, and Mund 2002), and mesh point and element numbering can be optimized to reduce processor cache misses and improve cache reuse (Dimitri Komatitsch, Labarta, and Michéa 2008). The SEM can also handle triangular (in 2D) or tetrahedral (in 3D) elements (Wingate and Boyd 1996; Taylor and Wingate 2000; D. Komatitsch et al. 2001; Gary Cohen 2002; Mercerat, Vilotte, and Sánchez-Sesma 2006) as well as mixed meshes, although with increased cost and reduced accuracy in these elements, as in the discontinuous Galerkin method.

Note that in many geological models in the context of seismic wave propagation studies (except for instance for fault dynamic rupture studies, in which very high frequencies or supershear rupture need to be modeled near the fault, see e.g. (Benjemaa et al. 2007, 2009; Puente, Ampuero, and Käser 2009; Tago et al. 2010)) a continuous formulation is sufficient because material property contrasts are not drastic and thus conforming mesh doubling bricks can efficiently handle mesh size variations (D. Komatitsch and Tromp 2002; Dimitri Komatitsch et al. 2004; Lee et al. 2008; Lee, Chan, et al. 2009; Lee, Komatitsch, et al. 2009).

For a detailed introduction to the SEM as applied to regional seismic wave propagation, please consult Peter et al. (2011; Tromp, Komatitsch, and Liu 2008; D. Komatitsch and Vilotte 1998; D. Komatitsch and Tromp 1999; Emmanuel Chaljub et al. 2007) and in particular Lee, Komatitsch, et al. (2009; Lee, Chan, et al. 2009; Lee et al. 2008; Godinho et al. 2009; Wijk et al. 2004; Dimitri Komatitsch et al. 2004). A detailed theoretical analysis of the dispersion and stability properties of the SEM is available in Gary Cohen (2002), Jonás D. De Basabe and Sen (2007), Géza Seriani and Oliveira (2007), G. Seriani and Oliveira (2008) and Melvin, Staniforth, and Thuburn (2012).

The SEM was originally developed in computational fluid dynamics (Patera 1984; Maday and Patera 1989) and has been successfully adapted to address problems in seismic wave propagation. Early seismic wave propagation applications of the SEM, utilizing Legendre basis functions and a perfectly diagonal mass matrix, include (G. Cohen, Joly, and Tordjman 1993), (Dimitri Komatitsch 1997), (Faccioli et al. 1997), (Casadei and Gabellini 1997), (D. Komatitsch and Vilotte 1998) and (D. Komatitsch and Tromp 1999), whereas applications involving Chebyshev basis functions and a non-diagonal mass matrix include (G. Seriani and Priolo 1994), (Priolo, Carcione, and Seriani 1994) and (G. Seriani, Priolo, and Pregarz 1995). In the Legendre version that we use in SPECFEM the mass matrix is purposely slightly inexact but diagonal (but can be made exact if needed, see (Teukolsky 2015)), while in the Chebyshev version it is exact but non diagonal.

Beware that, in a spectral-element method, some spurious modes (that have some similarities with classical so-called "Hourglass modes" in finite-element techniques, although in the SEM they are not zero-energy modes) can appear in some (but not all) cases in the spectral element in which the source is located. Fortunately, they do not propagate away from the source element. However, this means that if you put a receiver in the same spectral element as a source, the recorded signals may in some cases be wrong, typically exhibiting some spurious oscillations, which are often even non causal. If that is the case, an easy option is to slightly change the mesh in the source region in order to get rid of these Hourglass-like spurious modes, as explained in (Duczek et al. 2014), in which this phenomenon is described in details, and in which practical solutions to avoid it are suggested.

All SPECFEM2D software is written in Fortran2003 with full portability in mind, and conforms strictly to the Fortran2003 standard. It uses no obsolete or obsolescent features of Fortran. The package uses parallel programming based upon the Message Passing Interface (MPI) (Gropp, Lusk, and Skjellum 1994; Pacheco 1997).

This new release of the code includes support for GPU graphics card acceleration (Dimitri Komatitsch 2011; Michéa and Komatitsch 2010; Dimitri Komatitsch, Michéa, and Erlebacher 2009; Dimitri Komatitsch et al. 2010).

The code uses the plane strain convention when the standard P-SV equation case is used, i.e., the off-plane strain $\epsilon_{zz}$ is zero by definition of the plane strain convention but the off-plane stress $\sigma_{zz}$ is not equal to zero, one has $\sigma_{zz} = \lambda (\epsilon_{xx} + \epsilon_{yy})$. This implies, as in any plain strain software package, that the P-SV source is a line source along the direction perpendicular to the plane (see file discussion_of_2D_sources_and_approximations_from_Pilant_1979.pdf for more details).

Citation

You can find all the references below in format in file doc/USER_MANUAL/bibliography.bib.

If you use this code for your own research, please cite at least one article written by the developers of the package, for instance:

• (Tromp, Komatitsch, and Liu 2008),

• (Peter et al. 2011),

• (Vai et al. 1999),

• (Lee, Chan, et al. 2009),

• (Lee et al. 2008),

• (Lee, Komatitsch, et al. 2009),

• (Dimitri Komatitsch et al. 2010),

• (Dimitri Komatitsch, Michéa, and Erlebacher 2009),

• (Liu et al. 2004),

• (Emmanuel Chaljub et al. 2007),

• (D. Komatitsch and Vilotte 1998),

• (D. Komatitsch and Tromp 1999),

• (Dimitri Komatitsch et al. 2004),

• (Morency and Tromp 2008),

• (Blanc et al. 2016),

• and/or other articles from https://specfem.github.io/komatitsch.free.fr/publications.html.

If you use the C-PML absorbing layer capabilities of the code, please cite at least one article written by the developers of the package, for instance:

• (Xie et al. 2014),

• (Xie et al. 2016).

If you use the UNDO_ATTENUATION option of the code in order to produce full anelastic/viscoelastic sensitivity kernels, please cite at least one article written by the developers of the package, for instance (and in particular):

• (Dimitri Komatitsch et al. 2016).

More generally, if you use the attenuation (anelastic/viscoelastic) capabilities of the code, please cite at least one article written by the developers of the package, for instance:

• (Dimitri Komatitsch et al. 2016),

• (Blanc et al. 2016).

If you use the kernel capabilities of the code, please cite at least one article written by the developers of the package, for instance:

• (Tromp, Komatitsch, and Liu 2008),

• (Peter et al. 2011),

• (Liu and Tromp 2006),

• (Morency, Luo, and Tromp 2009).

If you use the SCOTCH / CUBIT non-structured capabilities, please cite:

• (Martin et al. 2008).

If you use axisymmetric geometries please also cite:

• (Bottero et al. 2016)

The corresponding BibTeX entries may be found in file doc/USER_MANUAL/bibliography.bib.

Support

This material is based upon work supported by the USA National Science Foundation under Grants No. EAR-0406751 and EAR-0711177, by the French CNRS, French Inria Sud-Ouest MAGIQUE-3D, French ANR NUMASIS under Grant No. ANR-05-CIGC-002, and European FP6 Marie Curie International Reintegration Grant No. MIRG-CT-2005-017461. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the USA National Science Foundation, CNRS, Inria, ANR or the European Marie Curie program.

References

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———. 2011. “Fluid-Solid Coupling on a Cluster of GPU Graphics Cards for Seismic Wave Propagation.” C. R. Acad. Sci., Ser. IIb Mec. 339: 125–35. https://doi.org/10.1016/j.crme.2010.11.007.

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Komatitsch, Dimitri, Jesús Labarta, and David Michéa. 2008. “A Simulation of Seismic Wave Propagation at High Resolution in the Inner Core of the Earth on 2166 Processors of MareNostrum.” Lecture Notes in Computer Science 5336: 364–77.

Komatitsch, Dimitri, Qinya Liu, Jeroen Tromp, Peter Süss, Christiane Stidham, and John H. Shaw. 2004. “Simulations of Ground Motion in the Los Angeles Basin Based Upon the Spectral-Element Method.” Bull. Seism. Soc. Am. 94 (1): 187–206. https://doi.org/10.1785/0120030077.

Komatitsch, Dimitri, and Roland Martin. 2007. “An Unsplit Convolutional Perfectly Matched Layer Improved at Grazing Incidence for the Seismic Wave Equation.” Geophysics 72 (5): SM155–67. https://doi.org/10.1190/1.2757586.

Komatitsch, Dimitri, David Michéa, and Gordon Erlebacher. 2009. “Porting a High-Order Finite-Element Earthquake Modeling Application to NVIDIA Graphics Cards Using CUDA.” Journal of Parallel and Distributed Computing 69 (5): 451–60. https://doi.org/10.1016/j.jpdc.2009.01.006.

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———. 2002. “Spectral-Element Simulations of Global Seismic Wave Propagation-I. Validation.” Geophys. J. Int. 149 (2): 390–412. https://doi.org/10.1046/j.1365-246X.2002.01653.x.

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Lee, Shiann Jong, Yu Chang Chan, Dimitri Komatitsch, Bor Shouh Huang, and Jeroen Tromp. 2009. “Effects of Realistic Surface Topography on Seismic Ground Motion in the Yangminshan Region of Taiwan Based Upon the Spectral-Element Method and LiDAR DTM.” Bull. Seism. Soc. Am. 99 (2A): 681–93. https://doi.org/10.1785/0120080264.

Lee, Shiann Jong, How Wei Chen, Qinya Liu, Dimitri Komatitsch, Bor Shouh Huang, and Jeroen Tromp. 2008. “Three-Dimensional Simulations of Seismic Wave Propagation in the Taipei Basin with Realistic Topography Based Upon the Spectral-Element Method.” Bull. Seism. Soc. Am. 98 (1): 253–64. https://doi.org/10.1785/0120070033.

Lee, Shiann Jong, Dimitri Komatitsch, Bor Shouh Huang, and Jeroen Tromp. 2009. “Effects of Topography on Seismic Wave Propagation: An Example from Northern Taiwan.” Bull. Seism. Soc. Am. 99 (1): 314–25. https://doi.org/10.1785/0120080020.

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Liu, Qinya, Jascha Polet, Dimitri Komatitsch, and Jeroen Tromp. 2004. “Spectral-Element Moment Tensor Inversions for Earthquakes in Southern California.” Bull. Seism. Soc. Am. 94 (5): 1748–61. https://doi.org/10.1785/012004038.

Liu, Qinya, and Jeroen Tromp. 2006. “Finite-Frequency Kernels Based on Adjoint Methods.” Bull. Seism. Soc. Am. 96 (6): 2383–97. https://doi.org/10.1785/0120060041.

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Martin, Roland, Dimitri Komatitsch, Céline Blitz, and Nicolas Le Goff. 2008. “Simulation of Seismic Wave Propagation in an Asteroid Based Upon an Unstructured MPI Spectral-Element Method: Blocking and Non-Blocking Communication Strategies.” Lecture Notes in Computer Science 5336: 350–63.

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