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Introduction

The software package SPECFEM3D Cartesian simulates seismic wave propagation at the local or regional scale and performs full waveform imaging (FWI) or adjoint tomography based upon the spectral-element method (SEM). The SEM is a continuous Galerkin technique (Jeroen 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).

In fluids, when gravity is turned off, SPECFEM3D 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, and/or see if you should turn gravity on. For more details on that see e.g. (Jensen et al. 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 2002a; 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; Jeroen 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).

Effects due to lateral variations in compressional-wave speed, shear-wave speed, density, a 3D crustal model, topography and bathymetry are included. The package can accommodate full 21-parameter anisotropy (see Chen and Tromp (2007)) as well as lateral variations in attenuation (Savage, Komatitsch, and Tromp 2010). Adjoint capabilities and finite-frequency kernel simulations are included (Jeroen Tromp, Komatitsch, and Liu 2008; Peter et al. 2011; Qinya Liu and Tromp 2006; Fichtner et al. 2009; Virieux and Operto 2009).

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 SPECFEM3D 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).

SPECFEM3D won the Gordon Bell award for best performance at the SuperComputing 2003 conference in Phoenix, Arizona (USA) (see Dimitri Komatitsch et al. (2003) and https://dl.acm.org/doi/10.1145/1048935.1050155). It was a finalist again in 2008 for a run at 0.16 petaflops (sustained) on 149,784 processors of the ‘Jaguar’ Cray XT5 system at Oak Ridge National Laboratories (USA) (Carrington et al. 2008). It also won the BULL Joseph Fourier supercomputing award in 2010.

It reached the sustained one petaflop performance level for the first time in February 2013 on the Blue Waters Cray supercomputer at the National Center for Supercomputing Applications (NCSA), located at the University of Illinois at Urbana-Champaign (USA).

This new release of the code includes Convolution or Auxiliary Differential Equation Perfectly Matched absorbing Layers (C-PML or ADE-PML) (Martin, Komatitsch, and Ezziani 2008; Martin, Komatitsch, and Gedney 2008; Martin and Komatitsch 2009; Martin et al. 2010; Dimitri Komatitsch and Martin 2007). It also 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 next release of the code will use the PT-SCOTCH parallel and threaded version of SCOTCH for mesh partitioning instead of the serial version.

SPECFEM3D Cartesian includes coupled fluid-solid domains and adjoint capabilities, which enables one to address seismological inverse problems, but for linear rheologies only so far. To accommodate visco-plastic or non-linear rheologies, the reader can refer to the GeoELSE software package (Casadei and Gabellini 1997; Stupazzini, Paolucci, and Igel 2009).

Announcements

  • FWI: SPECFEM3D can now perform full waveform inversion (FWI), i.e. invert for models in an iterative fashion, and it can also perform source inversions in a constant model; please refer to the two new directories, inverse_problem_for_model and inverse_problem_for_source, and the README files they contain. For FWI inversions for the model, also refer to the new examples provided in directory EXAMPLES.

  • External coupling: SPECFEM3D can now perform coupling with an external code (DSM, AxiSEM or FK) based on a database of displacement vectors and traction vectors on the outer edges of the mesh created once and for all (see (Monteiller et al. 2013, 2015; Wang et al. 2016; Tong, Chen, et al. 2014; Tong, Komatitsch, et al. 2014), and if you use that feature of the code please cite at least one of these papers).

    To use coupling with FK, just use the set of parameters that is in the DATA/Par_file input file of the code:

    #-----------------------------------------------------------
#
# Coupling with an injection technique (DSM, AxiSEM, or FK)
#
#-----------------------------------------------------------
COUPLE_WITH_INJECTION_TECHNIQUE = .false.
INJECTION_TECHNIQUE_TYPE        = 3       # 1 = DSM, 2 = AxiSEM, 3 = FK
MESH_A_CHUNK_OF_THE_EARTH       = .false.
TRACTION_PATH                   = ./DATA/AxiSEM_tractions/3/
FKMODEL_FILE                    = FKmodel
RECIPROCITY_AND_KH_INTEGRAL     = .false. # does not work yet

    That part (coupling with FK) is actively maintained and works fine. See e.g. GRL_Ping_Tong_2014 for some examples. There is also an example that is provided with the code: specfem3d/EXAMPLES/small_example_coupling_FK_specfem.

    Regarding coupling with DSM, that part is not actively maintained any more, but it is still included in the code, you may have to test it again and make minor adjustments if needed. The necessary tools are in directory specfem3d/EXTERNAL_PACKAGES_coupled_with_SPECFEM3D/DSM_for_SPECFEM3D, and there is a README file in specfem3d/EXTERNAL_PACKAGES_coupled_with_SPECFEM3D that should be more or less up-to-date (there are about four steps to follow in total, the first one being creating the database of DSM tractions and displacements on the edges of the coupling box). See e.g. GJI_Vadim_2013 and GJI_Vadim_2015 for some examples.

  • Gravity: SPECFEM3D can now perform gravity field calculations in addition (or instead of) seismic wave propagation only. See flag GRAVITY_INTEGRALS in file setup/constants.h. Please also refer to GJI_Martin_gravimetry_2017. And yes, that is the reason why Dimitri added a falling apple on the cover of the manual :-). Note that SPECFEM3D can also model transient gravity perturbations induced by earthquake rupture, as developed and explained in (Harms et al. 2015). These are two different things, and both are implemented and avaible in SPECFEM3D. To use the second feature, please refer to doc/how_to/README_gravity_perturbation.txt.

Citation

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

If you use SPECFEM3D Cartesian for your own research, please cite at least one of the following articles:

Numerical simulations in general
Forward and adjoint simulations are described in detail in Jeroen Tromp, Komatitsch, and Liu (2008; Peter et al. 2011; Vai et al. 1999; Dimitri Komatitsch, Michéa, and Erlebacher 2009; Dimitri Komatitsch et al. 2010; Emmanuel Chaljub et al. 2007; Madec, Komatitsch, and Diaz 2009; D. Komatitsch, Vinnik, and Chevrot 2010; Carrington et al. 2008; Jeroen Tromp et al. 2010; D. Komatitsch, Ritsema, and Tromp 2002; D. Komatitsch and Tromp 2002a, 2002b, 1999) or D. Komatitsch and Vilotte (1998). Additional aspects of adjoint simulations are described in Jeroen Tromp, Tape, and Liu (2005; Qinya Liu and Tromp 2006; Jeroen Tromp, Komatitsch, and Liu 2008; Q. Liu and Tromp 2008; Jeroen Tromp et al. 2010; Peter et al. 2011). Domain decomposition is explained in detail in Martin et al. (2008), and excellent scaling up to 150,000 processor cores is shown for instance in Carrington et al. (2008; Dimitri Komatitsch, Labarta, and Michéa 2008; Martin et al. 2008; Dimitri Komatitsch et al. 2010; Dimitri Komatitsch 2011),

GPU computing
Computing on GPU graphics cards for acoustic or seismic wave propagation applications is described in detail in Dimitri Komatitsch (2011; Michéa and Komatitsch 2010; Dimitri Komatitsch, Michéa, and Erlebacher 2009; Dimitri Komatitsch et al. 2010).

If you use this new version, which has non blocking MPI for much better performance for medium or large runs, please cite at least one of these six articles, in which results of non blocking MPI runs are presented: Peter et al. (2011; Dimitri Komatitsch et al. 2010; D. Komatitsch, Vinnik, and Chevrot 2010; Dimitri Komatitsch 2011; Carrington et al. 2008; Martin et al. 2008).

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:

  • (Jeroen Tromp, Komatitsch, and Liu 2008),

  • (Peter et al. 2011),

  • (Qinya Liu and Tromp 2006),

  • (Morency, Luo, and Tromp 2009).

If you work on geophysical applications, you may be interested in citing some of these application articles as well, among others:

Southern California simulations
Dimitri Komatitsch et al. (2004; Krishnan et al. 2006a, 2006b).

If you use the 3D southern California model, please cite Süss and Shaw (2003) (Los Angeles model), Lovely et al. (2006) (Salton Trough), and Hauksson (2000) (southern California). The Moho map was determined by Zhu and Kanamori (2000). The 1D SoCal model was developed by Dreger and Helmberger (1990).

Anisotropy
Chen and Tromp (2007; Ji et al. 2005; Chevrot, Favier, and Komatitsch 2004; Favier, Chevrot, and Komatitsch 2004; Ritsema et al. 2002; J. Tromp and Komatitsch 2000).

Attenuation
Savage, Komatitsch, and Tromp (2010; D. Komatitsch and Tromp 2002a, 1999).

Topography
Lee, Komatitsch, et al. (2009; Lee, Chan, et al. 2009; Lee et al. 2008; Godinho et al. 2009; Wijk et al. 2004).

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.

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Komatitsch, Dimitri, Gordon Erlebacher, Dominik Göddeke, and David Michéa. 2010. “High-Order Finite-Element Seismic Wave Propagation Modeling with MPI on a Large GPU Cluster.” J. Comput. Phys. 229 (20): 7692–7714. https://doi.org/10.1016/j.jcp.2010.06.024.

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.

Komatitsch, Dimitri, Seiji Tsuboi, Chen Ji, and Jeroen Tromp. 2003. “A 14.6 Billion Degrees of Freedom, 5 Teraflops, 2.5 Terabyte Earthquake Simulation on the Earth Simulator.” In Proceedings of the SC’03 ACM/IEEE Conference on Supercomputing, 4–11. Phoenix, Arizona, USA: ACM. https://doi.org/10.1145/1048935.1050155.

Komatitsch, Dimitri, Zhinan Xie, Ebru Bozdağ, Elliott Sales de Andrade, Daniel Peter, Qinya Liu, and Jeroen Tromp. 2016. “Anelastic Sensitivity Kernels with Parsimonious Storage for Adjoint Tomography and Full Waveform Inversion.” Geophys. J. Int. 206 (3): 1467–78. https://doi.org/10.1093/gji/ggw224.

Komatitsch, D., R. Martin, J. Tromp, M. A. Taylor, and B. A. Wingate. 2001. “Wave Propagation in 2-D Elastic Media Using a Spectral Element Method with Triangles and Quadrangles.” J. Comput. Acoust. 9 (2): 703–18. https://doi.org/10.1142/S0218396X01000796.

Komatitsch, D., J. Ritsema, and J. Tromp. 2002. “The Spectral-Element Method, Beowulf Computing, and Global Seismology.” Science 298 (5599): 1737–42. https://doi.org/10.1126/science.1076024.

Komatitsch, D., and J. Tromp. 1999. “Introduction to the Spectral-Element Method for 3-D Seismic Wave Propagation.” Geophys. J. Int. 139 (3): 806–22. https://doi.org/10.1046/j.1365-246x.1999.00967.x.

———. 2002a. “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.

———. 2002b. “Spectral-Element Simulations of Global Seismic Wave Propagation-II. 3-D Models, Oceans, Rotation, and Self-Gravitation.” Geophys. J. Int. 150 (1): 303–18. https://doi.org/10.1046/j.1365-246X.2002.01716.x.

Komatitsch, D., and J. P. Vilotte. 1998. “The Spectral-Element Method: An Efficient Tool to Simulate the Seismic Response of 2D and 3D Geological Structures.” Bull. Seism. Soc. Am. 88 (2): 368–92.

Komatitsch, D., L. P. Vinnik, and S. Chevrot. 2010. “SHdiff/SVdiff Splitting in an Isotropic Earth.” J. Geophys. Res. 115 (B7): B07312. https://doi.org/10.1029/2009JB006795.

Kopriva, D. A. 2006. “Metric Identities and the Discontinuous Spectral Element Method on Curvilinear Meshes.” Journal of Scientific Computing 26 (3): 301–27. https://doi.org/10.1007/s10915-005-9070-8.

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Krishnan, Swaminathan, Chen Ji, Dimitri Komatitsch, and Jeroen Tromp. 2006a. “Case Studies of Damage to Tall Steel Moment-Frame Buildings in Southern California During Large San Andreas Earthquakes.” Bull. Seism. Soc. Am. 96 (4A): 1523–37. https://doi.org/10.1785/0120050145.

———. 2006b. “Performance of Two 18-Story Steel Moment-Frame Buildings in Southern California During Two Large Simulated San Andreas Earthquakes.” Earthquake Spectra 22 (4): 1035–61. https://doi.org/10.1193/1.2360698.

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, 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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Madec, Ronan, Dimitri Komatitsch, and Julien Diaz. 2009. “Energy-Conserving Local Time Stepping Based on High-Order Finite Elements for Seismic Wave Propagation Across a Fluid-Solid Interface.” Comput. Model. Eng. Sci. 49 (2): 163–89.

Martin, Roland, and Dimitri Komatitsch. 2009. “An Unsplit Convolutional Perfectly Matched Layer Technique Improved at Grazing Incidence for the Viscoelastic Wave Equation.” Geophys. J. Int. 179 (1): 333–44. https://doi.org/10.1111/j.1365-246X.2009.04278.x.

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.

Martin, Roland, Dimitri Komatitsch, and Abdelaâziz Ezziani. 2008. “An Unsplit Convolutional Perfectly Matched Layer Improved at Grazing Incidence for Seismic Wave Equation in Poroelastic Media.” Geophysics 73 (4): T51–61. https://doi.org/10.1190/1.2939484.

Martin, Roland, Dimitri Komatitsch, and Stephen D. Gedney. 2008. “A Variational Formulation of a Stabilized Unsplit Convolutional Perfectly Matched Layer for the Isotropic or Anisotropic Seismic Wave Equation.” Comput. Model. Eng. Sci. 37 (3): 274–304. https://doi.org/10.3970/cmes.2008.037.274.

Martin, Roland, Dimitri Komatitsch, Stephen D. Gedney, and Émilien Bruthiaux. 2010. “A High-Order Time and Space Formulation of the Unsplit Perfectly Matched Layer for the Seismic Wave Equation Using Auxiliary Differential Equations (ADE-PML).” Comput. Model. Eng. Sci. 56 (1): 17–42. https://doi.org/10.3970/cmes.2010.056.017.

Melvin, Thomas, Andrew Staniforth, and John Thuburn. 2012. “Dispersion Analysis of the Spectral-Element Method.” Quarterly Journal of the Royal Meteorological Society 138 (668): 1934–47. https://doi.org/10.1002/qj.1906.

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Michéa, David, and Dimitri Komatitsch. 2010. “Accelerating a 3D Finite-Difference Wave Propagation Code Using GPU Graphics Cards.” Geophys. J. Int. 182 (1): 389–402. https://doi.org/10.1111/j.1365-246X.2010.04616.x.

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Monteiller, Vadim, Sébastien Chevrot, Dimitri Komatitsch, and Nobuaki Fuji. 2013. “A Hybrid Method to Compute Short Period Synthetic Seismograms of Teleseismic Body Waves in a 3-D Regional Model.” Geophys. J. Int. 192 (1): 230–47. https://doi.org/10.1093/gji/ggs006.

Monteiller, Vadim, Sébastien Chevrot, Dimitri Komatitsch, and Yi Wang. 2015. “Three-Dimensional Full Waveform Inversion of Short-Period Teleseismic Wavefields Based Upon the SEM-DSM Hybrid Method.” Geophys. J. Int. 202 (2): 811–27. https://doi.org/10.1093/gji/ggv189.

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Süss, M. P., and J. H. Shaw. 2003. “P Wave Seismic Velocity Structure Derived from Sonic Logs and Industry Reflection Data in the Los Angeles Basin, California.” J. Geophys. Res. 108 (B3): 2170. https://doi.org/10.1029/2001JB001628.

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Tong, Ping, Chin-Wu Chen, Dimitri Komatitsch, Piero Basini, and Qinya Liu. 2014. “High-Resolution Seismic Array Imaging Based on a SEM-FK Hybrid Method.” Geophys. J. Int. 197 (1): 369–95. https://doi.org/10.1093/gji/ggt508.

Tong, Ping, Dimitri Komatitsch, Tai-Lin Tseng, Shu-Huei Hung, Chin-Wu Chen, Piero Basini, and Qinya Liu. 2014. “A 3-D Spectral-Element and Frequency-Wave Number Hybrid Method for High-Resolution Seismic Array Imaging.” Geophys. Res. Lett. 41 (20): 7025–34. https://doi.org/10.1002/2014GL061644.

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Tsuboi, Seiji, Dimitri Komatitsch, Chen Ji, and Jeroen Tromp. 2003. “Broadband Modeling of the 2002 Denali Fault Earthquake on the Earth Simulator.” Phys. Earth Planet. Inter. 139 (3-4): 305–13. https://doi.org/10.1016/j.pepi.2003.09.012.

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Xie, Zhinan, René Matzen, Paul Cristini, Dimitri Komatitsch, and Roland Martin. 2016. “A Perfectly Matched Layer for Fluid-Solid Problems: Application to Ocean-Acoustics Simulations with Solid Ocean Bottoms.” J. Acoust. Soc. Am. 140 (1): 165–75. https://doi.org/10.1121/1.4954736.

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