% HOW TO CITE SPECFEM2D
% HowToCiteSPECFEM2D.bib
%
% This file includes the bibtex entries necessary for citing SPECFEM2D.
%
% Citing software gives proper credit to those who contribute to the
% development of the code and allows CIG to gather citation metrics
% for our community.
%
% There are 3 parts to citing software:
%   I. Acknowledge CIG
%  II. Cite the code
% III. Cite published papers
%
% I. ACKNOWLEDGE CIG
% Use the following text in your “Acknowledgements”:
% We thank the Computational Infrastructure for Geodynamics (geodynamics.org) which is funded by the National Science Foundation under award EAR-0949446, EAR-1550901, and EAR-2149126 for supporting the development of SPECFEM2D.

% II. CITE THE CODE
% Select the proper code version from the following:
%
@software{dimitri_komatitsch_2023_10415228,
  author       = {Dimitri Komatitsch and
                  Jeroen Tromp and
                  Hom Nath Gharti and
                  Daniel Peter and
                  Eduardo Valero Cano and
                  Etienne Bachmann and
                  Alexis Bottero and
                  Quentin Brissaud and
                  Bryant Chow and
                  Paul Cristini and
                  Congyue Cui and
                  Rene Gassmoeller and
                  Michael Gineste and
                  Felix Halpaap and
                  Eric Heien and
                  Jesus Labarta and
                  Matthieu Lefebvre and
                  Nicolas Le Goff and
                  Pieyre Le Loher and
                  Qiancheng Liu and
                  Qinya Liu and
                  Youshan Liu and
                  Zhaolun Liu and
                  David Luet and
                  Roland Martin and
                  Rene Matzen and
                  Ryan Modrak and
                  Christina Morency and
                  Masaru Nagaso and
                  Eric Rosenkrantz and
                  Herurisa Rusmanugroho and
                  Elliott Sales de Andrade and
                  Carl Tape and
                  Jean-Pierre Vilotte and
                  Zhinan Xie and
                  Zhendong Zhang},
  title        = {SPECFEM/specfem2d: SPECFEM2D v8.1.0},
  month        = dec,
  year         = 2023,
  publisher    = {Zenodo},
  version      = {v8.1.0},
  doi          = {10.5281/zenodo.10415228},
  url          = {https://doi.org/10.5281/zenodo.10415228}
}
%
@software{specfem3dglobe-dev,
  author       = {Komatitsch, D. and Vilotte, J.-P. and Cristini, P. and Labarta, J. and Le Goff, N. and Le Loher, P. and Liu, Q. and Martin, R. and Matzen, R. and Morency, C. and Peter, D. and  Tape, C. and Tromp, J. and Xie, Z.},
  title        = {SPECFEM2D [software]},
  year         = 9999,
  version      = {},
  doi          = {GITHASH8},
  url          = {specfem.org}
  organization = {Computational Infrastructure for Geodynamics},
  optkeywords  = {SPECFEM2D}
}
@software{dimitri_komatitsch_2022_7434516,
  author       = {Dimitri Komatitsch and
                  Daniel P{eter and
                  Bottero and
                  Xie Zhinan and
                  Ryan Modrak and
                  Christina Morency and
                  Etienne Bachmann and
                  David Luet and
                  Bryant Chow and
                  Eric Heien and
                  Qiancheng Liu and
                  Paul Cristini and
                  Rene Gassmoeller and
                  evcano and
                  Eric-Rosenkrantz and
                  herurisa and
                  Zhendong Zhang and
                  Zhaolun Liu and
                  Congyue Cui},
  title        = {SPECFEM/specfem2d: SPECFEM2D v8.0.0},
  month        = dec,
  year         = 2022,
  publisher    = {Zenodo},
  version      = {v8.0.0},
  doi          = {10.5281/zenodo.7434516},
  url          = {https://doi.org/10.5281/zenodo.7434516}
}
%
@software{specfem2d-v7.0.0,
  author       = {Komatitsch, D. and Vilotte, J.-P. and Cristini, P. and Labarta, J. and Le Goff, N. and Le Loher, P. and Liu, Q. and Martin, R. and Matzen, R. and Morency, C. and Peter, D. and  Tape, C. and Tromp, J. and Xie, Z.},
  title        = {SPECFEM2D v7.0.0 [software]},
  year         = 2012,
  version      = {v7.0.0},
  doi          = {NoDOI},
  organization = {Computational Infrastructure for Geodynamics},
  optkeywords  = {SPECFEM2D}
}
% III. CITE PUBLISHED PAPERS
% Cite the following:
%
@article{tromp2008spectral,
  Title   = {Spectral-element and adjoint methods in seismology},
  Author  = {Tromp, J. and Komatitsch, D. and Liu, Q.},
  Journal = {Communications in Computational Physics},
  Year    = {2008},
  Pages   = {1-32},
  Volume  = {3},
  Number  = {1}
}

% In addition, if you use any of the following features
%  please cite:

% If you use the kernel capabilities of the code, please cite at least one
% article written by the developers of the package:
@article{tromp2008spectral,
  title   = {Spectral-element and adjoint methods in seismology},
  author  = {Tromp, J. and Komatitsch, D. and Liu, Q.},
  journal = {Communications in Computational Physics},
  year    = {2008},
  pages   = {1-32},
  volume  = {3},
  number  = {1}
}
@article{,
  title   = {Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes: SPECFEM3D Version 2.0 'Sesame'},
  author  = {Peter, D. and Komatitsch, D. and Luo, Y. and Martin, R. and Le Goff, N. and Casarotti, E. and Le Loher, P. and Magnoni, F. and Liu, Q. and Blitz, C. and Nissen-Meyer, T. and Basini, P. and Tromp, J.},
  journal = {Geophysical Journal International},
  year    = {2011},
  pages   = {721-739},
  volume  = {186},
  number  = {2},
  optkeywords = {SPECFEM3D Cartesian; SPECFEM3D GLOBE},
  doi     = {10.1111/j.1365-246X.2011.05044.x},
  iSSN    = {0956540X},
  url     = {http://gji.oxfordjournals.org/cgi/doi/10.1111/j.1365-246X.2011.05044.x}
}
@article{,
  title   = {Finite-Frequency Kernels Based on Adjoint Methods},
  author  = {Liu, Q. and Tromp, J.},
  journal = {Bulletin of the Seismological Society of America},
  year    = {2006},
  pages   = {2383-2397},
  volume  = {96},
  number  = {6},
  optkeywords = {SPECFEM3D Cartesian},
  doi     = {10.1785/0120060041},
  iSSN    = {0037-1106},
  url     = {http://www.bssaonline.org/cgi/doi/10.1785/0120060041}
}
@article{,
  title   = {Finite-frequency kernels for wave propagation in porous media based upon adjoint methods},
  author  = {Morency, C. and Luo, Y. and Tromp, J.},
  journal = {Geophysical Journal International},
  year    = {2009},
  pages   = {1148-1168},
  volume  = {179},
  number  = {2},
  optkeywords = {SPECFEM2D},
  doi     = {10.1111/j.1365-246X.2009.04332.x},
  iSSN    = {0956540X},
  url     = {http://gji.oxfordjournals.org/cgi/doi/10.1111/j.1365-246X.2009.04332.x}
}

% If you use the SCOTCH / CUBIT non-structured capabilities, please also cite:
@InProceedings{martin2008simulation,
  title   = {Simulation of seismic wave propagation in an asteroid based upon an unstructured MPI spectral-element method: blocking and non-blocking communication strategies},
  author  = {Martin, R. and Komatitsch, D. and Blitz, C. and Le Goff, N.},
  journal = {International Conference on High Performance Computing for Computational Science},
  year    = {2008},
  pages   = {350-363}
}

%If you use this code for your own research, please cite at least one article written by the developers of the package, for instance:
@Article{CiCP-3-1,
author = {},
title = {Spectral-Element and Adjoint Methods in Seismology},
journal = {Communications in Computational Physics},
year = {2008},
volume = {3},
number = {1},
pages = {1--32},
abstract = {
We provide an introduction to the use of the spectral-element method (SEM) in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss in some detail how these equations may be solved numerically based upon the SEM to address the forward problem in seismology. Examples of synthetic seismograms calculated based upon the SEM are compared to data recorded by the Global Seismographic Network. Finally, we discuss the challenge of using the remaining differences between the data and the synthetic seismograms to constrain better Earth models and source descriptions. This leads naturally to adjoint methods, which provide a practical approach to this formidable computational challenge and enables seismologists to tackle the inverse problem.

},
issn = {1991-7120},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/cicp/7840.html}
}
@article{10.1111/j.1365-246X.2011.05044.x,
    author = {Peter, Daniel and Komatitsch, Dimitri and Luo, Yang and Martin, Roland and Le Goff, Nicolas and Casarotti, Emanuele and Le Loher, Pieyre and Magnoni, Federica and Liu, Qinya and Blitz, Céline and Nissen-Meyer, Tarje and Basini, Piero and Tromp, Jeroen},
    title = "{Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes}",
    journal = {Geophysical Journal International},
    volume = {186},
    number = {2},
    pages = {721-739},
    year = {2011},
    month = {08},
    abstract = "{We present forward and adjoint spectral-element simulations of coupled acoustic and (an)elastic seismic wave propagation on fully unstructured hexahedral meshes. Simulations benefit from recent advances in hexahedral meshing, load balancing and software optimization. Meshing may be accomplished using a mesh generation tool kit such as CUBIT, and load balancing is facilitated by graph partitioning based on the SCOTCH library. Coupling between fluid and solid regions is incorporated in a straightforward fashion using domain decomposition. Topography, bathymetry and Moho undulations may be readily included in the mesh, and physical dispersion and attenuation associated with anelasticity are accounted for using a series of standard linear solids. Finite-frequency Fréchet derivatives are calculated using adjoint methods in both fluid and solid domains. The software is benchmarked for a layercake model. We present various examples of fully unstructured meshes, snapshots of wavefields and finite-frequency kernels generated by Version 2.0 ‘Sesame’ of our widely used open source spectral-element package SPECFEM3D.}",
    issn = {0956-540X},
    doi = {10.1111/j.1365-246X.2011.05044.x},
    url = {https://doi.org/10.1111/j.1365-246X.2011.05044.x},
    eprint = {https://academic.oup.com/gji/article-pdf/186/2/721/6017323/186-2-721.pdf},
}
@article{VAI199911,
title = {Elastic wave propagation in an irregularly layered medium},
journal = {Soil Dynamics and Earthquake Engineering},
volume = {18},
number = {1},
pages = {11-18},
year = {1999},
issn = {0267-7261},
doi = {https://doi.org/10.1016/S0267-7261(98)00027-X},
url = {https://www.sciencedirect.com/science/article/pii/S026772619800027X},
author = {Rossana Vai and José Manuel Castillo-Covarrubias and FranciscoJ. Sánchez-Sesma and Dimitri Komatitsch and Jean-Pierre Vilotte},
keywords = {Irregularly layered viscoelastic media, Synthetic seismograms, Diffracted–scattered wavefields},
abstract = {The indirect boundary element method (IBEM) is used to simulate wave propagation in two-dimensional irregularly layered elastic media for internal line sources. The method is based on the integral representation for scattered elastic waves using single layer boundary sources. Fulfillment of the boundary conditions leads to a system of integral equations. Results are obtained in the frequency domain and seismograins are computed through Fourier synthesis. In order to test and validate the method we present various comparisons between our results and the time series obtained analytically for a buried line source in a half-space and by using the recently developed spectral element method (SEM).}
}
@article{10.1785/0120080264,
    author = {Lee, Shiann-Jong and Chan, Yu-Chang and Komatitsch, Dimitri and Huang, Bor-Shouh and Tromp, Jeroen},
    title = "{Effects of Realistic Surface Topography on Seismic Ground Motion in the Yangminshan Region of Taiwan Based Upon the Spectral-Element Method and LiDAR DTM}",
    journal = {Bulletin of the Seismological Society of America},
    volume = {99},
    number = {2A},
    pages = {681-693},
    year = {2009},
    month = {04},
    abstract = "{We combine light detection and ranging (LiDAR) digital terrain model (DTM) data and an improved mesh implementation to investigate the effects of high-resolution surface topography on seismic ground motion based upon the spectral-element method. In general, topography increases the amplitude of shaking at mountain tops and ridges, whereas valleys usually have reduced ground motion, as has been observed in both records from past earthquakes and numerical simulations. However, the effects of realistic topography on ground motion have not often been clearly characterized in numerical simulations, especially the seismic response of the true ground surface. Here, we use LiDAR DTM data, which provide two-meter resolution at the free surface, and a spectral-element method to simulate three-dimensional (3D) seismic-wave propagation in the Yangminshan region in Taiwan, incorporating the effects of realistic topography. A smoothed topographic map is employed beneath the model surface in order to decrease mesh distortions due to steep ground surfaces. Numerical simulations show that seismic shaking in mountainous areas is strongly affected by topography and source frequency content. The amplification of ground motion mainly occurs at the tops of hills and ridges whilst the valleys and flat-topped hills experience lower levels of ground shaking. Interaction between small-scale topographic features and high-frequency surface waves can produce unusually strong shaking. We demonstrate that topographic variations can change peak ground acceleration (PGA) values by ±50\\% in mountainous areas, and the relative change in PGA between a valley and a ridge can be as high as a factor of 2 compared to a flat surface response. This suggests that high-resolution, realistic topographic features should be taken into account in seismic hazard analysis, especially for densely populated mountainous areas.}",
    issn = {0037-1106},
    doi = {10.1785/0120080264},
    url = {https://doi.org/10.1785/0120080264},
    eprint = {https://pubs.geoscienceworld.org/ssa/bssa/article-pdf/99/2A/681/3674222/681.pdf},
}
@article{10.1785/0120070033,
    author = {Lee, Shiann-Jong and Chen, How-Wei and Liu, Qinya and Komatitsch, Dimitri and Huang, Bor-Shouh and Tromp, Jeroen},
    title = "{Three-Dimensional Simulations of Seismic-Wave Propagation in the Taipei Basin with Realistic Topography Based upon the Spectral-Element Method}",
    journal = {Bulletin of the Seismological Society of America},
    volume = {98},
    number = {1},
    pages = {253-264},
    year = {2008},
    month = {02},
    abstract = "{We use the spectral-element method to simulate strong ground motion throughout the Taipei metropolitan area. Mesh generation for the Taipei basin poses two main challenges: (1) the basin is surrounded by steep mountains, and (2) the city is located on top of a shallow, low-wave-speed sedimentary basin. To accommodate the steep and rapidly varying topography, we introduce a thin high-resolution mesh layer near the surface. The mesh for the shallow sedimentary basin is adjusted to honor its complex geometry and sharp lateral wave-speed contrasts. Variations in Moho thickness beneath Northern Taiwan are also incorporated in the mesh. Spectral-element simulations show that ground motion in the Taipei metropolitan region is strongly affected by the geometry of the basin and the surrounding mountains. The amplification of ground motion is mainly controlled by basin depth and shallow shear-wave speeds, although surface topography also serves to amplify and prolong seismic shaking.}",
    issn = {0037-1106},
    doi = {10.1785/0120070033},
    url = {https://doi.org/10.1785/0120070033},
    eprint = {https://pubs.geoscienceworld.org/ssa/bssa/article-pdf/98/1/253/3585562/253.pdf},
}
@article{10.1785/0120080020,
    author = {Lee, Shiann-Jong and Komatitsch, Dimitri and Huang, Bor-Shouh and Tromp, Jeroen},
    title = "{Effects of Topography on Seismic-Wave Propagation: An Example from Northern Taiwan}",
    journal = {Bulletin of the Seismological Society of America},
    volume = {99},
    number = {1},
    pages = {314-325},
    year = {2009},
    month = {02},
    abstract = "{Topography influences ground motion and, in general, increases the amplitude of shaking at mountain tops and ridges, whereas valleys have reduced ground motions, as is observed from data recorded during and after real earthquakes and from numerical simulations. However, recent publications have focused mainly on the implications for ground motion in the mountainous regions themselves, whereas the impact on surrounding low-lying areas has received less attention. Here, we develop a new spectral-element mesh implementation to accommodate realistic topography as well as the complex shape of the Taipei sedimentary basin, which is located close to the Central Mountain Range in northern Taiwan. Spectral-element numerical simulations indicate that high-resolution topography can change peak ground velocity (PGV) values in mountainous areas by ±50\\% compared to a half-space response. We further demonstrate that large-scale topography can affect the propagation of seismic waves in nearby areas. For example, if a shallow earthquake occurs in the I-Lan region of Taiwan, the Central Mountain Range will significantly scatter the surface waves and will in turn reduce the amplitude of ground motion in the Taipei basin. However, as the hypocenter moves deeper, topography scatters body waves, which subsequently propagate as surface waves into the basin. These waves continue to interact with the basin and the surrounding mountains, finally resulting in complex amplification patterns in Taipei City, with an overall PGV increase of more than 50\\%. For realistic subduction zone earthquake scenarios off the northeast coast of Taiwan, the effects of topography on ground motion in both the mountains and the Taipei basin vary and depend on the rupture process. The complex interactions that can occur between mountains and surrounding areas, especially sedimentary basins, illustrate the fact that topography should be taken into account when assessing seismic hazard.}",
    issn = {0037-1106},
    doi = {10.1785/0120080020},
    url = {https://doi.org/10.1785/0120080020},
    eprint =
     {https://pubs.geoscienceworld.org/ssa/bssa/article-pdf/99/1/314/3675351/314.pdf},
}
@article{KOMATITSCH20107692,
title = {High-order finite-element seismic wave propagation modeling with MPI on a large GPU cluster},
journal = {Journal of Computational Physics},
volume = {229},
number = {20},
pages = {7692-7714},
year = {2010},
issn = {0021-9991},
doi = {https://doi.org/10.1016/j.jcp.2010.06.024},
url = {https://www.sciencedirect.com/science/article/pii/S0021999110003396},
author = {Dimitri Komatitsch and Gordon Erlebacher and Dominik Göddeke and David Michéa},
keywords = {GPU computing, Finite elements, Spectral elements, Seismic modeling, CUDA, MPI, Speedup, Cluster},
abstract = {We implement a high-order finite-element application, which performs the numerical simulation of seismic wave propagation resulting for instance from earthquakes at the scale of a continent or from active seismic acquisition experiments in the oil industry, on a large cluster of NVIDIA Tesla graphics cards using the CUDA programming environment and non-blocking message passing based on MPI. Contrary to many finite-element implementations, ours is implemented successfully in single precision, maximizing the performance of current generation GPUs. We discuss the implementation and optimization of the code and compare it to an existing very optimized implementation in C language and MPI on a classical cluster of CPU nodes. We use mesh coloring to efficiently handle summation operations over degrees of freedom on an unstructured mesh, and non-blocking MPI messages in order to overlap the communications across the network and the data transfer to and from the device via PCIe with calculations on the GPU. We perform a number of numerical tests to validate the single-precision CUDA and MPI implementation and assess its accuracy. We then analyze performance measurements and depending on how the problem is mapped to the reference CPU cluster, we obtain a speedup of 20x or 12x.}
}

@article{KOMATITSCH2009451,
title = {Porting a high-order finite-element earthquake modeling application to NVIDIA graphics cards using CUDA},
journal = {Journal of Parallel and Distributed Computing},
volume = {69},
number = {5},
pages = {451-460},
year = {2009},
issn = {0743-7315},
doi = {https://doi.org/10.1016/j.jpdc.2009.01.006},
url = {https://www.sciencedirect.com/science/article/pii/S0743731509000069},
author = {Dimitri Komatitsch and David Michéa and Gordon Erlebacher},
keywords = {GPGPU, CUDA, Speedup, Finite elements, Spectral methods},
abstract = {We port a high-order finite-element application that performs the numerical simulation of seismic wave propagation resulting from earthquakes in the Earth on NVIDIA GeForce 8800 GTX and GTX 280 graphics cards using CUDA. This application runs in single precision and is therefore a good candidate for implementation on current GPU hardware, which either does not support double precision or supports it but at the cost of reduced performance. We discuss and compare two implementations of the code: one that has maximum efficiency but is limited to the memory size of the card, and one that can handle larger problems but that is less efficient. We use a coloring scheme to handle efficiently summation operations over nodes on a topology with variable valence. We perform several numerical tests and performance measurements and show that in the best case we obtain a speedup of 25.}
}

@article{10.1785/012004038,
    author = {Liu, Qinya and Polet, Jascha and Komatitsch, Dimitri and Tromp, Jeroen},
    title = "{Spectral-Element Moment Tensor Inversions for Earthquakes in Southern California}",
    journal = {Bulletin of the Seismological Society of America},
    volume = {94},
    number = {5},
    pages = {1748-1761},
    year = {2004},
    month = {10},
    abstract = "{We have developed and implemented an automated moment tensor inversion procedure to determine source parameters for southern California earthquakes. The method is based upon spectral-element simulations of regional seismic wave propagation in an integrated 3D southern California velocity model. Sensitivity to source parameters is determined by numerically calculating the Fréchet derivatives required for the moment tensor inversion. We minimize a waveform misfit function, and allow limited time shifts between data and corresponding synthetics to accommodate additional 3D heterogeneity not included in our model. The technique is applied to three recent southern California earthquakes: the 9 September 2001, ML 4.2 Hollywood event, the 22 February 2003, ML 5.4 Big Bear event, and the 14 December 2001, ML 4.0 Diamond Bar event. Using about half of the available three-component data at periods of 6 sec and longer, we obtain focal mechanisms, depths, and moment magnitudes that are generally in good agreement with estimates based upon traditional body-wave and surface-wave inversions.}",
    issn = {0037-1106},
    doi = {10.1785/012004038},
    url = {https://doi.org/10.1785/012004038},
    eprint = {https://pubs.geoscienceworld.org/ssa/bssa/article-pdf/94/5/1748/2719317/1748\_945\_04038.1748\_1761.pdf},
}

@article{chaljub2007spectral,
  title={Spectral-element analysis in seismology},
  author={Chaljub, Emmanuel and Komatitsch, Dimitri and Vilotte, Jean-Pierre and Capdeville, Yann and Valette, Bernard and Festa, Gaetano},
  journal={Advances in geophysics},
  volume={48},
  pages={365--419},
  year={2007},
  publisher={Elsevier}
}
@article{10.1046/j.1365-246x.1999.00967.x,
    author = {Komatitsch, Dimitri and Tromp, Jeroen},
    title = "{Introduction to the spectral element method for three-dimensional seismic wave propagation}",
    journal = {Geophysical Journal International},
    volume = {139},
    number = {3},
    pages = {806-822},
    year = {1999},
    month = {12},
    abstract = "{We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the flexibility of a finite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wavefield on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simplifies the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/reflectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.}",
    issn = {0956-540X},
    doi = {10.1046/j.1365-246x.1999.00967.x},
    url = {https://doi.org/10.1046/j.1365-246x.1999.00967.x},
    eprint = {https://academic.oup.com/gji/article-pdf/139/3/806/6006771/139-3-806.pdf},
}
@article{10.1785/BSSA0880020368,
    author = {Komatitsch, Dimitri and Vilotte, Jean-Pierre},
    title = "{The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures}",
    journal = {Bulletin of the Seismological Society of America},
    volume = {88},
    number = {2},
    pages = {368-392},
    year = {1998},
    month = {04},
    abstract = "{We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as ΔtC \\< O (nel−1/ndN−2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.}",
    issn = {0037-1106},
    doi = {10.1785/BSSA0880020368},
    url = {https://doi.org/10.1785/BSSA0880020368},
    eprint = {https://pubs.geoscienceworld.org/ssa/bssa/article-pdf/88/2/368/5344788/bssa0880020368.pdf},
}

@article{10.1785/0120030077,
    author = {Komatitsch, Dimitri and Liu, Qinya and Tromp, Jeroen and Süss, Peter and Stidham, Christiane and Shaw, John H.},
    title = "{Simulations of Ground Motion in the Los Angeles Basin Based upon the Spectral-Element Method}",
    journal = {Bulletin of the Seismological Society of America},
    volume = {94},
    number = {1},
    pages = {187-206},
    year = {2004},
    month = {02},
    abstract = "{We use the spectral-element method to simulate ground motion generated by two recent and well-recorded small earthquakes in the Los Angeles basin. Simulations are performed using a new sedimentary basin model that is constrained by hundreds of petroleum-industry well logs and more than 20,000 km of seismic reflection profiles. The numerical simulations account for 3D variations of seismic-wave speeds and density, topography and bathymetry, and attenuation. Simulations for the 9 September 2001 Mw 4.2 Hollywood earthquake and the 3 September 2002 Mw 4.2 Yorba Linda earthquake demonstrate that the combination of a detailed sedimentary basin model and an accurate numerical technique facilitates the simulation of ground motion at periods of 2 sec and longer inside the basin model and 6 sec and longer in the regional model. Peak ground displacement, velocity, and acceleration maps illustrate that significant amplification occurs in the basin.}",
    issn = {0037-1106},
    doi = {10.1785/0120030077},
    url = {https://doi.org/10.1785/0120030077},
    eprint = {https://pubs.geoscienceworld.org/ssa/bssa/article-pdf/94/1/187/2718258/187\_ssa03077.pdf},
}
@article{10.1111/j.1365-246X.2008.03907.x,
    author = {Morency, Christina and Tromp, Jeroen},
    title = "{Spectral-element simulations of wave propagation in porous media}",
    journal = {Geophysical Journal International},
    volume = {175},
    number = {1},
    pages = {301-345},
    year = {2008},
    month = {10},
    abstract = "{We present a derivation of the equations describing wave propagation in porous media based upon an averaging technique which accommodates the transition from the microscopic to the macroscopic scale. We demonstrate that the governing macroscopic equations determined by Biot remain valid for media with gradients in porosity. In such media, the well-known expression for the change in porosity, or the change in the fluid content of the pores, acquires two extra terms involving the porosity gradient. One fundamental result of Biot's theory is the prediction of a second compressional wave, often referred to as ‘type II’ or ‘Biot's slow compressional wave’, in addition to the classical fast compressional and shear waves. We present a numerical implementation of the Biot equations for 2-D problems based upon the spectral-element method (SEM) that clearly illustrates the existence of these three types of waves as well as their interactions at discontinuities. As in the elastic and acoustic cases, poroelastic wave propagation based upon the SEM involves a diagonal mass matrix, which leads to explicit time integration schemes that are well suited to simulations on parallel computers. Effects associated with physical dispersion and attenuation and frequency-dependent viscous resistance are accommodated based upon a memory variable approach. We perform various benchmarks involving poroelastic wave propagation and acoustic–poroelastic and poroelastic–poroelastic discontinuities, and we discuss the boundary conditions used to deal with these discontinuities based upon domain decomposition. We show potential applications of the method related to wave propagation in compacted sediments, as one encounters in the petroleum industry, and to detect the seismic signature of buried landmines and unexploded ordnance.}",
    issn = {0956-540X},
    doi = {10.1111/j.1365-246X.2008.03907.x},
    url = {https://doi.org/10.1111/j.1365-246X.2008.03907.x},
    eprint = {https://academic.oup.com/gji/article-pdf/175/1/301/5978976/175-1-301.pdf},
}
@article{10.1093/gji/ggw024,
    author = {Blanc, Émilie and Komatitsch, Dimitri and Chaljub, Emmanuel and Lombard, Bruno and Xie, Zhinan},
    title = "{Highly accurate stability-preserving optimization of the Zener viscoelastic model, with application to wave propagation in the presence of strong attenuation}",
    journal = {Geophysical Journal International},
    volume = {205},
    number = {1},
    pages = {427-439},
    year = {2016},
    month = {02},
    abstract = "{This paper concerns the numerical modelling of time-domain mechanical waves in viscoelastic media based on a generalized Zener model. To do so, classically in the literature relaxation mechanisms are introduced, resulting in a set of the so-called memory variables and thus in large computational arrays that need to be stored. A challenge is thus to accurately mimic a given attenuation law using a minimal set of relaxation mechanisms. For this purpose, we replace the classical linear approach of Emmerich \\& Korn with a nonlinear optimization approach with constraints of positivity. We show that this technique is more accurate than the linear approach. Moreover, it ensures that physically meaningful relaxation times that always honour the constraint of decay of total energy with time are obtained. As a result, these relaxation times can always be used in a stable way in a modelling algorithm, even in the case of very strong attenuation for which the classical linear approach may provide some negative and thus unusable coefficients.}",
    issn = {0956-540X},
    doi = {10.1093/gji/ggw024},
    url = {https://doi.org/10.1093/gji/ggw024},
    eprint = {https://academic.oup.com/gji/article-pdf/205/1/427/8036946/ggw024.pdf},
}

%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:
@article{10.1093/gji/ggu219,
    author = {Xie, Zhinan and Komatitsch, Dimitri and Martin, Roland and Matzen, René},
    title = "{Improved forward wave propagation and adjoint-based sensitivity kernel calculations using a numerically stable finite-element PML}",
    journal = {Geophysical Journal International},
    volume = {198},
    number = {3},
    pages = {1714-1747},
    year = {2014},
    month = {07},
    abstract = "{In recent years, the application of time-domain adjoint methods to improve large, complex underground tomographic models at the regional scale has led to new challenges for the numerical simulation of forward or adjoint elastic wave propagation problems. An important challenge is to design an efficient infinite-domain truncation method suitable for accurately truncating an infinite domain governed by the second-order elastic wave equation written in displacement and computed based on a finite-element (FE) method. In this paper, we make several steps towards this goal. First, we make the 2-D convolution formulation of the complex-frequency-shifted unsplit-field perfectly matched layer (CFS-UPML) derived in previous work more flexible by providing a new treatment to analytically remove singular parameters in the formulation. We also extend this new formulation to 3-D. Furthermore, we derive the auxiliary differential equation (ADE) form of CFS-UPML, which allows for extension to higher order time schemes and is easier to implement. Secondly, we rigorously derive the CFS-UPML formulation for time-domain adjoint elastic wave problems, which to our knowledge has never been done before. Thirdly, in the case of classical low-order FE methods, we show numerically that we achieve long-time stability for both forward and adjoint problems both for the convolution and the ADE formulations. In the case of higher order Legendre spectral-element methods, we show that weak numerical instabilities can appear in both formulations, in particular if very small mesh elements are present inside the absorbing layer, but we explain how these instabilities can be delayed as much as needed by using a stretching factor to reach numerical stability in practice for applications. Fourthly, in the case of adjoint problems with perfectly matched absorbing layers we introduce a computationally efficient boundary storage strategy by saving information along the interface between the CFS-UPML and the main domain only, thus avoiding the need to solve a backward wave propagation problem inside the CFS-UPML, which is known to be highly ill-posed. Finally, by providing several examples we show numerically that our formulation is efficient at absorbing acoustic waves for normal to near-grazing incident body waves as well as surface waves.}",
    issn = {0956-540X},
    doi = {10.1093/gji/ggu219},
    url = {https://doi.org/10.1093/gji/ggu219},
    eprint = {https://academic.oup.com/gji/article-pdf/198/3/1714/1583823/ggu219.pdf},
}
@article{doi:10.1121/1.4954736,
author = {Xie,Zhinan  and Matzen,René  and Cristini,Paul  and Komatitsch,Dimitri  and Martin,Roland },
title = {A perfectly matched layer for fluid-solid problems: Application to ocean-acoustics simulations with solid ocean bottoms},
journal = {The Journal of the Acoustical Society of America},
volume = {140},
number = {1},
pages = {165-175},
year = {2016},
doi = {10.1121/1.4954736},
URL = {   https://doi.org/10.1121/1.4954736},
eprint = {    https://doi.org/10.1121/1.4954736}
}
%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:
@article{10.1093/gji/ggw024,
    author = {Blanc, Émilie and Komatitsch, Dimitri and Chaljub, Emmanuel and Lombard, Bruno and Xie, Zhinan},
    title = "{Highly accurate stability-preserving optimization of the Zener viscoelastic model, with application to wave propagation in the presence of strong attenuation}",
    journal = {Geophysical Journal International},
    volume = {205},
    number = {1},
    pages = {427-439},
    year = {2016},
    month = {02},
    abstract = "{This paper concerns the numerical modelling of time-domain mechanical waves in viscoelastic media based on a generalized Zener model. To do so, classically in the literature relaxation mechanisms are introduced, resulting in a set of the so-called memory variables and thus in large computational arrays that need to be stored. A challenge is thus to accurately mimic a given attenuation law using a minimal set of relaxation mechanisms. For this purpose, we replace the classical linear approach of Emmerich \\& Korn with a nonlinear optimization approach with constraints of positivity. We show that this technique is more accurate than the linear approach. Moreover, it ensures that physically meaningful relaxation times that always honour the constraint of decay of total energy with time are obtained. As a result, these relaxation times can always be used in a stable way in a modelling algorithm, even in the case of very strong attenuation for which the classical linear approach may provide some negative and thus unusable coefficients.}",
    issn = {0956-540X},
    doi = {10.1093/gji/ggw024},
    url = {https://doi.org/10.1093/gji/ggw024},
    eprint = {https://academic.oup.com/gji/article-pdf/205/1/427/8036946/ggw024.pdf},
}