Combination of the discontinuous Galerkin method with finite differences for simulation of seismic wave propagation

We present an algorithm for the numerical simulation of seismic wave propagation in models with a complex near surface part and free surface topography. The approach is based on the combination of finite differences with the discontinuous Galerkin method. The discontinuous Galerkin method can be used on polyhedral meshes; thus, it is easy to handle the complex surfaces in the models. However, this approach is computationally intense in comparison with finite differences. Finite differences are computationally efficient, but in general, they require rectangular grids, leading to the stair-step approximation of the interfaces, which causes strong diffraction of the wavefield. In this research we present a hybrid algorithm where the discontinuous Galerkin method is used in a relatively small upper part of the model and finite differences are applied to the main part of the model.

[1]  Vadim Lisitsa,et al.  OPTIMAL DISCRETIZATION OF PML FOR ELASTICITY PROBLEMS , 2008 .

[2]  L. Thomsen.  Weak elastic anisotropy , 1986 .

[3]  Peter Monk,et al.  A Discontinuous Galerkin Method for Linear Symmetric Hyperbolic Systems in Inhomogeneous Media , 2005, J. Sci. Comput..

[4]  D. Komatitsch,et al.  The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures , 1998, Bulletin of the Seismological Society of America.

[5]  L. Thompson.  A review of finite-element methods for time-harmonic acoustics , 2006 .

[6]  Patrick Joly,et al.  A New Family of Mixed Finite Elements for the Linear Elastodynamic Problem , 2001, SIAM J. Numer. Anal..

[7]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[8]  J. Kristek,et al.  3D Heterogeneous Staggered-grid Finite-difference Modeling of Seismic Motion with Volume Harmonic and Arithmetic Averaging of Elastic Moduli and Densities , 2002 .

[9]  Azzam Haidar,et al.  Seismic wave modeling for seismic imaging , 2009 .

[10]  Patrick Joly,et al.  A Conservative Space-time Mesh Refinement Method for the 1-D Wave Equation. Part II: Analysis , 2003, Numerische Mathematik.

[11]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[12]  Romain Brossier,et al.  Parsimonious finite-volume frequency-domain method for 2-D P–SV-wave modelling , 2008 .

[13]  J. Virieux,et al.  An hp-adaptive discontinuous Galerkin finite-element method for 3-D elastic wave modelling , 2010 .

[14]  Yder J. Masson,et al.  Finite-difference modeling of Biot's poroelastic equations across all frequencies , 2010 .

[15]  Martin Galis,et al.  A brief summary of some PML formulations and discretizations for the velocity-stress equation of seismic motion , 2009 .

[16]  J. Carcione,et al.  Numerical experiments of fracture-induced velocity and attenuation anisotropy , 2014 .

[17]  Ludovic Métivier,et al.  A robust absorbing layer method for anisotropic seismic wave modeling , 2014, J. Comput. Phys..

[18]  Patrick Joly,et al.  Stability of perfectly matched layers, group velocities and anisotropic waves , 2003 .

[19]  John L. Tassoulas,et al.  CONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION , 2000 .

[20]  Kristel C. Meza-Fajardo,et al.  A Nonconvolutional, Split-Field, Perfectly Matched Layer for Wave Propagation in Isotropic and Anisotropic Elastic Media: Stability Analysis , 2008 .

[21]  Galina V. Reshetova,et al.  Local time-space mesh refinement for simulation of elastic wave propagation in multi-scale media , 2015, J. Comput. Phys..

[22]  A. Levander.  Fourth-order finite-difference P-SV seismograms , 1988 .

[23]  Mark Ainsworth,et al.  Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods , 2004 .

[24]  J. Carcione,et al.  Computational poroelasticity — A review , 2010 .

[25]  Elastic wave propagation in heterogeneous anisotropic media using the lumped finite‐element method , 2002 .

[26]  Ruey-Beei Wu,et al.  Hybrid finite-difference time-domain modeling of curved surfaces using tetrahedral edge elements , 1997 .

[27]  A. Samarskii.  The Theory of Difference Schemes , 2001 .

[28]  Liu Wei,et al.  A class of hybrid DG/FV methods for conservation laws II: Two-dimensional cases , 2012, J. Comput. Phys..

[29]  Dan Givoli,et al.  High-order Absorbing Boundary Conditions for anisotropic and convective wave equations , 2010, J. Comput. Phys..

[30]  J. Kristek,et al.  The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion , 2007 .

[31]  Thomas Bohlen,et al.  Paralel 3-D viscoelastic finite difference seismic modelling , 2002 .

[32]  Martin Galis,et al.  3‐D finite‐difference, finite‐element, discontinuous‐Galerkin and spectral‐element schemes analysed for their accuracy with respect to P‐wave to S‐wave speed ratio , 2011 .

[33]  C. Tsogka,et al.  Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media , 2001 .

[34]  Joakim O. Blanch,et al.  Modeling of a constant Q; methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique , 1995 .

[35]  Mauricio Hanzich,et al.  Mimetic seismic wave modeling including topography on deformed staggered grids , 2014 .

[36]  Z. Jianfeng.  Quadrangle‐grid velocity–stress finite difference method for poroelastic wave equations , 1999 .

[37]  K. Yee.  Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media , 1966 .

[38]  Mrinal K. Sen,et al.  The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion , 2008 .

[39]  Martin Galis,et al.  Stable discontinuous staggered grid in the finite-difference modelling of seismic motion , 2010 .

[40]  Qing Huo Liu,et al.  Higher-Order Numerical Methods for Transient Wave Equations , 2003 .

[41]  S. Tsynkov.  Numerical solution of problems on unbounded domains. a review , 1998 .

[42]  J. Virieux.  P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method , 1986 .

[43]  Zhang Jianfeng,et al.  Quadrangle-grid velocity-stress finite-difference method for elastic-wave-propagation simulation , 1997 .

[44]  Thomas Rylander,et al.  Stable FEM-FDTD hybrid method for Maxwell's equations , 2000 .

[45]  R. Mittra,et al.  Time-domain (FE/FDTD) technique for solving complex electromagnetic problems , 1998 .

[46]  M. Y. Hussaini,et al.  An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems , 1999 .

[47]  Thomas Hagstrom,et al.  Accurate Radiation Boundary Conditions for the Linearized Euler Equations in Cartesian Domains , 2002, SIAM J. Sci. Comput..

[48]  F. Muir,et al.  Modeling elastic fields across irregular boundaries , 1992 .

[49]  Patrick Joly,et al.  A Conservative Space-time Mesh Refinement Method for the 1-D Wave Equation. Part I: Construction , 2003, Numerische Mathematik.

[50]  V. Tcheverda,et al.  On the interface error analysis for finite difference wave simulation , 2010 .

[51]  Beatriz Otero,et al.  Low dispersive modeling of Rayleigh waves on partly staggered grids , 2014, Computational Geosciences.

[52]  V. Lisitsa.  Dispersion analysis of discontinuous Galerkin method on triangular mesh for elastic wave equation , 2016 .

[53]  Tada-nori Goto,et al.  An accuracy analysis of a Hamiltonian particle method with the staggered particles for seismic-wave modeling including surface topography , 2014 .

[54]  Henri Calandra,et al.  A review of the spectral, pseudo‐spectral, finite‐difference and finite‐element modelling techniques for geophysical imaging , 2011 .

[55]  Robert Laws,et al.  Rough seas and statistical deconvolution , 2006 .

[56]  Vadim Lisitsa,et al.  Numerical simulation of seismic waves in models with anisotropic formations: coupling Virieux and Lebedev finite-difference schemes , 2012, Computational Geosciences.

[57]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[58]  Julien Diaz,et al.  Energy Conserving Explicit Local Time Stepping for Second-Order Wave Equations , 2007, SIAM J. Sci. Comput..

[59]  Vadim Lisitsa,et al.  Numerical study of the interface errors of finite-difference simulations of seismic waves , 2014 .

[60]  Randall J. LeVeque,et al.  Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains , 2008, SIAM Rev..

[61]  James F. Doyle,et al.  The Spectral Element Method , 2020, Wave Propagation in Structures.

[62]  Sofia Davydycheva,et al.  An efficient finite‐difference scheme for electromagnetic logging in 3D anisotropic inhomogeneous media , 2003 .

[63]  Peter G. Petropoulos,et al.  Reflectionless Sponge Layers as Absorbing Boundary Conditions for the Numerical Solution of Maxwell Equations in Rectangular, Cylindrical, and Spherical Coordinates , 2000, SIAM J. Appl. Math..

[64]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms , 2006 .

[65]  Luc Giraud,et al.  New curvilinear scheme for elastic wave propagation in presence of curved topography , 2011 .

[66]  D. Komatitsch,et al.  An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation , 2007 .

[67]  Marcus J. Grote,et al.  Discontinuous Galerkin Finite Element Method for the Wave Equation , 2006, SIAM J. Numer. Anal..

[68]  S. Shapiro,et al.  Modeling the propagation of elastic waves using a modified finite-difference grid , 2000 .

[69]  Murthy N. Guddati,et al.  On Optimal Finite-Difference Approximation of PML , 2003, SIAM J. Numer. Anal..

[70]  Peter M. Pinsky,et al.  Finite element dispersion analysis for the three‐dimensional second‐order scalar wave equation , 1992 .

[71]  Vadim Lisitsa,et al.  Lebedev scheme for the numerical simulation of wave propagation in 3D anisotropic elasticity ‡ , 2010 .

[72]  M. Nafi Toksöz,et al.  Discontinuous-Grid Finite-Difference Seismic Modeling Including Surface Topography , 2001 .

[73]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[74]  Bent O. Ruud,et al.  2D finite‐difference viscoelastic wave modelling including surface topography , 2000 .

[75]  Martin Galis,et al.  On accuracy of the finite-difference and finite-element schemes with respect to P-wave to S-wave speed ratio , 2010 .

[76]  Ronnie Kosloff,et al.  Wave propagation simulation in a linear viscoacoustic medium , 1988 .

[77]  V. Tcheverda,et al.  Finite-difference algorithm with local time-space grid refinement for simulation of waves , 2011, Computational Geosciences.

[78]  Donald F. Winterstein,et al.  Velocity anisotropy terminology for geophysicists , 1990 .

[79]  Xi Chen,et al.  A hybrid Hermite-discontinuous Galerkin method for hyperbolic systems with application to Maxwell's equations , 2014, J. Comput. Phys..

[80]  S. Hestholm,et al.  3D free‐boundary conditions for coordinate‐transform finite‐difference seismic modelling , 2002 .

[81]  José M. Carcione,et al.  Hybrid modeling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures , 1997, Bulletin of the Seismological Society of America.

[82]  Erik H. Saenger,et al.  Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves , 2006 .

[83]  Martin Galis,et al.  A 3-D hybrid finite-difference—finite-element viscoelastic modelling of seismic wave motion , 2008 .

[84]  Tiegang Liu,et al.  Multidomain Hybrid RKDG and WENO Methods for Hyperbolic Conservation Laws , 2013, SIAM J. Sci. Comput..

[85]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes — II. The three-dimensional isotropic case , 2006 .