Coupling the spectral element method with a modal solution for elastic wave propagation in global earth models

SUMMARY We present a new method for wave propagation in global earth models based upon the coupling between the spectral element method and a modal solution method. The Earth is decomposed into two parts, an outer shell with 3-D lateral heterogeneities and an inner sphere with only spherically symmetric heterogeneities. Depending on the problem, the outer heterogeneous shell can be mapped as the whole mantle or restricted only to the upper mantle or the crust. In the outer shell, the solution is sought in terms of the spectral element method, which stem from a high order variational formulation in space and a second-order explicit scheme in time. In the inner sphere, the solution is sought in terms of a modal solution in frequency after expansion on the spherical harmonics basis. The spectral element method combines the geometrical flexibility of finite element methods with the exponential convergence rate of spectral methods. It avoids the pole problems and allows for local mesh refinement, using a non-conforming discretization, for the resolution of sharp variations and topography along interfaces. The modal solution allows for an accurate isotropic representation in the inner sphere. The coupling is introduced within the spectral element method via a Dirichlet-to-Neumann (DtN) operator. The operator is explicitly constructed in frequency and in generalized spherical harmonics. The inverse transform in space and time requires special attention and an asymptotic regularization. The coupled method allows a significant speed-up in the simulation of the wave propagation in earth models. For spherically symmetric earth model, the method is shown to have the accuracy of spectral transform methods and allow the resolution of wavefield propagation, in 3-D laterally heterogeneous models, without any perturbation hypothesis.

[1]  J. Woodhouse The coupling and attenuation of nearly resonant multiplets in the Earth's free oscillation spectrum , 1980 .

[2]  Wolfgang Friederich,et al.  COMPLETE SYNTHETIC SEISMOGRAMS FOR A SPHERICALLY SYMMETRIC EARTH BY A NUMERICAL COMPUTATION OF THE GREEN'S FUNCTION IN THE FREQUENCY DOMAIN , 1995 .

[3]  Robert J. Geller,et al.  Computation of synthetic seismograms and their partial derivatives for heterogeneous media with arbitrary natural boundary conditions using the Direct Solution Method , 1994 .

[4]  Toshiro Tanimoto,et al.  Waveforms of long-period body waves in a slightly aspherical earth model , 1993 .

[5]  F. Pollitz Scattering of spherical elastic waves from a small-volume spherical inclusion , 1998 .

[6]  Wei-jia Su,et al.  Degree 12 model of shear velocity heterogeneity in the mantle , 1994 .

[7]  Bengt Fornberg,et al.  A Pseudospectral Approach for Polar and Spherical Geometries , 1995, SIAM J. Sci. Comput..

[8]  E. R. Engdahl,et al.  Evidence for deep mantle circulation from global tomography , 1997, Nature.

[9]  KosloffDan,et al.  A modified Chebyshev pseudospectral method with an O(N1) time step restriction , 1993 .

[10]  B. Kennett,et al.  The velocity structure and heterogeneity of the upper mantle , 1990 .

[11]  David A. Randall,et al.  Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid. Part II. A Detailed Description of the Grid and an Analysis of Numerical Accuracy , 1995 .

[12]  Marcus J. Grote,et al.  On nonreflecting boundary conditions , 1995 .

[13]  K. Aki Attenuation and Scattering of Short-Period Seismic Waves in the Lithosphere , 1981 .

[14]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[15]  F. Gilbert Excitation of the Normal Modes of the Earth by Earthquake Sources , 1971 .

[16]  Faker Ben Belgacem,et al.  The Mortar finite element method with Lagrange multipliers , 1999, Numerische Mathematik.

[17]  B. Kennett Observational and theoretical constraints on crustal and upper mantle heterogeneity , 1987 .

[18]  A. Patera,et al.  Spectral element methods for the incompressible Navier-Stokes equations , 1989 .

[19]  Roel Snieder,et al.  The spectrum of tomographic earth models , 1998 .

[20]  D. Helmberger,et al.  Modelling D″ structure beneath Central America with broadband seismic data , 1997 .

[21]  J. Tromp,et al.  Variational principles for surface wave propagation on a laterally heterogeneous Earth—II. Frequency-domain JWKB theory , 1992 .

[22]  F. Dahlen The Normal Modes of a Rotating, Elliptical Earth , 1968 .

[23]  Joseph S. Resovsky,et al.  A degree 8 mantle shear velocity model from normal mode observations below 3 mHz , 1999 .

[24]  P. Silver,et al.  Constraints from seismic anisotropy on the nature of the lowermost mantle , 1996, Nature.

[25]  R. Snieder,et al.  A new formalism for the effect of lateral heterogeneity on normal modes and surface waves—I: isotropic perturbations, perturbations of interfaces and gravitational perturbations , 1988 .

[26]  G. Laske,et al.  A shear - velocity model of the mantle , 1996, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[27]  J. Sochacki Absorbing boundary conditions for the elastic wave equations , 1988 .

[28]  D. Rockmore,et al.  Computational Harmonic Analysis for Tensor Fields on the Two-Sphere , 2000 .

[29]  V. Rokhlin,et al.  A generalized one-dimensional fast multipole method with application to filtering of spherical harmonics , 1998 .

[30]  Robert J. Geller,et al.  Computation of complete synthetic seismograms for laterally heterogeneous models using the Direct Solution Method , 1997 .

[31]  H. Takeuchi,et al.  Seismic Surface Waves , 1972 .

[32]  Adam M. Dziewonski,et al.  Global seismic tomography of the mantle , 1995 .

[33]  R. Geller,et al.  Inversion for laterally heterogeneous upper mantle S-wave velocity structure using iterative waveform inversion , 1993 .

[34]  P. Lognonné,et al.  10 – Normal Modes of the Earth and Planets , 2002 .

[35]  Roger Penrose,et al.  Note on the Bondi-Metzner-Sachs Group , 1966 .

[36]  F. Dahlen The Normal Modes of a Rotating, Elliptical Earth—II Near-Resonance Multiplet Coupling , 1969 .

[37]  D. F. Johnston,et al.  Representations of the Rotation and Lorentz Groups and Their Applications , 1965 .

[38]  F. Mesinger,et al.  A global shallow‐water model using an expanded spherical cube: Gnomonic versus conformal coordinates , 1996 .

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

[40]  Bradley K. Alpert,et al.  A Fast Spherical Filter with Uniform Resolution , 1997 .

[41]  Heiner Igel,et al.  P‐SV wave propagation in the Earth's mantle using finite differences: Application to heterogeneous lowermost mantle structure , 1996 .

[42]  D. Vasco,et al.  Whole Earth structure estimated from seismic arrival times , 1998 .

[43]  Brian Kennett,et al.  Seismic wavefield calculation for laterally heterogeneous earth models—II. The influence of upper mantle heterogeneity , 1999 .

[44]  Bengt Fornberg,et al.  The pseudospectral method; accurate representation of interfaces in elastic wave calculations , 1988 .

[45]  Robert J. Geller,et al.  Complete synthetic seismograms for 3-D heterogeneous Earth models computed using modified DSM operators and their applicability to inversion for Earth structure , 2000 .

[46]  B. Romanowicz,et al.  Seismic anisotropy in the D″ layer , 1995 .

[47]  D. Kosloff,et al.  Solution of the equations of dynamic elasticity by a Chebychev spectral method , 1990 .

[48]  R. Sadourny Conservative Finite-Difference Approximations of the Primitive Equations on Quasi-Uniform Spherical Grids , 1972 .

[49]  B. Romanowicz,et al.  Modelling of coupled normal modes of the Earth: the spectral method , 1990 .

[50]  Jacobo Bielak,et al.  On absorbing boundary conditions for wave propagation , 1988 .

[51]  Robert J. Geller,et al.  DSM complete synthetic seismograms : SH, spherically symmetric, case , 1994 .

[52]  P. Lognonné Normal modes and seismograms in an anelastic rotating Earth , 1991 .

[53]  Emmanuel Chaljub,et al.  Sensitivity of SS precursors to topography on the upper‐mantle 660‐km discontinuity , 1997 .

[54]  Dan Kosloff,et al.  A modified Chebyshev pseudospectral method with an O(N –1 ) time step restriction , 1993 .

[55]  Thorne Lay,et al.  The core–mantle boundary layer and deep Earth dynamics , 1998, Nature.

[56]  A. Patera A spectral element method for fluid dynamics: Laminar flow in a channel expansion , 1984 .

[57]  W. Ames The Method of Weighted Residuals and Variational Principles. By B. A. Finlayson. Academic Press, 1972. 412 pp. $22.50. , 1973, Journal of Fluid Mechanics.

[58]  Saito Masanori,et al.  DISPER80; a subroutine package for calculation of seismic normal-mode solutions , 1988 .

[59]  K. R. Kelly,et al.  SYNTHETIC SEISMOGRAMS: A FINITE ‐DIFFERENCE APPROACH , 1976 .

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

[61]  Z. Alterman,et al.  Pulse Propagation in a Laterally Heterogeneous Solid Elastic Sphere , 1970 .

[62]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[63]  B. Engquist,et al.  Absorbing boundary conditions for acoustic and elastic wave equations , 1977, Bulletin of the Seismological Society of America.

[64]  J. Tromp,et al.  Theoretical Global Seismology , 1998 .

[65]  C. Bernardi,et al.  Approximations spectrales de problèmes aux limites elliptiques , 2003 .

[66]  Robert A. Phinney,et al.  Representation of the Elastic ‐ Gravitational Excitation of a Spherical Earth Model by Generalized Spherical Harmonics , 1973 .

[67]  Géza Seriani,et al.  Numerical simulation of interface waves by high‐order spectral modeling techniques , 1992 .

[68]  M. A. Dablain,et al.  The application of high-order differencing to the scalar wave equation , 1986 .

[69]  Mrinal K. Sen,et al.  Evidence for anisotropy in the deep mantle beneath Alaska , 1996 .

[70]  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.

[71]  Philip E. Merilees,et al.  The pseudospectral approximation applied to the shallow water equations on a sphere , 1973 .

[72]  T. Wu,et al.  Some Properties of Monopole Harmonics , 1977 .

[73]  P. Paolucci,et al.  The “Cubed Sphere” , 1996 .

[74]  B. Romanowicz,et al.  First-order asymptotics for the eigenfrequencies of the earth and application to the retrieval of large-scale lateral variations of structure , 1986 .

[75]  C. Young,et al.  Scale lengths of shear velocity heterogeneity at the base of the mantle from S wave differential travel times , 1997 .

[76]  W. Friederich Propagation of seismic shear and surface waves in a laterally heterogeneous mantle by multiple forward scattering , 1999 .

[77]  Ezio Faccioli,et al.  2d and 3D elastic wave propagation by a pseudo-spectral domain decomposition method , 1997 .

[78]  B. Romanowicz,et al.  The COSY Project: verification of global seismic modeling algorithms , 2000 .

[79]  Barbara Romanowicz,et al.  Coupling spectral elements and modes in a spherical Earth: an extension to the ‘sandwich’ case , 2003 .

[80]  Yvon Maday,et al.  A spectral element methodology tuned to parallel implementations , 1994 .

[81]  Eli Turkel,et al.  A fourth-order accurate finite-difference scheme for the computation of elastic waves , 1986 .

[82]  D. L. Anderson,et al.  Preliminary reference earth model , 1981 .

[83]  Sri Widiyantoro,et al.  Global seismic tomography: A snapshot of convection in the Earth: GSA Today , 1997 .

[84]  Mrinal K. Sen,et al.  Seismic anisotropy in the core–mantle transition zone , 1998 .

[85]  Eugene M. Lavely,et al.  Three‐dimensional seismic models of the Earth's mantle , 1995 .

[86]  Barbara Romanowicz,et al.  Global mantle shear velocity model developed using nonlinear asymptotic coupling theory , 1996 .

[87]  Hiroshi Takenaka,et al.  Parallel 3-D pseudospectral simulation of seismic wave propagation , 1998 .

[88]  B. Romanowicz Multiplet-multiplet coupling due to lateral heterogeneity: asymptotic effects on the amplitude and frequency of the Earth's normal modes , 1987 .

[89]  Barbara Romanowicz,et al.  Seismic waveform modeling and surface wave tomography in a three-dimensional Earth: asymptotic and non-asymptotic approaches , 2000 .