Elastic modelling in tilted transversely isotropic media with convolutional perfectly matched layer boundary conditions

To simulate wave propagation in a tilted transversely isotropic (TTI) medium with a tilting symmetry-axis of anisotropy, we develop a 2D elastic forward modelling algorithm. In this algorithm, we use the staggered-grid finite-difference method which has fourth-order accuracy in space and second-order accuracy in time. Since velocity-stress formulations are defined for staggered grids, we include auxiliary grid points in the z-direction to meet the free surface boundary conditions for shear stress. Through comparisons of displacements obtained from our algorithm, not only with analytical solutions but also with finite element solutions, we are able to validate that the free surface conditions operate appropriately and elastic waves propagate correctly. In order to handle the artificial boundary reflections efficiently, we also implement convolutional perfectly matched layer (CPML) absorbing boundaries in our algorithm. The CPML sufficiently attenuates energy at the grazing incidence by modifying the damping profile of the PML boundary. Numerical experiments indicate that the algorithm accurately expresses elastic wave propagation in the TTI medium. At the free surface, the numerical results show good agreement with analytical solutions not only for body waves but also for the Rayleigh wave which has strong amplitude along the surface. In addition, we demonstrate the efficiency of CPML for a homogeneous TI medium and a dipping layered model. Only using 10 grid points to the CPML regions, the artificial reflections are successfully suppressed and the energy of the boundary reflection back into the effective modelling area is significantly decayed. In this study, we developed a forward modelling algorithm with the staggered-grid finite-difference method. The elastic wave propagation in 2D tilted transversely isotropic media is well simulated using this algorithm. We also implement convolutional perfectly matched layer absorbing boundaries in our algorithm in order to handle the artificial boundary reflection efficiently.

[1]  J. Bancroft,et al.  Finite Difference Modeling In Structurally Complex Anisotropic Medium , 2005 .

[2]  Robert W. Graves,et al.  Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences , 1996, Bulletin of the Seismological Society of America.

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

[4]  Chrysoula Tsogka,et al.  Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic hete , 1998 .

[5]  Roland Martin,et al.  An unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media , 2008 .

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

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

[8]  R. Higdon.  Absorbing boundary conditions for elastic waves , 1991 .

[9]  F. Hu.  Absorbing Boundary Conditions , 2004 .

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

[11]  Jean Virieux,et al.  SH-wave propagation in heterogeneous media: velocity-stress finite-difference method , 1984 .

[12]  Tariq Alkhalifah,et al.  An acoustic wave equation for anisotropic media , 2000 .

[13]  John B. Schneider,et al.  Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation , 1996 .

[14]  John L. Volakis,et al.  Absorbing boundary conditions , 1995 .

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

[16]  M. Gunzburger,et al.  Boundary conditions for the numerical solution of elliptic equations in exterior regions , 1982 .

[17]  Heiner Igel,et al.  Anisotropic wave propagation through finite-difference grids , 1995 .

[18]  P. Stoffa,et al.  Finite‐difference modeling in transversely isotropic media , 1994 .

[19]  Jean Virieux,et al.  Finite-difference frequency-domain modeling of viscoacoustic wave propagation in 2D tilted transversely isotropic (TTI) media , 2009 .

[20]  Stephen D. Gedney,et al.  Convolution PML (CPML): An efficient FDTD implementation of the CFS–PML for arbitrary media , 2000 .

[21]  D. Komatitsch,et al.  Simulation of anisotropic wave propagation based upon a spectral element method , 2000 .

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

[23]  Jeroen Tromp,et al.  A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation , 2003 .

[24]  G. Mur.  Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations , 1981, IEEE Transactions on Electromagnetic Compatibility.

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

[26]  Moshe Reshef,et al.  A nonreflecting boundary condition for discrete acoustic and elastic wave equations , 1985 .

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

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

[29]  Changsoo Shin.  Sponge boundary condition for frequency-domain modeling , 1995 .

[30]  C. Juhlin.  Finite‐difference elastic wave propagation in 2D heterogeneous transversely isotropic media1 , 1995 .