Elastic wave-mode separation for TTI media |

where is the P-wave mode in the wavenumber domain, is the wavenumber vector, is the elastic wavefield in the wavenumber domain, and is the P-wave polarization vector as a function of the wavenumber .

The polarization vectors of plane waves for VTI media in the symmetry planes can be found by solving the Christoffel equation (Aki and Richards, 2002; Tsvankin, 2005):

where

For plane waves in the vertical symmetry plane of a TTI medium, since
*q*P and *q*SV modes are decoupled from the SH-mode and
polarized in the symmetry planes, one can set
and obtain

where

(4) | |||

(5) | |||

(6) |

Equation 3 allows one to compute the polarization vectors and (the eigenvectors of the matrix

Equation 1 represents the separation process for the P-mode in 2D homogeneous VTI media. To separate wave-modes for heterogeneous models, one needs to use different polarization vectors at every location of the model (Yan and Sava, 2009), because the polarization vectors change spatially with medium parameters. In the space domain, an expression equivalent to equation 1 at each grid point is

where indicates spatial filtering, and and are the filters to separate P waves representing the inverse Fourier transforms of and , respectively. The terms and define the ``pseudo-derivative operators'' in the and directions for a VTI medium, respectively, and they change according to the material parameters, , ( and are the P and S velocities along the symmetry axis, respectively), , and (Thomsen, 1986).

Elastic wave-mode separation for TTI media |

2013-08-29