|Elastic wave-mode separation for TTI media|
To obtain the polarization vectors for P and S modes in the symmetry
planes of TTI media, one needs to solve for the Christoffel equation 3
In anisotropic media, generally deviates from the wave vector direction , where is the angular frequency, is the phase vector. Figures 1(a) and fig:TTIpolar show the P-mode polarization in the wavenumber domain for a VTI medium and a TTI medium with a 30 tilt angle, respectively. The polarization vectors for the VTI medium deviate from radial directions, which represent the isotropic polarization vectors . The polarization vectors of the TTI medium are rotated 30 about the origin from the vectors of the VTI medium.
Figures and fig:dK_notaper_TTI show the components of the P-wave polarization of a VTI medium and a TTI medium with a 30 tilt angle, respectively. Figure shows that the polarization vectors in Figure rotated to the symmetry axis and its orthogonal direction of the TTI medium. Comparing Figures and fig:dK_notaper_rot_TTI, we see that within the circle of radius radians, the components of this TTI medium are rotated 30 from those of the VTI medium. However, note that the and components of the polarization vectors for the VTI medium (Figure ) are symmetric with respect to the and axes, respectively; in contrast, the vectors of the TTI medium (Figure ) are not symmetric because of the non-alignment of the TTI symmetry with the Cartesian coordinates.
To maintain continuity at the negative and positive Nyquist wavenumbers for Fourier transform to obtain space-domain filters, i.e. at radians, one needs to apply tapers to the vector components. For VTI media, a taper corresponding to the function (Yan and Sava, 2009)
For TTI media, due to the asymmetry of the Fourier domain derivatives (Figure ), one needs to apply a rotational symmetric taper to the polarization vector components to obtain continuity across Nyquist wavenumbers. A simple Gaussian taper
The value of determines the size of the operators in the space domain and also affects the frequency content of the separated wave-modes. For example, Figure 5 shows the component and operator for values of , , and radians. A larger value of results in more concentrated operators in the space domain and better preserved frequency of the separated wave-modes. However, one needs to ensure that the function at radians is small enough to assume continuity of the value function across Nyquist wavenumbers. When one chooses radian, the TTI components can be safely assumed to be continuous across the Nyquist wavenumbers.
For heterogeneous models, I can pre-compute the polarization vectors at each grid point as a function of the ratio, the Thomsen parameters and , and tilt angle . I then transform the tapered polarization vector components to the space domain to obtain the spatially-varying separators and . The separators for the entire model are stored and used to separate P- and S-modes from reconstructed elastic wavefields at different time steps. Thus, wavefield separation in TI media can be achieved simply by non-stationary filtering with spatially varying operators. I assume that the medium parameters vary slowly in space and that they are locally homogeneous. For complex media, the localized operators behave similarly to the long finite difference operators used for finite difference modeling at locations where medium parameters change rapidly.
Figure 1. The polarization vectors of P-mode as a function of normalized wavenumbers and ranging from radians to radians, for (a) a VTI model with km/s, km/s, and , and for (b) a TTI model with the same model parameters as (a) and a symmetry axis tilt . The vectors in (b) are rotated 30 with respect to the vectors in (a) around and .
Figure 2. The and components of the polarization vectors for P-mode in the Fourier domain for (a) a VTI medium with and , and for (b) a TTI medium with , , and . Panel (c) represents the projection of the polarization vectors shown in (b) onto the tilt axis and its orthogonal direction.
Figure 3. The wavenumber-domain vectors in Figure 2 are tapered by the function in equation 12 to avoid Nyquist discontinuity. Panel (a) corresponds to Figure 2(a), panel (b) corresponds to Figure 2(b), and panel (c) corresponds to Figure 2(c).
Figure 4. The space-domain wave-mode separators for the medium shown in Figure 1. They are the Fourier transformation of the polarization vectors shown in Figure 3. Panel (a) corresponds to Figure 3(a), panel (b) corresponds to Figure 3(b), and panel (c) corresponds to Figure 3(c). The zoomed views show samples out of the original samples around the center of the filters.
Figure 5. Panels (a)-(c) correspond to component (left) and operator (right) for values of , , and radians in equation 12, respectively. A larger value of results in more spread components in the wavenumber domain and more concentrated operators in the space domain.
|Elastic wave-mode separation for TTI media|