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
with

(8) | |||

(9) | |||

(10) |

Here, since the symmetry axis of the TTI medium does not align with the vertical axis , the TTI Christoffel matrix is different from its VTI equivalent. The stiffness tensor is determined by the parameters , , , , and the tilt angle .

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)

can be applied to the and components of the polarization vectors (Figure ), where represent the components and of the vector . This taper ensures that and are zero at radians and radians, respectively. The components and are continuous in the and directions across the Nyquist wave numbers, respectively, due to the symmetry of the VTI media. Moreover, the application of this taper transforms polarization vector components to 8 order derivatives. If the components of the isotropic polarization vectors are tapered by the function in equation 11 and then transformed to the space domain, one obtains the conventional 8 order finite difference derivative operators and (Yan and Sava, 2009). Therefore, the VTI separators reduce to conventional derivatives--the components of the divergence and curl operators--when the medium is isotropic.

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

can be used, where C is a normalizing constant. When one chooses a standard deviation of radian, the magnitude of this taper at radians is about 0.7% of the peak value, and therefore the TTI components can be safely assumed to be continuous across the Nyquist wavenumbers. Tapering the polarization vector components in Figure 2 with the function in equation 12, one obtains the plots in Figure 3. The panels in Figure 3, which exhibits circular continuity across the Nyquist wavenumbers, transform to the space-domain separators in Figure 4. The space-domain filters for TTI media is rotated from the VTI filters, also by the tilt angle .

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.

VTIpolar,TTIpolar
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 1. |
---|

dK-notaper-VTI,dK-notaper-TTI,dK-notaper-rot-TTI
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 2. |
---|

dK-VTI,dK-TTI,dK-rot-TTI
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 3. |
---|

dX-VTI,dX-TTI,dX-rot-TTI
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 4. |
---|

dzKX-sig0-TTI,dzKX-sig1-TTI,dzKX-sig2-TTI
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.
Figure 5. |
---|

Elastic wave-mode separation for TTI media |

2013-08-29