Elastic wave-mode separation for TTI media

Wave-mode separation for symmetry planes of TTI media

My separation algorithm for TTI models is similar to the approach used for VTI models. The main difference is that for VTI media, the wavefields consist of P- and SV-modes, and equations 1 and 7 can be used for separation in all vertical planes of a VTI medium. However, for TTI media, this separation only works in the plane containing the dip of the reflector, where P- and SV-waves are polarized, while other vertical planes contain SH-waves as well.

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

$$G_{11} = c_{11} n_x^2 +2c_{15}n_xn_z+c_{55} n_z^2 ,$$ (8)

$$G_{12} = c_{15} n_x^2+\left (c_{13}+c_{55}\right)n_xn_z+c_{35} n_z^2 ,$$ (9)

$$G_{22} = c_{55} n_x^2 +2c_{35}n_xn_z+c_{33} n_z^2 .$$ (10)

Here, since the symmetry axis of the TTI medium does not align with the vertical axis $k_z$ , the TTI Christoffel matrix is different from its VTI equivalent. The stiffness tensor is determined by the parameters $V_{P0}$ , $V_{S0}$ , $\epsilon$ , $\delta$ , and the tilt angle $\nu$ .

In anisotropic media, $W_P$ generally deviates from the wave vector direction ${\bf k}=\frac{\omega}{V}{\bf n}$ , where $\omega$ is the angular frequency, $V$ 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$^\circ$ tilt angle, respectively. The polarization vectors for the VTI medium deviate from radial directions, which represent the isotropic polarization vectors ${\bf k}$ . The polarization vectors of the TTI medium are rotated 30$^\circ$ 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$^\circ$ 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 $\pi$  radians, the components of this TTI medium are rotated 30$^\circ$ from those of the VTI medium. However, note that the $z$ and $x$ components of the polarization vectors for the VTI medium (Figure ) are symmetric with respect to the $x$ and $z$ 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 $k_x,k_z=\pm\pi$  radians, one needs to apply tapers to the vector components. For VTI media, a taper corresponding to the function (Yan and Sava, 2009)

$$f(k)= -\frac{8\sin\left ( k\right)}{5k} + \frac{2\sin\left (2 k\r... ...}{5k} -\frac{8\sin\left (3 k\right)}{105k} + \frac{\sin\left (4 k\right)}{140k}$$ (11)

can be applied to the $x$ and $z$ components of the polarization vectors (Figure ), where $k$ represent the components $k_x$ and $k_z$ of the vector ${\bf k}$ . This taper ensures that $U_x$ and $U_z$ are zero at $k_z=\pm\pi$  radians and $k_x=\pm\pi$  radians, respectively. The components $U_x$ and $U_z$ are continuous in the $z$ and $x$ 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$^{th}$ order derivatives. If the components of the isotropic polarization vectors ${\bf k}$ are tapered by the function in equation 11 and then transformed to the space domain, one obtains the conventional 8$^{th}$ order finite difference derivative operators $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial z}$  (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

$$g({\bf k})=C exp\left [-\frac{\left\vert{\bf k}\right\vert^2}{2\sigma^2}\right]$$ (12)

can be used, where C is a normalizing constant. When one chooses a standard deviation of $\sigma =1$ radian, the magnitude of this taper at $\left\vert{\bf k}\right\vert=\pi$  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 $\nu$ .

The value of $\sigma$ 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 $U_z$ and operator $L_z$ for $\sigma$ values of $0.25$ , $1.00$ , and $1.25$ radians. A larger value of $\sigma$ 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 $g({\bf k})$ at $\left\vert{\bf k}\right\vert=\pi$  radians is small enough to assume continuity of the value function across Nyquist wavenumbers. When one chooses $\sigma =1$ 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 $V_{P0}/ V_{S0}$ ratio, the Thomsen parameters $\epsilon$ and $\delta$ , and tilt angle $\nu$ . I then transform the tapered polarization vector components to the space domain to obtain the spatially-varying separators $L_x$ and $L_z$ . 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
Figure 1. The polarization vectors of P-mode as a function of normalized wavenumbers $k_x$ and $k_z$ ranging from $-\pi$  radians to $+\pi$  radians, for (a) a VTI model with $V_{P0}=3.0$  km/s, $V_{S0}=1.5$  km/s, $\epsilon=0.25$ and $\delta=-0.29$ , and for (b) a TTI model with the same model parameters as (a) and a symmetry axis tilt $\nu=30^\circ$ . The vectors in (b) are rotated 30$^\circ$ with respect to the vectors in (a) around $k_x=0$ and $k_z=0$ .

dK-notaper-VTI,dK-notaper-TTI,dK-notaper-rot-TTI
Figure 2. The $z$ and $x$ components of the polarization vectors for P-mode in the Fourier domain for (a) a VTI medium with $\epsilon=0.25$ and $\delta=-0.29$ , and for (b) a TTI medium with $\epsilon=0.25$ , $\delta=-0.29$ , and $\nu=30^\circ$ . Panel (c) represents the projection of the polarization vectors shown in (b) onto the tilt axis and its orthogonal direction.

dK-VTI,dK-TTI,dK-rot-TTI
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).

dX-VTI,dX-TTI,dX-rot-TTI
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 $24\times 24$ samples out of the original $64\times 64$ samples around the center of the filters.

dzKX-sig0-TTI,dzKX-sig1-TTI,dzKX-sig2-TTI
Figure 5. Panels (a)-(c) correspond to component $U_z$ (left) and operator $L_z$ (right) for $\sigma$ values of $0.25$ , $1.00$ , and $1.25$ radians in equation 12, respectively. A larger value of $\sigma$ results in more spread components in the wavenumber domain and more concentrated operators in the space domain.

Elastic wave-mode separation for TTI media