VTI migration velocity analysis using RTM

First-order Two-way VTI wave-equation

The first-order two-way VTI wave-equation can be derived from Hooke's law and Newton's law using Thomson anisotropy parameters ($\epsilon$ , $\delta$ ) and setting shear wave velocity $c_s = 0$ (Duveneck et al., 2008). The first-order system reads as follows:

$$\rho \partial_t v_x = - \partial_x p_H$$

$$\rho \partial_t v_y = - \partial_y p_H$$

$$\rho \partial_t v_z = - \partial_z p_V$$ (1)

$$\frac{1}{\rho c^2}\partial_t p_V = - \sqrt{(1+2\delta)}(\partial_x v_x + \partial_y v_y) - \partial_z v_z + f_V$$

$$\frac{1}{\rho c^2}\partial_t p_H = - (1+2\epsilon)(\partial_x v_x + \partial_y v_y) - \sqrt{(1+2\delta)} \partial_z v_z + f_H$$

where $\rho$ is the density, $c$ is the velocity, $(v_x, v_y, v_z)$ is the particle velocity vector, and $p_V$ and $p_H$ are pressure in the vertical and horizontal directions, respectively. The source term $f_V$ and $f_H$ are defined by the source wavelet $w(t)$ as follows:

$$f_V(t)=f_H(t)=\int_{-\infty}^{\tau}w(\tau)d\tau.$$ (2)

It is straightforward to see that when $\rho=1$ , $\epsilon=0$ and $\delta=0$ , the first-order system 1 is equivalent to the familiar isotropic acoustic second-order wave-equation:

$$\frac{1}{c^2} \partial_{t}^{2} p - \nabla p = w.$$ (3)

For simplicity, we can rewrite system 1 in a matrix-vector notation:

$${\bf L}(c) {\bf p} = {\bf f},$$ (4)

where ${\bf p} = (v_x, ~v_y, ~v_z, ~p_V, ~p_H)^T$ , ${\bf f} = (0, ~0, ~0, ~f_V, ~f_H)^T$ , and

$${\bf L} = \left \vert \begin{array}{ccccc} \partial_t & 0 & 0 & 0... ...2\delta} \partial_z & 0 & \frac{1}{c^2} \partial_t \\ \end{array} \right \vert.$$ (5)

VTI migration velocity analysis using RTM