Wavefield extrapolation in pseudodepth domain

Anisotropic extrapolation

Velocity variation with angle in anisotropic media allows more choices for the mapping velocity $v_m$ in Equation 2. It is natural to use, but not limited to, the vertical velocity, $v_v$ , as $v_m$ .

The kinematics of a quasi-P wave in an anisotropic acoustic medium can be characterized by three parameters: P-wave velocity in the direction of the axis of symmetry $v_v$ , NMO velocity $v = v_v\sqrt{1+2\delta}$ and anellipticity $\eta = (\epsilon-\delta) / (1 + 2\delta)$  (Alkhalifah, 2000), here $\epsilon$ and $\delta$ are Thomsen parameters (Thomsen, 1986).

The quasi-P wave motion in transversely isotropic media with vertical axis of symmetry (VTI) is described by the following first-order system (Duveneck and Bakker, 2011)

$$\frac{\partial p_H}{\partial t} = (1+2\eta)v^2 \left(\frac{\partial q_1}{\partial x_1} + \frac{\partial q_2}{\partial x_2}\right) + v v_v\frac{\partial q_3}{\partial x_3}$$

$$\frac{\partial p_V}{\partial t} = v v_v \left(\frac{\partial q_1}{\partial x_1} + \frac{\partial q_2}{\partial x_2}\right) + v_v^2 \frac{\partial q_3}{\partial x_3}$$ (20)

$$\frac{\partial q_i}{\partial t} = \frac{\partial p_H}{\partial x_i} \quad (i=1,2)$$

$$\frac{\partial q_3}{\partial t} = \frac{\partial p_V}{\partial x_3} ,$$

where $p_H$ and $p_V$ are horizontal and vertical stresses, $-\mathbf{q}$ is the particle momentum.

In the $\tau$ domain, the wave equation is obtained by applying the chain rule 9 to 20, the resulting system of equations is

$$\frac{\partial p_H}{\partial t} = (1+2\eta)v^2 \sum_{i=1}^2 \left(\frac{\partial q_i}{\partial \x... ...{\partial \xi_3}\right) + \frac{v v_v}{v_m} \frac{\partial q_3}{\partial \xi_3}$$

$$\frac{\partial p_V}{\partial t} = v v_v \sum_{i=1}^2 \left(\frac{\partial q_i}{\partial \xi_i} + ... ...{\partial \xi_3}\right) + \frac{v_v^2}{v_m} \frac{\partial q_3}{\partial \xi_3}$$ (21)

$$\frac{\partial q_i}{\partial t} = \frac{\partial p_H}{\partial \xi_i} + \sigma_i\frac{\partial p_H}{\partial \xi_3} \quad (i=1,2)$$

$$\frac{\partial q_3}{\partial t} = \frac{1}{v_m} \frac{\partial p_V}{\partial \xi_3}$$

Equation 20 has the second-order form

$$\frac{\partial^2 p_H}{\partial t^2} = (1+2\eta)v^2 \left(\frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2}\right) p_H + v v_v\frac{\partial^2 p_V}{\partial x_3^2}$$

$$\frac{\partial^2 p_V}{\partial t^2} = v v_v \left(\frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2}\right) p_H + v_v^2 \frac{\partial p_V}{\partial x_3^2} .$$ (22)

Similarly, the second order form of 21 is

$$\frac{\partial^2 p_H}{\partial t^2} = (1+2\eta)v^2 \left[ \left(\frac{\partial}{\partial \xi_1} + \si... ...\partial \xi_3} \left(\frac{1}{v_m} \frac{\partial p_V}{\partial \xi_3} \right)$$

$$\frac{\partial^2 p_V}{\partial t^2} = v v_v \left[ \left(\frac{\partial}{\partial \xi_1} + \sigma_1\f... ...\partial \xi_3} \left(\frac{1}{v_m} \frac{\partial p_V}{\partial \xi_3} \right)$$ (23)

In addition to the cost reduction, the vertical time axis also allows time processing in VTI media to be independent of the vertical velocity $v_v$ , which is usually unresolvable from surface seismic data.

In transversely isotropic media with tilted axis of symmetry (TTI), the symmetry plane and symmetry axis are rotated by tilt angle $\theta$ and azimuth $\phi$ . The two-way wave equation is obtained by substituting derivatives in Equation 20 by the following relations

$$\frac{\partial}{\partial x_1} \leftarrow \cos\theta\cos\phi \frac{\partial}{\partial x_1} + \co... ...in\phi \frac{\partial}{\partial x_2} - \sin\theta \frac{\partial}{\partial x_3}$$

$$\frac{\partial}{\partial x_2} \leftarrow -\sin\phi \frac{\partial}{\partial x_1} + \cos\phi \frac{\partial}{\partial x_2}$$ (24)

$$\frac{\partial}{\partial x_3} \leftarrow \sin\theta\cos\phi \frac{\partial}{\partial x_1} + \si... ...\phi \frac{\partial}{\partial x_2} + \cos\theta \frac{\partial}{\partial x_3} .$$

In $\tau$ domain, the TTI extrapolation equation is obtained by replacing each of the spatial derivatives $\partial / \partial x_i \; (i=1,2,3)$ on the right-hand side of Equation 24 by the expressions given by chain rule in Equation 9. The $\tau$ coordinate transformation does not affect stability of the extrapolation.

Wavefield extrapolation in pseudodepth domain