Extrapolation operator in laterally varying media

The phase-shift method is effective, but it is not suitable for a strongly heterogeneous medium, where strong lateral changes are present in velocity as well as in the anisotropy parameters, $\varepsilon$ and $\delta$.This can be remedied by PSPI Rousseau (1997), explicit operator Zhang et al. (2001), or reference anisotropic phase-shift with an explicit correction filter Baumstein and Anderson (2003).

In this paper, we use an explicit anisotropic correction filter in addition to the normal isotropic operator. For each z step, we first regard the medium as an isotropic medium and extrapolate the wavefield using an isotropic operator with the velocity in the direction parallel to the symmetry axis. The isotropic operator can be the split-step method Stoffa et al. (1990), the general screen propagator Huang and Wu (1996), or Fourier finite difference (FFD) Ristow and Ruhl (1994). Then we correct the wavefield with an explicit correction operator.

After we extrapolate the wavefield with an isotropic operator, the resulting error relative to anisotropic extrapolation is:  

$$F(k_x)=e^{i\Delta z\Delta \phi(k_x)},$$ (6)

where $\Delta \phi(k_x)$ is the difference between the isotropic wavenumber, k_zⁱ, and the anisotropic wavenumber, k_z^a, that satisfies

$$\Delta \phi(k_x)=k_z^a(V_{P0},\delta,\varepsilon,\varphi)-k_z^i(V_{P0}),$$ (7)

where,

$$k_z^i(V_{P0})=\pm\sqrt{\left(\frac{\omega}{V_{P0}}\right)^2-k_x^2},$$

and k_z^a is one of the four roots of the dispersion-relation equation (4). We design the explicit correction operator by weighted least squares. The obtained explicit operator is approximately the same as F(k_x) in the wavenumber domain for desired wavenumbers.