Plane-wave migration in tilted coordinates

Plane-wave source migration

Shot gathers can also be synthesized into a new dataset to represent a physical experiment that is not performed in reality. One of the most important examples is to synthesize shot gathers into plane-wave source gathers. A plane-wave source gather represents what would be recorded if a planar source were excited at the surface with geophones covering the whole area. It can also be regarded as the accurate phase-encoding of the shot gathers (Liu et al., 2002). Plane-wave source gathers can be generated by slant-stacking receiver gathers. The process can be described as follows:

$$R_p(p_x,r_x,z=0,\omega)=\int R(s_x,r_x,z=0,\omega)e^{i\omega s_xp_x}ds_x,$$ (6)

where $p_x$ is the ray parameter for the $x$-axis, $s_x$ is the source location, and $r_x$ is the receiver location at the surface. Its corresponding plane-wave source wavefield at the surface is

$$S_p(p_x,r_x,z=0,\omega)=e^{i\omega r_xp_x}.$$ (7)

As with the Fourier transformation, we can transform the plane-wave source gathers back to shot gathers by inverse slant-stacking (Claerbout, 1985) as follows:

$$R(s_x,r_x,z=0,\omega)= \int \omega R_p(p_x,r_x,z=0,\omega)e^{-i\omega s_xp_x} dp_x.$$ (8)

In contrast to the inverse Fourier transformation, the kernel of the integral is weighted by the angular frequency $\omega$. This inverse transformation weighting function is also called $\rho$ filter in Radon-transform literature.

As with shot-profile migration, there are two steps to migrate a plane-wave source gather by a typical plane-wave migration method. First, the source wavefield $S_p$ and receiver wavefield $R_p$ are extrapolated into all depths in the subsurface independently, using the one-way wave equations 2 and 3, respectively. Second, the image of a plane-wave source with a ray parameter $p_x$ is constructed by cross-correlating the source and receiver wavefields weighted with the angular frequency $\omega$:

$$I_{p_x}(x,z)=\int \omega {S_p}^*(p_x,x,z,\omega)R_p(p_x,x,z,\omega) d\omega,$$ (9)

where ${S_p}^*$ is the conjugate complex of the source wavefield $S_p$. The whole image is formed by stacking the images of all possible plane-wave sources:

$$I_{p}=\int\int I_{p_x}(x,z) dp_x.$$ (10)

Because both slant-stacking and migration are linear operators, the image of the plane-wave migration $I_p$ is equivalent to the image obtained by shot-profile migration (Liu et al., 2002; Zhang et al., 2005). In the discrete form, in practice we need a sufficient number of $p_x$ to make the two images equivalent.

Plane-wave migration in tilted coordinates