3D plane-wave migration

The original surface seismic data are usually shot gathers. A typical seismic shot gather (the receiver wavefield of a shot at the surface) is a five dimensional object: R(s_y,s_x,r_y,r_x,z=0,t), where (s_y,s_x) is the source location, (r_x,r_y) is the receiver location and t is the travel time. After a Fourier transformation in t, we have the receiver wavefield in the frequency domain $R(s_y,s_x,r_y,r_x,z=0,\omega)$, where $\omega$ is the angular frequency.

Each shot represents a real physical experiment. The most straight forward way to obtain the image of the subsurface is shot-profile migration, in which we obtain the local image of each experiment independently and form the final image of the subsurface by stacking all the local images. A typical shot-profile migration algorithm includes two steps. First, source and receiver wavefields are extrapolated into the subsurface using one-way wave equations. In isotropic media they are defined as follows:

$$\begin{eqnarray} \frac{\partial S}{\partial z}=-\frac{i\omega}{v}\sqrt{1+\frac{v... ...tial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2} \right) }R,\end{eqnarray}$$ (1) (2)

where v=v(x,y,z) is the velocity of the media, $S=S(s_y,s_x;x,y,z,\omega)$ is the source wavefield, which is an impulse at the surface and $R=R(s_y,s_x;r_y,r_x,z,\omega)$ is the receiver wavefield. Second, the image is formed by cross-correlating the source and receiver wavefields:

$$I(x,y,z)= \int \int \int \bar{S}(s_x,s_y;x,y,z,\omega)R(s_x,s_y;x,y,z,\omega)d\omega ds_xds_y.$$ (3)

 

• Plane-wave source migration
• Conical-wave source migration