Practice considerations

There are three steps to implement plane-wave migration in tilted coordinates for one plane-wave source. First, the source and receiver wavefields are rotated to the new coordinates. Second, for each frequency, source and receiver wavefields are extrapolated into the subsurface and the image is generated by cross-correlating the two wavefields in the new coordinates. Third, the image is rotated back to the original Cartesian coordinates after stacking the images of all the frequencies. In the first step, we rotate a 2D dataset for each frequency, this cost is trivial compared to the wavefield extrapolation. In the third step, we rotate the 3D image back to Cartesian coordinates after stacking all frequencies, the cost is also very small.

To further reduce the cost of rotations, the plane-wave sources with similar propagation directions share the same tilted coordinate system. This reduces the rotation number in the third step. Figure illustrates how to sample the ray parameters p_x and p_y and how to put plane-wave sources together to share a coordinate system. In Figure we sample (p_x,p_y) in Cartesian coordinates, where p_x,p_y are defined as:

$$\begin{eqnarray} p_x=-p_{max},-p_{max}+dp_x,\cdots,p_{max}-dp_x,p_{max},\ p_y=-p_{max},-p_{max}+dp_y,\cdots,p_{max}-dp_y,p_{max},\end{eqnarray}$$ (21) (22)

where p_max is defined by the maximum take-off angle and dp_x, dp_y are the sampling of the ray parameters. In Figure , each dot represents a plane-wave source with a ray parameter pair (p_x,p_y). Given the surface velocity, we calculate the azimuth angle $\alpha$ and take-off angle $\theta$ from (p_x,p_y) and vice versa using equations 16 and 17. Therefore, we can divide the whole area into cells using $(\alpha,\theta)$ as the coordinate system. Figure shows the cells in the coordinates $(\alpha,\theta)$.All the plane-wave sources whose ray parameter pair (p_x,p_y) fall in a cell share the same coordinate system. The dots in the smallest circle in Figure represent plane-wave sources whose take-off angle $\theta$ is smaller than $15^\circ$. For those plane-wave sources, we extrapolate its source and receiver wavefields in Cartesian coordinates. For all plane-wave sources whose ray parameter pair (p_x,p_y) is in a cell, we use their average take-off angle and azimuth angle to design the coordinate system for the migration.