Phase shift plus interpolation

The simplest extension to the phase shift algorithm is ``phase shift plus interpolation'' (PSPI) Gazdag and Sguazzero (1984). This algorithm approximates the extrapolation of the wavefield in a spatially varying velocity field by interpolating the results of a set of constant velocity extrapolations. The interpolation is performed in the space domain. This method therefore requires an inverse Fourier transform of each constant velocity panel back to the space domain (x or $\theta$), and a forward transform of the interpolated result to the wavenumber domain, for each extrapolation step in z or r.

I use the simplest form of the PSPI algorithm which uses only two constant velocity extrapolations. The two velocities are the maximum and minimum velocity at the current depth or radius.

In rectangular coordinates, In polar coordinates the algorithm has the same form with r replacing z.

Figure  shows the results of modeling in a medium which has a linear gradient in x and z. The velocity field in this model is $v(x,z) = v_0+ \alpha (x-x_0) + \beta (z-z_0).$ The plot on the left is computed in a rectangular coordinates and the one on the right in polar coordinates. They both show an asymmetric wavefront produced by the velocity gradient The polar coordinate result contains energy at higher dips that the rectangular coordinate result because it does not start with a dip filtered impulse.

 

grad-both
Figure 5 A time slice from modeling in a linear gradient medium. The velocity field has a gradient of 5 (ms^-1)/m in depth and 5 (ms^-1)/m in x. The result on the left is computed in rectangular coordinates, the one on the right is computed in polar coordinates.