THE PSPM OPERATOR

Prestack partial migration is a mapping of prestack data to zero-offset data. This mapping can be divided in two steps:

• Full prestack migration to a depth model. An input data point in a constant offset section is migrated into an ellipse.
• Zero-offset modeling. Each point on the ellipse with a local dip angle $\alpha$, is imaged in a zero-offset section.

As a result each sample in a constant-offset section is spread over the whole range of possible dips which could generate the given sample in a zero-offset section.

For zero-offset, the time and space coordinates are (t₀,x₀); v represents the velocity; t_h represents the source-receiver travel time for a constant offset; $\alpha$ represents the dip angle and h is the half-offset. The parameters V, a and b are

$$\left \{ \begin{array} {l} V={ {v}/ {2}}\\ {a} = t_hV \\ {b}=\sqrt{t^2_hV^2 - h^2}\end{array} \right.$$

For the constant velocity medium, the analytical formulation for the PSPM operator as expressed by Popovici and Biondi, (1989)is

 

$$\left \{ \begin{array} {l} t_0 = \displaystyle{{b^2 \over ... ...^2\sin^2{\alpha}+b^2\cos^2{\alpha}} } }} }\end{array} \right.$$ (1)

The equations (1) are the parametric equations (functions of the dip angle $\alpha$)of the DMO ellipse containing the velocity-time truncation. In the constant velocity case the PSPM operator is equivalent to NMO followed by DMO. The form of the PSPM operator is identical to the form of the DMO operator, as NMO introduces merely a time shift. Equations (1) are velocity dependent, and the velocity truncation has to be determined for each medium with a given velocity.