Stolt 1996 first introduced
prestack
residual migration.
Sava 1999 reformulated
Stolt residual migration in order to handle
prestack depth images.
This section presents the extension of
Sava (1999) for
two different wavefields, therefore, two
different velocities.
We present this
extension for converted waves data, where the
*P* to *S* conversion occurs at the reflector.
Although the formulation
involves *P*-velocities and *S*-velocities, its
application is not limited to converted waves
only. Rosales and Biondi (2001) present
a possible application for imaging under salt edges.

Residual prestack Stolt migration operates in the Fourier domain. Considering the representation of the input data in shot-geophone coordinates, the mapping from the data space to the model space takes the form

(1) |

In residual prestack Stolt migration for converted waves, we attempt to simultaneously correct the effects of migrating with two inaccurate velocity fields.

Supposing that the initial migration was done with
the velocities *v*_{0p} and *v*_{0s}, and that
the correct velocities are *v*_{mp} and *v*_{ms}, we
can then write

Solving for in the first equation of (2) and substituting it in the second equation of (2), we obtain the expression for prestack Stolt depth residual migration for converted waves [equations (3) and/or (4)] (see Appendix A, for details in derivation)

(2) |

(3) |

where is the transformation kernel and is defined as

and , , and .

In equation (2) it appears that
Stolt residual migration for converted waves
depends on four parameters: *v*_{0p}, *v*_{0s}, *v*_{mp}, *v*_{ms}.
These four degrees of freedom can be reduced to three
(, and ), as seen in
equations (3) and (4) and demonstrated in Appendix A.
This is important,
because a three parameters search for updating converted waves images
is simpler than a four parameters search.
However, it would be useful to further reduce
the number of parameter to two.
Assuming that the *v*_{p}/*v*_{s} ratio is the same after and
before the residual migration process, it is possible to simplify
equations (3) and/or (4) into a two
parameter equation:

(4) |

where the transformation kernel takes the form

If we just specify two different ratios in both square roots of Sava's 1999 formulation we have

(5) |

where the transformation kernel
has the same form as the one of *PP* waves:

Equation (6) shows another way of doing prestack residual
migration for converted waves.
Although equations (5) and
(6) may look similar because they depend on only two
parameters, the transformation kernels
( and )are different.
Equation
(6) has a similar transformation kernel as the conventional *PP*
prestack residual migration, while equation (5) presents
a kernel deduced for the case of converted waves.

It is important to note that all three equations (3), (5),
and (6) reduce to the same expressions in the limit
when of *v*_{s} tends to *v*_{p}, or tends to 1.
All of them reduce
to the simple case of prestack residual migration for
conventional *PP* data. Appendix B demonstrates this result.

9/18/2001