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The generalized phase-shift method is based on the solution of the temporally
transformed acoustic wave equation
| |
(1) |

where denotes the temporal transform of the pressure field and
*c*(*x*,*z*)
is velocity field. As suggested by Kosloff and Baysal(1983), it is convenient to
recast equation (1) as a set of two first-order coupled equations given by
| |
(2) |

The downward continuation in the migration consists of the solution of equation
(2) for each frequency at all depths under the initial conditions of the values of *P*
and at the surface *z*=0 (Kosloff and Baysal, 1983).
Equation (2) can be expressed in a more compact form
| |
(3) |

where denotes a column vector of length
2*N*_{x} containing first the *N*_{x} pressures and then the
*N*_{x} pressure derivatives ,
for *i*=0,...,*N*_{x-1}.
As with the ordinary phase-shift method, the solution here is propagated in
depth increments. Within each increment *z* to *z*+*dz*, the velocity is assumed
to be invariant in the vertical direction although it may vary horizontally.
The solution of equation (3) can then be written as
| |
(4) |

The solution (4) embodies a phase-shift of the eigenvector coefficients of A.
It would therefore seem that a matrix diagonalization would have to be
performed before each propagation.

** Next:** Horizontally Uniform Structures
** Up:** Ji: generalized phase-shift method
** Previous:** Introduction
Stanford Exploration Project

1/13/1998