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For laterally-variant media, phase-shift datuming can be
performed by using an adaptation of Gazdag and Sguazzero's (1984) phase
shift plus interpolation (PSPI) method.
In this method, the wavefield is extrapolated at multiple velocities
and a single upward-propagated wavefield is obtained by
interpolation.
Linear interpolation of two wavefields *P*_{1} and *P*_{2}
is an operation of the type
where the adjoint operator spreads a value with
two weights *w*_{1} and *w*_{2}:
The PSPI upward datuming operator for the geometry of Figure
can be written as

| |
(18) |

where the matrices *U*_{i} represent the
extrapolation operators for laterally-variant velocity.
The matrix *U*_{i} can be further decomposed into the sequence
| |
(19) |

The physical interpretation of equation ()
is that the wavefield, after Fourier
transformation, is split and upward continued with two different
velocities. The two wavefields are independently inverse Fourier
transformed and then interpolated. This sequence is repeated for
each depth level.
The adjoint algorithm is found by
transposing each matrix and reversing the multiplication order,
as follows:

| |
(20) |

The matrices *U*^{*}_{i} represent the operator
for downward continuation
of the wavefield to the depth level *i*.
The matrix *U*^{*}_{i} is obtained by taking the adjoint of equation
():
| |
(21) |

Popovici (1992) shows that the Split-Step formulation is similar, and that
the only difference is in the *U*_{i} matrices.

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Stanford Exploration Project

2/12/2001