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For laterally variant media, Gazdag and Sguazzero (1984) propose to downward
extrapolate the wavefield one depth interval at a time with several
velocities. We can apply the same idea to upward propagate the
wavefield with several velocities.
To obtain a single upward propagated
wavefield at each space point the value of the resulting wavefield
is interpolated between the two wavefields with
the closest velocities.
Linear interpolation between two values *P*_{1} and *P*_{2}
is an operation of the type
where the conjugate transpose operator is spreading a value with
two weights *w*_{1} and *w*_{2}
The PSPI upward datuming can be written as

| |
(9) |

where the matrices *U*_{i} represent the operator
for upward continuation
of the wavefield to the depth level *i*.
The matrix *U*_{i} can be further decomposed into the sequence
| |
(10) |

The matrix
contains two diagonal matrices *w*_{1} and *w*_{2} which have constant
coefficients along the diagonal. Multiplication by this matrix, which
is the transpose to linear interpolation, splits the data with two
different weights.
The matrix
contains two diagonal matrices,
which perform the upward extrapolation
with different velocities. Each element of the diagonal is a
complex exponential as in equation (7).
And finally, the matrix
contains two identity matrices which add
the two wavefields.
The conjugate transpose algorithm is found by
transposing each matrix and reversing the multiplication order:

| |
(11) |

The matrices *U*^{*}_{i} represent the operator
for downward continuation
of the wavefield to the depth level *i*.
The matrix *U*^{*}_{i} can be further decomposed into the sequence
| |
(12) |

The physical interpretation of equation (12)
is that the wavefield, after Fourier
transformation, is split and downward continued with two different
velocities. The two wavefields are independently inverse Fourier
transformed and then interpolated. This sequence is then repeated for
each depth level.
The Split-Step algorithm is very similar to the PSPI algorithm, with the difference
being that a single average velocity is used and an extra
phase-shift correction is applied.
Although the up-datuming sequence is exactly the same as in equation
(9), the values of *U*_{i} are different then in equation
(10).
The matrix *U*_{i} is decomposed in the sequence

| |
(13) |

The only new matrix here is the matrix *S*_{i} which has the form
| |
(14) |

and is responsible for a laterally varying phase-shift correction.
The conjugate transpose downward-datuming has the form given in equation
(11) except that the values of *U*_{i} are found by
transposing the matrices in equation (13)

| |
(15) |

** Next:** Wave-equation Datuming Models
** Up:** WAVE-EQUATION DATUMING ALGORITHM
** Previous:** Two conjugate transpose datuming
Stanford Exploration Project

11/17/1997