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# THE SPLIT-STEP FOURIER MODELING

Stoffa et al. (1990) present an alternative to the PSPI. Stoffa et al. replace the different velocities used in the downward continuation step with a single average velocity. A following phase shift of the wavefield with a perturbation term accounts for lateral velocity variations. The flow of Stoffa's migration is shown in Figure .

The Split-Step algorithm is based on splitting the space variant slowness () into a constant term and a perturbation term,

where s0(z) is a reference slowness defined as the average slowness in a depth interval. The wave equation (1) is Fourier transformed along the time axis to become
 (13)
After inserting the slowness split into the perturbation term and the average term the equation is transformed into
 (14)
Defining the right side of the equation (14)

we can write (14) as
 (15)
which is an inhomogeneous wave equation with a source term .

splitboth
Figure 2
Split-Step Fourier migration and the conjugate transpose Split-Step Fourier modeling algorithm.

Stoffa et al. show that equation (15) can be integrated over a thin depth layer by ignoring the contribution of the .This is done by Fourier transforming equation (15) in surface coordinates, dropping the second order term of the slowness perturbation and subsequently integrating over the depth layer . After inverse Fourier transforming into surface coordinates the solution for downward propagating the wavefield has the form
 (16)
where represents the wavefield downward continued with the average slowness s0(z). For upward continuation we just have to change the sign of to have
 (17)
Though the mathematical path is very different from the one Gazdag and Sguazzero (1984) followed, the solution is very similar if you consider the PSPI algorithm with a single velocity.

The phase addition and subtraction trick in the PSPI algorithm is replaced by multiplication with
 (18)
after the downward extrapolation. Compare the phase shift in equation (18) with the trick in PSPI modeling to phase shift with

followed by

The only difference is that in PSPI the phase shift is done after the inverse Fourier transform while in Split-Step it is done before the Fourier transform. The two modeling algorithms are compared in Figure .

pspisplitmod
Figure 3
PSPI modeling and Split-Step Fourier modeling.

Next: A 2-D example Up: Popovici : PSPI and Previous: THE PHASE SHIFT PLUS
Stanford Exploration Project
11/18/1997