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 ($s(x,z) = {1 \over v(x,z)}$) into a constant term and a perturbation term,

$$s(x,z)=s_0(z)+\Delta s(x,z),$$

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  

$${\partial^2 P(x,z,\omega) \over \partial z^2}+ {\partial^2 P... ...a) \over \partial x^2}+ {\omega^2}{s^2(x,z) P(x,z,\omega)}=0 .$$ (13)

After inserting the slowness split into the perturbation term and the average term the equation is transformed into  

$${\partial^2 P(x,z,\omega) \over \partial z^2}+ {\partial^2 P... ...omega^2[2 s_0(z) \Delta s(x,z)+\Delta s^2(x,z)] P(x,z,\omega) .$$ (14)

Defining the right side of the equation (14)

$$S(x,z,\omega)=\omega^2(2 s_0(z) \Delta s(x,z)+\Delta s^2(x,z)) P(x,z,\omega)$$

we can write (14) as  

$${\partial^2 P(x,z,\omega) \over \partial z^2}+ {\partial^2 P... ...tial x^2}+ {\omega^2}{{s_0}^2(z) P(x,z,\omega)}=-S(x,z,\omega)$$ (15)

which is an inhomogeneous wave equation with a source term $S(x,z,\omega)$.

 

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 $\Delta z$by ignoring the contribution of the $\Delta s^2(x,z)$.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 $\Delta z$. After inverse Fourier transforming into surface coordinates the solution for downward propagating the wavefield has the form  

$$P(x,z_0 + \Delta z,\omega)=P_0(x,z_0 + \Delta z,\omega) e^{i\omega\Delta s(x,z) \Delta z},$$ (16)

where $P_0(x,z_0 + \Delta z,\omega)$ represents the wavefield downward continued with the average slowness s₀(z). For upward continuation we just have to change the sign of $\Delta z$ to have  

$$P(x,z_0 - \Delta z,\omega)=P_0(x,z_0 - \Delta z,\omega) e^{- i\omega\Delta s(x,z) \Delta z},$$ (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  

$$e^{i ({1 \over v(x,z)} - {1 \over v_{med}} )\omega \Delta z}$$ (18)

after the downward extrapolation. Compare the phase shift in equation (18) with the trick in PSPI modeling to phase shift with

$$e^{-i{\omega \over {v_j}} \Delta z} ,$$

followed by

$$e^{i{\omega \over v(x,z)} \Delta z}.$$

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.