THE SPLIT-STEP FOURIER METHOD

Stoffa et al. (1990) present an alternative to the PSPI, replacing the downward continuation step with two or more velocities by a single average velocity followed by a phase shift of the wavefield with a perturbation term to account for lateral velocity variations. The Split-step algorithm is based on splitting the space variant slowness ($s= {1 \over v(x,z)}$) 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  

$${\partial^2 P(x,z,\omega) \over \partial z^2}+ {\partial^2 P... ...a) \over \partial x^2}+ {\omega^2}{s^2(x,z) P(x,z,\omega)}=0 .$$ (7)

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

$${\partial^2 P(x,z,\omega) \over \partial z^2}+ {\partial^2 P... ...omega^2[2 s_0(z) \Delta s(x,z)+\Delta s^2(x,z)] P(x,z,\omega) .$$ (8)

Noting the right side of the equation

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

equation (8) becomes  

$${\partial^2 P(x,z,\omega) \over \partial z^2}+ {\partial^2 P... ...tial x^2}+ {\omega^2}{{s_0}^2(z) P(x,z,\omega)}=-S(x,z,\omega)$$ (9)

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

Stoffa et al. show that equation (9) can be integrated over a thin depth layer $\Delta z$by ignoring the contribution of the $\Delta s^2(x,z)$.Equation (9) is Fourier transformed in surface coordinates after the second order term of the slowness perturbation is dropped and subsequently integrated over the depth layer $\Delta z$. After inverse Fourier transformation back into surface coordinates the solution has the form  

$$P(x,z_{n+1},\omega)=P_0(x,z_{n+1},\omega)e^{i\omega\Delta s(x,z) \Delta z},$$ (10)

where $P_0(x,z_{n+1},\omega)$ represents the wavefield downward continued with the average slowness s₀(z). Though the mathematical path is very different from the one Gazdag (1984) followed, the solution is very similar if you consider Gazdag's algorithm with a single velocity.

The phase subtraction and addition trick in Gazdag's algorithm is replaced by multiplication with

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

after the downward extrapolation. Compared with Gazdag's trick to phase shift with

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

followed by

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

the only difference is that Gazdag (1984) does the first phase shift before the Fourier transform while Stoffa et al. (1990) do it after the Fourier transform. Though analytically not identical in the general case, the qualitative idea is the same.

The two algorithms are compared in Figure 1.

 

gazsplit
Figure 1 The two Fourier migration algorithms. a) Phase Shift Plus Interpolation (PSPI). b) Split-step Fourier Migration.