Depth extrapolation using split-step Fourier method

The split-step Fourier extrapolation Stoffa and Fokkema (1990) consists of two-step extrapolation, which is a phase shifting in the $(k,\omega)$ domain followed by an additional phase shifting in the $(x,\omega)$ domain.

Let us consider a simple case that consists of two depth levels of extrapolation. Then the forward operator can be expressed as follows:

$$W = F^\ast_t \exp^{-ip_1\Delta z} F^\ast_x \exp^{-ip_2\Delta... ...F_x \exp^{-ip_1\Delta z} F^\ast_x \exp^{-ip_2\Delta z} F_x F_t,$$ (55)

where F_x and F_t represent the Fourier transform along the space and the time domains, respectively. The phases p₁ and p₂ quantify the amount of phase shift in the $(x,\omega)$ and in the $(k,\omega)$ domain, and are defined as follows:

$$p_1(x,\omega) = \omega \Delta u(x)$$ (56)

and

$$p_2(k,\omega) = \omega u_0 \sqrt{ 1-\left({k \over \omega u_0}\right)^2}$$ (57)

where u₀ represents the reference slowness and $\Delta u(x)$ is a variation of the slowness from the reference slowness. Then the adjoint of the forward operator becomes

$$W^\ast = F^\ast_t F^\ast_x \exp^{ip_2\Delta z} F_x \exp^{ip_... ...ta z} F^\ast_x \exp^{ip_2\Delta z} F_x \exp^{ip_1\Delta z} F_t.$$ (58)

If we apply the adjoint operator after the forward operator as follows:

$$W^\ast W = F^\ast_t F^\ast_x P_2^\ast F_x P_1^\ast F^\ast_x ... ...\ast F_t F^\ast_t P_1 F^\ast_x P_2 F_x P_1 F^\ast_x P_2 F_x F_t$$ (59)

with $P_1=\exp^{-ip_1\Delta z}$ and $P_2=\exp^{-ip_2\Delta z}$,where P₁ is a unitary operator, but P₂ is a pseudo-unitary operator because we substitute for the evanescent component when p₂ becomes imaginary. If I group the operators so that they are in the same domain, for example the $(k,\omega)$ domain, we get the following:

$$\begin{eqnarray} W^\ast W&=& F^\ast_t F^\ast_x (P_2^\ast) ( F_x P_1^\ast F^\ast_... ... I_p I I_p F_x F_t \nonumber \\ &=& F^\ast_t F^\ast_x I_p F_x F_t\end{eqnarray}$$ (60)

where I is the identity matrix and I_p is an idempotent matrix with some elements that are 1 and others that are zeros. Therefore, the split-step Fourier extrapolation is pseudo-unitary.