Reverse Time Migration of up and down going signal for ocean bottom data

The up-down separation operator

The derivation for decomposing over/under pressure waves into up-going and down-going signals is best done in the Fourier domain. For a thorough review of this method, please refer to Sonneland et al. (1986). Denote $P_1(\omega, k_x)$ and $P_2(\omega,k_x)$ to be the Fourier-transformed measurements of compressional waves at depths $z_1$ (over) and $z_2$ (under). Theoretically, $P_1(\omega, k_x)$ is a sum of the up-going $U_1(\omega, k_x)$ and down-going $D_1(\omega, k_x)$ components. Likewise for $P_2(\omega,k_x)$:

$$P_1(\omega,k_x) = U_1(\omega,k_x) + D_1(\omega,k_x),$$

$$P_2(\omega,k_x) = U_2(\omega,k_x) + D_2(\omega,k_x).$$ (5)

Down-going waves arrive at the under array ($D_2$) before the over ($D_1$) array. Therefore, shifting $D_2$ forward in time would match the signal $D_1$. Similarly, up-going waves visit the over array first. Therefore, shifting $U_1$ forward in time would match the signal $U_2$. This relationship is equivalent to a phase-shift in the Fourier domain:

$$e^{i k_z \Delta z } D_2 = D_1 ,$$

$$U_2 = e^{i k_z \Delta z } U_1 ,$$ (6)

where $\Delta z = z_2 - z_1$, and $k_z$ is the usual dispersion relation. Finally, substituting equation 6 into equation 5 yields the formula for the up-going and down-going waves at the receivers:

$$U_2 = \frac{P_2 - e^{i k_z \Delta z } P_1}{1 - e^{2 i k_z \Delta z }},$$

$$D_2 = \frac{e^{i k_z \Delta z } P_1 - e^{2 i k_z \Delta z } P_2}{1 - e^{2 i k_z \Delta z }}.$$ (7)

Over/under acquisition is used to eliminate receiver ghosting and water reverberation. Although over/under arrays are rarely placed on the sea floor in real seismic surveys, this technique allows easy generation of up-going and down-going waves at the sea bottom for synthetic examples or modeling. For the remaining of this paper, we will denote the operation that separates over/under data into up-down data in equation 7 as $\bold S$.

Reverse Time Migration of up and down going signal for ocean bottom data