Inversion of up and down going signal for ocean bottom data

Separation using over-under recordings

The derivation for decomposing over-under pressure waves into up-going and down-going signals is best done in the Fourier domain. Denote $S_1(\omega, k_x)$ and $S_2(\omega,k_x)$ to be the Fourier transformed measurement of compressional waves at depth $z_1$ (over) and $z_2$ (under). Theoretically, $S_1(\omega, k_x)$ can be viewed as a sum of the up-going $U_1(\omega, k_x)$ and down-going $D_1(\omega, k_x)$ components. Likewise for $S_2(\omega,k_x)$:

$$S_1(\omega,k_x) = U_1(\omega,k_x) + D_1(\omega,k_x),$$

$$S_2(\omega,k_x) = U_2(\omega,k_x) + D_2(\omega,k_x).$$ (3)

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

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

$$U_1 = e^{i k_z \Delta z } U_2 ,$$ (4)

where $\Delta z = z_2 - z_1$ and $k_z$ is the usual dispersion relation. Finally, substituting equation 4 into 3 yields the formula for the up-going and down-going waves at the receivers:

$$U_2 = \frac{S_2 - e^{i k_z \Delta z } S_1}{1 - e^{2 i k_z \Delta z }} ,$$

$$D_2 = \frac{e^{i k_z \Delta z } S_1 - e^{2 i k_z \Delta z } S_2}{1 - e^{2 i k_z \Delta z }}.$$

Data acquisition using over-under arrangement is often used to elimate receiver ghosts and water reverberation. For a thorough review of this method, please see Sonneland et al. (1986). Although over-under arrays are rarely placed at the sea floor in real seismic surveying, this technique allows easy generation of up- and down-going data at the sea bottom in synthetic examples using the simpler acoustic wave equation.

Inversion of up and down going signal for ocean bottom data