P/S separation of OBS data by inversion in a homogeneous medium

PZ summation

PZ summation involves summing the pressure data recorded by the hydrophone with the vertical particle velocity data recorded by the geophone, with some scaling factor:

$$U (z_r) = \frac{1}{2} \left[ P(z_r) - \beta V_z (z_r) \right] ,$$

$$D (z_r) = \frac{1}{2} \left[ P(z_r) + \beta V_z (z_r) \right] ,$$ (1)

where $P$ is the hydrophone data, $V_z$ is the vertical geophone, $U$ is the upgoing data, $D$ is the downgoing data and $z_r$ is the receiver depth. $\beta$ is a scaling factor, which is theoretically the acoustic impedance at the wave's incidence angle. In practice, as a result of frequency-dependent instrument response, and as a result of the two different impedances above and below the receivers, $\beta$ is frequency dependent, propagation-direction dependent, and wave-mode dependent.

Amundsen (1993) does the separation in the $f-k$ domain, and uses $\beta = \frac {\rho \omega}{k_z}$ , where $k_z = \sqrt {\frac {\omega^2}{v^2} - k^2_x -k^2_y}$ , and $\rho$ is the density. This implicitly requires a laterally invariant medium at the receiver level. Alternately in the $t-x$ space, the scaling can be determined by the ratio of the direct arrival's amplitude on the hydrophone and vertical geophone components at various offsets.

As emphasized above, this method assumes that all energy is pressure wave energy, and therefore everything recorded by the hydrophone should have its counterpart in the geophone data, with either positive polarity (upgoing) or negative polarity (downgoing).

P/S separation of OBS data by inversion in a homogeneous medium