AVO from pressure and vertical particle velocity

The near seafloor parameters can be determined through the reflection coefficient at the fluid/solid interface. Amundsen and Reitan 1995 have shown that the reflection coefficient can be calculated by spectral division of the pressure and vertical particle velocity of the direct wave and primary reflection transformed to the frequency-radial wavenumber ($\omega - k$)domain. The data can be transformed to the ($\omega - k$) domain by applying a Fourier transform with respect to time and a Hankel transform of order 0 with respect to offset Amundsen (1993). The reflection coefficient can then be expressed as:

 

$$R_{calc} \: = \: {{P(z_1) \: - \: {\rho_1 \over q_{v_{p1}}}\: V_z(z_1)}\over{P(z_1)\: +{\rho_1 \over q_{v_{p1}}}\: V_z(z_1)}}$$ (1)

where $P = P(\omega,k)$ is the pressure component and $V_z = V_Z(\omega,k)$ is the vertical particle velocity component. Both of them are measured at the seafloor, whose depth is given by z₁. The vertical slowness is given by $q_{v_{p1}} \: = \: \sqrt{v_{p1}^{-2} \: - \: p^2}$ and p is the radial slowness given as $p\: = \: {k/\omega} \: = {sin{\alpha_1}/v_{p1}}$. The angle of incidence is given by $\alpha_1$, and the horizontal wavenumber by k. Each combination of k and $\omega$ of the transformed data corresponds to a specific slowness p.

The density, P-wave, and S-wave velocity of the near seafloor can then be determined by inversion. This requires minimization of the difference between the calculated and theoretical reflection coefficient. The theoretical reflection coefficient for plane waves incident at a fluid/solid interface can be described as Berkhout (1987):

 

$$R_{pp}\:=\: {{A_1 \: \rho_2 \: q_{v_{p1}} \: + A_2 \: q_{v_{... ...} \: + A_2 \: q_{v_{p1}}\: q_{v_{p2}}\: + \rho_1\: q_{v_{p2}}}}$$ (2)

where $A_1 \: = \: (\: 1 \: - \: 2\: p^2 \: v_{s2}^2)^2$and $A_2 \:= \: 4\: p^2 \: v_{s2}^4 \:\rho_2 \: q_{v_{s2}}$are shear coefficients, $q_{v_{p1}} \: = \: \sqrt{\:{v_{p1}}^{-2}\: - \:p^2}$,$q_{v_{p2}} \: = \: \sqrt{\: {v_{p2}}^{-2} \: - \: p^2}$and $q_{v_{s2}} \: = \: \sqrt{\: {v_{s2}}^{-2} \: - \: p^2}$are the vertical P- and S-wave slownesses in the fluid and solid, and $p \: = \: {\sin{\alpha_1}/{v_{p1}}}$is the horizontal slowness.