Time domain methodology

The method that Barr and Sanders (1989) proposed to combine the hydrophone and geophone is simple and easy to implement. They simply add the hydrophone and the calibrated geophone in the shot domain. The calibrated geophone is computed with a constant factor equal to the ratio of the amplitudes of the hydrophone and geophone. We calculate a constant factor per trace, we average all of them, and finally apply the averaged constant factor to the entire shot gather.

This procedure not only calibrates the geophone but also eliminates the ghost reflection. The final combined signal (s(t)) is given by the following:  

$$s(t) = h(t) + \frac{\rho v_p}{\cos \gamma'_p}\frac{kt(1+kr)}{kt(1-kr)} z(t),$$ (1)

where h(t) and z(t) correspond to the hydrophone and geophone, respectively, $\rho$ is the water density, v_p is the P-wave water velocity, $\gamma'_p$ is the P-wave refraction angle in water, and kr, kt are the reflection coefficient and the refraction coefficient, respectively.

Figure 3 presents the physical model for this approach. Solving the boundary conditions for the elastic wave-equation at the water bottom (left panel on Figure 3) gives the amplitudes of the reverberations (right panel on Figure 3). This model explains that combining the hydrophone and the geophone components as in equation (1) results in a reverberation-free signal.

The right panel on Figure 3 also explains how to obtain the scale factor for equation (1). Comparing the amplitudes of the reverberations shows that the scale factor is just the absolute value of the ratio between the amplitudes of the hydrophone and the geophone. The final result, s(t), is a deghosted output.

 

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Figure 3 Physical model for the reverberations. It solves for the boundary conditions of the elastic wave field for the model on the left. On the right are shown the reverberations as a function of time for the hydrophone (h) and geophone (z) components. The first arrival corresponds to event a_h and a_z with an amplitude equal to a_h=kt, a_z=$\frac{kt}{\rho v_p} \cos \gamma'_p$. The first reverberation, b_h and b_z, has an amplitude of b_h=-kt(1+kr) and b_z=$\frac{kt(1-kr)}{\rho v_p} \cos \gamma'_p$. The second reverberation, c_h and c_z, has an amplitude of c_h=kt kr(1+kr) and c_z=$\frac{-kt kr(1-kr)}{\rho v_p} \cos \gamma'_p$. The third reverberation, d_h and d_z, has an amplitude of d_h=-kt kr²(1+kr) and d_z=$\frac{kt kr^2(1-kr)}{\rho v_p} \cos \gamma'_p$.

Figure 4 shows the result of this approach over the common-shot gather from Figure 1. Although it was possible to eliminate some of the multiples, the final result, s(t), is not totally multiple-free.

 

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Figure 4 Hydrophone-geophone summation. From left to right: hydrophone component, geophone component, summation.