SEPARATION OF P AND S WAVEFIELDS

To apply scattering theory we need to transform the pressure field into a displacement field. The pressure wavefield obeys the scalar wave equation

$$\nabla^2 \phi(x,z,\omega)= -{\omega^2 \over v^2} \phi(x,z,\omega) .$$

The P wave displacement $\vec{p}$ is related to the pressure by the following expression:

$$\phi = - K \: \nabla \! \cdot \! \vec{p} , \mbox{ \hspace{1... ...pace{0.6cm} $K$ \hspace{0.6cm} is the bulk modulus of water. }$$

Recalling that the medium (water) is homogeneous and that the displacement field is irrotational, we can use the relation $\nabla^2 = \nabla \cdot \nabla$ and combine the two previous equations to get

$$\vec{p} = {v^2 \over \omega^2 K} \nabla \phi .$$

This equation relates the displacement vector field to the scalar pressure field and can be rewritten in the following form:

$$\pmatrix{p_x \cr p_z } = {v^2 \over \omega^2 K} \pmatrix{{\p... ...partial x} \cr {\partial \over \partial z}} \phi(x,z,\omega) .$$

And after a double spatial Fourier transform, it will be expressed by

$$\pmatrix{P_x(k_x,k_z,\omega) \cr P_z(k_x,k_z,\omega) } = i {... ...r \omega^2 K} \pmatrix{ k_x \cr k_z} \: \Phi(k_x,k_z,\omega) .$$

Finally, the displacement amplitude $\acute{P}$ of the upcoming P wave on the water can be expressed as a function of the upcoming pressure field:

$$\acute{P} = i {v^2 \over \omega^2 K} \sqrt{k_x^2 + k_z^2} \: \acute{\Phi}$$

or

$$\acute{P} = {v \over -i \omega K} \acute{\Phi} .$$

and, as one should expect, the pressure field differs from the velocity field (time derivative of the displacement) by just a scale factor $v \over K$ .

Figure  shows how the upcoming P and S waves are scattered at the ocean bottom interface. The displacement amplitude of the upcoming pressure field just above the sea floor can be expressed as functions of the displacement amplitudes of all the incident waves:   

$$\acute{P_1} = [ \acute{P} \acute{P} ]_{2-1} \acute{P_2} + [ ... ...2-1} \acute{S_2} + [ \grave{P} \acute{P} ]_{1-1} \grave{P_1} ,$$ (6)

where $[ \acute{P} \acute{P} ]_{2-1}$ is the transmission coefficient for upcoming P into P waves, $[ \acute{S} \acute{P} ]_{2-1}$is the transmission coefficient for upcoming S into P waves , and $[ \grave{P} \acute{P} ]_{1-1}$is the reflection coefficient for downgoing P into P waves (Aki and Richards, 1980). The last term in the previous equation corresponds to the ocean floor primary reflection and all its associated multiples and peg-legs. The conditions most suitable for the generation of converted waves in marine surveys are the same ones that are responsible for the occurrence of strong multiple contamination. While the ocean floor primary will not interfere with any converted wave, the multiples and peg-legs must be removed in order to isolate the PSSP wavefield. The importance of this step and the method used for its accomplishment are the subjects of Appendix A.

The two remaining terms in equation (6) (which correspond to the P and S wavefields) can be reasonably separated by the use of the critical angle for P waves at the first layer as a discriminant factor (Cunha, 1989), as illustrated on Figure . Waves of the type PPPP, PPSP and PSPP will be found only with horizontal slownesses smaller than the slowness corresponding to the ocean floor critical angle for P waves (30 degrees on the figure), while only the PSSP mode will be found at higher horizontal slownesses. As a first approach we consider that only the dominant mode (PPPP) is present for pre-critical horizontal slownesses.

Under these assumptions, a ``P-sensible receiver" coupled at the sea bottom would record a displacement

 

$$\acute{P_2} ={1 \over [ \acute{P} \acute{P} ]_{2-1}} \acute{... ...}for \hspace{1.5cm}$p = {k_x \over \omega} < {1 \over V_p}$} ,$$ (7)

whereas an ``S-sensible receiver" would record  

$$\acute{S_2} ={1 \over [ \acute{S} \acute{P} ]_{2-1}} \acute{... ...for \hspace{1.5cm}$p={k_x \over \omega} \geq {1 \over V_p}$} ,$$ (8)

where V_p is the velocity of P waves at the first layer.