P-P reflection coefficient

In most practical cases only purely compressional modes can be resolved from conventional surface seismic, especially if the data were collected in a marine environment where shear waves cannot be generated or recorded. In such cases the only reliable reflection coefficient to image is the one associated to P to P reflection.

Reflection coefficients are often defined for either displacement-amplitude ratios or potential ratios. For a pure compressional wave the displacement field can be expressed as the gradient of a potential field. Taking the divergence of the displacement and using the wave-equation for the potential leads to  

$$\phi(x,z,\omega) \; = \; {v_p^2 \over \omega^2} \; \nabla^2 ... ... \; {v_p^2 \over \omega^2} \; \nabla \cdot {\bf u}(x,z,\omega).$$ (1)

It is clear that the above formulation is restricted to isotropic cases where P waves are described by a single velocity v_p, however the background media, used for the time extrapolation is considered to be anisotropic. This means that anisotropic behavior is correctly handled in the propagation through the overburden, but not at the target zone. The displacement reflection coefficient relates to the potential reflection coefficient through

$$\mbox{Displacement coefficient} \;\; = \;\; {v_{inc} \over v_{ref}} \;\; \mbox{Potential coefficient},$$

Aki and Richards (1980). Since the velocities for the incident and reflected wave, are the same, the imaging condition for obtaining the P-P displacement reflection coefficient can be formulated as  

$$R_{PP}(x,z,t) \; = \; {\phi^r(x,z,t) \over \phi^s(x,z,t)} \;... ...la \cdot {\bf u^r}(x,z,t) \over \nabla \cdot {\bf u^s}(x,z,t),}$$ (2)

where superscripts r and s correspond, respectively, to the reverse propagated recorded wavefield and to the forward propagated shot wavefield.

It is important to emphasize that the wavefield decomposition to separate the purely compressional field is performed only at the imaging step but the propagation (depth extrapolation) uses the full wavefield. Therefore, all elastic effects of mode conversion in all interfaces are taken into account.