P-S and S-P reflection coefficients

Similarly, a pure S_v shear mode can be described by the curl of a vector potential. To isolate the shear wave it is enough to apply a curl operator to the displacement field.

$$\mbox{\boldmath$\chi$}(x,z,\omega) \; = \; \nabla \times {\b... ...\nabla \times \nabla \times \mbox{\boldmath$\psi$}(x,z,\omega),$$ (3)

For this 2-D implementation the wavefields are invariant in the direction of the unit vector normal to the experiment plane ($\bf \hat{y}$). As a result, the only non vanishing component of $\chi$ will be in the $\bf \hat{y}$ direction. Defining

$$\chi(x,z,\omega) \; = \; \left(\nabla \times {\bf u}(x,z,\omega) \right) \cdot {\bf \hat{y}},$$

and using the same approach used for P-P, results in the following equation to describe the imaging condition for the P-S reflection coefficient  

$$R_{PS}(x,z,t) \; = \; {v_p \; \chi^r(x,z,t) \over v_s \; \ph... ...dot {\bf \hat{y}} \over v_p \; \nabla \cdot {\bf u^s}(x,z,t)},$$ (4)

and for the S-P reflection coefficient  

$$R_{SP}(x,z,t) \; = \; {v_s \; \phi^r(x,z,t) \over v_p \; \ch... ...ft(\nabla \times {\bf u^s}(x,z,t) \right) \cdot {\bf \hat{y}}}$$ (5)