CONCLUSION

I have posed the goal of elastic specular reflectivity estimation as a formal least-squares inverse problem, which I subsequently solve by stationary point analysis. After introducing the elastodynamic wave equation, a representation integral solution is presented by use of Betti's Theorem which is the vector analogy to the Kirchhoff-Rayleigh-Sommerfeld integrals for scalar waves. The representation integrals are evaluated by assuming WKBJ approximate Green's tensors, which are in turn evaluated by the ray theoretic eikonal and transport equations. The representation integrals are substituted back into the generalized least-squares solution to obtain expressions for elastic specular reflectivity. I give explicit integral expressions for the $\grave{P}\!\acute{P}$ and $\grave{P}\!\acute{S_1}$ reflectivities as surface integrals over the recorded vector displacement wavefields.