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 and
reflectivities as surface integrals over the
recorded vector displacement wavefields.