We have derived a new Kirchhoff forward and inverse theory for modeling and
estimating P-P angle-dependent reflectivity. We replaced conventional
reflecting surface excitations by *equivalent body force* volumetric
excitations, and then linearized the divergence of the elastic stress
tensor wavefield with respect to smooth background material properties.
The reflected wavefield is calculated by a volume integral over the
equivalent body force distribution, and represents a hybrid combination of
Zoeppritz plane-wave reflection and Rayleigh-Sommerfeld elastic diffraction.

We have posed the inverse problem as a least-squares optimization to estimate angle-dependent reflectivity, including the reflection angles, by minimizing the squared error between our forward theory and the observed seismograms. The resulting coupled normal equations are decoupled by the method of stationary phase, and then the uncoupled equations are solved by a classical Gauss-Newton gradient method with an approximate diagonal Hessian operator. The estimation for angle-dependent reflectivity requires a simultaneous calculation of four differently weighted Kirchhoff prestack depth migration images, which combine to compensate for limited acquisition aperture and enhanced reflection amplitude recovery.

11/16/1997