  An extension of a previously developed rock physics model is made that
quantifies the relationship between the ductile fraction of a brittle/ductile
binary mixture and the isotropic seismic reflection response. By making a weak
scattering (Born) approximation and plane wave (eikonal) approximation, with a
subsequent ordering according to the angle of incidence, singular value
decomposition analysis are done to understand the stack weightings, number of
stacks, and the type of stacks that will optimally estimate the two fundamental
rock physics parameters. Through this angle ordering, it is found that
effective wavelets can be used for the stacks up to second order. Finally, it
is concluded that the full PP stack and the "full" PS stack are the two optimal
stacks needed to estimate the two rock physics parameters. They dominate over
both the second order AVO "gradient" stack and the higher order (4th order) PP
stack (even at large angles of incidence). Using this result and model based
Bayesian inversion, the detectability of the ductile fraction (shown by others
to be the important quantity for the geomechanical response of unconventional
reservoir fracking) is demonstrated on a model characteristic of the Marcellus
shale play.
