Perturbations of the seismic reflectivity of a fluid-saturated   depth-dependent poro-elastic medium
  Analytical formulas are derived to compute the first-order effects produced by plane inhomogeneities on the point source seismic response of a fluid-filled stratified porous medium. The derivation is achieved by a perturbation analysis of the poro-elastic wave equations in the plane-wave domain using the Born approximation. This approach yields the Fr\'echet derivatives of the P -- SV - and SH-wave responses in terms of the Green's functions of the unperturbed medium. The accuracy and stability of the derived operators are checked by comparing, in the time-distance domain, differential seismograms computed from these analytical expressions with complete solutions obtained by introducing discrete perturbations into the model properties. For vertical and horizontal point forces, it is found that the Fr\'echet derivative approach is remarkably accurate for small and localized perturbations of the medium properties which are consistent with the Born approximation requirements. Furthermore, the first-order formulation appears to be stable at all source-receiver offsets. The porosity, consolidation parameter, solid density and mineral shear modulus emerge as the most sensitive parameters in forward and inverse modeling problems. Finally, the Amplitude-Versus-Angle response of a thin layer shows strong coupling effects between several model parameters. 
Louis De Barros (LGIT), M. Dietrich (LGIT, Ifp)