Angle-dependent reflectivity estimation by
Kirchhoff migration/inversion: Theory

David E. Lumley and Wafik B. Beydoun

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ABSTRACT

We derive a new Kirchhoff forward and inverse theory for modeling and estimating P-P angle-dependent reflectivity. We replace conventional reflecting surface excitations by equivalent body force volumetric excitations, and then linearize the divergence of the elastic stress tensor wavefield with respect to smooth background material properties. To obtain the reflected wavefield, we perform a volume integral over the equivalent body force distribution, and interpret this result as a hybrid of Zoeppritz plane-wave reflection and Rayleigh-Sommerfeld elastic diffraction. We pose the inverse problem as a least-squares optimization to estimate angle-dependent reflectivity, including the reflection angles, by minimizing the squared error between the forward theory predictions 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 migrations, which combine to compensate for limited acquisition aperture and enhanced reflection amplitude recovery.