High order approximation for scattering matrix in layered elastic medium and its application in pre-stack seismic inversion

• Give out the method to derive high order approximation for scattering matrix.
• Generalize the Aki–Richards approximation.
• Break the limitation due to the assumption of small elastic parameter contrasts.

Scattering matrix of Plane P- and SV-wave in a layered elastic medium is the theoretical basis for pre-stack seismic inversion. Since the exact expression of scattering matrix is too complicated to apply, the current pre-stack seismic inversion method uses the first order approximation to estimate the stratigraphic elastic parameters, assuming the elastic parameter contrasts are small. It only works in weak reflection interface situation. When the elastic parameter contrasts become large, the first order assumption will suffer reduction in accuracy. At this time, the low order approximation should be adjusted with higher order corrections. Using series expansion, we infer the expression of any order approximation for scattering matrix. These high order terms are corrective terms adjusted to the first order approximation. Through different layer models, the comparison between the first, second, third order approximation reflection coefficient with the exact results of scattering matrix is done, and the accuracy of different order approximations is analyzed. In the following, using a 1D model, the pre-stack inversion based on different order approximations via nonlinear variable metric algorithm is performed. Inversion results illustrate the pre-stack seismic inversion based on higher order approximation not only adapts situations with weak reflection interfaces, but also adapts situations with strong elastic parameter variations on reflection interface. It breaks the limitation of small elastic parameter contrast assumption in conventional pre-stack seismic inversion.