Second-order approximations of the reflection and transmission coefficients between two visco-elastic isotropic media

The plane-wave reflection and transmission (R/T) coefficients between two i sotropic visco-elastic media where the medium parameters are complex functi on of frequency, are found from the continuity of stress and displacement. Normalizing the R/T coefficients with respect to vertical energy-flux resul ts in important symmetry properties. By expanding the eigenvector matrices for up- and down-going waves into second-order Taylor series and using the symmetry properties of these matrices we derive simple second-order approxi mations for the R/T coefficients valid for pre-critical reflections. The st andard linear weak-contrast approximations are obtained by neglecting the s econd-order terms in the relative changes in the medium parameters at the i nterface. The two approximations are compared to the exact R/T coefficients in three numerical examples where the attenuation of P- and S-waves are given by the Kolsky-Futterman attenuation relation. In all cases, the new approximation s give a better fit to the exact R/T coefficients than the linearized appro ximations. Inversion of the PP and PS reflection coefficients into the rela tive changes in the medium parameters shows that the new approximations per form much better than using the linear approximations.