The Information Content of the Elastic Reflection Matrix

The P-SV reflection matrix for a plane interface between two elastic media depends on the density, P-wave velocity and S-wave velocity of the two media. When the reflection matrix is given as a function of slowness, five parameters—the ratio between the two densities, and P- and S-wave velocities for the two media—can theoretically be estimated. When the reflection matrix is given as a function of the angle of incidence for P waves, only the density ratio and three velocity ratios can be estimated.A numerical sensitivity analysis is performed on modelled data to determine how the different parameters contribute to the information content of the reflection matrix. It is confirmed that, theoretically, five parameters can be estimated from the PP reflection coefficient as a function of slowness, when the slowness varies from zero to some maximum value. However, certain linear combinations of the parameters are very poorly determined. These are the parameters associated with the eigenvectors of the Hessian matrix (or the normalized Fisher information matrix) corresponding to the smallest eigenvalues. A maximum of three parameters can be estimated from the PP reflection coefficients when only pre-critical data are used, and up to five when post-critical data are included. The analysis shows that the best-determined parameter from the PP-reflection coefficients is the gradient of the P-wave impedance, whereas the best-determined parameter from the PS-, SP- and SS-reflection coefficients is the gradient of the S-wave impedance. The second best parameters for all reflection coefficients are linear combinations of the two S-wave velocities and the density ratio. The sum of the two S-wave velocities is very poorly determined.