Sensitivities of seismic traveltimes and amplitudes in reflection tomography

Seismic traveltimes and amplitudes in reflection-seismic data show different dependences on the geometry of reflection interfaces, and on the variation of interval velocities. These dependences are revealed by eigenanalysis of the Hessian matrix, defined in terms of the Frechet matrix and its adjoint associated with different norms chosen in the model space. The eigenvectors and eigenvalues of the Hessian clearly show that for reflection tomographic inversion, traveltime and amplitude data contain complementary information. Both for reflector-geometry and for interval-velocity variations, the traveltimes are sensitive to the model components with small wavenumbers, whereas the amplitudes are more sensitive to the components with high wavenumbers. The model resolution matrices, after the rejection of eigenvectors corresponding to small eigenvalues, give us some insight into how the addition of amplitude information could potentially contribute to the recovery of physical parameters. In order to cooperatively invert seismic traveltimes and amplitudes simultaneously, we propose an empirical definition of the data covariance matrix which balances the relative sensitivities of different types of data. We investigate the cooperative use of both data types for, separately, interface-geometry and 2-D interval-velocity variations. In both cases we find that cooperative inversions can provide better solutions than those using traveltimes alone. The potential benefit of including amplitude-data constraints in seismic-reflection traveltime tomography is therefore that it may be possible to resolve the known ambiguity between the reflector-depth uncertainty and the interval-velocity uncertainty better.