A FORMALISM FOR INCLUDING AMPLITUDE DATA IN TOMOGRAPHIC INVERSIONS

Using first-order ray perturbation theory and the paraxial ray approximation, a linear relation is derived that describes the first-order perturbation in amplitude due to slowness perturbations along a ray. The method is valid for smooth media, without interfaces. The method allows for an efficient calculation of Frechet derivatives for body wave amplitudes. This makes it possible to add body wave amplitude data to large-scale tomographic inversions for slowness distributions in the Earth. Depending directly on the derivatives of the slowness field, body wave amplitudes provide a constraint on the velocity distribution independent of traveltime data. For a realistic slowness distribution in the upper mantle, examples show that the region of applicability of the linearized perturbation theory to the calculation of amplitude perturbations may become very small due to the non-linear dependence of amplitudes on slowness perturbations. Traveltimes show a more linear dependence on slowness perturbations. The examples suggest that when amplitudes and traveltimes are combined in an inversion for the velocity structure of the upper mantle, a non-linear traveltime inversion must be performed before amplitude data are included. This ensures that the non-linear behaviour of amplitudes due to ray shift induced by slowness perturbations are minimized.