On the relationship between ray theory and the banana-doughnut formulation (Q1944360)

The paper is devoted to clarify certain mathematical statements given in the paper [\textit{F. A. Dahlen} et al., ``Frechét kernels for finite-frequency traveltimes'', Geophys. J. Int. 141, 157--174 (2000)] devoted to ray theory and its extend to finite-frequency waves. They clarify the conclusion about the apparent lack of sensitivity along the ray, which is not a general statement about ray theory, but about the particular formulation of Dahlen et al. [loc. cit.], since their definition used for sensitivity depends derectly on whether or not a scatterer is on the ray. In particular, they note that there are mathematical problems in the derivation of some of the results of Dahlen et al. [loc. cit.] and they attempt to provide derivations that avoid these issues. In general, they maintain that their formulation is a contribution to the properties of ray theory, rather than its replacement or extension.