Proof of validity of first-order travel estimates

In the seminal paper by Dahlen et al. the authors formulate an important expression as a first-order estimate of traveltime delay. The authors left out a term which would at first glance seem nontrivial, on the basis that their intention was to derive the Fr\'echet derivative linking the observed delay to the model perturbation (Nolet 2009, pers. comm.). Here we show that the derivation by Dahlen et al. results in a first-order estimate even without anticipating a Fr\'echet derivative, but instead remaining deductively in their Taylor-series formulation. Although a mathematical technicality, this strengthens the result of Dahlen et al. by showing that it is intrinsically valid, requiring no external justification. We show also that ignoring the aforementioned term is not valid in general and needs to be supported by careful argument.