Proof of validity of first-order travel estimates

TL;DR

This paper validates the first-order travel time delay estimate by Dahlen et al., demonstrating its intrinsic correctness within a Taylor-series framework and clarifying the conditions under which additional terms are necessary.

Contribution

It proves that the first-order estimate is valid without relying on the Fréchet derivative, strengthening the original result and clarifying the importance of including all relevant terms.

Findings

The first-order estimate is valid within a Taylor-series framework.

Ignoring additional terms requires careful justification.

The result is mathematically strengthened and clarified.

Abstract

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.