Abstract

Seismic traveltime tomography is a nonlinear inverse problem wherein an unknown slowness model is inferred from the observed arrival times of seismic waves. Nonlinearity arises because the raypath connecting a given source and receiver depends on the slowness. Specifically, if L(s) designates a raypath through the slowness model s between two fixed endpoints, then the path integral for traveltime  
t(s)=L(s)s dl

is a nonlinear functional of s because it does not, in general, satisfy the superposition condition (i.e., t(s1, + s2) ≠ t(s1), + t(s2) where s1 and s2 are two different slowness models). The tomographic inverse problem can be solved after linearizing the traveltime expression about a known slowness model s0. This linearized expression is usually obtained by appealing to Fermat's principle (e.g., Nolet, 1987). Alternately, the required relation can be rigorously derived via ray-perturbation theory (Snieder and Sambridge, 1992). The purpose of this note is to present a straightforward derivation of the same result by linearizing the eikonal equation for traveltimes. Wenzel (1988) adopts this approach, but his method of proof cannot be generalized to heterogeneous 3-D media. A full 3-D treatment is given here. The proof is remarkably simple, and thus it is quite possible that others have discovered it previously.

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