Traveltime inverse problem

The traveltime inverse problem provides another typical example of application of the linearization method to the solution of inverse problems for wave phenomena. 

In the framework of the geometric optics approach to seismic problems, the traveltime r(r, r ) of the seismic ray can be related to the local seismic velocity c(r) by the relationship 

The last equation can be written in another form, using slowness s(r) = 

Note that this equation looks linear, but is actually nonlinear, because the raypath L(r, rj) also depends on the slowness s(r). 

However, we can calculate the variation of the traveltime using Fermat s principle, which states that the traveltime is stationary with respect to a variation of the raypath L(r, Vj)  

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