What information on the Earth model do reflection travel times provide?

Abstract

Inversion of reflection travel times (or reflection tomography or tomographic inversion) allows determination of the structure of the subsurface from arrival times of reflected waves. Because the solution of a tomographic inversion is often underdetermined, it is essential to carry out an uncertainty analysis on the solution, which is classically done by linearizing the problem at the vicinity of the computed solution. Earlier uncertainty analyses gave results which strongly depend on the chosen discretization interval: they have no physical meaning. Our goal is to compute numerical uncertainties that approximate the physical uncertainties. Limiting our study to Hilbertian model spaces, we theoretically derive a necessary and sufficient condition which yields the desired result: the norm chosen in model space has to bind the Fréchet derivative of the forward map. Physically, this requires the velocity distribution and the reflectors to be smooth functions, an assumption which is quite compatible with the smoothness imposed to make use of ray theory. The required degree of smoothness depends on the space dimension (two‐ or three‐dimensional). This theoretical study yields a new method which allows, in the classical framework of a linearization, calculation of uncertainties that are almost independent of the chosen model discretization interval: they are thus intrinsically related to travel time information. Hence we can predict how travel times determine the slowness distribution and the reflector geometries. Our method is validated for a simple example. As a result, a very fine parameterization of the model is needed for an accurate estimate of the uncertainties on the solution.