Ill‐posed and well‐posed formulations of the reflection travel time tomography problem

Many studies have shown the strong dependence of the solution of reflection tomography (or travel time inversion) in the model discretization interval: such solutions are purely numerical artifacts. The cause lies in the formulation of the reflection tomography which turns out to be an ill-posed problem unless properly regularized. Adequate regularizations are those which make use of derivatives of the functions describing the model (velocity distributions and reflector geometries). The order of the derivatives to use depends on the space dimension (two or three dimension). Of practical interest is curvature regularization: it works both in two and three dimensions with minimum a priori knowledge on the substratum under survey. We give theoretical arguments to prove that, with such a regularization, the computed solution is unique, stable, and almost independent of model discretization, provided that it is fine enough. These theoretical results are confirmed by numerical tests on a two-dimensional model with a velocity field varying both vertically and horizontally: they show the practical interest of the proposed regularization. However, mathematical stability is not sufficient, and a priori information has to be used in order to decrease the size of the set of possible solutions. As a consequence of our theoretical study of regularization techniques, only a priori information involving derivatives of the functions describing the model yields stability. In other words, except for some pathological examples, stability is obtained only if we try to determine a smooth model of the subsurface. These conclusions remain valid for three-dimensional models and for transmission tomography. As a by-product of our study, we suggest a preconditioning which makes much easier the solution of the linear systems involved in the Gauss-Newton optimization method.