# Travel time tomography with formally determined incomplete data in 3D

## TL;DR

This paper introduces a new globally convergent numerical method for 3D travel time tomography with incomplete data, providing stability estimates and a convergent algorithm.

## Contribution

It develops the first globally convergent method with stability estimates for 3D travel time tomography with incomplete data.

## Key findings

- Lipschitz stability estimate derived, ensuring uniqueness.
- A weighted convex functional is constructed for the inverse problem.
- A gradient projection method is proven to converge globally as data noise diminishes.

## Abstract

For the first time, a globally convergent numerical method is developed and Lipschitz stability estimate is obtained for the challenging problem of travel time tomography in 3D for formally determined incomplete data. The semidiscrete case is considered meaning that finite differences are involved with respect to two out of three variables. First, Lipschitz stability estimate is derived, which implies uniqueness. Next, a weighted globally strictly convex Tikhonov-like functional is constructed using a Carleman-like weight function for a Volterra integral operator. The gradient projection method is constructed to minimize this functional. It is proven that this method converges globally to the exact solution if the noise in the data tends to zero.