Travel time tomography with formally determined incomplete data in 3D
This solves a long-standing challenge in 3D travel time tomography by providing a globally convergent method for incomplete data, which is crucial for geophysical and medical imaging applications.
The paper develops the first globally convergent numerical method and Lipschitz stability estimate for 3D travel time tomography with formally determined incomplete data, proving global convergence of a gradient projection method as noise tends to zero.
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.