Sparse travel time tomography with adaptive dictionaries

We develop a 2D travel time tomography method which regularizes the inversion by modeling groups of slowness pixels from discrete slowness maps, called patches, as sparse linear combinations of atoms from a dictionary. We further propose to learn optimal slowness dictionaries using dictionary learning, in parallel with the inversion. This patch regularization, which we call the local model, is integrated into the overall slowness map, called the global model. Where the local model considers small-scale variations using a sparsity constraint, the global model considers larger-scale features which are constrained using $\ell_2$-norm regularization. This local-global modeling strategy with dictionary learning has been successful for image restoration tasks such as denoising and inpainting, where diverse image content is recovered from noisy or incomplete measurements. We use this strategy in our locally-sparse travel time tomography (LST) approach to model simultaneously smooth and discontinuous slowness features. This is in contrast to conventional tomography methods, which constrain models to be exclusively smooth or discontinuous. We develop a $\textit{maximum a posteriori}$ formulation for LST and exploit the sparsity of slowness patches using dictionary learning. We demonstrate the LST approach on densely, but irregularly sampled synthetic slowness maps.