  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 propose
to use dictionary learning during the inversion to adapt dictionaries to
specific slowness maps. This patch regularization, called the local model, is
integrated into the overall slowness map, called the global model. The local
model considers small-scale variations using a sparsity constraint and the
global model considers larger-scale features constrained using $\ell_2$
regularization. This strategy in a locally-sparse travel time tomography (LST)
approach enables simultaneous modeling of 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. The LST approach compares favorably
with smoothness and total variation regularization methods on densely, but
irregularly sampled synthetic slowness maps.
