  Travel time tomography is used to infer the underlying three-dimensional
wavespeed structure of the Earth by fitting seismic travel time data collected
at surface stations. Data interpolation and denoising techniques are important
pre-processing steps that use prior knowledge about the data, including
parsimony in the frequency and wavelet domains, low-rank structure of
matricizations, and local smoothness.
  We show how local smoothness structure can be combined with low rank
constraints using level-set optimization formulations, and develop a new
relaxation algorithm that can efficiently solve these joint problems. In the
seismology setting, we use the approach to interpolate missing stations and
de-noise observed stations. The new approach is competitive with alternative
algorithms, and offers new functionality to interpolate observed data using
both smoothness and low rank structure in the presence of data fitting
constraints.
