  Full waveform inversion (FWI) is a process in which seismic numerical
simulations are fit to observed data by changing the wave velocity model of the
medium under investigation. The problem is non-linear, and therefore
optimization techniques have been used to find a reasonable solution to the
problem. The main problem in fitting the data is the lack of low spatial
frequencies. This deficiency often leads to a local minimum and to
non-plausible solutions. In this work we explore how to obtain low frequency
information for FWI. Our approach involves augmenting FWI with travel time
tomography, which has low-frequency features. By jointly inverting these two
problems we enrich FWI with information that can replace low frequency data. In
addition, we use high order regularization, in a preliminary inversion stage,
to prevent high frequency features from polluting our model in the initial
stages of the reconstruction. This regularization also promotes the
non-dominant low-frequency modes that exist in the FWI sensitivity. By applying
a joint FWI and travel time inversion we are able to obtain a smooth model than
can later be used to recover a good approximation for the true model. A second
contribution of this paper involves the acceleration of the main computational
bottleneck in FWI--the solution of the Helmholtz equation. We show that the
solution time can be reduced by solving the equation for multiple right hand
sides using block multigrid preconditioned Krylov methods.
