  The inference of flows of material in the interior of the Sun is a subject of
major interest in helioseismology. Here we apply techniques of Full Waveform
Inversion (FWI) to synthetic data to test flow inversions. In this idealized
setup, we do not model seismic realization noise, training the focus entirely
on the problem of whether a chosen supergranulation flow model can be
seismically recovered. We define the misfit functional as a sum of L_2 norm
deviations in travel times between prediction and observation, as measured
using short-distance f and p_1 filtered and large-distance unfiltered $p$
modes. FWI allows for the introduction of measurements of choice and
iteratively improving the background model, while monitoring the evolution of
the misfit in all desired categories. Although the misfit is seen to uniformly
reduce in all categories, convergence to the true model is very slow, possibly
because it is trapped in a local minimum. The primary source of error is
inaccurate depth localization, which, owing to density stratification, leads to
wrong ratios of horizontal and vertical flow velocities (`cross talk'). In the
present formulation, the lack of sufficient temporal frequency and spatial
resolution makes it difficult to accurately localise flow profiles at depth. We
therefore suggest that the most efficient way to discover the global minimum is
to perform a probabilistic forward search, involving calculating the misfit
associated with a broad range of models (generated, for instance, by a
Monte-Carlo algorithm) and locating the deepest minimum. Such techniques
possess the added advantage of being able to quantify model uncertainty as well
as realization noise (data uncertainty).
