  Directionally sensitive dark matter (DM) direct detection experiments present
the only way to observe the full three-dimensional velocity distribution of the
Milky Way halo local to Earth. In this work we compare methods for extracting
information about the local DM velocity distribution from a set of recoil
directions and energies in a range of hypothetical directional and
non-directional experiments. We compare a model independent empirical
parameterisation of the velocity distribution based on an angular
discretisation with a model dependent approach which assumes knowledge of the
functional form of the distribution. The methods are tested under three
distinct halo models which cover a range of possible phase space structures for
the local velocity distribution: a smooth Maxwellian halo, a tidal stream and a
debris flow. In each case we use simulated directional data to attempt to
reconstruct the shape and parameters describing each model as well as the DM
particle properties. We find that the empirical parametrisation is able to make
accurate unbiased reconstructions of the DM mass and cross section as well as
capture features in the underlying velocity distribution in certain directions
without any assumptions about its true functional form. We also find that by
extracting directionally averaged velocity parameters with this method one can
discriminate between halo models with different classes of substructure.
