  Direct detection experiments searching for weakly interacting massive
particle (WIMP) dark matter typically use a simplified model of the Galactic
halo to derive parameter constraints. However, there is strong evidence that
this Standard Halo Model is not a good approximation to our Galaxy. We discuss
previous attempts to extract the WIMP mass, cross-section and speed
distribution from direct detection data and show that these lead to significant
biases in the reconstructed parameter values. We develop and test an
alternative model-independent method based on parametrising the momentum
distribution of the WIMPs. This allows us to limit the analysis only to those
regions of momentum space to which the experiments are sensitive. The method
can be applied to a single experiment to extract information about the
distribution function, as well as information on the degenerate WIMP mass and
interaction cross-section combined in a single parameter. This degeneracy can
be broken by including data from additional experiments, meaning that the WIMP
mass and speed distribution can be recovered. We test the momentum
parametrisation method using mock datasets from proposed ton-scale direct
detection experiments, showing that it exhibits improved coverage properties
over previous methods, as well as significantly reduced bias. We are also able
to accurately reconstruct the shape of the WIMP speed distribution but
distinguishing between different underlying distributions remains difficult.
