  The investigation of the statistical properties of maps of line centroids has
been used for almost 50 years, but there is still no general agreement on their
interpretation. We try to quantify which properties of underlying turbulent
velocity fields can be derived from centroid velocity maps, and we test
conditions under which the scaling behaviour of the centroid velocities matches
the scaling of the three-dimensional velocity field. Using fractal cloud models
we study systematically the relation between three-dimensional density and
velocity fields and the statistical properties of the produced line centroid
maps. We put special attention to cases with large density fluctuations
resembling supersonic interstellar turbulence. Starting from the Delta-variance
analysis we derive a new tool to compute the scaling behaviour of the
three-dimensional velocity field from observed intensity and centroid velocity
maps. We provide two criteria to decide whether the information from the
centroid velocities directly reflects the properties of the underlying velocity
field. Applying these criteria allows to understand the different results found
so far in the literature on the interpretation of the statistics of velocity
centroids. A new iteration scheme can be used to derive the three-dimensional
velocity scaling from centroid velocity maps for arbitrary density and velocity
fields, but it requires an accurate knowledge of the average density of the
considered interstellar cloud.
