  (Modified) The scaling of velocity fluctuation, dv, as a function of spatial
scale L in molecular clouds can be measured from size-linewidth relations,
principal component analysis, or line centroid variation. Differing values of
the power law index of the scaling relation dv = L^(g3D) in 3D are given by
these different methods: the first two give g3D=0.5, while line centroid
analysis gives g3D=0. This discrepancy has previously not been fully
appreciated, as the variation of projected velocity line centroid fluctuations
(dv_{lc} = L^(g2D)) is indeed described, in 2D, by g2D=0.5. However, if
projection smoothing is accounted for, this implies that g3D=0. We suggest that
a resolution of this discrepancy can be achieved by accounting for the effect
of density inhomogeneity on the observed g2D obtained from velocity line
centroid analysis. Numerical simulations of compressible turbulence are used to
show that the effect of density inhomogeneity statistically reverses the effect
of projection smoothing in the case of driven turbulence so that velocity line
centroid analysis does indeed predict that g2D=g3D=0.5. Using our numerical
results we can restore consistency between line centroid analysis, principal
component analysis and size-linewidth relations, and we derive g3D=0.5,
corresponding to shock-dominated (Burgers) turbulence. We find that this
consistency requires that molecular clouds are continually driven on large
scales or are only recently formed.
