  Peculiar velocity surveys have non-uniform spatial distributions of tracers,
so that the bulk flow estimated from them does not correspond to that of a
simple volume such as a sphere. Thus bulk flow estimates are generally not
strictly comparable between surveys, even those whose effective depths are
similar. In addition, the sparseness of typical surveys can lead to aliasing of
small scale power into what is meant to be a probe of the largest scales. Here
we introduce a new method of calculating bulk flow moments where velocities are
weighted to give an optimal estimate of the bulk flow of an idealized survey,
with the variance of the difference between the estimate and the actual flow
being minimized. These "minimum variance" estimates can be designed to estimate
the bulk flow on a particular scale with minimal sensitivity to small scale
power, and are comparable between surveys. We compile all major peculiar
velocity surveys and apply this new method to them. We find that most surveys
we studied are highly consistent with each other. Taken together the data
suggest that the bulk flow within a Gaussian window of radius 50 Mpc/h is 407
km/s toward l=287 and b=8. The large-scale bulk motion is consistent with
predictions from the local density field. This indicates that there are
significant density fluctuations on very large scales. A flow of this amplitude
on such a large scale is not expected in the WMAP5-normalized LCDM cosmology,
for which the predicted one-dimensional r.m.s. velocity is ~110 km/s. The large
amplitude of the observed bulk flow favors the upper values of the WMAP5
error-ellipse, but even the point at the top of the WMAP5 95% confidence
ellipse predicts a bulk flow which is too low compared to that observed at >98%
confidence level.
