From Cosmicflows distance moduli to unbiased distances and peculiar
  velocities
  Surveys of galaxy distances and radial peculiar velocities can be used to
reconstruct the large scale structure. Other than systematic errors in the
zero-point calibration of the galaxy distances the main source of uncertainties
of such data are errors on the distance moduli, assumed here to be Gaussian and
thus turn into lognormal errors on distances and velocities. Naively treated,
it leads to spurious nearby outflow and strong infall at larger distances. The
lognormal bias is corrected here and tested against mock data extracted from a
$\Lambda$CDM simulation, designed to statistically follow the grouped
Cosmicflows-3 (CF3) data. Considering a subsample of data points, all of which
have the same true distances or same redshifts, the lognormal bias arises
because the means of the distributions of observed distances and velocities are
skewed off the means of the true distances and velocities. Yet, the medians are
invariant under the lognormal transformation. That invariance allows the
Gaussianization of the distances and velocities and the removal of the
lognormal bias. This Bias Gaussianization correction (BGc) algorithm is tested
against mock CF3 catalogs. The test consists of a comparison of the BGC
estimated with the simulated distances and velocities and of an examination of
the Wiener filter reconstruction from the BGc data. Indeed, the BGc eliminates
the lognormal bias. The estimation of Hubble's ($H_{0}$) constant is also
tested. The residual of the BGc estimated $H_{0}$ from the simulated values is
$0.6 \pm 0.7 {\rm kms}^{-1}{\rm Mpc}^{-1}$ and is dominated by the cosmic
variance. The BGc correction of the actual CF3 data yields $H_{0} = 75.8 \pm
1.1 {\rm kms}^{-1}{\rm Mpc}^{-1}$ .
