  In this paper we show how to use data on Lyman-limit and Damped Lyman-alpha
absorption systems to derive the hydrogen ionization fractions and the
distribution of the face-on total gas column density. We consider axially
symmetric, randomly oriented absorbers, ionized by an external background
radiation field in order to relate the face-on total gas distribution to that
of the neutral hydrogen observed along the line of sight. We devise a
statistical procedure based on the Maximum Likelihood criterion, that is able
to treat simultaneously data coming from different surveys and statistically
recovers the "true" column densities in the presence of large uncertainties:
this is especially important for Lyman-limit systems which leave an
unmeasurable residual flux at wavelengths shorter than the Lyman break. We make
use of simulated data to look for possible observational biases and extensively
test our procedure. For a large statistical sample of real data in the redshift
range [1.75,3.25] (collected from all published surveys) our Maximum Likelihood
procedure gives a power-law slope for the total hydrogen distribution of -2.7.
All together Lyman-limit systems therefore contain more gas than Damped
Lyman-alpha systems. Analysis of data at other redshifts shows that more
observations are needed to reach a compelling evidence for a cosmological
evolution of the slope of the gas distribution.
