Using data on the H I column density distribution in the local Universe , f ( N _ { HI } ) , in this paper we show how to determine g ( N _ { H } ) , the distribution of the total gas ( H I +H II ) column density .
A simple power law fit to f ( N _ { HI } ) fails due to bendings in the distributions when N _ { HI } < 10 ^ { 20 } cm ^ { -2 } and H is no longer fully neutral .
If an ultraviolet background is responsible for the gas ionization , and g ( N _ { H } ) \propto N _ { H } ^ { - \alpha } , we find the values of \alpha and of the intensity of the background radiation which are compatible with the present data .
The best fitting values of \alpha , however , depend upon the scaling law of the the gas volume densities with N _ { H } and can not be determined unambiguously .
We examine in detail two models : one in which the average gas volume density decreases steadily with N _ { H } , while in the other it stays constant at low column densities .
The former model leads to a steep power law fit for g ( N _ { H } ) , with \alpha \simeq 3.3 \pm 0.4 and requires an ultraviolet flux larger than what the QSOs alone produce at z = 0 .
For the latter \alpha \simeq 1.5 \pm 0.1 and a lower ionizing flux is required .
The ambiguities about the modelling and the resulting steep or shallow N _ { H } distribution can be resolved only if new 21-cm observations and QSOs Lyman limit absorbers searches will provide more data in the H I -H II transition region at low redshifts .
Using the best fit obtained for higher redshift data we outline two possible scenarios for the evolution of gaseous structures , compatible with the available data at z \sim 0 .