**5.2. General Techniques**

The abundance analysis for absorption lines is, in principle,
much simpler than for the emission lines because the absorption
yields direct estimates of the ionic column densities.
One only has to apply an ionization correction to convert the column
density ratios into relative abundances. The logarithmic abundance
ratio of any two elements *a* and *b* is related to their column
densities by,

(1)

where (*b / a*)_{}
is the solar abundance ratio, and *N* and *f*
are respectively the column densities and ionization fractions
of elements *a* and *b* in ion states *i* and *j*.
If the gas is in photoionization equilibrium and optically thin at all
far-UV continuum wavelengths, the correction factors,
*f (b _{j}) / f (a_{i})*,
depend only on the shape of the ionizing spectrum and the ionization
parameter

Figure 6 shows theoretical ionization fractions,
*f*(M_{i}), for
various metals, M, in ion stage, *i*, as a function of the ionization
parameter, *U*, in optically thin photoionized clouds (from
HF99).
The HI fraction, *f*(HI), is shown across the top of
the figure. The calculations were performed using CLOUDY (version
90.04,
Ferland et al. 1998)
with a power-law ionizing spectrum
with index = -1.5, where
*f*_{} ^{}.
Note that the results in Figure 6 are not
sensitive to the specific
densities or abundances used in the calculations (within reasonable
limits, see
Hamann 1997).
Ideally, we would constrain the ionization
state (i.e. *U*) in Figure 6 by comparing
the column densities in
different ionization stages of the same element. We can also constrain the
ionization by comparing ions of different metals with some reasonable
assumption about their relative abundance. With *U* thus constrained,
Figure 6 provides the correction factors needed
to derive abundance ratios from Eqn. 1. Repeating this procedure
with calculations for different ionizing spectral shapes
yields estimates of the theoretical uncertainties
(Hamann 1997).

If the data provide no
useful constraints on *U* (because too few lines are measured)
or multiple constraints imply a range of ionization states
(as in the *z _{a}*