The first step in the determination of the line parameters is to compute both the level of the local continuum and the noise in the spectrum. Since all spectra had their continuum normalised to unity, the local continuum is essentially equal to one. When the uncertainties in the intensity of each spectral point were computed by error propagation during data reduction (see Section 3.4) the noise level is taken to be the root mean square (rms) of the uncertainties in each spectral point. Otherwise, an estimate for the noise in the spectrum is given by the rms of the deviations from the continuum in a region of the spectrum devoid of line features, i.e.
where 
is the intensity at the wavelength 
, corresponding to a region in the spectrum without line
features. The continuum level is taken to be the average intensity in
that region, i.e.
and the uncertainty 
is
There is an excellent agreement between the two estimates as already indicated in Section 3.4.
In order to determine the line peak and its position, a spectral region in the
vicinity of the line peak is chosen and a least squares fit of a second
degree polynomial is
performed to the data points in this region. Let 
be the
second order polynomial that fits the region in the vicinity of the line
peak, 
the parameters resulting from the fit and let 
be
the uncertainty in the parameters. Then,
is the resulting fit. The position of the line peak is found by
determining the maximum of the parabola. From differentiating
5.3 the peak position 
is
with uncertainty
The peak intensity is found by substituting 
into
5.3 to find
and its uncertainty is
Since the continuum is always essentially equal to unity
is the line to continuum ratio at line peak (L/C), tabulated
in Tables 5.3 and 5.4.
The next parameter to be computed is the FWHM. In order to do that, firstly one has to compute hpi, half of the maximum intensity of the line relative to the continuum, which is determined from
and its uncertainty is
Assuming that in the vicinity of the two positions corresponding to half
of the peak intensity the spectrum can be approximated by a polynomial
of degree one
(
), one can fit a
straight line to the data in each of the regions, using a least squares
fit. The resulting best parameters give
with each region having their own 
and 
. The solution of
for each region gives 
and 
,
resulting in
with
The uncertainty in 
and in 
results
from 
above (equation 5.9) but also from the fact that the straight line fits
are just an approximation to the local line profile. The latter is taken into
account by changing 
such that
and
where 
and 
are respectively the rms of the fit
residuals blueward of the line peak and the rms of the fit residuals
redward of the line peak.
The uncertainties in 
and 
are therefore
and
with 
being the appropriate value for each fit.
The computation of the HWZI (really a half width at 2% of the line maximum intensity relative to the continuum) follows along the same lines as that of the FWHM. The 2% level at which the HWZI is determined is
with uncertainty
One now assumes that a parabola (
) is a good approximation
for the spectrum in
the vicinity of the velocities corresponding to zl given by equation
5.17. A least squares fit of a second degree polynomial to each of these
regions results in
for each region, which, when solved for 
, yields
and 
, the velocities at which 'zero' intensity is
reached, implying
The uncertaity in the HWZI results from the uncertainties in 
and 
, which in turn results from 
but also from the fit, analogously to the cases of 
and 
. Following the same
philosophy as above (see equations 5.13 and 5.14),
and
where 
and 
are, respectively, the rms of the fit
residuals blueward of the line peak and the rms of the fit residuals
redward of the line peak.
The uncertainties in 
and in 
follow
from the solution of equation 5.19 for 
with the
respective parameters for each fit.
The Equivalent Width is defined by
where
with 
being the line profile and 
and
being the wavelengths blueward and redward of the line peak at which the
line reaches the level of the continuum.
From 5.24 and 5.2 and from the fact that the noise in the
spectrum is 
follows that the uncertainty in R is,
and thence
Finally, the Asymmetry factor is defined by
where
and
The uncertainty in Af is
with 
and 
computed using equation
5.26 with the appropriate integration limits.
The method described above to compute the line parameters was implemented in an IDL routine. The integrations in equations 5.23, 5.26, 5.28 and 5.29 were performed using an IDL built-in routine ( int_tabulated.pro), which integrates a tabulated set of data on a closed interval using a five-point Newton-Cotes integration formula.
