14
- EXTRACTING THE SPECTRAL PROFILE
The following figure shows how to extract the spectral profile: the
pixels under the star's spectrum are added along the columns (transverse
axis). In this manner an image with a single dimension is obtained, the
spectral
profile. This profile can then be visualized as a graph. For a better
interpretation of the profile along the dispersion axis the profile is
duplicated along the transverse axis, thus producing a new image with two
dimensions.
Of course, a spectral image is like any other image: it should be pre-processed
first, by removing the offset and the obscurity signal and by dividing
by the flat-field image. The following image shows a cross section
of a non processed spectrum: note the presence of the sky background signal
superimposed on the star's spectrum: it should be removed before usable
results are obtained.
The level of quality achieved in this numerical binning operation
has important implications on the final result. If this operation is not
properly conducted it may add noise to the spectrum and so limit the radiometric
precision. On the following image, showing a spectrum portion in 3 dimensions,
the rules to follow are obvious: one should not bin on too large
a width L else excessive noise from the sky background would
be integrated, but on the other hand L should be large enough that
most of the spectrum's signal be taken into account. There is a compromise
to be found here. Also the sky background level should be carefully measured
on either side of each point in the spectrum, then substracted from
the spectrum's value at this point.
Determining the optimal width L is not really obvious. In the following
example the blue part of Vega's spectrum (type A0V) is shown. Note that
this spectrum was obtained with an Audine camera featuring a KAF-0401E
CCD, which gives access to the H and K lines of ionized Calcium below 400
nm (the CaII-H line is practically indistinguishable from the H-epsilon
line in this spectrum). The spectrograph that has been used features ordinary
dioptric objectives (using classical lenses), there is some severe chromatism
in this part of the spectrum, which results in defocusing of the spectrum
as a function of the wavelength. Extracting the spectral profile
is not very easy, and may imply using algorithms more complex than a mere
addition of columns on a given width L.
The following figure shows a simplified algorithm that allows an optimized
extraction of the spectral profile. The logics are simple: before summing
up the pixels in a spectrum's column, a ponderation proportionnal to the
noise's variance is applied to them. In other words, the weaker the signal
for a pixel, the lesser its contribution to the line's profile computation.
There are several techniques for the weight function W. The one indicated
here is relatively simple because the value of the function does not depend
on the wavelength. However it is efficient.
However, improving the signal to noise ratio of the profile R (see next
figure) is only significant if the spectrum is very weak. An improvement
of about 10% of the signal to noise ratio can be reached, equivalent
to a reduction in observing time (for equivalent results) of some 20%.
Note that as in deep sky imaging one should composite many spectra to obtain
a better signal to noise ratio.
