.help approfiles Feb93 noao.twodspec.apextract

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Spectrum Profile Determinations


The foundation of variance weighted or optimal extraction, cosmic ray
detection and removal, two dimensional flat field normalization, and
spectrum fitting and modeling is the accurate determination of the
spectrum profile across the dispersion as a function of wavelength.
The previous version of the APEXTRACT package accomplished this by
averaging a specified number of profiles in the vicinity of each
wavelength after correcting for shifts in the center of the profile.
This technique was sensitive to perturbations from cosmic rays
and the exact choice of averaging parameters.  The current version of
the package uses two different algorithm which are much more stable.

The basic idea is to normalize each profile along the dispersion to
unit flux and then fit a low order function to sets of unsaturated
points at nearly the same point in the profile parallel to the
dispersion.  The important point here is that points at the same
distance from the profile center should have the nearly the same values
once the continuum shape and spectral features have been divided out.
Any variations are due to slow changes in the shape of the profile with
wavelength, differences in the exact point on the profile, pixel
binning or sampling, and noise.  Except for the noise, the variations
should be slow and a low order function smoothing over many points will
minimize the noise and be relatively insensitive to bad pixels such as
cosmic rays.  Effects from bad pixels may be further eliminated by
chi-squared iteration and clipping.  Since there will be many points
per degree of freedom in the fitting function the clipping may even be
quite aggressive without significantly affecting the profile
estimates.  Effects from saturated pixels are minimized by excluding
from the profile determination any profiles containing one or more
saturated pixels as defined by the \fIsaturation\fR parameter.

The normalization is, in fact, the one dimensional spectrum.  Initially
this is the simple sum across the aperture which is then updated by the
variance weighted sum with deviant pixels possibly removed.  This updated
one dimensional spectrum is what is meant by the profile normalization
factor in the discussion below.  The two dimensional spectrum model or
estimate is the product of the normalization factor and the profile.  This
model is used for estimating the pixel intensities and, thence, the
variances.

There are two important requirements that must be met by the profile fitting
algorithm.  First it is essential that the image data not be
interpolated.  Any interpolation introduces correlated errors and
broadens cosmic rays to an extent that they may be confused with the
spectrum profile, particularly when the profile is narrow.  This was
one of the problems limiting the shift and average method used
previously.  The second requirement is that data fit by the smoothing
function vary slowly with wavelength.  This is what precludes, for
instance, fitting profile functions across the dispersion since narrow,
marginally sampled profiles require a high order function using only a
very few points.  One exception to this, which is sometimes useful but
of less generality, is methods which assume a model for the profile
shape such as a gaussian.  In the methods used here there is no
assumption made about the underlying profile other than it vary
smoothly with wavelength.

These requirements lead to two fitting algorithms which the user
selects with the \fIpfit\fR parameter.  The primary method, "fit1d",
fits low order, one dimensional functions to the lines or columns
most nearly parallel to the dispersion.  While this is intended for
spectra which are well aligned with the image axes, even fairly large
excursions or tilts can be adequately fit in this
way.  When the spectra become strongly tilted then single lines or
columns may cross the actual profile relatively quickly causing the
requirement of a slow variation to be violated.  One thought is to use
interpolation to fit points always at the same distance from the
profile.  This is ruled out by the problems introduced by
image interpolation.  However, there is a clever method which, in
effect, fits low order polynomials parallel to the direction defined by
tracing the spectrum but which does not interpolate the image data.
Instead it weights and couples polynomial coefficients.  This
method was developed by Tom Marsh and is described in detail in the
paper, "The Extraction of Highly Distorted Spectra", PASP 101, 1032,
Nov. 1989.  Here we refer to this method as the Marsh or "fit2d"
algorithm and do not attempt to explain it further.

The choice of when to use the one dimensional or the two dimensional
fitting is left to the user.  The "fit1d" algorithm is preferable since it
is faster, easier to understand, and has proved to be very robust.  The
"fit2d" algorithm usually works just as well but is slower and has been
seen to fail on some data.  The user may simply try both to achieve the
best results.

What follows are some implementation details of the preceding ideas in the
APEXTRACT package.  For column/line fitting, the fitting function is a
cubic spline.  A base number of spline pieces is set by rounding up the
maximum trace excursion; an excursion of 1.2 pixels would use a spline of 2
pieces.  To this base number is added the number of coefficients in the
trace function in excess of two; i.e. the number of terms in excess of a
linear function.  This is done because if the trace wiggles a large amount
then a higher order function will be needed to fit a line or column as the
profile shifts under it.  Finally the number of pieces is doubled
because experience shows that for low tilts it doesn't matter but for
large tilts this improves the results dramatically.

For the Marsh algorithm there are two parameters to be set, the
polynomial order parallel to the dispersion and the spacing between
parallel, coupled polynomials.  The algorithm requires that the spacing
be less than a pixel to provide sufficient sampling.  The spacing is
arbitrarily set at 0.95 pixels.  Because the method always fits
polynomials to points at the same position of the profile the order
should be 1 except for variations in the profile shape with
wavelength.  To allow for this the profile order is set at 10; i.e. a
9th order function.  A final parameter in the algorithm is the number
of polynomials across the profile but this is obviously  determined
from the polynomial spacing and the width of the aperture including an
extra pixel on either side.

Both fitting algorithms weight the pixels by their variance as computed
from the background and background variance if background subtraction
is specified, the spectrum estimate from the profile and the spectrum
normalization, and the detector noise parameters.  A poisson
plus constant gaussian readout noise model is used.  The noise model is
described further in \fBapvariance\fR.

As mentioned earlier, the profile fitting can be iterated to remove
deviant pixels.  This is done by rejecting pixels greater than a
specified number of sigmas above or below the expected value based
on the profile, the normalization factor, the background, the
detector noise parameters, and the overall chi square of the residuals.
Rejected points are removed from the profile normalization and
from the fits.
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SEE ALSO
apbackground apvariance apall apsum apfit apflatten
.endhelp
