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<H3><A NAME="SECTION001853200000000000000">
Subtraction of dark and sky measurements</A>
</H3>
<P>
After interpolating any meteorological values that exist,
you may have to subtract dark current and/or sky values.
If there are no data for dark current, the reduction program will complain but
continue.
Sky values may have already been subtracted (e.g., in CCD measurements or
other data marked as RAWMAG instead of SIGNAL.)
<P>
DARK CURRENT:
Dark current is usually a strong function of detector temperature.
If you have regulated the detector temperature, then only a weak time
dependence might be expected -- perhaps only a small difference from one night
to the next, or a weak linear drift during each night.
You will have the choice of the model to be used in interpolating dark values.
<P>
If the detector temperature is measured, you should look for a temperature
dependence that is the same for all nights.
The program will show you all the dark data, with a separate symbol for each
night, as a function of temperature.
If all nights seem to show the same behavior, you can then fit a smoothing
function of temperature to the dark values.
You can choose a constant, a simple exponential (i.e., a straight line in the
plot of log<IMG
WIDTH="18" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img560.gif"
ALT="$\,($">DARK) vs. temperature), or the sum of two exponential terms.
<P>
Although in principle these ought to be of the form
<!-- MATH: $D = a \, exp \, (-b/T)$ -->
<IMG
WIDTH="175" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img561.gif"
ALT="$D = a \, exp \, (-b/T)$">,
the range of temperature available is usually insufficient
to distinguish between this and a simple
<!-- MATH: $a \, exp \, ( c T )$ -->
<IMG
WIDTH="101" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img562.gif"
ALT="$a \, exp \, ( c T )$">
term.
Furthermore, though the temperatures <I>ought</I> to be absolute temperatures in
Kelvins, you may have only some arbitrary numbers available, which might even
assume negative values.
In this case, an attempt to fit the correct physical form would blow up, but
the simple exponential term might still give reasonable results.
So the simpler form is actually used.
<P>
If the plot of log<IMG
WIDTH="9" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img563.gif"
ALT="$\,$">(DARK) vs. temperature bends up at the ends, or at least at
the right end, you should be able to get a good two-term fit.
If it looks linear, you can just fit a single line.
You also have the option of adopting a single mean value.
<P>
On the other hand, if the data are not consistent from night to night, or
show a temperature dependence that is different from the expected form, or if
you have no temperature information at all, you may have to interpolate the
dark data simply as a function of time.
As with the weather data, you have a choice of polygon, linear, or constant
fits.
Remember that a polygon fit uses every datum, right or wrong, and so is not
robust.
<P>
After removing a temperature-dependent fit, you will see the remaining
residuals plotted as a function of time.
This provides a double check on the adequacy of dark subtraction.
<P>
SKY SUBTRACTION:
Sky data must be treated separately for each night and passband.
Here, your options are more numerous.
You can choose the usual linear or constant fits; but those are likely to be a
poor representation of sky brightness.
<P>
More conventional choices are to use either the preceding or following sky for
each star observation, or the ``nearest'' sky (in which both time and position
separations are used to decide what ``nearest'' means).
Linear interpolation between successive sky measurements (i.e., a polygon fit)
is also an option.
These choices, while conventional, are not robust.
They are sensitive to gross errors in sky data, such as
star observations that have been marked as sky by mistake.
<P>
One might argue that bad sky data will stand out in the plots discussed below,
and that careful users will remove them and re-reduce their data.
One might also argue that really bad sky values will cause the stellar data to
be discarded or down-weighted in the later reductions, so that a robust fit at
this stage is not absolutely necessary.
However, such arguments are not completely convincing.
Therefore a more elaborate sky subtraction option is available, which tries to
model the sky brightness, discriminating
against outlying points in a robust regression.
<P>
To help you choose the best method, the program displays three plots of sky
brightness against different independent variables: time, airmass, and lunar
elongation.
In the time plot, the times of moonrise and moonset are marked,
and twilight data are marked t;
Figure <A HREF="node276.html#skyplot">13.1</A> shows an example.
In the other two plots, points with the Moon above the horizon are marked with
the letter M, points with the Moon below the horizon are marked by a minus
sign, and twilight data are marked t.
In these and other plots, the characters <TT>^ </TT> on the top line or <TT>v</TT>
on the lower edge indicate points
outside the plotting area; and $ indicates multiple overlapping points.
You can re-display the plots if you want to look at them again before deciding
which sky-subtraction method to use.
<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="skyplot"> </A><A NAME="12393"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure:</STRONG>
Plot of sky brightness as a function of time</CAPTION>
<TR><TD><IMG
WIDTH="1004" HEIGHT="834"
SRC="img564.gif"
ALT="\begin{figure}
\begin{center}
\begin{tex2html_preform}\begin{verbatim}u SKY on F...
... decimal of day) -->\end{verbatim}\end{tex2html_preform}\end{center}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<BR>
<P>
Note that no one method is best for all circumstances.
While modelling the sky should work well under good conditions, there
are certainly cases in which it will fail.
<P>
For example, when using DC or charge-integration equipment,
an observer commonly
uses the same gain setting for both (star+sky) and sky alone.
This is perfectly appropriate, as it makes any electrical zero-point error
cancel out in taking the difference.
But often the limited precision available -- for example, a 3-digit digital
voltmeter -- means that the sky brightness is measured with a precision of
barely one significant figure when bright stars are observed.
If a bright star reading is 782 and the sky alone is 3, one does not have much
information to use in modelling the sky.
<P>
Another case where one does better to subtract individual sky readings is
observations made during auroral activity.
While one would prefer not to use such data, because of the rapidity of sky
variations, they must sometimes be used.
Here again, subtraction of the nearest sky reading is better than using a
model, because the rapid fluctuations are not modelled.
Likewise, when terrestrial light sources around the horizon make the sky
brightness change rapidly with azimuth
and/or time, no simple sky model would be adequate.
<P>
If it is necessary to make measurements of some objects through two or more
different focal-plane diaphragms, these measurements cannot be combined
directly.
Ordinarily, all observations to be reduced together should be measured through
the same aperture, because the instrumental system changes in an unpredictable
way with aperture size.
Even the sky measurements are not exactly proportional to the diaphragm area.
However, it may be possible to reduce program objects observed with a
non-standard aperture <I>as if</I> they were measured through the standard
one, and then apply a suitable transformation after the fact.
This means that a sufficient number of calibration measurements of stars having
a considerable range in color <I>must</I> be taken, using both aperture sizes,
to determine the transformation between the two instrumental systems.
In such cases, individual sky readings taken through the same apertures must be
used in the reductions.
The reduction program will complain if you try to intermix data taken through
different diaphragms, and data taken with a peculiar aperture will be rejected
if there are no corresponding sky measurements.
<P>
Finally, when very faint stars are observed (as in setting up secondary
standards for use with a CCD), so that the sky is a large fraction of the star
measurement, it may be necessary to subtract individual sky readings simply
because the model used is not sufficiently accurate.
The model is reasonably good, but is not good enough to produce
estimates free of systematic error.
<P>
In any case, the plots of sky vs. time, airmass, and lunar elongation should
prove useful in assessing the quality of the sky data, and in choosing the
best subtraction strategy.
Furthermore, the residuals from the sky model may be useful in identifying bad
sky measures that should be removed; so it is a good idea to run the sky
model, even if you decide not to subtract its calculated values from the star
data.
<P>
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<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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