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<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>

<UL CLASS="ChildLinks">
<LI><A NAME="tex2html591"
  HREF="node7.html#SECTION00710000000000000000">CCD Camera Response Model</A>
<LI><A NAME="tex2html592"
  HREF="node7.html#SECTION00720000000000000000">Bias and Dark Frames</A>
<UL>
<LI><A NAME="tex2html593"
  HREF="node7.html#SECTION00720010000000000000">Dark frames</A>
<LI><A NAME="tex2html594"
  HREF="node7.html#SECTION00720020000000000000">Noise contribution of bias and dark frames.</A>
<LI><A NAME="tex2html595"
  HREF="node7.html#SECTION00721000000000000000">Working without Bias Frames</A>
</UL>
<BR>
<LI><A NAME="tex2html596"
  HREF="node7.html#SECTION00730000000000000000">Flat-field Frames</A>
<LI><A NAME="tex2html597"
  HREF="node7.html#SECTION00740000000000000000">Reducing the Data Frames</A>
<LI><A NAME="tex2html598"
  HREF="node7.html#SECTION00750000000000000000">Frame Combining Methods</A>
<UL>
<LI><A NAME="tex2html599"
  HREF="node7.html#SECTION00750010000000000000">Average</A>
<LI><A NAME="tex2html600"
  HREF="node7.html#SECTION00750020000000000000">Median</A>
<LI><A NAME="tex2html601"
  HREF="node7.html#SECTION00750030000000000000">Mean-Median</A>
<LI><A NAME="tex2html602"
  HREF="node7.html#SECTION00750040000000000000">Kappa-Sigma Clipping</A>
</UL>
<BR>
<LI><A NAME="tex2html603"
  HREF="node7.html#SECTION00760000000000000000">CCD Reduction with gcx</A>
<UL>
<LI><A NAME="tex2html604"
  HREF="node7.html#SECTION00761000000000000000">Loading and Selecting Image Frames</A>
<LI><A NAME="tex2html605"
  HREF="node7.html#SECTION00762000000000000000">Creating a Master Bias or Master Dark Frame</A>
<LI><A NAME="tex2html606"
  HREF="node7.html#SECTION00763000000000000000">Creating a Bias-Subtracted Master Dark Frame</A>
<LI><A NAME="tex2html607"
  HREF="node7.html#SECTION00764000000000000000">Creating a Master Flat Frame</A>
<UL>
<LI><A NAME="tex2html608"
  HREF="node7.html#SECTION00764010000000000000">Making a Superflat</A>
</UL>
<LI><A NAME="tex2html609"
  HREF="node7.html#SECTION00765000000000000000">Reducing the Data Frames</A>
<LI><A NAME="tex2html610"
  HREF="node7.html#SECTION00766000000000000000">Aligning and Stacking Frames</A>
<LI><A NAME="tex2html611"
  HREF="node7.html#SECTION00767000000000000000">Running CCD Reductions from the Command Line</A>
</UL></UL>
<!--End of Table of Child-Links-->
<HR>

<H1><A NAME="SECTION00700000000000000000">
CCD Reduction</A>
</H1>

<P>
Ideally, an image taked by a CCD camera through a telescope will give accurate information 
about the light flux distribution over a portion of the sky. Unfortunately, this is not 
generally the case. Instrument imperfections and the discrete nature of light itself
concur to introduce errors in the measured data. The errors (differences between 
the measured values and the ``true'' ones) are the result of several factors, some 
random in nature, and some deteministic. 

<P>
The goal of the CCD reduction process is to eliminate (or at least minimise) the contribution 
of deterministic factors in the errors, in other words to remove the <EM>instrument signature</EM>
from the data.

<P>
A second, but not less important, goal is to preserve information about the noise sources, so that 
users of the reduced data can evaluate the random errors of the data.

<P>
We begin this chapter by describing the way the general CCD reduction
process works, and in the process define bias, dark and flat
files. The second part of the chapter is devoted to the practical
implementation of the reduction tasks in the program.

<P>

<H1><A NAME="SECTION00710000000000000000">
CCD Camera Response Model</A>
</H1>
Raw pixel values for a CCD frame can be calculated as
follows:<A NAME="tex2html18"
  HREF="footnode.html#foot372"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">1</SPAN></SUP></A>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
s(x,y) = B(x,y) + t D(x,y) + t G(x, y) I(x, y) + {\rm noise}
\end{equation}
 -->
<A NAME="eq:ccd1"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:ccd1"></A><IMG
 WIDTH="398" HEIGHT="31" BORDER="0"
 SRC="img4.png"
 ALT="\begin{displaymath}
s(x,y) = B(x,y) + t D(x,y) + t G(x, y) I(x, y) + {\rm noise}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">1</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <SPAN CLASS="MATH"><IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.png"
 ALT="$B(x,y)$"></SPAN> is the <EM>bias</EM> value of each pixel, <SPAN CLASS="MATH"><IMG
 WIDTH="11" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img6.png"
 ALT="$t$"></SPAN> is the integration time, 
<SPAN CLASS="MATH"><IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img7.png"
 ALT="$D(x,y)$"></SPAN> is the <EM>dark current</EM>, 
<SPAN CLASS="MATH"><IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$G(x,y)$"></SPAN> is the <EM>sensitivity</EM> and <SPAN CLASS="MATH"><IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img9.png"
 ALT="$I(x,y)$"></SPAN> is the light flux reaching the pixel.
We cannot predict the instantaneous values of the noise component but some statistics 
about it can be calculated (Appendix&nbsp;<A HREF="node10.html#ap:noise">A</A>).

<P>
To estimate the flux values reaching the sensor from the raw frame, we need to
estimate <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img10.png"
 ALT="$B$"></SPAN>, <SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img11.png"
 ALT="$D$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.png"
 ALT="$G$"></SPAN>. After that, Equation <A HREF="#eq:ccd1">6.1</A> can be solved for <SPAN CLASS="MATH"><IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$I$"></SPAN>.
<SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img10.png"
 ALT="$B$"></SPAN>, <SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img11.png"
 ALT="$D$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.png"
 ALT="$G$"></SPAN> are calculated starting from calibration frames taken
under controlled conditions: 
<EM>bias</EM>, <EM>dark</EM> and <EM>flat</EM> frames.

<P>

<H1><A NAME="SECTION00720000000000000000">
Bias and Dark Frames</A>
</H1>

<P>
If we take very short exposures without opening the camera's shutter (<EM>bias frames</EM>
<SPAN CLASS="MATH"><IMG
 WIDTH="43" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img14.png"
 ALT="$t = 0$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.png"
 ALT="$I(x,y)=0$"></SPAN>; Equation&nbsp;(<A HREF="#eq:ccd1">6.1</A>) becomes: 
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
b(x,y) = B(x,y) + {\rm noise}
\end{equation}
 -->
<A NAME="eq:bias"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:bias"></A><IMG
 WIDTH="183" HEIGHT="31" BORDER="0"
 SRC="img16.png"
 ALT="\begin{displaymath}
b(x,y) = B(x,y) + {\rm noise}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">2</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
To obtain an estimate of <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img10.png"
 ALT="$B$"></SPAN>, we simply use <SPAN CLASS="MATH"><IMG
 WIDTH="12" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img17.png"
 ALT="$b$"></SPAN>:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{B}(x,y) = b(x,y)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="124" HEIGHT="31" BORDER="0"
 SRC="img18.png"
 ALT="\begin{displaymath}
\widetilde{B}(x,y) = b(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
We use the tilde to denote that we can only estimate <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img10.png"
 ALT="$B$"></SPAN>, because of the noise. If we average 
several bias frames together we can get arbitrarily close to <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img10.png"
 ALT="$B$"></SPAN>, as the relative noise 
contribution decreases with the square root of the number of frames averaged.
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{B}(x,y) = \frac{1}{N}\sum_i b_i(x,y)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="178" HEIGHT="51" BORDER="0"
 SRC="img19.png"
 ALT="\begin{displaymath}
\widetilde{B}(x,y) = \frac{1}{N}\sum_i b_i(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
We will call this the <EM>master bias frame</EM>.

<P>

<H4><A NAME="SECTION00720010000000000000">
Dark frames</A>
</H4> If we now take longer exposures with the shutter closed, 
we obtain <EM>dark frames</EM>:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
d(x,y) = B(x,y)+tD(x,y)+{\rm noise}
\end{equation}
 -->
<A NAME="eq:dark"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:dark"></A><IMG
 WIDTH="268" HEIGHT="31" BORDER="0"
 SRC="img20.png"
 ALT="\begin{displaymath}
d(x,y) = B(x,y)+tD(x,y)+{\rm noise}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
From this, we can simply subtract the bias and divide by the exposure time, and we get
our dark current estimate:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{D}(x,y) = \frac{d(x,y) - \widetilde{B}(x,y)}{t_{\rm dark}}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="206" HEIGHT="50" BORDER="0"
 SRC="img21.png"
 ALT="\begin{displaymath}
\widetilde{D}(x,y) = \frac{d(x,y) - \widetilde{B}(x,y)}{t_{\rm dark}}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">6</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Of course, to reduce the noise contribution we can also average several dark frames:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{D}(x,y) = \frac{1}{t_{\rm dark}}\frac{1}{M}\sum_i d_i(x,y) - \widetilde{B}(x,y)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="296" HEIGHT="51" BORDER="0"
 SRC="img22.png"
 ALT="\begin{displaymath}
\widetilde{D}(x,y) = \frac{1}{t_{\rm dark}}\frac{1}{M}\sum_i d_i(x,y) - \widetilde{B}(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">7</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
It is convenient to work with a different form of the dark current frame:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{D}'(x,y)=t_{\rm dark}\widetilde{D}(x,y)=\frac{1}{M}\sum_i d_i(x,y) - \widetilde{B}(x,y)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="375" HEIGHT="51" BORDER="0"
 SRC="img23.png"
 ALT="\begin{displaymath}
\widetilde{D}'(x,y)=t_{\rm dark}\widetilde{D}(x,y)=\frac{1}{M}\sum_i d_i(x,y) - \widetilde{B}(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">8</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
which will be called the <EM>bias subtracted master dark frame</EM>.

<P>
If we have a data frame with an integration time of <SPAN CLASS="MATH"><IMG
 WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img24.png"
 ALT="$t_{\rm data}$"></SPAN>, the first two
terms in (<A HREF="#eq:ccd1">6.1</A>) are estimated by the <EM>master dark</EM> frame <!-- MATH
 $\widetilde{D}_M$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="34" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img25.png"
 ALT="$\widetilde{D}_M$"></SPAN>:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{D}_M(x,y)
=\widetilde{B}(x,y) + t_{\rm data}\widetilde{D}(x,y)=
\widetilde{B}(x,y)+\frac{t_{\rm data}}{t_{\rm dark}}\widetilde{D}'(x,y)
\end{equation}
 -->
<A NAME="eq:mdark"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:mdark"></A><IMG
 WIDTH="452" HEIGHT="44" BORDER="0"
 SRC="img26.png"
 ALT="\begin{displaymath}
\widetilde{D}_M(x,y)
=\widetilde{B}(x,y) + t_{\rm data}\wide...
...e{B}(x,y)+\frac{t_{\rm data}}{t_{\rm dark}}\widetilde{D}'(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">9</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>

<P>

<H4><A NAME="SECTION00720020000000000000">
Noise contribution of bias and dark frames.</A>
</H4>
A detailed description of noise sources in CCD cameras is provided in Appendix&nbsp;<A HREF="node10.html#ap:noise">A</A>. 
If the dark current contribution is not very large, the <EM>noise</EM> terms of (<A HREF="#eq:bias">6.2</A>) and 
(<A HREF="#eq:dark">6.5</A>) are both equal to the camera <EM>read noise</EM>. 

<P>
If we use our estimated <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img27.png"
 ALT="$\widetilde{B}$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img28.png"
 ALT="$\widetilde{D}$"></SPAN> to reduce a data frame with
an integration time of <SPAN CLASS="MATH"><IMG
 WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img24.png"
 ALT="$t_{\rm data}$"></SPAN>, the bias and dark subtraction will contribute a
noise level of:<A NAME="tex2html19"
  HREF="footnode.html#foot453"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">2</SPAN></SUP></A>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{DB} = N_R\sqrt{\frac{1}{N}+\frac{1}{M}
  \left(\frac{t_{\rm data}}{t_{\rm dark}}\right)^2}
\end{equation}
 -->
<A NAME="eq:dbnoise"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:dbnoise"></A><IMG
 WIDTH="232" HEIGHT="55" BORDER="0"
 SRC="img29.png"
 ALT="\begin{displaymath}
\sigma_{DB} = N_R\sqrt{\frac{1}{N}+\frac{1}{M}
\left(\frac{t_{\rm data}}{t_{\rm dark}}\right)^2}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">10</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <SPAN CLASS="MATH"><IMG
 WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img30.png"
 ALT="$N_R$"></SPAN> is the camera read noise, <SPAN CLASS="MATH"><IMG
 WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img31.png"
 ALT="$N$"></SPAN> is the number of bias frames averaged, <SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img32.png"
 ALT="$M$"></SPAN> the number of 
dark frames averaged and <SPAN CLASS="MATH"><IMG
 WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img33.png"
 ALT="$t_{\rm dark}$"></SPAN> the integration time used for the dark frames.

<P>
We generally want to keep the square root in (<A HREF="#eq:dbnoise">6.10</A>) between <SPAN CLASS="MATH"><IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img34.png"
 ALT="$1/3$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img35.png"
 ALT="$1$"></SPAN>. 
A value lower than <SPAN CLASS="MATH"><IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img34.png"
 ALT="$1/3$"></SPAN> will provide a negligible improvement in the overall signal/noise 
ratio, while for values larger than 1, this term will dominate the camera read noise and 
become significant.

<P>

<H2><A NAME="SECTION00721000000000000000">
Working without Bias Frames</A>
</H2>
It is easy to observe from (<A HREF="#eq:mdark">6.9</A>) that we can obtain our master dark frame by simply
averaging dark frames taken with the same integration time as our data frames. In this
case, we don't need the bias frames at all:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{D}_M(x,y) = \frac{1}{M}\sum_i d_i(x,y)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="198" HEIGHT="51" BORDER="0"
 SRC="img36.png"
 ALT="\begin{displaymath}
\widetilde{D}_M(x,y) = \frac{1}{M}\sum_i d_i(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">11</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
As a bonus, the noise contribution of the master dark frame is reduced to:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{D} = N_R\sqrt{\frac{1}{M}}
\end{equation}
 -->
<A NAME="eq:dnoise"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:dnoise"></A><IMG
 WIDTH="108" HEIGHT="45" BORDER="0"
 SRC="img37.png"
 ALT="\begin{displaymath}
\sigma_{D} = N_R\sqrt{\frac{1}{M}}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">12</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>

<P>
In general, people using large telescopes and <SPAN CLASS="MATH"><IMG
 WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img38.png"
 ALT="$LN_2$"></SPAN> cooled cameras preffer using bias frames,
as they require less time than dark frames; The dark current of these cameras is 
very low and stable, and a single set of darks can be used to reduce many
observations. Users of thermoelectrically-cooled cameras, which have more significant 
dark currents, are more likely to use dark frames exclussively. 

<P>

<H1><A NAME="SECTION00730000000000000000">
Flat-field Frames</A>
</H1>

<P>
With <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img10.png"
 ALT="$B$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img11.png"
 ALT="$D$"></SPAN> out of the way, we need a way to estimate <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.png"
 ALT="$G$"></SPAN> in (<A HREF="#eq:ccd1">6.1</A>) before we can
recover the incident flux. To do this, we apply a flat-field (even)
illumination to 
the camera<A NAME="tex2html20"
  HREF="footnode.html#foot481"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">3</SPAN></SUP></A>and acquire several <EM>flat-field frames</EM>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
f(x,y)=B(x,y)+t_{\rm flat}D(x,y)+t_{\rm flat}G(x,y)L+{\rm noise}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="403" HEIGHT="31" BORDER="0"
 SRC="img39.png"
 ALT="\begin{displaymath}
f(x,y)=B(x,y)+t_{\rm flat}D(x,y)+t_{\rm flat}G(x,y)L+{\rm noise}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">13</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<SPAN CLASS="MATH"><IMG
 WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img40.png"
 ALT="$L$"></SPAN> is the light flux reaching each pixel, assumed equal across the frame.

<P>
We then calculate a master dark frame for the flat fields <SPAN CLASS="MATH"><IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img41.png"
 ALT="$D_M^F(x,y)$"></SPAN> and subtract it from the
flats, obtaining:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
f'(x,y)=f(x,y)-D_M^F(x,y)=t_{\rm flat}G(x,y)L+{\rm noise}
\end{equation}
 -->
<A NAME="eq:flat1"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:flat1"></A><IMG
 WIDTH="394" HEIGHT="31" BORDER="0"
 SRC="img42.png"
 ALT="\begin{displaymath}
f'(x,y)=f(x,y)-D_M^F(x,y)=t_{\rm flat}G(x,y)L+{\rm noise}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">14</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Again, to reduce the noise contribution, we usually average several flat frames, to arrive at a
<EM>master flat</EM> frame 
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{F}_M(x,y) = \frac{1}{N}\sum_i f'(x,y) = \frac{1}{N}\sum_i f(x,y)-D_M^F(x,y)
\end{equation}
 -->
<A NAME="eq:flat2"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:flat2"></A><IMG
 WIDTH="406" HEIGHT="51" BORDER="0"
 SRC="img43.png"
 ALT="\begin{displaymath}
\widetilde{F}_M(x,y) = \frac{1}{N}\sum_i f'(x,y) = \frac{1}{N}\sum_i f(x,y)-D_M^F(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">15</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
If we knew <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img44.png"
 ALT="$F$"></SPAN>, we could solve the above equation for <SPAN CLASS="MATH"><IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$G(x,y)$"></SPAN>. However, the aboslute value of 
<SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img44.png"
 ALT="$F$"></SPAN> is not known, and in many cases can vary between different flats. So, instead of 
calibrating the absolute value of <SPAN CLASS="MATH"><IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$G(x,y)$"></SPAN>, we only try to remove it's variation across
the frame. We write:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
G(x,y)=\bar{G}g(x,y)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="138" HEIGHT="31" BORDER="0"
 SRC="img45.png"
 ALT="\begin{displaymath}
G(x,y)=\bar{G}g(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">16</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where the average of <SPAN CLASS="MATH"><IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img46.png"
 ALT="$g(x,y)$"></SPAN> across the frame is 1. (<A HREF="#eq:flat2">6.15</A>) becomes:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{F}_M(x,y)=t_{\rm flat}\bar{G}g(x,y)L
\end{equation}
 -->
<A NAME="eq:flat3"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:flat3"></A><IMG
 WIDTH="191" HEIGHT="31" BORDER="0"
 SRC="img47.png"
 ALT="\begin{displaymath}
\widetilde{F}_M(x,y)=t_{\rm flat}\bar{G}g(x,y)L
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">17</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
We take the average of <!-- MATH
 $\widetilde{F}_M(x,y)$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="71" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img48.png"
 ALT="$\widetilde{F}_M(x,y)$"></SPAN> across the frame:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\bar{F}=\sum_x\sum_yt_{\rm flat}\bar{G}g(x,y)L=t_{\rm flat}\bar{G}L\sum_x\sum_yg(x,y)=
  t_{\rm flat}\bar{G}L
\end{equation}
 -->
<A NAME="eq:flat4"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:flat4"></A><IMG
 WIDTH="445" HEIGHT="49" BORDER="0"
 SRC="img49.png"
 ALT="\begin{displaymath}
\bar{F}=\sum_x\sum_yt_{\rm flat}\bar{G}g(x,y)L=t_{\rm flat}\bar{G}L\sum_x\sum_yg(x,y)=
t_{\rm flat}\bar{G}L
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">18</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Dividing equation <A HREF="#eq:flat3">6.17</A> by <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img50.png"
 ALT="$\bar{F}$"></SPAN>, we obtain:<A NAME="tex2html21"
  HREF="footnode.html#foot525"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">4</SPAN></SUP></A>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\frac{\widetilde{F}_M(x,y)}{\bar{F}}=g(x,y)
\end{equation}
 -->
<A NAME="eq:flat5"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:flat5"></A><IMG
 WIDTH="139" HEIGHT="47" BORDER="0"
 SRC="img51.png"
 ALT="\begin{displaymath}
\frac{\widetilde{F}_M(x,y)}{\bar{F}}=g(x,y)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">19</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>

<P>

<H1><A NAME="SECTION00740000000000000000">
Reducing the Data Frames</A>
</H1>
Armed with our master dark and master flat frames, we can proceed to reduce our data frame. 
Starting from:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
s(x,y) = B(x,y) + t_{data} D(x,y) + t_{\rm data} \bar{G} g(x, y) I(x, y)) + {\rm noise}
\end{equation}
 -->
<A NAME="eq:ccd2"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:ccd2"></A><IMG
 WIDTH="468" HEIGHT="31" BORDER="0"
 SRC="img52.png"
 ALT="\begin{displaymath}
s(x,y) = B(x,y) + t_{data} D(x,y) + t_{\rm data} \bar{G} g(x, y) I(x, y)) + {\rm noise}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">20</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
we subtract the master dark frame (<A HREF="#eq:mdark">6.9</A>) and divide by the normalised 
master flat (<A HREF="#eq:flat5">6.19</A>) and obtain:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\frac{\bar{F}}{\widetilde{F}_M(x,y)}[s(x,y) - \widetilde{D}_M(x,y)] = 
t_{\rm data} \bar{G} I(x,y) + {\rm noise}
\end{equation}
 -->
<A NAME="eq:ccd3"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:ccd3"></A><IMG
 WIDTH="396" HEIGHT="51" BORDER="0"
 SRC="img53.png"
 ALT="\begin{displaymath}
\frac{\bar{F}}{\widetilde{F}_M(x,y)}[s(x,y) - \widetilde{D}_M(x,y)] =
t_{\rm data} \bar{G} I(x,y) + {\rm noise}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">21</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{I}(x,y) = \frac{1}{\bar{G}\cdot t_{\rm data}}\frac{\bar{F}}{\widetilde{F}_M(x,y)}
[s(x,y) - \widetilde{D}_M(x,y)]
\end{equation}
 -->
<A NAME="eq:ccd4"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:ccd4"></A><IMG
 WIDTH="354" HEIGHT="51" BORDER="0"
 SRC="img54.png"
 ALT="\begin{displaymath}
\widetilde{I}(x,y) = \frac{1}{\bar{G}\cdot t_{\rm data}}\frac{\bar{F}}{\widetilde{F}_M(x,y)}
[s(x,y) - \widetilde{D}_M(x,y)]
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">22</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Or, in terms of the master bias and bias-subtracted master dark frames:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\widetilde{I}(x,y) = \frac{1}{\bar{G}\cdot t_{\rm data}}\frac{\bar{F}}{\widetilde{F}_M(x,y)}
[s(x,y) - \frac{t_{\rm data}}{t_{\rm dark}}\widetilde{D}'(x,y) - \widetilde{B}(x,y)]
\end{equation}
 -->
<A NAME="eq:ccd5"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:ccd5"></A><IMG
 WIDTH="457" HEIGHT="51" BORDER="0"
 SRC="img55.png"
 ALT="\begin{displaymath}
\widetilde{I}(x,y) = \frac{1}{\bar{G}\cdot t_{\rm data}}\fra...
... data}}{t_{\rm dark}}\widetilde{D}'(x,y) - \widetilde{B}(x,y)]
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">23</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>

<P>
We have obtained an estimate of the incident light flux up to a constant 
(<!-- MATH
 $\bar{G}t_{\rm data}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img56.png"
 ALT="$\bar{G}t_{\rm data}$"></SPAN>) which represents the average sensitivity of the camera
multiplied by the integration time, which is the best we can do without a reference 
source calibrated in absolute units. 

<P>

<H1><A NAME="SECTION00750000000000000000"></A><A NAME="sec:combining"></A>
<BR>
Frame Combining Methods
</H1>

<P>
Earlier in this chapter we mentioned that it is useful in many cases to ``average'' several frames
in order to reduce the noise. It turns out that while arithmetic averaging does indeed reduce
the resulting noise, it doesn't go very far in removing the effect of
devinat values that are far from the mean. When combining CCD frames, the most common causes of deviant values are
cosmic ray hits on all frames and unwanted star images on sky flats. 

<P>
When combining <SPAN CLASS="MATH"><IMG
 WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img31.png"
 ALT="$N$"></SPAN> CCD frames, we start with <SPAN CLASS="MATH"><IMG
 WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img31.png"
 ALT="$N$"></SPAN> values for each pixel and want to arrive
at a combined pixel value that uses as much as possible of the available information (so that 
we obtain the maximum noise reduction), while rejecting any values that are affected by artifacts.
<SMALL>GCX </SMALL>implements four combining methods, described below.

<P>

<H4><A NAME="SECTION00750010000000000000">
Average</A>
</H4>
Average is the ``baseline'' method of frame combining. The values are added together and the 
result divided by the number of values (arithmetic mean). For Gaussian-distributed data, averaging
produces the values with the least variance (so it has the maximal <EM>statistical efficiency</EM>).
Averaging is independent of the individual frame scaling, and
computationally efficient.

<P>
The deviant value rejection of averaging is only modest; deviant values are reduced by a factor
of <SPAN CLASS="MATH"><IMG
 WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img31.png"
 ALT="$N$"></SPAN>. For this reason, it is recommended that averaging only be used when we know that deviant 
values are not present (for example, when combining frames for which the outlier rejection has
already been done).

<P>

<H4><A NAME="SECTION00750020000000000000">
Median</A>
</H4>
A far more robust method of obtaining a combined value is the median (selecting the values which 
has an equal number of values greater and smaller than itself). The median is easily calculated 
and very little influenced by deviant values. It requires all frames 
to have the same intensity scale.

<P>
On the down side, the median's statistical efficiency for Gaussian distributed values is only 
0.65 of average's. Also, when combining integer values, the median does nothing to smooth out the
quantisation effects; the median of any number of integer values is also an integer.

<P>

<H4><A NAME="SECTION00750030000000000000">
Mean-Median</A>
</H4>

<P>
Mean-median is a variant of the median method intended to improve the statistical efficiency
of the median, and get around the quantisation problem. In the mean-median method, we compute the
standard deviation of the pixel values around the median, and discard all the values than are 
farther away than a specified number of standard deviations (usually 1.5 or 2). 
The remaining values are averaged together.

<P>
Mean-median is fast, and works well for large sets. With small sets, the fact that the deviant 
pixels increase the calculated standard deviation limits it's efficiency. It requires all frames 
to have the same intensity scale.

<P>

<H4><A NAME="SECTION00750040000000000000">
Kappa-Sigma Clipping</A>
</H4>

<P>
<SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img57.png"
 ALT="$\kappa\sigma$"></SPAN>-clipping is the most ellaborate combining method provided by <SMALL>GCX </SMALL>. It starts 
by calculating the median and the standard deviation around it. The values with large
deviations relative to the standard deviation are excluded. Then, the mean and standard 
deviation of the remaining values are computed. Again, the values that are away for the 
mean are excluded, and the process is repeated until there is no change in the mean or the
iteration limit is reached. 

<P>
<SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img57.png"
 ALT="$\kappa\sigma$"></SPAN>-clipping is very effective at removing cosmic rays and star images from sky
flats, even with a reduced number of frames. It is also capable of removing all the star 
images from a number of frames of different fields, leaving only the sky background, which makes
it possible to create a flat frame from the regular data frames (see below for <EM>super-flat</EM>).

<P>
<SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img57.png"
 ALT="$\kappa\sigma$"></SPAN>-clipping is the most computationally-intensive method of frame combining. 
It requires all frames to have the same intensity scale.

<P>

<H1><A NAME="SECTION00760000000000000000">
CCD Reduction with gcx</A>
</H1>

<P>
Like most tasks in <SMALL>GCX </SMALL>CCD Reductions can be performed either interactively or using 
the command-line interface. We'll discuss the interactive way first.

<P>

<H2><A NAME="SECTION00761000000000000000">
Loading and Selecting Image Frames</A>
</H2>

<P>
To open up the CCD reduction dialog, select <EM>Processing/CCD Reduction</EM> or press <B>L</B>.
In the <EM>Image Files</EM> tab, click on <EM>Add</EM> and in the file selector select any number of files
and click <EM>Ok</EM>. The selected files will appear in the file list.

<P>
To display any of the frames in the list, click on it (it will become selected) and then click
<EM>Display</EM>. The image will show in the main window. The loaded files can be viewed in sequence 
by clicking <EM>next</EM> or pressing <B>N</B> repeteadly. If for some reson we want to exclude a 
frame from being processed, clicking on <EM>Skip</EM> or pressing <B>S</B> will mark it to be skipped.

<P>
Skipped frames appear in the list with their names enclosed in square brackets. If we want 
to remove the skip mark, clicking <EM>Unskip</EM> while the frames are selected will accomplish 
that. Note that all reduction operations apply to all the frames in the file list that don't have 
the skip mark, regardless of which are selected.

<P>
Whenever a frame is selected, a status line at the bottom of the dialog shows the file's name 
and any operations already performed on it.<A NAME="tex2html22"
  HREF="footnode.html#foot596"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">5</SPAN></SUP></A>
<P>
To revert some frames to their original status, select them and click on <EM>Reload</EM>. Frames 
can be completely removed from the list by selecting them and the clicking on <EM>Remove</EM>.

<P>

<H2><A NAME="SECTION00762000000000000000">
Creating a Master Bias or Master Dark Frame</A>
</H2>

<P>
We will create a master bias frame by stacking several bias frames. If
we preffer to work without bias 
frames, we can use the same procedure to create a master dark frame.
First clear the file list 
(<EM>Select all</EM> then <EM>Remove</EM>) and add the bias frames to be
stacked to the list.

<P>
Then, in the <EM>CCD Reduction</EM> tab, check that all the ``enable'' ticks are off (we don't want 
to apply any operation to the bias frames). Same for the <EM>Aligment</EM> tab.

<P>
In the <EM>Stacking</EM> tab, select the desired parameters for the stacking operation (the method 
and method parameters). A good initial setting is kappa_sigma, 1.5 sigmas and an iteration limit
of 4. Set background matching to Off, and check ``Enable stacking''.

<P>
Finally, in the <EM>Run</EM> tab type in a file name<A NAME="tex2html23"
  HREF="footnode.html#foot606"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">6</SPAN></SUP></A><A NAME="tex2html24"
  HREF="footnode.html#foot1240"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">7</SPAN></SUP></A> (for example ``master_bias'') and click <EM>Run</EM>. The progress of the stacking operation 
can be followed in the text window. After stacking is complete, the resulting frame is
shown in the main window.

<P>

<H2><A NAME="SECTION00763000000000000000">
Creating a Bias-Subtracted Master Dark Frame</A>
</H2>

<P>
Suppose now that we have a number of dark frames taken with the same integration time 
and we want to create a bias subtracted master dark frame from them. We'll use the
master bias frame we just created.

<P>
Like above, clear the frame list and load the dark frames. Then, in the <EM>CCD Reduction</EM> tab
enter the name of the master bias frame we just created (or click on the ``...'' button and select
it in the file selector). The ``enable'' mark for bias subtraction should be on.

<P>
Then make sure that stacking is still enabled, enter an output file name and click on <EM>Run</EM>.
The program will subtract the master bias frame from each of the dark frames, and then combine 
the results. As before, the result will be shown in the main window.

<P>
Sometimes we need to scale the bias-subtracted dark frame, so it can be used to reduce data 
frames taken with a different integration time. We can also do that now. Suppose our
dark frames used 60 seconds on integration time, and we want to reduce 30-seconds data frames.
We will need out bias-subtracted master dark to be scaled by a factor of 0.5. 
In the <EM>CCD Reduction</EM> tab check the ``multiply enable'' box and enter 0.5 in the
multiply entry. Then click on <EM>Run</EM> again,<A NAME="tex2html25"
  HREF="footnode.html#foot1241"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">8</SPAN></SUP></A>of course not before changing the output 
file name into something meaningful, like ``bmdark-30'' or something similar.

<P>

<H2><A NAME="SECTION00764000000000000000">
Creating a Master Flat Frame</A>
</H2>

<P>
To create a master frame, we need either a master dark frame of suitable integration time for
the flats, or a master bias and suitably scaled bias-subtracted master dark frame.

<P>
Clear the file list and add the flat frames to it. In the <EM>CCD Reduction</EM> tab, set the
bias and dark frame file names,<A NAME="tex2html26"
  HREF="footnode.html#foot618"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">9</SPAN></SUP></A> and make sure their ``enable'' boxes are checked. 
Go to the <EM>Stacking</EM> tab and enable stacking, set your algorithm and parameters. Except 
if you are very sure that the flat frames have the same intensity (such as when using a
known stable light box), enable multiplicative background matching.
Finally, select an output file name, and run the flat generation.

<P>

<H4><A NAME="SECTION00764010000000000000">
Making a Superflat</A>
</H4> If we have enough data frames with a significant sky background and 
different fields, they can be combined and the sky background used as a flat. We proceed 
the same as for normal flats, except that we select data frames with a good background. 
It is recommended that frames with low sky level are excluded from the superflat set. 

<P>
We probably want to multiply the frames by a constant to make sure the relatively low level 
flat resulting is not affected by quatisation when the frame is saved in an integer format. 

<P>

<H2><A NAME="SECTION00765000000000000000">
Reducing the Data Frames</A>
</H2>

<P>
Reducing the data frames should is easy now; load them into the file list, specify the 
bias, dark and flat to be used, select an output directory name and click on <EM>Run</EM>.

<P>
<SMALL>GCX </SMALL>will never overwrite the original frames, so we will need to create a second set of 
reduced files. However, unless we have some use for the reduces frames, we can avoid saving
them and just run the reduction every time we use the frames. For instance, we can align 
and stack the frame, or run aperture photometry on the reduced frames by just specifying
the bias/dark/flats together with the required operation. 

<P>

<H2><A NAME="SECTION00766000000000000000">
Aligning and Stacking Frames</A>
</H2>

<P>
When combining data frames, it is often required that they are aligned before their values are 
combined. <SMALL>GCX </SMALL>contains an automatic alignment algorithm that works well with frames taken
with the same setup (for instance multiple exposures). In the current version, the alignment 
routine only performs image translations (it does not rotate or scale
the images).<A NAME="tex2html27"
  HREF="footnode.html#foot624"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">10</SPAN></SUP></A>
<P>
To align frames, they need to be in the file list. Then, in the <EM>Alignment</EM> tab select an
alignment frame file name. This is the frame used as a reference--all others will be shifted 
to match this one. Clicking the <EM>Show Alignment Stars</EM> button will detect suitable stars 
from the alignment frame and display them in the main
window.<A NAME="tex2html28"
  HREF="footnode.html#foot627"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">11</SPAN></SUP></A>
<P>
Clicking on <EM>Run</EM> will perform the actual alignment. Stars are detected from each frame, and
their position matched to the reference frame.<A NAME="tex2html29"
  HREF="footnode.html#foot1242"><SUP><SPAN CLASS="arabic">6</SPAN>.<SPAN CLASS="arabic">12</SPAN></SUP></A> The frame will then be translated the appropiate
amount. It is also possible to apply a gaussian smoothing filter here. Usually small FWHMs work
best (0.1-0.5 pixels). 

<P>
After the frames are aligned, we can use the <EM>Display</EM> and <EM>Next</EM> buttons in the file list
tab to browse the aligned files. Stars should show well centered inside the alignment star 
markings. This is also a good time to exclude any bad frames (such as frames with 
tracking problems).

<P>
When satisfied with the aligned frames, we can stack them and produce a final image. Just enable
the stack check box and click on <EM>Run</EM> again. 

<P>

<H2><A NAME="SECTION00767000000000000000">
Running CCD Reductions from the Command Line</A>
</H2>

<P>
All CCD reduction operations are also accessible from the command line, which makes it easy
to integrate <SMALL>GCX </SMALL>with other applications. The command-line options are described by calling
<BLOCKQUOTE>
<TT>gcx -help</TT>
</BLOCKQUOTE>
Here we'll provide some examples of use. Let's assume that we have a number of 20-second 
data frames, some 20-second dark frames, some 1-sec sky flats, and some bias frames.
We want to reduce the data frames and align and stack them together.

<P>
First, we check that the stacking method and parameters are properly set in the
<EM>CCD Reduction Options</EM> page or the <TT>&nbsp;/.gcxrc</TT> file.

<P>
We combine the bias frames to create a master bias file (<TT>m-bias.fits</TT>):
<BLOCKQUOTE>
<TT>gcx -s -o m-bias bias*</TT>
</BLOCKQUOTE>
Then we combine the dark frames to create the master dark frame for
the data frames (<TT>m-dark-20.fits</TT>):
<BLOCKQUOTE>
<TT>gcx -s -o m-dark-20 dark*</TT>
</BLOCKQUOTE> 
We create a bias-subtracted master dark frame and scale it to be used
on the flats (<TT>dark-1s.fits</TT>):
<BLOCKQUOTE>
<TT>gcx -b m-bias -s -M 0.05 -o dark-1s dark*</TT>
</BLOCKQUOTE> 
And then the master flat frame (<TT>m-flat.fits</TT>):
<BLOCKQUOTE>
<TT>gcx -b m-bias -d dark-1s -F -o m-flat flat*</TT>
</BLOCKQUOTE> 
Assuming that <TT>red</TT> is a directory, we reduce all frames and save them there:
<BLOCKQUOTE>
<TT>gcx -d m-dark-20 -f m-flat -o red data*</TT>
</BLOCKQUOTE> 
We align and stack the data files (with a 0.1 pixels FWHM gaussian blur):
<BLOCKQUOTE>
<TT>gcx -a red/data001 -s -G 0.1 -o stack1 red/data*</TT>
</BLOCKQUOTE>
As an alternative, we align and stack directly from the original files, this time without any
blur:
<BLOCKQUOTE>
<TT>gcx -d m-dark-20 -f m-flat -a data001 -s -o stack1 data*</TT>
</BLOCKQUOTE>
The order of the command-line options is not important. CCD reduction operations
are always performed in the order: bias, dark, flat, multiply by a constant, add a constant,
align and gaussian blur, stack. If no extension is provided on the file names, <SMALL>GCX </SMALL>will
append <TT>.fits</TT> or <TT>.fits.gz</TT> automatically on saved image files, and will look
for one of <TT>.fits</TT> or <TT>.fit</TT> with or without a <TT>.gz</TT> suffix.

<P>

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