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<H1><A NAME="SECTION00620000000000000000">&#160;</A>
 <A NAME="ccd:ccd-output">&#160;</A>
<BR>
Nature of CCD Output
</H1>
 The nominal output <I>X</I><SUB><I>ij</I></SUB> of a CCD-element to a quantum of light <I>I</I><SUB><I>ij</I></SUB> can be given as 
 <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
X_{ij} =  M_{ij} \times I_{ij} + A_{ij} +F{_{ij}}(I_{ij})
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-1">&#160;</A><IMG
 WIDTH="303" HEIGHT="41"
 SRC="img132.gif"
 ALT="\begin{displaymath}X_{ij} = M_{ij} \times I_{ij} + A_{ij} +F{_{ij}}(I_{ij})
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.1)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where the additive contribution <I>A</I><SUB><I>ij</I></SUB> is caused by the dark current,
 by pre-flashing, by charge that may have skimmed from columns having a 
 deferred charge (skim) and by bias added to the output electronically to 
 avoid problems with digitizing values near zero. Quantum and transfer
 efficiency of the optical system enter into the multiplicative term <I>M</I>. 
 The term <I>I</I> consist of various components:  object, sky and the photons
 emitted from the telescope structure. It is known that the response
 of a CCD can show non-linear effects that can be as large as 5-10%. 
 These effects are represented by the term <I>F</I><SUB><I>ij</I></SUB>.

<P>
In the following we ignore the pre-flash and skim term, and hence only take
 the bias and dark frames into account. The objective in reducing CCD frames 
 is to  determine the relative intensity <I>I</I><SUB><I>ij</I></SUB> of a science data frame. In 
 order to do this, at least two more frames are required in addition to the 
 science frame, namely:  
 <UL>
<LI>dark frames to describe the term <I>A</I><SUB><I>ij</I></SUB>, and
<LI>flat frames to determine the term <I>M</I><SUB><I>ij</I></SUB>.
 </UL>
<P>
The dark current <I>dark</I> is measured in absence of any external input signal.
 By considering a number of dark exposures a medium &lt;<I>dark</I>&gt; can be 
 determined: 
 <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\rm
\left< dark(i,j) \right> = dark(i,j) +  bias
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-2">&#160;</A><IMG
 WIDTH="273" HEIGHT="40"
 SRC="img134.gif"
 ALT="\begin{displaymath}\rm
\left< dark(i,j) \right> = dark(i,j) + bias
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.2)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
<P>
The method to correct the frame for multiplicative spatial systematics is 
 know as flat fielding. Flat fields are made by illuminating the CCD with 
 a uniformly emitting source. The flat field then describes the sensitivity 
 over the CCD which is not uniform. A mean flat field frame with a higher
 S/N ratio can be obtained buy combining a number of flat exposures. The 
 mean flat field and the science frame can be described by:
 <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\rm
\overline{flat(i,j)} = M(i,j) \times icons + dark(i,j) + bias
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-3">&#160;</A><IMG
 WIDTH="406" HEIGHT="40"
 SRC="img135.gif"
 ALT="\begin{displaymath}\rm
\overline{flat(i,j)} = M(i,j) \times icons + dark(i,j) + bias
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.3)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P> <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
science(i,j)=  M(i,j) \times intens(i,j) + dark(i,j)+ bias
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-4">&#160;</A><IMG
 WIDTH="525" HEIGHT="40"
 SRC="img136.gif"
 ALT="\begin{displaymath}science(i,j)= M(i,j) \times intens(i,j) + dark(i,j)+ bias
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.4)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where 
<!-- MATH: $intens(i,j)$ -->
<I>intens</I>(<I>i</I>,<I>j</I>) represents the intensity distribition on the sky,
 and <I>icons</I> a brightness distribution from a uniform source. If set 
 to the average signal of the dark corrected flat frame or a subimage thereof: 
 <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\rm
icons = \left< flat - dark\right>
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-6">&#160;</A><IMG
 WIDTH="194" HEIGHT="40"
 SRC="img137.gif"
 ALT="\begin{displaymath}\rm
icons = \left< flat - dark\right>
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.5)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
then the reduced intensity frame <I>intens</I> will have similar data values 
 as the original science frame <I>science</I>. 

<P>
Combining Eqs.(<A HREF="node33.html#ccd:ccd-2">3.2</A>), (<A HREF="node33.html#ccd:ccd-3">3.3</A>) and (<A HREF="node33.html#ccd:ccd-4">3.4</A>) we isolate: 
 <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\rm
intens(i,j) = {{science(i,j) - \left< dark(i,j) \right> }
                \over 
                {\overline{flat(i,j)} - \left< dark(i,j)_{F} \right> }} 
                \times icons 
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-5">&#160;</A><IMG
 WIDTH="433" HEIGHT="64"
 SRC="img138.gif"
 ALT="\begin{displaymath}\rm
intens(i,j) = {{science(i,j) - \left< dark(i,j) \right> ...
...e{flat(i,j)} - \left< dark(i,j)_{F} \right> }}
\times icons
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.6)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
<P>
Here <I>icons</I> can be any number, and term 
<!-- MATH: $\left< dark(i,j) \right>$ -->
<IMG
 WIDTH="112" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img139.gif"
 ALT="$ \left< dark(i,j) \right> $">
now 
 denotes a dark frame obtained by <I>e.g.</I> applying a local median over a 
 stack of single dark frames. The subscript in 
<!-- MATH: $\left< dark(i,j)_{F}\right>$ -->
<IMG
 WIDTH="127" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img140.gif"
 ALT="$ \left< dark(i,j)_{F}\right> $">
 denotes that this dark exposures may necessarily be the same frame 
 used to subtract the additive spatial systematics from the raw science frame.

<P>
The mean absolute error of 
<!-- MATH: $intens(i,j)$ -->
<I>intens</I>(<I>i</I>,<I>j</I>) yields with <I>icons</I> = 1 (only the 
 first letter is used for abbreviations):
  <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
(\Delta I)^2 = \left({\partial I \over \partial S}\right)^2 (\Delta S)^2 +
\left({\partial I \over \partial D}\right)^2 (\Delta D)^2 +
                 \left({\partial I \over \partial F}\right)^2 (\Delta F)^2  
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-7">&#160;</A><IMG
 WIDTH="550" HEIGHT="62"
 SRC="img141.gif"
 ALT="\begin{displaymath}(\Delta I)^2 = \left({\partial I \over \partial S}\right)^2 (...
...+
\left({\partial I \over \partial F}\right)^2 (\Delta F)^2
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.7)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
Computing the partial derivatives we get
  <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
(\Delta I)^2 = {(F - D)^2(\Delta S)^2 +
(S - F)^2(\Delta D)^2 +
                   (S - D)^2(\Delta F)^2 \over (F - D)^4}  
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-8">&#160;</A><IMG
 WIDTH="584" HEIGHT="63"
 SRC="img142.gif"
 ALT="\begin{displaymath}(\Delta I)^2 = {(F - D)^2(\Delta S)^2 +
(S - F)^2(\Delta D)^2 +
(S - D)^2(\Delta F)^2 \over (F - D)^4}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.8)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
<P>
A small error <IMG
 WIDTH="35" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="img143.gif"
 ALT="$\Delta I$">
is obtained if  <IMG
 WIDTH="39" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img144.gif"
 ALT="$\Delta S$">,
<IMG
 WIDTH="42" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="img145.gif"
 ALT="$\Delta D$">
and 
 <IMG
 WIDTH="42" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="img146.gif"
 ALT="$\Delta F$">
are kept small. This is achieved by averaging Dark, Flat and 
 Science frames. <IMG
 WIDTH="35" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="img147.gif"
 ALT="$\Delta I$">
is further reduced if <I>S</I>=<I>F</I>, then
 Equation&nbsp;(<A HREF="node33.html#ccd:ccd-8">3.8</A>) simplifies to 
 <BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
(\Delta I)^2  = {(\Delta S)^2 + (\Delta F)^2 \over (F - D)^2}
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ccd:ccd-9">&#160;</A><IMG
 WIDTH="234" HEIGHT="63"
 SRC="img148.gif"
 ALT="\begin{displaymath}(\Delta I)^2 = {(\Delta S)^2 + (\Delta F)^2 \over (F - D)^2}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.9)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
<P>
This equation holds only at levels near the sky-background and is relevant
 for detection of low-brightness emission. In practice however it is difficult
 to get a similar exposure level for the <I>flatfrm</I> and <I>science</I> since the
 flats are usually measured inside the dome. From this point of view it is
 desirable to measure the empty sky (adjacent to the object) just before or
 after the object observations. In the case of infrared observations this is
 certainly advisable because of variations of the sky on short time
scales.

<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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