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

<UL CLASS="ChildLinks">
<LI><A NAME="tex2html649"
  HREF="node9.html#SECTION00910000000000000000">Color Transformation Coefficients</A>
<UL>
<LI><A NAME="tex2html650"
  HREF="node9.html#SECTION00911000000000000000">Transforming the Stars</A>
</UL>
<BR>
<LI><A NAME="tex2html651"
  HREF="node9.html#SECTION00920000000000000000">All-Sky Reduction</A>
<UL>
<LI><A NAME="tex2html652"
  HREF="node9.html#SECTION00921000000000000000">Extinction Coefficient Fitting</A>
<LI><A NAME="tex2html653"
  HREF="node9.html#SECTION00922000000000000000">Calculating Zero Points</A>
</UL>
<BR>
<LI><A NAME="tex2html654"
  HREF="node9.html#SECTION00930000000000000000">Running Multi-Frame Reduction</A>
<UL>
<LI><A NAME="tex2html655"
  HREF="node9.html#SECTION00931000000000000000">Specifying Reduction Bands</A>
<LI><A NAME="tex2html656"
  HREF="node9.html#SECTION00932000000000000000">Loading Report Files</A>
<UL>
<LI><A NAME="tex2html657"
  HREF="node9.html#SECTION00932010000000000000">Frames</A>
<LI><A NAME="tex2html658"
  HREF="node9.html#SECTION00932020000000000000">Stars</A>
<LI><A NAME="tex2html659"
  HREF="node9.html#SECTION00932030000000000000">Bands</A>
</UL>
<LI><A NAME="tex2html660"
  HREF="node9.html#SECTION00933000000000000000">Fitting Individual Zero Points</A>
<LI><A NAME="tex2html661"
  HREF="node9.html#SECTION00934000000000000000">Plots</A>
<LI><A NAME="tex2html662"
  HREF="node9.html#SECTION00935000000000000000">Fitting Color Transformation Coefficients</A>
<LI><A NAME="tex2html663"
  HREF="node9.html#SECTION00936000000000000000">All-Sky Reduction</A>
</UL>
<BR>
<LI><A NAME="tex2html664"
  HREF="node9.html#SECTION00940000000000000000">Reporting</A>
</UL>
<!--End of Table of Child-Links-->
<HR>

<H1><A NAME="SECTION00900000000000000000"></A><A NAME="ch:multiframe"></A>
<BR>
Multi-Frame and All-Sky Reduction
</H1>

<P>
If we want to determine star colors, calculate transformation
coefficients to transform data to a standard system or obtain 
magnitudes of stars for which we don't have standards in the same
field, we must reduce multiple observation frames together.

<P>
People fortunate enough to observe in photometric conditions can use
a number of packages to reduce their data. For low altitude dwellers,
the selection is not that large. For them, <SMALL>GCX </SMALL>implements
multiple-frame 
reduction routines that are designed to
work in less than perfect conditions. 

<P>
Input data to the multi-frame reduction consists of observation
reports as produced by the aperture photometry routine. For color
coefficient fitting and transformation to a standard system, we need
frames of the target objects taken in enough bands. For
all-sky reductions, the observation reports need to have accurate time
and airmass information (which implies that the original frames need
to have enough information for the airmass determination).

<P>

<H1><A NAME="SECTION00910000000000000000">
Color Transformation Coefficients</A>
</H1>

<P>
To keep notation simple, let's assume that we reduce data taken in B
and V. We'll use <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img10.png"
 ALT="$B$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img68.png"
 ALT="$V$"></SPAN> for the standard magnitudes, and <SPAN CLASS="MATH"><IMG
 WIDTH="12" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img17.png"
 ALT="$b$"></SPAN> and
<SPAN CLASS="MATH"><IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img69.png"
 ALT="$v$"></SPAN> for the instrumental magnitudes. The expressions for the standard
magnitudes are:<A NAME="tex2html40"
  HREF="footnode.html#foot794"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">1</SPAN></SUP></A>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
B_j = b_j + Z_k + k_B(B_j-V_j)
\end{equation}
 -->
<A NAME="eq:transform"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:transform"></A><IMG
 WIDTH="212" HEIGHT="32" BORDER="0"
 SRC="img70.png"
 ALT="\begin{displaymath}
B_j = b_j + Z_k + k_B(B_j-V_j)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">1</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
V_j = v_j + Z_k + k_V(B_j-V_j)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="208" HEIGHT="32" BORDER="0"
 SRC="img71.png"
 ALT="\begin{displaymath}
V_j = v_j + Z_k + k_V(B_j-V_j)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">2</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The <SPAN CLASS="MATH"><IMG
 WIDTH="13" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img72.png"
 ALT="$j$"></SPAN> subscripts go over all individual star observations in a given
band, while the <SPAN CLASS="MATH"><IMG
 WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img73.png"
 ALT="$k$"></SPAN> subscripts iterate over the observation frames.
For standard stars, the <SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img74.png"
 ALT="$B_j$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img75.png"
 ALT="$V_j$"></SPAN> are known while <SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img76.png"
 ALT="$b_j$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="20" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img77.png"
 ALT="$v_j$"></SPAN> are
measured from the frames themselves. We have to fit the zeropoints
<SPAN CLASS="MATH"><IMG
 WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img78.png"
 ALT="$Z_k$"></SPAN> and the transformation coefficients <SPAN CLASS="MATH"><IMG
 WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img73.png"
 ALT="$k$"></SPAN>. Because we have chosen
to express the transformation coefficients in function of the standard
magnitudes, we can proceed with one band at a time. We'll assume it is 
V. The steps are:

<OL>
<LI>Set an starting value of the transformation coefficient. We can
  start with 0 without problems, as the coefficients are generally
  small numbers.
</LI>
<LI>For each V frame, fit the zeropoint using the robust algorithm
  in Appendix&nbsp;<A HREF="node11.html#ap:robust">B</A>, with the difference that the current color
  transformation is applied when calculating the residuals, so
  Equation&nbsp;<A HREF="node11.html#eq:residuals">B.5</A> becomes:
  <BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\rho_j=y_j - \widetilde{Z_k} - k_V (B_j - V_j)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="209" HEIGHT="32" BORDER="0"
 SRC="img79.png"
 ALT="\begin{displaymath}
\rho_j=y_j - \widetilde{Z_k} - k_V (B_j - V_j)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Note that in this equation, the <SPAN CLASS="MATH"><IMG
 WIDTH="13" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img72.png"
 ALT="$j$"></SPAN> subscripts iterate over the
  stars in frame <SPAN CLASS="MATH"><IMG
 WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img73.png"
 ALT="$k$"></SPAN>.
</LI>
<LI>Now, for all the stars in all the V frames, estimate the
  ``tilt'' of the residuals, and adjust the transformation coefficient
  and zeropoints accordingly. We use the weights from the individual
  frames' fits when estimating the tilt:
  <BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\theta = \frac{\sum_j\rho_j(B_j-V_j) W'_j}{\sum_j(B_j-V_j)^2 W'_j}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="172" HEIGHT="53" BORDER="0"
 SRC="img80.png"
 ALT="\begin{displaymath}
\theta = \frac{\sum_j\rho_j(B_j-V_j) W'_j}{\sum_j(B_j-V_j)^2 W'_j}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
  <BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
k_V = k_V + \theta
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="93" HEIGHT="29" BORDER="0"
 SRC="img81.png"
 ALT="\begin{displaymath}
k_V = k_V + \theta
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
  <BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
Z_k = Z_k + \theta\frac{\sum_j(B_j-V_j) W'_j}{\sum_jW'_j}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="214" HEIGHT="53" BORDER="0"
 SRC="img82.png"
 ALT="\begin{displaymath}
Z_k = Z_k + \theta\frac{\sum_j(B_j-V_j) W'_j}{\sum_jW'_j}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">6</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
</LI>
<LI>Iterate the last two steps until the solution converges (the
  transformation coefficient doesn't change significantly).
</LI>
</OL>
We have now obtained the transformation coefficient for the V
magnitudes, and also adjusted all the frame zeropoints so that their
dependence on the color of the standard stars in each frame is
eliminated.<A NAME="tex2html41"
  HREF="footnode.html#foot817"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">2</SPAN></SUP></A>The above is repeated for each band we want to reduce. Note that we
have obtained the transformation coefficients without assuming any
relation between the zero points of various frames--just differential 
photometry.

<P>
We can choose any color index for a given band. For instance, there is
nothing stopping us from calculating a <SPAN CLASS="MATH"><IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img83.png"
 ALT="$(B-V)$"></SPAN> transformation
coefficient for <SPAN CLASS="MATH"><IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$I$"></SPAN> or <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img84.png"
 ALT="$R$"></SPAN> magnitudes. In fact, if we have more
standards data in B and V, it may prove better to do so. In general,
Equation&nbsp;(<A HREF="#eq:transform">8.1</A>) can be written for any band <SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img32.png"
 ALT="$M$"></SPAN> as:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
M_j = m_j + Z_k + k_M(C_1^M - C_2^M)\\
\end{equation}
 -->
<A NAME="eq:transform2"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:transform2"></A><IMG
 WIDTH="247" HEIGHT="32" BORDER="0"
 SRC="img85.png"
 ALT="\begin{displaymath}
M_j = m_j + Z_k + k_M(C_1^M - C_2^M)\\
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">7</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <SPAN CLASS="MATH"><IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img86.png"
 ALT="$C_1^M$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img87.png"
 ALT="$C_2^M$"></SPAN> are any bands for which we have standards
data. We can fit the transformation coefficient <SPAN CLASS="MATH"><IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img88.png"
 ALT="$k_M$"></SPAN> using only the
<SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img32.png"
 ALT="$M$"></SPAN> observations. However, when we want to transform the stars, we
will need observations in <SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img32.png"
 ALT="$M$"></SPAN>, <SPAN CLASS="MATH"><IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img86.png"
 ALT="$C_1^M$"></SPAN>, <SPAN CLASS="MATH"><IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img87.png"
 ALT="$C_2^M$"></SPAN> and all the bands that
these depend on. 

<P>

<H2><A NAME="SECTION00911000000000000000">
Transforming the Stars</A>
</H2>
To calculate the transformed standard magnitudes of our target stars,
all we have to do is to write Equation&nbsp;(<A HREF="#eq:transform2">8.7</A>) for each
band, and solve the resulting system of linear equations for the
standard magnitudes. The system is very well behaved (it's matrix is
close to unity) so <SMALL>GCX </SMALL>uses the simple Gauss-Jordan elimination method to
solve it.

<P>

<H1><A NAME="SECTION00920000000000000000">
All-Sky Reduction</A>
</H1>

<P>
When the field of our intended target doesn't contain any suitable
standard stars, we have to determine their magnitudes by comparing to
stars in a different field. To do this, we need to determine a
relation between the zeropoints of different frames.

<P>
Under photometric conditions, we can consider that the <EM>atmospheric
extinction</EM><A NAME="tex2html42"
  HREF="footnode.html#foot826"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">3</SPAN></SUP></A>depends only on the thickness of the atmosphere along the light
path. The ratio between the thickness of the atmosphere in the
direction of field and it's thickness towards zenith is called the
<EM>airmass</EM> of the field. The airmass depends on the zenital angle
<EM>z</EM> of the field, and is close to <SPAN CLASS="MATH"><IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img89.png"
 ALT="$\sec(z)$"></SPAN> when far from the
horison. The formula used by <SMALL>GCX </SMALL>is the following:<A NAME="tex2html43"
  HREF="footnode.html#foot1248"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">4</SPAN></SUP></A>
<BR>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{eqnarray}
s = \sec(z) - 1;\\
A=1 + s[0.9981833 - s(0.002875+0.0008083 s)]
\end{eqnarray}
 -->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
 WIDTH="115" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img90.png"
 ALT="$\displaystyle s = \sec(z) - 1;$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">8</SPAN>)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
 WIDTH="376" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img91.png"
 ALT="$\displaystyle A=1 + s[0.9981833 - s(0.002875+0.0008083 s)]$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">9</SPAN>)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
The zenital angle of a frame can be determined given it's equatorial
coordinates, the geographical coordinates of the observing site, and time.

<P>
If the extinction is unform in all directions, we can define an <EM>  extinction coefficient</EM> <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img92.png"
 ALT="$E$"></SPAN>, so that for any frame:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
Z(A) = Z_0 - EA
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="130" HEIGHT="31" BORDER="0"
 SRC="img93.png"
 ALT="\begin{displaymath}
Z(A) = Z_0 - EA
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">10</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <SPAN CLASS="MATH"><IMG
 WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img94.png"
 ALT="$Z_0$"></SPAN> is the zeropoint outside the atmosphere, and A is the
frame's airmass. If we have two frames with airmasses <SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img95.png"
 ALT="$A_1$"></SPAN>
and <SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img96.png"
 ALT="$A_2$"></SPAN>, their zeropoints can be related by:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
Z_2 = Z_1 - E(A_2-A_1)
\end{equation}
 -->
<A NAME="eq:ext2"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:ext2"></A><IMG
 WIDTH="170" HEIGHT="31" BORDER="0"
 SRC="img97.png"
 ALT="\begin{displaymath}
Z_2 = Z_1 - E(A_2-A_1)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">11</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Under photometric conditions, it is customary to determine the
extinction coefficient by observing the same field at different
airmasses and then fitting E from (<A HREF="#eq:ext2">8.11</A>). This is only
possible when <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img92.png"
 ALT="$E$"></SPAN> doesn't change (or changes in a smooth, linear
fashion) over a period of the order of hours.

<P>
Because the <SMALL>GCX </SMALL>all-sky routine is targeted at less-than-perfect 
conditions, we will choose another strategy in determining the
extinction coefficient. We use several standard fields located 
relatively near our target fields. Then we we try to ``chop'' the
extinction coefficient as much as possible by alternating between the 
standard and target fields.<A NAME="tex2html44"
  HREF="footnode.html#foot840"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">5</SPAN></SUP></A>
<P>
We end up with a series of observations from different fields, all in
the same general airmass range. By examining the standard fields'
zeropoints variation with time and airmass, we can determine if there
were any ``windows'' during which the extinction was stable.  

<P>
Once a stable window was found, we can fit the extinction coefficient
from the observations in that window. It is unlikely that the
observations will span a wide range of airmasses, which will make the
fitted value of the extinction coefficient somewhat imprecise. But
this is offset by the fact that the airmass of the target frames is the
same range, so the contribution of the extinction term is not very
large. As long as the airmass of the standard fields brackets those of
the target fields, we are <EM>interpolating</EM> rather than
extrapolating the extinction.<A NAME="tex2html45"
  HREF="footnode.html#foot842"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">6</SPAN></SUP></A>
<P>

<H2><A NAME="SECTION00921000000000000000">
Extinction Coefficient Fitting</A>
</H2>

<P>
Before attempting to fit the extinction coefficient, the zeropoints
and color transformation coefficient of all frames must be fitted. It
is highly recommended to examine plots of the resulting zeropoints
versus time and airmass to see if it's worth trying to do any all-sky
reduction at all (more on this below).

<P>
With these precautions, the program will proceed to fit the extinction
coefficients using a variant of the algorithm described in
Appendix&nbsp;<A HREF="node11.html#ap:robust">B</A>,<A NAME="tex2html46"
  HREF="footnode.html#foot845"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">7</SPAN></SUP></A>with the initial weights assigned based on the calculated errors of 
the zeropoints. The fitted model is:<A NAME="tex2html47"
  HREF="footnode.html#foot846"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">8</SPAN></SUP></A>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
Z = \bar{Z} + E (A - \bar{A})
\end{equation}
 -->
<A NAME="eq:zmodel"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:zmodel"></A><IMG
 WIDTH="143" HEIGHT="31" BORDER="0"
 SRC="img98.png"
 ALT="\begin{displaymath}
Z = \bar{Z} + E (A - \bar{A})
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">12</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <SPAN CLASS="MATH"><IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img99.png"
 ALT="$\bar{A}$"></SPAN> is the mean airmass of the standard frames, while
<SPAN CLASS="MATH"><IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img100.png"
 ALT="$\bar{Z}$"></SPAN> is the zeropoint of a mean airmass frame.

<P>
A different extinction coefficient is fitted for each
band. Frames that are outliers of the fit (their standard error
exceeds the threshold set in <EM>Multi-Frame Photometry
  Options/Zeropoint outlier threshold</EM>) are marked as such.

<P>

<H2><A NAME="SECTION00922000000000000000">
Calculating Zero Points</A>
</H2>

<P>
After fitting the extinction coefficient, we can apply
Equation&nbsp;<A HREF="#eq:zmodel">8.12</A> and calculate the zeropoint of any target
frame. The program tries to filter the frames for which such a
determination would likely be in error. It will only calculate a
zeropoint for frames which satisfy the following:

<OL>
<LI>The frame has to be both preceded and succeded in time by
  non-outlier standard frames;
</LI>
<LI>The frame's airmass has to be in the same range as the standard
  frames from which the extinction coefficient was fitted. 

<P>
If <SPAN CLASS="MATH"><IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img101.png"
 ALT="$A_m$"></SPAN> is the minimum and <SPAN CLASS="MATH"><IMG
 WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img102.png"
 ALT="$A_M$"></SPAN> is the maximum standard airmass,
  and <SPAN CLASS="MATH"><IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img103.png"
 ALT="$r$"></SPAN> is the value of <EM>Multi-Frame Photometry
    Options/Airmass range</EM>
  the zeropoint is only calculated for frames with airmasses between 
  <BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\frac{A_M + A_m}{2} - \frac{r}{2}(A_M - A_m)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="198" HEIGHT="42" BORDER="0"
 SRC="img104.png"
 ALT="\begin{displaymath}
\frac{A_M + A_m}{2} - \frac{r}{2}(A_M - A_m)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">13</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
and 
  <BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\frac{A_M + A_m}{2} + \frac{r}{2}(A_M + A_m)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="199" HEIGHT="42" BORDER="0"
 SRC="img105.png"
 ALT="\begin{displaymath}
\frac{A_M + A_m}{2} + \frac{r}{2}(A_M + A_m)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">14</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
</LI>
</OL>

<P>

<H1><A NAME="SECTION00930000000000000000">
Running Multi-Frame Reduction</A>
</H1>

<P>
This section is a step-by-step tour of the multi-frame reduction
tool. An realistically-sized example input file is provided in the
distribution data directory (<TT>cygs-aug19.out</TT>). This file was generated by
<SMALL>GCX </SMALL>aperture photometry from 143 frames taken in B, V, R and I in a
single night, all in Cygnus. The standards data is from
Henden sequence files, which were converted into <SMALL>GCX </SMALL>recipies with
the import function.
The file consists of individual aperture photometry reports appended
together.

<P>

<H2><A NAME="SECTION00931000000000000000">
Specifying Reduction Bands</A>
</H2>

<P>
Before we can reduce data, we have to define which color indices are
used for each band. The <EM>Multi-Frame Photometry Options/Bands
  setup</EM> option specifies this. It contains a list of specifiers of
the form: <code>&lt;band&gt;(&lt;c1&gt;-&lt;c2&gt;)</code> separated by spaces. Each specifier
tells the program to use the color index ``<TT>&lt;c1&gt;-&lt;c2&gt;</TT>'' to reduce
frames taken in ``<TT>band</TT>''. For example, the default setting:
<DIV ALIGN="CENTER"><BLOCKQUOTE><TT>b(b-v) v(b-v) r(v-r) i(v-i)</TT></BLOCKQUOTE></DIV>
will set the following transformation model:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{eqnarray}
B&=&b+Z_B+k_B(B-V)\\
V&=&v+Z_V+k_V(B-V)\\
R&=&r+Z_R+k_R(V-R)\\
I&=&i+Z_I+k_I(V-I)
\end{eqnarray}
 -->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img106.png"
 ALT="$\displaystyle B$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img107.png"
 ALT="$\textstyle =$"></TD>
<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG
 WIDTH="162" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img108.png"
 ALT="$\displaystyle b+Z_B+k_B(B-V)$"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">15</SPAN>)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img109.png"
 ALT="$\displaystyle V$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img107.png"
 ALT="$\textstyle =$"></TD>
<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG
 WIDTH="163" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img110.png"
 ALT="$\displaystyle v+Z_V+k_V(B-V)$"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">16</SPAN>)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img111.png"
 ALT="$\displaystyle R$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img107.png"
 ALT="$\textstyle =$"></TD>
<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG
 WIDTH="161" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img112.png"
 ALT="$\displaystyle r+Z_R+k_R(V-R)$"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">17</SPAN>)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
 WIDTH="13" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img113.png"
 ALT="$\displaystyle I$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img107.png"
 ALT="$\textstyle =$"></TD>
<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG
 WIDTH="147" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img114.png"
 ALT="$\displaystyle i+Z_I+k_I(V-I)$"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">18</SPAN>)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
This model is appropiate if we reduce BV, BVI, BVR or BVRI
data. However, it will have to be changed if for instance we want to
reduce VI data, as using it will require B, V and I observations,
because of the V dependence on B. An appropiate model for VI data
would be:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{eqnarray}
V&=&v+Z_V+k_V(V-I)\\
I&=&i+Z_I+k_I(V-I)
\end{eqnarray}
 -->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img109.png"
 ALT="$\displaystyle V$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img107.png"
 ALT="$\textstyle =$"></TD>
<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG
 WIDTH="158" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img115.png"
 ALT="$\displaystyle v+Z_V+k_V(V-I)$"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">19</SPAN>)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
 WIDTH="13" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img113.png"
 ALT="$\displaystyle I$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img107.png"
 ALT="$\textstyle =$"></TD>
<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG
 WIDTH="147" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img114.png"
 ALT="$\displaystyle i+Z_I+k_I(V-I)$"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">20</SPAN>)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
which is specified by setting:
<DIV ALIGN="CENTER"><BLOCKQUOTE><TT>v(v-i) i(v-i)</TT></BLOCKQUOTE></DIV>
in the <EM>Bands setup</EM> option.

<P>
The same option can be used to set initial transformation coefficients
and their errors, by appending ``<TT>=&lt;coeff&gt;/err</TT>'' to each band
specifier like for example:
<BLOCKQUOTE>
<TT>b(b-v)=0.12/0.001 v(b-v)=-0.07/0.02</TT>
</BLOCKQUOTE>

<P>

<H2><A NAME="SECTION00932000000000000000">
Loading Report Files</A>
</H2>

<P>
The data to be reduced can reside in one or more files. 
To load data, open the the multi-frame reduction dialog using 
<EM>Processing/Multi-frame reduction</EM> or <B>Ctrl-M</B>, and select
<EM>File/Add to Dataset</EM>. Select the file name and press <EM>Ok</EM>.
The data from the frames contained in the file will load, and the
frames will appear in the ``Frames'' tab of the dialog.

<P>
More observations can be added by using <EM>Add to Dataset</EM>
repeteadly.<A NAME="tex2html48"
  HREF="footnode.html#foot1251"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">9</SPAN></SUP></A> We'll assume the example file (<TT>cygs-aug19.out</TT>) is loaded for the next
steps.

<P>

<H4><A NAME="SECTION00932010000000000000">
Frames</A>
</H4> The ``Frames'' tab contains a list with all the frames in the
dataset, one per line. It will display increasing amounts of
information as the fit progresses. The columns that are filled up 
right after loading the data files should be self-explanatory. The
other columns show:
<TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT">Zpoint</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The fitted zero point of the frame;</TD>
</TR>
<TR><TD ALIGN="LEFT">Err</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The calculated error of the zero point;</TD>
</TR>
<TR><TD ALIGN="LEFT">Fitted</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The number of standard stars used in the fit;</TD>
</TR>
<TR><TD ALIGN="LEFT">Outliers</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The number of standard stars that are considered 
outliers of the fit (have large standard errors);</TD>
</TR>
<TR><TD ALIGN="LEFT">MEU</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The mean error of unit weight for the zeropoint fit of
the frame.</TD>
</TR>
</TABLE>

<P>
Clicking on the column headers will make the program sort the list by
the respective column. Clicking again will reverse the sort order. 
One or more frames can be selected in the list. All operations apply
to the selected frames or, if none are selected, to the whole list.

<P>

<H4><A NAME="SECTION00932020000000000000">
Stars</A>
</H4> When a frame line is clicked, the stars from that frame are displayed
in the ``Stars'' tab. Like the frames, the stars can be
sorted on various columns by clicking on the column headers. The
columns of the stars table show:
<TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT">Name</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The star's name. A star is identified across multiple frame by
it's name;</TD>
</TR>
<TR><TD ALIGN="LEFT">Type</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>Star type (standard or target);</TD>
</TR>
<TR><TD ALIGN="LEFT">Band</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The band the magnitudes are in;</TD>
</TR>
<TR><TD ALIGN="LEFT">Smag</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The standard magnitude for the star. If the contents of this
field are calculated by the program, as for target stars, the
magnitude appears in square brackets;</TD>
</TR>
<TR><TD ALIGN="LEFT">Err</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>Error of the standard magnitude (either taken from the report
file, or calculated by the program);</TD>
</TR>
<TR><TD ALIGN="LEFT">Imag</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>Instrumental magnitude in this observation;</TD>
</TR>
<TR><TD ALIGN="LEFT">Err</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>Error of the instrumental magnitude, taken from the report file;</TD>
</TR>
<TR><TD ALIGN="LEFT">Residual</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>The residual in the last fit of the frame. Only appears for
standard stars;</TD>
</TR>
<TR><TD ALIGN="LEFT">Std Error</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>Standard error of the star (the residual divided by the
estimated error). Only for the standard stars;</TD>
</TR>
<TR><TD ALIGN="LEFT">Outlier</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>''Y'' or ``N'' depending on whether the star has a large
standard error or not;</TD>
</TR>
<TR><TD ALIGN="LEFT">R.A, Dec</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>Star catalog position;</TD>
</TR>
<TR><TD ALIGN="LEFT">Flags</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=255>A list of flags that apply to the star. Some are taken from the
report file, some are added by the fitting routines.</TD>
</TR>
</TABLE>

<P>

<H4><A NAME="SECTION00932030000000000000">
Bands</A>
</H4> The bands tab shows the currently configured bands,
and the various transformation coefficients relating to these
bands. They only show the fitted coefficients as they resulted from
the last fit operation. If only some frames were selected in that
operation, then these values may only apply to those frames. 

<P>

<H2><A NAME="SECTION00933000000000000000">
Fitting Individual Zero Points</A>
</H2>

<P>
The simplest type of fit we can do is fit the zeropoints of each frame
individually, without taking the other frames into consideration (like
the last step in the aperture photometry routine). Even though the
report files likely contained the individual fit information, it was
discarded when the report was loaded. We need to perform at least
this step before we can generate any plots for the data.

<P>
There are two variants of this command: one zeroes all the
transformation coefficients before doing the fit (<EM>Fit Zero
  Points with Null Coefficients</EM>), while the other will apply the
current transformation coefficients to the standard stars first.
(<EM>Fit Zero Points with Current Coefficients</EM>).

<P>
Make sure the frames you want to fit are selected before applying the
command (if no frames are selected, the command will apply to all
frames).

<P>
After the fit, examine the MEU column, which will show the quality of
the fit (the number should be around 1.0). Since we only fitted the
zeropoint, and not the color coefficients the values are slightly
larger than the best than can be obtained.

<P>

<H2><A NAME="SECTION00934000000000000000">
Plots</A>
</H2>

<P>

<DIV ALIGN="CENTER"><A NAME="fig:v-res-mag-zponly"></A><A NAME="910"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.1:</STRONG>
Residuals vesus standard magnitudes for all frames in the
  example data set, after zero point fitting without color
  transformation.</CAPTION>
<TR><TD><IMG
 WIDTH="523" HEIGHT="380" ALIGN="BOTTOM" BORDER="0"
 SRC="img116.png"
 ALT="\includegraphics[width=\textwidth]{v-res-mag-zponly}"></TD></TR>
</TABLE>
</DIV>

<P>
At this point, we can generate various plots, which are instrumental
in judging the quality of the data, especially when we consider the more
ellaborate fits. The program generates data file for the <TT>gnuplot</TT>
utility, and will run <TT>gnuplot</TT> directly if the option <EM>File
  and Device Options/Gnuplot command</EM> is correctly set.<A NAME="tex2html50"
  HREF="footnode.html#foot916"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">10</SPAN></SUP></A> If the <EM>Plot to File</EM> option in the <EM>Plot</EM> menu is 
selected, the program will generate a data file instead of running 
gnuplot directly.<A NAME="tex2html51"
  HREF="footnode.html#foot919"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">11</SPAN></SUP></A>
<P>
Let's select the V frames (click on the band column header twice to
bring the V band at the top, then click on the first V frame, and
finaly shift-click on the last V frame). Now run <EM>Plot/Residuals
  vs Magnitude</EM>. A plot should appear that is similar to the
one in Figure&nbsp;<A HREF="#fig:v-res-mag-zponly">8.1</A>.

<P>

<DIV ALIGN="CENTER"><A NAME="fig:aucyg-res-std"></A><A NAME="924"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.2:</STRONG>
Residuals versus standard magnitudes of one AU CYG frame.</CAPTION>
<TR><TD><IMG
 WIDTH="524" HEIGHT="380" ALIGN="BOTTOM" BORDER="0"
 SRC="img117.png"
 ALT="\includegraphics[width=\textwidth]{aucyg-res-std}"></TD></TR>
</TABLE>
</DIV>

<P>
The plot generally has the familiar shape of photon-shot
noise dominated observations, with random errors increasing as the
stars become fainter. An additional feature of this dataset are the 
``branches'' going up starting at around mag 12 and 11. These are
caused by saturated standard stars (the standards we used are not
reliable above mag 12.5 or so). If the stars would be saturated in our
observations, the ``branches'' would go downward.<A NAME="tex2html53"
  HREF="footnode.html#foot1254"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">12</SPAN></SUP></A>
<P>

<DIV ALIGN="CENTER"><A NAME="fig:aucyg-wres-std"></A><A NAME="930"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.3:</STRONG>
Standard errors versus magnitudes of one AU CYG frame.</CAPTION>
<TR><TD><IMG
 WIDTH="523" HEIGHT="380" ALIGN="BOTTOM" BORDER="0"
 SRC="img118.png"
 ALT="\includegraphics[width=\textwidth]{aucyg-wres-std}"></TD></TR>
</TABLE>
</DIV>

<P>
To investigate the matter further, we select a frame with a large
number of outliers, which is likely to contain such a ``branch''. For
example, let's select aucyg. The <EM>Residuals vs
  Magnitude</EM> plot for this frame is shown in
Figure&nbsp;<A HREF="#fig:aucyg-res-std">8.2</A>. The bright stars branching up are
obvious in this plot. However, the importance of the errors is
difficult to judge, as the ``normal'' error changes with the stars'
magnitudes (and fainter stars show similar residuals). The
<EM>Standard Errors vs Magnitude</EM> plot comes handy in this
situation. It is similar to the previous plot, only the residuals are
divided by the expected error of the respective stars. We expect all
stars to show similar standard errors, all within a 6 units wide band
around zero. This plot is shown in Figure&nbsp;<A HREF="#fig:aucyg-wres-std">8.3</A>.
We can clearly see that the relatively large residuals to the right of
the plot are within normal limits (also indicated by the value of the 
MEU fit parameter). The ``branch'' is clearly deviant (with standard
errors going up to 30 and more).<A NAME="tex2html55"
  HREF="footnode.html#foot937"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">13</SPAN></SUP></A> Fortunately, the robust fitting
algorithm has downweighted the deviant points significantly, so the
``good'' values still spread symetrically around zero.<A NAME="tex2html56"
  HREF="footnode.html#foot938"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">14</SPAN></SUP></A>
<P>

<H2><A NAME="SECTION00935000000000000000">
Fitting Color Transformation Coefficients</A>
</H2>

<P>

<DIV ALIGN="CENTER"><A NAME="fig:v-se-color-zponly"></A><A NAME="942"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.4:</STRONG>
Standard errors vs color for the V frames, before
  transformation coefficient fitting.</CAPTION>
<TR><TD><IMG
 WIDTH="497" HEIGHT="375" ALIGN="BOTTOM" BORDER="0"
 SRC="img121.png"
 ALT="\includegraphics[width=\textwidth]{v-se-color-zponly}"></TD></TR>
</TABLE>
</DIV>

<P>

<DIV ALIGN="CENTER"><A NAME="fig:v-se-color"></A><A NAME="947"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.5:</STRONG>
Standard errors vs color for the V frames, after
  transformation coefficient fitting.</CAPTION>
<TR><TD><IMG
 WIDTH="497" HEIGHT="375" ALIGN="BOTTOM" BORDER="0"
 SRC="img122.png"
 ALT="\includegraphics[width=\textwidth]{v-se-color}"></TD></TR>
</TABLE>
</DIV>

<P>
With the V frames selected, let's plot the standard errors again, this
time against the star's color index. For this, select <EM>Plot/Standard Errors
vs Color</EM>. The output should look similar to
Figure&nbsp;<A HREF="#fig:v-se-color-zponly">8.4</A>. Even given the scatter of the
individual observations, the plot shows a clear sloping (making the
residuals proportional to the color index).<A NAME="tex2html59"
  HREF="footnode.html#foot952"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">15</SPAN></SUP></A> To remove this slope and 
at the same time calculate the color transformation coefficient, we
use <EM>Reduce/Fit Zero Points and Transformation Coefficients</EM>. 
After the fit is done, the slope is removed, as shown in 
Figure&nbsp;<A HREF="#fig:v-se-color">8.5</A>. The title of the figure shows the
transformation used. In our case, the resulting tranformation
coefficient is 0.062, a rather small figure indicating a good fit
between the filters used and the standard ones. If we check the MEU 
fields for each frame, we will see that they have decreased, showing 
that the data more closely matches the standard magnitudes after the
color transformation.

<P>
After the fit, the list in the ``Bands'' tab is updated to show the
fitted transformation coefficients and their expected errors. Note
that the error is quite small in our case (0.002),<A NAME="tex2html60"
  HREF="footnode.html#foot955"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">16</SPAN></SUP></A> even though the
data seemed to spread a lot. The large number of stars used in the fit
helped reduce the error considerably.

<P>
A good sanity check for the transformation coefficient fit is to run
the same routine on subsets of the initial data set and compare the 
resulting transformation coefficients. They should match within the
reported error figures.

<P>
Before proceeding, let's do the transformation coefficient fit for the
whole dataset: <EM>Edit/Unselect All</EM>, then <EM>Reduce/Fit Zero 
Points and Transformation Coefficients</EM>. 

<P>

<H2><A NAME="SECTION00936000000000000000">
All-Sky Reduction</A>
</H2>

<P>

<DIV ALIGN="CENTER"><A NAME="fig:zp-am-1"></A><A NAME="961"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.6:</STRONG>
Zero points vs airmass for frames with standards data.</CAPTION>
<TR><TD><IMG
 WIDTH="537" HEIGHT="375" ALIGN="BOTTOM" BORDER="0"
 SRC="img125.png"
 ALT="\includegraphics[width=\textwidth]{zp-am-1}"></TD></TR>
</TABLE>
</DIV>

<DIV ALIGN="CENTER"><A NAME="fig:zp-t-1"></A><A NAME="966"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.7:</STRONG>
Zero points vs time for frames with standards data.</CAPTION>
<TR><TD><IMG
 WIDTH="536" HEIGHT="377" ALIGN="BOTTOM" BORDER="0"
 SRC="img126.png"
 ALT="\includegraphics[width=\textwidth]{zp-t-1}"></TD></TR>
</TABLE>
</DIV>

<P>
The example data set contains BVRI frames for all fields. However,
only some of the fields have R and I standards data. The night was
clear, but conditions were changing. Let's see what we can do about
the R and I frames that need all-sky reduction.

<P>
We can examine the frame zero points versus the airmass, expecting
them to fall on a down-sloping line.<A NAME="tex2html63"
  HREF="footnode.html#foot969"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">17</SPAN></SUP></A> Using <EM>Plot/Zeropoints vs Airmass</EM> will produce the plot in
Figure&nbsp;<A HREF="#fig:zp-am-1">8.6</A>, which shows all the bands' zeropoints on the
same graph. We see that most of the frames do indeed lie on
down-sloping lines with a scatter consistent with their expected
errors as shows by the error bars, but there are some outliers. So the
conditions weren't photometric. If we now plot the same zeropoints
against time (<EM>Plot/Zeropoints vs Time</EM>), Figure&nbsp;<A HREF="#fig:zp-t-1">8.7</A> we
can see what has happened: the transparency has improved starting at
MJD 53236.95, to the point where we can use the all-sky method for
frames taken after that point. 

<P>

<DIV ALIGN="CENTER"><A NAME="fig:zp-am-2"></A><A NAME="976"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.8:</STRONG>
Zero points vs airmass for all frames, after the extinction fit.</CAPTION>
<TR><TD><IMG
 WIDTH="537" HEIGHT="375" ALIGN="BOTTOM" BORDER="0"
 SRC="img127.png"
 ALT="\includegraphics[width=\textwidth]{zp-am-2}"></TD></TR>
</TABLE>
</DIV>

<DIV ALIGN="CENTER"><A NAME="fig:zp-t-2"></A><A NAME="981"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.9:</STRONG>
Zero points vs time for all frames, after the extinction fit.</CAPTION>
<TR><TD><IMG
 WIDTH="536" HEIGHT="377" ALIGN="BOTTOM" BORDER="0"
 SRC="img128.png"
 ALT="\includegraphics[width=\textwidth]{zp-t-2}"></TD></TR>
</TABLE>
</DIV>

<P>
Let's run the all-sky reduction (<EM>Reduce/Fit Extinction and
  All-Sky Zero Points</EM>) and generate the plots again. As we can see in 
Figures&nbsp;<A HREF="#fig:zp-am-2">8.8</A> and&nbsp;<A HREF="#fig:zp-t-2">8.9</A>, the program has
  selected the frames which are bracketed by other ``good'' frames,
<A NAME="tex2html66"
  HREF="footnode.html#foot987"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">18</SPAN></SUP></A> and calculated their all-sky zeropoints. The all-sky frames are
  shown with different colors. These plots should be carefully
  examined and any suspicious frames removed from the all-sky
  reduction. In our case however, it seems that the program has made a
  good choice of frames. 

<P>
The calculated extinction coefficients and their errors are displayed
in the ``Bands'' tab. The errors of the exctinction coefficients are
relatively large. In this case, this is due to the fact that the
frames are taken in a narrow range of airmasses. The same narrow range
of airmasses will however reduce the impact of the errors on the
calculated zeropoints. this can be seen on the graphs, where the error
bars of the all-sky frames, which take the extinction coefficient
errors into account, are of the same order as those of the ``normal''
frames.<A NAME="tex2html67"
  HREF="footnode.html#foot988"><SUP><SPAN CLASS="arabic">8</SPAN>.<SPAN CLASS="arabic">19</SPAN></SUP></A>
<P>

<H1><A NAME="SECTION00940000000000000000">
Reporting</A>
</H1>

<P>
After the fits are done, the complete dataset can be saved in the
native format using <EM>File/Save Dataset</EM>. The native format
preserves all the information in a future-proof fashion, but importing
it into other applications can be a little involved.

<P>
The <TT>-rep-to-table</TT> or <TT>-T</TT> command-line option allows the native format
to be converted into a table with fixed-width columns. The format and 
content of the columns are fully programmable by changing the <EM>File
and Device Options/Report converter output format</EM> option. The
following command will convert dataset.out from the native format to a
table (dataset.txt) with the format as set in the option:
<BLOCKQUOTE>
<TT>gcx -T dataset.out <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img65.png"
 ALT="$&gt;$"></SPAN>dataset.txt</TT>
</BLOCKQUOTE>
Alternatively, the table format can be specified on the command
line. For example, to create a table with the stars' name, mjd of
observation, and V magnitudes and errors use:
<BLOCKQUOTE>
<TT>gcx -T dataset.out -S ".file.tab_format=name jdate
  <SPAN CLASS="MATH"><IMG
 WIDTH="13" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img66.png"
 ALT="$\backslash$"></SPAN>
<BR>
smag 'v' serr 'v'" <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img65.png"
 ALT="$&gt;$"></SPAN>dataset.txt</TT>
</BLOCKQUOTE>
The complete format string specification can be found in
Appendix&nbsp;<A HREF="node14.html#ap:repconv">E</A>.

<P>
Finally, it is possible to list the target stars in the AAVSO
format. If a validation file location is set in the <EM>File and
  Device Options/Aavso validation file</EM>, it will be searched 
for the designation of the stars. The observer code field will be
filled in from the <EM>general Observation Setup Data/Observer code</EM>
option.

<P>

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