%+
% Name:
%   bias_repair_4.4.tex
%
% Notes:
%   1. Describe what to do if a pixel has a value of 4096.
%       - It should be much like the repair of portions of columns.
%       - Steps 4.1.e and 4.1.f will have to be changed.
%
% History:
%   2009 Aug 12, created (v1.0), Glenn E. Allen
%   2009 Aug 26, added a description of the criteria used to identify a
%     replacement bias; added a reference to Peter Ford's Perl script
%     search4bias; added a change to the header keywords INITOCLx; added
%     a note to include suitable HISTORY keywords (v2.0), GEA
%   2009 Oct 27, added bias repair (v3.0), GEA
%   2009 Nov 04, added some header keywords (v3.1), GEA
%   2009 Nov 09, added repair of partial columns; changed the header
%     procedure for bias replacement (v4.0), GEA
%   2009 Nov 13, changed language to reflect that a partial column is part
%     of a trapezoid, not part of a rectangle (v4.1), GEA
%   2009 Dec 07, minor changes to address Jonathan McDowell's comments
%     (v4.2), GEA
%   2010 Feb 02, removed 4096 (v4.3), GEA
%   2010 Apr 07, removed an obsolete parenthetical statement (v4.4), GEA
%-

\documentclass{article}

\usepackage{changebar}
\usepackage{cxo-memo-logo}
\usepackage[dvips]{graphics}
\usepackage{epsfig}
\usepackage{gea}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 1. Header

\memobasic{
   Jonathan McDowell, SDS Group Leader }{
   Glenn E.\ Allen, SDS }{
   Bias repair }{
   4.4 }{
   http://space.mit.edu/CXC/docs/docs.html\#bias\_rep }{
   /nfs/cxc/h2/gea/sds/docs/memos/bias\_repair\_4.4.tex }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 2. Description

\section{ Introduction }

Biases can be adversely affected for a number of reasons, including the
interaction of charged particles in the detector, the transmission of
optical light through the optical-blocking filter (\eg\ Fig.~\ref{fig01}),
and the loss of part or all of the bias telemetry.  In order to produce the
most accurate ACIS event data possible, biases may have to be repaired or
replaced. This spec describes how to use the bias for one observation to
replace or repair the bias for another observation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 3. Replacement

\section{ Replacement }
\label{repl}

If an entire bias file has problems, then the bias file should be replaced.
The following steps should be performed, in sequence, to replace bias file
$B$ for an observation (\eg\ Fig.~\ref{fig01}) with a modified version of a
bias file $B'$ from another observation (\eg\ Fig.~\ref{fig02}).

\begin{enumerate}

  \item
    Create a replacement bias file $B''$.

    \begin{itemize}

      \item[a.]
        Identify a suitable bias file $B'$.\footnote{Peter Ford's Perl
	script {\tt search4bias.pl}
	(/nfs/acis/h4/tools/bin/sun4/search4bias.pl on the MIT CXC network)
	can be used to identify a suitable replacement file.\label{foo02}}
	The file $B'$ is suitable if the following conditions are satisfied.
        %
        \begin{itemize}

          \item[i.]
            The bias file $B'$ is for the same CCD as the bias file $B$.

          \item[ii.]
            The parameter-block keywords
	    %
            \begin{eqnarray}
              {\tt BEP\_MODE}' & = & {\tt BEP\_MODE}, \label{eqn01} \\
              {\tt DEALOAD}' & = & {\tt DEALOAD}, \\
              {\tt DTYCYCLE}' & = & {\tt DTYCYCLE}, \\
              {\tt EXP\_SKIP}' & = & {\tt EXP\_SKIP}, \\
              {\tt EXPTIMEA}' & = & {\tt EXPTIMEA}, \\
              {\tt EXPTIMEB}' & = & {\tt EXPTIMEB}, \\
              {\tt FEPLOAD}' & = & {\tt FEPLOAD}, \\
              {\tt FEP\_MODE}' & = & {\tt FEP\_MODE}, \\
              {\tt NOBADCOL}' & = & {\tt NOBADCOL}, \\
              {\tt NOBADPIX}' & = & {\tt NOBADPIX}, \\
              {\tt OCLKPAIR}' & = & {\tt OCLKPAIR}, \\
              {\tt ORC\_MODE}' & = & {\tt ORC\_MODE}, \\
              {\tt ROWCNT}' & = & {\tt ROWCNT}, \\
              {\tt STARTROW}' & = & {\tt STARTROW},\ {\rm and} \\
              {\tt SUM\_2X2}' & = & {\tt SUM\_2X2}. \label{eqn15}
	    \end{eqnarray}
	    %
	    The keywords on the left-hand sides of
	    equations~\ref{eqn01}--\ref{eqn15} are from the parameter-block
	    file (acisf*pbk0.fits) for the same observation as the
	    replacement bias file $B'$.  The keywords on the right-hand
	    sides are from the parameter-block file for the same observation
	    as the bias file $B$ being replaced.

          \item[iii.]
            The tabulated parameter-block data
	    %
            \begin{eqnarray}
              {\tt BIASALG}' & = & {\tt BIASALG}, \label{eqn16} \\
              {\tt BIASARG0}' & = & {\tt BIASARG0}, \\
              {\tt BIASARG1}' & = & {\tt BIASARG1}, \\
              {\tt BIASARG2}' & = & {\tt BIASARG2}, \\
              {\tt BIASARG3}' & = & {\tt BIASARG3}, \\
              {\tt BIASARG4}' & = & {\tt BIASARG4},\ {\rm and} \\
              {\tt VIDRESP}' & = & {\tt VIDRESP} \label{eqn22}
	    \end{eqnarray}
	    %
	    for the CCD associated with $B'$ and $B$. Again, the left- and
	    right-hand sides of equations~\ref{eqn16}--\ref{eqn22} are from
	    the parameter-block files associated with $B'$ and $B$,
	    respectively.

          \item[iv.] The time difference between $B'$ and $B$,
            %
            \begin{equation}
	      \Delta t \equiv \left| t' - t \right|,
	    \end{equation}
	    %
	    should be minimized. (Ideally $\Delta t < 5$~days.)

          \item[v.]
	    The focal-plane temperature difference between $B'$ and $B$,
            %
            \begin{equation}
	      \Delta T \equiv \left| T' - T \right|,
	      \label{eqn24}
	    \end{equation}
	    %
	    should be minimized. (Ideally $\Delta T < 1$~$^{\circ}$C.)
	    Here, $T'$ and $T$ are the mean temperatures during the
	    intervals over which the data for the bias files $B'$ and $B$,
	    respectively, were obtained.

        \end{itemize}
        %

      \item[b.]
        Copy $B'$ to $B''$, where $B''$ will become the replacement bias
	file. The files $B$ and $B'$ remain unchanged.

      \item[c.]
        For the first node of the bias $B''$ (\ie\ columns 1--256), modify
	the values of every pixel in the node so that
        %
        \begin{equation}
          B'' = B' - {\tt INITOCLA}' + {\tt INITOCLA},
        \end{equation}
        %
        where ${\tt INITOCLA}'$ and {\tt INITOCLA} are keywords in the
	headers of the bias files $B'$ and $B$, respectively.

      \item[d.]
        If the value of $B' = 4094$, 4095, or 4096 for a pixel on the first
	node, then $B'' = B'$ for that pixel instead of $B'' = B' - {\tt
	INITOCLA}' + {\tt INITOCLA}$.

      \item[e.]
        If the value of $B = 4094$ or 4095 for a pixel on the first node,
	then $B'' = B$ for that pixel instead of $B'' = B' - {\tt INITOCLA}'
	+ {\tt INITOCLA}$.

      \item[f.]
        Repeat steps~2.1.c--2.1.e for the other three nodes, using {\tt
	INITOCLB}, {\tt INITOCLC}, and {\tt INITOCLD}, instead of {\tt
	INITOCLA}, for columns 257--512, 513--768, and 769--1024,
	respectively.

      \item[g.]
        Copy the header from $B$ to $B''$.

      \item[h.]
        Update the values of the keywords {\tt CHECKSUM}, {\tt DATASUM}, and
	{\tt DATE} in $B''$.

      \item[i.]
        Add appropriate history keywords to the header of the modified bias
	file $B''$.  For example:

        { \tt
          \hspace*{0.15in}HISTORY\ \ YYYY-MM-DD:HH:MM:SS \\
          \hspace*{0.15in}HISTORY\ \ The name and version number of the code
            used to produce the file \\
          \hspace*{0.15in}HISTORY\ \ CXC contact person \\
          \hspace*{0.15in}HISTORY \\
          \hspace*{0.15in}HISTORY\ \ This bias file was created by replacing
	    the file \\
          \hspace*{0.15in}HISTORY\ \ acisf?????????N???\_?\_bias0.fits with a
            modified version of the file \\
          \hspace*{0.15in}HISTORY\ \ acisf?????????N???\_?\_bias0.fits.
        }

      \end{itemize}

  \item
    Reprocess the ACIS event data using the bias file $B''$.

  \item
    Examine the pulse-height information of the event data after it has been
    processed with the replacement bias (\eg\ Fig.~\ref{fig03}).  If ${\tt
    DATAMODE} = {\tt FAINT}$, then prepare a histogram of the pulse heights
    of the 4 corner pixels for every event on the first node.  If ${\tt
    DATAMODE} = {\tt VFAINT}$, then prepare a histogram using the pulse
    heights of the outer 16 pixels instead of the 4 corner pixels.  The peak
    of the distribution should be at a pulse height of 0 adu.  If it is not,
    then add the difference between the location of the peak and 0~adu to
    every pixel on the node for the bias file $B''$.\footnote{The value of
    the keyword ${\tt INITOCLA}''$ should not be changed in the same way.
    Only the data in the bias file $B''$ is modified in this step.} Repeat
    this process for the other nodes.

  \item
    Repeat steps~2.2 and 2.3 until the condition specified in step~2.3 is
    satisfied.

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 4. Repair of full columns

\section{ Repair of full columns }
\label{repa1}

If only a portion of a bias file has problems, then it may be more
appropriate to repair the bias instead of replacing it. For example, suppose
that every pixel in the columns from $l$--$n$ of bias file $B$ for an
observation has excess charge (\eg\ Fig.~\ref{fig04}). In this case, the
following steps should be performed, in sequence, to replace the data for
these columns with the data from the same columns in bias file $B'$ less the
difference(s) between the node medians of $B'$ and $B$ (\eg\
Fig.~\ref{fig05}).

\begin{enumerate}

  \item
    %
    Create a replacement bias file $B''$.

    \begin{itemize}

      \item[a.]
        Identify a suitable bias file $B'$.$^{\ref{foo02}}$ The file $B'$ is
	suitable if the conditions described in item~1a of
	section~\ref{repl} are satisfied.

      \item[b.]
        Copy $B$ to $B''$, where $B''$ will become the repaired bias file.
	The files $B$ and $B'$ remain unchanged.

      \item[c.]
        %
        Suppose that the range of columns from $l$--$n$ straddles a node
	boundary between columns $m$ and $m+1$ (\ie\ $l \le m < n$). For the
	columns from $l$--$m$ of bias $B''$, replace the bias values of
	every pixel $(i,j)$ of the region such that
        %
        \begin{equation}
          B_{ij}'' = B_{ij}' - M' + M,
        \end{equation}
        %
        where the column number $i$ (\ie\ {\tt CHIPX}) is in the range from
	$l$--$m$ and the row number $j$ (\ie\ {\tt CHIPY}) is in the range
	from 1--1024.  $M'$ is the median of the bias values $B_{ij}'$ where
	$i$ includes all of the columns on the same node as columns $l$--$m$
	and $j$ is in the range from 1--1024.  The columns $i$ from
	$l$--$m$, the pixels $(i,j)$ for which $B_{ij}' = 4094$, 4095, or
	4096, and the pixels $(i,j)$ for which $B_{ij} = 4094$, 4095, or
	4096 are excluded from the computation of $M'$. $M$ is computed in
	the same manner as $M'$, except that the bias $B$ is used instead of
	$B'$.  The same set of pixels are used to compute $M$ and $M'$.  If
	the range from $l$--$n$ does not straddle a node boundary, then $m$
	should be replaced by $n$ in the description of the computation of
	$M'$ and $M$. If the file $B$ is for an observation that used one of
	the standard subarrays or some other user-specified subset of the
	rows of a CCD, then the appropriate range of $j$ is not 1--1024.

      \item[d.]
        The process described in step~3.1.c is repeated for the columns from
	$m+1$ to $n$.  However, if the range from $l$--$n$ does not straddle
	a node boundary, then step~3.1.d is skipped.

      \item[e.]
        If the value of $B_{ij}' = 4094$, 4095, or 4096 for a pixel in the
	region being repaired, then $B_{ij}'' = B_{ij}'$ for that pixel
	instead of $B_{ij}'' = B_{ij}' - M' + M$.

      \item[f.]
        If the value of $B_{ij} = 4094$ or 4095 for a pixel in the region
	being repaired, then $B_{ij}'' = B_{ij}$ for that pixel instead of
	$B_{ij}'' = B_{ij}' - M' + M$.

      \item[g.]
        Copy the header from $B$ to $B''$.

      \item[h.]
        Update the values of the keywords {\tt CHECKSUM}, {\tt DATASUM}, and
	{\tt DATE} in $B''$.

      \item[i.]
        Add appropriate history keywords to the header of $B''$. For
	example:

        { \tt
          \hspace*{0.15in}HISTORY\ \ YYYY-MM-DD:HH:MM:SS \\
          \hspace*{0.15in}HISTORY\ \ The name and version number of the code
            used to produce the file \\
          \hspace*{0.15in}HISTORY\ \ CXC contact person \\
          \hspace*{0.15in}HISTORY \\
          \hspace*{0.15in}HISTORY\ \ This bias file was created by repairing
	    the file \\
          \hspace*{0.15in}HISTORY\ \ acisf?????????N???\_?\_bias0.fits using
	    data from the file \\
          \hspace*{0.15in}HISTORY\ \ acisf?????????N???\_?\_bias0.fits.
        }

    \end{itemize}

  \item
    Reprocess the ACIS event data using the bias file $B''$.

  \item
    Examine the pulse-height information of the event data after it has been
    processed with the repaired bias (\eg\ Fig.~\ref{fig06}).  If ${\tt
    DATAMODE} = {\tt FAINT}$, then prepare a histogram of the pulse heights
    of the 4 corner pixels for every event in the region being repaired. If
    ${\tt DATAMODE} = {\tt VFAINT}$, then prepare a histogram using the
    pulse heights of the outer 16 pixels instead of the 4 corner pixels.
    The peak of the distribution should be at a pulse height of 0 adu.  If
    it is not, then add the difference between the location of the peak and
    0~adu to every pixel in the repaired region of $B''$.$^{2}$

  \item
    Repeat steps~3.2 and 3.3 until the condition specified in step~3.3 is
    satisfied.

\end{enumerate}

If a subset of the columns from $l$--$n$ includes all of the columns in a
node, then follow the process described in section~\ref{repl} for that
\begin{changebar} subset. For \end{changebar} the remaining columns, use the
process described in section~\ref{repa1}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 5. Repair of portions of columns

\section{ Repair of portions of columns }
\label{repa2}

If every pixel in a region that includes portions of the columns from
$l$--$n$ of bias file $B$ has excess charge (\eg\ Fig.~\ref{fig07}), then
the following steps should be performed, in sequence, to replace the data
for this region with the data from the same region in bias file $B'$ less
the difference(s) between the node medians of $B'$ and $B$ (\eg\
Fig.~\ref{fig08}).

\begin{enumerate}

  \item
    Create a replacement bias file $B''$.

    \begin{itemize}

      \item[a.]
        Identify a suitable bias file $B'$.$^{\ref{foo02}}$ The file $B'$ is
	suitable if the conditions described in item~1a of
	section~\ref{repl} are satisfied.

      \item[b.]
        Copy $B$ to $B''$, where $B''$ will become the repaired bias file.
	The files $B$ and $B'$ remain unchanged.

      \item[c.]
        Suppose that the range of columns from $l$--$n$ straddles a node
	boundary between columns $m$ and $m+1$ (\ie\ $l \le m < n$). For the
	columns from $l$--$m$ of bias $B''$, replace the bias values of
	every pixel $(i,j)$ of the region such that
        %
        \begin{equation}
          B_{ij}'' = B_{ij}' - M' + M,
        \end{equation}
        %
        where the column number $i$ (\ie\ {\tt CHIPX}) is in the range from
	$l$--$m$ and the row number $j$ (\ie\ {\tt CHIPY}) for the region
	being repaired is in the range from $p(i)$--$q(i)$, where $p$ and
	$q$ can vary from column to column.  $M'$ is the median of the bias
	values $B_{ij}'$ where $i$ includes the columns $l$--$m$ and $j$ is
	in the range from 1--1024. The pixels $(i,j)$ in the region being
	repaired, the pixels $(i,j)$ for which $B_{ij}' = 4094$, 4095, or
	4096, and the pixels $(i,j)$ for which $B_{ij} = 4094$, 4095, or
	4096 are excluded from the computation of $M'$. $M$ is computed in
	the same manner as $M'$, except that the bias $B$ is used instead of
	$B'$.  The same set of pixels are used to compute $M$ and $M'$.  If
	the range from $l$--$n$ does not straddle a node boundary, then $m$
	should be replaced by $n$ in the description of the computation of
	$M'$ and $M$. If the file $B$ is for an observation that used one of
	the standard subarrays or some other user-specified subset of the
	rows of a CCD, then the appropriate range of $j$ for the computation
	of $M$ and $M'$ is not 1--1024.

      \item[d.]
        The process described in step~4.1.c is repeated for the columns from
	$m+1$ to $n$.  However, if the range from $l$--$n$ does not straddle
	a node boundary, then step~4.1.d is skipped.

      \item[e.]
        If the value of $B_{ij}' = 4094$, 4095, or 4096 for a pixel in the
	region being repaired, then $B_{ij}'' = B_{ij}'$ for that pixel
	instead of $B_{ij}'' = B_{ij}' - M' + M$.

      \item[f.]
        If the value of $B_{ij} = 4094$ or 4095 for a pixel in the region
	being repaired, then $B_{ij}'' = B_{ij}$ for that pixel instead of
	$B_{ij}'' = B_{ij}' - M' + M$.

      \item[g.]
        Copy the header from $B$ to $B''$.

      \item[h.]
        Update the values of the keywords {\tt CHECKSUM}, {\tt DATASUM}, and
	{\tt DATE} in $B''$.

      \item[i.]
        Add appropriate history keywords to the header of $B''$. For
	example:

        { \tt
          \hspace*{0.15in}HISTORY\ \ YYYY-MM-DD:HH:MM:SS \\
          \hspace*{0.15in}HISTORY\ \ The name and version number of the code
            used to produce the file \\
          \hspace*{0.15in}HISTORY\ \ CXC contact person \\
          \hspace*{0.15in}HISTORY \\
          \hspace*{0.15in}HISTORY\ \ This bias file was created by repairing
	    the file \\
          \hspace*{0.15in}HISTORY\ \ acisf?????????N???\_?\_bias0.fits using
	    data from the file \\
          \hspace*{0.15in}HISTORY\ \ acisf?????????N???\_?\_bias0.fits.
        }

    \end{itemize}

  \item
    Reprocess the ACIS event data using the bias file $B''$.

  \item
    Examine the pulse-height information of the event data after it has been
    processed with the repaired bias (\eg\ Fig.~\ref{fig09}).  If ${\tt
    DATAMODE} = {\tt FAINT}$, then prepare a histogram of the pulse heights
    of the 4 corner pixels for every event in the region being repaired. If
    ${\tt DATAMODE} = {\tt VFAINT}$, then prepare a histogram using the
    pulse heights of the outer 16 pixels instead of the 4 corner pixels.
    The peak of the distribution should be at a pulse height of 0 adu.  If
    it is not, then add the difference between the location of the peak and
    0~adu to every pixel in the repaired region of $B''$.$^{2}$

  \item
    Repeat steps~4.2 and 4.3 until the condition specified in step~4.3 is
    satisfied.

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 6. Caveats

\section{ Caveats }

\begin{enumerate}

  \item
     The algorithms described in this spec were designed for timed-exposure
     mode observations and may not be appropriate for continuous-clocking
     mode observations.

  \item
     If a portion of a bias has excess charge, then events in that region
     whose pulse heights are near the event threshold may have been lost and
     cannot be recovered.

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 7. TBD

\section{ TBD }

\begin{enumerate}

  \item
    This spec does not include an algorithm describing how to identify the
    regions of a bias that should be repaired.  Since these regions are
    readily identifiable by eye (\eg\ Figs.~\ref{fig04} and \ref{fig07}),
    visual inspection may suffice.

  \item
    The algorithms described in this spec do not address systematic offsets
    at the sub-adu level.

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 8. Figures

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig01,angle=0,height=8.32in}
  \end{center}
  \vspace*{-0.25in}
  \caption{%
    An image of the bias residuals for the five CCDs used for {\tt OBS\_ID}
    11799. The median of each column has been subtracted from the bias
    values for the column.  Excess charge from an optical light leak is
    evident, particularly on ACIS-S2, -I2, and -I0.  The labels for each CCD
    are located near the coordinates $({\tt CHIPX},{\tt CHIPY}) = (1,1)$.
    %
    \label{fig01}
    }
\end{figure}

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig02,angle=0,height=8.32in}
  \end{center}
  \vspace*{-0.25in}
  \caption{%
    An image of the bias residuals for the five CCDs used for {\tt OBS\_ID}
    11799 after the bias files were replaced with modified versions of the
    bias files for {\tt OBS\_ID} 9345. The median of each column has been
    subtracted from the bias values for the column.  The labels for each CCD
    are located near the coordinates ${({\tt CHIPX},{\tt CHIPY}) = (1,1)}$.
    %
    \label{fig02}
    }
\end{figure}

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig03,angle=270,width=6.5in}
  \end{center}
  \caption{%
    Histograms of the pulse heights of the outer 16 pixels of the 5~pixel
    $\times$ 5~pixel event islands for {\tt OBS\_ID} 11799.  Only events on
    the first node of ACIS-I2 are included.  The red histogram is the
    distribution obtained using the data processed with the biases shown in
    Figure.~\ref{fig01}.  The peak of the red histogram is at $-5$~adu,
    instead of zero.  The black histogram is the distribution obtained after
    the data were reprocessed using the biases shown in Figure~\ref{fig02}.
    The peak of the black histogram is at 0~adu, as desired.
    %
    \label{fig03}
    }
\end{figure}

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig04,angle=0,width=6.5in}
  \end{center}
  \vspace*{-0.30in}
  \caption{%
    An image of the ACIS-I1 bias residuals for {\tt OBS\_ID} 9336. The
    median of each node has been subtracted from the bias values for the
    node.  Excess charge is evident in the columns from 913--961.
    %
    \label{fig04}
    }
\end{figure}

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig05,angle=0,width=6.5in}
  \end{center}
  \vspace*{-0.30in}
  \caption{%
    An image of the ACIS-I1 bias residuals for {\tt OBS\_ID} 9336 after the
    data in the columns from 913--961 were repaired using data from a bias
    file for {\tt OBS\_ID} 10811. The median of each node has been
    subtracted from the bias values for the node.
    %
    \label{fig05}
    }
\end{figure}

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig06,angle=270,width=6.5in}
  \end{center}
  \caption{%
    Histograms of the pulse heights of the outer 16 pixels of the 5~pixel
    $\times$ 5~pixel event islands for {\tt OBS\_ID} 9336.  Only events in
    the columns from 913--961 of ACIS-I1 are included.  The red histogram is
    the distribution obtained using the data processed with the bias shown
    in Figure~\ref{fig04}. The peak of the red histogram is at $-2$~adu,
    instead of zero.  The black histogram is the distribution obtained using
    the bias shown in Figure~\ref{fig05}. The peak of the black histogram is
    at 0~adu, as desired.
    %
    \label{fig06}
    }
\end{figure}

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig07,angle=0,width=6.5in}
  \end{center}
  \vspace*{-0.30in}
  \caption{%
    An image of the ACIS-S2 bias residuals for {\tt OBS\_ID} 4111. The
    median of each node has been subtracted from the bias values for the
    node.  Excess charge is evident in a trapezoidal-shaped region that
    includes portions of the columns from 624--887.
    %
    \label{fig07}
    }
\end{figure}

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig08,angle=0,width=6.5in}
  \end{center}
  \vspace*{-0.30in}
  \caption{%
    An image of the ACIS-S2 bias residuals for {\tt OBS\_ID} 4111 after the
    data in the trapezoidal-shaped region were repaired using data from a
    bias file for {\tt OBS\_ID} 3920. The median of each node has been
    subtracted from the bias values for the node.
    %
    \label{fig08}
    }
\end{figure}

\begin{figure}
  \begin{center}
    \epsfig{file=bias_repair_figs/fig09,angle=270,width=6.5in}
  \end{center}
  \caption{%
    Histograms of the pulse heights of the outer 16 pixels of the 5~pixel
    $\times$ 5~pixel event islands for {\tt OBS\_ID} 4111.  Only events in
    the trapezoidal-shaped region of ACIS-S2 are included. The red histogram
    is the distribution obtained using the data processed with the bias
    shown in Figure~\ref{fig07}. The peak of the red histogram is at
    $-2$~adu, instead of zero.  The black histogram is the distribution
    obtained using the bias shown in Figure~\ref{fig08}. The peak of the
    black histogram is at 0~adu, as desired.
    %
    \label{fig09}
    }
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 9. Finish

\end{document}

