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<H2><A NAME="SECTION003633000000000000000"> </A>
<A NAME="sec-Man-wlccal"> </A>
<BR>
Fitting the dispersion curve
</H2>
<P>
Now you have several possibilities to perform your wavelength
calibration. At first there are three different modes to identify
the calibration lines (
<!-- MATH: $\fbox{{\small \tt WLCMET(1)}}$ -->
<IMG
WIDTH="108" HEIGHT="29" ALIGN="BOTTOM" BORDER="0"
SRC="img1197.gif"
ALT="\fbox{{\small \tt WLCMET(1)}}">
(<B>F</B>)):
<DL>
<DT><STRONG>Identify:</STRONG>
<DD>You identify at least 2 arc lines in one slitlet
with the command <TT>IDENTIFY/MOS</TT>. The command <TT>CALIBRATE/MOS</TT> then
performs a first fit for the CCD row with the identified lines.
<P>
<DT><STRONG>Linear:</STRONG>
<DD>You know the central wavelength and the mean linear
dispersion of your grism. These values are used as first fit of the first
selected CCD row.
You have to correct the value of this central wavelength
if you used the command
<TT>OFFSET/MOS</TT> to determine the offsets of your slitlets:
<DL>
<DT><STRONG>Example:</STRONG>
<DD>Your reference slitlet has an offset of -100 relative to the
center of the CCD in x-direction and you have a mean dispersion of 2Å/pixel
and a central wavelength of 5500Å. This central wavelength will always
lie at the x-position of the respective slitlet, which is in this case
-100 pixels (i.e. -200Å) from the center of the CCD. This
means that you have a wavelength of 5700Å at the center of the CCD
within the reference slitlet. This wavelength should be used as <TT>wcenter</TT>,
since the program assumes that <TT>:xoffset = 0</TT> means that the slitlet is
at the center of the CCD in x-direction.
</DL><DT><STRONG>Recall:</STRONG>
<DD>Method Linear is performed in the first slitlet.
The dispersion coefficients of the this slitlet is recalled to calibrate the
remaining slitlets. This identification is more stable in this case than for
method Linear and the convergence of the fit is reached faster.
</DL>The first fit is used to identify as many lines as possible in the
corresponding CCD row by comparing the fitted line positions to wavelength
catalogue
<!-- MATH: $\fbox{{\small \tt LINECAT}}$ -->
<IMG
WIDTH="87" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1198.gif"
ALT="\fbox{{\small \tt LINECAT}}">.tbl (<B>hear</B>).
For the identified lines a polynomial fit of chosen order is
performed (using Legendre or Chebyshev polynomials - the type being selected
with the keyword
<!-- MATH: $\fbox{{\small \tt POLTYP}}$ -->
<IMG
WIDTH="77" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1199.gif"
ALT="\fbox{{\small \tt POLTYP}}">
(<B>CHEBYSHEV</B>)).
The line identification criterion will identify a computed wavelength
(<IMG
WIDTH="27" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img1200.gif"
ALT="$\lambda_c$">)
with a catalog wavelength (
<!-- MATH: $\lambda_{cat}$ -->
<IMG
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img1201.gif"
ALT="$\lambda_{cat}$">), if the residual
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\delta \lambda = |\lambda_c - \lambda_{cat}|
\end{displaymath} -->
<IMG
WIDTH="145" HEIGHT="40"
SRC="img1202.gif"
ALT="\begin{displaymath}\delta \lambda = \vert\lambda_c - \lambda_{cat}\vert \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
is small compared to the distances to the next neighbours (in the arc spectrum
as well as in the catalog):
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\delta \lambda < min(\delta \lambda_c, \delta \lambda_{cat})*\alpha
\end{displaymath} -->
<IMG
WIDTH="229" HEIGHT="40"
SRC="img1203.gif"
ALT="\begin{displaymath}\delta \lambda < min(\delta \lambda_c, \delta \lambda_{cat})*\alpha\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where
<!-- MATH: $\delta\lambda_{cat}$ -->
<IMG
WIDTH="53" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img1204.gif"
ALT="$\delta\lambda_{cat}$">
(
<!-- MATH: $\delta\lambda_c)$ -->
<IMG
WIDTH="45" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img1205.gif"
ALT="$\delta\lambda_c)$">
is the distance to the next
neighbour in the catalog (arc spectrum) and <IMG
WIDTH="20" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img1206.gif"
ALT="$\alpha$">
is the tolerance
parameter (0 ...0.5) given by
<!-- MATH: $\fbox{{\small \tt ALPHA}}$ -->
<IMG
WIDTH="66" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1207.gif"
ALT="\fbox{{\small \tt ALPHA}}">
(<B>0.2</B>).
The automatic line identification is repeated with this polynomial
fit in order to identify additional lines
to further improve the dispersion curve. <BLOCKQUOTE>
<DIV ALIGN="CENTER">
<B>Note</DIV>
<I>For very low dispersion spektroscopy one would expect that a linear
guess will cause line mismatchs at the edge of the detector. One can avoid
this, if more then two lines are identified with method Identify.</I></B></BLOCKQUOTE>After the polynomial fit the residual of each line are checked and the line
is thrown out, if it the residual exceeds the tolerance parameter
<!-- MATH: $\fbox{{\small \tt TOL}}$ -->
<IMG
WIDTH="45" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1208.gif"
ALT="\fbox{{\small \tt TOL}}">
(<B>2</B>) (> 0 - in pixels; < 0 - in units of the wavelength).
One of three fitting methods can be selected by keyword
<!-- MATH: $\fbox{{\small \tt WLCMET(2)}}$ -->
<IMG
WIDTH="108" HEIGHT="29" ALIGN="BOTTOM" BORDER="0"
SRC="img1209.gif"
ALT="\fbox{{\small \tt WLCMET(2)}}">
(<B>C</B>):
<DL>
<DT><STRONG>Constant fit in spatial direction:</STRONG>
<DD>the dispersion coeficiants are constant for the whole slitlet. This method is
typically appropriate for small slits.
<P>
<DT><STRONG>Variable fit in spatial direction:</STRONG>
<DD>bad lines are thrown out. Dispersion coeficiants are calculated for any row.
The dispersion relation of the first fitted row is used as estimate for all
following rows. A large number of arc-lines is required for this method.
If there are only a few lines identified at the edge of the detector, small
oscillations at the edge of the detector may occur in spatial direction.
<P>
<DT><STRONG>Two dimensional fit over the slitlet:</STRONG>
<DD>A two-dimensional fit is performed in spatial and dispersion direction over the
slitlet. In spatial direction a ``normal polynom'' is fitted but a polynom
as specified in keyword
<!-- MATH: $\fbox{{\small \tt POLTYP}}$ -->
<IMG
WIDTH="77" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1210.gif"
ALT="\fbox{{\small \tt POLTYP}}">
in dispersion direction.
The dispersion coeficiants may smoothly evolve over the slitlet.
This method is the most accurate for most applications, although the resulting
residuals are typically larger than for a variable fit.
<P>
</DL>
<P>
The iteration is repeated until
a stable solution is obtained (and the minimum number of iterations
<!-- MATH: $\fbox{{\small \tt WLCNITER(1)}}$ -->
<IMG
WIDTH="129" HEIGHT="29" ALIGN="BOTTOM" BORDER="0"
SRC="img1211.gif"
ALT="\fbox{{\small \tt WLCNITER(1)}}">
(<B>3</B>) is exceeded) or the maximum number of cycles
(
<!-- MATH: $\fbox{{\small \tt WLCNITER(2)}}$ -->
<IMG
WIDTH="129" HEIGHT="29" ALIGN="BOTTOM" BORDER="0"
SRC="img1212.gif"
ALT="\fbox{{\small \tt WLCNITER(2)}}">
(<B>20</B>)) is reached.
The resulting dispersion coefficients are stored in table
<!-- MATH: $\fbox{{\small \tt LINFIT}}$ -->
<IMG
WIDTH="77" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1213.gif"
ALT="\fbox{{\small \tt LINFIT}}">.tbl (<B>coerbr</B>),
together with the r.m.s. error of the fit, the slitlet
and the y-coordinates (world and pixel coordinates).
Also a plot option for the resulting residuals (
<!-- MATH: $\fbox{{\small \tt PLOTC}}$ -->
<IMG
WIDTH="67" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1214.gif"
ALT="\fbox{{\small \tt PLOTC}}">,
(<B>N</B>))
and various degrees of display (
<!-- MATH: $\fbox{{\small \tt DISP}}$ -->
<IMG
WIDTH="56" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1215.gif"
ALT="\fbox{{\small \tt DISP}}">,
(<B>0</B>))
are available.
<P>
After fitting all rows of the respective slitlet with polynomials
of the chosen order
the program performs at last a linear fit to get the central wavelength and the
mean linear dispersion necessary to derive a starting wavelength
for the next slitlet from its known offset (modes Linear/Recall). In the mode
Ident it tries to match the manually identified lines in the next slitlet,
using the known offsets and a maximum allowed shifting tolerance stored
in
<!-- MATH: $\fbox{{\small \tt SHIFTTOL}}$ -->
<IMG
WIDTH="98" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1216.gif"
ALT="\fbox{{\small \tt SHIFTTOL}}">
(<B>10</B>).
Rows where no fit could be achieved are stored in the table
<!-- MATH: $\fbox{{\small \tt LINFIT}}$ -->
<IMG
WIDTH="77" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1217.gif"
ALT="\fbox{{\small \tt LINFIT}}">.tbl
with the slit number -1.
<P>
Any selection of slitlets made in the table
<!-- MATH: $\fbox{{\small \tt LINPOS}}$ -->
<IMG
WIDTH="77" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1218.gif"
ALT="\fbox{{\small \tt LINPOS}}">.tbl will be taken
into account, but all selections of the table
<!-- MATH: $\fbox{{\small \tt MOS}}$ -->
<IMG
WIDTH="45" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1219.gif"
ALT="\fbox{{\small \tt MOS}}">.tbl will be ignored.
If you want those to be respected, too, redo the search for the wavelength
calibration lines with the chosen selection in
<!-- MATH: $\fbox{{\small \tt MOS}}$ -->
<IMG
WIDTH="45" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img1220.gif"
ALT="\fbox{{\small \tt MOS}}">.tbl.
<P>
After the wavelength calibration you may rebin your frame
two-dimensionally to constant wavelength steps with <TT>REBIN/MOS</TT>.
Point sources are normally wavelength calibrated after extraction (see below).
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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