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<H3><A NAME="SECTION00841200000000000000">
FIT/ROMAFOT</A>
</H3>
<TT>FIT/ROMAFOT</TT>
<FONT SIZE="-1"><TT>frame [int_tab] [thres,sky] [sig,sat,tol,iter]
[meth[,beta]] <BR>[fit_opt] [mean_opt]</TT></FONT>
<P>
This command determines the characteristics (position, width, height)
of each selected stellar object through a non-linear best fit to the
data. It assumes that a Gaussian or a Moffat function is adequate
to describe the PSF and that a (possibly tilted) plane is a good
approximation of the sky background.
<P>
This command can be used for many purposes. For instance, the shape
of the object can be determined by performing a best fit with all
parameters allowed to vary; alternatively, a complex object
(e.g. a blend of ten or more objects) can be reconstructed using
some a priori knowledge, such as the width of the PSF or the positions.
In the first case, an object with an informative content which is as
high as possible is necessary to settle the parameters involved; in the
latter case this information is added to the data having a low
degree of information.
<P>
Experience has shown that a Moffat function with appropriate parameters
is always able to follow the actual profile of the data satisfactorily;
a Gaussian is adequate in case of poor seeing. In general the fitting
function can be described by the expression:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
F(a_{i},p_{i})\,=\,a_{1}x+a_{2}y+a_{3}+\sum_{k=1}^{36}f_{k}(x,y,p_{i,k}),
\end{displaymath} -->
<IMG
WIDTH="435" HEIGHT="74"
SRC="img190.gif"
ALT="\begin{displaymath}F(a_{i},p_{i})\,=\,a_{1}x+a_{2}y+a_{3}+\sum_{k=1}^{36}f_{k}(x,y,p_{i,k}),
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where the <I>a</I><SUB><I>i</I></SUB>'s are the sky background coefficients, the <I>p</I><SUB><I>i</I></SUB>'s
the parameters of the <I>k</I> elementary components and the <I>f</I><SUB><I>k</I></SUB> given by:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
f_{k}\Longrightarrow
\left\{ \begin{array}{ll}
I=p_{1}\exp -4 \ln 2\{{(x-p_{2})^{2}+(y-p_{3})^2\over
p_{4}^{2}}\}, & \mbox{Gaussian} \\
I=p_{1}\{1+{(x-p_{2})^{2}+(y-p_{3})^2\over
p_{4}^2}\}^{-\beta}, & \mbox{Moffat}
\end{array}
\right.
\end{displaymath} -->
<IMG
WIDTH="533" HEIGHT="77"
SRC="img191.gif"
ALT="\begin{displaymath}f_{k}\Longrightarrow
\left\{ \begin{array}{ll}
I=p_{1}\exp ...
...r
p_{4}^2}\}^{-\beta}, & \mbox{Moffat}
\end{array} \right.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
As has be mentioned in the Introduction, in the both expressions above,
the <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img192.gif"
ALT="$\sigma$">
is <B>NOT</B> the sigma in the statistical sense (the standard
deviation). For the Gaussian function <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img193.gif"
ALT="$\sigma$">
refers to the Full Width
Half Maximum (FWHM) of the distribution; in case of the Moffet distribution,
<IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img194.gif"
ALT="$\sigma$">
is a function of the parameter <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img195.gif"
ALT="$\beta$">.
Suppose at the beginning of the session some isolated objects have been
selected in order to derive the PSF. The number of components per window,
<I>k</I>, is set to 1 and the command runs with
<!-- MATH: $p_{1}, p_{2}, p_{3}, p_{4}$ -->
<I>p</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB>, <I>p</I><SUB>4</SUB>
and <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img196.gif"
ALT="$\beta$">
all allowed to vary. However, experience shows that <I>p</I><SUB>4</SUB> (hereafter often referred to as <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img197.gif"
ALT="$\sigma$">)
and <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img198.gif"
ALT="$\beta$">
are not totally
independent. Therefore, it is preferable to fix <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img199.gif"
ALT="$\beta$">
at the typical
value of <IMG
WIDTH="14" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img200.gif"
ALT="$\beta=4$">,
to derive the corresponding <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img201.gif"
ALT="$\sigma$">
and to check
the quality of the fit interactively. If the fit is unsatisfactory,
change <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img202.gif"
ALT="$\beta$">
and derive the new <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img203.gif"
ALT="$\sigma$">.
Since profiles are not
a strong function of <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img204.gif"
ALT="$\beta$">
the parameter can be changed by a couple
of units. Remember that if the fitted profile is wider than
the object, <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img205.gif"
ALT="$\beta$">
should increase and vice versa.
Typically, <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img206.gif"
ALT="$\beta$">
must be kept greater than 1 and it should not exceed
10. Naturally, these considerations do not apply if the Gaussian
function is used. However, in case of stellar photometry the use of the
Moffat function is strongly recommended.
<P>
Besides the best fit, <TT>FIT/ROMAFOT</TT> also computes the quality of
the fit by the <IMG
WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img207.gif"
ALT="$\chi^{2}$">
test and the <I>semi-interquartile interval</I>
for each individual object. These data are stored together with the fit
parameters and will be used by other commands (see <TT>EXAMINE/ROMAFOT</TT>).
<P>
During the execution the user will realise that the command occasionally
makes several trials on the same window. This happens when the command
is requested to fit a window with several objects and when one or more
of these falls into the category ``NO CONVERGENCY''. In this case the
command continues by ignoring such objects. When finally the convergency
is found, the objects so far ignored are added and a new trial will start.
The program will never delete objects on its own. The only exception
is if an object falls under the threshold selected by the user,
and after the background has been properly calculated considering, for
instance, ``tails'' of nearby stars.
<P>
It should be emphasised that, even if the window is marked ``NO
CONVERGENCY'', some objects in that window (in general the most luminous
ones) have been found with adequate convergency. These objects will
be flagged ``1'', while the objects flagged ``3'', ``4'' or ``5''
will be those responsible for ``NO CONVERGENCY''. Finally, objects
flagged ``0'' are those under the photometry threshold. These flags
should not worry the user; they will be used by subsequent commands.
Now, the trial values contained in the intermediate table have been
substituted by the result of fit in each record. To check their quality
interactively in order to define the PSF, the user should execute the
next command.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
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