%LaTeX2e <1996/12/01> patch level 1

\leftskip=0pt \parskip=8pt \parindent=4pt
\documentclass[11pt]{article}

\begin{document}
\parindent=0pt
\newcommand{\half}{1 \over 2}
\section*{Count Rate Modeling Strategy}

In the {\bf unpixelized} form of {\it Forward Fitting} we will 
compute a model count rate profile directly from the parameters
of the basic Gaussian sources.  To do this, we need to get the
following quantities of the {\it calibrated event list} for:

%\begin{enumerate}
\begin{itemize}
 \item   $\Phi $ =  phase at map center
 \item   $\tau$ =  gridtran 
 \item   $T = $  livetime
 \item   $\theta $ =  roll angle
 \item   $A $ =  modamp 
\end{itemize}
%\end{enumerate}


Each one of these is a vector of length equal to the number of 
timebins used for the map.
We run the simulation of  the {\it calibrated event list}
exactly {\bf once} for a point source at the center of the map, throw
away extraneous tags (such as the counts), and save
the above vectors for subsequent iterations.  \medskip

For each of the Gaussian sources, we have 6 free parameters:
\begin{itemize}
 \item  $a_0$ = Flux 
 \item  $a_1$ = Long axis width
 \item  $a_2$ = short axis width 
 \item  $a_3$ = Tilt angle
 \item  $a_4$ =  X offset 
 \item  $a_5$ =  Y offset
\end{itemize}


Then we use these and the {\it calibrated-eventlist} parameters
together to compute the model count rate profile for each Gaussian and
each subcollimator.  The count rates can be written as a function of
the {\it phase at map center} $\Phi$:

\begin{equation}
f_{mod} = a_0 \tau (1 + c_1\ cos\Phi_{m} \quad+ {\rm higher\ harmonics} )
\end{equation}

The phase term $\Phi_{m}$ is the sum of the {\it phase at map center}
($\Phi$) and the phase offset of a point source at $a_4, a_5$
relative to map center:


\begin{equation}
\Phi_{m} = \Phi + 2\pi\ (a_4\ cos\theta\ +\ a_5\ sin\theta)/p,
\end{equation}

where $p$ is the angular pitch of the particular subcollimator.
In practice,  $\Phi$ need be computed only once, while the
offset terms change during the iteration and must be recomputed for
each trial source position.  The fundamental amplitude term
$c_1$ for the Gaussian source is computed from {\it modamp} ($A$)
and $a_1, a_2$ and $a_3$ using:

\begin{equation}
 c_1=  A \cdot e^{-\pi/p)^2\{cos^2(\theta-a_3)a_1^2+sin^2(\theta-a_3)a_2^2\}},
\end{equation}

At each step of the iterative search for a model which matches
the data, we compute
the model count rates $f_{mod}$ for each Gaussian using
equation (1), and add them up
to produce the total count rate.  This is done for each 
subcollimator separately.
\medskip

\subsection*{Questions}

In applying this to a real program, several questions arise:

\begin{itemize}
 \item How do we keep $f_{mod}$ positive at large radii? (3 harmonics?)
 \item Is there a way to use $c_1$ alone and not $f_{mod}$?
 \item How do we handle low count rates? 
  \begin{itemize}
     \item Model-Ordered Statistics?
     \item Cash Statistic?
     \item Regularization?
  \end{itemize} 
 \item What do we do about extraneous local minima? (Simulated annealing?)
\end{itemize}


\end{document}
\end
