[mmaimcal]Primary Beam Accuracy

[Mark Holdaway]

Dear Fellows,

Without a header to this pseudo-laTex file, you will just need
to use your innate ability to dcode laTex math and Tables.

Comments are welcome (I am basically on course, am I wrong...?
Take care,

	-Mark
-----------------------------------------------------------------



Primary Beam Accuracy	 (for addition to the ALMA Project Book)
=====================		draft Feb 26, MAH


Knowledge of the primary beam is essential for mosaicing.
Exactly how well do we need to know the primary beam?

The mosaicing algorithms take the dirty images from each field and
multiply each by the primary beam model, followed by a normalization:
\begin{equation}

I_{mosaic}(\bf{x}) = \frac{ \sum_p A_{p}(\bf{x} - \bf{x_p}) I_{p}(\bf{x})}
			  { \sum_p A_{p}(\bf{x} - \bf{x_p})^{2} },
\end{equation} where $p$ represents the index of the various
pointings, $A$ is the primary beam, and $I_{p}$ are dirty images from
each pointing.  The images $I_{p}$ represent the true sky multiplied
by the true primary beam (convolved with the PSF in the case of a
dirty image).  It is clear from this equation that our knowledge of
the primary beam must be more accurate where the primary beam is high,
and it can be less accurate where the primary beam is low.  For
example, an error at the half power point of the primary beam is not
so important because there will be another pointing which samples the
sky with the center of it's primary beam, where the error will be
negligible.

We have quantified this effect with numerical calculations.  We will
start with the assumption that we need 1\% image accuracy (ie, as in
the goal for the image fidelity).  We took an annulus of the primary
beam of width equal to 1/5 of the FWHM, and within that annulus, gave
an additive error to the primary beam model, formed the mosaic image
(without deconvolution), and looked at the RMS error in the final
image.  We now invert the problem and ask the question "How much primary
beam error in that annulus will result in a 1\% mosaic image error?" 
The results of this simple test are shown in Table~\ref{tab1}.

\begin{table}
\begin{center}
\begin{tabular}{c|lrr}
PB     & Additive	& Fractional  & Frac. Error \\
level  & Error		& Error	      & Goal    	\\  \hline

0.92	& 0.008		& 0.9\%	& 0.5\%	\\
0.81	& 0.009		& 1.1\%	& 0.5\%	\\
0.52	& 0.015		& 2.8\%	& 1.4\%	\\
0.25	& 0.036		& 14\%	& 7\%	\\
0.10	& 0.125		& 100+\% & 50\%	\\ \hline
\end{tabular}
\end{center}
\caption{How much of a primary beam error is permitted to result
in a 1\% error?  This table is good for relative error levels
among the different primary beam levels, but not for absolute
beam errors.  See the text for a discusion.}  \label{tab1}
\end{table}

This is a very simple minded test in that it piled all the errors into
the annulus around the radius in question.  The results indicated here
are fairly good in terms of what relative accuracy the beam
measurements must have as you go out with radius, but the overall
scaling of this required accuracy depends upon how systematic the beam
errors are.  For example, errors in all annuli at once (ie, across the
entire beam) would tend to increase the mosaic error.  However, not
all annuli would have the same sign of error, nor will the error in
the model primary beam be of the same sign all the way around a given
annulus.  If all annuli had errors of the same sign, that basically
means that the size of the primary beam has been poorly determined.
Hence, any realistic case will have errors of both positive and
negative signs.  With a giant wave of the hands, we could assume that
these two effects (underestimate of mosaic errors by restricting
ourselves to one of five annuli, and overestimating errors by picking
a systematic case which will not average down) will to first order
cancel out, making these beam error estimates good for ballpark
limits.  At most we are off by a factor of about 2, so a more
conservative approach would be to take the errors in Table~\ref{tab1}
and divide them by 2 and use that error profile as a goal for ALMA
beam measurement.  Notice that very large errors at the primary beam
edge can be tolerated.  This is especially true if Sault weighting
is used, downweighting the mosaic image at the edge of the 
mosaic sensitivity pattern.

Another important point is the integral of the beam.  Total power
measurements will be significantly in error if the beam integral is
incorrect.  This is very important for homogeneous array mosaicing
which relies upon deconvolving the total power data.  If an extra
large single dish or the ACA plus 12~m single dishes were used to
measure short spacings, this would not be such a strong constraint.
Accurately measuring the beam volume will also ensure that the
detailed errors in the primary beam will be less systematic and will
therefore average out more.  The peak of the primary beam is already
constrained to be 1.0, with any errors being converted to gain errors.

Another factor to consider in the study of primary beam accuracy is
the image dynamic range.  So far, we have just focused on how 
errors in the beam model affect the image fidelity.  Under our
guidelines for primary beam model errors, a given pixel will on average
be in error by about 1\%.  As adjacent pointings will disagree about
how bright a given pixel is, flux will be scattered like the PSF
sidelobes across the image, potentially limiting the dynamic range.
Typically, the level of the scattered flux is a factor of 10-100 lower
than the on-source pixel errors.  Hence, dynamic range limitations of
order 1000:1 - 10000:1 are expected.  Other errors, such as pointing,
will be more problematic than the errors caused by the inaccurate primary
beam model at the levels we are talking about.

In Summary

An accurate primary beam model with the correct integral is very important
for homogeneous array imaging.

Errors in the primary beam model of 0.5\% out to the 80\% power point,
1.4\% at the half power point, and 7\% at the 25\% power point should
ensure that inaccuracies in the primary beam model don't limit
mosaicing accuracy.