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\slugcomment{ALMA memo 450 - 21mar2003}

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\title{Heterogeneous Imaging with the ALMA Compact Array}
\medskip

\author{M. C. H. Wright}
\affil{Radio Astronomy laboratory, University of California, Berkeley,
    CA, 94720}

\begin{abstract}

The ALMA Compact Array will be a powerful telescope at sub-millimeter
wavelengths for imaging the large scale structures which are not well
sampled by ALMA.  In this memo, we propose an observing mode for the
ACA as a heterogeneous array of 7 and 12~m antennas. We compare three
compact configurations:  the ACA (12 x 7~m),  a heterogeneous ACA with
4 x 12~m antennas, and the most compact ALMA configurations with 60 x
12~m antennas.

We present tables of the synthesised beam FWHM, brightness sensitivity
and sidelobe levels for source declinations +30, 0 and --30 degrees.
Using the ACA as a heterogeneous array with four 12~m antennas doubles
the collecting area of the ACA and provides synthesised beams and
brightness sensitivities which fill a useful parameter space between
the ACA alone and the most compact configurations of the ALMA
telescope.  Heterogeneous observing provides more short spacings and a
good cross calibration of the 7 and 12~m antennas.

Calibration of the bandpass and polarization characteristics can be
improved by scheduling a few more 12~m antennas for the ACA at times
when these calibrations are needed. The location of the ACA in
relation to the ALMA array, and the correlator configuration should be
considered with these possibilities.


\end{abstract}

\section{Introduction}

The ALMA Compact Array (ACA) is being designed to provide short spacing
and single dish data for the ALMA array. The current plan is for twelve
7~m antennas to provide short spacing interferometer data, and four 12~m
antennas to provide single dish data and additional sensitivity for
calibrating the 7~m interferometer array.  In this memo we explore the
imaging properties of the ACA using the 7 and 12~m antennas together in
a heterogeneous array.  This heterogeneous array is very similar to
compact configurations for the CARMA array with nine 6.1 m and six 10.4 m
antennas. The ACA 7 m antenna diameters are 15.4\% larger than the 6.1 m
antennas, and the ALMA 12 m antenna diameters are 14.8\% larger than the
10.4 m antennas.  Thus, we expect that the imaging properties of the ACA
heterogeneous array are very similar to those of the CARMA array with
an observing frequency 15\% higher, or a source size 15\% smaller.
The imaging properties of the CARMA array are discussed in BIMA memo 91.

In this memo we present simulations for the ACA using simple PYTHON
scripts to control MIRIAD tasks.  The scripts are similar to those used
in ALMA memo 428 simulations, but are slightly more complex because of
the different antenna sizes used in the heterogeneous ACA telescope,
and use more flexible PYTHON scripts instead of CSH.

\section{Simulations}

For this memo, we generate uv-data for a point source with thermal and
atmospheric phase noise, do a phase calibration, and make an image
and beam. A Gaussian fit is made to the synthesised beam, and the
results written into a table.  The brightness sensitivity, beam FWHM,
and residual sidelobe level after the fit are calculated. Timing
information for each step is given. The script takes $\sim$ 10s run
time on a PC or laptop.

Figure 1 shows a trial ACA configuration with 12 x 7 m antennas and 4 x
12 m antennas.  The 7 m antennas are arranged in two nested Reuleaux
triangles (see Keto, 1997). 
The 12 m antennas are placed asymetrically outside the Reuleaux
triangles to give an approximately Gaussian $uv$ sampling with as
many short spacings as possible without excessive synthesised beam
sidelobes or antenna shadowing. This configuration was
optimized using Boone's code (Boone 2001) with a target Gaussian FWHM=40m
for the $uv$ distribution. Other configurations should be explored.
%Two alternative starting points for ACA configurations are, 
For example,
a hexagonal close packed configuration offers the most short spacings;
the high sidelobes can be reduced by randomizing and optimizing.
Logarithmic spirals offer a central high density of $uv$
samples with an approximately Gaussian $uv$ distribution providing good
beam sidelobes.  For comparison,
some logarithmic spiral and ring configurations were evaluated for the
15-antenna CARMA telescope in BIMA memo 91.

The simulations exclude all antenna shadowing which was calculated from
the projected $uv$ spacing. In a heterogeneous array with different
antenna structures, shadowing depends on the height and relative
placement of the 7 and 12 m antennas and cannot be calculated simply
from the projected $uv$ spacing.  In practice, flagging the shadowed
data is probably best done on-line from a model of the antenna
structure and configuration.
Each configuration was sampled from HA --2 to +2 hours, with an
elevation limit of 10 degrees.  We used natural weighting of the
uv-data with a double sideband receiver temperature 80 K, an
atmospheric model with zenith opacity 0.08 at 230 GHz, and a bandwidth
8 GHz. Cross correlations are made between all antennas.
Assuming an aperture efficiency of 60\%, we used antenna gains of 40,
120, and 70 Jy per K for the 12, 7, and cross correlations between 12
and 7 m antennas respectively. The weighting and sensitivities were
calculated from the resulting system temperatures and antenna gains.
Placing the larger antennas at the longest radii in the array
configurations provides a more uniform sensitivity in the $uv$ data,
and reduces the required $uv$ data sample rate ($\delta uv =
D/2\lambda$, where D is the antenna diameter).

\section{Results}

We compare 4 compact configurations: 
(1) the ACA (12 x 7 m antennas), 
(2) the heterogeneous ACA with 4 x 12 m antennas, 
(3) the ALMA compact configurations 1 and 2 with all $uv$ spacings, and
(4) the ALMA compact configurations with $uv$ spacings tapered to a
FWHM=40~m and a maximum spacing of 48m to match the ACA heterogeneous configuration.

Table 1 lists the synthesized beam FWHM, sensitivity $RMS[\mu Jy]$ and
$T_b rms[\mu K]$, sidelobes levels (\% RMS, Max \& Min), and fraction
of the data remaining after shadowed data are omitted, Nvis[\%], for
each configuration and source declinations +30, 0 and --30 $\deg$.  The
heterogeneous ACA with an 8 hour track is also listed to show the
effects of larger hour angles (the last 3 rows in Table 1).

\subsection{Sensitivity}

The collecting area of the heterogeneous ACA with 4 x 12 m antennas
$\sim$ 2x that of the ACA alone (12 x 7 m antennas) as seen in the RMS point
source sensitivity.  
The synthesized beam area $\sim$ 2x that of the
ACA alone, so the brightness sensitivity is about the same, 
even though the resolution is increased.  
For comparison, the most compact
configurations of the ALMA telescope give about the same brightness
sensitivity at $\sim$ 4x the resolution.
When convolved to the $\sim 4.5''$ resolution of the heterogeneous ACA,
the ALMA configurations 1 and 2 have about 4x the brightness sensitivity,
and use $\sim$ 25\% of the $uv$ data.
If one is interested in source structures larger than $\lambda/D$, where
D is the antenna diameter, then 
the ACA, and the heterogeneous ACA with
4 x 12~m antennas nicely extend the range of spatial frequencies sampled
by interferometer measurements with about the same brightness
sensitivity.

\subsection{$uv$ Sampling}

Figures 2--4 show the $uv$ coverage for a heterogeneous with 12 x 7~m
and 4 x 12~m antennas for declination  +30, 0, and --30 degrees. The
$uv$ tracks from the correlations of the different sized antennas are
intermixed, allowing good cross calibration of the 7 and 12~m antennas.
The $uv$ coverage is reasonably complete from the 7~m central hole to
about 25~m, with an approximately Gaussian distribution with a FWHM =
40~m. The sampling shown is for HA from --2 to +2 hours.  Extending the
HA coverage from --4 to + 4 hours give better $uv$ coverage, in
particular completing the uvtracks down to the 7~m antenna shadowing
from all directions, but suffers from more antenna shadowing and higher
system temperatures at low elevations (see Table 1). 
More closely packed antenna configurations may give more short spacings,
but suffer from more antenna shadowing, and higher sidelobes.

\subsection{Calibration}

Table 2 gives the sensitivity per baseline for each combination of 7
and 12~m antennas assuming 60\% aperture efficiency, Tsys=200 K, and 1
minute integration. The table lists the sensitivity per baseline for
calibrating the antenna gains (amplitude and phase) in an 8 GHz
bandwidth, the bandpass with a 1 MHz spectral resolution, and the
brightness sensitivity at $5''$ angular resolution at 230 GHz.  For
antenna based calibration, the RMS is reduced by the {\it sqrt} of the
number of reference antennas (i.e. a factor $\sim 2-4$).  The
calibration sensitivity is greatly enhanced by using correlations with
12~m antennas. This is especially useful for calibration of the
bandpass and instrumental polarization for the 7 m antennas at
sub-millimeter wavelengths where strong calibrators are hard to find.

Usually, the bandpass and polarization calibrations are slowly time
variable and need only be made at infrequent intervals. The signal to
noise can be greatly improved by allocating more 12~m antennas to the
ACA array for these calibrations.  This can readily be done within the
framework of dynamic scheduling.  The ACA can be thought of as a
sub-array with special characteristics (mixed antenna sizes, perhaps
with nutating subreflectors on some of the antennas to enhance the
single dish observing). More 12 m antennas can be allocated to the ACA
for bandpass and polarization calibration observations. Antennas which
are closer to the ACA may suffer from less atmospheric phase
fluctuations, so the location of the ACA in relation to the ALMA array,
and the requirements for the correlator should be considered with these
possibilities.


\section{Conclusions}

A heterogeneous observing mode for ACA has $\sim$ twice the sensitivity
of the ACA alone, and will be a powerful telescope at sub-millimeter
wavelengths for imaging the large scale structures which are not well
sampled by ALMA. If one is only interested in sensitivity at $uv$
spacings greater than $\sim$ 12~m, then the ALMA configurations with 60
x 12~m antennas clearly provide much greater sensitivity. The design of
the ACA configuration is a compromise between antenna shadowing and the
most short spacings.  Heterogeneous observing provides a useful mode
for ACA observations.  ALMA is much more sensitive, so the focus should
be on getting short spacings to complement ALMA. Heterogeneous
observing provides more short spacings and a good cross calibration of
the 7 and 12~m antennas.  The signal to noise for bandpass and
polarization calibrations can be greatly improved by allocating more
12~ m antennas to the ACA array for these calibrations.  This can
readily be done within the framework of dynamic scheduling. The
location of the ACA in relation to the ALMA array,  and the
configuration for the correlator should be considered with these
possibilities.

\section{Acknowledgements}

I thank Peter Teuben for converting the CSH scripts to PYTHON scripts
including writing a number of python procedures which can be reused. 
Thanks also to Frederic Boone for providing his configuration optimization
code, and help with its use for these compact configurations.


\section{References}

"Interferometric array design: Optimizing the locations of the antenna pads", Frederic Boone, 2001, 
A\&A 377, 368.

"The shapes of cross-correlation Interferometers", Eric Keto, 1997, ApJ, 475, 843

"Miriad Users Guide", 1999, Bob Sault \& Neil Killeen, http://www.atnf.csiro.au/computing/software/miriad
 
"Compact Configuration Evaluation for CARMA", Melvyn Wright, BIMA memo 91, http://bima.astro.umd.edu/memo/memo.html

"Compact Configuration Evaluation", Melvyn Wright, ALMA memo 428, http://www.alma.nrao.edu/memos/index.html
 
\clearpage

\begin{table}
\tablenum{1}
\centering
\caption{Compact Configurations}
\smallskip 
\begin{tabular}{ccccccccc}  
\hline   
\hline 
Config  &  DEC & $RMS[\mu Jy]$& Beam [arcsec] & $T_b rms[\mu K]$ & Sidelobe[\%]:RMS & Max & Min & Nvis[\%] \\ \hline

%ACA only with harange=-2,2,.01
 
ACA & -30 &   95 & 6.34 x 5.02 & 69 & 2.3 & 15.3 & -10.3 & 100 \\
ACA &   0 &   99 & 6.60 x 5.21 & 67 & 3.6 & 43.3 & -14.3 & 99 \\
ACA &  30 &  174 & 9.02 x 4.97 & 90 & 3.2 & 21.2 & -12.9 & 50 \\ \hline

% heterogeneous ACA with harange=-2,2,.01
 
ACA+4 & -30 & 48 & 4.41 x 4.11 & 61 & 2.1 & 15.7 &  -9.5 &  100 \\
ACA+4 &   0 & 50 & 4.68 x 4.20 & 59 & 3.3 & 32.9 & -14.0 &  99 \\
ACA+4 &  30 & 75 & 6.72 x 4.01 & 64 & 2.6 & 18.4 & -11.5 &  60 \\ \hline


% ALMA configurations with harange=-2,2,.01

config1 & -30 &    6 & 1.63 x 1.47 & 58 & 0.5 & 2.6 & -5.0 & 100 \\
config1 &   0 &    6 & 1.64 x 1.59 & 53 & 0.7 & 6.1 & -4.6 & 100 \\
config2 &  30 &    8 & 1.61 x 1.55 & 74 & 0.6 & 4.1 & -4.3 & 89 \\ \hline

% ALMA configurations with harange=-2,2,.01
% 161 ns = 48.3 m max spacing FWHM=40m == 4.5" at 230 GHz
% set select = '-shadow(12),uvnrange(0,161)'
 
config1 $4.5''$ &  -30 &  15 & 4.47 x 4.40 & 18 & 1.5 & 10.6 & -13.0 & 23 \\
config1 $4.5''$ &    0 &  15 & 4.52 x 4.48 & 17 & 1.8 & 12.6 & -12.2 & 25 \\
config2 $4.5''$ &   30 &  20 & 4.50 x 4.40 & 23 & 1.3 & 13.1 & -12.2 & 22 \\ \hline
 
% heterogeneous ACA with harange=-4,4,.01

ACA+4 8h & -30 & 37 & 4.65 x 4.20 & 44 & 1.5 & 11.2 &  -8.9 &  91 \\
ACA+4 8h &   0 & 40 & 4.72 x 4.37 & 45 & 2.7 & 32.6 & -12.0 &  79 \\
ACA+4 8h &  30 & 67 & 6.39 x 4.19 & 58 & 2.3 & 14.2 & -12.0 &  39 \\ \hline

\end{tabular}
\raggedright\smallskip
\end{table}


\begin{table}
\tablenum{2}
\centering
\caption{Sensitivity per baseline}
\smallskip
\begin{tabular}{ccccccccc}
\hline
\hline
Antennas & Equivalent diameter & JyperK & RMS (8 GHz) & RMS in 1 MHz & RMS in 5'' \\ \hline
m x m     &  m   &   Jy/K &  [mJy] & [Jy]  & [K] \\ \hline
12 x 12   & 12   &    41  &    9   & 0.8   & 0.8 \\
12 x  7   & 9.2  &    70  &   16   & 1.5   & 1.3 \\
7  x  7   & 7    &   120  &   28   & 2.5   & 2.3 \\ \hline
\end{tabular}
\raggedright\smallskip
\end{table}

\clearpage

\begin{figure}
\psfig{figure=ant_positions.cps,angle=-90,width=6.0in}
\caption{Heterogeneous ACA configurations. 
The larger symbols show the positions of the 12 m antennas.}
\end{figure}

\begin{figure}
\psfig{figure=opt5.30.4hr.cps,angle=-90,width=6.0in}
\caption{$uv$ coverage for heterogeneous Alma Compact Array with 12 x 7~m and 4 x 12~m antennas at declination +30 deg.
The $uv$ data were sampled from HA --2 to +2 hours. 12 m x 12 m correlations in red, 7~m x 7~m in black, and 7~m x 12~m in blue.}
\end{figure}

\begin{figure}
\psfig{figure=opt5.0.4hr.cps,angle=-90,width=6.0in}
\caption{$uv$ coverage for heterogeneous Alma Compact Array with 12 7~m and 4 12~m antennas at declination 0 deg.
The $uv$ data were sampled from HA --2 to +2 hours. 12~m x 12~m correlations in red, 7~m x 7~m in black, and 7~m x 12~m in blue.}
\end{figure}

\begin{figure}
\psfig{figure=opt5.-30.4hr.cps,angle=-90,width=6.0in}
\caption{$uv$ coverage for heterogeneous Alma Compact Array with 12 7~m and 4 12~m antennas at declination --30 deg.
The $uv$ data were sampled from HA --2 to +2 hours. 12 m x 12 m correlations in red, 7 m x 7 m in black, and 7 m x 12 m in blue.}
\end{figure}

\end{document}

\clearpage

\begin{figure}
\psfig{figure=/falcon/wright/carma/config/mosaicing/spiral5.30.cas.0.05.imcat.ps,angle=-90,width=4.0in}
\caption{Maximum Entropy Image [upper panel] and Cas A model [lower panel]
scaled to $32''$ diameter, imaged with CARMA compact configuration (spiral5)
 at declination +30 deg. Contours at 12.5\%.}
\end{figure}

\begin{figure}
\psfig{figure=/falcon/wright/carma/config/mosaicing/spiral5.30.cas.0.05.resid.ps,angle=-90,width=4.0in}
\caption{Residual Image for spiral5 configuration at declination +30 deg.
using joint deconvolution of heterogeneous array and 12 m single dish data.
Contours at 0.2\% of Maximum Entropy Image.}
\end{figure}



\begin{table}
\tablenum{1}
\centering
\caption{Primary Beam at 230 GHz}
\smallskip
\begin{tabular}{ccccccccc}
\hline
\hline
Antennas & Equivalent diameter & FWHM & Nyquist interval \\ \hline
m x m     &  m   &   arcsec  &  arcsec \\ \hline
12 x 12   & 12   &   24      &  10.9   \\
12 x  7   & 9.2  &   31      &         \\  
7  x  7   & 7    &   41      &  18.6   \\
\end{tabular}
\raggedright\smallskip
\end{table}

