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\title{\bf ALMA Memo \#289.1\\
Atmospheric Noise in Single Dish Observations }
\medskip

\author{Melvyn Wright\\
Radio Astronomy Laboratory, University of California, Berkeley}
\date{29 February 2000}
\maketitle

\begin{abstract}

Atmospheric noise and pointing fluctuations will severely impact the
signal-to-noise and image fidelity of ALMA observations.  Single dish
observations are required for extended sources.  For wideband total
power observations, the maximum integration time $\sim$ 0.1 s in order
to keep atmospheric fluctuations less than the thermal noise. A gain
stability $\sim$ $10^{-4}$ is required.  For OTF mapping, the
noise is limited by the atmospheric fluctuations between the ON and OFF
positions.  For sources which are only a few beamwidths across,
OTF mapping is inefficient. The single dish capability of ALMA could be
improved by instrumenting a few antennas to scan the beam pattern,
either with  x-y nutating subreflectors, or better, with array
receivers.  A number of experiments are possible to check the
conclusions of atmospheric modeling. These could be done on existing
submillimeter telescopes such as the JCMT, and might impact the ALMA
design.

\end{abstract}

\section{Preamble}

The ALMA telescope will be the largest telescope at sub-millimeter
wavelengths, both for single dish and interferometer observations.
Whilst interferometer measurements are better suited for observations
of compact sources, single dish observations, and single dish
techniques, are needed for extended objects. This is especially true at
the highest frequencies where the primary beam width is so small.
Single dish observations are required to image the large scale
structure of many astronomical sources. Single dish observations
provide low spatial frequency data for mosaiced images, and may limit
the overall image fidelity (Holdaway \& Woody 1998).  In this memo we
discuss the errors in a wideband total power measurement.  Noise
contributions come from thermal noise in the radiometer and the
atmosphere, from gain variations through the radiometer and atmosphere,
and from atmospheric fluctuations both in position and in time. Ideally
we would like to maximize the signal-to-noise for both interferometer
and single dish observations using all the antennas efficiently.
Spectral line observations are easier than total power observations
because one may remove a spectral baseline caused by total power
fluctuations from the instrument and atmosphere, and because the
thermal noise is higher in the narrower bandwidth channels. Spectral
line observations may be limited by bandpass variations rather than
total power fluctuations.  Since the atmospheric noise varies at
different sites, we need to consider whether experience and tests done
at existing sites are relevant to Chajnantor.

\section{Error analysis}

We consider total power measurements with a single dish radiometer.
The measured total power, $p[K]$ =  $ g \times T[K]$, where $T[K]$
includes noise power from the radiometer and atmosphere, and
$g$ is the normalized gain though the system so that both $p$ and $T$
are expressed in units of temperature.  There are error terms,  $\delta
p$, given by $ g \times \delta T$ and  $\delta g \times T$.

Thermal noise from the radiometer and the atmosphere contributes an RMS
noise, $\delta T = T_{sys} / \sqrt(2B \times t)$, where $T_{sys}$ is
the system temperature, $B$ is the bandwidth, and $t$ is the
integration time. The zenith emission from water vapor is  $\sim$ 30 K at
100 GHz with 10 mm of precipitable water.  A similar value is obtained
for observations at 300 GHz with 1 mm of precipitable water, so we will
adopt a total system temperature of 100 K in a 1 GHz bandwidth as a
reference.

$\delta T = 2 mK \times T_{sys}/[100 K] / \sqrt(B[GHz] \times t[secs])$ ------ (1)

Gain variations give an error $\delta g$/$g \times$ the total noise power.
A gain stability of $ 10^{-4}$ in 0.1s with a $1/freq$ spectrum gives:

$\delta T = 1 mK \times T_{sys}/[100 K] \times \delta g/g [10^{-5}] \times t[sec]$ ------ (2)

An additional source of noise comes from atmospheric fluctuations as
discussed by Holdaway, Owen \& Emerson (1995).  We can estimate the
fluctuations in the atmospheric emission from the path length
variations measured with millimeter wavelength interferometers and from
satellite phase monitors.  At millimeter wavelengths, refractive index
fluctuations are expected to be dominated by water vapor variations.
We can calculate the atmospheric emission from atmospheric profiles.
For Hat Creek at 100 GHz a calculated value of 0.44 mK per micron of
path length is in good agreement with measured values 0.4 to 0.5
mK per micron.  For the VLA site and Chajnantor at 230 GHz we expect
$\sim$ 1.7 mK per micron of path length.

Consider the atmospheric path length variations for observations over
an angle $\theta$, and a time interval $\delta t$.  At a height, $h$,
an offset, $\theta$, between two positions in the sky gives an
displacement $d = h \times airmass \times \theta$.  There is
considerable evidence for fluctuations in atmospheric boundary layers,
e.g. cross correlation of water vapor radiometer fluctuations at Hat
Creek yields heights of 2 - 3 km.  An offset, $\theta$  = $1'$,
gives a displacement $d$ = 1 m over a path of 3.4 km.
 The fluctuations over a distance $d$ are given by the structure
function; the RMS scales as $\sim$ $d^{\beta/2}$, where $\beta/2$
$\sim$ 0.6 over distances of less than $\sim$ 100 m.  The median RMS
path length fluctuation, $\sigma$,  measured on a 100 m baseline at Hat
Creek in good weather is $\sim$ 290 microns. Scaling this using
$\beta/2$ = 0.6, the path length fluctuations over 1 m are
$\sim$ 18 microns giving $\delta T \sim$ 8 mK over 1 arcmin.

For Chajnantor at 230 GHz we expect

$\delta T = 1.7 mK \times \sigma[\mu] \times (\theta[arcmin] \times h[km] \times airmass / 340)^{\beta/2}$ ------ (3) 

where $\sigma$ is the RMS path length measured on a 100 m baseline.

The time interval, $\delta t$, between observations also contributes
atmospheric noise corresponding to a displacement $ v \times \delta t$,
where the velocity of atmospheric water vapor fluctuations, $v$
$\sim$ 10 m/s. Over a time interval $\sim$ 0.1 s, this contributes a
similar fluctuation in brightness temperature as a $1'$ angular
offset.

$\delta T = 1.7 mK  \times \sigma[\mu] \times (v[m/s] \times \delta t[sec] / 100)^{\beta/2}$ ------ (4)

For the ALMA site, the median seeing is 254 micron measured at 36
degree elevation on a 300 m baseline. Scaling this to zenith on a 100 m
baseline using $\beta/2$ = 0.6, we obtain a median $\sigma$ $\sim$ 80
micron.  There are seasonal and diurnal variations in both $\sigma$ and
$\beta$ (Sramek 1990, Olmi \& Downes 1992, Wright 1996). For example,
Sramek measured a median seeing for the VLA which scales to 150 microns
for winter nights and 400 micron for summer days, on a 100 m baseline
with $\beta/2$ = 0.6.  The median seeing at the ALMA site is about a
factor of two better than the VLA site in winter.  On spatial scales
smaller than the thickness of the turbulent region, $\beta$ should tend
towards the the value 5/3 expected for three dimensional, Kolmogorov
turbulence. This is observed on short baselines at Hat creek on
turbulent summer afternoons.  If $\beta$ is steeper, then the
extrapolations to short baselines are reduced.


\section{Possible calibration schemes}

Assuming that the above error terms add in quadrature, evidently we
must calibrate the atmospheric fluctuations on time scales $\sim$ 0.1
sec in order that atmospheric temporal fluctuations be less than the
thermal noise.  This is usually done by beam switching (using focal
plane choppers or nutating subreflectors), or by slewing the antenna
across the source using on-the-fly (OTF) observing.  In either case we
must have a gain stability of $\sim$ $ 10^{-4}$ between ON and OFF
positions in order to be limited by thermal noise.  Any calibration
must also be stable to better than $10^{-4}$.

\subsection{On-the-fly observing}

Sources larger than several beam widths are usually observed by
scanning the antenna across the source, establishing a reference at
either ends of the scan. Beam switching during the scan limits the
errors from spatial and temporal atmospheric fluctuations, but the
errors are propagated as $\sqrt(N_{throws})$ (see Holdaway, Owen \&
Emerson 1995).  If the sample rate is faster than $\sim$ 10 Hz, then
atmospheric fluctuations are smaller than thermal noise. In this case
the gain and atmospheric spatial fluctuations over the length of the
scan may become the limiting errors.  The OFF positions at the ends of
each scan across the source also suffer from atmospheric noise
fluctuations since they are observed at different times and
displacements from the ON positions.  The shortest time for a scan
across the source is determined by the antenna acceleration. In
practice this is likely to be 1-2 seconds resulting in significant
atmospheric noise in the OFF positions.  For a raster scan across an
extended source, there will also be atmospheric fluctuations between
the OFF positions for each scan.

\subsection{Beam switching}

Sources which are smaller than a few beamwidths across are not
efficiently observed using on-the-fly techniques.  These sources need
single dish observations, but the antenna switching time is about 10x
longer than the maximum time on source before the noise is
dominated by either atmospheric or gain fluctuations. In this case, one
might do better with beam switching.  The usual beam throws of a few
arcmin at $\sim$ 5 Hz suffice to keep atmospheric fluctuations smaller
than the thermal noise in a 10 GHz bandwidth.  Larger throws or slower
switching may increase the noise.

%Experiments could be made with existing submillimeter telescopes such
%as the JCMT to test these calculations.

\section{Taylor hypothesis}

Under the Taylor hypothesis, the bulk velocity of the turbulent pattern
is larger than the internal velocities, and the temporal and spatial
fluctuations are simply related through the bulk velocity.
For sufficiently high scanning speeds the turbulence may be considered
to be frozen.  An antenna scan at 1 degree/sec, or a beam scan of $6'$
at 10 Hz, corresponds to 17 m/s at a height of 1 km, so the Taylor
criteria is approximately satisfied for water vapor fluctuations less
than $\sim$ 10 m/s.

If the atmospheric turbulence is considered to be a spatially frozen
pattern which moves across the telescope then there is a preferred
direction in which beam switching or telescope scans will cancel out
noise from atmospheric fluctuations. In practice, the
atmospheric turbulence may be re-arranging itself with similar
velocities to any drift velocity. At Hat Creek, cross correlations of
interferometer phase, total power, or WVR fluctuations on different
antennas, have multiple peaks; i.e. there is not usually a unique
velocity which dominates the temporal fluctuations.  Taylor's
hypothesis may be more valid on large open sites such as for the VLA or
ALMA.  If a frozen turbulence screen is drifting past the antenna then
we should be able to obtain better sky subtraction by scanning the
antenna or subreflector in the direction of the drift velocity. This is
an experiment which should be tried.

\section{Atmospheric refraction}

Anomalous refraction can change the antenna pointing by more than the
beamwidth at submillimeter wavelengths. We could simply say that we
should not schedule single dish observing under poor conditions, but
this might be too restrictive. Atmospheric pointing errors will
severely limit the fidelity of mosaic images (Holdaway 1997), and
radiometric correction of anomalous refraction (Lamb \& Woody 1998) has
been proposed.  One can devise experiments to measure and
correct for antenna pointing errors induced by atmospheric refraction.
For example, Church \& Hills (1990) measured atmospheric pointing
errors at the JCMT by scanning the subreflector in a small circle.
Atmospheric pointing errors which persist over larger times and
distances can be measured by observing a nearby calibrator with the
same or adjacent antennas (analogs of rapid switching and paired
antenna phase correction). Pointing self-calibration may be possible
for mosaic observations (Holdaway 1997). Similarly, by scanning the
beam whilst scanning the antenna in a larger basket weave, one could
correct for pointing errors and knit together the rows and columns of a
mosaic.  A subreflector which allows nutation in both Az and El
provides great flexibility for optimizing single dish observing
techniques.  An orthogonal sinusoidal nutation in both Az and El gives
a circular motion.  A sinusoidal nutation has more gentle accelerations
than a beam switch, and so can be driven faster.  A nutation in the
direction of a frozen turbulence drift could be used to deconvolve the
sky brightness from the atmospheric emission.

\section{How many antennas are used for single dish observing?}

For mosaic observations, the single dish data fill a similar
area of the u-v plane as the shortest interferometer baselines.  If
there is no special equipment for single dish observations, then we can
use all the antennas.  If we assume that the atmosphere is uncorrelated
over each antenna, we can average the data, and need only a
fraction of the time for single dish observations to complement
interferometer mosaics. However systematic atmospheric pointing errors
may limit the image fidelity (Holdaway 1997).

Alternatively we could instrument some antennas to do single dish
observations in an optimum way whilst the rest of the array is used for
interferometer observations. This would be particularly useful for
imaging transient phenomena, such as comets.  With the usual
signal-to-noise penalty for single dish observations, this implies that
a few antennas are needed for single dish observations.

In addition to providing the missing spatial frequencies for
interferometric mosaics, the ALMA array will provide the largest
submillimeter wavelength telescope on a superb site.  To exploit this
potential it is desirable that a significant collecting area be
instrumented for the best possible single dish observations.  The use
of heterodyne or bolometer array receivers should be considered. As
well as speeding up the observations, both on-the-fly and beam switched
observations are improved using array receivers.  A heterodyne receiver
array can be electrically phased to steer the beam pattern much faster
than a nutating subreflector. Bolometer arrays installed on existing
telescopes should provide some valuable tests of atmospheric modeling
and single dish techniques which are relevant to the ALMA design.

\clearpage
{\bf References}

Carilli, C.L., \& Holdaway, M.A., 1999, Millimeter Array Memo 262

Church, S. \& Hills, R., 1990, IAU symp on Radio Astronomical Seeing, J.E.Baldwin \& Wang Shouguan (eds). page 75.

% Removal of atmospheric emission from total power observations.
Holdaway, M.A., Owen, F.N., \& Emerson, D.T., 1995, NRAO. Millimeter Array Memo 137

% anomalous refraction
Holdaway, M.A. \& Woody, D., 1998, NRAO. Millimeter Array Memo 223

% pointing errors with 8 12 and 15 m dishes.
Holdaway, M.A., 1997, NRAO. Millimeter Array Memo 178

% radiometric correction of anomalous refraction
Lamb, J.W. \& Woody, D., 1998, NRAO. Millimeter Array Memo 224

Olmi, L. \& Downes, D., 1992, A\&A 262, 634.

% Atmospheric phase stability at the VLA.
Sramek,R.A., 1990, in IAU symp on Radio Astronomical Seeing, J.E.Baldwin \& Wang Shouguan (eds). p.21.

Wright, M.C.H.,1996, PASP, 108, 520


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