degree per second, On-The-Fly (OTF) continuum single dish observations
of sources up to a few degrees across will usually not be limited by
atmospheric subtraction.
M.A. Holdaway, F.N. Owen, and Darrel T. Emerson
National Radio Astronomy Observatory
September 20, 1995
On the Chajnantor site, switching on time scales of 1 s will usually
increase the continuum total power noise over the optimum value by
about 50%. If switching can be done on time scales of 0.2 s, the
atmosphere will only rarely limit the noise. Using the ``fast
switching'' capabilities which appear to be required for phase
calibration, it should be possible to switch the primary by a few
arcminutes on 1 s time scales. However, with slew speeds of
degree per second, On-The-Fly (OTF) continuum single dish observations
of sources up to a few degrees across will usually not be limited by
atmospheric subtraction.
These calculations indicate that a nutating subreflector may not be required for the MMA if fast OTF mapping can be used for measuring the total power. Scanning faster than about 1 degree per second will not improve the atmospheric subtraction since the errors will then be dominated by the differing paths through the atmosphere at opposite ends of the extended target source. In order to accommodate 1 degree per second scanning, the correlator will need to record integration times as short as 0.003 s, which may be possible for total power data alone, but will likely not be possible for the visibility data. This will require that the total power and interferometric data be taken separately for many continuum observations.
Water vapor in the atmosphere, especially inhomogeneously distributed water vapor, is a bad thing for millimeter wavelength telescopes. Single dish observations are mainly affected by the opacity of the water vapor and by the variable emission from the inhomogeneously distributed water vapor. The problem of variable atmospheric emission has been solved to a large extent by the dual beam, or beam switching, observational technique of Emerson, Klein, and Haslam (1979). Two beams are formed on the sky, either by observing with two physical feeds simultaneously or by switching the sky position of the beam by chopping the secondary reflector or by actually changing the pointing of the telescope. It is assumed that the atmospheric emission in the ``off'' beam and the ``on'' beam are similar, so when the ``off'' power is subtracted from the ``on'' power, only the astronomical emission remains. Success of the atmospheric emission subtraction depends on how similar the atmosphere really is in the ``off'' and ``on'' beams, which depends upon how close the ``off'' and ``on'' columns are to each other in the atmosphere: beam switching works best when small angular beam throws and high switching frequencies are used. However, there has not been a good analysis of the switching speed and angular distance requirements for total power observations, primarily because the atmospheric fluctuations have not been well characterized at the sites of interest until recently.
For extended sources, both the ``on'' and ``off'' beams will fall on
the source of interest, so some astronomical emission is subtracted as
well as the atmospheric emission. Various deconvolution algorithms
can reconstruct the image using the information from the ``on'' and
``off'' beams (see Emerson, 1995 for a review). Because we must
difference many ``on'' regions to get to the edge of the source, the
signal to noise of a large reconstruction falls off like
, where 
is the number of beam throws
across the target source. Imaging very large regions of the sky in
single dish continuum at high sensitivity then becomes very
problematic.
Recently, David Woody has suggested that the MMA antennas might not
require a nutating subreflector if the MMA were built on a superior
site and the antennas were able to position switch a few arcminutes at
1 Hz. If the atmosphere is stable enough, On-The-Fly continuum
observing may be able to produce good images of very large regions
without the signal to noise degradation which the beam switching
technique suffers from. This memo performs an error analysis of the
beam switching and OTF single dish continuum observing techniques
using the atmospheric stability data from our Chilean site test
interferometer. We compare the observing techniques at a frequency of
230 GHz and address the question of whether a nutating subreflector is
required for the MMA.
The atmospheric emission as seen by a single dish will depend upon the opacity and the temperature of the atmosphere:
The rate of evolution of the atmosphere will be slow compared to the times of interest pertinent to the beam switching problem, so we adopt the frozen turbulence model in which all temporal fluctuations are assumed to result from spatial fluctuations blowing past the dish.
Spatial fluctuations in either the opacity 
or sky temperature
will cause fluctuations in the atmospheric emission received
by the single dish. We will find later on in this work that the
errors in the sky emission subtraction due to temporal and spatial
fluctuations of 
will be of the order of 0.01 K. Will the
temperature microstructure of the atmosphere be able to contribute a
comparable error? If so, then these calculations are a lower limit to
the atmospheric errors in continuum single dish observing. If the
turbulent water vapor is distributed uniformly through a 
1 km
atmosphere, isotropic temperature variations of 2 K over 10 m and an
opacity of 
will result in variations in the sky
brightness of about 0.01 K without any opacity fluctuations. During
some conditions, it appears that a narrow layer is responsible for the
phase fluctuations seen by the site test interferometer as the phase
structure function exponent is seen to be about 0.33. This just
requires that most of the water vapor fluctuations reside in a
thin layer. However, if a large fraction of the water vapor
also is in a thin layer, even modest temperature fluctuations will be
enough to dominate the sky brightness fluctuations. Measurements of
the atmospheric temperature structure function near the earth's
surface indicate that 0.5 K over 10 m is not uncommon (Tatarski,
1961). If the temperature microstructure decreases with height in the
atmosphere, temperature fluctuations will probably not dominate the
fluctuations in the sky brightness.
This is one of the issues involved in correcting for atmospheric phase errors using water vapor radiometry. Foster et al. (1995, in preparation) demonstrate that sky brightness fluctuations measured by the 225 GHz site test radiometer imply rms path length fluctuations which agree with the phase fluctuations measured by the 11.7 GHz site test interferometer on the Chajnantor site. This correlation between brightness fluctuations and path length fluctuations indicates that we are justified in scaling the phase fluctuations to estimate the magnitude of the sky emission fluctuations as seen by a single dish.
We assume the microstructure of the opacity dominates the sky brightness fluctuations and the microstructure of the physical temperature of the sky is insignificant. Then the errors made in the atmospheric emission subtraction are similar to the errors made in fast switching phase calibration:
where 
is the opacity structure function, v is the
atmospheric velocity, t is the switching cycle time, and d is the
typical distance between the centers of the ``on'' and ``off'' near
field columns. To use Equation 2 requires a model of the
height of the turbulent water vapor to determine d. If the
turbulent water vapor is primarily a surface phenomenon, then d is
essentially zero. For water vapor at 1 km height above the telescope
and a 1 arcminute beam throw, 
meters.
If the distance between the ``on'' and ``off'' lines of site is much larger than the distance the atmosphere moves in half a cycle time, as for very fast switching with a large beam throw or for fast OTF mapping of a large source, Equation 2 approaches
If the distance the atmosphere moves in half a cycle time is much larger that the distance between the ``on'' and ``off'' lines of site, as for slow switching with a small beam throw, Equation 2 approaches
We have techniques to estimate v from the site test data, but we can
bypass the need for knowledge of v entirely by expressing
4 in terms of the temporal structure function
, which can be derived from the phase time series:
Under the assumptions that the opacity fluctuations and the differential path length fluctuations are solely due to water vapor, the opacity structure function is simply related to the phase structure function. The opacity fluctuations are given by
where PWV is the precipitable water vapor in millimeters and
is assumed constant over the observed bandpass. For 230 GHz,
at the elevation of the Chajnantor, Chile site (Schwab, private
communication)1
If the path length fluctuations seen by our site test interferometer
are caused solely by inhomogeneously distributed water vapor,
(Thompson, Moran, and Swenson, 1986). Then the opacity structure function is related to the path length structure function, which is measured by the site test interferometer, as
Inserting this relationship into Equations 2 through 5, we can use the site test data to estimate the errors in atmospheric emission cancellation for different observing strategies. In order to convert the atmospheric noise from Kelvins to Janskies for the MMA antennas in total power, we multiply by 46.
The site test interferometer measures the rms path length fluctuations over a 300 m baseline, but the total power problem requires knowledge of fluctuations on baselines of 1-10 m. We produce a temporal structure function from the interferometer's phase time series (Holdaway et al. 1995), which provides information about the path length variations on time scales down to 1 s. Our site test data shows no evidence for deviations from a constant power law down to 1 s. However, at 1 s we are often affected by the interferometer's instrumental function, and it is not clear if we could see a change in the structure function. We will assume that the structure function which we derive on time scales between 1 s and the turnover time, typically 20-60 s, is representative of the structure function on time scales of 0.1 s to a few seconds.
We are now prepared to analyze these observing modes:
Since the beam throw distance will be limited by the optics in
beam switching, it will take multiple throws to reach the
zero-emission level when imaging a large source, increasing
the noise by 
. This becomes quite
problematic for very large sources.
. Since
position switching does not require any off-axis observing,
the beam shapes for the ``on'' and ``off'' will be more nearly
identical than in the beam switching case. The error analysis
for position switching is identical to beam switching, but with
lower switching frequencies.
At 230 GHz, the MMA dishes will have a 40 arcsecond beam, and the beam
throw might be 80 arcseconds. At 1 km height above the MMA, the
centers of the ``on'' and ``off'' near field columns will be about
0.4 m when observing overhead. Hence, with wind speeds of 10 m/s,
for cycle times of of 0.4 s or greater. By ignoring the
spatial term, we can perform our error analysis in terms of the
temporal structure function and completely ignore the issue of wind
velocity. Using the temporal structure functions derived from the
June 1995 site test data from Chajnantor near San Pedro in Chile and
Equation 5, we can derive the distribution of the
error in atmospheric cancellation for different switching frequencies.
Figure 1 illustrates the distributions of atmospheric
cancellation error for beam switching at 0.5, 1, 2, and 5 Hz. At 5 Hz,
the spatial term will be comparable to the temporal, so the errors
will likely be somewhat larger than the graph portrays. As the
switching frequency increases, we decrease the atmospheric cancellation
error and increase the system noise per cycle time2, both working against
limitation by the atmosphere. The atmosphere will limit the continuum
total power sensitivity about half the time when switching at 2 Hz,
and much less than half the time when switching at 5 Hz.
David Woody has suggested that position switching by several arcminutes at 1 Hz should be possible with the MMA antennas, and that this should be considered as an alternative to beam switching for single dish continuum observations. While Figure 1 indicates the MMA in single dish mode will not often reach its ultimate sensitivity with 1 Hz switching, it will often be close to its ultimate sensitivity. Hence, we should seriously consider position switching as a means for the MMA to remove atmospheric emission in continuum total power observations, eliminating the requirement for a nutating subreflector. Note that position switching will almost never limit the sensitivity for spectral line observations.
Since On-The-Fly mapping generally deals with large sources, certainly
much larger than the ~1 arcminute beam throw suggested in the
previous section, we must consider spatial variations in the
atmosphere's brightness temperature, but if the telescope slews fast
enough, we don't need to consider temporal fluctuations caused by the
atmosphere moving while we observed. This allows us to use
Equation 3 to calculate the errors in cancellation of
the atmospheric emission, which will depend only on source size. For
the case of OTF mapping, the distance d is given by the half the
distance between the lines of sight to opposite ends of the target
source, at the mean height of the atmospheric turbulence. It doesn't
make sense to slew so fast that 
. Equating the temporal
and the spatial terms, we find the slew rate at which the two terms
contribute equally:
degrees per second, where h is the typical height of the atmosphere
and 
is the elevation angle. Hence, a slew rate of one
degree per second for 1000 m water vapor scale height and 10 m/s wind
velocity will be fast enough so that the atmospheric cancellation error
is dominated by the distance between the lines of site to the two
different ends of the target source, rather than by the temporal
variation in the atmosphere over the observations.
Figure 2 indicates the distribution of error in
atmospheric cancellation for four different source sizes. Assuming a
slew speed of 1 degree/s for all sources, the integration time per
Nyquist sample will be 0.005 seconds, implying a system noise per
Nyquist sample of about 2.0 Jy, considering that the system noise in
the ``off'' signal will not increase the noise by 
because
of its higher SNR due to longer integration on the ``off'' position
during telescope reacceleration. Hence, very fast OTF mapping
will seldom be dominated by errors in atmospheric cancellation untill
observed sources are much larger than a degree. OTF mapping will
perform significantly better than beam switching or position switching
for small sources, though more time will be wasted off source. For
sources smaller than a degree across, a much slower scan rate will still
result in OTF observations which are not limited by the atmosphere.
These results have a large impact on several areas of the MMA:
better SNR on a
one degree source than traditional beam switching (49 beam throws
across the image degrades the traditional beam switching image SNR by 7,
and differencing from a very low noise off in OTF mapping improves the
OTF image SNR by 
).
These methods represent a major change in the way total power continuum measurements would be made. In order to test these ideas fully, we would need to build the MMA, as no other instrument is capable of performing these observations as quickly. Some atmospheric testing at slower slew rates and with conventional beam switching with existing telescopes, in concert with phase monitor observations, will undoubtedly add to our understanding. In addition to bolstering the basic theory, we will have to address a number of technical concerns:
numbers every 0.003 s?
We would like to thank David Woody for suggesting this work, and Michael Rupen and Phil Jewell for their insightful comments on this work. Also, we thank the entire MMA site testing group, especially Simon Radford, Scott Foster, and Jerry Petincin, without whom this work would not have been possible.
References
Emerson, D.T., 1995, ``Multi-beam Data Analysis'', in Multi-Feed Systems for Radio Telescopes, ed. Darrel T. Emerson and John M. Payne, Astronomical Society of the Pacific, San Fransisco.
Emerson, Klein, and Haslam, 1979. A&A, 76, 92.
Holdaway, M.A., Radford, Simon J.E. , Owen, F.N., and Foster, Scott M., 1995, MMA Memo 129, ``Data Processing for Site Test Interferometers''.
Tatarski, 1961. Wave Propagation in a Turbulent Medium, McGraw-Hill, New York.
Thompson, Moran, and Swenson, 1986, Interferometry and Synthesis in Radio Astronomy, John Wiley & Sons, New York.
Figure 1: Distribution of the atmospheric emission cancellation error
for beam switched continuum observations at 230 GHz for four
different switching frequencies, assuming the temporal
variations dominate the spatial variations. At 5 Hz, the
spatial and temporal fluctuations will be comparable, so this
approximation breaks down. Also listed on each curve is the
thermal noise obtained from 4 GHz total bandwidth (two stokes
from two 1 GHz IFs) per half cycle time. As the switching
frequency increases, the cancellation errors decrease and the
system noise per half cycle increases. In order for the
atmosphere not to limit the sensitivity, the atmospheric
errors must be less than the system noise.
Figure 2: Distribution of the atmospheric emission cancellation error
for On-The-Fly single dish continuum mapping at 230 GHz for
four different source sizes, assuming the spatial variations
dominate the temporal variations, as is met of the slewing
speed is greater than about 1 degrees/s. At a slew rate of 1
degree/s, the noise per beam is about 1.4 Jy, so even sources
as large as 4 degrees can often be mapped without being limited by
the atmosphere. Smaller sources will not require such fast slew
rates and short integrations.
. We have dropped
the quadratic term to simplify our analysis.
, losing one factor of 
by observing the source only half the time and another factor of
by differencing two noisy signals.
