O. P. Lay
Radio Astronomy Laboratory, University of California, Berkeley
olay@astro.berkeley.edu
Radiometry can be used to correct the wavefront distortions of millimeter and submillimeter signals introduced by the irregular distribution of water vapor in the Earth's troposphere. By measuring the emission due to water vapor along a line of sight it is possible to estimate the corresponding electrical path length. A correction system based on the strong water vapor transition at 183 GHz is considered.
Simulations of the line profile show that this transition is at least
partially saturated under even the driest conditions expected at the
Chajnantor site in Chile. Potential errors in the radiometric
correction process, including gain fluctuations, uncertainty in the
altitude of the water vapor fluctuations, clouds, temperature
fluctuations and spillover are analyzed. It is shown that the line
saturation is actually beneficial, as it allows the spectral signature
of a water vapor fluctuation to be discriminated from the various
error components. Three or more channels are needed for this, but only
moderate gain stability (
%) is required, greatly easing the
level of calibration necessary. Correcting the path to 50 
m is
much easier to achieve when the column of water vapor is low (1 mm of
Precipitable Water Vapor) than when it is high (PWV > 4 mm); the
latter case requires a cooled radiometer to achieve the necessary
sensitivity.
Various practical implementations are also considered, including the relative benefits of cooled and uncooled systems, and the use of dedicated radiometers versus the astronomical receivers for measuring the water line.
Fluctuations in the electrical path length through the atmosphere limit the spatial resolution and sensitivity that can be achieved by millimeter- and submillimeter-wave arrays. For the MMA to realize its full potential, it is vital that these fluctuations can be corrected.
The fluctuations are due predominantly to water vapor, which is poorly
mixed in the atmosphere and has a high refractive index - both a
result of the polar nature of the water molecule. The general
distibution of water vapor falls off exponentially with altitude, with
a scale height of 
km (compared to 8 km for the dry air
component), although it is likely that this varies with convective
activity in the troposphere. The precipitable water vapor (PWV) column
at the Chajnantor site in Chile, measured with a 225 GHz tipping
radiometer, varies from 0.5 mm under the very best conditions to over
4 mm. These are for zenith and must be scaled by the appropriate
airmass at other elevations. A water vapor column of 8 mm is
certainly possible for observations at low elevations, and will be
adopted here as the extreme case.
Each 1 mm of precipitable water vapor increases the electrical path
length through the atmosphere by about 6 mm, although the exact
conversion depends on the temperature and pressure of the water vapor.
Following Carilli, Lay & Sutton (1998), the benchmark accuracy
adopted here for the path correction is 50 m on each baseline. This
gives a coherence of 95% at 300 GHz, and 85% at 600 GHz. A column of
8 mm PWV adds approximately 50 mm to the propagation path, so that
50 m represents 0.1% of the total water vapor path in this extreme
case. With the high opacity of a large water vapor column, however, it
is likely that longer wavelength observations would be made under
these conditions and the 50 m goal could be relaxed somewhat.
The technique of water vapor radiometry uses the emission from water
vapor present in the antenna beams to estimate the excess path that
this water vapor introduces and compensate for the resulting phase
fluctuations at the output of the correlator. The rotational
transition at 183 GHz is a prime candidate for radiometric phase
correction on the MMA (Carilli, Lay & Sutton 1998).
Figure 1 shows simulations of the line profile for lines
of sight containing different amounts of water vapor. Also shown is
the increment to the spectral line when a small extra amount of water
is present in the line of sight; the extra quantity is such that it
adds 50 m to the electrical path. The 5 curves in each plot show
how this incremental response depends on the altitude at which this
fluctuation is introduced, as well as the PWV.
Figure 1: Thick lines: brightness temperature profiles of the 183 GHz
water line for different columns of precipitable water vapor along the
line of sight. Thin lines: increment in brightness temperature
resulting from 50 m of extra path at each of 5 altitudes (100 m to
4100 m above ground in steps of 1000 m).
Note the change in scales for this incremental emission: case (a) has
over 10 times more incremental emission than case (e).
Ground level is at an altitude
of 5000 m (16250 ft), the atmosphere structure is based on the
U.S. Standard Atmosphere (1962) and the spectral line profiles are
calculated from formulae given by Waters (1976).
It is apparent that there is saturation at the line center for even 0.5 mm PWV, and that this gives rise to a characteristic double peaked spectrum for an incremental amount of water vapor (the response from the line center is reduced because of the saturation).
A single channel radiometer produces an electrical signal (a voltage
V, for example) that is proportional to the total effective
brightness temperature 
, where g is the
instrumental gain factor. The total effective brightness temperature
is the sum of the brightness temperature of the radiation entering the
input horn 
and the internal noise temperature of the
radiometer 
. Extra water vapor in the beam of the
radiometer adds electrical path 
and increases the
brightness temperature by 
. The
temperature-to-path conversion factor 
is given by
where 
is the center frequency of the radiometer channel, and h
is the height of the fluctuation above the ground. The sets of 5
curves in Fig. 1 correspond to 50 m/C. For a baseline
in an array of antennas, we wish to measure fluctuations in the
difference of the electrical path lengths to the two antennas A and B,
in which case 
.
Since we actually measure 
, it is important to
minimize fluctuations in both the radiometer gains and noise
temperatures, which are indistinguishable from fluctuations due to
water vapor.
With multi-channel radiometers we measure 
over a number of channels with center frequencies 
. The estimator of path fluctuations is now some
linear combination:
where 
is the temperature-to-path conversion factor that is
appropriate for the set of weights 
. The goal is to
find a set of channels and weights 
that
is sensitive to extra path, but insensitive to the various errors that
complicate the measurement. These errors are described in the next section.
In addition to the desired signal from water vapor,
there are contributions in the measurement of the excess path from a
number of errors and uncertainties. Those considered here are:
(1) Thermal noise;
(2) Radiometer gain changes;
(3) Passband changes;
(4) Height of the fluctuations;
(5) Clouds;
(6) Temperature fluctuations;
(7) Spillover;
and (8) Uncertainties in atmospheric structure.
Figure 2 shows the spectral signature of the water vapor signal and each error (except for passband changes and atmospheric structure uncertainties) for different water vapor columns. Figure 3 compares the signal and errors for three different water vapor columns. The particular parameters adopted to calculate the amplitude of these errors are given at the end of each subsection below. They are guesses at realistic values, and are really only intended for illustration. The relative importance of the various contributions might be quite different in practice.
Figure: 2 Spectral signature of various errors: a) thermal
noise; b) radiometer gain variations; c) altitude dependence;
d) clouds; e) temperature fluctuations in the atmosphere.
f) spillover. The different curves in (a) to (e) show
the error spectrum for 0.5, 1, 2, 4 and 8 mm of precipitable water
vapor. The particular parameters used to derive these curves are
listed in §2.
Figure: 3 Comparison of different error contributions for three
different water vapor columns: a) 0.5 mm PWV (excellent conditions);
b) 2.0 mm PWV (average conditions); c) 8.0 mm PWV (bad conditions).
Key: w = water vapor (for 50 m extra path 1 km above the ground),
g = gain change, c = cloud, h = height of fluctuations,
t = temperature fluctuation. The particular parameters used to derive
these curves are listed in §2.
For a single channel of bandwidth 
and an
integration time of t, the rms uncertainty in the excess path length
due to thermal noise is given by
The factor of 
accounts for the fact that the phase
correction on a baseline is derived from the difference of
measurements made at two antennas.
The thermal noise fluctuations are not correlated over frequency
(unlike the errors discussed below), so that for a linear combination
the resulting rms error is given by
Figure 2a shows the rms temperature fluctuations
expected for a single channel with 
MHz, t = 1 s
and 
K (i.e. cooled radiometers). The temperature
uncertainty is plotted in preference to the path uncertainty, so that
the curves are also relevant to the multi-channel case where the
conversion factor depends on the channels and weights
chosen.
The gain g of a radiometer or astronomical receiver is subject to
fluctuations 
arising from instrument instabilities.
For a one channel system, the
error in the estimated path due to a gain fluctuation at one antenna is
given by
This accounts for changes in the gain that are common to all channels,
e.g. those originating in the front-end mixer or IF amplifiers.
Changes in the relative gains of the different channels (i.e. the
passband shape) are discussed separately below. The calibration of a
radiometer is simplified if as much of the gain as possible is common
to all channels.
For a multi-channel system,
Figure 2b shows the spectral signature of a 1%
gain fluctuation, assuming K. In contrast to the
thermal noise, the fluctuation will be common to all channels; a 1%
increase in the gain will increase the measured brightness temperature
in all channels by 1%.
A change in the relative gains of the channels can introduce significant errors. Since arbitrary changes can be introduced to the measured spectrum, no attempt has been made to simulate the effect. A change of 0.1% in a channel with a system temperature of 200 K would introduce a 200 mK of apparent extra brightness. Having channels as close together as possible is likely to be preferable, but this has not been taken into account in the analysis of §3.1. Calibration of the passband requires measuring the response to two loads with different temperatures.
It is clear from Fig. 1 that the temperature-to-path
conversion factor C is a function of altitude. Consider a path
length fluctuation 
. If the fluctuation is close to the
ground, there is generally more emission than if it is at higher
altitude (this is because the hotter water vapor molecules closer to
the ground have a higher random component to their rotation than the
cooler molecules higher in the troposphere, resulting in a lower
refractivity). For more than 1 mm of precipitable water vapor, 
, for example.
The difference in measured brightness temperature for the same
path length fluctuation at altitudes 
and 
is given by
This is shown in Fig. 2c for 
km, 
km and 
m. The latter value is a typical rms path
fluctuation level measured by the 12 GHz atmospheric phase monitor at
the Chile site. The problem can be turned around to give the
uncertainty in the path length estimate:
i.e. if the real fluctuation is at altitude , but we
assume that it is at , then our estimate of the fluctuation based
on the change in brightness temperature will be in error by 
.
For the single channel case, it is desirable to operate at a frequency
where C is independent of altitude. These are called `hinge
points'. There are examples in Fig. 1a at 182.2 GHz and
184.3 GHz, which correspond to the zero crossing points in
Fig. 2c for 
mm. These hinge points are
not present for higher water vapor columns, however.
For a multi-channel radiometer,
and we can synthesize a hinge point by choosing a set of channels and
weights that give a which is
independent of h. This is equivalent to finding a set of that give zero response to the error spectra of
Fig. 2c.
Liquid water, in the form of cloud droplets, generates a brightness
temperature spectrum that basically scales as 
, but does not
introduce significant excess path delay (Sutton and Hueckstaedt
1997). Clouds are visible above the Chajnantor site at least part of
the time, and even when there appear to be no clouds, it is possible
that there are diffuse clumps of small droplets
present. Figure 2d shows the increment to the
spectral brightness temperature profile due to a cloud that has an
opacity of 0.005 at 183 GHz. The basic contribution is
modified by the higher opacity near 183 GHz due to the water vapor
component.
Variations in temperature act on both the `wet' and `dry' atmospheric components. If there is a patch of atmosphere that has higher temperature than the surrounding air, then there will be extra emission from the water vapor without a significant increase in the wet refractivity. The spectra in Fig. 2e show the extra emission component from a 100 m column of air that is 1 K warmer than average.
The refractivity of dry air is also a function of temperature, and it
is temperature fluctuations that limit the seeing at optical
frequencies (where the refractivity of water vapor is only 5% of its
value at radio wavelengths). A 1 m column of air with density
1 kg m
has an electrical path length about 200 m longer than
1 m of vacuum. Raising the temperature decreases the density and
reduces the extra path. If the atmosphere is in hydrostatic
equilibrium, the total excess path is only a function of the surface
pressure (Thompson, Moran and Swenson 1987), but under turbulent
conditions this is no longer the case. An rms path length fluctuation
of 12 m was measured on a 13 m baseline of the Infrared Spatial
Interferometer on the summit of Mount Wilson, operating at a
wavelength of 11 m (Treuhaft et al. 1995). Fluctuations of 44 m were
reported for the position of the white light fringe from a 40 m
baseline of the IOTA optical interferometer on Mount Hopkins (Coudé
du Foresto et al. 1998). It is not known how these measurements
should be extrapolated to the size scales relevant to the MMA, but at
least some of the fluctuations must be due to water vapor.
Although the rms level of dry fluctuations is likely to be much less
than that due to water vapor, it may come to dominate once the wet
component has been corrected. Radiometric correction of dry
fluctuations is not currently possible, and if the rms is substantially
larger than 50 m, then fast switching must be used for phase
correction.
Spillover past the edge of the primary and spillover past the edge of the secondary must both occur to some extent. In the first case, parts of the beam will be terminated on the ground and/or the sky, depending on the geometry. Spillover past the secondary is likely to be terminated on the sky, but at a different elevation from the main beam.
Spillover onto the ground produces a flat error spectrum, as shown in Fig. 2f. Since we are considering pairs of antennas, it is only the difference in the coupling of the spillover patterns to the ground that is important. For example, if antenna 1 has 0.5% of its beam terminated on the ground at 280 K, compared to 0.4% for antenna 2, the error in the differential brightness temperature is 280 mK. This error contribution will only vary slowly with time as the elevation is changed, and should be mostly removed by regular phase calibration on a nearby source.
Spillover onto the sky at a lower elevation introduces an extra
component of water vapor which will change when slewing from the
target source to the calibrator. For example, consider the case where
antenna 1 has 0.1% of extra spillover past the top edge of the
primary compared to antenna 2, which is terminated on an 8 mm column
of water vapor at 
elevation. Moving 
in elevation
from target to calibrator changes this spillover water column by about
10%, which would change the measured brightness temperature profile
by about 5 mK. This is not large enough to be significant.
Measuring a total excess path of 50 mm (from 8 mm of PWV) to an
accuracy of 50 m requires a very accurate model of the temperature,
pressure and water vapor distribution as a function of altitude, since
the refractive index of the water vapor depends on temperature and
pressure (Carilli et al. 1998). Such an accurate model is not
required for phase correction on the MMA, however, since the broad
distribution of water vapor (i.e. excluding the fluctuations) is
common to all antennas and cancels out when differences between the
antenna beams are calculated.
The vertical distribution of this common component does influence
the temperature-to-path conversion factor C, and is discussed in the
next section.
With only one frequency channel there is no way to distinguish between
fluctuations due to water vapor and the various sources of error
discussed above. Even with multiple channels, separating the signal
from the errors can be difficult. This is true for radiometers that
measure emission from the 22 GHz water line, where the spectral
signature of extra water vapor is very similar to the spectral
signature from a gain change. This is because the line is optically
thin and the line amplitude is proportional to the water vapor
column. The design of a 22 GHz radiometer is therefore driven by the
need to minimize or calibrate out gain fluctuations, requiring tight
thermal regulation of components and frequent calibration with
standard loads. Even a cooled radiometer requires a gain stability of
better than 0.1%, corresponding to 50 m out of 50 mm; room
temperature 22 GHz instruments need to be better than 0.01%.
The saturation of the 183 GHz line is actually a big advantage in this situation, since the spectral signature of extra water vapor is easily distinguished from the various error contributions (Fig. 3). This concept is explored quantitatively in the next section with a simple 3 channel system.
Consider a radiometer that has 3 channels at frequencies that can be
adjusted to suit the prevailing conditions. There are 6 free
parameters: the center frequencies of the channels 
and the corresponding weights 
. In order to
restrict the parameter space, the weights are chosen so that the
linear combination has a zero response to any brightness temperature
spectrum of the form 
. Choosing
has the desired result. Figure 4 illustrates graphically
the response of the linear combination 
to an arbitrary spectral profile. The response is
proportional to the net curvature between 
and 
. A
computer fitting routine takes a set of 32 channels spaced by 0.25 GHz
in the range 183.3 GHz (the line center) to 191.3 GHz, and finds the
combination of 3 channels that minimizes the total residual
error. This is repeated for different columns of water vapor and the
results are tabulated in Table 1.
Figure 5 shows how this approach effectively
discriminates between the signal and error terms. From
Table 1 it can be seen that the optimum frequencies
shift further away from the line center as the column of water vapor
increases. For each water vapor column there are several channel
combinations that give very similar residual errors, but only the best
one has been shown. There is no fixed set of 3 channels that satisfies
the 50 m criterion under all conditions, even if the 
mm
case is ignored. In practice, the optimum combination of frequencies
will also depend on the relative importance of the different error terms.
Figure 4: The response of the 3-channel linear combination described in
§3.1 to an arbitrary spectral profile. The linear combination
is sensitive to the deviation from a
straight line.
Figure 5: The response of a linear combination of 3 channels to extra
water vapor and to the various error contributions (1 mm PWV). Since
the weights of the channels are chosen to have a null response to a
straight line, the 3 frequencies can be chosen to reject the main
error terms while giving a significant response to the signal.
Key: w = water vapor (for 50 m extra path 1 km above the ground),
g = gain change, c = cloud, h = height of fluctuations,
t = temperature fluctuation. The particular parameters used to derive
these curves are listed in §2.
| Water vapor column | 0.5 mm | 1.0 mm | 2.0 mm | 4.0 mm | 8.0 mm |
 GHz |
 |  |  |  |  |
 for 50 m path
| 205 mK | 256 mK | 189 mK | 59 mK | 23 mK |
| Thermal noise | 4 m | 4 m | 6 m | 16 m | 43 m |
| Gain change | 3 m | 1 m | 1 m | 5 m | 21 m |
| Altitude uncert. | 2 m | 3 m | 17 m | 35 m | 44 m |
| Cloud emission | 2 m | 1 m | 4 m | 5 m | 8 m |
| Temperature | 1 m | 2 m | 11 m | 25 m | 61 m |
| Ground spillover | 0 | 0 | 0 | 0 | 0 |
| Total uncert. | 6 m | 6 m | 21 m | 46 m | 89 m |
The numbers in Table 1 clearly demonstrate that
estimating the excess path accurately becomes more difficult as the
water vapor column increases, primarily because of the reduced signal
(Fig. 1). For 0.5 mm PWV, the predicted path correction
error of 6 m is well below the benchmark goal of 50 m; at 8 mm PWV
the error has risen to 89 m. The uncertainty resulting from changes
in the passband, and from possible dry fluctuations has not been
included.
The vertical structure of the atmosphere is likely to deviate from the
US Standard Atmosphere used in these calculations. To investigate
this, the optimum channels listed in Table 1 were used
with different atmospheric models to calculate new values for the
conversion factor . It was found that neither the surface
pressure nor the temperature profile have much impact, but that the
scale height of the water vapor distribution is more important. The
effect was most marked for mm, where decreased
by 16% if the scale height was increased from 2.0 to 2.5 km.
This problem is related to the uncertainty in the altitude of the
fluctuations (§2.4) and emphasizes the need for
empirical determinations of from measurements of the phase
fluctuations of bright astronomical sources.
The 3 channel example discussed above was chosen to illustrate how the signal spectral signature may be discriminated from the various error terms. The main conclusions are:
m.
A radiometer with many channels will, in general, be superior to one
with few channels, since there is more information available for
making the correction. Consider an instrument with many channels,
equally spaced over the line. The path correction is estimated by
where 
is the difference in brightness
temperature measured by two radiometers in the channel centered on
. In practice, an algorithm is required that takes all of
the available information (e.g. the measured water line profile;
temperature, pressure and humidity at ground level; best estimates of
the importance of the various error terms) and calculates the optimum
set of weights 
appropriate for the conditions. The most
important term will be the column of water vapor along the line of
sight which can be determined from the line profile; the other factors
provide refinements. Although theoretical lineshapes provide a
starting point for such an algorithm, it is likely that empirical
measurements of the phase fluctuations measured for strong
astronomical sources will provide continuous adaptation and
improvement.
Several options are considered for the 183 GHz radiometer front-end:
High thermal noise contribution (see below).
Demanding calibration requirements (see below).
Pick-off mirror needed to couple the radiometer to the sky via the secondary and primary mirrors. Must not interfere with any of the astronomical beams.
A typical receiver temperature for a room temperature Schottky system
is of order 2000 K, giving an rms thermal noise of 90 mK for a 1 s
integration and 0.5 GHz of bandwidth. This is not sufficient to
measure the values of 
given in Table 1 for
4 mm or 8 mm of water vapor. An integration time of 10 s would reduce
the noise by a factor of 3, but it is likely that some fraction of the
time would have to be spent on calibration against reference
loads. With a system temperature of 2000 K, small changes in the
passband (the gain of one channel relative to the others) can
introduce significant errors. The relative gains of the channels need
to be known to within approximately 0.002%. This is a very demanding
level of calibration, requiring two loads, one of which must be cold
(in order to achieve a significant temperature differential).
Has sufficient sensitivity when the water vapor column is large.
Calibration requirements are more tractable than uncooled system (although passband stability of 0.01% is required for 8 mm of water vapor).
Extra expense, space and complication of another dewar in the antenna cabin.
Pick-off mirror needed.
Extra expense and maintenance.
Has sufficient sensitivity when the water vapor column is large.
Calibration requirements are more tractable than uncooled system (although passband stability of 0.01% is required for 8 mm of water vapor).
Difficult to have a warm load calibration that is independent of the astronomy system. A warm load could be shared, but the calibration interval would be driven by the radiometry requirement, rather than the astronomy.
No extra dewar space used for radiometry.
Has sufficient sensitivity when the water vapor column is large.
Calibration requirements are more tractable than uncooled system (although passband stability of 0.01% is required for 8 mm of water vapor).
A sideband separating receiver might make passband calibration much easier.
Difficult to have a warm load calibration that is independent of the astronomy system. A warm load could be shared, but the calibration interval would be driven by the radiometry requirement, rather than the astronomy.
There are at least two possible ways in which the receivers might be used:
Continuous coverage of the atmospheric fluctuations.
A fraction of the potential astronomical observing time is lost.
Requires no extra LO system.
IF signals can be sent to a central location using the existing IF optical fiber.
Requires that receivers and LO can be switched rapidly.
A spectrometer covering approximately 8 GHz of IF bandwidth with a
resolution of order 0.5 GHz is needed for each radiometer. The
passband should be very stable. Either a filterbank or an analog lag
autocorrelator (e.g. Harris, Isaak and Zmuidzinas 1998) could provide
16 equally spaced channels across the band.
Alternatively, a fewer number of tunable channels could be used. In the
extreme case one channel could be switched rapidly between several
different frequencies. This would be at the expense of the
signal-to-noise ratio, but with a potentially very stable passband.
16 fixed 500 MHz channels going out to 8 GHz from
the line center, it should be possible to make a very good correction
for water vapor. It is possible, however, that temperature
fluctuations acting on the dry air component (responsible for poor
seeing at optical wavelengths) will determine the residual error. Little
is currently known about the structure function of these dry fluctuations
on large scales.
m requires cooled
radiometers when the water vapor column is 4 mm or more.
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