**Figure 6:** **a-c)** Plots showing how small gain differences in the radiometers
can lead to a bad path correction. **a)** Propagation delays (in arbitrary units) to two
antennas as a function of time, alternating between the calibrator (C)
and target object (X). The abrupt changes are the result of a
difference in elevation between C and X (C has lower elevation in this
case). Also shown is the difference between these quantities, which is
the required path correction. This also has small discontinuities
(arrowed), since the path correction is different for the lines of
sight to X and C. **b)** The derived path correction if . **c)** The resulting error in the path
correction. The average error level is removed by phase referencing, but the
steps remain

Any radiometry system, whether it measures fluctuations in the noise level of existing receivers, or consists of separate,
dedicated instruments, measures a signal that is related to the amount
of emission *E* from the water vapor. This is converted into the path
excess *S* through a gain factor *g*, i.e. *S* = *g E*. The fluctuations
in the amount of water vapor along a line of sight can be typically of
order a few percent of the total water vapor column (Paper I), so that
the fluctuating part of *S* is small compared to its absolute value. The latter scales linearly with the airmass and in the Owens Valley,
site of the millimeter array run by Caltech, ranges from to
100 mm in the zenith direction.

The path excess derived from a single radiometer is given by , where is an estimate of the true gain factor: . The error includes uncertainties associated with the instrument (gain, bandpass, spillover, etc.), and also uncertainties in the atmospheric model used to convert emission into path delay (e.g. temperature profile, altitude of the turbulence). Ideally, the path excess should be measured to within ; when observing at 230 GHz (), the required accuracy is therefore m. Since can be (50 mm zenith delay at elevation), it is necessary that for this absolute measurement.

For an interferometer with radiometers on antennas 1 and 2, the
differential path correction is estimated by . The calibration requirement in this case is that , i.e. the gain factors have to *agree
with each other* to within 0.03%. The errors in the gain factors,
and , can be substantial, so long as they are
the same for all radiometers. It follows that the uncertainty in the
atmospheric model is largely unimportant for this differential
measurement. However, a small quantity (the difference in the
effective path lengths) is being estimated by subtracting two large
numbers ( and ) which are measured independently, and the
level of calibration needed is still daunting. These ideas are
illustrated schematically in Fig. 6. The difference signal in Fig. 6a is small compared to the individual measurements, and a small error in the relative calibration of the radiometer gains generates a path correction (Fig. 6b) that is in error by
the amount shown in Fig. 6c.

In practice, this stringent calibration requirement can be relaxed slightly, since it is only necessary to reference the phase on the target source X to the phase on the calibrator C, i.e. it is sufficient to measure the quantity as an estimate of the phase change that water vapor introduces between observations of the target and calibrator. Phase referencing then removes the error that is common to both target and calibrator measurements, and only the steps in Fig. 6c remain.

The difference in the water vapor column between the lines of sight to
the target and the calibrator can still be substantial, however. The
difference in path excess, , to an antenna for
sources at elevation but separated by a small elevation
offset is given by

where is the path excess in the zenith direction. The worst case
is for low and high . For example, for
, and mm, mm. The requirement is now that the
radiometers must be calibrated to the level of with respect to each other, i.e. all
radiometers in the system must give the same value for the path delay from a given column of water vapor to within 0.1%. Two methods that
attempt to achieve this are outlined in Sect. 3.4 (click here).

The value of chosen for is typical (in many cases optimistic) for existing millimeter arrays. Arrays proposed for the future (e.g. the National Radio Astronomy Observatory Millimeter Array and the Japanese Large Millimeter and Submillimeter Array) have substantially higher sensitivity, and therefore have access to a much larger number of calibrator sources. If is used in the example above, then only 2% precision is required in the agreement between the radiometers.

The method just described assumes that radiometry is used to correct the phase changes due to water vapor introduced in moving from the target to the calibrator and back (the small discontinuities in in Fig. 6a), in addition to correcting the fluctuations during each on-source period. It is the former that demands tight calibration of the radiometers, and so it is instructive to investigate the level of phase errors remaining if only the on-source fluctuations are removed. In this case, the average phase correction for each on-source period is made equal to zero.

Figure 7 (click here) shows the result of doing this, for a case where the time spent on the target is four times longer than the time spent on the calibrator. The plot is similar to Fig. 4, except that the radiometry corrects all fluctuations with for the target and for the calibrator. The residual rms phase for this specific example can be estimated from the diagram: the uncorrected (aliased) area is approximately 40% of the total for the calibrator, corresponding to an rms phase error of at 230 GHz.

It is also possible to estimate the level of calibration needed for the radiometers in this case. The rms phase with that is being corrected by the radiometry while observing the target is , i.e. an rms path correction of m. Path corrections of mm would not be uncommon, so that an accuracy of 50 m requires that %.

By reducing decorrelation and increasing the coherence time of the interferometer, this partial radiometric correction offers a substantial improvement over phase referencing alone. Using radiometry to its full potential, however, requires a method of achieving the much more stringent calibration requirements outlined in 3.2 (click here).

**Figure 7:** Phase power plots showing the result of correcting on-source
fluctuations only. This example has s and s. The residual phase error is dominated by the aliased contribution from the calibrator

To ensure that the full radiometric phase correction does not do more
harm than good, all radiometers need to be calibrated to have the same
sensitivity to water vapor to within 1 part in
(Sect. 3.2 (click here)), i.e. if all the radiometers were to look at the
same column of water vapor (difficult to arrange in practice) then they must give the same reading to
within 0.1%. Although the response of radiometers to hot and cold
loads can be determined to this level of precision, small differences
in the bandpass shapes, spillover, scattering, etc., will inevitably
introduce systematic uncertainties in the response to water vapor. The errors
introduced will be a function of elevation and the prevailing
atmospheric conditions (e.g. temperature profile, altitude of
turbulence), both of which vary with time. There may also be a residual
gain variation of the radiometers themselves that is not removed by a
hot and cold load calibration. The relative calibration of the
radiometers should therefore be made *against a column of water vapor on
the sky*, frequently enough to follow variations in the response. Two possible
calibration schemes have been identified.

For a given baseline, windspeed and thickness of the turbulent layer, there is a timescale beyond which there is very little fluctuation power. The data presented in Paper I indicate that for a 100 m baseline, during typical conditions in the Owens Valley, there is very little power for hour. This timescale is longer for the 500 m model shown in Fig. 1 (click here), but this does not include the effects of an outer scale to the turbulence, which will reduce the power on long timescales.

The radiometer gains , , etc., should therefore be scaled to ensure that the phase corrections derived for each baseline (Fig. 6b) average to zero, over periods exceeding (Fig. 2 (click here) for factor of 2.5). This effectively calibrates out all variations in , , etc. with period greater than . By also monitoring over a long period how the difference between radiometer measurements on a baseline vary as a function of elevation and azimuth, the effects of different spillover patterns can also be reduced.

This method assumes that the antennas are at the same altitude and that the atmosphere can be considered planar over the area of the array. If these conditions are not satisfied, the long term average of the atmospheric phase fluctuations will be non-zero, but it may still be possible to estimate the offsets to the required level of precision. The long averaging times needed for long baselines means that this technique is likely to be more practical for short baselines.

Another possibility is to observe two bright calibrators whose positions are well known and well separated in elevation (thereby emphasizing the steps in Fig. 6). In the absence of atmospheric fluctuations the phase measured by an interferometer should be the same for each source, assuming that each is at the phase center and that baseline errors are negligible. This should also be the case after radiometry has been used to correct for atmospheric fluctuations, and the radiometer gain factors can be scaled to ensure this. Because of the ambiguity in phase, there are many scalings that satisfy this condition. The correct relative scaling can be found by measuring several bright calibrators at different elevations initially, and then subsequently correcting for drifts by using just two calibrators.

An observing cycle that uses two calibrators (in addition to the target source) has the advantage that while one is used as a phase reference to remove instrumental drifts, the other one can be mapped in parallel with the target, as a useful check on the phase correction and imaging procedure.

This method can be used to remove variations in , , etc., with periods exceeding twice the observing cycle time. The disadvantages are that time must be spent observing an object that is not of direct interest, and the method will not correct errors introduced by different spillover patterns.

There are several other possibilities that could improve phase correction without the stringent radiometer calibration described above.

The technique of phase self-calibration (e.g. Readhead et al. 1980) can be used to remove phase errors that are not measured directly. The use of radiometry for correcting fluctuations during each on-source period means that self-calibration can be applied to periods of data up to the on-source time . If the instrumental drift is slow, then can be made large, and it will be possible to self-calibrate on much fainter objects than was possible without radiometry; radiometry effectively increases the coherence time of the interferometer.

Another possibility is to filter out the frequencies in the derived
phase correction that result from the error pattern shown in
Fig. 6c, before the correction is applied to the data in post processing.
The power spectrum of this error signal consists of peaks centered on
frequencies of (where *n* is a positive integer), widths that depend on the variation in
over time, and strengths proportional to
. By removing those frequencies
where the error phase power exceeds the atmospheric phase power, the
phase error can be reduced substantially. This technique will be more
successful for shorter baselines where there is less power on long
timescales, and when the gain factors of the radiometers vary only
slowly with time, giving narrower peaks in the error power spectrum. It
is also important that the observations of the target and calibrator
are as regular as possible.

An increase in the array sensitivity through larger antennas, a greater number of antennas, or more sensitive receivers, for example, allows fainter calibrators to be used that can be found much closer to the target. This reduces the size of the steps in and in Fig. 6a and therefore reduces the error introduced by differences in the radiometer gains. The same is true for a site that has a lower column of water vapor above it. Phase correction using radiometry will therefore be much easier for the future generation of millimeter arrays, with many antennas at a drier, high altitude site, than for the existing arrays.

If an average radiometric path correction is derived and applied to the incoming signals for contiguous time intervals of duration , then all fluctuations with can be corrected. In the ideal system, the correction would be made in real-time, either by applying appropriate phase offsets to the local oscillators at each antenna, or by applying the offsets in the correlator. The limiting value of is then set by the sensitivity of the radiometry system, and could easily be less than 1 second. This form of correction is irreversible, however, so that a bad radiometer measurement could do more harm than good to the astronomical data. Until water vapor radiometry becomes a reliable technique, it would be prudent to apply as much of the correction as possible offline. Offline corrections will also be needed to account for gain changes in the radiometers, as might be determined from the two calibration procedures described above.

At the Owens Valley Millimeter Array, continuum data are recorded every 10 s, so that with a purely offline correction, fluctuations with s go uncorrected. Reference to Fig. 7 of Paper I indicates that there is generally little phase power on these timescales in the Owens Valley, so that an offline correction every 10 s would be adequate in most cases for the continuum data. The much larger quantity of spectral line data is typically recorded at intervals of 1 to 5 minutes, however. A purely offline correction is much less effective in this case, unless the sampling time can be reduced substantially.

**Figure 8:** **a-d)** The effect of residual phase errors on the ability to resolve
a small object. **a)** Phase power distributions as a function of fluctuation period *t* for four different
baseline lengths. The hatched region indicates the fluctuations
contributing to the phase error for each integration on the
calibrator. **b)** Visibility amplitude from one integration as a
function of baseline length for an unresolved point source of unit
flux, with (dashed line) and without (solid line) phase fluctuations. The shaded region shows the
distribution of amplitudes that might be expected, scattered about the
average value. **c)** Amplitude for one integration, corrected for
decorrelation by dividing by the corresponding calibrator amplitude.
**d)** The amplitude expected after vector averaging over many
integrations, with radiometry used to correct the decorrelation in each
integration

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