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3. Water vapor radiometry


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 tex2html_wrap_inline1576. 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 tex2html_wrap_inline1588 to 100 mm in the zenith direction.

3.1. The absolute water vapor column


The path excess derived from a single radiometer is given by tex2html_wrap_inline1590, where tex2html_wrap_inline1592 is an estimate of the true gain factor: tex2html_wrap_inline1594. The error tex2html_wrap_inline1596 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 tex2html_wrap_inline1598; when observing at 230 GHz (tex2html_wrap_inline1600), the required accuracy is therefore tex2html_wrap_inline1602m. Since tex2html_wrap_inline1604 can be tex2html_wrap_inline1606 (50 mm zenith delay at tex2html_wrap_inline1608 elevation), it is necessary that tex2html_wrap_inline1610 for this absolute measurement.

3.2. The differential water vapor column


For an interferometer with radiometers on antennas 1 and 2, the differential path correction tex2html_wrap_inline1612 is estimated by tex2html_wrap_inline1614. The calibration requirement in this case is that tex2html_wrap_inline1616, i.e. the gain factors have to agree with each other to within 0.03%. The errors in the gain factors, tex2html_wrap_inline1618 and tex2html_wrap_inline1620, 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 (tex2html_wrap_inline1622 and tex2html_wrap_inline1624) 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 tex2html_wrap_inline1626 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, tex2html_wrap_inline1628, to an antenna for sources at elevation tex2html_wrap_inline1630 but separated by a small elevation offset tex2html_wrap_inline1632 is given by
where tex2html_wrap_inline1634 is the path excess in the zenith direction. The worst case is for low tex2html_wrap_inline1636 and high tex2html_wrap_inline1638. For example, for tex2html_wrap_inline1640, tex2html_wrap_inline1642 and tex2html_wrap_inline1644 mm, tex2html_wrap_inline1646 mm. The requirement is now that the radiometers must be calibrated to the level of tex2html_wrap_inline1648 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 tex2html_wrap_inline1650 chosen for tex2html_wrap_inline1652 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 tex2html_wrap_inline1654 is used in the example above, then only 2% precision is required in the agreement between the radiometers.

3.3. Correcting on-source fluctuations only


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 tex2html_wrap_inline1656 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 tex2html_wrap_inline1658 spent on the target is four times longer than the time tex2html_wrap_inline1660 spent on the calibrator. The plot is similar to Fig. 4, except that the radiometry corrects all fluctuations with tex2html_wrap_inline1662 for the target and tex2html_wrap_inline1664 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 tex2html_wrap_inline1666 at 230 GHz.

It is also possible to estimate the level of calibration needed for the radiometers in this case. The rms phase with tex2html_wrap_inline1668 that is being corrected by the radiometry while observing the target is tex2html_wrap_inline1670, i.e. an rms path correction of tex2html_wrap_inline1672m. Path corrections of tex2html_wrap_inline1674 mm would not be uncommon, so that an accuracy of 50 tex2html_wrap_inline1676m requires that tex2html_wrap_inline1678%.

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 tex2html_wrap_inline1680 s and tex2html_wrap_inline1682 s. The residual phase error is dominated by the aliased contribution from the calibrator

3.4. Calibrating the radiometers


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 tex2html_wrap_inline1684 (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.

3.4.1. Monitoring the difference between the derived phase corrections

For a given baseline, windspeed and thickness of the turbulent layer, there is a timescale tex2html_wrap_inline1686 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 tex2html_wrap_inline1688 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 tex2html_wrap_inline1690, tex2html_wrap_inline1692, etc., should therefore be scaled to ensure that the phase corrections derived for each baseline (Fig. 6b) average to zero, over periods exceeding tex2html_wrap_inline1694 (Fig. 2 (click here) for factor of 2.5). This effectively calibrates out all variations in tex2html_wrap_inline1696, tex2html_wrap_inline1698, etc. with period greater than tex2html_wrap_inline1700. 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.

3.4.2. Two phase calibrators

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 tex2html_wrap_inline1702 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 tex2html_wrap_inline1704, tex2html_wrap_inline1706, 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.

3.5. Alternative schemes

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 tex2html_wrap_inline1708. If the instrumental drift is slow, then tex2html_wrap_inline1710 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.

3.5.2. Filter out the error signal:

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 tex2html_wrap_inline1712 (where n is a positive integer), widths that depend on the variation in tex2html_wrap_inline1716 over time, and strengths proportional to tex2html_wrap_inline1718. 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.

3.5.3. Increase sensitivity and move to drier site:

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 tex2html_wrap_inline1720 and tex2html_wrap_inline1722 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.

3.6. How often should a radiometric correction be applied?

If an average radiometric path correction is derived and applied to the incoming signals for contiguous time intervals of duration tex2html_wrap_inline1724, then all fluctuations with tex2html_wrap_inline1726 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 tex2html_wrap_inline1728 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 tex2html_wrap_inline1730 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 tex2html_wrap_inline1734 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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