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2. Traditional gain calibration

The usual observing procedure for millimeter arrays is to interleave observations of the target object with a point-like calibrator. This is known as phase referencing. An observing cycle usually consists of several integrations of duration tex2html_wrap_inline1398 on the target, followed by a single integration on the calibrator. For example, there might be four 5-minute integrations on the target followed by a 5-minute integration on the calibrator, giving a cycle time tex2html_wrap_inline1400 of 25 minutes. The amplitude and phase measured on each baseline for the calibrator change with time, as a result of instrumental drifts and atmospheric fluctuations. A complex gain curve is fitted to the calibrator observations as a function of time, and is then removed from the measurements of the target object.

The next two sections quantify how phase fluctuations get folded into this observing procedure. Some contribute to a decorrelation (i.e. reduction in the visibility amplitude) within each integration, some are effectively removed, while others result in phase errors in the complex visibility measurements.

2.1. Decorrelation within an integration


The complex visibility sampled by an interferometer can be written as tex2html_wrap_inline1402 where the tex2html_wrap_inline1404 is the visibility intrinsic to the brightness distribution on the sky, and tex2html_wrap_inline1406 is the variation in the phase introduced by the atmosphere and instrumental drifts. Variations in the amplitude gain have been ignored. After integrating for time tex2html_wrap_inline1408, the averaged visibility is described by tex2html_wrap_inline1410, where f (tex2html_wrap_inline1414) reflects a reduction in the amplitude through decorrelation, and tex2html_wrap_inline1416 is a phase error.

Rapid fluctuations tend to get averaged out and contribute little to tex2html_wrap_inline1418 but dominate in f; conversely, fluctuations with periods much longer than tex2html_wrap_inline1422 have very little effect on f and a large impact on tex2html_wrap_inline1426. It can be shown (see Appendix) that the total phase power remaining after averaging the fluctuations over tex2html_wrap_inline1428 is given by
where tex2html_wrap_inline1430 is the spectral density of the phase fluctuations at the output of the correlator (units: radtex2html_wrap_inline1432 Hztex2html_wrap_inline1434). Figure 2 (click here) shows the sinctex2html_wrap_inline1436 function plotted on a logarithmic horizontal scale, and how it is well approximated by a step function that becomes zero for tex2html_wrap_inline1438. The rest of the phase power goes into the decorrelation of the measured amplitude:
where tex2html_wrap_inline1440 represents the phase fluctuations with period tex2html_wrap_inline1442, the angled brackets denote an ensemble average over all values of tex2html_wrap_inline1444, and the approximation is valid for tex2html_wrap_inline1446. The second integral gives the power of tex2html_wrap_inline1448, and can be expressed as an integral over the relevant frequencies of the power spectrum, so that
Changing the variable of integration from tex2html_wrap_inline1450 to tex2html_wrap_inline1452 gives
which shows that the area of the distribution in Fig. 1 (click here) with tex2html_wrap_inline1454 is a measure of the decorrelation, 1-f.

Figure 2: tex2html_wrap_inline1458 is well approximated by a step function with the transition at tex2html_wrap_inline1460

The visibility amplitudes measured for the calibrator also exhibit decorrelation, which can be quantified as a function of time for each baseline and then used to correct the corresponding amplitudes on the target object. Although this restores the average amplitude to the correct level, the amount of decorrelation is somewhat random from one integration to the next, and the uncertainty in the corrected amplitude is increased (see Figs. 8 (click here)a and b). Note also that decorrelation is fundamentally a baseline-based quantity and cannot be expressed as a combination of antenna-based contributions in the same way that is possible for phase errors.

2.2. Phase referencing


The phase measured on the target for a given baseline is referenced to the corresponding phase measured on the calibrator, a technique that eliminates most slow drifts in the instrumental response. The calibrator is chosen to be both bright enough to give sufficient signal-to-noise in a single integration, and as close to the target as possible to reduce position-dependent errors (e.g. from uncertainty in the relative locations of the antennas).

The cycling time between observations of the calibrator is tex2html_wrap_inline1462. This sampling rate can follow any component of tex2html_wrap_inline1464 (the phase variation remaining after integration) with a period longer than the Nyquist limit of tex2html_wrap_inline1466. These fluctuations constitute the true "slow" component. A slow component fitted to the calibrator phases will not represent the true slow component, since there are errors introduced to each calibrator measurement by a "fast" component, comprising those fluctuations with a frequency exceeding the Nyquist limit. This is illustrated in Fig. 3 (click here). The error caused by the fast component (shaded) looks like a slow component. This is the result of aliasing. The error component is white phase noise up to a high frequency cut-off of tex2html_wrap_inline1468.

Figure 3: Illustration of how phase fluctuations from the fast component (tex2html_wrap_inline1470) cause an error in the fitting of the slow component (tex2html_wrap_inline1472). The fast component is effectively aliased into a slow error component

Figure 4: a) Phase power plot showing the contributions from the atmosphere and the instrument (schematic only). The upper and lower halves represent the calibrator and target measurements, respectively. The phase variance in the diagonally hatched region causes decorrelation of the amplitudes measured in time tex2html_wrap_inline1474. Phase fluctuations in the "bricked" zone are correlated between the target and calibrator and are removed by phase referencing. The residual phase fluctuation power in the target observations after phase referencing comprises equal contributions from the target and the calibrator (blank regions in lower and upper halves, respectively). b) Spectrum of the residual phase errors. The contribution from the calibrator is actually aliased to lower frequencies, so that it has a white noise spectrum up to a cut-off frequency of tex2html_wrap_inline1476 (the sharp distribution is a result of the logarithmic horizontal scale)

The fitted slow component is subtracted from the phases measured on the target object as a function of time. This removes the effects of all phase fluctuations with a period longer than tex2html_wrap_inline1478, but leaves the phase fluctuations from the target and calibrator for which tex2html_wrap_inline1480, as illustrated in Fig. 4a. After phase referencing, there remain equal contributions to the phase error from the target and calibrator measurements. Since the calibrator contribution is aliased to lower frequencies, the spectrum of the phase errors has the appearance depicted in Fig. 4b.

The specific example shown in Fig. 4 uses the atmospheric model of Fig. 1 and has tex2html_wrap_inline1482 s and tex2html_wrap_inline1484 s. For an observing frequency of 230 GHz, the phase variance from fluctuations with tex2html_wrap_inline1486 on each of the target and the calibrator is tex2html_wrap_inline1488, corresponding to tex2html_wrap_inline1490 50% decorrelation. The total phase error tex2html_wrap_inline1492 is tex2html_wrap_inline1494, consisting of equal contributions from the calibrator (the aliased component) and target measurements.

One method for reducing the aliased component is to increase the shortest period tex2html_wrap_inline1496 of the curve fitted to the calibrator phases (Fig. 3 (click here)), so that it becomes over-sampled and no longer passes through all of the measurements. Only fluctuations with period tex2html_wrap_inline1498 are removed, so that the blank area in Fig. 4a corresponding to the uncorrected component is increased. Careful consideration of the effect of over-sampling, however, shows that the area of the aliased component on the calibrator is reduced by a factor tex2html_wrap_inline1500, which can potentially more than offset the original increase.

The optimum choice of tex2html_wrap_inline1502, tex2html_wrap_inline1504 and tex2html_wrap_inline1506 depends on the phase power distribution (which depends in turn on the baseline length and orientation, windspeed, thickness of the turbulent layer, etc.) as well as instrumental constraints. Two cases are discussed briefly.

2.3. Regular phase referencing

When tex2html_wrap_inline1508 is of order 1000 s, the spatial offset between the target and calibrator is unimportant for the cancellation of atmospheric fluctuations; the calibrator might as well be coincident with the target, as long as the elevation is not too low. This point is illustrated in Fig. 5 (click here). The wavelength of any fluctuation that is part of the slow component is much larger than the separation tex2html_wrap_inline1510 of the lines of sight to target and calibrator as they pass through the turbulent layer. It is then sufficient to subtract the fitted phase as a function of time on the calibrator from the measured target phases without worrying about the time offset tex2html_wrap_inline1512, so that for the slow component of the phase error, tex2html_wrap_inline1514.

Phase referencing becomes less effective at low elevations. There are two reasons for this: tex2html_wrap_inline1516 becomes larger, and, more significantly, the amplitude of large-scale phase fluctuations is proportional to the airmass tex2html_wrap_inline1518. Although fluctuations with tex2html_wrap_inline1520 are correlated between target and calibrator, they are stronger for the object at lower elevation. The fractional error in the phase referencing correction is tex2html_wrap_inline1522, where tex2html_wrap_inline1524 is the average of the two elevations and tex2html_wrap_inline1526 is the difference.

Another important point is that the impact of phase fluctuations does not increase indefinitely as the baseline gets longer. This is discussed further in Sect. 4 (click here).

Figure 5: A water vapor fluctuation (part of the slow component) at average height tex2html_wrap_inline1528 with wavelength tex2html_wrap_inline1530. The distance between the lines of sight to the target (X) and calibrator (C) at height tex2html_wrap_inline1532 is tex2html_wrap_inline1534. For w=5 m stex2html_wrap_inline1538, tex2html_wrap_inline1540 s, tex2html_wrap_inline1542, tex2html_wrap_inline1544 and tex2html_wrap_inline1546 km, values of tex2html_wrap_inline1548 km and tex2html_wrap_inline1550 km are obtained

2.4. Fast phase referencing

In principle, tex2html_wrap_inline1552 (and therefore also tex2html_wrap_inline1554 and tex2html_wrap_inline1556) can be reduced to timescales of order 10 s, below which there is very little phase fluctuation power. In this regime, the separation of the lines of sight to the target and calibrator, tex2html_wrap_inline1558, is comparable to the wavelength of the smallest fluctuation that can be sampled, tex2html_wrap_inline1560. For such fluctuations, it is no longer a good approximation to say that tex2html_wrap_inline1562; instead, a time lag needs to be introduced: tex2html_wrap_inline1564, where in the one-dimensional case tex2html_wrap_inline1566. The two-dimensional reality requires that tex2html_wrap_inline1568 and w are treated as vectors, with tex2html_wrap_inline1572. There is then a residual, uncorrected phase error associated with the separation of the target and calibrator perpendicular to the wind direction.

The time overhead associated with rapid switching between sources is generally prohibitive for existing arrays - calibrators can be over tex2html_wrap_inline1574 from the target and the antennas are not designed to be agile - but fast phase calibration is being seriously considered for future arrays (e.g. Holdaway 1992; Holdaway & Owen 1995).

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