As radio interferometers push to higher frequencies and longer baselines, phase fluctuations resulting from the irregular distribution of water vapor in the troposphere become the dominant limitation on spatial resolution.
The technique of phase referencing, where observations of a calibrator object are interleaved with observations of the target, has been used to compensate for drifts in the instrumental response and also corrects for slow atmospheric variations. In its traditional "slow" form, with calibrator observations every 20 minutes or so, phase referencing is insufficient for dealing with most atmospheric fluctuations, which are on shorter timescales. The possibility of very fast phase referencing, with a cycle time of order 10 seconds, is now being considered for new arrays (Holdaway 1992; Holdaway & Owen 1995), but sets stringent requirements on antenna agility and data aquisition rates.
Water vapor radiometry has long been proposed as a solution to the atmospheric phase problem (e.g. Westwater 1967; Schaper et al. 1970). As well as causing a propagation delay, water vapor along the line of sight emits radiation; the more water vapor present along the line of sight, the greater the emission, and the greater the propagation delay. By monitoring the emission from water vapor in an antenna beam as a function of time, it is possible to derive a phase correction that can be applied to the data. There are two basic approaches being developed to measure the water vapor emission: (1) monitoring fluctuations with the same receivers used for the astronomical observations (e.g. Welch 1994; Bremer 1995), or (2) mounting a dedicated radiometer on each antenna, a technique being pursued at the Owens Valley Millimeter Array and the submillimeter interferometer comprising the Caltech Submillimeter Observatory and the James Clerk Maxwell Telescope on Mauna Kea. The algorithms for obtaining a path correction from an emission measure are not considered here (see e.g. Sutton & Hueckstaedt 1996), although the accuracy needed is addressed.
The aim of this paper is to describe and quantify the impact of atmospheric phase fluctuations on astronomical observations, including phase referencing and water vapor radiometry. The analysis centers on the spectral density distribution of the phase fluctuations, which shows graphically the amount of phase fluctuation power on each timescale for a given baseline. The model is based on Kolmogorov turbulence "frozen" in a layer of given thickness, and is presented with supporting data in Lay (1997), hereafter referred to as Paper I.
Figure 1: Model for the frequency distribution of phase fluctuation
power from a 1 km thick turbulent layer, measured on a 500 m baseline
perpendicular to a 
wind. 
is the spectral density
of the phase variations at the output of the interferometer. The axes are
chosen so that the area under the curve is proportional to the variance
of the phase. For typical conditions at the Owens Valley Radio
Observatory, the total area corresponds to a phase variance of
when observing at 230 GHz
Figure 1 (click here) shows an example for a 500 m baseline,
where the turbulent layer is 1 km thick and blows perpendicular to the
baseline at 
. Most of the fluctuation power is on
timescales of order 600 s (10 minutes). The distribution depends on a
number of parameters, e.g.: longer baselines will show more power on
longer timescales; the orientation of the baseline with respect to the
wind direction affects the shape; a stronger wind
shifts the curve to the right (doubling the windspeed halves the
timescales); and the vertical scaling depends on the strength of the
turbulence and is proportional to the square of the observing frequency. The effect of an outer scale to
the turbulence has not been included; in this example, an outer scale
size of 10 km will reduce the phase power on timescales longer than
2000 s. The total area under the curve corresponds to a phase variance
of 
when observing at 230 GHz. Approximately one quarter
of the area is accounted for by fluctuations with a period less than
400 s, so that these fluctuations alone would give an rms phase of
55
.
The next section examines the ability of phase referencing to correct for phase fluctuations. Section 3 (click here) shows how radiometric corrections can be incorporated into the phase referencing framework, and the demands this places on the precision of the correction. Section 4 (click here) illustrates how the different levels of phase correction improve the response to a point-like object. The summary is followed by an appendix that shows how only the fast fluctuations contribute to decorrelation in an integration.