|Annu. Rev. Astron. Astrophys. 2014. 54:
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For parameters typical of cm-wavelength VLBI observations seeking to measure relative positions between two sources separated by ≈ 1° with ≈ 10 μas accuracy, phase reference sources should have coordinates known to 5 mas, antenna locations known to 1 cm, and electronic, clock and propagation delays known to 0.05 nsec.
5.1. Tropospheric Delay Calibration
Tropospheric (non-dispersive) delays are usually calibrated by one of two methods: 1) using geodetic-like observing blocks or 2) using GPS data.
5.1.1 GEODETIC BLOCK CALIBRATION: One can use observations of radio sources spread over the sky to measure broad-band (group) delays. For sources with positions known to better than ≈ 1 mas, group delay residuals will generally be dominated by the effects of tropospheric (and at low frequencies ionospheric) mis-modeling, provided the geometric model for the array is accurate to better than ± 1 cm (including antenna locations, earth orientation parameters and solid-earth tides). By observing ≈ 10 sources over a range of source azimuths and, most importantly, elevations in rapid succession ( 30 min), one can estimate residual "zenith" tropospheric delays at each telescope. These observing blocks are similar to those used for geodetic VLBI observations to determine source and telescope positions, as well as Earth's orientation parameters and, hence, are called "geodetic blocks."
The observing sequence can be determined by Monte Carlo simulations of large numbers of blocks and choosing the block with the lowest expected zenith-delay uncertainties. Operationally, it may be best to choose sources above an elevation of ≈ 8° and available at a minimum of about ~ 60% of the telescopes in the array. Geodetic blocks should be placed before the start, during (roughly every 2 hours), and at the end of phase-reference observations. This allows one to monitor slow changes in the total tropospheric delay for each telescope. In order to measure group delays accurately, the observations should span the maximum IF bandwidth, spaced in a "minimum redundancy pattern" to uniformly sample, as best as possible, all frequency differences. For a system with 8 IF bands, this can be accomplished with bands spaced by 0, 1, 4, 9, 15, 22, 32 and 34 units. If the recording system limited to 500 MHz IF bandwidth, the unit separation would be ≈ 14.7 MHz. With such a setup, estimated zenith delays are generally accurate to ≈ 0.03 nsec (≈ 1 cm of propagation path-delay). Comparisons of the geodetic block technique with those using Global Positioning System (GPS) data and an image-optimization approach confirm this accuracy (Honma, Tamura & Reid 2008).
Residual multi-band (group) delays and fringe rates are modeled as owing to a vertical tropospheric delay and delay-rate, as well as a clock offset and clock drift rate, at each antenna. Note that the geodetic blocks need not be observed at the same frequency as the phase-referencing observations. If one has independent calibration of ionospheric (dispersive) delays (see Section 5.2), then it is simplest to observe the geodetic blocks at a frequency (ν) above ≈ 20 GHz to avoid contamination of the group delays by residual ionospheric effects, which decline approximately as 1/ν2. However, if one is limited to one (wide-band) receiver, then for observing frequencies below about ≈ 10 GHz, it is important to remove unmodeled ionospheric contributions to the geodetic group delays, since correction of interferometer phases has a different sign for dispersive compared to non-dispersive delays. This can be accomplished with "dual-frequency" geodetic blocks as outlined in Section 5.2.
5.1.2 GPS TROPOSPHERIC DELAY CALIBRATION: The GPS is a navigation system used to determine an accurate three-dimensional position for an observer on the Earth. The system consists of more than 20 satellites constituting an artificial "constellation" of reference sources. Accurate positions of the satellites are broadcast by radio transmission to receivers on Earth. The basic principles for position determination using GPS signals is similar to geodetic VLBI: the receiver position can be accurately determined based on delay measurements from multiple satellites. Since the broadcasts are at low frequencies (1.2 to 1.6 GHz), propagation delays in the ionosphere strongly affect the data. Dual frequencies allow removal of the ionospheric (dispersive) delay, and by observing several GPS satellites at different slant-paths through the atmosphere simultaneously, one can solve for the tropospheric delay as well as the receiver position. As such, one can use GPS data to calibrate tropospheric delays. Steigenberger et al. 2007 and Honma, Tamura & Reid 2008 have conducted detailed comparisons of tropospheric delay measurements by GPS and geodetic-mode VLBI (see Section 5.1.1) observations; both studies conclude that the difference between the two methods are small, 2 cm, with no systematic differences.
5.2. Ionospheric Delay Calibration
In principle, the effects of ionospheric (dispersive) delays can be largely removed by using global models of the total electron content (Walker & Chatterjee 2000), by direct use of GPS data at each antenna, or from dual-frequency geodetic block observations. Because the ionosphere is confined to a shell far above the Earth's surface, rays from the source to a telescope can penetrate the ionosphere far from the telescope location on the Earth's surface. This generally favors using global models, based on smoothed results from a grid of GPS stations, over measurements made only at the VLBI telescopes. However, this is an area of current development and the methods of dispersive delay calibration may improve.
If one observes at frequencies below ≈ 10 GHz, one can in principle obtain both dispersive and non-dispersive delays simultaneously from "wide-band" geodetic blocks. For example, with receivers covering 4 to 8 GHz, one can generate two separate sets of geodetic block data: spanning 500 MHz centered at the low end of the band (νL = 4.25 GHz) and at the high end of the band (νH = 7.75 GHz). This gives observables τ(νL) and τ(νH) , which contain both dispersive and non-dispersive delays (eg, tropospheric delays and clock terms). Differencing τ(νL) and τ(νH) values for all sources and baselines yields the dispersive contribution. The dispersive contribution can be used to model the ionospheric contribution and, importantly, can be scaled and subtracted from the τ(νH) delays to produce pure non-dispersive delays. The non-dispersive delays can then be modeled as described in Section 4.2.1 to solve for pure tropospheric and clock terms. Note, that if one uses observations of a strong continuum source (fringe finder) to remove electronic delay differences among IF sub-bands (i.e., zeroing the delay on the fringe finder scan), this must be taken into account when modeling the dispersive delays.
An alternative to direct measurement is to use global maps of total ionospheric electron content generated from GPS data. Global Ionosphere Maps (GIMs) are produced on a regular basis by several groups, such as NASA, EAS, CODE, and UCP, who provide GIMs every two hours. Basic trends in these global maps are generally consistent with each other, although there are differences in scale for TEC values. Also note that currently GIMs have a typical angular resolution of 2.5 deg, and thus small scale fluctuations in the ionosphere may not be well traced. Hernández-Pajares et al. (2009) compared TEC values obtained by GPS and direct measurements (such as with JASON and TOPEX) and found vertical TEC values among the methods differ at the ≈ 20% level. For a 50 TECU zenith column and an error of ± 20%, the uncertainty in the ionospheric delay correction would be ~ 1 cm at an observing frequency of 22 GHz, which is slightly smaller than the delay error caused by the troposphere. On the other hand, at frequencies below 10 GHz the ionosphere is the dominant contributor to the delay error budget, even after corrections with GIMs are applied.
5.3. Dual-beam Systems
For a dual-beam receiving system, an additional calibration is required, because the propagation paths through the antenna and electronics are independent for the target and reference sources. At VERA, the horn-on-dish method is utilized (Honma et al. 2008a), in which an artificial noise source is mounted on an antenna's main reflector and a common signal is injected into both receivers. During observations, cross-correlation of the noise source between the two receivers is monitored in real-time, so that one can trace the time variation of the instrumental delay difference. The measured delay difference between the two receivers can be applied in post-processing. Honma et al. (2008a) have conducted tests of the horn-on-dish calibration and found that the VERA system can measure instrumental (path) delay differences to an accuracy of ± 0.1 mm, more than adequate for 10 μas relative astrometry.
5.4. Reference Source Position
The error in the position of the source used as the phase reference causes an error in its delay/phase measurements, and these errors are then propagated to the target source. The effect of a reference source position offset differs from other errors previously discussed. The first order effect is that the position offset of the reference source is transferred to the target source position. If the reference source has negligible proper motion and parallax, this only introduces a constant term, which is absorbed when fitting for parallax and proper motion. This is why we do not include such a term in Equation (3).
However, if the reference source position offset becomes large ( 10 mas), there are additional second-order effects which come from the fact that the target and reference sources are separated on the sky. Because they are at different positions, the interferometer phase response to a position offset at one position on the sky does not exactly mimic the response at the other position. These second order effects lead to a small position error and to degraded image quality. For an interferometric fringe spacing of 1 mas and a pair separation of 1°, one radian of second order phase-shift results from a calibrator position offset of ~ 10 mas (this offset, δθ, can be estimated by 2π θsep (δθ / θbeam) ~ 1). In general, for phase-referencing astrometry, one needs to know the calibrator position with an accuracy of 10 mas. For a finer beam size or larger separation angle, the required positional accuracy is even higher.
There are several methods for determining the absolute position of a phase reference source. If one can find a compact ICRF radio source (with δθ ≈ 1 mas) close to the target source, one can transfer the position accuracy to the phase-reference source. However, ICRF sources are sparsely distributed on the sky and some are heavily resolved on long interferometer baselines. In such cases, one should include an ICRF source in the phase-referenced observations, even if offset by up to about ≈ 5° from the other sources used for differential astrometry, in order to determine their positions. Even if using a distant ICRF source as a reference produces poor images for other sources, one can usually obtain ~ 1 mas accurate positions (and then discard the ICRF source data).
Alternatively, one can do preparatory observations to measure the position of a phase reference source. Usually connected-element interferometers (e.g., JVLA, eMERLIN, or the ATCA), which are limited in baseline length to 100 km, can yield absolute positions with accuracies between 0.01 and 1 arcsec. While this is usually adequate for VLBI data correlation, it is not good enough for high accuracy VLBI astrometry. Instead, VLBI observations using broad-band group-delay observables (as done for ICRF campaigns) are preferable.
If a maser source is to serve as the phase reference, then broad-band group-delays cannot be measured, since spectral lines are intrinsically very narrow, leading to large uncertainty in group-delay estimates. In this case, it is best to use an ICRF source to transfer positional accuracy via phase-referencing. As a last resort, if no qualified source is available, one can attempt to synthesize a maser spot image without phase-referencing. If the VLBI data have been correlated (or later corrected) with a model accurate to a few wavelengths of path delay, then such an image will resemble optical speckles (caused by short-term phase fluctuations induced by tropospheric water fluctuations and instabilities in the frequency standard). However, the centroid of these speckles can often be determined to ~ 10 mas, which may be sufficient for the position of the phase reference.
5.5. Source Elevation Limits
Often the dominant source of systematic error is uncompensated tropospheric delays. At any telescope, if one misestimates the zenith (i.e., z = 0) tropospheric delay by δτ0, this error is magnified at larger zenith angles by a factor of ≈ secz. The differential effect between the target and a background source, with a zenith angle difference of Δz, is then given by Δτ(z) = δτ0 ∂secz / ∂ z Δz = δτ0secz tanz Δz. Note that Δτ(z) increases dramatically at large zenith angles where both secz and tanz tend to diverge. For example, the ratio of differential delay error Δτ(z) at z = 70° to z = 45° is a factor of 8.0. This suggests that astrometric observations should be limited, whenever possible, to small zenith angles. Of course, when restricting observations to small zenith angles, one must balance the loss of interferometer (u, v)-coverage, which affects both sensitivity and image fidelity, against increasing systematic errors that come with large zenith angle observations. In practice, it may prove valuable to fully simulate interferometric observations in order to estimate an optimum zenith angle limit for the declination of a given target source and the orientation and separation on the sky of the background source.
5.6. Measuring Positions
Perhaps the simplest method of measuring a position from radio interferometric data is to work in the image domain. This involves placing the visibility (u, v) data on a grid, performing a 2-dimensional Fourier transformation to sky coordinates, (u, v) → (x, y), deconvolving the resulting "dirty" map using the "dirty" beam (point source response) with the CLEAN algorithm, and fitting a 2-dimensional Gaussian brightness distribution to compact emission components. This provides offsets, (Δx, Δy), from the map center, whose absolute position is defined by the position used when cross-correlating the interferometer data and any subsequent shifts applied during calibration.
Care must be taken in both shifting positions and interpreting the measured offsets from the map center, to properly account for the effects of precession, nutation, aberration and even general relativistic ray bending near the Sun. For example, position shifts applied to (u, v)-data are best done by calculating the full interferometric delay and phase for the desired source position and subtracting those values for the original (correlator) position. Simply calculating a differential correction can lead to significant astrometric errors. Similarly, one should not simply add map offsets, (Δx, Δy) (which "know nothing" of physical effects) to the map-center coordinates, without correcting for differential precession, nutation, and aberration, unless the map offsets are very small. For example, over time scales of less than a few years, neglecting differential effects generally leads to position errors of ~ 10-4(Δx, Δy). So, to ensure astrometric accuracies of ~ 1 μas, (Δx, Δy) values should be 10 mas. Therefore, if one measures large (Δx, Δy) values in a map, one should (accurately) shift positions in the (u, v)-data, re-image, and re-measure the offsets.