Issue 
A&A
Volume 527, March 2011



Article Number  A108  
Number of page(s)  12  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201116435  
Published online  04 February 2011 
Revisiting the radio interferometer measurement equation
III. Addressing directiondependent effects in 21 cm WSRT observations of 3C 147
Netherlands Institute for Radio Astronomy (ASTRON) PO Box 2,
7990AA
Dwingeloo,
The Netherlands
email: smirnov@astron.nl
Received:
5
November
2010
Accepted:
5
January
2011
Context. Papers I and II of this series have extended the radio interferometry measurement equation (RIME) formalism to the fullsky case, and provided a RIMEbased description of calibration and the problem of directiondependent effects (DDEs).
Aims. This paper aims to provide a practical demonstration of a RIMEbased approach to calibration, via an example of extremely highdynamic range calibration of WSRT observations of 3C 147 at 21 cm, with full treatment of DDEs.
Methods. A version of the RIME incorporating differential gains has been implemented in MeqTrees, and applied to the 3C 147 data. This was used to perform regular selfcal, then solve for interferometerbased errors and for differential gains.
Results. The resulting image of the field around 3C 147 is thermal noiselimited, has a very high dynamic range (1.6 million), and none of the offaxis artefacts that plague regular selfcal. The differential gain solutions show a high signaltonoise ratio, and may be used to extract information on DDEs and errors in the sky model.
Conclusions. The differential gain approach can eliminate DDErelated artefacts, and provide information for iterative improvements of sky models. Perhaps most importantly, sources as faint as 2 mJy have been shown to yield meaningful differential gain solutions, and thus can be used as potential calibration beacons in other DDErelated schemes.
Key words: methods: analytical / methods: data analysis / methods: numerical / techniques: interferometric / techniques: polarimetric
© ESO, 2011
Introduction
The field around the bright radio source 3C 147 is a favourite showcase for dynamic range (DR) demonstrations. 3C 147 itself is very bright (22 Jy at 21 cm), which ensures a high SNR for selfcal solutions, while the surrounding field boasts a spectacular collection of mostly pointlike fainter sources. The absence of significant extended emission has allowed very accurate sky models to be constructed. It is then not surprising that 3C 147 was the first field to break the 10^{6} dynamic range barrier (de Bruyn 2006; de Bruyn et al. 2011, in prep.), on a single 12h synthesis. This spectacular result was achieved with regular selfcal implemented in the NEWSTAR package. A major contributing factor is the relatively low level of beamrelated DDEs at the WSRT, as discussed in Paper II (Smirnov 2011, Sect. 2.1)^{1}. An image of the 3C 147 field at 1 600 000:1 dynamic range is shown in Fig. 1. The making of this image is the subject of Sect. 1.
Fig. 1 “Showcase” image of the field around the bright radio source 3C 147, produced after reduction with MeqTrees. The image is noiselimited, and has a dynamic range of 1.6 million. This DR was already achieved by de Bruyn using regular selfcal in NEWSTAR, but the resulting images contained artefacts around offaxis sources (left inset) due to DDEs. A MeqTrees reduction incorporating differential gains, as described in this paper, has completely eliminated the artefacts (right inset). This image also appears in Noordam & Smirnov (2010). 
The very same benign properties that allow the WSRT to achieve record DR also make it a perfect instrument for studying DDEs. The latter are prominent enough to clearly show up in highDR maps (see e.g. left inset of Fig. 1), but not severe enough to hinder the building up of very deep sky models during normal selfcal^{2}. The 3C 147 observations by de Bruyn et al. (2011, in prep.) are a perfect example of this. I have therefore decided to reprocess these data using MeqTrees, to see if DDEs can be eliminated through the use of a suitable form of the RIME.
In the presence of a dominant source (in this case 3C 147 itself), selfcal solutions will tend to subsume all effects in the direction of that source. DDEs will then manifest themselves as artefacts around other sources, which need to be at a certain distance from the dominant source for the effect to become apparent. Since the dominant source is usually placed at or near the pointing centre (i.e. onaxis), DDErelated artefacts are also called offaxis artefacts. The artefacts themselves are quite clear (Fig. 1, left inset): the nature of the effects responsible for them, far less so. At least four possibilities have been postulated:

Pointing errors (Paper II, Smirnov 2011, Sect. 2.1.4);

Differential tropospheric refraction (ibid., Sect. 2.2.4);

Errors in antenna positions, including noncoplanarity;

Other systematic coordinate errors.
One of the objects of this study was to narrow down these possibilities. Section 2 therefore analyses the differential gain solutions obtained during this calibration, with a view to characterizing the DDEs.
1. Calibration approaches and results
1.1. Observations and NEWSTAR reduction
The observational data in question were obtained by de Bruyn in 2003. A single 12h synthesis was taken, using 8 × 20 MHz bands (of 64 channels each) from 1300 to 1460 MHz, with 30 s integration time. Due to a backend problem, one of the crosscorrelations was corrupted, so only the XX and YY correlations were usable. De Bruyn then successfully reduced the observation using the NEWSTAR package, achieving a world record 1 600 000:1 dynamic range (de Bruyn et al. 2011, in prep.). Only regular selfcal was done and no peeling was attempted, so the resulting image showed DDErelated artefacts around offaxis sources (Fig. 1, left inset). One result of the reduction was a very deep NEWSTARformat sky model for the field, containing about 300 point sources. This provided a fantastic platform from which to begin my DDE study with MeqTrees. I had a readymade sky model that was known to be good enough to reach the thermal noise with this particular dataset, and I had intermediate images from de Bruyn’s reduction that could be used as checkpoints.
The same observation was repeated in 2006 with somewhat different correlator settings (see de Bruyn et al. 2011, in prep., for details), and reduced in a similar manner.
For his NEWSTAR reduction, de Bruyn selfcalibrated each channel independently, rather than explicitly calibrating for a bandpass (see below). This procedure is described in Paper II (Smirnov 2011, Sect. 1.1). It uses the following implicit form of the RIME: (1)and consists of the following steps:

1.
Find that minimizes  X_{pq} − D_{pq}  in a leastsquares sense. Compute “corrected data” as (The solution interval here was one timeslot, one frequency channel. Only 56 channels in each band were usable; these were further averaged down using Hanning tapering, so in the end only 28 frequency points per band were used.)

2.
Find that minimizes . Compute “corrected data” as , where “ ÷ ” is elementbyelement division – the inverse of “ ∗ ”. (The solution interval here was the full 12 h, one band.)

3.
Compute “residual data” as .
This procedure was repeated for each band. Residual visibilities were imaged and summed across all 8 bands, then deconvolved using Högbom CLEAN. The sky model was then added back in using a Gaussian restoring beam.
1.2. Calibration in MeqTrees: an overview
In broad terms, selfcal in MeqTrees also consists of a leastsquares fit of an equation such as (1) to the data. However, the following features are different:

The structure of the equation is not fixed: arbitrary forms of theRIME may be constructed. Crucially for my purposes, these mayinclude DDE terms.

The elements of the Jones matrices are not necessarily simple solvable parameters (though they may be), but can be represented by arbitrary functions. For example, rather than solve for EJones (or ZJones) elements directly, MeqTrees can derive them from some model of the primary beam (or ionosphere) and solve for the parameters of the model. An example of this is given in Sect. 1.6.

Different solvables may have different solution intervals, even in a simultaneous solution.
Because of the inherent flexibility of MeqTrees, calibration can be pursued in a great variety of ways. During this study, a specific methodology was narrowed down, implemented and tested. This became the basis of the “Calico” calibration framework that is now included with MeqTrees. The steps (and terminology) of calibration with Calico are as follows:
A desired form of the RIME is constructed, by selecting a skymodel, and picking a series of Jones terms (plus, optionally,interferometerbased errors). For example, a form similar toNEWSTAR’s implicit RIME (Eq. (1)), would be: V_{pq} is usually called the “corrupted predict”.
 2.
“Corrupted predict” is fitted to “data”. That is, the MeqTrees solver is instructed to find values of solvable parameters that minimize  D_{pq} − V_{pq}  in a leastsquares sense. This can be (and usually is) done in multiple stages, e.g. first, followed by , etc.
Output visibilities are computed as either “corrected data” or as “corrupted residuals” R_{pq} = D_{pq} − V_{pq}, or as “corrected residuals”
While Calico can produce “corrected data” (for purposes of imaging, etc.) at any step of the reduction, it does not use it as input for subsequent stages like NEWSTAR and other 2GC packages do. The fitting at step 2 is always done using the original^{3} observed data D_{pq}, and calibration consists of building up the RIME until the “corrupted predict” fits the observations. Once DDEs enter the picture, “corrected data” (in the conventional understanding of data corrected for instrumental errors) do not really exist any more, since visibilities can only be corrected for the value of a DDE in a particular direction. Hence the Calico philosophy is to work with the original data at all times.
1.3. Bandpass selfcal
A perchannel, pertimeslot solution for G_{p} in Eq. (1) has 2N complex unknowns, and N(N − 1) complex measurements, where N is the number of stations (14 for the WSRT). While an acceptable ratio (this is, after all, why selfcal works in the first place!), it does not leave a lot of room for introducing DDErelated parameters. In general, we want to allow our solutions as few DoF’s as possible, and making the solution intervals (in time and/or frequency) larger is one way of ensuring this.
The WSRT has a reasonably stable bandpass, so an obvious way to reduce the parameter count is to separate G_{p} into a bandpass component to capture the frequency structure (with little to none variation in time), and a frequencyindependent, rapidly varying complex gain. Per each station/receiver, this replaces N_{chan} × N_{time} parameters with only N_{chan} + N_{time} of them.
For the MeqTrees reduction, I therefore started with the following RIME: (2)Here, X_{spq}, E_{s}, G_{p} and M_{pq} have the same meaning as in the NEWSTAR equation above (including a similar smearing correction term in X_{spq}). The B_{p} term is a second diagonal Jones matrix representing the bandpass. Note that MeqTrees itself makes no special distinction between G and B. Both are generic diagonal complex Jones matrices, with solvable real and imaginary parts. It is only when we specify the solution intervals that these Jones terms acquire their intended meanings:

The G_{p} solution interval is 30 s, all channels (thus, one independent solution every timeslot, per the entire band).

The B_{p} solution interval was initially set to 30 min, and one channel (thus, one independent solution every channel, per 60 timeslots). Note that as the bandpass has significant structure, I did not attempt to fit it with any smooth function, but rather allowed each channel to be fitted as an independent parameter, with a timescale of 30 min. The latter interval was intended to accommodate slow drift in the bandpass.
Fig. 2 Singleband residual images produced via bandpass selfcal with different solution intervals for B_{p}: 30 min (upper left), 15 min (upper right), 7.5 min (lower left), and with the 7.5 min solution smoothed using splines (lower right). Even in the bestcase image, the dominant source 3C 147 was not subtracted out perfectly, leaving behind DRlimiting artefacts. 
The initial result of this calibration was profoundly unsatisfactory (Fig. 2, upper left). The residual image was dominated by spokelike artefacts centred on 3C 147, at about 10 000:1 level (relative the flux of 3C 147 itself). These spokes correspond to edges of the 30min solution intervals (being an EW array, WSRT has a onedimensional instantaneous PSF). The obvious explanation for the error is shortterm bandpass instability. Decreasing the solution interval of B_{p} to 15 and 7.5 min reduced the artefacts to levels of 50 000:1 to 100 000:1, but did not eliminate them entirely. Smoothing the 7.5 min solution with a spline (along the time axis, per each channel independently) produced a marginal improvement (Fig. 2, lower right). Subtraction artefacts are still plainly visible in the map, although at a level not significantly above thermal noise.
At this stage I had to conclude that the WSRT bandpass exhibits some very lowlevel, but extremely shortterm jitter, precluding a separate bandpass selfcal at extreme dynamic ranges. On the other hand, this result also shows that where a singleband dynamic range of no more than 100 000 is expected (as is the case for many other observations), bandpass selfcal provides perfectly adequate results, and can cut down on the number of solvable parameters significantly. In the meantime, for the 3C 147 study I had to revert to the perchannel selfcal approach of de Bruyn.
1.4. Perchannel selfcal
The RIME for perchannel selfcal is just Eq. (2) without the bandpass term. It is, in fact, very similar to NEWSTAR’s implicit Eq. (1)^{4}: (3)Perchannel selfcal is achieved by setting the solution interval of G_{p} to one channel and one timeslot. The resulting singleband residual images are dominated by closure errors at a level of ~100 μJy (or 1:200 000 relative to 3C 147 itself); these go away after an M_{pq} solution (Fig. 3). These images are qualitatively very similar to those obtained by de Bruyn during his NEWSTAR calibration (which is to be expected, given the similarity of the equations).
Fig. 3 Singleband residual images produced via perchannel selfcal. The left image is the result of solving for G_{p}. It is dominated by concentric rings centred on 3C 147 (designated as “A150” here). These are caused by closure errors, and go away once a solution for interferometerbased errors M_{pq} is done (right image). The remaining artefacts are associated with offaxis sources B, C and D, and are due to DDEs. 
The remaining artefacts in Fig. 3 are associated with the three nextbrightest sources^{5} in the field: B (42 mJy), C (52 mJy), and D (22 mJy). The furthest of these (B) is about 20′ away from centre. The artefacts themselves are under 50 μJy (thermal noise in one band being ~30 μJy), or at a level of 1:1000 relative to the associated sources. This is why de Bruyn (2006) talks about an “offaxis dynamic range” of a 1000: while 3C 147 itself (22 Jy) is subtracted without a trace (down to the thermal noise), the offaxis sources are only subtracted to a precision of about 1000. Some of the artefact structure is doubtlessly due to slightly under or overestimated sky model fluxes (this produces regular rings matching the WSRT PSF), which can be taken care of during subsequent deconvolution. Most of it, however, is due to DDEs and does not deconvolve, producing artefacts in the final 8band images such as the one shown in the left inset of Fig. 1 (the inset is, in fact, a closeup of source B).
1.5. Interferometerbased errors
It is not clear what causes closure errors at the WSRT. Common sense suggests the analogue part of the signal chain is to blame, but there is no hard evidence either way. What is evident is that highDR images exhibit concentric ringlike artefacts such as those in the left image of Fig. 3, and that these go away once a solution for an interferometerbased multiplicative error – the M_{pq} term of Eq. (3) – is applied. A single solution per band, per the entire 12 h (per correlation and interferometer) is sufficient. The M_{pq} solutions are usually within 10^{3}−10^{4} of unity (as is the case here), but can be much higher in some observations, for reasons that remain mysterious.
The latter fact suggests an intrinsic time variability, but solving for M_{pq} on short time intervals is very dangerous. Any solution for M_{pq} will also try to compensate for observed flux that is not present in the sky model. Unless the solution interval is sufficiently long, there will be unmodelled sources with a fringe rate slow enough that their vector average visibility over the solution interval will be significantly nonzero. These sources will then tend to be attenuated by the M_{pq} solutions. The 3C 147 observations provide a perfect example (Fig. 4). On the left is an 8band residual image with a 12 h M_{pq} solution; on the right is one with a 30 min solution. Model sources are indicated by crosses. Attenuation of unmodelled sources towards phase centre is clearly visible in the right image.
Fig. 4 Source suppression through interferometerbased error solutions. On the left is a deconvolved 8band residual image of the centre of the field, with 12 h solutions for M_{pq}. On the right is the same image with 30 min solutions. The positions of (subtracted) model sources are indicated by crosses. Suppression of unmodelled sources is evident in the right image. 
This implies that closure errors cannot be reliably solved for on observations shorter than 12 h, unless a “complete” (i.e. down to the noise) sky model of the center of the field is available.
1.6. Pointing selfcal
Since the main cause of artefacts in WSRT images is commonly considered to be pointing error (see Paper II, Smirnov 2011, Sect. 2.1.4), I decided to implement a form of the pointing selfcal algorithm suggested by Bhatnagar et al. (2004). This proved to be a straightforward exercise in MeqTrees, since only a small modification of the RIME of Eq. (3) was required: (4)Instead of a persource beam gain E_{s} = E(l_{s},m_{s}) (where E(l,m) is the cos^{3} approximation given by Paper II, Smirnov 2011, Sect. 2.1.1), which I had been using in the previous equations, here I used a perantenna, persource beam gain E_{sp}, defined as follows: (5)Perantenna pointing offsets δl_{p},δm_{p} were then treated as solvable parameters.
The results of this proved inconclusive. Even though the solution converged to some physicallysensible offsets (on the order of arcmin), no tangible reduction of artefacts was observed in residual images. This could be due to a number of reasons:

1.
The cos^{3} approximation is not good enough – unlikely, as it has been independently verified at least for the core of the main lobe, which the sources in question sources are well within.

2.
With only a few sources sufficiently bright to exhibit DDEs, we simply don’t have enough constraints for a pointing solution on this field.

3.
The model fluxes/positions for the sources in question are not sufficiently accurate.

4.
The dominant DDE affecting this observation is not pointing error. This will be elaborated on further in Sect. 2.

5.
There is something wrong with my implementation of pointing selfcal, especially since the figures in Sect. 2 suggest that mispointings are detectable.
Trying to get a better grip on the problem, I eventually settled on a “controlled experiment”: locating a field containing multiple bright offaxis sources, and observing it with deliberately exaggerated mispointings, to see if these can be more readily recovered from the data. This experiment became known as the “QMC Project” (in honour of the longdefunct Quality Monitoring Committee of WSRT), and was successfully carried out. The results of this are still being processed, and will be presented in a followup paper. In the meantime, I had to look for alternative approaches to DDEs in the 3C 147 field.
1.7. Differential gains: the “flyswatter”
In the spirit of “phenomenological” equations discussed in Paper II (Smirnov 2011, Sect. 1.3), I decided to introduce a differential gain Jones (ΔEJones) into my form of the RIME: (6)The ΔE_{sp} term is meant to subsume all DDEs associated with source s and antenna p (with the exception of the nominal primary beam gain, which is already represented by E_{s}). Solving for this term requires some caution, lest too many DoF’s be introduced into the equation. I approached this as follows:

ΔE was fixed at unity for all sources except B, C, and D;

For B, C and D, ΔE was set to a diagonal matrix with solvable complex elements;

The solution intervals for the ΔE elements were set to one solution per 30 min, per entire band (and per source, antenna, receptor).
Fig. 5 Results if the flyswatter. On the left is a singleband residual image after G_{p} and M_{pq} solutions only. On the right is the same image with differential gain solutions for sources B, C, and D. 
Once 8band residual images were created, the increased sensitivity made it apparent that four more sources were exhibiting a small amount of DDEs. A more indepth look at the sky model also showed that all seven sources were in fact slightly extended; the NEWSTAR model represented each by a tight cluster of point sources. Since all “sources” in such a cluster are subject to the same DDE, it seemed sensible to make sure the same ΔE_{sp} term was applied to all of them. This was easily done using Tigger, a sky model manager/viewer tool included with MeqTrees. Tigger automatically detects source clusters and assigns source identifiers appropriately (e.g. B232, B232a, ... B232e) It was then a simple matter to tell the Calico framework to use the same ΔE_{s0p} term for all sources s associated with cluster s_{0}.
Fig. 6 Positions (relative to nominal pointing centre) and aggregate fluxes (apparent) of the seven offaxis source clusters for which ΔE solutions were obtained. Circles are at a radius of 30′ and 1°. For reference, the FWHM of the WSRT voltage beam is ~50′ at 1.4 GHz. 
The seven source clusters for which differential gain solutions were eventually obtained are summarized in Fig. 6. Two of them are somewhat noteworthy. Source C270 is very close to centre, and therefore shouldn’t be affected by DDEs as much as the other sources. It is, however, a complicated and highly polarized source, so perhaps the artefacts it exhibits after regular selfcal are primarily due to sky model inaccuracies, which the ΔE solutions absorb (see discussion in Sect. 2.1). Source ae317 is almost the opposite: it is very faint, but far enough offaxis to be in a sidelobe of the primary beam, and so subject to especially severe DDEs.
1.8. The showcase result
The ultimate result of my calibration of the 2003 observation is shown in Fig. 1. This image is a true showcase for the differential gains approach. The precise steps leading to this image were as follows:
 1.
Each of the 8 bands was independently calibrated usingperchannel selfcal, interferometerbased errors, andΔE solutions on seven source clusters, as described above. Corrected residuals were generated.
 2.
The residuals for all 8 bands were imaged together (in MFS mode) to produce a single residual image. This revealed a large number of fainter sources not visible in the perband maps.
 3.
The 8band image was deconvolved using CottonSchwab CLEAN (Schwab 1984).
 4.
The sky model was added back into the deconvolved image, using a Gaussian restoring beam.
1.9. Flyswatter limitations
Fig. 7 Differential gainamplitudes (ΔE) as a function of time for the 2003 (top) and 2006 (bottom) observations. Rows correspond to sources, columns to antennas. The vertical plot scale is fixed within each row, but differs from row to row. Horizontal lines indicate the  ΔE  = 1 level. 
While it can help produce spectacular images, the flyswatter has some serious caveats and drawbacks that need to be explored. First of all, it is a bruteforce approach, in the sense that it squashes all effects into a single ΔE term. This includes inaccuracies in the sky model! Indeed, any missing source flux or error in source position can be accommodated with a suitable ΔE. Even unmodelled source structure can “leak” into differential gain solutions (Sect. 2.1). Thus, differential gains are good for subtracting sources, but at the cost of mashing up information on the source per se. (One does not use a flyswatter to probe a fly’s anatomy!)
Secondly, solvable differential gains can lead to a proliferation of DoF’s. Perchannel selfcal has N_{ant} unknowns per measurements, or unknowns per measurement; differential gains add unknowns per measurement, where N_{time} and N_{freq} are the sizes of the solution interval for ΔE. This ratio remains favourable for small N_{src} and large N_{time} and/or N_{freq} (as is the case for my 3C 147 reduction), but one must be careful.
The third caveat is processing cost. While usually not as expensive in terms of I/O or CPU as peeling (which, in addition to the solutions themselves, requires repeated subtraction and phase shifting steps), the flyswatter is not free. Every source with a differential gain solution adds 4N_{ant} unknowns (assuming a diagonal complex ΔE_{sp} term, hence 4 real values per matrix) to the equations. As the number of unknowns (N_{unk} = 4N_{ant}N_{src}) grows, inversion of the normal matrix within the leastsquares solver becomes a CPU bottleneck, since it scales as . This makes it impractical to solve for ΔE’s for more than a handful^{6} of sources at a time.
One way to mitigate the solver bottleneck is to decompose the ΔE_{sp}’s into nearlyorthogonal sets of unknowns. For example, we can treat the set of ΔE_{sp}’s associated with one source s as independent from all other sources. The (4N_{ant}N_{src})^{2} normal matrix inside the solver then becomes blockdiagonal, composed of N_{src} blocks of size (4N_{ant})^{2}. Inversion of this matrix then scales as O. This scheme was tested in MeqTrees, and it was found that the tradeoff is slower convergence, requiring more iterations. For large numbers of sources, however, this becomes very favourable.
2. Analysis of differential gain solutions
It is time to see whether any useful information can be gleaned from the differential gains solutions themselves. As a result of the reduction, I had obtained: per each source direction (7 of these: see Fig. 6), per each antenna (14), per each band (8), per 30min interval (24 of these in a 12h synthesis), two complex numbers representing the apparent differential gain of the X and Y dipole (“differential” being relative to the gain in the direction of 3C 147 – almost at the centre of the field – which had been taken care of by regular selfcal).
I then adopted the following approach. Given the relatively low fractional bandwidth, I didn’t expect much variation with frequency in ΔE. I therefore treated each set of 8 perband solutions as independent samples of the same variable. The mean of the 8 samples was used as an estimator of variable, and the standard deviation of the 8 samples as an estimator of the error (i.e. the error bar).
Since it quickly became apparent that the ΔE solutions were exhibiting some very interesting behaviour, I applied exactly the same procedure to the 2006 observations, so that comparisons could be made. The plots below show the results from both observations.
Figure 7 is a summary of the differential gainamplitudes per source, per antenna. The precise quantity plotted here was computed as follows. First, I computed the norm of the ΔE matrix (diagonal by construction) as
I then normalized (divided) this value by the mean value per source (that is, the mean across all time intervals, bands, and antennas), which was meant to take out the effect of incorrect model fluxes (see below). The resulting “normalized norm” was then plotted as a function of time, per source, per antenna.
Figure 8 is a similar plot of the differential gainphases, computed as
The most striking feature of Figs. 7 and 8 is the high SNR. They show a high degree of temporal continuity in the solutions, and statistically significant structure. This strongly suggests that the solutions represent real physical or numerical effects. As to the nature of these effects, I still do not have satisfactory answers, though it is hoped that the “QMC Project” mentioned earlier will shed some more light on the issues. The rest of this section discusses some of the more prominent questions, and proposes some rather speculative explanations.
2.1. Absorbing errors in the sky model
Fig. 8 Differential gainphases (argΔE, in degrees) as a function of time for the 2003 (top) and 2006 (bottom) observations. Rows correspond to sources, columns to antennas. The vertical plot scale is fixed within each row, but differs from row to row. Horizontal lines indicate the argΔE = 0 level. 
As already mentioned, a major caveat of the flyswatter approach is that ΔE solutions will tend to absorb inaccuracies in the source model. By analogy (and for exactly the same reasons), classic selfcal alone cannot solve for absolute positions or fluxes. Indeed, if the true position of a source l = (l,m) is offset from the model position l^{(mod)} = l + δl = (l + δl,m + δm), while the true brightness differs from the model brightness by a multiplicative matrix factor: (the latter being a straightforward generalization of a scalar factor a^{2}), then the coherency term of the RIME for the model source may be written out as If a solvable differential gain is then assigned to the source, the model can be made to fit the data by absorbing the K_{p}(δl)A factor into the ΔE_{ps} solutions.
Even more insidiously, differential gains can absorb some source structure. Consider a source that is slightly extended in one direction, enough to be resolved on the longest baselines. An EW array like the WSRT has a onedimensional instantaneous fan beam. It will “see” the source as a point source when the fan beam is aligned with the source orientation, and start resolving it when the fan beam becomes perpendicular to the source. In other words, the apparent flux of the source will remain constant in time on short baselines, and vary in time on the long baselines as the source resolves. If such a source is represented by a point source in the sky model, the model flux will be constant on all baselines. Now, if some antennas are predominantly involved in long baselines (RTC and RTD, in the case of WSRT), ΔE solutions can compensate for some of the flux discrepancy by changing the gainamplitudes of these antennas. I would expect to see a variation of  ΔE  with a 12h period. Since most of the baselines to RTC and RTD are mutually redundant (0C equals 1D, etc.), their variation in  ΔE  should be very similar.
This is exactly what we’re seeing in Fig. 7! The plots very strongly suggest that the top three sources (B, C and D) are indeed slightly extended (more so than in the model, that is). If this is the case, then the dominant contribution to ΔE on antennas RTC and RTD is due to source structure rather than any actual DDE.
2.2. Amplitude behaviour
Fig. 9 “Rogues gallery” plot for the 2003 observation. This shows the 12h average ΔE per source, as seen by each antenna. Blue circles correspond to values of ΔE > 1, red circles to values of ΔE < 1, and areas are proportional to   ΔE −1  . Line thickness indicates the statistical significance of   ΔE  −1 ; filled circles are for detections of over 3σ. The large grid circle is at radius 30′. 
Quite a few plots in Fig. 7 do show (mostly) static offsets. It can be illuminating to present ΔE in a format I call the “rogues gallery” (Figs. 9 and 10). This shows, for each of the 14 antennas, a 12h average ΔE per source, using circles of varying size placed at the position of the source. The magnitude of (  ΔE −1) is indicated by circle size, and the sign by colour. A static mispointing in, e.g., a Northern direction would show up as blue circles in the top half of the plot (i.e. sources appearing brighter), and red circles in the bottom half.
The galleries show exactly such a pattern for antennas RT5 through RT8 (and perhaps RT4), for both the 2003 and, to a lesser extent, the 2006 observations. It is a little bit strange that 4 (or even 5) adjacent antennas would so consistently mispoint North, and do the same three years later. Perhaps this is another, poorly understood consequence of unmodelled source structure. Some pointers to this are that antennas RT4–8, being in the middle of the array, form up predominantly shorter baselines, and that the longbaseline antennas RTC and RTD exhibit the opposite behaviour. (What hinders such an analysis is the unfortunate fact that the three brightest offaxis sources all exhibit some structure, and all three lie in the bottom half of the field.) Another puzzling feature is the consistently low ΔE for sources F, H and K on antenna RTC in 2003 (and to a far lesser extent in 2006). If due to source structure, why does it not repeat on RTD? Perhaps RTC is mispointing to the South?
Antennas RT0–2, RT9 and RTA, on the other hand, show completely different patterns, with little to no similarity between 2003 and 2006. Some of these are consistent with a static mispointing. Some antennas (RT8 and RT9, and RTB especially) also show a hint of time variability in ΔE .
In any case, it is clear that the complicated interaction between source structure and differential gainamplitudes makes the latter extremely difficult to interpret. Note also that my (or rather de Bruyn’s) source model was built by NEWSTAR based on regular selfcal, so there’s bound to be some contamination from DDErelated artefacts in the source parameters. Truly robust methods for disentangling source structure from DDEs have yet to be developed. It is also clear that an approach that parametrizes the DDEs in a “global” way, such as pointing selfcal, is the way forward – what is not yet clear is how much the global solution itself can be affected by unmodelled structure in the brighter sources, and what to do about it.
2.3. Phase behaviour
Fig. 11 Phase slopes over the array as a function of time (in deg/km) in the direction of the seven sources for the 2003 (top) and 2006 observations (bottom). The green lines indicate phase slopes corresponding to the fitted position offsets (Fig. 12), the red lines – phase slopes corresponding to an overall field rotation of 45″. 
Fig. 12 Fitted position offsets corresponding to the phase slopes of Fig. 11 (2003 observation on the left, 2006 on the right). The length of the arrows is exaggerated by a factor of 1200: the biggest offset is in fact just under 1″. 
A rather prominent feature of the phase plots in Fig. 8 is their continuity from antenna to antenna, and the fact that the phases at the two ends of the array exhibit opposite temporal trends. This is suggestive of an evolving phase slope over the array. Fitting such a slope (per source) produces some very striking results (Fig. 11). The dominant phase effect is clearly global rather than antennabased, and is extremely consistent across both observations.
A phase slope over the array can be interpreted as an apparent position offset. It appears that the slope behaviour in Fig. 11 can be fitted quite well by constant position offsets. The bestfitting position offsets are indicated in Fig. 12, and the corresponding slope curves are plotted in green in Fig. 11.
Figure 12 immediately suggests a field rotation. And indeed, the entire collection of phase slopes (for both the 2003 and 2006 observation), is, to first order, consistent with a rotation of 45″ around the phase centre. The corresponding slope curves are plotted in red. While there are some significant differences in the brighter sources, it seems clear that the dominant effect is not an instrumental DDE at all, but a systematic rotation of the sky model. The model positions are derived by NEWSTAR from direct fits to the visibilities, and de Bruyn (priv. comm.) has independently crosschecked the positions of distant sources against the NVSS, which seems to preclude a rotation in the model itself. Note that a 45″ rotation can also be introduced by a clock error of about 2.9 s, or a corresponding rotational error in conversion of uvw coordinates from apparent to J2000. Since NEWSTAR and MeqTrees use completely different tool chains and visibility data formats, I cannot exclude a coordinate conversion error somewhere along the line. This needs to be urgently investigated. If indeed the entire sky model is slightly rotated, then perhaps the image of Fig. 1 can even be improved upon!
2.4. Feeding differential gains back into the sky model
The results above suggested that I could improve my sky model by feeding back in some information extracted from the ΔE solutions. In the previous section, I obtained a correction to the model positions of the seven sources^{7}. Following the discussion of Sect. 2.1, I could also provide corrections for the I and Q fluxes by applying the persource average ΔE amplitudes: where t_{i} and ν_{j} represent the time and frequency solution intervals of ΔE. In terms of the I and Q fluxes, the correction becomes: I therefore applied these corrections for I, Q, and position to my sky models (independently for the 2003 and 2006 observations), and repeated the calibration procedure. An improvement in singleband residuals was immediately apparent (Fig. 13) – after G_{p} and M_{pq} solutions, the seven offaxis sources subtracted noticeably better. Residuals after ΔE_{sp} solutions, on the other hand, looked pretty much the same (this is not surprising, since differential gains had already taken care of the visible offaxis errors in the original reduction), with a very slight improvement around 3C 147 itself, which can be explained by improved G_{p} solutions due to the more accurate sky model.
Fig. 13 Calibration with an improved sky model. This shows singleband residual images after G_{p} and M_{pq} solutions. The left image is from the original reduction, the right image uses a sky model improved via my ΔE analysis. Crosses indicate the positions of sources for which the model was improved, plus 3C 147 itself (source A). 
2.5. Phase behaviour II
Presumably, the remaining residual structures in Fig. 13 are more representative of the instrumental DDEs per se, since inaccuracies in the sky model have been significantly reduced. We should also expect the differential gain solutions to be more indicative of the actual DDEs (apart from the issue of resolved sources affecting ΔE on antennas RTC and RTD, which the improved sky models do not address at all). Of particular interest is the effect that the improved model has on the differential gainphases. As for the gainamplitudes, we would expect them to differ by only an overall persource scaling factor. Indeed, making the same ΔE plot as in Fig. 7 confirms this – it is, to all intents and purposes, identical (and omitted here to save space), since the plotted amplitudes are renormalized by the persource average ΔE .
The phases, on the other hand, show a marked difference, since the formerly dominant effect – that of position offsets – has been taken out. The argΔE solutions themselves are shown in Fig. 14. Phase slopes are still very much in evidence, as can be seen in Fig. 15. Somewhat surprisingly, these slopes indicate that some residual position offsets remain, at a level of 15% to 20% of the original offsets (Fig. 16). This suggests that my procedure of fitting phase slopes to argΔE solutions, followed by fitting position offsets to the slopes, systematically underestimates the true position offsets. This is possibly an effect of the complex averaging implicit in having one ΔE solution per a relatively large solution interval (20 MHz by 30 min). If so, this could perhaps be incorporated as a multiplicative correction factor in the model update procedure. Further work is required to fully understand the effect.
The brighter sources B, C, and (to a lesser extent) D show clear secondorder phase effects, both in the phase slopes, and in the phase solutions themselves. The temporal continuity in the phase slopes can be interpreted as a timevariable position offset. I can speculatively offer two explanations for such an offset:

Unmodelled source structure (again!) For any given hour angle,an EW array only sees an integrated crosssection through thesource in a given direction. If the source is slightly resolved withan asymmetric “hotspot”, the zeroorder moment of each suchcrosssection will be slightly different.

Differential tropospheric or largescale ionospheric refraction, including perhaps apparent change of baseline caused by refraction (the Anderson effect).
Another puzzling feature of the argΔE solutions in Fig. 14 are the significant and (to first degree) constant phase offsets of some sources (e.g. H, K, ae) on some antennas. The offsets are mostly (though not completely) consistent between the 2003 and 2006 observations. None of the explanations offered above are consistent with a constant phase offset! Could this be the phase component of the primary beam? There are too few sources in this reduction to infer any sort of directional dependence, but perhaps the “QMC Project” can provide more insights on this effect.
2.6. The lurking errors
The two calibrations (with the original and the improved sky models) described above have produced what appear to be identical final maps. This shows that the “flyswatter” can accommodate for significant errors in the sky model. On the other hand, the detailed structures in Fig. 15 suggest that (even in the very benign case of 21 cm WSRT observations!) moderately bright offaxis sources still require some form of DDE correction even if the model is perfect. If this is the case, then a legitimate question is: why worry about getting the sky model right, if we need to do ΔE solutions anyway, which will absorb any imperfections? (Besides the obvious caveats of the “flyswatter” discussed in Sect. 1.9, that is.)
Fig. 14 Differential gainphases (argΔE, in degrees) as a function of time, using improved sky models for the 2003 (top) and 2006 (bottom) observations. Compare to Fig. 8. 
The rather striking image of Fig. 17 shows that the final maps are not in fact identical, although the difference is buried in the noise. This image was produced by subtracting the originalmodel 8band residual map from the improvedmodel map (2003 observation). Since the noise term in both maps is the same, subtraction reveals very faint structures that would normally be hidden in the noise. We’re beginning to see more limitations of the “flyswatter”. In the original reduction, apparent position offsets of the offaxis sources caused phase gradients in t,νspace in the differential gainphases. These were approximated by a stepwise ΔE solution (since I solved for only one ΔE term per 30 min, per entire band), which proved to be good enough to drive offaxis errors to a level below the thermal noise. Improving the model positions has effectively “flattened out” these gradients, reducing the error made by a stepwise approximation even further. Figure 17 demonstrates the improvement. The radial spokes correspond to “jumps” at the boundaries of the solution intervals, but the other structures (especially the halfcircles) are rather more difficult to explain, and will have to be addressed in followup work.
The implications of this result is that any errors in the sky model (or uncorrected DDEs) will propagate into the selfcal solutions, and result in faint but highly coherent structures in the residual maps. We may think we are reaching the thermal noise, but in the process, we are producing “submerged” calibration artefacts at levels below the noise, where we can’t even see that something is still going wrong! This is of particular concern to ongoing work on detection of the Epoch of Reionization (EOR) signature, which relies on statistical analysis of residual images to find subnoise artefacts of astrophysical origin (Harker et al. 2010; Morales et al. 2006). Such analysis will have to reckon with these lurking selfcal artefacts.
Fig. 15 Phase slopes over the array as a function of time (in deg/km), using improved sky models for the 2003 (top) and 2006 observations (bottom). The green lines indicate phase slopes corresponding to the fitted position offsets (Fig. 16). Compare to Fig. 11. 
Fig. 16 Fitted position offsets corresponding to the phase slopes of Fig. 15 (2003 observation on the left, 2006 on the right). The length of the arrows is exaggerated by a factor of 1200: the biggest offset is in fact just under 0.2″. Compare to Fig. 12. 
3. Conclusions
One of the biggest selling points of the RIME formalism is the flexibility it offers for describing observational effects. Unfortunately, to date only three software packages have exploited the power of the RIME (CASA, MeqTrees, and the LOFAR BBS system). Of these, only MeqTrees allows for truly arbitrary forms of the RIME. This paper has explored some practical applications of one such form of the RIME: a form that includes differential gain terms. I have demonstrated that the differential gain approach (the “flyswatter”) can be a powerful way of dealing with DDEs on a sourcebysource basis. This has been used with WSRT data to produce artefactfree maps of 3C 147 at record dynamic ranges of well over a milliontoone. While the differential gain solutions themselves absorb inaccuracies in the sky model as well as the DDEs themselves, I have demonstrated that at least flux and positions corrections can be recovered, so iterative improvements to the sky model are possible.
The latter may also prove be necessary: I have demonstrated that even a perfectlooking map produced using differential gains contains a large number of selfcal artefacts hidden in the thermal noise, which can be significantly reduced by improving the sky models. These “invisible” artefacts have hitherto been ignored, but they should be of particular concern to projects relying on statistical signal extraction, such as the ongoing search for the EOR signature.
Fig. 17 Calibration with an improved sky model. This shows the difference between the 8band residual maps (2003 observation) produced with the original and the improved sky models. Structures around 3C 147 itself and the offaxis sources are mostly due to “selfcal contamination” in the original model caused by incorrect offaxis source positions. These are well within the noise: the intensity range of this image is ± 2 μJy, while the 8band maps have a thermal noise of 13.5 μJy. 
The nature of the remaining DDEs (as seen in the differential gain solutions) has not yet been adequately explained. Some of the amplitude effects are consistent with pointing error. The phase behaviour is even more difficult to understand, but may be due to unmodelled source structure. Further work is required on the subject.
I have shown that differential gainphase solutions can be used to detect position shifts to within small fractions of the synthesized beam size. Offsets of less than 0.05″ (well under 0.01 of the PSF size!) have been reliably detected. There is a very clear indication of a systematic rotational offset of ~45″ in the sky model generated by NEWSTAR, when interpreted using MeqTrees. This is may be due to a coordinate conversion error somewhere in the visibility data processing tool chain, and needs to be investigated further.
Finally, I should consider some wider implications of my results. All currently mooted schemes of DDE calibration for LOFAR (Nijboer & Noordam 2007), the MWA (Mitchell et al. 2008) and the ionosphere in general (Intema et al. 2009; Cotton et al. 2004) revolve around the use of “beacon sources” to probe the ionosphere and/or the primary beam. It is rather difficult to envisage a closedloop scheme without beacons (how else would one sample a DDE?), so future telescopes such as the SKA will most likely need to use something very similar. Any such scheme predicates on there being a sufficient number of sufficiently bright inbeam beacons for any direction on the sky. This is not a problem at the LOFAR and MWA end of the spectrum, since the lowfrequency sky is so much brighter, but it has been a bit of a worry for the higher frequencies, where FoVs are narrower and sources are fainter.
My 3C 147 results suggest that calibration beacons can be a lot fainter than previously thought. What has been established
is that for this particular configuration of the WSRT, sources as faint as 2 mJy can provide meaningful DDE solutions. This result can be scaled to future telescope designs by comparing their expected sensitivity with that of the 3C 147 observation.
Strictly speaking, this is only true in continuum mode. In spectral line mode, the 17 MHz “ripple” discussed in Paper II (Smirnov 2011, Sect. 2.1.1) becomes a very troublesome DDE. Further work is required on this subject.
With the exception of the position of the term, which is on the inside of Eq. (1), and on the outside here. For this particular dataset it makes no difference: since only the and correlations are used, all matrices in Eq. (1) are diagonal, and for diagonal matrices the “” operator is equivalent to matrix multiplication, and commutes. For the fullpolarization case, the two approaches are not equivalent.
The source IDs used here follow the “COPART” (Clustering, Order, Position Angle, Radius, Type) convention, as implemented by the Tigger sky model management tool (available with MeqTrees). A COPART source ID starts with an alphabetic designator (A, B, ... Z, aa, ab, ...) assigned to sources in order of decreasing brightness. This is already a unique identifier, and is sometimes used by itself for brevity, as in the paragraph above. A full ID also encodes approximate position relative to field centre: two digits for the position angle (in units of ), and one digit for radial distance (in units of ). Optional suffixes indicate source type and clustering. Thus the full ID of source B is B232; being slightly extended, in this particular sky model it is actually represented by a cluster of six delta functions: B232, B232a, ... B232e.
Acknowledgments
I would like to thank a succession of managers for putting up with me all these years, and Jan Noordam for making this process considerably easier (especially for the managers), and for many other things besides. Ger de Bruyn has been more than generous with data, models and wisdom. Johan Hamaker started it all, and Wim Brouw has provided an avalanche of insights. Last but certainly not least, the rest of the MeqTrees team has been instrumental in making everything work.
References
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All Figures
Fig. 1 “Showcase” image of the field around the bright radio source 3C 147, produced after reduction with MeqTrees. The image is noiselimited, and has a dynamic range of 1.6 million. This DR was already achieved by de Bruyn using regular selfcal in NEWSTAR, but the resulting images contained artefacts around offaxis sources (left inset) due to DDEs. A MeqTrees reduction incorporating differential gains, as described in this paper, has completely eliminated the artefacts (right inset). This image also appears in Noordam & Smirnov (2010). 

In the text 
Fig. 2 Singleband residual images produced via bandpass selfcal with different solution intervals for B_{p}: 30 min (upper left), 15 min (upper right), 7.5 min (lower left), and with the 7.5 min solution smoothed using splines (lower right). Even in the bestcase image, the dominant source 3C 147 was not subtracted out perfectly, leaving behind DRlimiting artefacts. 

In the text 
Fig. 3 Singleband residual images produced via perchannel selfcal. The left image is the result of solving for G_{p}. It is dominated by concentric rings centred on 3C 147 (designated as “A150” here). These are caused by closure errors, and go away once a solution for interferometerbased errors M_{pq} is done (right image). The remaining artefacts are associated with offaxis sources B, C and D, and are due to DDEs. 

In the text 
Fig. 4 Source suppression through interferometerbased error solutions. On the left is a deconvolved 8band residual image of the centre of the field, with 12 h solutions for M_{pq}. On the right is the same image with 30 min solutions. The positions of (subtracted) model sources are indicated by crosses. Suppression of unmodelled sources is evident in the right image. 

In the text 
Fig. 5 Results if the flyswatter. On the left is a singleband residual image after G_{p} and M_{pq} solutions only. On the right is the same image with differential gain solutions for sources B, C, and D. 

In the text 
Fig. 6 Positions (relative to nominal pointing centre) and aggregate fluxes (apparent) of the seven offaxis source clusters for which ΔE solutions were obtained. Circles are at a radius of 30′ and 1°. For reference, the FWHM of the WSRT voltage beam is ~50′ at 1.4 GHz. 

In the text 
Fig. 7 Differential gainamplitudes (ΔE) as a function of time for the 2003 (top) and 2006 (bottom) observations. Rows correspond to sources, columns to antennas. The vertical plot scale is fixed within each row, but differs from row to row. Horizontal lines indicate the  ΔE  = 1 level. 

In the text 
Fig. 8 Differential gainphases (argΔE, in degrees) as a function of time for the 2003 (top) and 2006 (bottom) observations. Rows correspond to sources, columns to antennas. The vertical plot scale is fixed within each row, but differs from row to row. Horizontal lines indicate the argΔE = 0 level. 

In the text 
Fig. 9 “Rogues gallery” plot for the 2003 observation. This shows the 12h average ΔE per source, as seen by each antenna. Blue circles correspond to values of ΔE > 1, red circles to values of ΔE < 1, and areas are proportional to   ΔE −1  . Line thickness indicates the statistical significance of   ΔE  −1 ; filled circles are for detections of over 3σ. The large grid circle is at radius 30′. 

In the text 
Fig. 10 “Rogues gallery” plot for the 2006 observation, using the same scale as Fig. 9 

In the text 
Fig. 11 Phase slopes over the array as a function of time (in deg/km) in the direction of the seven sources for the 2003 (top) and 2006 observations (bottom). The green lines indicate phase slopes corresponding to the fitted position offsets (Fig. 12), the red lines – phase slopes corresponding to an overall field rotation of 45″. 

In the text 
Fig. 12 Fitted position offsets corresponding to the phase slopes of Fig. 11 (2003 observation on the left, 2006 on the right). The length of the arrows is exaggerated by a factor of 1200: the biggest offset is in fact just under 1″. 

In the text 
Fig. 13 Calibration with an improved sky model. This shows singleband residual images after G_{p} and M_{pq} solutions. The left image is from the original reduction, the right image uses a sky model improved via my ΔE analysis. Crosses indicate the positions of sources for which the model was improved, plus 3C 147 itself (source A). 

In the text 
Fig. 14 Differential gainphases (argΔE, in degrees) as a function of time, using improved sky models for the 2003 (top) and 2006 (bottom) observations. Compare to Fig. 8. 

In the text 
Fig. 15 Phase slopes over the array as a function of time (in deg/km), using improved sky models for the 2003 (top) and 2006 observations (bottom). The green lines indicate phase slopes corresponding to the fitted position offsets (Fig. 16). Compare to Fig. 11. 

In the text 
Fig. 16 Fitted position offsets corresponding to the phase slopes of Fig. 15 (2003 observation on the left, 2006 on the right). The length of the arrows is exaggerated by a factor of 1200: the biggest offset is in fact just under 0.2″. Compare to Fig. 12. 

In the text 
Fig. 17 Calibration with an improved sky model. This shows the difference between the 8band residual maps (2003 observation) produced with the original and the improved sky models. Structures around 3C 147 itself and the offaxis sources are mostly due to “selfcal contamination” in the original model caused by incorrect offaxis source positions. These are well within the noise: the intensity range of this image is ± 2 μJy, while the 8band maps have a thermal noise of 13.5 μJy. 

In the text 
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