Revisiting the radio interferometer measurement equation. III. Addressing direction-dependent effects in 21 cm WSRT observations of 3C 147

Papers I and II of this series have extended the radio interferometry measurement equation (RIME) formalism to the full-sky case, and provided a RIME-based description of calibration and the problem of direction-dependent effects (DDEs). This paper aims to provide a practical demonstration of a RIME-based approach to calibration, via an example of extremely high-dynamic range calibration of WSRT observations of 3C 147 at 21 cm, with full treatment of DDEs. 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 interferometer-based errors and for differential gains. The resulting image of the field around 3C 147 is thermal noise-limited, has a very high dynamic range (1.6 million), and none of the off-axis artefacts that plague regular selfcal. The differential gain solutions show a high signal-to-noise ratio, and may be used to extract information on DDEs and errors in the sky model. The differential gain approach can eliminate DDE-related 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 DDE-related schemes.