Imaging and internal calibration

Once the data had been averaged and edited of bad points and the visibility amplitudes calibrated it was used to produce an estimate of the surface brightness distributions of the radio sources, partly via the equations listed in 2.1.

A major obstacle in applying the equations of aperture synthesis to VLBI data is that while the visibility amplitudes can usually be calibrated with an accuracy of better than 10%, it is not possible to calibrate the visibility phases in an absolute sense. The external methods of calibrating phase with a connected element interferometer do not apply to VLBI observations, as each antenna possesses a completely independent frequency standard.

Thus, the phases from the fringe-fitting process are not absolute phases and appear quasi-random in time. The only well behaved properties of the phases are their closure relations [Jennison 1958; Rogers 1974]. The relation between the measured baseline phase and the true baseline phase is [Pearson & Readhead 1984]

where is the observed phase, is the true phase, and are the antenna dependent phase errors, and is the noise in the measurement on the m-n baseline. If the closure phase,

is formed then it can be seen that

depends upon the true phases, degraded only by noise. A somewhat similar closure relation also exists for visibility amplitudes [Pearson & Readhead 1984].

A method for the internal calibration of the visibility phases is generally utilised so that the visibility function can be reconstructed and used to produce images of the source structure. This internal calibration method relies heavily on the visibility closure phases and is known as self-calibration [Pearson & Readhead 1984]. The method proceeds as follows: an approximate model for the source structure is guessed, usually from an inspection of the visibility amplitudes. The closure phase information from the data, augmented by phases predicted from the approximate model are used to form a complete set of visibility phases which, along with the measured visibility amplitudes, can be inverted to produce a dirty image of the source. A deconvolution algorithm is then used to subtract copies of the dirty beam from the dirty image, producing a set of point sources which is a better representation of the source structure. This new representation of the source structure can then be used to generate a new set of visibility phases which, in turn, can be used to produce a new dirty image, a new source model, new visibility phases, etc. Thus, imaging using self-calibration is an iterative procedure, eventually converging to the point where new inversions of the data do not improve the quality of the image. Details of self-calibration imaging, the various deconvolution techniques and image restoration techniques can be found in the review by Pearson and Readhead [1984].

For the SHEVE data presented in this thesis, self-calibration techniques, as implemented in the DIFMAP imaging software [Shepherd, Pearson & Taylor 1994] have been used to produce the VLBI images. DIFMAP uses a difference mapping algorithm which de-convolves regions of the dirty image specified by the user and accumulates a list of point sources which represents the source. As the dirty image is de-convolved the list of point sources grows and the user is left with a residual dirty image, the difference between the original dirty image and the accumulated point sources. DIFMAP allows the dirty image to be de-convolved and self-calibrated in a piecewise manner, in a way which is easy to visualise and which allows the outcomes of different choices of imaging procedure to be explored efficiently.

This last point is very important for the imaging analysis of SHEVE data. The SHEVE array generally operates with between 4 and 7 antennae. The inner u-v coverage of the array is generally concentrated in a north-south direction, due to the geographic locations of the antennae within Australia, and large gaps can exist within the outer u-v coverage if Hartebeesthoek data is retained. This can make imaging difficult. To ensure that the images in this thesis are good representations of the source structures, careful analyses were made of all of the data sets, according to a basic imaging philosophy; the images should be as simple as possible and still model the data very well, and each dataset should be imaged on its merits, with any a priori knowledge of the source structure not influencing the choices made during imaging. The imaging choices made in producing each image in this thesis were, therefore, as follows:

• A uniform weighting scheme was employed for data in the u-v plane.
• Pixel sizes between 0.1 and 2 mas and map sizes between 512 and 1024 pixels were used throughout. The particular combination of pixel and map size depended on the extent of the u-v coverage for the observation.
• Initially the visibility phases were self-calibrated with a 1 Jy point source.
• The dirty image and consequently the residual dirty images were de-convolved using iterations of 100 cleans with a loop gain of 0.1. Windows were initially set tightly around the brightest region of emission.
• Phase self-calibration and cleaning proceeded. Gradually the number of clean windows was increased to incorporate more of the source emission, until the total flux density in the source model was comparable to the visibility amplitudes.
• An amplitude self-calibration was performed, generating a single amplitude correction for each antenna and applying it to the data. Generally the amplitude self-calibration significantly reduced noise in the residual image due to errors in the a priori flux density calibration, allowing more emission to be cleaned from the clean windows and further phase self-calibration to be performed.
• Eventually, more amplitude self-calibration was performed over shorter time-scales, so as to remove short time-scale variations in the amplitude gains. These further amplitude self-calibrations were always accompanied by one or more iterations of phase-only self-calibration. The last amplitude self-calibration was performed on the data at each u-v point.

Many trial maps of each dataset were produced, exploring slightly different sequences of phase and amplitude self-calibration, and windowing. In all cases it was found that the results of imaging were repeatable, and not sensitive to small changes.

Having obtained the final images for each of the sources, specific analyses of the images could be made. The analyses particular to a given source are discussed in the relevant chapter.