Self-Calibration for the LOFAR Radio Astronomical Array

LOFAR is a low-frequency radio astronomical array currently under development in The Netherlands. It is designed to produce synthesis images of the most distant celestial objects yet observed. Due to high redshift levels, observations must be at unusually low frequencies (30-240 MHz), over large apertures (100 km), using thousands of antennas. At these frequencies, Earth's ionosphere acts as a random refractive sheet which over the large aperture induces source direction dependent gain and phase errors that must be estimated and calibrated out. Current radio astronomy "self-calibration" algorithms do not address direction dependence and will not work in the LOFAR environment. This paper presents a formal study of the parameter estimation problem for LOFAR calibration. A data model is proposed, and a Cramer-Rao lower bound (CRB) analysis is developed with a new general formulation to easily incorporate a variety of constraining signal models. It is shown that although the unconstrained direction dependent calibration problem is ambiguous, physically justifiable constraints can be applied in LOFAR to yield viable solutions. Use of a "compact core" of closely spaced array elements as part of the larger array is shown to significantly improve full array direction dependent calibration performance. Candidate algorithms are proposed and compared to the CRB.