Bayesian estimation for ionospheric calibration in radio astronomy

Radio astronomical observations at low frequencies (< 250 MHz), can be severely distorted by fluctuations in electron density in the ionosphere. The free electrons cause a phase change of electromagnetic waves traveling through the ionosphere. This effect increases for lower frequencies. For this reason observations at low frequencies have been limited to short baselines and hence poor angular resolution. Most radio astronomical observations today are done at higher frequencies. The lower frequency bands however contain signals that are of great scientific value. Due to the expansion of the universe signals from distant objects are redshifted, i.e. shifted to lower frequencies. The more distant an observed object is the further we look back in time. An important period in the history of the universe is the ”Epoch of Reionization” (EoR). A few hundred thousand years after the Big Bang the universe has cooled enough to allow the formation of neutral hydrogren and helium. When the universe was a few hundred million years old the EoR started and the almost completely neutral gas was ionized again. Probably the only method to trace the neutral gas in this period is through the 1420 MHz spectral line of neutral hydrogen. For the EoR this line is redshifted to somewhere probably 100 and 200 MHz. Recently it has in principle become possible to observe with high resolution at low frequencies because the ever increasing computing power of digital data processing devices has made it possible to correct for the effect of the ionosphere. Determining the necessary corrections to the effect of the ionosphere is calles ionospheric calibration. The reason that ionospheric calibration is difficult is that the gain is direction dependent and rapidly varying over time. This greatly increases the number of degrees of freedom which causes two problems. First, the estimation of a large number of parameters is computationally costly. Second, the more parameters need to be estimated, the larger the estimation error will be. Without further constraints the signal to noise ratio of the calibrator sources is too low to accurately estimate the free parameters. The first problem can in principle be tackled by a brute force approach by simply increasing the data processing capacity. In practice an efficient algorithm is needed. The second problem is more fundamental in nature. A good model of the ionosphere, including as much prior knowledge about the ionosphere as possible, is needed to reduce the number of degrees of freedom. The first problem is addressed in this thesis by an analysis of a proposed calibration method called “Peeling”. This is a calibration technique whereby the Least Squares (LS) optimization problem is sequentially solved for different calibrator sources. This can be computationally more efficient than joint estimation. Our analysis by simulation of a realistic target field finds that “Peeling” reaches the theoretically optimal result in a few iterations. The second problem is addressed by proposing a stochastic ionospheric model based on a single layer of Kolmogorov turbulence. The stochastic model consists of a parametric description of the spatial power spectral density of the ionospheric electron density fluctuations. This model is verified by GPS observations and low frequency observations from the Very Large Array (VLA). An optimal estimator for this model is the Bayesian Minimum Mean Square Error (MMSE) estimator. This estimator is impractical due to the necessary numerical integration of high dimensional integrals. The Maximum a Posteriori (MAP) estimator is an approximation of theMMSE estimator which leads to a Least Squares (LS) problem that can be solved efficiently by standard techniques. Simulations show that the MAP estimator based on the power law model performs better than estimation based on a Zernike polynomial model for the ionosphere. The MAP estimator has been incorporated into the software package SPAM (Source Peeling and Atmospheric Modeling). SPAM has been used on three test cases, a simulated visibility data set and two selected 74 MHz VLA data sets. This resulted in significant improvements in image background noise (5–75 percent reduction) and source peak fluxes (up to 25 percent increase) as compared to the existing self-calibration and field-based calibration methods. The improved image quality indicates a significant improvement in ionospheric phase calibration accuracy. For this particular single layer ionospheric model the results are encouraging. It is indicated how the MAP estimator can be applied to possible extensions of the model including the addition of a third spatial dimension and the time dimension