Investigations of Faraday Rotation Maps of Extended Radio Sources ...

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74 CHAPTER 4. PACMAN – A NEW RM MAP MAKING ALGORITHM Taking this into account, a least squares fit to calculate RM ij and ϕ 0 ij at each pixel has to be applied by allowing all possible combinations of n ij (k)π while determining the n ij (k) for which the χ 2 ij is minimal. The presence of observational noise might cause a standard least squares fit, as suggested by Vallée & Kronberg (1975), Haves (1975) or Ruzmaikin & Sokoloff (1979), to choose a spurious RM value especially for areas of low signal-to-noise. The idea of Pacman is to reduce the number of pixels for which the nπ-ambiguity has to be individually solved. This is done by splitting the solution of the nπ-problem into two problems, a local and a global one, n ij (k) = ñ ij (k) + n(k), (4.2) where ñ ij (k) is the local solution, linking polarisation angles of neighbouring pixels within a frequency map, and n(k) is the global solution to the problem, linking polarisation angles of the different frequencies. The local part ñ ij (k) is determined by construction of absolute polarisation angle maps for each frequency with respect to a high signal-to-noise reference pixel being defined to possess a unique polarisation angle. The term absolute polarisation angle is to be understood as a value determined relative to the reference pixel by adding ±nπ to the measured polarisation angles in order to remove jumps of the order of π in the measured polarisation angle map of each observed frequency. The global nπ ambiguity is solved for a high signal-to-noise area surrounding the reference pixel resulting in n(k). This is then also the solution of the global nπ-problem for all spatially connected points which are assigned absolute polarisation angles with respect to this reference area. The splitting of the problem in a local and a global one is possible if the real polarisation angle ¯ϕ ij (k) ( ¯ϕ ij (k) = RM ij λ 2 k + ϕ0 ij ) is a smooth quantity which does not change more than ±π/2 between neighbouring pixels. In Sect. 4.2.2, it is described how Pacman ensures that only pixels fulfilling this condition are used. The source might consists of several spatially independent areas of polarisation, to which hereafter is refered to as patches. The nπ-ambiguity has to be solved for each of these patches separately, requiring separate reference pixels to be defined for the construction of absolute polarisation angles. The advantage of Pacman is that the global nπ-ambiguity is solved only for pixels having the highest signal-to-noise ratios. Therefore, noisier pixels which are situated at the margin of the source profit from an already defined global solution n(k) to the nπ-ambiguity, making a reliable determination of RM ij and the intrinsic polarisation angle ϕ 0 ij for these pixels possible. 4.2.2 The Basic Algorithm The algorithm starts with reading the maps of polarisation angles and of its errors, processes them, and when finished, it saves the different calculated maps. A flow chart of the Pacman algorithm is exhibited in Fig. 4.1, which shows schematically the procedure the algorithm follows in order to determine the solution to the nπ-ambiguity and to calculate the various maps. After loading the various polarisation data for the different frequencies, the algorithm calculates a quality measure map. Different actions are involved in the processing of the data. One of the processes is building a patch which involves the construction
4.2. THE NEW PACMAN ALGORITHM 75 ϕ k load and maps σ k calculate RM quality map search best quality pixel in unprocessed map initialise new patch around reference pixel update border list (BL) pick best (updated) quality pixel in BL existing and sufficiently good? yes solve local n π−ambiguity in each frequency map by comparing to patch pixels no select superiour quality pixels in patch (voters) solve n π−ambiguity across frequency maps for voters individually add pixel to patch pick majority vote as patch global n π−solution calculate and store patch RM and ϕ 0maps yes good pixels still unprocessed ? no save complete maps Figure 4.1: A flow chart representing the individual steps involved in the calculation of RM and ϕ 0 maps as performed by the algorithm Pacman.
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Investigations of Faraday Rotation
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”Vom Himmel lächelte der Herr zu
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vi CONTENTS 3.4.2 Real Space Analys
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List of Figures 1.1 nπ-ambiguity .
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x LIST OF FIGURES
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xii ABSTRACT correlation function a
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xiv ABSTRACT
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2 CHAPTER 1. INTRODUCTION in heat c
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4 CHAPTER 1. INTRODUCTION Maxwell
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6 CHAPTER 1. INTRODUCTION broadenin
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8 CHAPTER 1. INTRODUCTION This sync
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10 CHAPTER 1. INTRODUCTION fluctuat
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12 CHAPTER 1. INTRODUCTION where a
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14 CHAPTER 1. INTRODUCTION where L
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20 CHAPTER 1. INTRODUCTION Figure 1
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22 CHAPTER 1. INTRODUCTION standard
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124 BIBLIOGRAPHY Simard-Normandin,
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126 ACKNOWLEDGEMENTS in his long em
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Publications Journals 1. A Bayesian
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