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

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80 CHAPTER 4. PACMAN – A NEW RM MAP MAKING ALGORITHM frequencies. In cases of k min < f, Pacman will start to determine the solution to global and local nπ-ambiguity only for the pixels fulfilling the quality requirements at all f frequencies. After finishing that, the algorithm proceeds to include pixels satisfying the quality criteria in less than f frequencies. For these pixels, the same patch building procedure as described in Sect. 4.2.2 applies with some modification in order to prevent the final maps from heavy fragmentation. The best quality pixel among the remaining pixel is picked but before starting a new patch, Pacman tests if the pixel under consideration adjoins a patch which has already been processed before. If the pixel adjoins such a patch, Pacman tries to solve the local nπ-ambiguity following Eq. (4.6) and applies the patch solution of the global nπ-ambiguity to the pixel. If the pixel is neither adjoined to an already processed patch nor the local nπ-ambiguity solvable (i.e. σij ∆ exceeds σ∆ max) then a new patch is initialised by this pixel. Another possibility is to force Pacman to use certain frequencies in all circumstances. Thus, if the quality requirements are not fulfilled for these particular frequencies, there will be no RM value determined for the pixel under consideration. This has the advantage that one can use relatively close frequencies as a basis and then include other frequencies at points when a reliable polarisation signal is detected. The advantage of this technique is discussed in Sect. 4.5. 4.2.8 Additional Information Apart from the resulting RM and ϕ 0 maps, Pacman provides sets of additional information about the data in order to estimate the reliability of the results. A patch map which contains all patches used is one example. Such a map is very useful, especially if one requires a minimal number of pixels in a patch in order to accept any calculated RM values from a particular patch. A map which includes rejected and thus flagged pixels, can also be obtained. Probably more important are the final χ 2 - and σ RM - maps which are also provided by Pacman. This information allows one to understand the reliability of the RM maps obtained and can be used for further evaluation and analysis of the RM maps. 4.3 Testing the Algorithm In order to demonstrate the ability of Pacman to solve the nπ-ambiguity, artificially generated maps were used. To generate them, it is started from a polarisation data set from Abell 2255 (Govoni et al. 2002), which was kindly provided by Federica Govoni. The RM and ϕ 0 maps were calculated from this data set and it was assumed that these two maps consist of exact values. Then polarisation maps were generated at four different frequencies which would exactly result from the RM and ϕ 0 maps. As error maps, Gaussian deviates were generated for each frequency which were multiplied by the original error maps provided by Federica Govoni. The so generated error maps were added to the generated frequency maps in order to provide realistic mock observations. Pacman and the standard fit algorithm were used to calculate the corresponding ’observed’ RM and ϕ 0 maps which then were compared pixel-by-pixel to the initial
4.4. STATISTICAL CHARACTERISATION OF IMPROVED RM MAPS 81 Figure 4.2: Comparison between the simulated RM sim values for an artificially generated set of polarisation angle maps of four frequencies and values derived by Pacman RM pacman and the standard fit algorithm RM stdfit . The scatter is due to noise which was added to the polarisation angle maps. Note the wrong solutions of the nπambiguity results which were calculated by the standard fit indicated by the points around ± 1000 rad m −2 . exact maps. The result of this comparison is shown in Fig. 4.2. On the right panel, the comparison between the RM pacman values of the Pacman map and the values RM sim of the initial map is shown. The scatter in the data is due to the noise which was added to the frequency maps. On the left panel of Fig. 4.2, the pixel by pixel comparison between the values of the standard fit RM stdfit map and the initial RM sim map is shown. Again the scatter is due to the added noise. However, one can clearly see the points at ± 1000 rad m −2 which deviate from the initial data and are due to the wrongly solved nπ-ambiguity in the case of the standard fit. Thus, this test demonstrates that the Pacman algorithm yields reliable results to the nπ-ambiguity. 4.4 Statistical Characterisation of Improved RM Maps 4.4.1 Gradient Vector Product Statistic In Chapter 2, a gradient vector product statistic V was introduced to reveal correlated noise in the data. Observed RM and ϕ 0 maps will always have some correlated fluctuations on small scales, since they are both calculated from the same set of polarisation angle maps, leading to correlated fluctuation in both maps. The noise correlation between ϕ 0 and RM errors is an anti-correlation of linear shape and can be detected by comparing the gradients of these quantities. A suitable quantity to detect correlated noise between ϕ 0 and RM is therefore V = ∫ d 2 x ⃗ ∇RM(⃗x) · ⃗∇ϕ 0 (⃗x) ∫ d 2 x | ⃗ ∇RM(⃗x)|| ⃗ ∇ϕ 0 (⃗x)| , (4.9)
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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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10 CHAPTER 1. INTRODUCTION fluctuat
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12 CHAPTER 1. INTRODUCTION where a
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28 CHAPTER 1. INTRODUCTION where n
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