Memoria histórica y Educación

The present work presents the first application of the LITMUS test for magnetic helicity proposed by Junklewitz & Enßlin (2011) to actual data. The application of the test in- volves thoroughly reconstructing a map of the Faraday depth distribution, calculating its transformed gradient field G, creating a map of GP∗, averaging over this map, shifting the two fields with respect to each other to see whether any signal vanishes, and filtering out large-scale contributions for a better detection of small-scale helicity. This procedure, applied to observations of the Faraday depth and polarization properties of the synchrotron radiation within our own galaxy in Sect. 2.3, does not show any signs of helicity in the Milky Way’s magnetic field.

In order to assess the significance of this, the applicability of the test was probed in different artificial settings. The complexity of these settings was increased bit by bit to find out under what circumstances exactly the LITMUS test yields reliable results. It was found that meaningful results can be achieved if the electron densities do not vary on the scales of the magnetic field, both in the regime of magnetic field structures whose distance from the observer is much greater than their extension, as shown in Sect. 2.4.1, and in the regime of magnetic fields surrounding the observer, as shown in Sect. 2.4.2. We showed that the performance of the LITMUS test with regard to small-scale helicity is further improved by dropping the first few terms in Eq. (2.26). However, indications of helicity on large scales are unreliable, as shown in Sect. 2.4.2. Furthermore, it was demonstrated in Sect. 2.4.2 that any non-trivial electron density may distort the outcome of the test to a point where even small-scale helical structures fail to be detected. This is not too surprising since e.g. a variation in the thermal electron density will introduce a gradient in the Faraday depth that is not caused by the magnetic field structure. Therefore the nondetection of helicity for the Galactic magnetic field does not necessarily mean that the field is nonhelical on small scales. It may be the case that small-scale fluctuations of the electron density introduce effects in the observational data that prevent the detection of helicity.

So, as a natural next step, the hunt for helicity in astrophysical magnetic fields should focus on a region that is small and/or homogeneous enough for the assumption of constant electron densities to hold at least approximatively. Although our work has shown that the helicity test that we studied is not suitable for all astrophysical settings, we are confident that it may nevertheless yield useful results if applied in a setting with constant electron

2.5 Discussion and conclusion 53

densities.

As a side effect of this paper, it was demonstrated in Sect. 2.3.2 that the method proposed by Enßlin & Frommert (2011) to reconstruct a Gaussian signal with unknown power spectrum is very well suited for practical application.

Chapter 3

Reconstructing signals from noisy

data with unknown signal and noise

covariance

Note: This chapter, as well as Appendix C, has been published in Physical Review E

(Oppermann et al. 2011b).

3.1

Introduction

The problem of signal inference consists of reconstructing a set of parameters or even a continuous field s from some data set d, which is influenced in some way by the signal,

d=f(s) +n. (3.1)

Two problems will arise. First, the functionf may not be invertible and, second, the noise term n will not be known. In the Bayesian framework, one uses prior information on the signal and the noise term to calculate a best estimate for the true signal realization or, ideally, the whole probability distribution for the signal given the prior information and the information contained in the data.

Symmetry considerations and knowledge about the underlying physics of the signal and the measurement process may restrict the class of priors that one has to consider. They might, however, still contain some free parameters that then become part of the inference problem. The case in which the signal covariance contains uncertain parameters was tackled by Enßlin & Frommert (2011), producing a whole class of filters for this problem. The filter that we extend in this work was reproduced by Enßlin & Weig (2010), where the principle of minimum Gibbs free energy was introduced (cf. also Sect. 3.3), and successfully applied in an astrophysical setting by Oppermann et al. (2011a).

Here, we focus on the case where we can assume zero-mean Gaussian priors both for the signal and for the noise. The priors are therefore completely characterized by the respective covariance matrices. Our goal is to extend the study of Enßlin & Weig (2010) to the case

56 3. Reconstructions with unknown signal and noise covariance

in which both the signal covariance and the noise covariance contain parameters that are not known a priori. This is motivated mainly by applications from the field of astrophysics. The theory and resulting filter formulas, however, are of general applicability. Gaussian noise, e.g., is omnipresent in nearly every area of the natural sciences and the situation in which its variance is not precisely known should be a rather common one.

Previous work dealing with the problem of unknown noise variance has mainly dealt with specific applications. One of these applications is the field of image reconstruction. Here, it is usually assumed that the measured picture is the sum of the underlying signal and a white Gaussian noise term. Often, it is further assumed that the noise level, i.e. its variance, is the same in every image pixel. A comparison of different algorithms for noise estimation under these assumptions was conducted e.g. by Olsen (1993). An example for an algorithm allowing for inhomogeneous noise was presented by Starck & Murtagh (1998), where a wavelet transform of the image is applied and the lack of correlated noise is exploited. Most of these algorithms, however, are not derived by rigorous statistical calculations but rather by a combination of intuition and experience.

From a mathematical viewpoint, the problem of an unknown noise prior has received some attention in the theory of density deconvolution, which deals with the inference of the probability density for a signal from measurements with additive noise. Here, the signal is usually assumed to consist of independent identically distributed variables. The case of Gaussian noise with unknown variance has been considered e.g. by Koltchinskii (2000) and Schwarz & van Bellegem (2010).

In this work, we create a general setting with well defined assumptions and a traceable derivation of a general filter formula within a Bayesian framework, not loosing sight of its applicability. Our result can accomodate a host of different assumptions and models, such as correlated or uncorrelated noise. It allows for a distinction between the data space and the signal space, with possibly different numbers of degrees of freedom.

The remainder of the paper is organized as follows. In Sect. 3.2 we introduce our model for the measurement process and the notation that is required. The derivation of the filter formulas follows in Sect. 3.3. We then demonstrate the usefulness of our filter by applying it in a set of mock observational situations in Sect. 3.4 and discuss the implications in Sect. 3.5.