Measuring solar magnetic fields with artificial neural networks

Abstract

The quantification of the solar magnetic field is a crucial step in modern solar physics to understand the dynamics, activity and variability of our star. Presently, a reliable inference of these fields is only possible by means of a computer-intensive process that has so far limited scientists to the analysis of observations from small regions of the solar disk, and/or very crude spatial and temporal resolution. This work presents a different approach to the problem, in which a multilayer perceptron, trained with known synthetic profiles, is able to recognize the profiles and return the magnetic field used to synthesize them. The network is then confronted with real observations of a sunspot which had been previously inverted using traditional inversion techniques. A quantitative comparison between these two procedures shows the reliability of the network when applied to points having magnetic filling factors larger than approximately 70%. The dramatic decrease in the required computing time presents an opportunity for the routine analysis of large-scale, high-resolution solar observations.

Introduction

The interaction between the ionized plasma and the magnetic fields confined in the sun gives rise to the rich variety of phenomena generally known as solar activity, including sunspots, plages, filaments, prominences, spikes, flares, coronal mass ejections, etc. This interaction is also responsible for the long-term variations (on time scales from minutes to years) of the solar irradiance, and its potential impact on the interplanetary medium.

In order to infer the magnetic configuration of a particular point on the solar surface one needs to observe the polarization state of the light (and its wavelength dependence) along the profile of one or more spectral lines. The information obtained is usually characterized in terms of the so-called Stokes profiles (Chandrasekhar, 1960), which are denoted by I, Q, U and V. In this notation, Stokes I represents the intensity of the light at each wavelength. Stokes V represents the amount of circular polarization. Finally, Stokes Q and U represent the amount of linear polarization along two reference axes. Fig. 1 represents the four Stokes parameters observed with the advanced stokes polarimeter (ASP, see Elmore et al., 1992) at a particular point on a sunspot.

But observing the polarization profiles does not suffice. One also needs a suitable procedure to determine the physical conditions in an atmosphere that produces profiles like those observed. For most applications, theories exist that allow us to calculate what the Stokes profiles formed in a given medium look like (the forward problem). But what we need is exactly the opposite. The determination of the physical conditions (including the magnetic field) that give rise to a given set of Stokes profiles is a non-linear, often ill-conditioned, inverse problem. In the best case, physicists are able to use computer-intensive inversion algorithms, whereby the observed profile is fitted with synthetic ones emergent from known model atmospheres (for recent reviews on inversion techniques, see del Toro Iniesta and Ruiz Cobo, 1996, Socas-Navarro, 2001).

The computing times involved in the inversion of Stokes profiles have so far limited the ability of scientists to work with large data sets. As a consequence, reliable magnetic field determinations are carried out only on small regions of the solar disk with very limited spatial and temporal resolution. As an example, let us consider a typical ASP map like the one shown in Fig. 2, with nearly 200 spatial pixels along the x and y axes and a pixel size of about 0.4 arc s.¹ The fraction of the visible solar disk covered by this map is approximately 2×10⁻³. An inversion of this data set using the fastest inversion code presently available (the HAO M–E code, developed by Skumanich and Lites (1987) takes approximately 10 h running on a modern workstation.

Furthermore, the demand for larger area coverage and higher temporal and spatial resolution, which are needed in order to study the sun as a whole, is pushing the new instrumental developments towards even larger data flows. Perhaps the best examples of this trend are the upcoming SOLIS (Keller, 1998) and Solar-B (Lites & Elmore, 2001) spectro-polarimeters. While these polarimeters will achieve unprecedented spatial coverage and resolution, respectively, it is not yet clear how to deal with such enormous amounts of data.

The solution to this situation may be a new generation of diagnostic techniques that have just recently been introduced. Socas-Navarro, López Ariste, & Lites (2001) (see also Rees, López Ariste, Thatcher, & Semel (2000)) used the principal component analysis (PCA) of the Stokes profiles to develop an inversion procedure based on a look-up table (FATIMA: fast analysis technique for the inversion of magnetic atmospheres). They showed that this method is robust and suitable to be applied on Stokes observations. It has two main advantages over traditional least-squares fitting codes. First, since it performs a global enumerative search over the whole model space, it never settles on secondary minima. Second, it is about two orders of magnitude faster. But FATIMA has also its drawbacks, related to the discrete nature of the database. Perhaps the most important is that the accuracy of the inferred parameters is limited by the size of the database.

A different approach was taken by Carroll and Staude (2001), who used artificial neural networks (ANNs) for the first time to invert synthetic Stokes profiles. They show that ANNs may be a powerful tool for fast diagnostics of the solar plasma. However, going from synthetic to real observations is more complicated in this case, due to the presence of features in the observations that do not exist in the synthetic profiles (e.g. blends, line asymmetries, molecular lines in the umbra of sunspots, etc). These features “confuse’ the ANN, which has never seen them before.

This paper presents a novel strategy where PCA and ANNs are used together to overcome their respective difficulties. We will consider only the relatively simpler case in which the magnetic structures are nearly resolved in the observations. If we denote by the term filling factor the fraction of the resolution element occupied by the magnetic field, we will restrict ourselves to filling factors equal to or greater than 0.7.

The structure of this paper is as follows. Section 2 describes the physical model used for the solar atmosphere, and how the radiative transfer problem is solved to synthesize the Stokes profiles. Section 3 details how a multiplayer perceptron is trained to recognize the PCA coefficients given as input and return an approximate value for the magnetic field as output. Section 4 explains the problems inherent to actual observations and a workaround to overcome them. The performance of the ANN is tested in this section by comparing its results with those obtained by the HAO M–E code. Finally, the most relevant conclusions of this paper are discussed in Section 5, along with some perspectives for future work.