4.1 How to derive vector magnetograms?

NLFFF extrapolations require the photospheric magnetic field vector as input. Before discussing how this vector can be extrapolated into the solar atmosphere, we will address known problems regarding the photospheric field measurements. Vector magnetographs are being operated daily at NAOJ/Mitaka (Sakurai et al., 1995), NAOC/Huairou (Ai and Hu, 1986), NASA/MSFC (Hagyard et al., 1982), NSO/Kitt Peak (Henney et al., 2006Jump To The Next Citation Point), and U. Hawaii/Mees Observatory (Mickey et al., 1996), among others. The Solar Optical Telescope (SOT; Tsuneta et al., 2008) on the Hinode mission has been taking vector magnetograms since 2006. Full-disk vector magnetograms are observed routinely since 2010 by the Helioseismic and Magnetic Imager (HMI; Scherrer et al. (2012)) onboard the Solar Dynamics Observatory (SDO). Measurements with these vector magnetographs provide us eventually with the magnetic field vector on the photosphere, say B z0 for the vertical and Bx0 and By0 for the horizontal fields. Deriving these quantities from measurements is an involved physical process based on the Zeeman and Hanle effects and the related inversion of Stokes profiles (e.g., LaBonte et al., 1999). Within this work we only outline the main steps and refer to del Toro Iniesta and Ruiz Cobo (1996), del Toro Iniesta (2003), and Landi Degl’Innocenti and Landolfi (2004) for details. Actually measured are polarization degrees across magnetically sensitive spectral lines, e.g., the line pair Fe i 6302.5 and 6301.5 Å as used on Hinode/SOT (see Lites et al., 2007) or Fe i 6173.3 Å as used on SDO/HMI (see Schou et al., 2012). The accuracy of these measurements depends on the spectral resolution, for example the HMI instruments measures at six points in the Fe i 6173.3 Å absorption line. In a subsequent step the Stokes profiles are inverted to derive the magnetic field strength, its inclination and azimuth. One possibility to carry out the inversion (see Lagg et al., 2004) is to fit the measured Stokes profiles with synthetic ones derived from the Unno–Rachkovsky solutions (Unno, 1956; Rachkovsky, 1967). Usually one assumes a simple radiative transfer model like the Milne–Eddington atmosphere (see, e.g., Landi Degl’Innocenti, 1992) in order to derive the analytic Unno–Rachkovsky solutions. The line-of-sight component of the field is approximately derived by B ℓ ∝ V ∕I, where V is the circular polarization and I the intensity (the so-called weak-field approximation). The error from photon noise is approximately δB ℓ ∝ δIV-, where δ corresponds to noise in the measured and derived quantities. As a rule of thumb, δV ∕I ∼ 10− 3 and δB ∼ ℓ a few gauss (G) in currently operating magnetographs. The horizontal field components can be approximately derived from the linear polarization Q and U as 2 ∘ --------- B t ∝ Q2 + U2∕I. The error in δBt is estimated as QδQ+U δU BtδBt ∝ √Q2+U2I-- from which the minimum detectable Bt(δBt ∼ Bt) is proportional to the square root of the photon noise ∘ ---2-----2- ∘ ----- ≈ δQ + δU ∕I ≈ δV∕I, namely around a few tens of G, one order of magnitude higher than δB ℓ. (Although δBt scales as 1∕Bt and gives much smaller δBt for stronger Bt, one usually assumes a conservative error estimate that δBt ∼ a few tens of G regardless of the magnitude of B t.)

Additional complications occur when the observed region is far away from the disk center and consequently the line-of-sight and vertical magnetic field components are far apart (see Gary and Hagyard, 1990, for details). The inverted horizontal magnetic field components Bx0 and B y0 cannot be uniquely derived, but contain a 180° ambiguity in azimuth, which has to be removed before the fields can be extrapolated into the corona. In the following, we will discuss this problem briefly. For a more detailed review and a comparison and performance check of currently available ambiguity-removal routines with synthetic data, see Metcalf et al. (2006Jump To The Next Citation Point).

To remove the ambiguity from this kind of data, some a priori assumptions regarding the structure of the magnetic field vector are necessary, e.g., regarding smoothness. Some methods require also an approximation regarding the 3D magnetic field structure (usually from a potential field extrapolation); for example to minimize the divergence of magnetic field vector or the angle with respect to the potential field. We are mainly interested here in automatic methods, although manual methods are also popular, e.g., the AZAM code. If we have in mind, however, the huge data stream from SDO/HMI, fully automatic methods are desirable. In the following, we will give a brief overview on the ambiguity removal techniques and tests with synthetic data.

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