An early experiment with combined stereoscopy and magnetic field modeling has been conducted with multi-frequency radio maps observed with the Owens Valley Radio Observatory (Aschwanden et al., 1995). For a given 3D magnetic field, the gyroresonance layers at each frequency and harmonic form a curved 2D surface that are layered like onion shells in the lower corona above sunspots, as shown for three frequencies () in Figure 10. Some features (e.g., the brightest locations or centroids) of these gyroresonance layers can be either stereoscopically triangulated (using the solar rotation) or be modeled by matching the contours of their brightness distribution for a given projection. Such a method successfully demonstrated that in a strongly sheared region only a nonlinear force-free field could explain the observed radio brightness distribution, while a potential-field model failed (Lee et al., 1999). Also the polarization of gyroresonance emission in the theory of wave mode coupling in quasi-transverse regions was tested with combined radio mapping and magnetic field modeling (Ryabov et al., 2005).
The method of using the observed projections of coronal EUV loops to constrain a magnetic field model was proposed by Wiegelmann and Neukirch (2002) in the pre-STEREO area, who varied the -parameter in a force-free magnetic field extrapolation to optimize the match to observed loop locations. Magnetic modeling was then also used to constrain tomographic reconstruction of the coronal density distribution (Wiegelmann and Inhester, 2003; Ruan et al., 2008). A variety of coronal magnetic field models, such as potential, linear, and nonlinear force-free field (NLFFF) models, were used to resolve ambiguities in stereoscopic triangulation of loops (Wiegelmann and Inhester, 2006; Feng et al., 2007b; Conlon and Gallagher, 2010), reviewed in Wiegelmann et al. (2009). Comparisons of potential-field and NLFFF models with STEREO and Hinode images were also calculated for the global corona (Petrie et al., 2011; Schrijver and Title, 2011).
The success of magnetic stereoscopy depends on the quality of theoretical magnetic field models. A critical assessment of NLFFF models identified a substantial mismatch between theoretical magnetic field models extrapolated from photospheric magnetograms and stereoscopically triangulated loops, in the order of a 3D misalignment angle of (DeRosa et al., 2009; Sandman et al., 2009). A comparison of a force-free field model with STEREO-triangulated loops is shown in Figure 11. Some parameterization of the theoretical field models is required to minimize the mismatch. Such a parameterization was implemented in potential-field models in terms of unipolar magnetic charges (Aschwanden and Sandman, 2010) or magnetic dipoles (Sandman and Aschwanden, 2011), which enabled to improve the misalignment angle to . The principle is illustrated in Figure 12. A caveat of these first attempts to adjust theoretical magnetic field models to observed 3D loop coordinates (obtained from stereoscopic triangulation) is that the optimization algorithm takes only the coronal misalignment into account, but neglects the constraints of the observed photospheric magnetograms. The justification for this neglect is the fact that the magnetic field in the photosphere and lower chromosphere is not force-free (Metcalf et al., 1995), but ultimately we would like to have magnetic field models that can handle both the non force-free extrapolation in the chromosphere plus fitting of the force-free field in the corona to observed 2D projections or 3D triangulations of loops. Once we have an optimized (matching) coronal 3D magnetic field model, we can populate each field line with a magnetic flux tube and perform hydrodynamic modeling of entire coronal sections, constrained by the observed EUV and soft X-ray images in different temperature filters, a technique called instant stereoscopic tomography of active region (ISTAR) (Aschwanden et al., 2009c).
Living Rev. Solar Phys. 8, (2011), 5
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