4.7 Preprocessing

Wiegelmann et al. (2006b) developed a numerical algorithm in order to use the integral relations (31View Equation) – (35View Equation) to derive suitable NLFFF boundary conditions from photospheric measurements. To do so, we define the functional:
Lprep = μ1L1 + μ2L2 + μ3L3 + μ4L4, (38 )
⌊ ⌋ ( ∑ )2 ( ∑ )2 ( ∑ )2 L = ⌈ B B + B B + B2 − B2 − B2 ⌉ , (39 ) 1 x z y z z x y ⌊ p p p ( ∑ )2 ( ∑ )2 L2 = ⌈ x (B2 − B2 − B2 ) + y(B2 − B2 − B2) p z x y p z x y ⌋ (∑ )2 + yBxBz − xByBz ⌉, (40 ) p [ ] ∑ 2 ∑ 2 ∑ 2 L3 = (Bx − Bxobs) + (By − Byobs) + (Bz − Bzobs) , (41 ) p p p [ ∑ ] L = (ΔB )2 + (ΔB )2 + (ΔB )2 . (42 ) 4 x y z p
The first and second terms (L1, L2) are quadratic forms of the force and torque balance conditions, respectively. The L3 term measures the difference between the measured and preprocessed data. L4 controls the smoothing, which is useful for the application of the data to finite-difference numerical code and also because the chromospheric low-β field is smoother than in the photosphere. The aim is to minimize Lprep so that all terms Ln are made small simultaneously. The optimal parameter sets μn have to be specified for each instrument separately. The resulting magnetic field vector is then used to prescribe the boundary conditions for NLFFF extrapolations. In an alternative approach Fuhrmann et al. (2007) applied a simulated annealing method to minimize the functional. Furthermore they removed the L3 term in favor of a different smoothing term L4, which uses the median value in a small window around each pixel for smoothing. The preprocessing routine has been extended in Wiegelmann et al. (2008) by including chromospheric measurements, e.g., by minimizing additionally the angle between the horizontal magnetic field and chromospheric Hα fibrils. In principle, one could add additional terms to include more direct chromospheric observations, e.g., line-of-sight measurements of the magnetic field in higher regions as provided by SOLIS. In principle, it should be possible to combine methods for ambiguity removal and preprocessing in one code, in particular for ambiguity codes which also minimize a functional like the Metcalf (1994) minimum energy method. A mathematical difficulty for such a combination is, however, that the preprocessing routines use continuous values, but the ambiguity algorithms use only two discrete states at each pixel. Preprocessing minimizes the integral relations (31View Equation35View Equation) and the value of these integrals reduces usually by orders of magnitudes during the preprocessing procedure. These integral relation are, however, only necessary and not sufficient conditions for force-free consistent boundary conditions, and preprocessing does not make use of condition (37View Equation). Including this condition is not straight forward as one needs to know the magnetic field line connectivity, which is only available after the force-free configuration has been computed in 3D. An alternative approach for deriving force-free consistent boundary conditions is to allow changes of the boundary values (in particular the horizontal field) during the force-free reconstruction itself, e.g., as recently employed by Wheatland and Régnier (2009Jump To The Next Citation Point), Amari and Aly (2010Jump To The Next Citation Point), and Wiegelmann and Inhester (2010Jump To The Next Citation Point). The numerical implementation of these approaches does necessarily depend on the corresponding force-free extrapolation codes and we refer to Sections 6.2 and 6.4 for details.
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