PREPROCESSING MAGNETIC FIELDS WITH CHROMOSPHERIC LONGITUDINAL FIELDS

Wiegelmann et al. (2006) first proposed preprocessing magnetic field data. In order to quantitatively perform preprocessing, they applied a function (L) as follows:

$$\begin{eqnarray} L &=& \mu _1 L_1 +\mu _2 L_2 +\mu _3 L_3 +\mu _4 L_4, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} L_1 &=& \left(\sum _p B_x B_z \right)^2 + \left(\sum _p B_y B_z \right)^2\nonumber\\ &&+ \left(\sum _p\big(B_x^2 + B_y^2 - B_z^2\big) \right)^2, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} L_2 &=& \left(\sum _p x\big(B_x^2 + B_y^2 - B_z^2\big) \right)^2 + \left(\sum _p y\big(B_x^2 + B_y^2 - B_z^2\big) \right)^2\nonumber\\ &+& \left(\sum _p (y B_x B_z - x B_y B_z) \right)^2, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} L_3 &=& \sum _p \big[\big(B_x - B_x^{\rm obs}\big)^2 + \big(B_y - B_y^{\rm obs}\big)^2 + \big(B_z - B_z^{\rm obs}\big)^2\big],\nonumber\\ \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} L_4 &=& \sum _p \big[(\Delta _{xy}B_x)^2 + (\Delta _{xy}B_y)^2 + (\Delta _{xy}B_z)^2\big], \end{eqnarray} \tag{ 5 }$$

where μ₁ to μ₄ are coefficients of L₁ to L₄, respectively; L₁ is the square of the total magnetic force; L₂ is the square of the total magnetic torque; L₃ is the difference between the preprocessed magnetogram and the observed one; L₄ is the amount of small-scale irregularities in the magnetogram; and the superscript "obs" indicates the observed magnetic field.

Here, L₁ and L₂ come from area integrals of the Lorentz force on a volume (Wiegelmann et al. 2006). In the following, we describe derivations of the area integrals (Fuhrmann et al. 2007). If J × B =0 in a volume (V), a volume integral of the Lorentz force is equal to zero, and we can obtain the following derivation using Gaussian's theorem:

$$\begin{eqnarray} \mbox{{\boldmath $0$}} &=& \int _V \mbox{{\boldmath $J$} $\times $ {\boldmath $B$}} {dV} \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &=& \int _V \nabla\, {{\bf T}} {dV}, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &=& \int _S {\bf T} {dS}, \end{eqnarray} \tag{ 8 }$$

where S is the entire closed surface on the volume and T is magnetic stress tensor,

$$\begin{equation} {\bf T}_{ij}=-\frac{\mbox{{\boldmath $B$}}^2}{2}\delta _{ij}+B_iB_j. \end{equation} \tag{ 9 }$$

As for a volume integral of the torque, we can also perform a similar derivation:

$$\begin{eqnarray} \mbox{{\boldmath $0$}} &=& \int _V \mbox{{\boldmath $r$} $\times $ ({\boldmath $J$} $\times $ {\boldmath $B$})} {dV}, \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} &=& \int _V \nabla {\bf T^{\prime }} {dV}, \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} &=& \int _S {\bf T^{\prime }} {dS}, \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} {\bf T^{\prime }}_{ij} &=& \epsilon _{ikl} r_k {{\bf T}}_{lj}. \end{eqnarray} \tag{ 13 }$$

In preprocessing, we consider this closed surface to be six surfaces of a computational cube. If the contributions of the side and top boundaries of the cube are small, one integrates only over the bottom boundary. Finally, the following equations are derived:

$$\begin{eqnarray} 0&=&-\int _S B_x B_z {dxdy}, \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} 0&=&-\int _S B_y B_z {dxdy}, \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} 0&=&\int _S \big(B_x^2+B_y^2-B_z^2\big) {dxdy}, \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} 0&=&\int _S y \big(B_x^2+B_y^2-B_z^2\big) {dxdy}, \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} 0&=&\int _S x\big(-B_x^2-B_y^2+B_z^2\big) {dxdy}, \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} 0&=&\int _S \big(yB_xB_z-xB_yB_z\big) {dxdy}. \end{eqnarray} \tag{ 19 }$$

In the solar atmosphere, this bottom boundary corresponds to the photosphere or to the chromosphere. Note that this is reasonable for an isolated and flux-balanced active region. Using these equations, we can expect that lower values of L₁ and L₂ indicate better degrees of force- and torque-freeness of magnetic fields. For original derivations of the area integrals, see Molodenskii (1969) and Aly (1989).

L₃ is a parameter to obtain preprocessed magnetic fields similar to the observed fields. L₄ is a parameter controlling the smoothness of magnetic fields. These parameters are independent of L₁ and L₂ and control the appearance of magnetic fields. We will discuss how combinations of μ₁ to μ₄ control preprocessing.

In this study, we add a term into the function as follows:

$$\begin{equation} \mu _5 L_5=\mu _5 \sum _p \big(B_z-B_z^{\rm chr}\big)^2, \end{equation} \tag{ 20 }$$

where B^chr_z is the longitudinal magnetic field in the chromosphere and μ₅ is a coefficient. In this study, we use a longitudinal magnetogram observed by SOLIS. We will describe these data later.

Note that, in our function, free parameters are μ₃ to μ₅. μ₁ and μ₂ are constants in this study. μ₁ is set to be 10. μ₂ is determined by μ₁ as follows (Wiegelmann et al. 2006):

$$\begin{equation} \mu _2=\mu _1/D^2, \end{equation} \tag{ 21 }$$

where D is the pixel size of our data. In this paper, D is 256.

In previous studies, Wiegelmann et al. (2006) used the Newton-Raphson method to obtain local minima, and Fuhrmann et al. (2007) used the simulated-annealing method. For precise descriptions of these methods, see Press et al. (1992). In this study, in order to obtain minima, we apply the simulated-annealing method. One reason why we apply this method is that the simulated-annealing method provides the opportunity to obtain minimum values far from the initial conditions. On the other hand, the Newton-Raphson method obtains local minimum values near the initial condition.

The simulated-annealing method assumes the pseudo-temperature (T) and its cooling rate (dT) of the system. T decreases step by step as T_new = T × dT. In this method, perturbation of the magnetic field is given by random numbers on each pixel. Concerning variations of L, a decrease of L is always allowed. On the other hand, an increase of L is allowed if a random number is smaller than the probability (P). P is defined as P = exp (− dL/T), where dL is the variation of L derived from perturbed magnetic fields. If T is much lower, L hardly increases in calculations. In other words, if T is higher, L can increase, and we can find minimum values far from the initial conditions.

In our calculations, the initial value of T is defined as T_init = t₀L₀, where t₀ is the initial temperature coefficient and L₀ is the initial value of L. In addition to t₀ and dT, the seed of the random numbers is also a free parameter. Including μ₃, μ₄, and μ₅, we perform a parameter survey of these six parameters.

We finish our calculations when values of L converge or pseudo-temperatures T achieve the threshold temperature. Convergence of L is reached when variations of L, |L_new − L_old|/(L_new + L_old), are less than 10⁻⁶ through five steps. The threshold temperature (T_stop) is set to be T_stop = T_init10⁻¹⁰⁰. Therefore, we can obtain enough steps for the temperature cooling. If pseudo-temperature T achieves T_stop, we consider the results to be non-converging.

In this study, the target region is AR 10953. We analyze a vector magnetogram on the photosphere and a longitudinal magnetogram in the chromosphere. The vector magnetogram is observed by the Hinode/Solar Optical Telescope (SOT; Kosugi et al. 2007; Tsuneta et al. 2008; Suematsu et al. 2008; Ichimoto et al. 2008; Shimizu et al. 2008), and the longitudinal magnetogram is observed by the SOLIS/Vector Spectro Magnetograph (VSM; Jones et al. 2002). These data are observed around 18:00 UT on 2007 May 2. The top panels of Figure 1 show magnetic field components observed by Hinode/SOT. Figure 2 shows longitudinal fields observed by SOLIS/VSM.

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The vector magnetogram is SOT/SP level 2 data ("20070502_173005.fits"). This file can be downloaded from the LMSAL Web site.³ These level 2 data are inverted with the MERLIN code from full Stokes data observed in the Fe i λ6301.5 and λ6302.5 lines. To solve the 180° ambiguity of transverse field azimuth, we apply the minimum energy method (for more precise information, see Leka et al. 2009 and their Web site⁴).

The longitudinal magnetogram in the chromosphere is observed in Ca ii 8542 Å line. The file name is "svsm_m2100_S2_20070502_1832.fts." This file can be found on the NSO Web site.⁵

Magnetic field data are interpolated to have a number of pixels, 256 × 256. The original number of pixels in the vector magnetogram is 512 × 512. Pixel widths are set to be about 0.6 arcsec. Alignment of coordinates between the vector magnetogram and the longitudinal magnetogram is calculated from weighted centers of longitudinal fields.

Figure 3 shows a scatter plot of the parameter combinations. r₁ and r₅ are ratios (initial values/final values) of L₁ and L₅, respectively. These values are summarized in the online version of Table 1. In this figure, larger values of r₁ indicate magnetic fields closer to force free, and those of r₅ indicate longitudinal fields closer to the chromospheric field. Therefore, larger values in both r₁ and r₅ indicate more ideal boundary conditions of NLFFF extrapolations.

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Figure 4 shows an enlarged view of the rectangle region in Figure 3. In this figure, numbers in gray show non-converging results. This non-converging indicates L did not converge within iteration limits. In our calculation, maximum iteration limits depend on t₀ and dT. We discuss these iteration limits in a later section.

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As a result, Figure 4 shows that parameter numbers 60, 63, and 66 show the largest values of r₁ and r₅. μ₄ is the difference between these parameter combinations (μ₄ = 10³ for 60, 10⁴ for 63, and 10⁵ for 66). The bottom panels of Figure 1 show preprocessed magnetic field maps of parameter number 63. Figure 5 shows variations of the function. In the following, we describe differences in the preprocessed field strengths of these parameter combinations. Magnetic field maps of these parameter combinations are similar to each other.

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The bottom panels of Figure 1 show that the preprocessed map of B_z is very similar to the longitudinal fields shown in Figure 2. It is found that, in 98% of the pixels, unsigned differences of B^chr_z − B_z⁶³ are lower than 5 G, where superscript 63 shows the parameter number. More precisely, the standard deviation of B^chr_z − B_z⁶³ is 1.9 G. As for parameter numbers 60 and 66, standard deviations of B⁶³_z − B_zⁱ (i  = 60 and 66) are 0.1 and 0.2 G, respectively.

Figure 6 shows histograms of B⁶³_t − B_t⁶⁰, where B_t = (B²_x + B²_y)^0.5. This figure shows that 90% of all pixels have unsigned differences of B_t less than 15 G. In pixels where |B⁶³_z| is larger than 1000 G (a black area in the negative polarity in the bottom right panel of Figure 1), 50% of the pixels have unsigned differences of B_t less than 15 G. Between parameter numbers 63 and 66, 76% of all pixels have unsigned differences of B_t less than 15 G. In pixels where |B⁶³_z| is larger than 1000 G, 29% of pixels have unsigned differences of B_t less than 15 G.

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For comparison, here we describe results of parameter number 11. This parameter combination lies at the bottom right end of Figure 3 and shows the largest r₁ (namely the smallest L₁). In this parameter combination, only coefficients of force- and torque-freeness are emphasized. Figures 7 and 8 show preprocessed field maps and profiles of the function. As shown in Figure 7, the resultant fields are very noisy. What is important in this figure is the possibility that noise fields are candidates of local minimum values. This will be discussed in a later section.

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Figure 3 is the scatter plot of all parameter combinations. The open circle in this figure shows the standard parameter combination (parameter number 0). As the first step, we irregularly change free parameters. These are parameter numbers 1–14. From these parameter combinations, we find that the initial temperature factor and the seed of random numbers have minor effects on calculation results.

Second, we check variations of temperature cooling rate (dT), μ₄, and μ₅ in parameter numbers 0, 2–4, and 15–58. Parameters are dT = [0.995, 0.999, 0.9995], μ₄, and μ₅ = [10⁵, 10⁶, 10⁷, 10⁸]. Figure 9 shows a scatter plot of these parameter combinations. This figure shows a clear dependence of r₅ on μ₅. Parameter combinations of μ₅ = 10⁸ show larger values of r₅, which means that parameter combinations of μ₅ = 10⁸ are more favorable than the other parameter combinations. Figure 10 is a scatter plot of μ₅ = 10⁸. Table 2 shows dT and μ₄ of the parameter combinations shown in Figure 10. From Figure 10 and Table 2, the following two points are found: (1) lower μ₄ and larger dT lead to larger r₁; and (2) in lower μ₄, cooling rates (dT) result in smaller differences in r₁. Hereafter, we consider only one cooling rate, 0.995.

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Table 2. Parameter Combinations Shown in Figure 10

	No.	dT	log(μ4)	log(r1)
$\Diamond$	3	0.995	7	7.08
	47	0.999	7	7.31
	31	0.9995	7	7.38
*	18	0.995	6	7.68
	49	0.999	6	7.74
	33	0.9995	6	7.76
+	19	0.995	5	7.89
	50	0.999	5	7.88
	34	0.9995	5	7.88

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As the third step, we check variations of μ₃, μ₄, and μ₅ in parameter numbers 59–85. In these parameter combinations, parameters are set as follows: μ₃ = [10⁵, 10⁶, 10⁷], μ₄ = [10³, 10⁴, 10⁵], and μ₅ = [10⁸, 10⁹, 10¹⁰]. As shown in Figure 4, calculations of some parameter combinations converge, while others do not. Therefore, positions of parameter combinations in Figure 4 strongly depend on μ₃ and μ₅. In the right side of log(r₁) = 10.3, μ₃ is 10⁵. In the left side, μ₃ is 10⁶. In the upper part of log(r₅) = 4.25, μ₅ is 10⁹ and 10¹⁰. In the lower part, μ₅ is 10⁸. As described in the previous subsection, μ₄ of parameter numbers 60, 63, and 66 are 10³, 10⁴, and 10⁵, respectively.

Finally, concerning parameter number 63, we change dT, t₀, and the seed of the random numbers: dT = [0.995, 0.999, 0.9995], t₀ = [10⁻⁴, 10⁻¹, 1]. From these calculations, values of log(r₁) are distributed from 9.86 to 9.90, and those of log(r₅) are distributed from 4.167 to 4.194. In these calculations, dT and t₀ determine r₁ and r₅, although the ranges of r₁ and r₅ are small. Seeds of random numbers have minor effects on these results.

We find in the third step that, if μ₅/μ₃ is greater than 10², the calculations do not converge. As for parameter numbers 60, 63, and 66, values of μ₅/μ₃ are 10². As for the other parameter combinations shown in Figure 4, values of μ₅/μ₃ are equal to or larger than 10³. We will discuss non-converging parameters in a later section.

In Section 3.1, we point out that noisy maps can take smaller force- and torque-freeness, namely smaller L₁ and L₂, in preprocessing. Here, we quantitatively investigate this tendency of L₁ in the wave number space from three magnetic field maps: one is the original vector magnetogram (the top panels of Figure 1), one is the preprocessed map of parameter number 63 (the bottom panels of Figure 1), and another is the noisy map of parameter number 11 (Figure 7).

In order to investigate L₁ in the wave number space, we calculate values in the wave number space via Fourier transform as follows:

$$\begin{eqnarray*} L_{1a} & = & \sum _PB_xB_z, \nonumber \\ & = & \sum _P \left(\sum _{m,n} \tilde{B}_x(m,n) E(m,n) \right)\nonumber \\ &&\times\left(\sum _{m^{\prime },n^{\prime }} \tilde{B}_z(m^{\prime },n^{\prime }) E(m^{\prime },n^{\prime })\right),\nonumber\\ & = & P \sum _{m,n} \tilde{B}_x(m,n) \tilde{B}_z(-m,-n), \nonumber \\ & = & \sum _{m,n} F_a(m,n), \nonumber \\ L_{1b} & = & \sum _PB_yB_z, \nonumber \\ & = & P \sum _{m,n} \tilde{B}_y(m,n) \tilde{B}_z(-m,-n), \nonumber \\ & = & \sum _{m,n} F_b(m,n), \nonumber \\ L_{1c} & = & \sum _P\big(B_x^2+B_y^2-B_z^2\big),\nonumber \\ & = & P \sum _{m,n} \tilde{B}_x(m,n) \tilde{B}_x(-m,-n)+\cdots , \nonumber \\ & = & \sum _{m,n} F_c(m,n),\nonumber \end{eqnarray*}$$

where $\tilde{B}$ is the field strength in the wave number space, E(m, n) indicates exp (2πi[mx/N_x + ny/N_y]), and m and n are integer numbers (m ⩽ |N_x/2| and n ⩽ |N_y/2|). For precise derivation, see the Appendix.

In the following, we compare summations of F_a, F_b, and F_c in four wave number regions. The powers of four wave number regions are defined as p₁ to p₄ and are shown in Table 3. Four wave number regions are defined in Table 4. p₀ is the total value of p₁ to p₄. In Table 3, p₁ to p₄ of the original vector magnetogram shows the single sign (either positive or negative), and p₁ indicates that the small wave number component is larger than the other values. On the other hand, p₁–p₄ of the preprocessed magnetograms show both positive and negative signs, and p₁ is not necessarily larger than the other values. As for the preprocessed maps, unsigned values of p₀ of parameter number 11 are smaller than those of parameter number 63, while unsigned values of p₁–p₄ of parameter number 11 are larger than those of parameter number 63. Therefore, in these preprocessed magnetograms, total values of the large wave number components cancel out those of the small wave number components. This tendency is emphasized in the noisy magnetogram (parameter number 11). In Figure 7, noise (namely large wave number components) destructs the magnetic field structures and seem to develop only for decreasing L₁ (canceling p₁ and p₄) in this preprocessing.

Table 4. Wave Number Ranges of p₀–p₄

p	m	n
0	$|m|\le \frac{N_x}{2}$	$|n|\le \frac{N_y}{2}$
1	$|m|\le \frac{N_x}{4}$	$|n|\le \frac{N_y}{4}$
2	$\frac{N_x}{4}<|m|\le \frac{N_x}{2}$	$|n|\le \frac{N_y}{4}$
3	$|m|\le \frac{N_x}{4}$	$\frac{N_y}{4}<|n|\le \frac{N_y}{2}$
4	$\frac{N_x}{4}<|m|\le \frac{N_x}{2}$	$\frac{N_y}{4}<|n|\le \frac{N_y}{2}$

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From these results, we conclude the ideal goals of preprocessing as follows: (1) values of p₁–p₄ in a preprocessed magnetogram have a single sign; (2) p₁ is larger than the other values; and (3) values of p₁–p₄ are smaller than those in an original vector magnetogram. One cause of the noisy map is clearly perturbations given to magnetic fields in the simulated-annealing method. How to give perturbations is one of our future subjects.

In this study, we limit iteration steps for calculation times. Iteration step limits are set by T and strongly depend on t₀ and dT. In the non-converging parameter combinations shown in Figure 4, these limits are about 4.6 × 10⁴ steps. In order to check converging results of these parameter combinations, we perform calculations of three parameter combinations until convergence. Three parameter combinations are 59, 68, and 77, and are shown at the right end in Figure 4.

Calculations of parameter number 59 and 68 converge in 8.3 × 10⁴ and 1.4 × 10⁵ steps, respectively. Concerning parameter number 77, calculation does not converge even in one week. Resultant maps of B_z of parameter numbers 59 and 68 are very similar to Figure 7. However, as for B_x and B_y, there are noisy regions similar to noisy preprocessed maps of parameter number 11 (Figure 7). Excluding log (r₅) of parameter number 68, log (r₁) and log (r₅) of parameter numbers 59 and 68 are similar values, shown in Figure 4. log (r₅) of parameter number 68 becomes 4.56.

From these results and the results described in Section 3.2, we conclude that a larger μ₅/μ₃ ratio in parameter numbers 59–85 leads to noisy maps, especially in B_x and B_y. The threshold ratio is μ₅/μ₃ = 10². In these cases, μ₃ is relatively so small that preprocessed B_x and B_y could end up far from the original ones shown in the top panels of Figure 1.

Magnetic structures broadening in the chromosphere are a necessary goal of improved preprocessing methods. For example, the upper right panel of Figure 1 shows B_z in the photosphere, and Figure 2 shows it in the chromosphere. Typical spatial scales in these magnetic structures are clearly different. Therefore, one of our subjects of preprocessing is how to broaden photospheric magnetic structures. For this subject, what we need to know is how magnetic structures broaden in the chromosphere.

Here, we smoothed B_z of the photosphere and calculated correlations between this smoothed B_z and B_z of the chromosphere. B_z of the photosphere is smoothed by the Gaussian function as follows:

$$\begin{equation} B^{\prime }_z(x_0,y_0) = \sum B_z(x,y) G(x-x_0,y-y_0), \end{equation} \tag{ 22 }$$

where G is the two-dimensional Gaussian function. We changed 1σ of the Gaussian function from 1 to 21 pixels. Figure 11 shows these correlations. In this figure, we find that correlations take large values around 5–7 pixels. The width of six pixels is about 3.6 arcsec.

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This magnetic structure broadening of 3.6 arcsec should be included in our future preprocessing method. One idea is to set broadening perturbations from this length, and another idea is to include this length as the L₄ term that decides smoothness of a preprocessed map. We also need to investigate magnetic structure broadening in other active regions.

One of the ideas to improve preprocessing methods is to add other pieces of information from the chromosphere. In addition to L₁ to L₄ terms, Wiegelmann et al. (2008) propose an additional term measuring differences of directions between chromospheric fibrils and transverse fields (B_x and B_y). Directions of fibrils can be measured from observed Hα images. Minimizing this additional term, we can obtain expected directions of B_x and B_y in the chromosphere. Wiegelmann et al. (2008) show results of preprocessing including this term and results of NLFFFs extrapolated with preprocessed magnetic fields. Preprocessing including this additional term shows better results than preprocessing without the additional term. It would be promising to add both our and Wiegelmann's additional terms into Equation (1). Multi-wavelength observations including Hα observation would be useful in deciding directions of magnetic fields. Other additional information from the chromosphere could further improve preprocessing.