Casting the Coronal Magnetic Field Reconstruction Tools in 3D Using the MHD Bifrost Model

The simulation covers a physical extent of 24 × 24 × 16.8 Mm, with a grid of 504 × 504 × 496 cells, extending from −2.4 Mm below the photosphere, which corresponds to Z = 0 Mm, to 14.4 Mm above the photosphere. It encompasses the upper convection zone, photosphere, chromosphere, and the lower corona. The horizontal axes have an equidistant grid spacing of 48 km; the vertical grid spacing is nonuniform, with a spacing of 19 km between Z = −1 Mm and Z = 5 Mm. The spacing increases toward the top of the computational domain to a maximum of 98 km. The magnetic topology is defined by two opposite-polarity patches separated by 8 Mm, with an average unsigned strength of ∼50 G in the photosphere, representing two patches of the quiet-Sun network. The simulation includes optically thick radiative transfer in the photosphere and low chromosphere; parameterized radiative losses in the upper chromosphere, transition region, and corona; thermal conduction along magnetic field lines; and an equation of state that includes the effects of non-equilibrium ionization of hydrogen. The Bifrost code solves the equations of resistive MHD on a staggered Cartesian grid but the published data cubes for en024048_hion have the variables specified at the cell centers.

One of the simulation snapshots, snapshot 385, has already been used in a series of papers to investigate the formation of the IRIS diagnostics (Leenaarts et al. 2013a, 2013b; Pereira et al. 2013, 2015; Lin & Carlsson 2015; Rathore & Carlsson 2015; Rathore et al. 2015) and of other chromospheric lines (Leenaarts et al. 2012, 2015; Štěpán et al. 2012, 2015; de la Cruz Rodríguez et al. 2013), as well as to model continuum free–free emission from the solar chromosphere (Loukitcheva et al. 2015a) with the perspective of ALMA observations (Loukitcheva et al. 2015b).

As has been mentioned, the available en024048_hion data cubes are specified on a nonuniform grid in the vertical direction, although most of the NLFFF approaches employ a regular grid, which has a number of computational advances, including uniform precision of the spatial derivatives. In order to make the en024048_hion model compatible with the extrapolated data cubes, the original model grid was transformed to a uniform grid (with a 48 km step along all three axes) by linear interpolation in the vertical direction.

Given that the applicability of the force-free magnetic modeling relies on the dominant role of the Lorentz force in the overall balance of forces, we start with an analysis of the height distribution of the plasma parameter β = p_kin/p_B in the model volume. In Figure 1, we plot the range of β values as a function of the model height using the original nonuniform grid and demarcate a number of characteristic levels for further references. Level H = 0 corresponds to the nominal photosphere in the model data cube. Since the model was created to describe an enhanced network on the quiet Sun rather than a typical active region (AR), the β values at the level of the nominal photosphere are, not surprisingly, much larger than those for an AR (Gary 2001). Such a boundary condition, having an overall weak magnetic field with small-scale inserts of a stronger field, represents a big challenge for the currently available methods of coronal force-free modeling designed for ARs with relatively large-scale areas of strong magnetic field, where the plasma parameter β, although larger than one, is not exceptionally large.

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Therefore, to emulate a photospheric boundary representative of a typical AR, we select a higher level where the distribution of the plasma β is similar to that in a typical AR (Gary 2001); we call this level a "typical AR β photosphere," or just the β-photosphere for short. Specifically, we define this layer as the lowest layer above the nominal photosphere where 95% of the nodes have β < 10². This level roughly corresponds to ${H}_{\beta \mbox{-} \mathrm{ph}}\approx 1$ Mm; the exact height varies between the data cubes obtained with different binning factors (see Section 2.2). Then, we define a chromospheric interface as the lowest layer where 95% of the nodes have β < 10⁻¹, which is supposed to offer an ideal boundary condition for  force-free modeling. This level roughly corresponds to the height H_chr ≈ 2.5 Mm. We return to a more specific definition of these levels in the next section, while introducing rebinned data cubes with a lower spatial resolution.

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From the original magnetic data cube, we initially created a new uniform data cube with full resolution over the x, y coordinates and a regular grid in the z direction, with the voxel height of 48 km equal to the pixel sizes in the x, y directions, while entirely discarding the subphotospheric part of the data cube. This yields a new full-resolution data cube with a regular grid having cubic voxels, convenient for further manipulation and analysis,  which we refer to as the "original regular" data cube hereafter.

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This new original data cube was then rebinned to produce lower-resolution data cubes⁵ with binning factors n = 2, 3, 4, 6, 7, 9. Apparently, a lower-resolution voxel includes n³ original voxels. Therefore, for each new voxel, we use the mean values of the magnetic field components to represent these values in the center of the new voxel such that

$$\begin{eqnarray}&&{\overline{B}}_{\alpha }=\displaystyle \frac{1}{{n}^{3}}\displaystyle \sum _{i=1}^{{n}^{3}}{B}_{\alpha }[i],\qquad \alpha =x,\,y,\,\mathrm{or}\,z,\end{eqnarray} \tag{ 1 }$$

and the standard deviations δB_α is

$$\begin{eqnarray}&&\delta {B}_{\alpha }^{2}=\displaystyle \frac{1}{{n}^{3}-1}\displaystyle \sum _{i=1}^{{n}^{3}}{({B}_{\alpha }[i]-{\overline{B}}_{\alpha })}^{2},\qquad \alpha =x,\,y,\,\mathrm{or}\,z.\end{eqnarray} \tag{ 2 }$$

This set of the data cubes with different resolutions represents the input for our tests. The distributions of z-component of the magnetic field for different binning factors at the nominal and β-photospheres and at the chromosphere are shown in Figures 2–4.

There are a number of numerical characteristics, similar to those introduced by Wheatland et al. (2000) and Schrijver et al. (2006), used to evaluate to what extent the reconstructed magnetic field matches the force-free criterion:

$$\begin{eqnarray}\theta & = & \arcsin \left(\displaystyle \frac{{\displaystyle \sum }_{i}^{N}{\sigma }_{i}}{N}\right),\quad {\theta }_{j}=\arcsin \left(\displaystyle \frac{{\displaystyle \sum }_{i}^{N}{| {\boldsymbol{j}}| }_{i}{\sigma }_{i}}{{\displaystyle \sum }_{i}^{N}{| {\boldsymbol{j}}| }_{i}}\right),\\ {\sigma }_{i} & = & \displaystyle \frac{{| {\boldsymbol{j}}\times {\boldsymbol{B}}| }_{i}}{{| {\boldsymbol{j}}| }_{i}{| {\boldsymbol{B}}| }_{i}},\end{eqnarray} \tag{ 3 }$$

and the divergence-free criterion:

$$\begin{eqnarray}&&f=\displaystyle \frac{1}{N}\displaystyle \sum _{i}^{N}\displaystyle \frac{{| {\rm{\nabla }}{\boldsymbol{B}}| }_{i}}{6{| {\boldsymbol{B}}| }_{i}}{dx},\end{eqnarray} \tag{ 4 }$$

where ${\boldsymbol{B}}$ is the model field; ${\boldsymbol{j}}$ is the corresponding electric current density, with the summation performed over the voxels of the computational grid (excluding boundaries); dx is the grid spacing; σ_i is the sine of the angle between the magnetic field and the current density at the ith node of the computational grid; θ is the angle averaged over all nodes, where in the ideal case of a force-free field it must be zero; θ_j is a similar metric but weighted with the electric current, which means that contributions coming from subdomains with strong currents dominate this metric; and f is a parameter characterizing the accuracy of the magnetic field being divergence-free.

Here we compute these metrics for the original regular data cube and for the data cubes obtained after the rebinning. Table 1 presents the metrics calculated for the data cube set. For each data cube, we present three sets of metrics. The first one is calculated for the entire data cube volume starting from the nominal photosphere. The other two are for subdomains that start at either the β-photosphere or chromosphere levels. Independently of the binning factor, each metric displays the same trend as the function of the subdomain used for the computation: from the full data cube to the chromosphere-limited subdomain, θ shows a minor decrease, while θ_j decreases significantly, which is not surprising. Indeed, θ_j is weighted with the current density, while the strongest currents are concentrated near the bottom boundary. If this bottom boundary is located below the chromosphere, then the contribution from the non-force-free field region dominates. Above the chromosphere, the field corresponds to a force-free configuration with a reasonably high accuracy, though not exactly. It is interesting that the divergence-free parameter f also improves when only the subdomain located above the chromosphere is taken into account, which, perhaps, originates from regridding the original nonuniform vertical grid to the uniform one used here. In addition, the prominent small-scale structure of the magnetic field at the lower levels results in more significant errors in the differential schemes used here to compute the spatial derivatives.

Table 1.  Numerical Characteristics of Different Domains Depending on Binning Factor

Binning Factor	Volume Above	Layer #	θ°	${\theta }_{j}^{^\circ }$	f × 106
1	Nominal photosphere	0	19.23	48.60	568
	β-photosphere	17	17.21	18.65	158
	chromosphere	42	16.23	5.59	107
2	Nominal photosphere	0	18.59	46.00	1573
	β-photosphere	8	16.65	18.57	401
	chromosphere	21	15.55	5.97	245
3	Nominal photosphere	0	18.48	43.93	2594
	β-photosphere	6	16.32	14.70	609
	chromosphere	14	15.37	6.41	410
4	Nominal photosphere	0	18.62	42.41	3315
	β-photosphere	4	16.67	16.53	934
	chromosphere	11	15.49	6.84	583
6	Nominal photosphere	0	19.12	39.53	3690
	β-photosphere	3	16.94	13.53	1315
	chromosphere	7	15.87	7.60	951
7	Nominal photosphere	0	19.40	38.24	3671
	β-photosphere	3	16.93	11.37	1415
	chromosphere	6	16.12	7.94	1135
9	Nominal photosphere	0	20.03	36.13	4057
	β-photosphere	2	17.82	13.63	1939
	chromosphere	5	16.64	8.58	1482

Note. For each particular layer, the "layer" column shows its level in the corresponding rebinned grid. The BIFROST photosphere always corresponds to zero level.

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All metrics get worse with increasing binning factor. The reason for this is the unavoidable increase of numerical errors of the derivative computation. Indeed, rebinning the model data causes distortions and thus, an increase of the divergence-free metrics f.

A number of disambiguation methods, which differ in their approaches, performance, and speed, have been proposed (Metcalf 1994; Georgoulis 2005; Crouch 2013; Gosain & Pevtsov 2013; Rudenko & Anfinogentov 2014). Detailed comparisons of different methods can be found in comprehensive reviews (Metcalf et al. 2006; Leka et al. 2009). For our tests, we selected the minimum energy (ME) solution (Metcalf 1994); the so-called super fast and quality disambiguation (SFQ) method (Rudenko & Anfinogentov 2014), also known as the new disambiguation method (NDA); and the acute angle (AA) method, which makes a choice by comparing the azimuth angle with that of a "reference" field (Metcalf et al. 2006). The reference field can be a potential or linear force-free field, for example, but we employ a simple version of the AA method, which utilizes a naive approach where the true transverse field direction is selected to be closer to the potential field.

The ME disambiguation (http://www.cora.nwra.com/AMBIG/) is based on simulated annealing (Metropolis et al. 1953) and is regarded as providing the highest disambiguation quality over all methods. However, this method is rather computationally expensive. Our implementation of the ME method follows that of Metcalf (1994), with an improvement suggested by Leka et al. (2009), namely with the initial "temperature" varying from pixel to pixel depending on the maximum temperature among all possible combinations of the transverse field orientation of the neighbor pixels. In addition, in this implementation, for any pixel keeping its transverse field orientation for a reasonably long time (typically, during 100 consecutive iterations), the ambiguity is treated as "resolved" and the algorithm does not change its state over the remaining iterations. Thus, the number of analyzed pixels decreases with the number of iterations, which reduces the annealing time.

The SFQ method (Rudenko & Anfinogentov 2014) includes two steps: a preliminary disambiguation and a "cleaning" procedure. The former is a local comparison of the observed ambiguous field with the reference (potential) one. The comparison procedure is similar to that in the AA method. The discontinuities in this first-order approximate solution are then cleaned out with an iterative procedure. On each iteration, the transverse magnetic field is compared with its local average and inverted in those pixels where the alternative value is closer to the smoothed field.

To test the quality of the π-disambiguation methods, we used the following metrics (Metcalf et al. 2006), pixel error E_pix and flux error E_flux :

$$\begin{eqnarray*}&&{E}_{\mathrm{pix}}=\displaystyle \frac{{N}_{\mathrm{err}}}{N},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{E}_{\mathrm{flux}}=\displaystyle \frac{{\sum }_{\mathrm{err}}{B}_{\tau }(x,y)}{{\sum }_{\mathrm{all}}{B}_{\tau }(x,y)},\end{eqnarray*}$$

where N is the total number of pixels and N_err is the number of pixels where the disambiguation method fails, B_τ(x, y) is the absolute value of the transverse magnetic field in pixel (x, y), and ${\sum }_{\mathrm{err}}$ and ${\sum }_{\mathrm{all}}$ are the summations over those pixels where the disambiguation failed and over all pixels in the magnetogram, respectively, as well as the typical computation time.

The π-disambiguation methods described above have been applied to all reference layers and all rebinned data cubes. Here we summarize their performance.

For the nominal photosphere (see Table 2), all tested methods fail to provide acceptable results. Both flux and pixel errors are of the order of 20%–30%, with the only exceptions being the large bin factors and SFQ method, where the errors are about 10%; see Figure 5. In most cases (but not always), both methods work better in areas of stronger magnetic field but fail in areas of weak small-scale magnetic field. This failure originates from the fact that none of the methods is capable of correctly processing the very small magnetic elements with the size of the order of one pixel (salt and pepper patterns). This might have important implications for the π-disambiguation of the magnetic field in the quiet-Sun areas, which is, in particular, a substantial part of the full-disk disambiguation in the SDO/HMI pipeline.

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Table 3 shows the test results for the level of a typical AR β-photosphere. At this level, the magnetic field becomes noticeably smoother than that at the nominal photosphere, and all methods improve their recovery success rate by at least an order of magnitude. AA pixel errors exceed 5% and the flux errors exceed 2%. SFQ and ME results are much better. On the finest grid, the best quality is provided by SFQ, while the output of the ME method contains an extended area where the transverse flux is inverted. For the grids with lower resolutions (bin 6–9), this artifact disappears and ME outperforms SFQ. However, both SFQ (all bins) and ME (bins 3–9) give flux errors of 1% or less and thus, their results can be used as boundary conditions for NLFFF extrapolations. In principle, it is tempting to use the π-disambiguation mismatches between the different disambiguation methods to reprocess those pixels, where two methods give opposite results, to improve the solution. However, as clearly seen from Figure 6, there are a number of pixels where both methods fail to recover the field azimuth, even though the total number of those "bad" pixels is very small. Figure 6 demonstrates that the failures are not random but occur in the areas where the field is either weak or almost vertical or both.

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At the chromospheric level (Table 4), the magnetic field is more horizontal and smooth enough, so it is disambiguated perfectly by ME (there are errors in several pixels only for bins 4 and 6). The SFQ method also works excellently and fails only in several tens of pixels giving flux errors from 0.0001% to 0.01%, depending on the binning. AA also performs better on the smooth chromospheric (rather than the photospheric) field with the flux error about 1.5% for all bins.

These tests validate both ME and SFQ disambiguation methods at the β-photosphere and the chromosphere. This permits us to assume that the π-ambiguity has been resolved perfectly while performing further tests on the field preprocessing and extrapolation. Note that in this present article we do not study the dependence of the disambiguation accuracy on the position of the data cube at the solar disk. Most of the disambiguation approaches tend to generate more errors toward the solar limb. Here, the SFQ method has an advantage. According to tests on real magnetograms observed close to the disk center and artificially rotated toward the limb, it maintains a high accuracy even in those areas (Rudenko & Anfinogentov 2014), while other methods often produce artifacts in the transverse field components.

Wiegelmann et al. (2006) proposed preprocessing the photospheric vector magnetogram data to drive the observed non-force-free photospheric data toward a suitable boundary condition in the chromosphere by minimizing a functional L_prep:

$$\begin{eqnarray}&&{L}_{\mathrm{prep}}={\mu }_{1}{L}_{1}+{\mu }_{2}{L}_{2}+{\mu }_{3}{L}_{3}+{\mu }_{4}{L}_{4},\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray*}&&{L}_{1}={({{\rm{\Sigma }}}_{p}{\tilde{B}}_{x}{\tilde{B}}_{z})}^{2}+{({{\rm{\Sigma }}}_{p}{\tilde{B}}_{y}{\tilde{B}}_{z})}^{2}+{({{\rm{\Sigma }}}_{p}({\tilde{B}}_{z}^{2}-{\tilde{B}}_{x}^{2}-{\tilde{B}}_{y}^{2}))}^{2},\end{eqnarray*}$$

$$\begin{eqnarray*}{L}_{2} & = & {({{\rm{\Sigma }}}_{p}x{({\tilde{B}}_{z}^{2}-{\tilde{B}}_{x}^{2}-{\tilde{B}}_{y}^{2}))}^{2}+({{\rm{\Sigma }}}_{p}y({\tilde{B}}_{z}^{2}-{\tilde{B}}_{x}^{2}-{\tilde{B}}_{y}^{2}))}^{2}\\ & & +{({{\rm{\Sigma }}}_{p}(y{\tilde{B}}_{x}{\tilde{B}}_{z}-x{\tilde{B}}_{y}{\tilde{B}}_{z}))}^{2},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{L}_{3}={{\rm{\Sigma }}}_{p}{({\tilde{B}}_{x}-{B}_{x})}^{2}+{{\rm{\Sigma }}}_{p}{({\tilde{B}}_{y}-{B}_{y})}^{2}+{{\rm{\Sigma }}}_{p}{({\tilde{B}}_{z}-{B}_{z})}^{2},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{L}_{4}={{\rm{\Sigma }}}_{p}{({\rm{\Delta }}{\tilde{B}}_{x})}^{2}+{{\rm{\Sigma }}}_{p}{({\rm{\Delta }}{\tilde{B}}_{y})}^{2}+{{\rm{\Sigma }}}_{p}{({\rm{\Delta }}{\tilde{B}}_{z})}^{2}.\end{eqnarray*}$$

Here, minimizing the L₁ and L₂ terms ensures that the final magnetogram corresponds, as closely as possible, to the force-free and torque-free conditions, respectively. The L₃ term is responsible for the similarity of the preprocessed data to the original magnetogram, while the L₄ term is responsible for the smoothing. The summations Σ_p represent the surface integrals over all available grid nodes p at the bottom boundary; Δ stands for the two-dimensional (2D) Laplace operator. The idea is to minimize L_prep at once, so that all L_n terms are getting smaller simultaneously. The weights μ_n are unknown a priori; Wiegelmann et al. (2006) tested various choices for μ_n and came up with a strategy on how to choose these weights; the default set of the weights is μ = [1, 1, 10⁻³, 10⁻²]. Minimization of the functional is performed iteratively using the Newton-Raphson scheme⁶ (Press et al. 2007). Hereafter, we refer to our implementation of the preprocessing method developed by Wiegelmann et al. (2006) as TW preprocessing.

Fuhrmann et al. (2007) proposed another form of functional (5) to be minimized. The distinctions are in the form of the L₄ term (the smoothing was performed using a median filter instead of the differential Laplace operator in Wiegelmann et al. 2006), in varying the field only inside a given range defined by a field threshold value instead of considering L₃, and in the minimization method selected—annealing simulation. Fuhrmann et al. (2011) have shown that the Wiegelmann et al. (2006) method gives a smoother preprocessed field. Since both methods have similar natures and show comparable final results, we only consider the Wiegelmann et al. (2006) method for further testing.

Jiang & Feng (2014) proposed a slightly different approach to the preprocessing that consists of two distinct steps. The first step produces a potential extrapolation starting from the photospheric vertical magnetic field component B_z. Then, at a certain level, typically one pixel above the photosphere, the B_z component is taken from the potential extrapolation, while a functional similar to Equation (5) is formed for the transverse field components only and is being minimized at the second step of the method. An apparent advantage of this approach is the ability to control the height level, to which the preprocessed magnetogram must correspond. For our tests, we used the preprocessing code provided by the authors Jiang & Feng (2014), with the default set of the weights μ = [1, 1, 10⁻³, 1] and with a few cosmetic modifications needed to ease the running of the code. In what follows, we refer to the preprocessing method of Jiang & Feng (2014) as JF preprocessing.

Preprocessing methods are supposed to modify the magnetic field vector measured at the photosphere to make it more compatible with the force-freeness of the magnetic field model in the corona. The corresponding modification of the photospheric field could either result in the removal of the force component from the magnetic field distribution without any noticeable change in the field strength, or, in addition to this removal, may yield a corresponding decrease of the field strength emulating an effective "elevation" of the field distribution up to a higher chromospheric level where the magnetic field does become a force-free one.

Thus, we assess the preprocessing methods in the following three respects:

• 1.  
  Has the magnetic field become significantly more force-free after preprocessing?
• 2.  
  How close is the preprocessed field to the magnetic field in the BIFROST model?
• 3.  
  Does the preprocessing cause an effective "elevation" of the field distribution up to a higher level?

The force-freeness of the preprocessed magnetic field is assessed using the L₁ and L₂ terms from Equation (5). In this work, we quantitatively measure the preprocessing efficiency by calculating the logarithmic ratio of the L₁ and L₂ parameters before and after the preprocessing: ${\mathrm{log}}_{10}\tfrac{{L}_{1}^{0}}{{L}_{1}^{p}}$ and ${\mathrm{log}}_{10}\tfrac{{L}_{2}^{0}}{{L}_{2}^{p}}$. The higher values of these parameters indicate a more efficient removal of the force-carrying magnetic field component.

The consistency of the preprocessed magnetic field with the model magnetic field at a given level of the data cube can straightforwardly be assessed using the L₃ metrics in Equation (5). To assess if any effective elevation has happened, we calculate this metric using the Bifrost data from the same level and a few higher levels. We also use a few more useful metrics defined below:

• 1.  
  Normalized discrepancy between the preprocessed and observed field components:

  $$\begin{eqnarray}&&\tilde{{L}_{3}}=\sqrt{\displaystyle \frac{{L}_{3}}{{\sum }_{p}{B}^{2}}}.\end{eqnarray} \tag{ 6 }$$
• 2.  
  Normalized discrepancy between the absolute value of the preprocessed and observed fields:

  $$\begin{eqnarray}&&{E}_{B}=\sqrt{\displaystyle \frac{{\sum }_{p}{(\tilde{B}-B)}^{2}}{{\sum }_{p}{B}^{2}}}.\end{eqnarray} \tag{ 7 }$$
• 3.  
  Average angle between the preprocessed and observed fields:

  $$\begin{eqnarray}&&A=\arccos \left({\sum }_{p}\displaystyle \frac{\tilde{{\boldsymbol{B}}}\cdot {\boldsymbol{B}}}{\tilde{B}B}/N\right).\end{eqnarray} \tag{ 8 }$$
• 4.  
  Slope of the regression dependence $\tilde{B}={SB}+b$:

  $$\begin{eqnarray}&&S=\displaystyle \frac{{\sum }_{p}\tilde{B}B-{\sum }_{p}\tilde{B}{\sum }_{p}B/N}{{\sum }_{p}{B}^{2}-{({\sum }_{p}B)}^{2}/N}.\end{eqnarray} \tag{ 9 }$$
• 5.  
  Correlation coefficient:

  $$\begin{eqnarray}&&R=\displaystyle \frac{{\sum }_{p}\tilde{B}B-{\sum }_{p}\tilde{B}{\sum }_{p}B/N}{\sqrt{{\sum }_{p}{\tilde{B}}^{2}-{({\sum }_{p}\tilde{B})}^{2}/N}\sqrt{{\sum }_{p}{B}^{2}-{({\sum }_{p}B)}^{2}/N}}.\end{eqnarray} \tag{ 10 }$$

A better preprocessing method is expected to give lower values of ${\tilde{L}}_{3}$, while the preferable values of the slope S and the correlation coefficient R are those closer to unity. If the preprocessed and the original fields are substantially different, ${\tilde{L}}_{3}$ will have a larger value. For instance, if $\tilde{{\boldsymbol{B}}}\equiv 0$, the ${\tilde{L}}_{3}$ metric will be equal to unity for any ${\boldsymbol{B}}$. Therefore, we will interpret the preprocessing results with ${\tilde{L}}_{3}\geqslant 0.5$ as unacceptable. Additionally, we calculate metrics (6)–(10) for the model magnetic field taken at the height of one voxel to nail down a possible effective elevation that can be a side or intended effect of the preprocessing.

Tables 5–7 show the metrics describing how force-free are the boundary conditions produced by the two codes. For the nominal photosphere, all results have ${\tilde{L}}_{3}\gt 0.5$, meaning that the preprocessed field is strongly different from the original magnetogram. Therefore, we interpret the preprocessing results obtained for the nominal photosphere as incorrect and do not analyze them in what follows. For all levels and binning factors, our implementation of the TW preprocessing (Wiegelmann et al. 2006) outperforms the JF preprocessing code (Jiang & Feng 2014), providing significantly more force-free results. However, the L₃ metrics are nearly the same for both codes, meaning that both solutions are comparably close to the input magnetic field.

Table 5.  Results of the Preprocessing Test at the Nominal Photosphere

	${\mathrm{log}}_{10}\tfrac{{L}_{1}^{o}}{{L}_{1}^{p}}$		${\mathrm{log}}_{10}\tfrac{{L}_{2}^{o}}{{L}_{2}^{p}}$		L3
Bin	JF	TW	JF	TW	JF	TW
2	4.55	11.26	5.42	11.28	0.60	0.72
3	4.50	10.68	5.42	10.80	0.67	0.79
4	4.35	10.42	5.32	10.89	0.70	0.83
6	4.20	9.80	4.97	10.16	0.74	0.85
7	4.11	9.59	4.72	9.96	0.74	0.85
9	3.99	9.24	4.29	9.58	0.74	0.85

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Table 6.  Results of the Preprocessing Tests at the β-photosphere

	${\mathrm{log}}_{10}\tfrac{{L}_{1}^{o}}{{L}_{1}^{p}}$		${\mathrm{log}}_{10}\tfrac{{L}_{2}^{o}}{{L}_{2}^{p}}$		L3
Bin	JF	TW	JF	TW	JF	TW
2	2.64	11.02	2.65	10.28	0.24	0.19
3	2.07	10.79	2.09	9.59	0.24	0.20
4	2.38	10.33	2.41	8.94	0.31	0.29
6	1.79	9.61	1.82	8.35	0.30	0.30
7	1.30	9.35	1.29	8.51	0.27	0.27
9	1.60	9.56	1.60	7.81	0.34	0.37

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Table 7.  Results of the Preprocessing Tests at the Chromosphere

	${\mathrm{log}}_{10}\tfrac{{L}_{1}^{o}}{{L}_{1}^{p}}$		${\mathrm{log}}_{10}\tfrac{{L}_{2}^{o}}{{L}_{2}^{p}}$		L3
Bin	JF	TW	JF	TW	JF	TW
2	0.67	9.37	0.57	9.22	0.08	0.039
3	0.67	10.25	0.57	11.04	0.10	0.060
4	0.64	10.31	0.55	10.53	0.12	0.088
6	0.66	9.67	0.57	9.81	0.16	0.12
7	0.66	9.47	0.56	9.55	0.17	0.13
9	0.62	9.24	0.53	9.11	0.20	0.15

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Detailed quantitative comparisons of the preprocessing results and the magnetic field in the Bifrost model are given in Tables 8 and 9. We separately assess the full vector and the longitudinal and transverse components at two levels: the same level as the initial magnetogram and one level higher. The E_B metric (Equation (7)) indicates the discrepancy between the absolute values of the preprocessed and modeled magnetic fields, while the difference in the directions of the fields is shown by the A metric (Equation (8)). This metric shows that there is an effective field elevation caused by either of the JF (all bins) or TW (bins 4–9) preprocessing methods. The effect is more pronounced for JF preprocessing especially at low spatial resolution (bin 9). However, the mean angle between the preprocessed and Bifrost fields shows a different behavior. A one-level elevation does not improve the mean angle metric for JF preprocessing, while the TW method demonstrates a slight improvement of this metric at bins 7 and 9. Being applied to the chromospheric level (even though this might not be needed in practice), JF preprocessing again demonstrates effective elevation in E_B, while the TW method shows no improvement at a higher level in either absolute value discrepancy or in angle.

The JF and TW methods preprocess longitudinal and transverse magnetic field components differently. Therefore, we assess the field components separately. For both longitudinal and transverse components, we calculate the slope (Equation (9)) and correlation (Equation (10)) metric. The former characterizes the absolute value of the field while the latter assesses the spatial structuring of the field.

Both slope and correlation metric show the presence of an effective elevation in the longitudinal component preprocessed using the JF method. Indeed, such a behavior is rather expected because the JF preprocessing fixes the longitudinal component to be equal to the potential field extrapolation results and does not optimize it. The transverse component is, however, elevated by a height exceeding one voxel size (but less than two voxel sizes). Therefore, there is a mismatch in the heights to where the longitudinal and transverse components are elevated in the JF method. This mismatch is primarily responsible for the mismatch in the angle between the preprocessed and the (one level up) model field mentioned above.

After the TW preprocessing, the value of the longitudinal field becomes underestimated for the original level (S < 1) but overestimated for the +1 voxel level (S > 1), meaning that some effective elevation is present, but the elevation height is somewhere between 0 and 1 voxel. In contrast, the transverse component displays an elevation very close to the one-voxel height; thus, the TW preprocessing also results in a mismatch in the elevation heights for the longitudinal and transverse components, the same as with the JF method.

The JF preprocessing demonstrates a better cross-correlation metric R for the longitude component for all levels and binning factors, both at the photospheric and chromospheric heights. The cross-correlation coefficient for the transverse field is worse for both methods at all levels and heights. At high resolutions (bins 2–4), JF and TW show very similar results, while at low resolution (bins > 4), JF demonstrates a higher correlation with the modeled field. This behavior is also demonstrated in Figure 7 in the form of two-dimensional histograms showing the scatter of the preprocessed field components versus the modeled ones.

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Our tests show that both methods have their own advantages and disadvantages. The TW preprocessing improves the force-freeness metrics by many orders of magnitude, but significantly changes the field structure, especially for lower spatial resolutions; see Figure 8. It also effectively elevates the field, but the elevation height seems to differ from one voxel and differs for the longitudinal and transverse components. The JF preprocessing saves the field structure for low-resolution grids better and more distinctly elevates the longitudinal field to the height of one voxel, but the output is not as force-free as that for the TW method; in addition, the transverse component is effectively elevated by a height larger than the size of one voxel. These mismatches in the elevation heights for the field components revealed in both methods will then have a negative impact on the fidelity of the NLFFF extrapolations.

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In both implementations used in this paper, the NLFFF reconstructions are performed following the optimization method (Wheatland et al. 2000). The main idea of the optimization method is to transform some trial configuration of the magnetic field (usually a potential extrapolation from the bottom boundary) to a final force-free field configuration. This is achieved by minimization of the following, positively defined, functional:

$$\begin{eqnarray}&&L={\int }_{V}[{B}^{-2}{[[{\rm{\nabla }}\times {\boldsymbol{B}}]\times {\boldsymbol{B}}]}^{2}+| {\rm{\nabla }}\cdot {\boldsymbol{B}}{| }^{2}]w(x,y,z){dV},\end{eqnarray} \tag{ 11 }$$

where w(x, y, z) is a "weight" function defined below.

It is obvious that if the magnetic field ${\boldsymbol{B}}$ has a force-free configuration everywhere in the volume of interest V then L must be zero. In practice, the solution with L = 0 is hardly achievable, so minimization of the functional leads to an approximate force-free solution for the magnetic configuration. Solving the minimization problem (see appendices in Wheatland et al. 2000 and Wiegelmann 2004 for the detailed math) yields two equations, which optimize the magnetic field in the inner volume and on the boundaries of the computational domain.

The two implementations of the optimization method used in our study are different in how the boundary zone is treated. The IM implementation takes into account both optimization equations as detailed by Rudenko & Myshyakov (2009), such as the magnetic field being allowed to reconfigure everywhere in the volume of interest including its top and side boundaries; w ≡ 1 in this implementation. The magnetic field at the bottom boundary, which represents the input magnetogram, remains fixed during the optimization. Having a variable magnetic field at the top and side boundaries is potentially helpful, because, given that the NLFFF reconstruction is initiated with a potential configuration, fixing the potential field at these boundaries may have a negative impact on the force-freeness of the reconstructed magnetic field.

The alternative, the AS implementation, follows the Wiegelmann (2004) approach in selecting the weight function: w(x, y, z) = w_x(x)w_y(y)w_z(z), where the factors w_x(x) and w_y(y) are both equal to 1 in the internal area of the simulation cube (0.1–0.9) · L_x,y. In the buffer zone outside this internal area, the factors w_x(x) and w_y(y) decrease toward zero with the cosine functions. The other factor w_z(z) = 1 for the heights (0–0.9) · L_z and then goes to zero with the cosine function. The magnetic field at the top and side boundaries is frozen to be the potential one, the same one used to initialize the optimization. After each successful iteration, the step is increased by a factor of 1.03, while after each unsuccessful iteration the step is reduced by a factor of 0.8. The optimization ends when the step has become less than 0.01 of the initial step.

One modification of the AS implementation relative to the original method is the selection of the initial approximation for the magnetic field (which follows a multigrid extension proposed by Metcalf et al. 2008): for the sparsest grid (bin 9 in our tests), the initial approximation is the potential field, while for each subsequent denser grid, the initial field is taken as an appropriate interpolation of the final (NLFFF) state of the previous grid. The mentioned modifications optimize the simulation time but have almost no effect on the final metrics of the reconstructed NLFFF cube.

Alternative implementations have different time performances. Both codes spend comparable times on a single iteration. For example, in the case of bin 9, the IM code does ≈770 iterations per second, while the speed of the AS code is ∼200 iterations per second (calculations were performed on a four-core 3.4 GHz processor). However, the AS code has to make only about 2,500 iterations in total to get the final result for bin = 9 due to a dynamically changed time step; the number of required iterations becomes significantly smaller (several hundreds) for the higher-resolution grids as they use the interpolated solution of the scarcer grid as the initial condition, which is a much better approximation of reality than the potential extrapolation used for the bin = 9 case. The IM code has a fixed time step, makes 10⁵ iterations in total and even more for lower bin factors. Therefore, the AS implementation requires a considerably shorter computational time.

There are two questions we want to answer about the performance of our NLFFF extrapolation codes: (1) how close the final data cubes are to the targeted force-free state and (2) how well they reproduce the original field in the entire 3D domain. To address the first question, we use the same metrics as we have used to evaluate the force-freeness of the original model field itself, Equations (3)–(4).

To assess how close the NLFFF extrapolated data cube is to the corresponding model data cube, we use "angular" metrics similar to Equation (3):

$$\begin{eqnarray}{\theta }_{m} & = & \arccos \left(\displaystyle \frac{{\displaystyle \sum }_{i}^{N}{\tau }_{i}}{N}\right),\quad {\theta }_{{mj}}=\arccos \left(\displaystyle \frac{{\displaystyle \sum }_{i}^{N}{| {\boldsymbol{j}}| }_{i}{\tau }_{i}}{{\displaystyle \sum }_{i}^{N}{| {\boldsymbol{j}}| }_{i}}\right),\\ {\tau }_{i} & = & \displaystyle \frac{{{\boldsymbol{B}}}_{\mathrm{NLFFF},i}\cdot {{\boldsymbol{B}}}_{i}}{{| {{\boldsymbol{B}}}_{\mathrm{NLFFF}}| }_{i}{| {\boldsymbol{B}}| }_{i}},\end{eqnarray} \tag{ 12 }$$

where ${\boldsymbol{j}}$ is the electric current density computed for the reconstructed field, the summation is performed over the voxels of the analyzed volume subdomain, τ_i is the cosine of the angle between the restored and model magnetic field at the ith node of the computational grid, θ_m is the angle averaged over all nodes which in the ideal case of the force-free field must be zero, θ_mj is a similar metric but weighted with the restored electric current which ensures that the contribution from nodes with a strong electric current dominates this metrics. For a voxel-to-voxel inspection, we compute the local error (residual) ${{\rm{\Delta }}}_{\alpha }[i]$, the local relative error δ_α[i], and the local normalized residual ${\chi }_{\alpha }^{2}[i]$ as

$$\begin{eqnarray}{{\rm{\Delta }}}_{\alpha }[i] & = & {B}_{\mathrm{NLFFF},\alpha }[i]-{\overline{B}}_{\alpha }[i],\\ {\delta }_{\alpha }[i] & = & \displaystyle \frac{{B}_{\mathrm{NLFFF},\alpha }[i]-{\overline{B}}_{\alpha }[i]}{\langle {\overline{B}}_{\alpha }[i]\rangle },\qquad \alpha =x,\,y,\,\mathrm{or}\,z,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\chi }_{\alpha }^{2}[i]=\displaystyle \frac{{({B}_{\mathrm{NLFFF},\alpha }[i]-{\overline{B}}_{\alpha }[i])}^{2}}{\delta {B}_{\alpha }^{2}[i]},\qquad \alpha =x,\,y,\,\mathrm{or}\,z,\end{eqnarray} \tag{ 14 }$$

where i is the number of a given voxel, $\langle {\overline{B}}_{\alpha }\rangle =\sqrt{{\overline{B}}_{\alpha }^{2}+\delta {B}_{\alpha }^{2}}$, ${\overline{B}}_{\alpha }[i]$ and $\delta {B}_{\alpha }^{2}$ are defined for each binning factor by Equations (1) and (2), while ${B}_{\mathrm{NLFFF},\alpha }$ is the corresponding component of the magnetic field obtained from the extrapolation. Here, to compute the relative error, we take into account that after cube rebinning, the magnetic field in each voxel is only known to an accuracy of ${\overline{B}}_{\alpha }\pm \delta {B}_{\alpha }$. Thus, in the denominators of ${\delta }_{\alpha }[i]$ in Equation (13), we use $\langle {\overline{B}}_{\alpha }\rangle$ rather than ${\overline{B}}_{\alpha };$ otherwise, in "singular" points, where ${\overline{B}}_{\alpha }$ is very close to zero in the denominator, such a metric would artificially underestimate the accuracy. However, to compute a similar metric for the absolute value of the magnetic field vector, we do not add any δB because the absolute value is never very close to zero in the analyzed volume.

To characterize the extrapolation performance in a given subdomain, which can be, for example, a given layer or the entire data cube, we use the normalized rms residual Δ_rms, the normalized rms error δ_rms, and the "effective χ²" metrics defined as

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{\mathrm{rms},\alpha }=\sqrt{\displaystyle \frac{{\sum }_{i=1}^{{N}_{\mathrm{vox}}}{{\rm{\Delta }}}_{\alpha }^{2}[i]}{{\sum }_{i=1}^{{N}_{\mathrm{vox}}}{\overline{B}}_{\alpha }^{2}[i]}},\qquad \alpha =x,\,y,\,\mathrm{or}\,z,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\delta }_{\mathrm{rms},\alpha }=\sqrt{\displaystyle \frac{1}{{N}_{\mathrm{vox}}}{\sum }_{i=1}^{{N}_{\mathrm{vox}}}{\delta }_{\alpha }^{2}[i]},\qquad \alpha =x,\,y,\,\mathrm{or}\,z,\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\chi }_{\mathrm{eff},\alpha }^{2}=\displaystyle \frac{1}{{N}_{\mathrm{vox}}}{\sum }_{i=1}^{{N}_{\mathrm{vox}}}{\chi }_{\alpha }^{2}[i],\qquad \alpha =x,\,y,\,\mathrm{or}\,z,\end{eqnarray} \tag{ 17 }$$

where the summation is performed over the subdomain of the data cube⁸ used for the analysis; N_vox is the total number of voxels in the selected subdomain. The normalized rms residual, Equation (15), is similar to metric (7), which has been used to evaluate the preprocessing performance; this metric gives more weight to the voxels, where the magnetic field is strong. In contrast, metric (16) gives equal weight to any voxel, with either strong or weak field. The χ² metric weights the voxel in accordance with the uncertainty to which the magnetic field is known in the given voxel.

5.2.1. NLFFF Extrapolations from the Chromospheric Boundary

We expect that any extrapolation approach will perform best if the bottom boundary condition is close to being force-free. Thus, testing the extrapolation performance from the nearly force-free chromospheric layer will characterize the true potential of the given extrapolation code itself. For this reason, we begin our tests from the NLFFF data cubes extrapolated from the chromospheric level.

Table 10 presents the angular metrics that characterize the force-freeness of the reconstructed field (the angles θ and θ_j) and its closeness to the original model field (the angles θ_m and θ_mj) computed for the entire 3D volume above the bottom boundary and excluding the top and side buffer zones. It is interesting that the IM code provides a more force-free magnetic field than the one in the model: the corresponding values of θ and θ_j in Table 10 are systematically lower than those in Table 1. This implies that the dynamics and the finite gas pressure in the coronal volume described by the MHD model produce a measurable deviation of the coronal magnetic field from the force-free state, while the NLFFF optimization drives the magnetic data cube toward another, more force-free, solution. Nevertheless, the extrapolated field is reasonably close to the model one; see the θ_m and θ_mj metrics. In contrast, the field restored with the AS code is less force-free than the model one, which is likely a negative effect of the fixed top and side boundary conditions and the buffer zone employed in this method.

Table 10.  Performance of the Magnetic Field Reconstruction Methods

Bin	Impl	Chromosphere				β-photosphere				β-photosphere, JF Preprocessed				β-photosphere, TW Preprocessed
		θ°	${\theta }_{j}^{^\circ }$	${\theta }_{m}^{^\circ }$	${\theta }_{{mj}}^{^\circ }$	θ°	${\theta }_{j}^{^\circ }$	${\theta }_{m}^{^\circ }$	${\theta }_{{mj}}^{^\circ }$	θ°	${\theta }_{j}^{^\circ }$	${\theta }_{m}^{^\circ }$	${\theta }_{{mj}}^{^\circ }$	θ°	${\theta }_{j}^{^\circ }$	${\theta }_{m}^{^\circ }$	${\theta }_{{mj}}^{^\circ }$
3	IM	9.3	4.5	17.8	11.4	16.4	11.1	30.5	20.9	11.5	13.7	28.2	19.5	12.4	13.7	26.7	18.8
	AS	24.4	8.7	22.2	12.7	33.8	14.9	24.5	20.6	35.9	19.0	24.5	19.5	33.8	18.2	24.3	20.1
4	IM	10.7	5.4	18.1	11.5	18.9	13.4	34.3	22.3	13.0	16.5	28.5	21.2	13.7	16.9	27.1	20.3
	AS	24.9	9.4	21.9	12.4	34.5	16.3	24.5	20.7	36.6	21.3	24.3	20.0	34.5	20.8	24.1	20.5
6	IM	10.6	5.9	16.0	10.0	18.2	13.4	27.2	17.4	13.7	16.9	23.4	15.8	15.1	18.9	22.6	16.4
	AS	26.8	10.7	21.6	12.0	34.3	18.1	23.4	18.2	39.8	25.5	23.4	15.4	34.2	24.2	23.0	16.9
7	IM	11.0	6.3	15.8	9.8	14.3	10.6	21.3	13.2	12.5	14.2	20.2	13.5	14.0	17.9	19.4	14.2
	AS	28.3	11.5	21.6	11.8	39.1	21.1	23.0	17.1	42.0	25.9	23.0	14.4	38.8	27.7	22.4	15.1
9	IM	11.8	7.0	15.1	9.4	17.7	14.7	24.2	15.2	16.2	20.2	20.7	13.9	19.4	26.2	20.0	14.6
	AS	30.1	13.3	22.1	12.2	43.0	26.1	24.0	18.9	46.1	32.6	24.1	14.9	42.8	35.0	23.2	15.8

Note. "Bin" is the binning factor. "Impl" is the implementation of the optimization method. The following four wide columns contain the numerical characteristics of the reconstructed field. The column title provides information on the starting layer and whether preprocessing was applied.

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Likewise in the preprocessing tests, we are looking whether the restored values of the magnetic field components (or the absolute values) correlate with the model ones and what the scatter is around the cross-correlation curves. Figures 9–11, two left columns, display these cross-correlation plots in the form of 2D histograms superimposed on the y = x diagonal line. These plots suggest that the performance of the extrapolations improves for higher binning factors; in particular, the "clouds" in the area of poorer restored weak field values (around zero) are bigger for the smaller binning factors. Comparison between the two methods suggests that the IM approach works better for small magnetic field values, while for a large magnetic field, both methods perform comparably, although sometimes the AS method performs marginally better. We will return to this comparison later.

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Figures 12 and 13 give a clear visual idea of the optimization code performance in three layers: one close to the bottom of the cube, another one in a middle height, and the last one close to the top buffer zone in the AS code (for a fair comparison of the methods, we exclude the same buffer zone in both AS and IM cases); see animated figure for all layers. The error (second and fourth columns) of the magnetic field reconstruction slightly increases with height: although the scale of the error variation is almost the same at all of the three presented levels, −2 G ≲ Δ_z ≲ 2 G, the areas occupied by blue or red colors (indicating bigger errors) increase with height. As a result, the relative error (third and fifth columns) increases with height noticeably: the green area, where the relative error is within ±10%, becomes smaller with height. The relative error becomes large at neutral lines (i.e., where the magnetic field is about zero) at any layer, even at the one closest to the bottom.

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At low heights, the two competing methods perform comparably well: although the AS methods shows a smaller error in the middle of the plot where the magnetic field is reasonably large, it also produces some artifacts close to the boundaries, which is not surprising given that the method employs a buffer zone at the boundary regions. It is, however, interesting, that at intermediate heights the AS method provides a bigger green area in the plot, where the relative error of the field reconstruction is within 10%, than the IM method, although the IM method outperforms the AS one at higher heights. A qualitatively similar picture is observed for two other components of the magnetic field: B_y (Figure 13) and B_x (not shown).

The fact that we created our model data cubes by rebinning the original data cube implies that the magnetic field value ${\overline{B}}_{\alpha }$ in a given voxel is only known to an accuracy of ${\overline{B}}_{\alpha }\pm \delta {B}_{\alpha }$, defined by Equations (1) and (2); thus, even the most precise field reconstruction would only recover the field to this accuracy. Stated another way, an ideally perfect reconstruction would have ${\chi }_{\alpha }^{2}[i]\sim 1$ and ${\chi }_{\mathrm{eff},\alpha }^{2}\sim 1.$ We evaluated the ${\chi }_{\mathrm{eff},\alpha }^{2}$ metrics for our extrapolation data cubes and found them to be much larger than 1 in all cases. Inspection of the inputs used to compute the ${\chi }_{\mathrm{eff},\alpha }^{2}$ metric shows that the δB_α from Equation (2) are very small, implying that the magnetic field in the model volume is known with a very high accuracy even after rebinning. No reconstruction can be performed with such high accuracy; so the formal χ² test fails.

However, the reconstruction often provides an accuracy of 10%–30% (see Figures 12, 13), which is fully acceptable for most practical applications, even though it is not as perfect as the accuracy of the original model field. To further quantify that, Figures 14(a) and (b) show the height dependence for the normalized rms error of the absolute value and all components of the magnetic field for a representative set of three different binnings. A few observations can be made from this figure. First, the performance of both methods improves from small to large bin factors. Given that Figures 12 and 13 contain domains with local errors larger than 70%, the rms error can be rather large, especially at greater heights. Second, reconstruction of the B_y component is not as good as for the B_x or B_z components. This is an outcome of the asymmetry of the original model, in which the B_y values are systematically smaller than other components. Third, the absolute value of the magnetic field is recovered much better than any single component of the field, which looks unexpected, but as we show below, this is an outcome of the large errors coming from the weak field contributions. Finally, we find that the IM code works on average better than the AS code, while extrapolation is being made starting from the chromospheric magnetogram; see Table 11 for a level-by-level comparison of the normalized rms error in the case of bin = 9. We believe that this is a direct outcome of taking into account the equations for the force-free field boundaries of the modeling cube: having an almost force-free field bottom boundary condition at the chromospheric level is more consistent with force-free boundaries (IM) than with the buffer zone with the potential boundaries (AS).

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Table 11.  Normalized rms Error at a Given Level for Bin Factor 9

	δrms(B)				δrms(Bx)				δrms(By)				δrms(Bz)
	Chromo		β-photo		Chromo		β-photo		Chromo		β-photo		Chromo		β-photo
Level, Mm	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS
0.86	⋯	⋯	0.00	0.00	⋯	⋯	0.00	0.00	⋯	⋯	0.00	0.00	⋯	⋯	0.00	0.00
1.29	⋯	⋯	13.22	15.80	⋯	⋯	41.50	51.53	⋯	⋯	48.41	64.89	⋯	⋯	28.98	32.92
1.71	⋯	⋯	11.83	19.44	⋯	⋯	54.58	77.71	⋯	⋯	68.40	122.66	⋯	⋯	30.94	55.48
2.14	0.00	0.00	11.28	17.82	0.00	0.00	52.36	83.10	0.00	0.00	76.79	150.87	0.00	0.00	34.28	37.58
2.57	1.43	2.21	11.27	16.83	12.60	13.75	45.84	73.84	18.60	45.44	98.33	177.45	11.36	11.21	32.92	35.95
3.00	2.34	4.38	11.26	17.54	19.70	27.23	46.90	79.76	31.38	144.01	147.28	259.00	22.30	20.17	33.03	42.88
3.43	3.25	6.40	11.43	16.50	24.16	33.32	46.40	79.76	48.00	184.75	190.45	280.19	32.75	28.91	43.18	56.51
3.86	4.31	7.09	12.03	17.02	25.46	38.36	47.57	91.95	67.50	157.94	172.51	230.89	33.39	32.08	49.91	65.25
4.29	5.33	8.55	12.84	16.35	29.48	48.58	55.01	102.57	71.85	125.21	147.31	178.25	35.67	50.82	57.52	76.18
4.71	6.25	8.98	13.75	16.65	33.59	57.48	65.65	113.74	75.95	129.38	156.30	184.49	39.86	53.36	72.54	83.33
5.14	7.08	9.97	14.79	16.05	33.71	68.16	62.19	125.21	85.61	169.44	185.57	221.02	45.60	63.24	82.46	84.68
5.57	7.93	10.28	16.08	16.36	35.51	84.52	61.14	138.04	114.93	203.77	233.19	252.55	53.54	70.00	94.02	91.71
6.00	8.72	11.19	17.46	16.22	37.03	81.69	60.85	139.94	138.18	226.76	273.98	261.13	55.65	74.83	98.34	92.08
6.43	9.63	11.73	18.86	16.85	40.70	93.39	68.74	153.62	160.61	257.38	319.05	290.81	55.29	77.39	95.58	94.24
6.86	10.47	12.36	19.81	16.87	44.69	106.27	77.61	165.73	138.80	221.37	285.29	243.29	56.07	86.16	89.36	101.62
7.29	11.34	13.21	21.22	18.02	46.95	119.11	84.25	177.37	148.52	228.13	309.66	251.31	58.98	95.69	85.24	111.87
7.71	12.30	14.66	22.99	19.37	49.63	132.08	92.43	189.03	170.13	246.65	344.42	266.38	64.64	113.54	83.16	129.34
8.14	13.37	16.07	24.90	21.19	53.43	145.35	100.12	198.74	189.50	260.63	370.92	282.46	74.22	137.59	82.77	154.39
8.57	14.41	17.74	27.06	22.95	57.27	158.31	109.08	210.81	200.99	267.05	392.14	287.85	83.27	156.08	85.29	173.00
9.00	15.50	19.49	29.60	25.21	61.65	171.14	119.76	220.42	217.77	287.73	431.59	311.45	84.57	155.05	82.42	171.96
9.43	16.92	21.85	32.90	27.76	66.06	181.68	127.53	228.34	243.57	325.13	497.96	348.70	90.25	159.59	84.62	176.12
9.86	18.70	24.49	37.03	30.87	68.89	189.89	136.12	231.95	258.39	339.71	539.04	364.20	107.26	177.86	96.88	195.19
10.29	20.69	27.21	41.73	33.76	72.72	197.35	147.00	236.25	258.66	316.44	528.81	336.67	129.45	197.34	108.42	215.54
10.71	23.21	30.01	47.38	36.83	79.75	210.62	162.30	245.62	282.73	328.77	564.24	348.96	160.20	220.26	130.77	238.96
11.14	26.26	32.92	54.07	39.79	87.27	225.24	179.17	255.57	315.55	345.40	591.32	368.48	198.86	246.45	159.97	265.59

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The absolute values of the normalized rms errors are rather large, which is the outcome of the large errors recovering small (close to zero) values of the field components. To explicitly demonstrate this, we turn to the normalized rms residual, which is weighted with the strong field contributions. Figures 15(a) and (b) display the height dependence of the normalized rms residuals for the IM and AS codes respectively; see Table 12 for a level-by-level comparison of the normalized rms residual in the case of bin = 9. This plot and the table confirm that the magnetic field is recovered more accurately in the voxels with the large magnetic field. In particular, the absolute value of the field and its components are now restored with comparable accuracies (although the transverse components often display a bigger error than the longitudinal component). This metric only slightly depends on the grid resolution because the most numerous voxels with a weak field (which are more numerous for higher-resolution data cubes) only have a weak contribution to this metric. The two methods recover the absolute value and B_z component comparably well, but the IM code outperforms the AS code in recovering the transverse components (especially B_x). In all cases, the normalized rms residuals for the absolute value $| B|$ and the vertical component B_z are within 20% while for the transverse components, these are within 40%. Overall, we can conclude that these extrapolations from the force-free chromospheric level work rather well.

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Table 12.  Normalized rms Residual at a Given Level for Bin Factor 9

	Δrms(B)				Δrms(Bx)				Δrms(By)				Δrms(Bz)
	Chromo		β-photo		Chromo		β-photo		Chromo		β-photo		Chromo		β-photo
Level, Mm	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS
0.86	⋯	⋯	0.00	0.00	⋯	⋯	0.00	0.00	⋯	⋯	0.00	0.00	⋯	⋯	0.00	0.00
1.29	⋯	⋯	5.25	6.07	⋯	⋯	5.52	6.27	⋯	⋯	6.82	7.52	⋯	⋯	2.89	3.30
1.71	⋯	⋯	4.40	6.14	⋯	⋯	6.79	9.16	⋯	⋯	9.64	12.00	⋯	⋯	4.29	6.27
2.14	0.00	0.00	4.06	5.04	0.00	0.00	7.35	10.83	0.00	0.00	11.37	14.58	0.00	0.00	5.12	5.83
2.57	0.82	0.81	4.25	4.67	1.13	1.06	7.67	11.16	1.70	1.59	13.43	16.84	0.75	0.62	6.13	6.14
3.00	1.60	1.56	4.78	4.73	2.21	2.45	8.39	12.57	3.13	3.51	14.55	17.98	1.81	1.36	7.27	6.74
3.43	2.43	2.13	5.56	4.72	3.19	4.09	9.20	14.17	4.84	5.53	15.85	19.49	2.64	2.07	8.53	7.37
3.86	3.28	2.70	6.53	5.08	3.91	5.22	10.21	15.88	6.65	7.40	18.00	21.90	3.18	2.76	9.67	8.17
4.29	4.14	3.30	7.63	5.32	4.51	6.86	11.50	18.44	6.70	8.86	18.90	22.59	3.78	3.50	10.99	8.88
4.71	5.01	3.86	8.82	5.90	5.21	8.71	13.16	21.11	7.36	10.31	21.13	24.60	4.45	4.25	12.49	9.91
5.14	5.91	4.50	10.08	6.30	6.10	10.66	15.01	24.28	8.33	12.92	24.77	28.08	5.07	5.02	14.01	10.78
5.57	6.78	5.08	11.33	6.96	6.96	12.62	16.85	27.32	9.47	15.42	29.16	31.63	5.62	5.78	15.48	11.81
6.00	7.66	5.67	12.56	7.36	7.87	14.53	18.87	30.52	11.11	18.67	34.59	35.54	6.19	6.64	16.90	12.77
6.43	8.55	6.16	13.72	7.94	8.88	16.59	21.09	33.55	13.02	21.43	40.24	38.31	6.80	7.54	18.28	13.94
6.86	9.40	6.66	14.84	8.34	9.95	18.61	23.25	36.47	15.17	23.80	46.41	40.48	7.43	8.49	19.56	15.11
7.29	10.17	7.11	15.95	9.06	10.85	20.63	24.97	38.90	18.07	26.43	53.50	43.78	8.03	9.47	20.73	16.36
7.71	10.85	7.58	17.06	9.71	11.64	22.56	26.41	41.22	20.42	28.56	59.51	46.37	8.62	10.36	21.88	17.48
8.14	11.47	8.11	18.19	10.71	12.40	24.69	28.04	43.80	22.46	30.11	64.21	48.21	9.12	11.20	22.99	18.51
8.57	12.04	8.78	19.45	11.71	13.20	26.71	29.95	46.53	25.04	32.39	68.59	49.82	9.48	11.82	24.07	19.34
9.00	12.61	9.61	20.94	13.09	13.95	28.73	31.88	49.22	27.16	33.97	71.74	50.31	9.99	12.60	25.46	20.40
9.43	13.30	10.71	22.75	14.52	14.68	30.73	33.73	51.53	28.73	34.87	74.45	50.28	10.82	13.52	27.39	21.84
9.86	14.16	12.06	25.00	16.28	15.40	32.88	35.33	53.28	29.72	34.67	77.87	50.43	11.92	14.72	29.99	23.73
10.29	15.22	13.68	27.81	18.13	16.23	35.15	37.03	54.73	30.41	33.88	81.51	49.90	13.31	16.13	33.44	26.09
10.71	16.64	15.52	31.34	20.20	17.12	37.58	38.81	55.97	31.75	33.21	86.29	49.49	15.10	17.93	38.12	29.01
11.14	18.58	17.64	35.72	22.40	18.19	39.55	40.66	56.64	34.63	33.28	93.10	49.36	17.50	20.17	44.62	32.77

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5.2.2. NLFFF Extrapolations from the $\beta$-photospheric Boundary

5.2.3. NLFFF Extrapolations from a Preprocessed Boundary

Finally, we tested extrapolations from the photospheric level after its preprocessing; specifically, we used both preprocessed methods for both IM and AS extrapolation codes. Now, unlike the cases without preprocessing, we include the bottom (preprocessed) layer in the metrics computation given that it was modified compared to the original model field. Table 10 shows that there is no coherent behavior of the metrics versus model grid resolution. Sometimes, but not always, the overall field becomes more force-free than without preprocessing (better θ metric), while becoming less force-free in the sub-volume with a strong electric field (worse θ_j metric).

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For the IM code, the use of any preprocessing improves agreement between the restored and model field for smaller binning factors (higher grid resolution): the TW preprocessing works better for bins 3 and 4, while the JF one works better for larger bin factors, where the extrapolation with the preprocessings only shows a marginal improvement (if any) compared with the extrapolation without preprocessing.

However, the price we pay for these marginally improved angular metrics is the corrupted height scale, which is vividly demonstrated by Figures 17–19. Specifically, the magnetic field extrapolated from the JF preprocessed boundary condition shows a nice y = x pattern for the B_z component, while it deviates from that for both B_x and B_y components, which underestimate the magnetic field in the volume. In contrast, the use of the TW preprocessed boundary condition overestimates the magnetic field components, but not equally: the B_z component is clearly more strongly overestimated than the transverse ones, which almost follow the y = x regression in some cases.

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Figure 20 reveals one more problem with the field reconstruction starting from a preprocessed boundary: the field restoration error is large starting from the very bottom layer with the normalized rms residual exceeding 10%–15% in many cases, especially for the case with TW preprocessing. We believe that this happens because the magnetic field becomes overly smooth due to the preprocessing (this smoothing is stronger for the TW preprocessing—see Section 4 and specifically, Figure 8; the amount of smoothing can be reduced by lowering the μ₄ parameter; however, it is well beyond our study to optimize the choice of the preprocessing parameters). At somewhat higher layers, where the field in the model is itself smooth, the metric decreases and thus, the field is more accurately reconstructed at intermediate heights compared with low heights; see Tables 13 and 14 for quantitative level-by-level comparisons of the normalized rms error and residual in the case of bin = 9 reconstruction from the preprocessed boundaries. Overall, the entire range of tests does not justify preprocessing the photospheric boundary given that a marginal or no improvement in the volumetric metrics is achieved at the expense of the height scale corruption.

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Table 13.  Normalized rms Error after Preprocessing at a Given Level for Bin Factor 9

	Δrms(B)				Δrms(Bx)				Δrms(By)				Δrms(Bz)
	JF		TW		JF		TW		JF		TW		JF		TW
Level, Mm	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS
1.29	11.74	11.74	18.52	18.52	49.39	49.39	59.91	59.91	53.99	53.99	62.80	62.80	43.53	43.53	56.00	56.00
1.71	10.20	10.62	15.88	16.19	57.55	60.68	63.15	67.46	58.04	75.16	67.31	81.91	42.61	40.57	49.65	47.24
2.14	9.26	11.65	14.66	17.41	57.71	63.22	66.38	72.56	63.11	112.86	71.07	128.83	40.51	42.27	47.20	53.71
2.57	8.86	12.26	13.99	18.46	55.86	62.28	68.84	71.48	66.53	157.49	76.49	174.11	37.28	42.30	43.73	52.01
3.00	8.69	12.49	13.30	18.31	59.76	64.16	73.87	67.36	86.77	219.79	99.08	244.61	36.86	43.26	45.01	45.72
3.43	8.79	12.83	12.71	18.49	63.75	67.47	79.79	71.74	115.00	264.09	123.13	295.16	42.79	58.30	54.18	57.00
3.86	9.24	12.91	12.32	18.34	68.76	76.11	87.46	74.34	119.52	209.52	125.65	238.25	44.34	58.69	57.04	56.78
4.29	9.81	13.12	11.95	18.28	77.17	86.33	101.38	84.83	119.39	166.58	125.54	192.68	48.39	78.24	59.27	78.25
4.71	10.40	13.17	11.42	18.37	92.92	98.20	125.24	95.05	120.45	166.99	128.14	193.75	51.69	75.26	57.69	71.12
5.14	11.06	13.13	10.92	18.17	86.43	105.57	119.66	101.92	136.51	201.81	147.75	237.17	58.54	82.58	58.61	84.14
5.57	11.84	13.23	10.69	18.36	82.65	117.65	116.60	112.03	173.13	229.95	178.85	268.30	70.34	86.36	64.69	89.18
6.00	12.71	13.46	10.75	18.53	76.01	120.13	109.49	115.10	206.64	241.49	205.23	285.48	76.77	88.02	69.53	92.58
6.43	13.74	13.87	11.15	19.07	78.13	133.96	106.72	129.77	243.42	268.11	245.12	312.53	77.84	89.20	71.81	95.07
6.86	14.67	14.01	11.54	19.34	84.33	146.48	104.80	141.80	210.74	223.94	204.64	263.12	78.33	97.22	74.96	103.99
7.29	15.73	14.64	12.25	20.48	87.35	158.28	100.91	154.88	225.33	229.43	217.68	266.84	79.19	105.43	74.85	113.87
7.71	16.96	15.73	13.23	21.98	92.34	169.14	108.47	166.54	254.39	243.45	251.71	284.09	86.63	122.39	77.93	132.37
8.14	18.36	16.89	14.46	23.76	95.86	179.78	111.08	179.02	274.37	257.53	280.08	299.48	101.12	144.44	84.03	157.78
8.57	19.88	18.29	15.87	25.53	100.29	190.13	113.97	189.64	287.52	262.93	300.73	305.69	115.77	162.48	90.98	177.24
9.00	21.59	19.83	17.45	27.68	106.53	200.27	121.84	200.94	306.10	283.20	335.13	327.94	116.86	160.11	89.20	175.69
9.43	23.76	21.92	19.41	30.11	112.49	206.55	129.09	208.11	328.16	318.05	374.39	367.12	123.33	163.78	90.70	179.46
9.86	26.38	24.33	21.73	33.08	119.47	211.15	135.16	215.00	325.05	331.21	380.54	379.92	143.46	180.35	104.73	198.94
10.29	29.27	26.70	24.18	35.67	127.75	215.23	143.72	220.97	303.74	306.72	356.55	350.54	168.32	198.96	123.37	219.53
10.71	32.69	29.16	27.12	38.61	136.22	224.39	164.62	233.03	321.87	317.83	380.88	361.37	204.36	219.73	157.59	243.95
11.14	36.57	31.54	30.78	41.03	150.26	233.82	196.81	246.27	330.97	336.39	402.63	380.18	248.93	243.80	202.33	270.82

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Table 14.  Normalized rms Residual after Preprocessing at a Given Level for Bin Factor 9

	Δrms(B)				Δrms(Bx)				Δrms(By)				Δrms(Bz)
	JF		TW		JF		TW		JF		TW		JF		TW
Level, Mm	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS	IM	AS
1.29	6.71	6.71	14.38	14.38	12.35	12.35	15.86	15.86	12.07	12.07	15.43	15.43	2.56	2.56	6.31	6.31
1.71	6.47	6.56	13.10	12.64	7.15	7.19	7.54	7.14	7.26	7.18	9.18	9.21	3.60	3.60	7.62	7.46
2.14	6.43	6.56	12.14	12.07	8.98	9.20	9.28	8.84	9.91	10.47	11.50	12.54	4.42	4.69	9.16	9.21
2.57	6.60	6.57	11.31	11.37	10.28	10.90	10.50	10.51	12.41	13.58	13.51	15.35	4.96	5.43	9.48	9.73
3.00	6.84	6.64	10.49	11.01	11.31	12.06	11.11	11.00	14.28	15.62	15.28	17.54	5.47	6.05	9.41	9.90
3.43	7.21	6.79	9.66	10.46	12.57	13.48	11.79	11.98	15.76	17.38	17.16	19.58	6.11	6.84	9.37	10.16
3.86	7.68	6.93	8.85	10.26	13.88	15.30	12.21	13.11	17.01	19.08	19.12	22.04	6.70	7.47	9.32	10.44
4.29	8.19	7.13	8.10	10.05	15.08	16.87	12.43	14.19	16.70	19.07	19.00	22.31	7.39	8.39	9.36	10.91
4.71	8.84	7.34	7.42	10.14	16.25	19.25	12.91	16.70	17.31	20.47	19.44	23.89	8.16	9.13	9.51	11.41
5.14	9.55	7.66	6.88	10.21	17.54	21.30	13.47	18.79	19.23	22.77	21.31	26.23	8.81	10.04	9.69	12.05
5.57	10.26	7.89	6.55	10.54	18.80	23.77	14.02	21.51	22.13	26.27	23.83	29.70	9.29	10.72	9.86	12.59
6.00	11.08	8.23	6.50	10.74	20.37	26.06	14.80	23.87	25.41	29.23	26.63	32.71	9.72	11.59	10.14	13.30
6.43	12.03	8.52	6.78	11.10	22.34	28.70	15.84	26.58	28.94	31.92	29.53	35.56	10.15	12.40	10.56	14.08
6.86	13.02	8.84	7.37	11.41	24.49	31.06	17.11	28.90	32.90	33.65	32.79	37.61	10.58	13.39	11.09	15.05
7.29	14.04	9.06	8.21	12.03	26.38	33.33	18.36	31.35	37.65	36.44	37.07	40.37	11.04	14.25	11.65	16.07
7.71	15.11	9.32	9.20	12.65	27.95	35.11	19.52	33.33	42.28	38.81	41.49	42.62	11.64	15.13	12.23	17.11
8.14	16.19	9.49	10.26	13.61	29.49	37.07	20.78	35.69	47.00	41.23	45.97	44.62	12.30	15.72	12.62	18.10
8.57	17.39	9.76	11.41	14.55	31.64	39.14	22.49	38.08	51.25	43.25	50.21	46.46	13.06	16.18	12.73	18.96
9.00	18.92	10.13	12.76	15.89	34.66	41.35	24.62	40.58	53.99	43.98	53.18	47.18	14.37	16.65	12.94	20.05
9.43	20.97	10.78	14.43	17.12	38.38	43.51	27.02	43.03	55.70	43.69	55.03	46.94	16.51	17.47	13.53	21.37
9.86	23.62	11.68	16.47	18.82	42.74	45.43	29.65	45.29	57.09	42.94	56.39	46.11	19.58	18.58	14.57	23.21
10.29	26.86	12.83	18.91	20.27	47.67	46.98	32.54	47.25	58.92	42.37	58.11	45.57	23.65	20.05	16.15	25.15
10.71	30.75	14.26	21.92	22.29	53.28	48.30	35.83	49.05	61.39	41.95	60.32	45.00	28.95	22.00	18.44	27.88
11.14	35.14	15.86	25.78	23.83	59.84	49.27	39.93	50.64	65.55	42.02	64.00	44.76	35.67	24.46	21.70	30.62

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