Issue 
A&A
Volume 591, July 2016



Article Number  A64  
Number of page(s)  13  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201525769  
Published online  13 June 2016 
Fast inversion of Zeeman line profiles using central moments
II. Stokes V moments and determination of vector magnetic fields
^{1} LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
email: Pierre.Mein@obspm.fr
^{2} National Solar Observatory, Sacramento Peak, PO Box 62, Sunspot, NM 88349, USA
^{3} Observatoire de Paris, 5 place Jules Janssen, 92195 Meudon, France
^{4} UMR 6525 H. Fizeau, Université de Nice Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, Campus Valrose, 06108 Nice, France
Received: 30 January 2015
Accepted: 30 March 2016
Context. In the case of unresolved solar structures or stray light contamination, inversion techniques using four Stokes parameters of Zeeman profiles cannot disentangle the combined contributions of magnetic and nonmagnetic areas to the observed Stokes I.
Aims. In the framework of a twocomponent model atmosphere with filling factor f, we propose an inversion method restricting input data to Q , U, and V profiles, thus overcoming ambiguities from stray light and spatial mixing.
Methods. The Vmoments inversion (VMI) method uses shifts S_{V} derived from moments of Vprofiles and integrals of Q^{2}, U^{2}, and V^{2} to determine the strength B and inclination ψ of a magnetic field vector through leastsquares polynomial fits and with very few iterations. Moment calculations are optimized to reduce data noise effects. To specify the model atmosphere of the magnetic component, an additional parameter δ, deduced from the shape of Vprofiles, is used to interpolate between expansions corresponding to two basic models.
Results. We perform inversions of HINODE SOT/SP data for inclination ranges 0 <ψ< 60° and 120 <ψ< 180° for the 630.2 nm Fe i line. A damping coefficient is fitted to take instrumental line broadening into account. We estimate errors from data noise. Magnetic field strengths and inclinations deduced from VMI inversion are compared with results from the inversion codes UNNOFIT and MERLIN.
Conclusions. The VMI inversion method is insensitive to the dependence of Stokes I profiles on the thermodynamic structure in nonmagnetic areas. In the range of Bf products larger than 200 G, mean field strengths exceed 1000 G and there is not a very significant departure from the UNNOFIT results because of differences between magnetic and nonmagnetic model atmospheres. Further improvements might include additional parameters deduced from the shape of Stokes V profiles and from large sets of 3DMHD simulations, especially for unresolved magnetic flux tubes.
Key words: line: profiles / Sun: magnetic fields
© ESO, 2016
1. Introduction
Most of the inversion techniques processing Zeeman line profiles derive magnetic field components and thermodynamical parameters from leastsquares fits of I ± S profiles, where S represents the Stokes parameters Q,U,V successively. For weak fields, algorithms using central moments (Semel 1967; Uitenbroek 2003; Criscuoli et al. 2013) can provide measurements of magnetic field components. In a previous paper (Paper I; Mein et al. 2011), we extended the inversion by moment calculations of I ± S profiles to the general case of any magnetic field strength.
Stray light and unresolved structures. Zeeman line profiles can be disturbed by stray light effects from the atmosphere of the Earth, the telescope, and the spectrometer. To eliminate such effects, it was proposed to take neighbouring pixels or the quietest parts of the data set into account to correct the observed profiles in each solar point (Skumanich & Lites 1987; Orozco Suarez et al. 2007; Asensio Ramos 2009; Del Toro Iniesta et al. 2010).
If magnetic structures are unresolved, magnetic and nonmagnetic areas may contribute simultaneously to the same pixel. The more usual solution to this problem is to define a filling factor f and to assume that the contributions of the magnetic area to the Stokes parameters are If, Qf, Uf, and Vf, while the contribution of nonmagnetic areas is limited to the nonpolarized contribution I′(1−f). The intensities I and I′ can be different. The ratio between areas of magnetic and nonmagnetic components is f/ (1−f).
For unresolved magnetic structures, it is well known that products Bf can be measured more easily than B and f measured separately (Bommier et al. 2007). For twocomponent models with magnetic and nonmagnetic areas, independent measurements of f were proposed (Bommier et al. 2009; Bommier 2011) to deduce the magnetic strength B. In the special case of the quiet Sun, methods using two lines simultaneously with different Lande factors were used by Stenflo (1973; 2010). Complex threecomponent models have also been proposed with the MISMA code to take different models and different magnetic strengths into account (Viticchié et al. 2011; Viticchié & Sánchez Almeida 2011). Very low data noise levels are of course required to obtain accurate results with models that depend on many parameters.
New inversion code insensitive to nonmagnetic model atmosphere. In this paper, we propose an inversion method based on a twocomponent model using only the polarized parts of the profiles, independent of intensities I and I′. Magnetic strength B and inclination ψ are derived from moments of Vprofiles and ratios of integrals of Q^{2}, U^{2}, and V^{2}. We do not investigate nearly transverse magnetic fields, but similar methods using Qprofiles might be developed for magnetic vector inclinations near 90°. We use moments of Vprofiles to determine barycenters. These are very different from the moments used in Taylor expansions of absorption coefficients by Mathys & Stenflo (1987) and Solanki et al. (1987).
We begin with the definition of the observable quantities used in the Vmoments inversion (VMI). We do not take into account the vector field azimuth φ, which might be deduced independently from Stokes Q and U and magnetooptical effects. In our simulations, at first we assume a given model atmosphere for magnetic areas. Later we propose a procedure to specify unknown models in the case of observations (Sect. 7).
We describe the inversion by deducing B and ψ from two quantities S_{V} and R_{V}, defined in Sect. 4. We derive polynomials necessary to extract B and ψ through a brief iteration process via synthetic spectra computed with the RH radiative transfer code (Uitenbroek 2001, 2003). We extend the method to the case of an unknown underlying solar model for the magnetic component using a third quantity D_{V}.
Coefficients of polynomials used in the inversion also depend on the instrumental broadening profile and, possibly, on unresolved transverse gradients of the magnetic field. We determine a broadening coefficient in the particuliar case of HINODE SOT/SP data and investigate expected data noise effects for standard models. We process data and compare results of VMI with maps deduced from inversions with UNNOFIT (Bommier et al. 2007) and the HAO MERLIN code (CSAC^{1}).
2. The RH radiative transfer code
As in Paper I, we derive synthetic spectra of the 630.25 nm line with the RH radiative transfer code (Uitenbroek 2001; 2003), which is based on the multilevel accelerated lambda iteration scheme (Rybicki & Hummer 1991, 1992) to define the required polynomial coefficients for inversion. In the transfer solution, the coupled equations of statistical equilibrium and radiative transfer were solved for a 23level, 33line atomic model of Fe i, including the 630.25 and 630.15 nm lines. NonLTE iterations were performed in the polarization free approximation to account for the effect of the splitting of the line profile on the radiative rates (Bruls & Trujillo Bueno 1996). More details can be found in Paper I.
We employed four different onedimensional hydrostatic solar atmospheric models in the calculations: FALA, FALC, FALF (Fontenla et al. 1993), and MALTM (Maltby et al. 1986). These models represent a quiet cell interior, the averaged quiet sun, the solar network, and a sunspot umbra, respectively. To compute coefficients for the inversion process, we use synthetic profiles with small wavelength steps. However, to simulate noise effects on real observations (Sect. 10), we use synthetic profiles with the same spectral resolution as SOT/SP spectra (spectral step 2.147 pm). In all of the cases, we performed computation of moments used in the inversion process after profile interpolations by third degree spline functions.
3. Vprofiles and vector magnetic field strength B
Fig. 1 Vprofiles for FALC and MALTM models at disk center (full and dashed lines, respectively) with magnetic field strengths 500 G (top) and 2000 G (bottom). Profiles are scaled to their own maximum value. 
Figure 1 shows the Vprofiles of the 630.25 nm line computed at disk center for FALC and MALTM models with magnetic field strengths B = 500 G and B = 2000 G (inclination ψ = 30°).
We denote with s the Zeeman splitting corresponding to the field strength B, for an effective Lande factor and a linecenter wavelength λ_{0} as follows: (1)with (2)where s and λ_{0} are expressed in nm and B in Gauss.
In both cases, vertical dotted lines are drawn at wavelengths λ = λ_{0} + s and λ = λ_{0}−s.
As expected, in the stronger field case (B = 2000 G), the two lobes of the Vprofiles approximately coincide at the same positions as the dotted lines. This means that the field strength can be easily deduced from the shifts of the Vprofiles alone. In the weaker field case (B = 500 G), however, the lobes are at larger distances than the dotted lines from line center and the distances depend on the width of the I profile and, thus, on the underlying corresponding atmosphere.
4. Inversion of Q, U, V profiles: Observable quantities S_{V}, R_{V}, and D_{V}
To deduce B and the inclination ψ from Stokes V, Q, and U profiles, we propose two observable quantities that are mainly sensitive to B and ψ, respectively.
The first quantity, S_{V}, is the halfshift between both lobes of the V^{2}profile. We use V^{2} instead of V to reduce the weights of line wings, which are very sensitive to data noise. We checked that using V^{2} instead of V significantly increases the accuracy of inversion results.
The second quantity, R_{V}, characterizes the ratio between integrals of Q^{2} + U^{2} and V^{2} + Q^{2} + U^{2}.
To specify the model atmosphere of magnetic areas inside each solar pixel, we use departures between the halfshift of Vprofiles and the halfshift S_{V} of V^{2}profiles that are sensitive to line wing shapes and noted as D_{V}.
We note that S_{V}, R_{V}, and D_{V} are very independent of global Doppler shifts of profiles.
4.1. Shift S_{V} and magnetic field strength B
In the following, moment calculations are performed with profiles interpolated by spline functions leading to a wavelength step divided by four with respect to the original spectral resolution.
Before calculating the halfshift S_{V} between the blue and red lobes of V^{2}profiles, we must determine the central wavelength of the line. To take the possible asymmetry of profiles into account, we separately compute the moments of the positive and negative parts V_{+} and V_{−} of the Vprofile. V_{+} and V_{−} are set to zero where V is negative and positive, respectively, (3)We have to compute central wavelengths λ_{b} and λ_{r} of the blue and red lobes. In the case of observed data, noise may strongly disturb moment calculations of V^{2}(λ) because the mean signal is not zero in the far line wings. To avoid this problem, we keep the signal sign and we replace V^{2} by V ×  V  as follows: The shift S_{V} is half the wavelength distance between both converted into magnetic field units (6)Figures 2a and 3a show theoretical values of S_{V} at disk center for different values of B and ψ for the FALC and MALTM models. The line styles depend on the values of ψ: full lines to 30° and dashed lines to 60°.
Fig. 2 S_{V}, R_{V}, D_{V}, and A_{V} functions for FALC model (Sect. 4). 
Fig. 3 S_{V}, A_{V}, R_{V}, and A_{V} functions for MALTM model. 
We can see that S_{V} depends mainly on B and slightly on ψ.
4.2. Ratio R_{V} and inclination angle ψ
In this section dealing with synthetic spectra, we consider ψvalues only between 0 and 90°. The extension to 180° is easily obtained for observed data, owing to the signs of the Vprofile lobes.
The ratio of integrated Q^{2} + U^{2} and Q^{2} + U^{2} + V^{2} profiles is 0 for ψ = 0 and 1 for ψ = 90°. It does not depend on the magnetic vector azimuth and can be used to determine the inclination. The use of the quantity R_{V} seems more convenient for forward polynomial inversion than the unlimited Q/V ratio used, for example, for quiet Sun analysis (Stenflo 2010).
We define the ratio R_{V} expressed in degrees and characterizing the inclination angle ψ through the equation (7)Figures 2b and 3b show theoretical values for FALC and MALTM models. R_{V} increases with ψ, regardless of the value of B. The line styles depend on the values of B: full lines from 100 G to 1000 G, dashed lines from 1100 G to 2000 G, dashdotted from 2100 G to 3000 G, and dotted from 3100 G to 4000 G.
We note that data noise strongly affects values of R_{V} for small B for real observations. If we assume that noise levels are similar for Q, U, and V, the expected limit of R_{V} for zero magnetic strength is the high value (8)To reduce noise effects, we estimate noise in each solar point by the root mean square (RMS) of fluctuations in some wavelength points at the beginning and end of the available wavelength interval of the line. Then we subtract the result from the corresponding Q^{2} and U^{2} values. We do not modify V^{2} to avoid undetermined R_{V} values. Results are shown in Sect. 10.
4.3. , D_{V}, and the model atmosphere
The difference between shifts of Vprofiles and S_{V} of V^{2}profiles, expressed in Gauss and named D_{V}, is used to specify the model atmosphere. The equations defining are very similar to Eqs. (4)−(6), i.e., Figures 2c and 3c show theoretical values of (12)The line styles depend on the values of ψ: full lines to 30°, dashed lines to 60°.
5. A_{V}, a nearly linear function of B for a given model atmosphere
The function S_{V} is too far from a linear function of B to be approximated accurately by a polynomial with a small number of nonzero coefficients. If we call S_{0,M} the limit at zero field of the function S_{v} relative to model M, we can define a new function, (13)which is nearly a linear function of B. Figures 2d and 3d show plots of A_{V} functions. Line styles are the same as in Fig. 2a.
6. Polynomials P_{A}(A_{V},ψ) and P_{R}(R_{V},B): inversion for a given model atmosphere
Because A_{V} is a monotonic function of B for all ψ values, it can be inverted by a polynomial leastsquares fit, i.e.,(14)In a similar way, ψ can be expressed as a function of R_{V} and B through the equation (15)The upper values of i and j are typically 7 and 5 for P_{A} and P_{R}, respectively.
For a given model atmosphere, the coefficients for polynomials P_{A} and P_{R} can be computed and B and ψ can be recovered from any set of Stokes profiles through a very fast iterative loop, initialized with B_{0} = 0 and ψ_{0} = R_{V}. The number of iteration steps is typically 3.
We do not discuss the determination of magnetic field azimuth φ, as it is detailed in Paper I; polynomial expansions of magnetooptical effects can be used to deduce directly φ from B, ψ, and Stokes Q and U.
7. Polynomials P_{D}(B,ψ): inversion for an unknown model atmosphere
For a given model atmosphere, the differences D_{V} can be expanded as functions of B and ψ, i.e., (18)The upper values of i and j are typically 5.
We performed simulations for four model atmospheres: FALA, FALC, FALF, and MALTM. They show that S_{V} , R_{V}, and D_{V} are very similar for models FALC, FALA, and FALF. Details are given in Sect. (14). As a consequence, the accuracy of the inversion process is not degraded by selecting two models, for example, FALC and MALTM, and by interpolating inversion coefficients between both. For any set of V, Q, and U profiles, we define an interpolation coefficient δ_{V} in the following way: (19)If (20)or (21)the inversion process is given up for the corresponding solar pixel. This may occur for small Bf products mainly because of noise. We see later (Figs. 11 and 13) that very few points are involved.
We replace S_{0,M} and polynomials P_{A} and P_{R}, defined in Sects. 5 and 6, by Inside the iteration loop (Eqs. (16), (17)) we introduce Eqs. (18)−(24).
8. Broadening function for instrumental effects in SOT/SP data
We analyze SOT/SP data of the active region NOAA10958 observed on 17 May 2007 at 13:01 UT. The SOT/SP instrument (Lites et al. 2013) of the Solar Optical Telescope (Tsuneta et al. 2008) is on board the Hinode mission (Kosugi et al. 2007). Stokes profiles and level 2 outputs from inversions using the HAO “MERLIN” inversion code developed under the Community Spectropolarimetric Analysis Center are available online^{2}.
As in any set of observations, a damping function can be used to mimic instrumental effects and to adjust synthetic profiles to observed data. In the case of unresolved solar structures, an additional effect should be taken into account. Across the same pixel, transverse gradients of the magnetic field may occur and, thereby, broaden Stokes profiles, especially for strong magnetic fields.
Figure 4 shows an example of observed Vprofile (stars) corresponding to 2500 G (according to UNNOFIT inversion) together with the synthetic MALTM profile (dashed lines). The observed profile is clearly wider than the synthetic profile. It is approximately matched by the synthetic MALTM profile (full line) broadened with the Lorentz function plotted in Fig. 5 and corresponding to the damping coefficient γ = 3 pm.
To find a unique broadening coefficient that is valid for all data, we turned to the inversion results obtained from UNNOFIT and MERLIN codes. The value γ = 3 pm appears to produce a very good agreement in both cases inside a wide magnetic field range corresponding to filling factors close to 1, as we show in Figs. 14 and 17. All further calculations use that damping coefficient.
The μ value is near 0.90, close to 1. We can also note that the use of a broadening function automatically adjusts the profile widths to the widths corresponding to the μ value of observations. So, before broadening, we can use synthetic profiles at disk center.
Using a single coefficient to correct instrumental effects, model fitting, magnetic unresolved fluctuations, and centertolimb effects is obviously a crude approximation. Further detailed investigations are needed, especially for future data exhibiting even higher resolutions and even lower noise levels. We return to this question in Sect. 15.
Fig. 4 Vprofile from SOT/SP data (stars) for magnetic field strengh 2500 G. Normalized synthetic profiles are plotted for MALTM with (full lines) and without (dashed lines) additional broadening. 
Fig. 5 Lorentz broadening function for synthetic Vprofiles. 
9. VMI inversion and model atmosphere selection
Fig. 6 S_{V}, A_{V}, D_{V}, and A_{V} functions for FALC model with additional broadening (Sects. 8, 9). 
Fig. 7 S_{V}, A_{V}, R_{V}, and D_{V} functions for MALTM model with additional broadening. 
The synthetic functions S_{V}, R_{V}, D_{V}, and A_{V} corresponding to models FALC and MALTM are shown in Figs. 6 and 7.
For a given solar point, the calculation of A_{V} through Eq. (13) is possible only if S_{V}>S_{0,M}, where M stands for the model atmosphere that must be selected. We start the iteration by using the δ_{V} value that corresponds to the model MALTM providing the lowest S_{0,M} value. By starting with the MALTM function we avoid eliminating some points of low magnetic field strengths.
The steps of the inversion are the following:

Iteration step n (n = 1, 2, 3): Eq. (22) →S_{0,M,n}Eq. (13) →A_{V,n}Eqs. (18)−(24) →δ_{V,n}, P_{A,n},P_{R,n}Eqs. (14), (15) →B_{n}, ψ_{n}
10. SOT/SP data: Estimates and reduction of noise effects
10.1. Data noise level
As mentioned in Sect. 4.2, we can estimate noise levels in Stokes parameters using RMS values ϵ over two wavelength intervals: at the beginning and the end of the available spectrum, where Q, U, and V are negligible.
Figure 8 shows results obtained across the full set of SOT/SP data with two 5 point intervals corresponding roughly to −52 < Δλ< −42 pm and 42 < Δλ< 52 pm, where Δλ is the wavelength distance from line center. To disentangle results from small and large magnetic fields B or filling factors f, we plot the results versus the product B_{U}f_{U} deduced from UNNOFIT inversion. We note that UNNOFIT and MERLIN inversions lead to almost the same results insofar as the product Bf is concerned (see Sect. 12, Fig. 16). The values are normalized by the continuum intensity of quiet Sun I_{Q}, obtained by averaging continuum intensities over pixels such that B_{U}f_{U}< 20 G. Results are presented with circles for Q, triangles for U, and squares for V.
For B_{U}f_{U}< 1000 G, Q, U, and V decrease strongly for  Δλ  > 42 pm and ϵ/I_{Q} is close to 1.2 × 10^{3}.
Fig. 8 RMS of Stokes profiles in far wings of the 630.2 nm line ( Δλ  > 0.042 nm) in SOT/SP data, divided by the quiet Sun continuum intensity, for noise level determination. Circles for Q, triangles for U, and squares for V. 
10.2. Noise effects reduction for R_{V} computations
According to Fig. 8, ϵ/I_{Q} remains very near 1.2 × 10^{3} for B_{U}f_{U}< 2500 G for Q and U. This shows that Q and U remain negligible in both 5point wavelength intervals in a very wide magnetic field range. As mentioned in Sect. 4.2, for each pixel, we subtract from Q^{2} and U^{2} the average values over the two 5point intervals to reduce noise effects on the R_{V} expression (7). Of course, more accurate results would be obtained with observations including a larger line profile that allows larger intervals.
10.3. Expected noise effects on VMI inversion
Fig. 9 Inversion of synthetic FALC model data for the case of data noise for filling factors 1 and 0.1. 
Fig. 10 Inversion of synthetic MALTM model data for the case of data noise for filling factors 1 and 0.7. 
We use the 1.2 × 10^{3}I_{Q} noise level to predict effects on VMI inversion results. We can assimilate the continuum intensity to the quiet Sun continuum I_{Q} for synthetic FALC profiles.
However, the ratio between synthetic continuum level intensities of FALC and MALTM is very small, i.e., (25)mainly because of temperature differences between both model atmospheres. Moreover, for unresolved structures, we must take into account a geometrical filling factor f.
Therefore, to simulate data noise effects, we introduce the stochastic noise at the RMS level of 1.2 × 10^{3} relative to the continuum intensity in the FALC model in synthetic profiles FALC and MALTM. Because the noise effects depend only on the signaltonoise ratio, we keep the local continuum (FALC or MALTM) as a reference, but we divide this level by 0.126 for MALTM and by f if the filling factor is not 1. Of course, different stochastic values are added to quantities Q, U, and V for the same point.
We can see expected errors and the RMS of departures in B and ψ in Figs. 9 and 10 for the FALC and MALTM models, with filling factors 1 and 0.1 for FALC, and 1 and 0.7 for MALTM. Triangles mean that owing to noise, B and ψ computations are not always possible in the useful range, mainly because of observed values of S_{V} that are too low.
For inversion of magnetic strengths (upper plots), errors are small in the case of the FALC model and in the MALTM model for f = 1. They are larger for MALTM model if f = 0.7 and B< 1400 G. In all of the cases, however, errors on B decrease strongly for large magnetic fields. Mean expected ψ errors (lower plots) are computed in the range 500 <B< 4000 G. They are reduced by noise subtraction from Q^{2} and U^{2} integrals (Sect. 4.2).
11. Comparison between VMI and UNNOFIT results
We discuss now the results of VMI inversion of SOT/SP data, notably B_{V} and ψ_{V}. We compare them to results of UNNOFIT inversion (Landolfi et al. 1984; Bommier et al. 2007), namely B_{U} and ψ_{U}. The range of magnetic strengths is limited by the condition S_{V}< 4500 G, corresponding roughly to B_{V}< 4000 G. For very large magnetic fields, Vprofiles cannot be neglected outside the available wavelength range (±52 pm). Additional inversion processes should be developed, for example, using Vprofile extrapolations or maximum value determinations.
Most of the comparisons are presented versus the product B_{U}f_{U}. Indeed, because unresolved structures prevail with similar B values at low magnetic fluxes, B_{U}f_{U} is a good criterion to disentagle ranges of quiet Sun, faculae, and spot penumbrae. Mean values are plotted with a 100 G step in the case of a number of values higher than 10 in each step. Dispersion is materialized by error bars.
11.1. Model selection and the δ_{V} coefficient
Figure 11 indicates the plot of δ_{V} versus the product B_{U}f_{U}, in the range of inclination angles 0 <ψ_{U}< 60° and 120°<ψ_{U}< 180°. Mean values are black points and the RMS of departures are indicated with vertical lines. Equations (20) and (21) imply −2 <δ_{V}< 3. The standard deviations, however, are generally smaller than 0.5, which show that very few points are lost because of this constraint.
We see that, in the range 200 <B_{U}f_{U}< 1000 G, δ_{V} is near 0 (FALC), while for higher values it increases up to 1 or more for spot umbrae (MALTM). This shows that the criterion D_{V}, which is only the difference between widths of V and V^{2}profiles, provides an estimate of the model atmosphere selection between FALC and MALTM, and that these two models together match well the whole set of data by representing two extreme cases.
In Fig. 12 we plot the ratio between continuum intensity I_{C} and continuum quiet Sun intensity I_{Q} that is obtained by averaging pixels defined by B_{U}f_{U}< 20 G. As expected, this ratio, close to 1 in the range B_{U}f_{U}< 1000 G, decreases down to 0.3 for B_{U}f_{U}> 2000 G. The discrepancy with the expected MALTM value 0.126 may be due to scattered light.
Fig. 11 SOT/SP data: δ_{V} versus UNNOFIT product B_{U}f_{U}. 
Fig. 12 SOT/SP data: continuum intensity versus UNNOFIT product B_{U}f_{U}. 
Fig. 13 Numbers of pixels so that ψ_{U}< 60° or ψ_{U}> 120° (squares) and numbers of pixels computed by VMI (black points) divided by the total number of available pixels in UNNOFIT inversion. 
Fig. 14 Magnetic strengths from VMI inversion (black points) and UNNOFIT inversion (circles) in the ranges 0 <ψ_{U}< 30° and 150°<ψ_{U}< 180° (top) and 0 <ψ_{U}< 60° and 120°<ψ_{U}< 180° (bottom). 
11.2. Validity range of VMI inversion for SOT/SP data
Figure 13 shows the number N of available solar pixels divided by the total number of pixels N_{max} (when N_{max}> 10) in two different assumptions. Squares correspond to pixels where ψ_{U}< 60° or ψ_{U}> 120° and black points to pixels available in VMI inversion. We see that black points are well centered in the corresponding squares, except for very large fields (B_{U}f_{U}> 2600 G) or points such that B_{U}f_{U}< 200 G.
Below this limit, lost pixels correspond generally, either to effects of noise lowering S_{V} values or to the fact that the assumed atmospheric model of magnetic areas is not valid so that the lowest S_{V} value of the model is higher than the observed value. The relative numbers of lost pixels correspond to the departures between the centers of squares and black points. They are very small for 200 <B_{U}f_{U}< 300 G and practically zero for 300 <B_{U}f_{U}< 2600 G.
11.3. Magnetic field strengths B
In Fig. 14, we plot B_{V} (black points) and B_{U} (circles) for B_{U}f_{U}> 100 G. Each field strength corresponds to an average value across a 100 G B_{U}f_{U} interval including more than 10 pixels. Vertical lines indicate RMS of departures. The data are restricted to points where ψ_{U}< 30° or ψ_{U}> 150° (upper diagram) and ψ_{U}< 60° or ψ_{U}> 120° (lower diagram). Dashed lines indicate limits corresponding to filling factors f_{U} equal to 1, 0.7, and 0.1, corresponding to values used in Sect. 10 to estimate noise effects.
To discuss the validity of VMI results in spite of noise effects, we assume that f_{U} values are near the ratio between magnetic and nonmagnetic areas inside each pixel. We consider three cases:

(a)
In the range 200 <B_{U}f_{U}< 1000 G, Fig. 11 shows that magnetic areas are mainly relevant to the FALC model. Because all B_{V} computed values are generally higher than 1000 G with f> 0.1, results appear to be reliable according to Fig. 9. We can note that if the noise level was reduced by a factor 2, Fig. 9b would apply to the filling factor f = 0.05. Then Fig. 14 shows that all values such that B_{U}f_{U}> 100 G would become reliable.
Fig. 15 Inclinations from VMI versus UNNOFIT.

(b)
In the range 1000 <B_{U}f_{U}< 2000 G, all B_{V} values correspond approximately to f> 0.7 with B_{V}> 1000G, which indicates reliable results for the FALC model and approximate results for MALTM model, according to Figs. 9 and 10. Anyway, they are in good agreement with B_{U} values.

(c)
For higher magnetic fields, some VMI results are a little larger than UNNOFIT results. The magnetic model with only one value of magnetic field is perhaps not relevant to such strong fields. Because VMI uses moments of V ×  V  that are less sensitive to noise than moments of V, departures can be due to the different weights assigned to different parts of the profiles. We can expect that magnetic field transverse gradients, which are larger for strong fields than for weak fields, account for observed discrepancies, increasing with magnetic fields. As for the MISMA code, more complex models, including more than two columns or vertical and horizontal gradients and asymmetries, might be investigated.
11.4. Inclinations ψ
In Fig. 15 we plot inclinations from VMI inversion versus UNNOFIT results in the range B_{U}f_{U}> 200 G. As expected, the relative behavior of the results is opposite for angles symmetrical versus 90°. We restrict the discussion to ψ< 90°. We can see from Figs. 6 and 7 that for a given R_{V} value, ψ decreases when B increases. Because VMI B values are a little higher than UNNOFIT values in some ranges, as we show in Fig. 14 (bottom), we may expect ψ_{V} to be somewhat lower than ψ_{U}, as it is observed. However, for low ψ values, noise effects still appear to be present in spite of the correction mentioned in Sect. 4.2.
12. Comparison between VMI and MERLIN results
Fig. 16 Bf products from MERLIN inversion versus Bf products from UNNOFIT inversion. 
Fig. 17 Magnetic strengths from VMI (black points) and MERLIN inversion (squares) versus B_{M}f_{M} values in the ranges 0 <ψ_{M}< 60° and 120°<ψ_{M}< 180°. 
12.1. Magnetic field strengths B
Figure 16 shows that, as expected, B_{M}f_{M} products, extracted from the MERLIN inversion, are very similar to B_{U}f_{U} products used in Sect. 11.
Magnetic field strengths from VMI and MERLIN inversions are plotted in Fig. 17 as functions of B_{M}f_{M} in the ranges 0 <ψ_{M}< 60° and 120°<ψ_{M}< 180°. Three cases can be considered again:

(a)
In the range B_{M}f_{M}< 700 G, VMI values are higher, especially for very low B_{M}f_{M}. Departures are simply because MERLIN code deals with stray light effects, but not with unresolved structures.

(b)
For 700 <B_{M}f_{M}< 2000 G, the agreement is good. It can be noted also that the dispersion of B is very small for each B_{M}f_{M} value.

(c)
For B_{M}f_{M}> 2000 G, VMI values appear to be a little higher than MERLIN values, especially for very large magnetic fields. As in Sect. 11.3, we can note that unresolved transverse magnetic field gradients, increasing with magnetic strengths, may account for the observed discrepancies.
12.2. Inclinations ψ
Inclinations are plotted in Fig. 18 in the range B_{M}f_{M}> 200 G. They are in rather good agreement with MERLIN results, even for small inclinations.
Fig. 18 Inclinations from VMI versus MERLIN inversion code. 
13. VMI quicklook without iteration
Fig. 19 Same caption as for Fig. 14 (bottom), but for VMI quicklook without iteration. Magnetic strengths from VMI inversion (black points) and UNNOFIT inversion (circles) in the ranges 0 <ψ_{U}< 60° and 120°<ψ_{U}< 180°. 
As we show in Figs. 6a and 7a, S_{V} values are similar for FALC and MALTM for instrumental broadening, especially for small magnetic strengths. So we can try to replace the interpolation between both basic models by a mean model atmosphere. Moreover, we can use a further simplification by suppressing iterations and replacing curves A_{V} and R_{V} by the first bisector in plots (b) and (d) of Figs. 6 and 7, to get a socalled quicklook VMI ignoring the coupling between B and ψ. It can be noted indeed that R_{V} is close to ψ especially for B values near 1000 G, that is, in the full range B_{U}f_{U}< 1000 G.
Because we know the broadening function and the mean values S_{V0} of S_{V} for FALC and MALTM near B = 0, we can reduce the inversion to two equations (26)and (27)where S_{V0} is equal to the average between S_{0,FALC} and S_{0,MALTM}.
Results plotted in Fig. 19 are not very different from results plotted in Fig. 14b in the range 200 <Bf< 2700 G. This is a clear indication of the small dependency of inversion results on the assumed model atmospheres.
14. Magnetic field strength dependency on model atmospheres
14.1. Synthetic S_{V} and D_{V} for four model atmospheres
Fig. 21 Synthetic S_{V} values as functions of B for ψ = 0 and instrumental broadening γ = 3 pm for model atmospheres FALC, FALA, FALF, and MALTM. Curves correponding to FALC and FALA are almost superimposed (departures around 2 G). 
As we show in Figs. 6a and 7a, S_{V} quantities deviate from B values especially in the weak field regime, so that inversion results are expected to depend more strongly from the assumed model atmospheres in this range. It is, therefore, interesting to estimate the accuracy of the full inversion process using D_{V} quantities and interpolations between model atmospheres.
Figure 21 shows synthetic S_{V} values as functions of B for instrumental broadening γ = 3 pm, as in Figs. 6a and 7a. Results from model atmospheres FALA and FALF were added to FALC and MALTM. For simplification, the plots are limited to low values of B with ψ = 0. Similarly, synthetic D_{V} values are plotted in Fig. 22, as in Figs. 6c and 7c.
According to the results from SOT/SP data (Fig. 14), the useful range of B magnetic strength is roughly B> 1000 G. In this range, we note that FALC, FALA, and FALF curves are almost superimposed. The S_{V} departures between FALC, FALA, and FALF are always smaller than 10 G, while departures between MALTM and FALC (or FALA and FALF) exceed 50 G. This shows that the accuracy of inversion results is practically not reduced by eliminating FALA and FALF from the interpolation process between FALC and MALTM.
Fig. 22 Synthetic D_{V} values as functions of B for ψ = 0 and instrumental broadening γ = 3 pm for model atmospheres FALC, FALA, FALF, and MALTM. Curves correponding to FALC, FALA and FALF are almost superimposed (departures around 2 G). 
14.2. Inversion of SOT/SP data in the range Bf > 1000 G
Fig. 23 Magnetic field strength output from UNNOFIT and VMI inversions (circles and black points, respectively). Full and dotted lines correspond to VMI inversions assuming only one given model atmosphere (F0 and M0 without instrumental broadening, Fγ and Mγ with broadening γ = 3 pm). 
For observational data such as the SOT/SP data, magnetic strengths (Fig. 14, top and bottom), depend only slightly on the range of inclination angles. We consider the most simple case of small inclinations ψ_{U} < 30° and ψ_{U} > 150°, which also takes advantage of a higher signaltonoise ratio. In Fig. 23, we plot the mean results of SOT/SP data VMI inversion with model interpolation between FALC and MALTM (black points) along with UNNOFIT results (circles), as in the top of Fig. 14. We add by comparison the curves noted Fγ (full lines) and Mγ (dotted lines) corresponding to results obtained by assuming only one model atmosphere, without interpolations, for FALC and MALTM, with the assumed instrumental broadening γ = 3 pm.
The VMI and UNNOFIT results behave similarly for B_{U}f_{U} > 1000 G. These results leave the Fγ line to reach the Mγ line, as is expected from the δ_{V} function plotted in Fig. 11. In the range B_{U}f_{U} > 1300 G, the use of a simple linear interpolation (Eqs. (19)−(24)) leads to departures less than 50 G between VMI and UNNOFIT.
14.3. Inversion of SOT/SP data in the range 200 < Bf < 1000 G
The agreement is a little less good in the range B_{U}f_{U} < 1000 G.
According to δ_{V} values (Fig. 11), FALC is roughly the best model atmosphere, so that results do not depend very much on the interpolation process. Moreover, although filling factors are small (around 0.15 for Bf = 200 G), we expect low data noise effects (even for f = 0.1, according to Fig. 9). However, we must remind ourselves that for small filling factors, we can expect departures between inversion methods that are using Iprofiles or not.
It can be noted as well that inversion results using FALC and MALTM separately (curves Fγ and Mγ ) lead to B values that are always higher than 1000 G in the range Bf > 200 G. According to Fig. 21, they correspond to S_{V} values higher than 1500 G and, therefore, much higher than S_{0} for any of the four models FALC, FALA, FALF, and MALTM. We see also in Fig. 23 that departures between curves Fγ and Mγ never exceed 150 G. This accounts for the relatively small dependency on model atmospheres in this range.
14.4. Prospects for a better model selection
To improve the model selection, further investigations are necessary to provide a better understanding of the relationship between thermodynamical parameters and the shape of Vprofiles. More specifications might be extracted from additional measurements based not only on differences between lobe shifts of V and V^{2}profiles. They would permit the use of nonlinear interpolation between more than one parameter and more than two model atmospheres. The magnetic strength itself might be included in atmosphere specifications during the iteration process. We return to the possibility of new simulations of unresolved magnetic flux tubes through 3DMHD simulations in Sect. 16.
15. Magnetic field strength dependency on instrumental broadening
In Fig. 23 we also plotted the mean results of SOT/SP data inversions without model interpolation for the FALC and MALTM models, but this time without instrumental broadening (γ = 0). They are noted F0 and M0.
The resulting mean magnetic strengths B are always larger than 1500 G. Because values that are too high are obtained in both cases, we assumed that the error was not due to model atmospheres, but mainly to instrumental broadening. The γ coefficient is adjusted to get the best agreement with UNNOFIT and MERLIN results especially in the range Bf > 1000 G, where filling factors are close to 1 and where disturbances due to mixing between magnetic and nonmagnetic spectra do not affect Iprofiles.
Of course, it would be better in the future to get direct estimates of instrumental effects to check whether the perceived broadening is not in part due to the assumed model atmospheres.
16. Stokes V amplitude and filling factors
In the weak field approximation, the Bf products and, more generally, the magnetic flux in the lineofsight direction can be deduced approximately from the maximum Stokes V amplitude by (28)according to Landi Degl’Innocenti et al. (2004) and Bommier et al. (2009). The parameter λ_{0} is the line wavelength in nm, the effective Lande factor, Δλ_{D} the Doppler width, and I_{C}−I_{0} the difference between continuum and line center intensity in magnetic areas.
For very small filling factors, the observed continuum, line center intensity, and Doppler width correspond to the spectrum of nonmagnetic areas, outside magnetic flux tubes. However, I_{C}−I_{0} decreases approximately by a factor 8 between the extreme cases of FALC and MALTM (Sect. 10.3), while Δλ_{D} decreases by a much smaller factor. Hence, assuming the same model atmosphere inside and outside magnetic areas may lead to errors on determinations of Bfcosψ and filling factors f.
The same problem may also arise in the general case of any magnetic strength for inversions using the same thermodynamical parameters in magnetic and nonmagnetic areas. Further investigations might be able to connect flux tube model atmospheres more accurately with various moments (barycenters and widths) of Vprofiles. This should lead to more accurate values of filling factors and magnetic fluxes.
Such investigations could be carried out with synthetic polarization profiles obtained from realistic 3DMHD simulations of the solar photosphere obtained, for example, from the MURaM code (Vögler et al. 2005). These simulations would allow us to estimate and model the effect of unresolved magnetic structures on the observed Vprofile shapes (see Shelyag et al. 2007). With that prospect, a database giving access to synthetic Stokes spectra of the most commonly used magnetic sensitive lines, computed for various magnetic regimes from 3DMHD simulations would be a very valuable tool for testing inversion methods.
17. Conclusions
VMI specificity and speed. The VMI inversion method can help to make progress in the analysis of unresolved structures by providing magnetic field vectors independently of Iprofiles. It determines magnetic field vectors for inhomogeneous solar structures in the context of twocomponent solar models, i.e., magnetic and nonmagnetic. In the same way, results remain reliable if intensity profiles are disturbed by scattered light. The specific point is that VMI does not depend on any nonmagnetic component, since it is independent of the Stokes I profile.
The number of iteration steps typically does not exceed 3 and VMI iterations are very fast. The computing time is less than 3 × 10^{4} s per pixel with a fourprocessor Xeon computer (8 cores, 2.4 GHz). The useful inclination range is typically 0 < ψ < 60° and 120 < ψ < 180°. It might be complemented by Qmoments inversions for ψ angles around 90°.
Comparison with UNNOFIT and MERLIN inversions. SOT/SP data in the 630.2 nm FeI line have been processed.
Mean VMI B values are always higher than 1000 G and in rather good agreement with UNNOFIT results for 200 < Bf < 2000 G. No really significant departures can be associated with differences between model atmospheres in magnetic and nonmagnetic areas (Sects. 1.2 and 11.3).
Mean VMI values are also in good agreement with MERLIN results for 700 < Bf < 2000 G. In the range Bf < 700 G, departures are present because unresolved structures are not included in the MERLIN code.
For very strong fields, VMI B values are slightly higher than those derived with UNNOFIT or MERLIN. Stokes profiles should probably be represented by more complex models with several columns or transverse gradients of magnetic field.
Model atmospheres. We emphasized the importance of model atmosphere selection from the shape of Vprofiles in Sect. 14. This paper uses the simplest way to accommodate thermodynamical variation, namely with an interpolation between only two model atmospheres, using the difference between lobe shifts in V and V^{2}profiles. Relationships between thermodynamical parameters and shape of Vprofiles might be investigated in more detail with new sets of theoretical 3DMHD simulations
Line broadening and data noise effects: Possible improvements. The accuracy of results should be improved via a direct determination of the γ coefficient characterizing instrumental line broadening.
Data noise effects are already reduced in Vmoments calculations with the use of V ×  V  instead of V^{2} (Sect. 4.1) and with noise subtraction in Q^{2} and U^{2} (Sect. 4.2). Estimates of errors (Figs. 9 and 10), however, show that some reduction of noise level should provide reliable B results for Bf values that are much lower than the present limit, which is approximately 200 G. A factor 2 should be sufficient to lower the limit down to 100 G.
Data noise might also be reduced by longer exposure times on the condition that spatial resolution is not degraded. Wider wavelength intervals across the line profile should also help, not only to increase the accuracy of high magnetic field measurements, but also to allow a better determination and a better correction of noise in the far wings of the line, where Stokes parameters are expected to be negligible.
Profiles of infrared lines, which are more sensitive to low magnetic fields because of larger Zeeman shifts, could also be investigated.
Acknowledgments
We thank the referee for fruitful comments. HINODE is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in cooperation with ESA and NSC (Norway). Hinode SOT/SP Inversions were conducted at NCAR under the framework of the Community Spectropolarimtetric Analysis Center (CSAC; http://www.csac.hao.ucar.edu).
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All Figures
Fig. 1 Vprofiles for FALC and MALTM models at disk center (full and dashed lines, respectively) with magnetic field strengths 500 G (top) and 2000 G (bottom). Profiles are scaled to their own maximum value. 

In the text 
Fig. 2 S_{V}, R_{V}, D_{V}, and A_{V} functions for FALC model (Sect. 4). 

In the text 
Fig. 3 S_{V}, A_{V}, R_{V}, and A_{V} functions for MALTM model. 

In the text 
Fig. 4 Vprofile from SOT/SP data (stars) for magnetic field strengh 2500 G. Normalized synthetic profiles are plotted for MALTM with (full lines) and without (dashed lines) additional broadening. 

In the text 
Fig. 5 Lorentz broadening function for synthetic Vprofiles. 

In the text 
Fig. 6 S_{V}, A_{V}, D_{V}, and A_{V} functions for FALC model with additional broadening (Sects. 8, 9). 

In the text 
Fig. 7 S_{V}, A_{V}, R_{V}, and D_{V} functions for MALTM model with additional broadening. 

In the text 
Fig. 8 RMS of Stokes profiles in far wings of the 630.2 nm line ( Δλ  > 0.042 nm) in SOT/SP data, divided by the quiet Sun continuum intensity, for noise level determination. Circles for Q, triangles for U, and squares for V. 

In the text 
Fig. 9 Inversion of synthetic FALC model data for the case of data noise for filling factors 1 and 0.1. 

In the text 
Fig. 10 Inversion of synthetic MALTM model data for the case of data noise for filling factors 1 and 0.7. 

In the text 
Fig. 11 SOT/SP data: δ_{V} versus UNNOFIT product B_{U}f_{U}. 

In the text 
Fig. 12 SOT/SP data: continuum intensity versus UNNOFIT product B_{U}f_{U}. 

In the text 
Fig. 13 Numbers of pixels so that ψ_{U}< 60° or ψ_{U}> 120° (squares) and numbers of pixels computed by VMI (black points) divided by the total number of available pixels in UNNOFIT inversion. 

In the text 
Fig. 14 Magnetic strengths from VMI inversion (black points) and UNNOFIT inversion (circles) in the ranges 0 <ψ_{U}< 30° and 150°<ψ_{U}< 180° (top) and 0 <ψ_{U}< 60° and 120°<ψ_{U}< 180° (bottom). 

In the text 
Fig. 15 Inclinations from VMI versus UNNOFIT. 

In the text 
Fig. 16 Bf products from MERLIN inversion versus Bf products from UNNOFIT inversion. 

In the text 
Fig. 17 Magnetic strengths from VMI (black points) and MERLIN inversion (squares) versus B_{M}f_{M} values in the ranges 0 <ψ_{M}< 60° and 120°<ψ_{M}< 180°. 

In the text 
Fig. 18 Inclinations from VMI versus MERLIN inversion code. 

In the text 
Fig. 19 Same caption as for Fig. 14 (bottom), but for VMI quicklook without iteration. Magnetic strengths from VMI inversion (black points) and UNNOFIT inversion (circles) in the ranges 0 <ψ_{U}< 60° and 120°<ψ_{U}< 180°. 

In the text 
Fig. 20 Same caption as for Fig. 15, but for VMI quicklook without iteration. 

In the text 
Fig. 21 Synthetic S_{V} values as functions of B for ψ = 0 and instrumental broadening γ = 3 pm for model atmospheres FALC, FALA, FALF, and MALTM. Curves correponding to FALC and FALA are almost superimposed (departures around 2 G). 

In the text 
Fig. 22 Synthetic D_{V} values as functions of B for ψ = 0 and instrumental broadening γ = 3 pm for model atmospheres FALC, FALA, FALF, and MALTM. Curves correponding to FALC, FALA and FALF are almost superimposed (departures around 2 G). 

In the text 
Fig. 23 Magnetic field strength output from UNNOFIT and VMI inversions (circles and black points, respectively). Full and dotted lines correspond to VMI inversions assuming only one given model atmosphere (F0 and M0 without instrumental broadening, Fγ and Mγ with broadening γ = 3 pm). 

In the text 
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