Issue 
A&A
Volume 535, November 2011



Article Number  A45  
Number of page(s)  9  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201117633  
Published online  31 October 2011 
Fast inversion of Zeeman line profiles using central moments
^{1}
Laboratoire d’Études Spatiales et d’Instrumentation en Astrophysique (LESIA), Observatoire de Paris, CNRS, UPMC Paris 6, Université Paris Diderot, 5 place Jules Janssen, 92195 Meudon, France
email: pierre.mein@obspm.fr
^{2}
National Solar Observatory 4, Sacramento Peak, PO Box 62, Sunspot, NM 88349, USA
^{3}
UMR 6525 H. Fizeau, Université de Nice Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, Campus Valrose, 06108 Nice, France
Received: 5 July 2011
Accepted: 13 September 2011
Context. Many inversion techniques derive vector magnetic fields and other parameters of the solar atmosphere from Stokes profiles with an iterative process.
Aims. We propose a new inversion method, using functions derived from central moments (ICM), to determine magnetic field vectors with very few iterations.
Methods. Two quantities A_{1} and A_{2} that combine moments of profiles I ± S (S = Q,U,V) are proposed. They are nearly linear functions of the longitudinal and transverse components of the magnetic field, and lead to estimates of the field components through a leastsquares polynomial fit. A third quantity A_{D} can be used to interpolate between expansions that correspond to two basic models. Exponents β_{1} and β_{2} in the moment expressions are adjusted to minimize the sensitivity to data noise.
Results. Inversion coefficients are computed for magnetic fields up to 3000 G in the case of the 630.2 Fe i line by forward modeling in two selected 1D model atmospheres (FALC and MALTM). After inversion of synthetic profiles computed with four models at disk center (FALA, FALC, FALF, MALTM), the mean standard deviations with respect to the input fields do not exceed 5 G for both components over the full range 0–3000 G. A comparison of ICM results with inversion by the UNNOFIT code of profiles observed with THEMIS/MTR shows good agreement. The typical computing time for a solar map of 100 000 points is less than 30 s.
Conclusions. The ICM inversions are almost insensitive to thermodynamic properties and solve for vector magnetic fields in a wide range of solar conditions, ranging from plage to spot, with very little computational effort. They are, therefore, extremely suitable for large data sets. Further improvements should take into account instrumental profiles and effects of limited spatial resolution by using filling factors. Extensions using more parameters and models with large departures from the Milne Eddington approximation could also be considered.
Key words: magnetic fields / line: profiles
© ESO, 2011
1. Introduction
Many inversion codes are available at present to derive magnetic field vectors from Zeeman Stokes profiles of solar lines. At the same time, these codes determine thermodynamical parameters of the solar atmosphere, which are coupled with magnetic fields in line formation. In general they employ leastsquares inversion techniques, in which atmospheric parameters are modified by iterations until synthetic profiles match the observed ones (see for example the review by Bellot Rubio 2006). Some codes, SIR (Ruiz Cobo & del Toro Iniesta 1992) for example, use response functions to iteratively determine parameters at all altitudes across the atmosphere. NLTE extensions are possible (SocasNavarro 2007). Some other codes such as UNNOFIT (Bommier et al. 2007) derive synthetic profiles with analytic expressions derived from the MilneEddington approximation and UnnoRachkowsky solution (Harvey et al. 1972; Landolfi & Landi Degl’Innocenti 1982; Skumanich & Lites 1987). Multicomponent atmospheres can also be investigated, with filling factors or microstructures (MISMA code Sanchez Almeida 1997). Some other inversion codes are based on a direct comparison of observed profiles with a number of synthetic ones. They can be strongly accelerated by a principal component analysis (Rees et al. 2000; SocasNavarro et al. 2001).
Central moments of unpolarized line profiles have been used in the past to determine velocity fields across the solar atmosphere (see for example Marmolino & Severino 1981). In the case of Zeeman sensitive lines, first order moments can be used to determine the center of gravity of the left and righthand circularly polarized profiles and estimate the longitudinal magnetic field (Semel 1967; Uitenbroek 2003). In this paper we expand on the use of moments, and propose the use of several quantities derived from central moments of the Stokes profiles I ± Q,U in addition to those of I ± V. These quantities turn out to be nearly linear funtions of the magnetic field components along and perpendicular to the line of sight, and are mostly insensitive to other atmospheric parameters. They can be used to determine vector magnetic fields with short polynomial expansions that can be properly calibrated with forward modeling in the context of a small number of onedimensional model atmospheres.
In Sect. 2 we define the moments that we use in Sect. 3 to find the basic quantities to be used in the inversions. In Sect. 4 we present the radiative transfer code and the four model atmospheres used to compute synthetic profiles and inversion coefficients. In Sects. 5 and 6, polynomial expansions are specified and tested in the case of one model atmosphere. Noise effects are estimated in Sect. 7. To satisfy simultaneously a great number of solar conditions, an interpolation method is proposed and tested with different models and centertolimb positions (Sect. 8). It is compared to UNNOFIT inversion of THEMIS/MTR data (Sect. 9). Because of the low sensitivity to thermodynamical parameters, the capabilities of a simpler method using one average model only are also investigated (Sect. 10). We end with conclusions in Sect. 11.
2. Moments of line profiles
We denote with s the Zeeman splitting corresponding to the longitudinal field B_{∥}, for an effective Lande factor and a linecenter wavelength λ_{0}: (1)with (2)where wavelengths are expressed in nm and B_{∥} in Gauss. The depth of a line profile P(λ) can be written (3)where I_{c} is the continuum intensity. We note with (4)the first order moment of R, expressed in Gauss. The exponent β_{1} is used to assign different weights to line wings and line center and to reduce noise effects (Sect. 8).
To characterize the widths of line profiles, we define a second moment of R similar to the classical secondorder central moment: (5)where λ_{2} is the wavelength corresponding to the first moment of R(λ) to the power β_{2}: (6)We use the modulus of λ − λ_{2} instead of (λ − λ_{2})^{2} to obtain the exact shifts of blue and red components in the case of strong transverse magnetic fields (see next section).
3. Quasilinear functions A_{1} and A_{2} of magnetic field components, from weak to strong fields
For a weak line formed in a MilneEddington atmosphere without velocity and magnetic field, the line depth can be written as (7)by neglecting damping effects. The coefficient c and the mean Dopplerwidth d depend on the line and model atmosphere.
Let us now assume a constant vector magnetic field throughout the atmosphere, and denote with B_{∥} and B_{⊥} the longitudinal and the transverse components of that field. Below we show that one can express the moments of line depths corresponding to polarized profiles I ± S, (S = Q,U,V) as functions of B_{∥} and B_{ ⊥ }. In this we restrict our calculations to the case of a Zeeman triplet.
3.1. B_{∥} and the function A_{1}
Let us assume B_{⊥} = 0. As before, we denote with s the wavelength shift corresponding to the longitudinal field B_{∥} and λ_{0} the wavelength at line center. The line depths of I − V and I + V are two Gaussian functions shifted by ± s from λ_{0}: The firstorder moments M_{1}(I + V, β_{1}) and M_{1}(I − V, β_{1}) provide the centers of gravity of line depths R_{I + V}(λ) and R_{I − V}(λ) so that the quantity (10)is equal to B_{∥}. This is the wellknown center of gravity method for determining the lineofsight magnetic field (Rees & Semel 1979). The result in Eq. (10) is independent of c, d and β_{1}.
3.2. B_{⊥} and the function A_{2}
Let us assume now that B_{∥} = 0 and that the azimuth of B_{ ⊥ } is parallel to the polarization direction of Q. The line depths of I − Q and I + Q can be written with (13)
3.2.1. Weak transverse components
If the transverse component is weak (w ≪ d), we can write (14)and (15)so that (16)and (17)From Eq. (15) we see that (18)To minimize crosstalk effects between the determination of the longitudinal and transverse field components, we replace M_{2}(I + Q, β_{2}) by the average value W of the widths of I + V and I − V (see Eq. (5)): (19)so that (from Eqs. (13), (17) and (18)) (20)Combining Eqs. (19) and (20) we see that, for weak transverse magnetic fields, (21)is a linear function of the transverse component B_{ ⊥ }, with a proportionality coefficient of 1/.
3.2.2. Strong transverse components
In the case of strong transverse fields (w ≫ d) (22)so that the component B_{ ⊥ } is provided directly by the moment M_{2}(I − Q, β_{2}). But at the same time M_{2}(I + Q, β_{2}) becomes much smaller than M_{2}(I − Q, β_{2}). We will see later that in realisitic cases (Sect. 5, Fig. 2) the width W of profiles I + V and I − V increases for large B_{ ⊥ }, in such a way that A_{2} remains close to a linear function of B_{ ⊥ } for strong fields as well as for weak fields (see Eq. (21)). It thus becomes possible to use A_{2} in the full range of magnetic field strengths. In the following sections, we will express A_{2} in Gauss, regardless of the (dimensionless) proportionality coefficient, which is close to (πβ_{2})^{ − 1/2} for weak fields.
3.3. Magnetooptical effects
We call Φ the angle characterizing magnetooptical effects: (23)with (24)(25)and (26)where U(λ_{0}) and Q(λ_{0}) are the values of the Stokes parameters U and Q at line center of the intensity profile.
We must note that while magnetic components are related to moments integrated over full profiles, the magnetooptical effects are determined by Stokes parameters observed in one single wavelength. This leads to a higher sensitivity to noise, especially for weak transverse magnetic components, corresponding to low Q and U values. Although other more sophisticated algorithms might be investigated in this respect, we restrict the present calculations to this most simple and least timeconsuming method.
4. The RH radiative transfer code
To calibrate the inversion method described here and to test its efficiency, we obtained forward solutions of the polarized radiative transfer equations with the RH radiative transfer code (Uitenbroek 2001, 2003) which is based on the Rybicki & Hummer (1991, 1992) multilevel accelerated lambda iteration scheme. The coupled equations of statistical equilibrium and radiative transfer were solved for a 23 level, 33 lines atomic model of Fe i, including the 630.25 and 630.15 nm lines. These nonLTE iterations were performed in the polarization free approximation to account for the effect of the line profile splitting on the radiative rates (Bruls & Trujillo Bueno 1996). After convergence the four Stokes parameters were calculated with the quasi parabolic DELO method developed by Trujillo Bueno (2003). In the calculation of the absorption and emission coefficients for the four Stokes parameters, the code fully accounts for the actual Zeeman splitting patterns of the atomic levels.
Four different onedimentional hydrostatic solar atmospheric models were employed in the calculations: FALA, FALC, FALF (Fontenla et al. 1993) and MALTM (Maltby et al. 1986), representing the stratification of the quiet cell interior, averaged quiet sun, solar network and a sunspot umbra, respectively.
5. Functions A_{1}, A_{2}, W and Φ for the Fe i 630.2 nm line with the FALC model at disk center
Fig. 1 Functions A_{1}, A_{2} and Φ (Eqs. (10), (21), (23)), measured in the case of the Fe i 630.2 nm line in the FALC model at disk center. The line styles depend on the values of B_{ ⊥ } in panels a) and c), and the values of B_{∥} in panel b): full lines until 1000 G, dashed lines until 2000 G, dasheddotted until 3000 G. 
We computed values for A_{1}, W , A_{2} and Φ for the Fe i 630.2 nm line in the FALC model at disk center for magnetic fields in the range of −3000 < B_{∥} < 3000 G and 0 < B_{ ⊥ } < 3000 G, generally suitable for quiet Sun and solar active regions, with steps of 200 G for both components. The step of wavelength sampling was 2 pm after interpolation of the synthetic profiles. The exponent values in the moment computations (Eqs. (4) and (5)) are according to optimizations that were chosen to minimize noise effects, as detailed in Sect. 8. Figure 1, panels (a) and (b), shows that A_{1} and A_{2} are nearly linear functions of B_{∥} and B_{ ⊥ } over all ranges of considered field strengths. Magnetooptical effects are plotted in panel (c) of Fig. 1. In Fig. 2 we note the increase of W, which accounts for the almost linear asymptotic behavior of A_{2} for strong transverse fields, and motivated the choice of substituting W for w in the definition of A_{2}. The increase in W is mainly caused by the increasing depth of the central core, because of the crosstalk of I ± V with the transverse component. Note that the obtained linearity of function A_{1} with B_{∥} provides clear evidence of the accuracy of the centerofgravity determination of the lineofsight component of the magnetic field for a large selection of longitudinal and transverse field component strentghs.
6. Polynomials P_{1, FALC}, P_{2, FALC}, and inversion of Fe i 630.2 nm profiles from FALC at disk center
For all synthetic profiles we have computed A_{1} and A_{2} after azimuth rotation over Φ, as indicated in Sect. 3.3. We first analyze the functions A_{1}, A_{2} and Φ that were computed with the FALC model as functions of B_{∥} and B_{⊥} in the full range of useful magnetic field components.
A_{1} and Φ always have the same sign as B_{∥} and A_{2} does not depend on the sign of B_{∥}. To reduce the number of coefficients, the following equations are supposed to be applied to the absolute value of A_{1}. If A_{1} < 0, the final values of B_{∥} and Φ must be replaced by −B_{∥} and −Φ.
Because A_{1} is a monotonic, almost linear function of B_{∥}, we can expand B_{∥} as a lowdegree polynomial in A_{1}, and obtain the coefficients with a leastsquares fit. The coefficients depend on B_{⊥} (see Fig. 1), so that
(29)We use polynomials with m ≤ 7. Because the coefficients p_{1,m}(B_{ ⊥ }) are known for all values of (B_{ ⊥ }), they can also be expanded as a polynomial in B_{ ⊥ }, (30)again with n ≤ 7 and the coefficients determined by a leastsquares fit from the results of the forward modeling in the FALC model. Combining both expansions, we find (31)In the same way, because A_{2} is a monotonic function of B_{ ⊥ }, we can determine coefficients p_{2,m}(B_{∥}) so that (32)and we can expand p_{2,m}(B_{∥}) as a function of B_{∥}(33)so that (34)In the following, we denote with P_{1,FALC} and P_{2,FALC} the polynomials of coefficients C_{1,m,n} and C_{2,m,n}: The longitudinal and transverse field components B_{∥} and B_{ ⊥ } can be derived from the measured quantities A_{1} and A_{2} by the following fast twostep iteration scheme. In the first step, B_{∥} is initialized with A_{1} and substituted into Eq. (36). Next the values of the two field components are derived by successive backsubstitutions: Magnetooptical effects can be deduced from the synthetic function Φ that we can also expand as a polynomial in B_{∥} and B_{ ⊥ }: Because Φ is not defined for B_{⊥} = 0, extrapolated values of P_{3,FALC}(B_{∥},0) are used before computing the coefficients C_{3,m,n}.
Fig. 2 Width W (Eq. (19)) in the case of the Fe i 630.2 nm line in the FALC model at disk center. Line styles are the same as in Fig. 1, panel b). 
Fig. 3 Comparison of longitudinal (panel a)) field strength, transverse field strength (panel b)), and azimuth (panel c)) recovered with the inversion with input values. 
To test the inversion code based on moments, we deduce vector magnetic fields from the values of A_{1}, A_{2} calculated with the forward modeling in model FALC and the procedure outlined above in Eqs. (37) to (42). Figure 3 shows the residual errors of B_{∥} and B_{⊥} resulting from the inversion process. The standard deviations σ_{∥} and σ_{⊥} are only 1 G and 4 G, respectively, over the full range of vector magnetic fields smaller than 3000 G, when leastsquares polynomials are extended to 6th order. For results of the azimuth φ, we restrict the calculation to transverse magnetic fields larger or equal to 200 G, because the azimuth is not defined for zero fields. The standard deviation σ_{φ} is smaller than 1°.
7. Effects of noise
To test the effects of noise on the results, we added noise to the synthetic profiles at a level of 1.5 × 10^{3} relative to the continuum intensity. This is the level assumed for typical THEMIS/MTR data (Bommier et al. 2007). This noise is simulated by a random number taken between −3 × 10^{3}I_{c} and 3 × 10^{3}I_{c}. Four different stochastic values are added to quantities I,Q,U and V.
Inversion errors are sensitive not only to the noise level, but also to the wavelength sampling of data. To mimic real observations, we used the wavelength sampling 2 pm, which is also a typical THEMIS/MTR value.
To minimize noise effects in the far line wings for moment calculations, we defined a new continuum value through the intensity at the highest and lowest wavelength in each profile. All data points with higher intensity are set to this value. In a similar way, any negative value of (Eq. (21)), which may appear because of noise in the case of low transverse magnetic field, is set to zero. Because of the limited number of points in the profiles, and because the results depend on the sets of stochastic numbers used to modify the synthetic data, the standard deviations are obtained by averaging over 100 successive inversions. The results which we present in Fig. 4 have standard deviations of 21 G and 29 G for both magnetic components, and 1.6° for azimuths. As expected, errors in the derived transverse component increase for very weak magnetic fields. The results are similar to the results derived from the UNNOFIT inversion of theoretical profiles (Bommier et al. 2007, their Fig. 4).
Fig. 4 Inversion of synthetic profiles for the FALC model atmosphere at disk center with noise a level of 1.5 × 10^{3}I_{c} and a sampling of 2 pm. Line styles are the same as in Fig. 1. 
8. Data from several models: interpolation using A_{D} functions; optimized β_{1} and β_{2} exponents
8.1. Definition of the function A_{D}
The widths of the I + V and I − V profiles deduced with Eq. (19) and included in the A_{2} functions (Eq. (21)) vary with the different employed model atmospheres. This leads to slight variations of the coefficients in Eqs. (35) and (36), depending on which atmospheric model is used for the polynomial fits, and this leads to variations in the values of the field that are recovered by the inversions. This model dependency needs to be accounted for. So far, we used the choice exponent β_{2} = 2 in Eq. (19), which was optimized to decrease the sensitivity of the inversion to noise. Setting the exponent β_{2} = 1 appears to yield a quantity that is more sensitive to the model.
To deal with model dependencies more adequately, we define a new quantity A_{D} similar to a Doppler width: (43)where we set (44)For each model M, A_{D,M} is a monotonic function of A_{2}. It can be deduced from A_{2} and B_{∥} by an equation similar to Eq. (34): Figure 5 shows the A_{D}functions corresponding to the two models FALC and MALTM. They look like the curves of Fig. 2 plotted for FALC with respect to B_{ ⊥ }, except that A_{2} varies roughly from 0 to 2000 G, while B_{ ⊥ } varies from 0 to 3000 G. Accordingly, the relevant model of any solar point can be specified by the comparison between the observed value of A_{D} and the value derived from Eq. (46), where B_{∥} can be replaced by A_{1} as a first approximation.
Fig. 5 Widths A_{D} of profiles I ± V computed with two selected models for inversion addressing several models. 
8.2. Inversion method
In our experiments we found that our inversion procedure yields sufficiently accurate results when we used a linear combination of polynomials P′ of only two selected models: FALC and MALTM. We therefore propose an inversion procedure using linear combinations of the polynomials P_{FALC} and P_{MALTM} determined by fitting the functions A_{1}, A_{2}, Φ and A_{D} for these models so that with the fractional coefficient given by (50)according to Eq. (46), where A_{1} approximates B_{∥}.
We show below that these linear interpolations are accurate enough because the inversion polynomials are relatively unsensitive to model parameters (Sect. 8.6).
With these modifications the inversion scheme now looks as follows:

1.
determine λ_{0} (Eq. (4)) and the angle Φ (Eqs. (23), (25) and (26));

2.
determine new Stokes Q parameter after rotation over Φ;

3.
calculate A_{1}, A_{2}, A_{D} and δ (Eqs. (10), (21), (46) and (50));

4.
evaluate polynomials P_{1,δ} and P_{2,δ} with known coefficients and δ (Eqs. (47) and (48));

5.
determine B_{∥} and B_{ ⊥ } (Eqs. (37) to (40)), with P_{δ} instead of P_{FALC};

6.
determine azimuth φ taking into account magnetooptical effects: (51)where P_{3,δ} is deduced fromp Eq. (49).
Fig. 6 Optimization of exponents β_{1} and β_{2} for a low sensitivity to data noise (see Sect. 8.3). 
Standard deviations of ICM results for magnetic components and azimuth angles with interpolations between FALC and MALTM model atmospheres using A_{D} functions, without and with noise.
Standard deviations of ICM results for magnetic components and azimuth angles with average model atmosphere (δ = 0.5), without and with noise.
8.3. Optimized exponents β_{1} and β_{2}
The accuracy of inversion from data not only depends on data noise, but also on the values of β_{1} and β_{2}. If the exponents are too small, the largest weight is assigned to the line wings, which are noisegenerating, while not very sensitive to magnetic fields. On the other hand, if they are too large, only a few points of the line core are effectively involved in the moments calculations, and the effects of noise increase.
To optimize the exponents β_{1} and β_{2}, we inverted the synthetic profiles of models FALC and MALTM, disturbed by a noise level 1.5 × 10^{3}I_{c}, using different values of β_{1} and β_{2}. We characterize the errors by standard deviations σ_{∥} and σ_{ ⊥ } corresponding to the mean results from inversions in both models, for field vectors smaller than 3000 G. Figure 6 presents the standard deviations for the parallel (panel a) and perpendicular (panel b) components, respectively, as functions of the parameters β_{1} and β_{2} for 100 successive realizations in each case. Noise effects are roughly minimized by the choice of integer values β_{1} = 1 and β_{2} = 2 used in previous sections. In Fig. 6 the results corresponding to β_{1} are plotted for β_{2} = 2 and the results corresponding to β_{2} are plotted for β_{1} = 1.
8.4. Results
Table 1 presents the standard deviations between input and recovered values of magnetic field components and azimuths in the cases of synthetic profiles from models FALC, MALTM, FALA and FALF, with and without noise. In all cases the wavelegth step is 2 pm. Averages are extended to all vector magnetic fields less than 3000 G, except for azimuth starting from transverse magnetic fields larger than 200 G.
Mean values ⟨ δ ⟩ characterizing the widths of line profiles are also presented. As expected, they are close to 0 and 1 for FALC and MALTM respectively. For a given model, they increase generally with noise. The reason is probably that according to the definition of the continuum (Sect. 7), the noise decreases the mean far wing intensities and increases A_{D}. This increase of line widths through noise may account for the result obtained for the deviation σ_{∥} of FALC data with δ = 0.05, which is better than in the case δ = 0 (Fig. 4a).
The standard deviation increases significantly with noise, showing that the errors caused by the inversion process itself can be generally neglected for real observed data.
It must be noted that the wavelength sampling used for the computation of inversion coefficients and for the inverted profiles is exactly the same as in Sect. 5. If additional interpolations are used between both sets of profiles, the σvalues without noise increase by a few Gauss, all the more because δ is also slightly affected through the A_{D} value. However, the results for noisy profiles are almost unchanged.
8.5. Centertolimb effects
In Table 1 additional σvalues are given for the inversion of synthetic profiles in the FALC model corresponding to different centertolimb positions. They are higher than values computed at disk center, but still acceptable. For any μvalue across the solar disk it is always possible to calibrate the exact coefficients of P_{1} and P_{2} from synthetic profiles at corresponding viewing angles and to obtain very low σvalues.
8.6. Sensitivity to model parameters
To test the sensitivity to the input models used to calibrate the expansion coefficients for the polynomials P, it is interesting to degrade the inversion code and to replace the interpolation between P_{FALC} and P_{MALTM} by the mean value (P_{FALC} + P_{MALTM})/2. Corresponding calculations have been performed and results similar to results of Table 1 are presented in Table 2.
For data without noise (lefthand parts of the tables), the higher accuracy of inversions with interpolations is obvious. The only exception is the last line of the tables which shows that the inversion of FALC data from μ = 0.5 is more accurate with the average of P_{FALC} and P_{MALTM} from the disk center than with P_{FALC} from the disk center.
For noisy data (righthand parts) the average standard deviations σ_{∥} and σ_{ ⊥ } for disk center increase roughly by factors 1.7 and 1.4 only from Tables 1 to 2. This means that for this noise level and this set of models, interpolations that are more sophisticated than the linear one used in Sect. 8.2 do not seem to be required.
9. Comparison with UNNOFIT inversion of THEMIS/MTR data^{5}
To test our inversion code with real data, we used THEMIS/MTR observations obtained by Bommier on 2005 September 13, which were already presented in a paper dealing with azimuth ambiguity solution (Bommier et al. 2011). They concern the 630.2 Fe i line and an active region located at L = −3.8° and b = −11.7°. Taking into account the slow variation of inversion coefficients around the disk center, as indicated in Tables 1 and 2, we can use the same diskcenter polynomials as before to invert the observed profiles.
Before computing the moments for the THEMIS data, we needed to suppress the telluric blend in the red wing of the line, because we deal with groundbased data. We replaced nine points of the observed profile by a leastsquares cubic interpolation. The corresponding wavelength interval is roughly 20 pm.
The exponents in the calculations of A_{1}, A_{2} and A_{D} are β_{1} = 1, β_{2} = 2 and β_{D} = 1 respectively. The investigated field of view is 90″ × 60″.
The UNNOFIT inversion code is based on the LevenbergMarquardt algorithm applied to the theoretical profile derived from the UnnoRachkovsky solution of the polarized radiative transfer equation through a MilneEddington atmosphere (Landolfi et al. 1984) The best way to use UNNOFIT inversions is to assume inhomogeneous fields with filling factors f different from 1, and to determine the products f ∗ B (Bommier et al. 2007). But our ICM inversion does not include this model so far. Because UNNOFIT codes do exist also with f = 1, we made our test by comparing results obtained in that case.
Fig. 7 THEMIS/MTR data: comparison between ICM and UNNOFIT results assuming a filling factor f = 1. 
The ICM computing time is very short, roughly 3 × 10^{4} s per solar point with a Dell R710 computer (2.67 GHz). The results are plotted in Fig. 7. In each interval of 100 G including more than 10 points, the mean ICM values are compared to values deduced from UNNOFIT inversion (Bommier et al. 2007). Error bars of Fig. 7 correspond to standard deviations in each interval. For azimuths, the 180°ambiguity was taken into account to compare results between − 90 and + 90 degrees. Intervals of 10 degrees are used in Fig. 7c, and magnetic fields smaller than 200 G are excluded as in previous sections.
Table 3 shows departures between vector magnetic field amplitudes resulting from both inversion methods in five different ranges of increasing magnetic fields. ϵvalues are mean differences and σvalues standard deviations around them. ϵvalues specify mainly systematic differences and σvalues mainly noise effects.
For weak magnetic fields, the two methods (UNNOFIT and ICM) give results that agree well, taking into account the inaccuracy resulting from the polarimetric noise.
For strong magnetic fields, the mean differences ϵ do not exceed about 50 G for vector amplitudes up to 2500 G. But the σ values are higher. They are probably partly caused by the correction of the telluric blend. Some tests have shown that the results are very sensitive to the wavelength interval used for the cubic interpolation. This might be checked by an inversion of satellite data.
In addition to blend effects, there are two other important points that have not been addressed in our ICM inversion so far:

1.
instrumental profiles were not taken into account. Theyincrease the effective width of line profiles: the computed δvaluesare between 0 and 4 across the fullfield of view, while the expected values arebetween 0 (quiet regions, FALC)and 1 (spots, MALTM);

2.
limitations caused by spatial resolution and seeing effects are neglected. They also increase the line widths and disturb the magnetic field determinations by mixing solar points of different Bvalues. A good way to eliminate part of them should be to take into account fillingfactor models.
Differences between ICM and UNNOFIT inversions of THEMIS/MTR data for five ranges of vector magnetic field amplitudes.
Differences between ICM and UNNOFIT inversions of THEMIS/MTR data for an ICM with an average model atmosphere.
10. Shorter ICM inversion with average model atmosphere
Because of instrumental profiles and limited spatial resolution, the differences between UNNOFIT and ICM inversions of THEMIS data given in Table 3 are larger than the expected standard deviations from noise effects, even in the case of inversion by average polynomials (P_{FALC} + P_{MALTM})/2 (Sect. 8.6, Table 2).
It is interesting to compare the UNNOFIT inversion with the ICM one in the case of average polynomials. The obtained results are quite similar to the previous ones (Table 4). This suggests that before any more sophisticated way to take into account instrumental profiles and spatial resolution limitations is considered, this short ICM inversion with average coefficients may be useful to invert the observed data. The inversion scheme of Sect. 8.2 is reduced: in point 3, A_{D} and δ were not calculated, and point 4 is suppressed. Only three 7 × 7 arrays of coefficients C_{1,m,n}, C_{2,m,n} and C_{3,m,n} are necessary to perform this fast inversion for a given solar line.
11. Conclusion
Inversion methods based on central moments (ICM) determine vector magnetic fields with very few iterations in realistic model atmospheres.
They use three observable quantities, A_{1}, A_{2} and Φ. A_{1} and A_{2} are nearly linear functions of magnetic field components in the full range of weak and strong fields. They are independent of mean Doppler velocities, almost independent of other thermodynmical parameters, and optimized to be also almost independent of data noise.
The extremly low number of iterations is possible because when a quantity a is a known function of two quantities b and c, and when a is a monotonic function of b, it is possible to expand b as a function of a and c: (52)which is the case for A_{1}(B_{∥},B_{⊥}) and A_{2}(B_{⊥},B_{∥}).
The ICM inversions can simultaneously address a wide range of models by using additional quantities A_{D}. In a shorter version, they can be used with an average model atmosphere and three fixed polynomials only.
The ICM methods are very fast (typically less than 30 s for 100 000 solar points) and very suitable for large data sets. It is possible to improve them in several ways. Instrumental profiles can be included. Solar models and an inversion process that include filling factors can also significantly increase the accuracy of the inversions. The magnetooptical effects in weak magnetic fields could be determined in more sophisticated ways that are less sensitive to noise.
We must remember that ICM methods do not require the restriction on the solution that is imposed by the MilneEddington inversion, namely that quantities be constant in height. Therefore, ICM methods could be very successful in determining the magnetic field structures in atmospheric conditions where the MilneEddington condition is not met. For example, while the ICM method presented in this paper uses the exponents β_{1} = 1 and β_{2} = 2, the method could be extended by varying these values, which would correspond to probing the height variation of the magnetic field, because higher βvalues give more weight to the line center, which forms higher than the line wings. More generally, extended ICM inversions using more parameters may provide promising new opportunities.
Finally, we note that the present method is free of any “first guess” of the solution to initiate the iteration, contrary to most usual inversion codes that are based on the LevenbergMarquardt algorithm for minimization.
Acknowledgments
We thank the referee for fruitful comments. Han Uitenbroek gratefully acknowledges financial support of the University of Nice Sophia Antipolis during the course of this work. This collaboration has also been supported by the CNRS through a grant of its Programme National SoleilTerre.
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All Tables
Standard deviations of ICM results for magnetic components and azimuth angles with interpolations between FALC and MALTM model atmospheres using A_{D} functions, without and with noise.
Standard deviations of ICM results for magnetic components and azimuth angles with average model atmosphere (δ = 0.5), without and with noise.
Differences between ICM and UNNOFIT inversions of THEMIS/MTR data for five ranges of vector magnetic field amplitudes.
Differences between ICM and UNNOFIT inversions of THEMIS/MTR data for an ICM with an average model atmosphere.
All Figures
Fig. 1 Functions A_{1}, A_{2} and Φ (Eqs. (10), (21), (23)), measured in the case of the Fe i 630.2 nm line in the FALC model at disk center. The line styles depend on the values of B_{ ⊥ } in panels a) and c), and the values of B_{∥} in panel b): full lines until 1000 G, dashed lines until 2000 G, dasheddotted until 3000 G. 

In the text 
Fig. 2 Width W (Eq. (19)) in the case of the Fe i 630.2 nm line in the FALC model at disk center. Line styles are the same as in Fig. 1, panel b). 

In the text 
Fig. 3 Comparison of longitudinal (panel a)) field strength, transverse field strength (panel b)), and azimuth (panel c)) recovered with the inversion with input values. 

In the text 
Fig. 4 Inversion of synthetic profiles for the FALC model atmosphere at disk center with noise a level of 1.5 × 10^{3}I_{c} and a sampling of 2 pm. Line styles are the same as in Fig. 1. 

In the text 
Fig. 5 Widths A_{D} of profiles I ± V computed with two selected models for inversion addressing several models. 

In the text 
Fig. 6 Optimization of exponents β_{1} and β_{2} for a low sensitivity to data noise (see Sect. 8.3). 

In the text 
Fig. 7 THEMIS/MTR data: comparison between ICM and UNNOFIT results assuming a filling factor f = 1. 

In the text 
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