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  Appendix A: Examples of inverted spectra

 \begin{figure} \par\includegraphics[width=18.4cm,clip]{11727fa1.eps} \end{figure} Figure A.1:

Examples of observed ( black line) and best-fit spectra ( red) in Op. 001 ( 1st and 3rd row). The dash-dotted horizontal lines in QUV indicate three times the rms noise level, and the solid horizontal line the zero level. The vertical solid line denotes the rest wavelength. The 2nd and 4th row show the corresponding temperature stratifications of the magnetic component ( solid), the field-free component ( dash-dotted), and the HSRA atmosphere that is used as initial model ( dashed). Field strength and LOS inclination and their respective errors are given in the plot of Stokes I of 1564.8 nm ( upper left in each panel); the number of each profile is given in the upper left corner of each panel.
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 \begin{figure} \par\includegraphics[width=18.4cm,clip]{11727fa2.eps} \end{figure} Figure A.2:

Same as Fig. A.1 for TIP Op. 005.
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 \begin{figure} \par\includegraphics[width=18.4cm,clip]{11727fa3.eps} \end{figure} Figure A.3:

Polarization degree of 1564.8 nm for the profiles shown in Figs. A.1 ( upper two rows) and A.2 ( lower two rows). The dash-dotted and solid horizontal lines denote the inversion and final rejection threshold, respectively.
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Figures A.1 and A.2 show several profiles taken from the first and second long-integration observation of 2008 May 21 (Op. 001 and Op. 005), respectively. The positions of the profiles are marked by consecutive numbers in Fig. 1. The profiles shown were selected to have a small polarization degree that in some cases was barely sufficient to meet the inversion threshold (e.g., profiles Nos. 6 and 7). Below the spectra, the temperature stratifications used in the generation of the best-fit spectra are shown. With 3 nodes in temperature, the SIR code can use a parabola for changing the stratification; the parabola shape appears quite prominent for many of the locations. We note, however, that the IR lines at 1.56 $\mu$m are not sensitive to the temperature in the atmosphere above log $\tau\sim -1.5$ (Cabrera Solana et al. 2005). Only one profile corresponds to a kG field (Fig. A.2, top middle, No. 8). Figure A.3 shows the polarization degree of 1564.8 nm for all profiles of the previous figures. Profile No. 9 exceeds the inversion threshold of 0.001 of $I_{\rm c}$ near +750 m${\AA}$ with a spike that is presumably not of solar origin, but is instead noise in the Stokes U profile. The final rejection threshold of 0.0014 is, however, only reached by signals clearly related to the Zeeman effect (multiple double or triple lobes).

Reliability of the inversion results.

As discussed in Sect. 3, we used a constant value for the magnetic field strength (B), inclination ($\gamma $), azimuth ($\psi$), and the LOS velocity. This inversion setup cannot reproduce the antisymmetric Stokes Q or U or symmetric Stokes V profiles, which would require gradients in the magnetic field strength and the velocities along the LOS. The inversion was initialized with the same model atmosphere on all pixels (B=0.9 kG, $\psi=65$ deg), only the inclination being modified to 10 or 170 deg depending on the polarity. In the inversion process, the equal weight used for QUV in the calculation of $\chi^2$ naturally favors the component of higher polarization signal. For example, in profile no. 5 in Fig. A.1 the Q and U signals are larger than the V signal by almost an order of magnitude, leading to a better fit quality for Q and U than for V. In polarimetric data of low S/N, a difference of this order usually implies that the weaker signal is not seen at all.

SIR calculates an error estimate for the free fit parameters using the diagonal elements of the covariance matrix, expressed by the response functions (Bellot Rubio et al. 2000; Bellot Rubio 2003). The error estimate depends on the number of degrees of freedom in each variable; for parameters constant with optical depth thus a single value is returned. The error estimate, however, provides only information about the reliability of the best-fit solution for the corresponding $\chi^2$-minimum inside the chosen inversion setup. The estimated errors in the inversion of the profiles shown in Figs. 4, A.1, and A.2 are noted on the Stokes I panel for the Fe I 1565.2 nm line. The average uncertainties in the calculated magnetic field strength and inclination angle given by SIR are $\pm 50~$G and $\pm 10$ deg, respectively. The values agree with a previous error estimate in Beck (2006) derived from a direct analysis of the profile shape of the 1.56 $\mu$m lines (Table 3.2 on p. 47; $\delta B \sim 50$ G and $\delta \theta \sim 5$ deg).

  Appendix B: Calibration of Ltot to transversal flux

We tried to follow the procedure described in LI08 to calibrate the linear polarization signal into a transversal magnetic flux estimate that is independent of the inversion results. To reduce the influence of noise, LI08 first determine the ``preferred azimuth frame'', where the linear polarization signal is concentrated in Stokes Q. To achieve this, we determined the azimuth angle from the ratio of U to Q, and rotated the spectra correspondingly to maximize the Stokes Q signal. The scatter plot in Fig. B.1 compares the previously used total linear polarization, $L_{\rm tot} = \int \sqrt{Q^2+U^2}(\lambda) {\rm d}\lambda$, with the corresponding $Q_{\rm tot}({\rm rotated}) = \int \vert Q(\lambda)\vert{\rm d}\lambda$ as a measure of the linear polarization. The rotation of the spectra reduces the noise contribution by a constant amount, but the old and new values otherwise have a linear relationship with a slope close to unity.

We then averaged the rotated Q spectra over all spatial positions exceeding the polarization threshold for the inversion. The average Stokes Q spectrum was used as a spectral mask by LI08, but unfortunately their method fails for the infrared lines. The wavelengths around the line core have negative values in the average Q profile (Fig. B.2), which prevents to use it in the same way as in LI08. We thus used $Q_{\rm tot}({\rm rotated})$ as defined above instead, which we understand to be equivalent to the approach of LI08 despite not using a (somewhat arbitrary) spectral mask.

The plot of $Q_{\rm tot}({\rm rotated})$ versus transversal flux (Fig. 12, middle upper panel) showed considerable scatter that places the use of a single calibration curve in doubt. We thus not only tried to obtain a calibration curve, but also to quantify the effect of various parameters on the obtained relation. The upper part of Fig. B.3 shows calibration curves of $Q_{\rm tot}$ versus field strength for different field inclinations $\gamma $. The uppermost curve for $\gamma = 90^\circ$ corresponds to the one used by LI08. With the assumption that the field inclination does not necessary equal $90^\circ$, one already finds that one and the same value of $Q_{\rm tot}$ can be obtained for a range of around 200-550 G in field strength. The same effect is shown in the middle part, where the magnetic flux, $\Phi=B\sin \gamma$, was kept constant at $1.8\times 10^{16}$ Mx, B was varied, and $\gamma $ was derived accordingly from $\gamma={\rm arcsin} (\Phi/B)$. Again a range of around 200-500 G in B corresponds to the same value of $Q_{\rm tot}$. As a final test, we chose to investigate the influence of the temperature stratification on the resulting $Q_{\rm tot}$-value. We retained the magnetic flux, field strength and field inclination constant at ( $1.8\times 10^{16}$ Mx, 20 G, $75^\circ$), and synthesized spectra for different temperature stratifications. We used 10 000 temperature stratifications that were derived for the magnetic component in the inversion, and thus can be taken to be an estimate of the range of temperatures expected in the quiet Sun. The histogram of the resulting $Q_{\rm tot}$-values is displayed in the bottom part of Fig. B.3. The value of $\sqrt {Q_{\rm tot}}$ ranges from nearly zero up to 0.01, which also roughly corresponds to the scatter in $\sqrt {Q_{\rm tot}}$ in Fig. 12. We thus conclude that the largest contribution to the scatter comes from temperature effects. We remark that we used a magnetic filling factor of unity in all calculations. Any additional variation in the filling factor due to unresolved magnetic structures would increase the scatter in $Q_{\rm tot}$ even more.

We conclude that the usage of a calibration curve for a derivation of transversal magnetic flux from $L_{\rm tot}$ or $Q_{\rm tot}$, regardless of the exact calculation of the wavelength integrated quantities, is not reliable for a solid estimate, mainly because the strong influence of the thermodynamical state of the atmosphere on the weak polarization signals.

 \begin{figure} \par\includegraphics[width=8.9cm,clip]{11727fb1.ps} \end{figure} Figure B.1:

Scatter plot of the integrated Stokes Q signal in the preferred reference frame versus the total linear polarization without rotation. Solid line: unity slope; dashed line: unity slope with an offset of 0.0001.
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 \begin{figure} \par\includegraphics[width=8.9cm,clip]{11727fb2.ps} \end{figure} Figure B.2:

The average Stokes Q profile.
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 \begin{figure} \par\includegraphics[width=8.9cm,clip]{11727fb3.ps} \end{figure} Figure B.3:

Top: calibration curves from $Q_{\rm tot}$ into field strength B for field inclinations $\gamma $ from 10 to 90 deg ( bottom to top). The horizontal dotted line is at $Q_{\rm tot}=0.007$; the solid part of it denotes a range in B that gives the same $Q_{\rm tot}$ at different $\gamma $. Middle: $\sqrt {Q_{\rm tot}}$ versus field strength for constant magnetic flux. Dotted line and solid part as above for $\sqrt {Q_{\rm tot}}=0.0063$. Bottom: histogram of $\sqrt {Q_{\rm tot}}$ for constant flux but varying temperature stratifications T. The vertical line denotes the value resulting from the HSRA atmosphere model.
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