A Study on Correcting the Effect of Polarization Crosstalk in Full-Disk Solar Photospheric Magnetic Fields Observations

The measurement of the solar magnetic field is an important part in the study of solar activities. The magnetic fields of active regions mostly contribute to the local energy release, such as solar flares, filament eruptions and small-scale activities. The full-disk magnetic field observation is very necessary to understand magnetic field creation, large-scale magnetic connections, large-scale properties of magnetic field and coronal mass eruptions. The full-disk magnetograph onboard the Advanced Space based Solar Observatory (ASO-S/FMG) is a new instrument to observe full-disk magnetic fields, which will surely contribute to studies of solar activity. FMG uses a magnetic sensitive line of Fe I $\lambda$5234.19 Å to observe full-disk magnetic field. FMG is designed to be a filter type instrument to obtain full polarized Stockes parameters ($I$, $Q$, $U$, $V$), which are used to calibrate magnetic field on the solar surface. During the operation of FMG, a few limited spectral positions inside the line are selected to observe Stockes parameters directly.

Many observation factors (the accuracy of wavelength position, wavelength shift due to filter properties or solar rotation, magnetic saturation, scattered light and so on) can affect the accuracy of magnetic field measurements, while for polarimetric instruments, the quality of Stockes parameters affect most directly the accuracy of the magnetic field derived. As a result, Stokes parameters with high-accuracy are the first step to obtain accurate magnetic fields. However, there is a very difficult problem to solve that there exist cross-talk effects among Stokes parameters, which means that individual Stokes signals can be contaminated by the others more or less (West & Hagyard 1983; West & Balasubramaniam 1992; Su & Zhang 2007).

Horizontal magnetic fields with a field strength above the equipartition value (400 G) in the solar photosphere are unstable, which then be pushed by the force of buoyancy upright with a vertical orientation relative to the solar surface (Spruit & Zweibel, 1979). The lateral expansion of magnetic flux concentrations (e.g. magnetic elements and sunspots) can only be set in the upper photosphere and chromosphere, where the density of gas has dropped sufficiently (Steiner et al., 1986). As result, a dominance of vertical magnetic fields in the solar photosphere, where the current measurements are taken. The amplitude of the Stokes parameters QU and V has a different dependence on the inclination $\gamma$ between the line of sight (LOS) and the magnetic field orientation. The amplitude of circular polarization Stokes $V$ is proportional to cos($\gamma$), while the amplitudes of linear polarization $Q$ and $U$ are proportional to sin²($\gamma$) (Jefferies et al., 1989). Because of the LOS over most of the disk and the magnetic field orientation are close to the solar vertical, the observed Stokes V amplitudes are usually much larger than those of Stokes Q or U on most locations. Consequently, once circular polarization $V$ contaminates the linear ones $Q$ and $U$, the influence of $V$ to $Q$ and $U$ would become serious and the most important one to be corrected urgently.

The inevitable cross-talk effects among Stokes parameters $Q$, $U$ and $V$ can not be removed completely. The standard method for removal of cross-talk is a polarimetric calibration of the telescope and instrument optics (Skumanich, et al., 1997; Beck, et al., 2005; Beck, et al., 2005; Socas-Navarro, et al., 2011; Harrington, et al., 2017;). The term $cross-talk$ used in this paper is only used for the removal of residual cross-talk (i.e., the residuals left over after the regular polarimetric calibration). The methods mentioned here are based on the assumption that V_obs $\approx$ V_true, which usually needs to be reached by a polarimetric calibration before. As for the removal of these residual cross-talk after polarimetric calibration, especially the cross-talk $V$ to $Q$ and $U$, which is the most important one, some effective methods or algorithms are proposed occasionally, especially for new developed instruments. West & Balasubramaniam (1992) developed and verified a method to correct crosstalk of circular-to-linear $V-Q/U$, under the assumption that $Q$ and $U$ profiles are completely symmetric and the asymmetries are results from errors of instruments. This kind of method has been used and tested in active region magnetic field measurements ( Mickey, 1985; West & Balasubramaniam, 1992). Su & Zhang (2007) extended the application of this method to full-disk magnetic field observation. Su & Zhang (2007) employ directly observations of two special spectral position, where $V$ profile is approximately symmetric around the position of their center. This symmetrical observation in pairs is also based on the symmetry and asymmetry of spectral profile in theory for circular and linear polarization signals, respectively.

In this paper, we focus on the removal of cross-talk from V to Q/U, which affects the accuracy of transverse field mainly, with the purpose to develop and test methods correcting crosstalk effect for the Full-disk solar vector MagnetoGraph (FMG) onboard the Advanced Space-based Solar Observatory (ASO-S). The followings parts are observations in sections 2, the descriptions of methods in section 3, the results with error analysis in section 4, the discussions in section 5, whereas the last section 6 gives the conclusions.

FMG (Deng et al., 2019) uses filters of the same type and the spectral line of Fe I $\lambda$5234.19 Å as the Solar Magnetism and Activity Telescope at Huairou Solar Observing Station (SMAT/HSOS: Zhang et al., 2007; Su & Zhang, 2007; Lin et al., 2013; Deng et al., 2019; Gan et al., 2019; Su et al., 2019) to observe the full-disk magnetic field, so the observations obtained from SMAT with scanning mode are used as experimental data to verify the effectiveness of the methods to remove cross-talk effects. The dataset used here was observed on 2017-09-07 from 01:33:05 to 02:12:09 UT, the scaning mode set: -0.30 Å to 0.30 Å with steps of 0.02 Å, $\sim$9 s exposure time with 256 frames. The observed $I$, $Q$, $U$ and $V$ images are 992 $\times$ 992 with the pixel size of 1.983 arcsec/pixel. Figure 1 shows the $I$/max($I$), $Q$/$I$, $U$/$I$, $V$/$I$ (hereafter $Q$, $U$, $V$ indicates $Q$/$I$, $U$/$I$, $V$/$I$ if there is no special points), polarization degree and Doppler velocity images of full-disk observations. For Doppler velocity map, it is used for modification of real spectral position. Figure 2 shows the local enlarged image indicated by the rectangle labelled in Figure 1 with strong polarization signals. To check the properties of observations, rms noise level in $Q$/$I$, $U$/$I$, $V$/$I$ are calculated at quiet region without significant polarization signal, which are $1\times 10^{-4}$, $1\times 10^{-4}$, $3\times 10^{-4}$ for $Q/I$, $U/I$ and $V/I$ for integration of all wave observations, respectively. While they are $7\times 10^{-4}$, $7\times 10^{-4}$, $15\times 10^{-4}$ for $Q/I$, $U/I$ and $V/I$ at one fixed wavelength location. Figure 3 shows an example of normalized $I$/max($I$), $Q$/$I$, $U$/$I$, $V$/$I$ spectra from a location labelled by the small squre 1 in Figure 2 with significant polarization signals. In panel a, the transmission profile of the Lyot filter using green dashed line is overplotted to compare spectral bandpass with the observed line width. The passband of the spectral line of Fe I $\lambda$5234.19 Å is about 0.125 Å  and the observed line width is about 0.3 Å, which can also refer to the paper of Deng et al. (2019) and Zhang et al. (2007).

A group of quasi pure $I$, $Q$, $U$ and $V$ images observed taken by the Near Infra-Red Imaging Spectropolarimeter (NIRIS: Cao et al., 2012) installed on Goode Solar Telescope (GST: Goode & Cao, 2012) at Big Bear Solar Observatory (BBSO) are used in this paper. NIRIS can obtain photospheric magnetic fields by spectropolarimetric observations of the Fe I 1.56 $\mu_{m}$ line (0.25 Å bandpass). $I$/max($I$), $Q$/$I$, $U$/$I$, $V$/$I$ images of NOAA 12600 used here are observed on 2016-10-15 from 19:18:53 to 19:1925 UT, the spectra is set from -3.0 Å to 3.0 Å  with a step of 0.11 Å. The image is 720 $\times$ 720 with the pixel size of 0.0244 arcsec/pixel, giving a total field of view of 17.5 $\times$ 17.5 arcsec. Figure 4 shows $I$/max($I$), $Q$/$I$, $U$/$I$, $V$/$I$ images observed by NIRIS, and the correlations between $V$ and $Q$/$U$ are calcuated and labeled with the values of $1.09\times 10^{-4}$ and $1.25\times 10^{-4}$, respectively. It is noted that polarization signal in the umbra is lower than in the penumbra, this maybe due to polarization intensity characteristics (’magnetic saturation’), when image is obtained at one wavelength position. For this dataset, the rms noise level are $3\times 10^{-4}$, $3\times 10^{-4}$, $1\times 10^{-4}$ for $Q/I$, $U/I$ and $V/I$ for integration of all wave observation, respectively, while they are $10\times 10^{-4}$, $9\times 10^{-4}$, $6\times 10^{-4}$ for $Q/I$, $U/I$ and $V/I$ at one fixed wavelength location. Figure 5 shows an example of normalized $I$/max($I$), $Q$/$I$, $U$/$I$, $V$/$I$ spectra from a location labelled by the small square in Figure 4 with significant polarization signal. On the whole these patterns of spectra are normal ones, since the spectra of $V$ keep good symmetry property, while $Q$ and $U$ has good antisymmetry property basically.

Figure 8 shows the distributions of cross-talk coefficents of $C_{q}(x,y)$ and $C_{u}(x,y)$ on solar full-disk obtained by method I and II labelled in panels (a), (b), (c) and (d), while the histogram plots in panels (e), (f), (g) and (h) are calculated from the black slits on full-disk for $C_{q}(x,y)$ and $C_{u}(x,y)$ by method I and II. It can be found that the distributions of cross-talk coefficents of $C_{q}$/$C_{u}$ have the similar patterns for method I and II (here the restrictions is added to the locations with polarization degree $>$ 3 $\sigma$, $\sigma$ is the stddev of polarization degree at the quiet region), however there should exist some differences between two methods that can be seen from histgram plots in panels (e), (f), (g) and (h). The top row of Figure 9 shows the scatter-plots of crosstalk coefficents of $C_{q}$ and $C_{u}$ labelled on solar full-disk for both of method I and II, at the mean time the correlations of $C_{q}$ and $C_{u}$ calculated by method I and II are labelled in each panels. It is found that the correlations of $C_{q}(x,y)$ and $C_{u}(x,y)$ calculated by methodI and II is about 40% on the whole. The bottom row in Figure 9 shows the differences between $C_{q}1$ and $C_{q}2$ ($C_{u}1$ and $C_{u}2$), where subscript 1 and 2 means method I and II. where the stddev of (C${}_{q}1$-$C_{q}2$) and (C${}_{u}1$-$C_{u}2$) calculated are 0.077 and 0.074, respectively, it can be found that the deviations between $C_{q}1$ and $C_{q}2$ ($C_{u}1$ and $C_{u}2$) of the region with strong polarization are relatively small, which can be see from the regions with significant polarization signals labelled by two white dotted rectangles 1 and 2. Then, an example of the images of $Q$ and $U$ observed at -0.12Å before and after cross-talk corrected by cross-talk coefficents of $C_{q}(x,y)$ and $C_{u}(x,y)$ in Figure 8 are shown in Figure 10. In this Figure, the white rectangles highlight the regions with evident polarization signal. The signals of $V$ are weakened in image of $Q$ and $U$ after cross-talk removal through comparisons of highlight region in detail, which can be seen more clearly in the magnification images in Figure 11. In Figure 11, the magnification images of the cut region labelled by rectangle 1 in Figure 10 are highlighted, additionally the image of $V$ and polarization degree are appended in the left column, and the correlations of $V$ and $Q$/$U$ before and after correction are printed in each panel individually. It is found that the contaminations of $V$ in $Q$ and $U$ are weakened obviously after cross-talk effects removal, which can also be seen from the lower values of correlations between $Q/U$ and $V$ after than those before the cross-talk removal for both method I and II. Figure 12 shows the spectra of $V$ and $Q$/$U$ before and after correction by method I (top row) and II (bottom row) labelled for the point indicated in Figure 2 by the black square 2, where the dotted lines are the observed spectrum and the solid ones are the corrected spectrum after crosstalk removal. It can be found that the modifications are evident for this point, the aymmetry properties of $Q$ and $U$ after corrections become better than those before corrections, which means that the effects of profile of $V$ in original profile of $Q$ and $U$ are significantly weakened. While the Figure 13 shows the spectra of a slit labelled in Figure 2 by the white dotted lines of $Q$ and $U$ before and after the corrections by methodI and II, and the correlations of images after corrections by methodI and II are printed in panels (d) and (h), It is found that the modification of pattern of $Q$ and $U$ are evident, and the correlations between $Q$/$U$ after correction by methodI and II exceed 90% that means the consistency of methodI and II for this slits selected.

For method I and II, it is under the assumption that the the true Stokes V is needed, but only the observed Stokes V is available, both methods should only work if V_obs $\approx$ V_true ($\pm$ 10 % relative deviation). More than this for method I is that the spectrum of $Q$, $U$ and $V$ should keep their symmetry properties. The presence of asymmetries caused by gradients in velocity and magnetic field violates the main assumption of method I on the symmetry, which maybe not so relevant for the current application due to the spatial resolution with order of $>$1 arcsec/pixel. For method II the basic idea is that Stokes $Q$, $U$ and $V$ should be independent variables, there should be low or zero correlation among $Q$, $U$ and $V$ signal. it is not important what the value of the correlation at the minimum is, but that it is a minimum. In addition the methods here should not be applied, instead of the standard method for removal of cross-talk, to calibrate polarimetric data in general, only for final clean-up of residual cross-talk. Before these applications, the better results from calibrate polarimetric data the better symmetry properties of Stokes parameters can be keeped and making V_obs more close to V_true, so the improvement of polarimetric calibration of the telescope and instrument optics is the most important task before these applications to remove the overall cross-talk effects.

There exist evidently large cross-talk values (e.g. $>$0.4) for the SMAT data, which can be results from the regions with high level noise or without significant polarization signals. In application of method I to spectra without significant polarization signal is not useful. If QU and V are exhibiting only noise, the application of Eqs. (1) and (2) will only give spurious results, which will contribute to large cross-talk values of $>$40% definitely. When we restricted to places with significant polarization signals, the large values would be reduced, however there are some large values that cannot be removed. Although the methods employed can provide a cross-talk value, but that the actual application of that cross-talk value multiplied with V_obs instead of V_true can worsen the cross-talk easily. For such large values, V_obs will differ so much from V_true that both methods cannort work anymore. The derived values of $C_{qu}$ and the correction with -$C_{qu}*V_{obs}$ will all be off, because V_obs deviate from V_true very far.Additionally, the ratio between $C_{qu}^{true}$ and $C_{qu}^{cal.}$ can be $>>$ 1 for small cross-talk values $<$ 0.1 seen from Figure 14, which cases the application of the correction can actually significantly worsen the problem instead of improving it.

These two methods have obvious correction effects in the sunspot region of strong magnetic field, which can be seee from the Figure 10 in detail (e.g. region 2 marked) and magnification image of Figure 11. It seems to show that the spatial distribution of the polarization signal in $Q$ and $U$ does not resemble anymore that of $V$ after the correction, additionally there is a consistency for methodI and II to some extent (e.g. bottom row of Figure 10). It is noted that the plage regions have strong kG magnetic fields, so the magnetic fields are mainly vertical, which leads to weak $QU$ signals. That can make $V->QU$ cross-talk stand out prominently. Although full-disk solar magnetic field is observed, we still focus on the sunspot region mainly. If the two methods in the sunspot region are effective, then they are still worth to be used in the actual observation for FMG.

Due to there is no true $Q$, $U$ and $V$ without cross-talk, even the polarimetric calibration of the telescope and instrument optics is good enough. We only do error analysis tentatively based on theories or certain assumptions. We find that once the symmetry of the spectral line is destroyed, ther errors will appear, some of which are very obvious especially for small values of cross-talk. Fortunately, it is note that although the relative errors do increase as the cross-talk coefficient decrease, the eventual effects from crosstalk are decreased because of actual weak contaminations (small values of crosstalk coefficient). While there are relatively larger crosstalk, the calculated crosstalk coefficients is more accurate which means the serious contaminations can be remove more effectively

In this paper, two methods with the aim of correcting the effect of crosstalk Stokes $V$ to $Q$ and $U$ are developed and tested for full-disk magnetic field observations, which should be one of the main magnetic field data processing process for FMG/ASO. Method I extends from Su & Zhang (2007), which is designed to deal with full-disk magnetic field observation of SMAT. Method II is a new developed one for full-disk magnetic field observation, which uses correlation information in images directly to correct the effect of crosstalk. Both methods are under the assumption of V_obs $\approx$ V_true.

The main ideas for method I is that the spectral profile can keep their symmetry, which can be possible due to the existence of wave shifts correcting data. Through the test that use SMAT special scanning observation designed, it is found that method I can be considered effective to correct the effects of crosstalk especially for sunspot regions. Method I would be used to process magnetic field data when FMG/ASO operates at special scanning magnetic field observation mode.

Method II is a novel way to correct the effect of crosstalk for full-disk magnetic field observations. The operation of method II travels on the whole full-disk using a reasonable small enough region to ensure the accuracy of local data, so this process leads to a huge consumption of computing resources. On the whole, method II can also correct the effect of crosstalk in sunspot regions that is basicall the same as method I, which can be seen from almost consistent correcting results of strong magnetic field regions in Figures 10 and 11. Additionally, only one set of $V$ to $Q$ and $U$ obtained at one position of spectral line is needed and enough for method II, which is the routine observation mode for FMG/ASO and is the evident advantage of method II.

Both of method I and II work well at the sunspot regions, which are our main areas of concern for the full-disk observations. While the methods maybe work off at the region without significant polarization signals or the region with high level noise. However, two method I and II tested for FMG magnetic field observation can complement and authenticate each other, and the two methods can backup each other under some specific conditions.

This work is supported by the Strategic Priority Research Program on Space Science, the Chinese Academy of Sciences (Grant No. XDA15320301, XDA15320302, XDA15052200), Natural Science Foundation of China (Grant No 11203036,11703042, U1731241,11427901, 11473039, 11427901 and 11178016), the Young Researcher Grant of National Astronomical Observations, Chinese Academy of Sciences, and the Key Laboratory of Solar Activity National Astronomical Observations, Chinese Academy