Automatic Detection and Correction Algorithms for Magnetic Saturation in the SMFT/HSOS longitudinal Magnetograms
Haiqing Xu, Suo Liu, Jiangtao Su, Yuanyong Deng, Andrei Plotnikov, Xianyong Bai, Jie Chen, Xiao Yang, Jingjing Guo, Xiaofan Wang, Yongliang Song
aa r X i v : . [ a s t r o - ph . S R ] S e p Research in Astronomy and Astrophysics manuscript no.(L A TEX: saturation˙R1.tex; printed on September 9, 2020; 0:30)
Automatic Detection and Correction Algorithms for MagneticSaturation in the SMFT/HSOS longitudinal Magnetograms *Hai-qing, Xu , Suo Liu , *Jiang-tao Su , , Yuan-yong Deng , , Andrei Plotnikov , Xian-yongBai , , Jie Chen , Xiao Yang , Jing-jing Guo , Xiao-fan Wang and Yong-liang Song Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese Academy of Sciences,Beijing, 100101, China; [email protected]; [email protected] School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, 19 A YuquanRoad, Shijingshan District, Beijing 100049, China Crimean Astrophysical Observatory, Russian Academy of Sciences, Nauchny, Crimea, 298409, Russia
Received 20xx month day; accepted 20xx month day
Abstract longitudinal magnetic field often suffers the saturation effect in strong magneticfield region when the measurement performs in a single-wavelength point and linear calibra-tion is adopted. In this study, we develop a method that can judge the threshold of saturationin Stokes
V /I observed by the Solar Magnetic Field Telescope (SMFT) and correct for itautomatically.
The procedure is that first perform the second-order polynomial fit to theStokes
V /I vs I/I m ( I m is the maximum value of Stokes I ) curve to estimate the thresholdof saturation, then reconstruct Stokes V /I in strong field region to correct for saturation.The algorithm is proved to be effective by comparing with the magnetograms obtained by theHelioseismic and Magnetic Imager (HMI). The accurate rate of detection and correction forsaturation is ∼ ∼
88% respectively among 175 active regions. The advantages anddisadvantages of the algorithm are discussed.
Key words:
Sun: sunspots — Sun: magnetic field — Methods: data analysis
The study of solar magnetic field has always been a core topic in solar physics. Some major unsolvedscientific problems in the study of solar physics, such as the generation of solar cycle, coronal heating,the origin of solar eruptions and so on, are all related to the solar magnetic field. The magnetic field ofsunspots was first investigated by Hale (1908). It is known that generally, the present magnetographsmeasure solar polarized light (present as Stokes parameters I , Q , U , and V ) rather than magnetic fields.Under a certain atmospheric models and assumptions, the solar magnetic field is obtained through inver-sion according to the radiation transfer theory. The solar magnetic field has been observed for one century H.-Q. Xu et al. and many interesting results have been presented from these observations.
Yet, there are still some basicquestions on the measurements of solar magnetic fields waiting to be carefully analyzed (Zhang 2019).Svalgaard et al. (1978) found the magnetograph was saturated when magnetic field is very strong. Ulrichet al. (2002) discussed reasons and treatment of saturation effects in the Mount Wilson 150 foot towertelescope system in detail. They pointed out that most spectral lines used for magnetic measurements aresubject to this saturation effect for at least some parts of their profile. Liu et al. (2007) found another typeof saturation in sunspot umbrae observed by the Michelson Doppler Imager on the Solar and HeliosphericObservatory (MDI/SOHO) caused by the 15-bit on-board numerical treatment used in deriving the MDImagnetograms. The saturation effect can be eliminated by using the information on spectral line, e.g., theSpectro-polarimeter (SP) onboard Hinode (Kosugi et al. 2007) obtains spectral profile with a wide spectralrange, and the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory (HMI/SDO,Schou et al. 2012) obtains spectral profile with six wavelength points. The saturation effect needs to be cor-rected by some supplementary methods if the polarized light is measured in a single-wavelength point andlinear calibration is adopted. Chae et al. (2007) performed cross-calibration of Narrow-band Filter Imager(NFI) Stokes
V /I and longitudinal magnetic field acquired by the SP, and proposed to use two different lin-ear relationships of longitudinal magnetic field and Stokes
V /I from Hinode/NFI to correct for saturation .Moon et al. (2007) used a pair of MDI intensity and magnetogram data simultaneously observed, and therelationship from the cross-comparison between the SP and MDI flux densities to correct for saturation in magnetic field obtained by MDI. Guo et al. (2020) explored a nonlinear calibration method to deal withthe saturation problem, which used a multilayer perceptron network.The Solar Magnetic Field Telescope (SMFT) at the Huairou Solar Observing Station (HSOS) of theNational Astronomical Observatories of China is a 35 cm vacuum telescope equipped with a birefringentfilter for wavelength selection and KD*P crystals to modulate polarization signals. The Fe I I, Q, U and V maps. The center wavelength of the filter can be tuned and is normally at − . ˚A forthe measurements of longitudinal magnetic fields and at the line center for the transversal magnetic fields(Ai & Hu 1986). It has been observing vector magnetic fields for more than 30 years. The theoreticalcalibration for SMFT vector magnetogram was first made by Ai et al. (1982). Several different methodsof the magnetic field calibration under the weak-field assumption have been done since then. Wang etal. (1996) used an empirical calibration and a velocity calibration methods to calibrate the longitudinalmagnetograms. Su & Zhang (2004) used 31 points of the Fe I of the profile of Fe I 5324.19 ˚A line,and the analytical Stokes profiles under the Milne-Eddington atmosphere model, adopting the Levenberg-Marquardt least-squares fitting algorithm. However, the routine measurements of Stokes I, Q, U and V parameters by SMFT are being performed in a single-wavelength point. The longitudinal magnetic field isreconstructed by equation 1: B SMF TL = C L VI (1) etection and Correction for Magnetic Saturation 3 where C L is the calibration coefficient inferred from the aforementioned calibration methods. This linearcalibration will result in the saturation when magnetic field is strong. Plotnikov et al. (2019) made anattempt to improve the routine magnetic field measurements of SMFT by introducing non-linear relationshipbetween the Stokes V /I and longitudinal magnetic field. They performed cross-calibration of SMFT dataand magnetograms provided by HMI to determine the form of the relationship. They found that the magneticfield saturation inside sunspot umbra can be eliminated by using non-linear relationship between Stokes
V /I and longitudinal magnetic field. They also discussed the influence of saturation effect on solving the180 degree ambiguity of the transversal magnetic field. They manually chose the threshold for separatingpixels into two subsets of strong and weak magnetic field, which is not convenient for dealing with largedata sample.In this paper, we attempt to develop a method which can judge the threshold of saturation in SMFTlongitudinal field and correct for saturation automatically. One purpose of this study is to correct forsaturation in longitudinal magnetic field obtained by SMFT since 1987. Another purpose is to preparecalibration technique for the Full-disk vector MagnetoGraph (FMG, Deng et al. 2019) which is one payload onboard the Advanced Space-based Solar Observatory (ASO-S, Gan et al. 2019) that will be launchedin early 2022. The routine observations for the FMG will be taken at one wavelength position of the Fe I5324.179 ˚A (Su et al. 2019). The magnetic field will be suffered saturation effect if the linear calibration isadopted.
The raw data registered by SMFT are left and right polarized light. The Stokes
V /I and I are calculated asfollowing: VI = V l − V r V l + V r I = V l + V r (2)where V l = I + V and V r = I − V represent modulated filtergrams. After this process, the influence offlat field is eliminated. The pixel size of SMFT data is approximately . ′′ × . ′′ since 2012 and the spatial resolution is approximately ′′ produced by local seeing effect. We selected 9 active regions (AR) inbetween 2013 and 2015 for case study and 175 ARs in 2013 for statistical study. The data were performed by × pixels median filtering to reduce the noise.To check the effectiveness of correction method for saturation, we downloaded the co-temporal magne-tograms of the selected 9 ARs from HMI/SDO. HMI is a full-disk filtergraph that measures the profile ofphotospheric Fe I 6173 ˚A line at six wavelength positions in two polarization states to derive the longitu-dinal magnetic field. The spatial resolution of the instrument is ′′ with . ′′ × . ′′ pixel size. In order toperform a detailed pixel by pixel comparison, HMI magnetograms were rotated for the p -angle correctionand reduced in spatial resolution to ′′ by a . ′′ × . ′′ . Then the same region that includes the maximized size of sunspots are selected andshifted each other to determine the optimal registration. H.-Q. Xu et al.
Fig. 1: Panels (a) and (b) are maps of Stokes I and V /I observed on 2014 September 10 by SMFT. Therectangle region in panel (a) is used to calculate I c . Panels (c) and (d) show the distribution of Stokes I and V /I along the red line; The green lines indicate the saturation locations marked by asterisks.
The studies show that there is a relationship between the continuum intensity and magnetic field, and thesmallest intensity always corresponding to the largest magnetic field ( e.g. , Mart´ınez & V´azquez 1993;Norton & Gilman 2004; Leonard & Choudhary 2008 ). Figure 1(a) and (b) show the maps of Stokes I and V /I for active region NOAA12158 observed on 2014 September 10 by SMFT. Figure 1(c) and (d) show thedistribution of Stokes I and V /I along the red line. The Stokes I decreases to the minimum in the sunspotcenter, but the Stokes V /I stops increasing at the points marked by green lines (asterisks) correspondingto sunspot umbrae. This phenomenon is called magnetic saturation. If performing linear calibration, thelongitudinal magnetic field will get weakened in sunspot umbrae in comparing with its surroundingarea .Next, we will show two examples to give a detail description for the detection and correction methodfor saturation effect.
The relationship between | V /I | and I/I c ( I/I m ) for NOAA12158 is given in Figure 2(a) ((b)). I c is themedian value of Stokes I within the rectangle region in Figure 1(a). I m is the maximum value of Stokes etection and Correction for Magnetic Saturation 5 Fig. 2: Scatter plots of | V /I | vs I/I c (panel (a)) and I/I m (panel (b)) for NOAA12158. The red line is thebest-fit second-order polynomial and I/I c ( I/I m ) marked in panel is corresponding to the apex. The redand blue contours in panels (c) and (d) represent I/I c =0.496 and I/I m =0.444. V /I uses the absolute valuesin the plots, similarly hereinafter. I within the whole active region. | V /I | first increases with Stokes I decreasing (going to sunspot center),then decreases. It is found that the second-order polynomial (red line) gives a well fit when | V /I | > I/I c ≤ I/I m ≤ in magnetic saturation regions , and used the second-order polynomial to separate the strong andweak field area. The pixels are separated into two parts by the apex. It is easy to calculate coordinates ofthe apex by fitting coefficient. We find that the corresponding I/I c is 0.496 and I/I m is 0.444, which aredenoted in Figure 2(c) and (d) by blue and red contours. Although the value of I/I c and I/I m is different,the region in Stokes I and | V /I | maps is the same. When I/I c < . ( I/I m < | V /I | suffered from saturation. So we may use this value as the thresholdto detect the saturation regions in longitudinal magnetic field.We performed the same analysis for NOAA12305 which includes multiple sunspots. We used the obser-vation on 2015 Mar 27. The similar relationship was found between | V /I | and I/I c ( I/I m ) as NOAA12158when | V /I | > I/I c ≤ I/I m ≤ I/I c ( I/I m ) is 0.577 (0.529) corresponding to theapex of the second-order polynomial (red line in Figure 3(a) and (b)), which are denoted in Figure 3(c) and(d) by blue and red contours. The region of saturation can also be detected accurately for multiple sunspots. H.-Q. Xu et al.
Fig. 3: Similar as Figure 2, but for NOAA12305.Table 1: Threshold of detecting magnetic saturation for 9 ARs observed by SMFT NOAA Date Position
I/I c I/I m We listed the thresholds for detecting saturation obtained by the above method for 9 active regions inTable 1. It can be seen that the thresholds are different for each active region. So it is necessary to calculatethe threshold for individual active region. There is no difference in detecting the saturation region using
I/I c and I/I m , but I/I m has more advantages than I/I c in automatic detection. etection and Correction for Magnetic Saturation 7 We take the above two active regions as examples to show how to correct saturation effect. We re-plot | V /I | vs I/I m in Figure 4. The correction procedure for magnetic saturation is as following:(1) The threshold I for occurrence of magnetic saturation is determined by the above algorithm, whichcorresponds to green asterisks in Figure 4(a) and (d).(2) The pixels are separated into two parts by threshold I . Those with
I/I m < I are suffered fromsaturation effect. Both linear (green lines in Figure 4(b) and (e)) and the second-order polyno-mial functions (yellow lines in Figure 4(b) and (e)) are used to fit the scatter plots for saturationdata. The green and yellow lines almost overlap. For its simplicity, we finally choose the linearfunctions to fit the scatter plots for both saturation and good data.
The fitting coefficients are( a , c ) and ( a , c ) corresponding to green and blue lines in Figure 4(b) and (e), respectively. a and a are slopes, c and c are constants. Using equation 3 to calculate Stokes V /I for pixels where
I/I m < I , VI s = (cid:12)(cid:12)(cid:12)(cid:12) a a (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) VI (cid:12)(cid:12)(cid:12)(cid:12) − c (cid:19) + c (cid:12)(cid:12)(cid:12)(cid:12) · sign (cid:18) VI (cid:19) . (3)(3) After re-calculating, the Stokes V /I maps are shown in Figure 5(a) and (c). It can be seen that thesaturation in sunspot umbrae has been eliminated. But the discontinuity at the boundary of umbrae andpenumbrae can be seen. To eliminate this discontinuity, we calculate the ± σ uncertainty of I correspond-ing to the cyan ( I + σ ) and blue ( I − σ ) asterisks in Figure 4(a) and (d). For pixels where I − σ < I/I m < I + σ , V /I is calculated by interpolation. Then smooth the data by Gauss-smooth function. The new
V /I mapsare shown in Figure 5(b) and (d). It can be seen that the discontinuity has been eliminated. The scatter plotsof | V /I | vs I/I m are shown in Figure 4(c) and (f). The relationship is approximate linear. The SMFT longitudinal magnetic field B SMF TL can be re-calibrated from equation 4: B SMF TL = C L VI s , II m < I C L VI , II m ≥ I (4)where C L is calibration coefficient. We adopt 8381 G as proposed by Su & Zhang (2004).The comparison of longitudinal magnetic field observed by SMFT and HMI for NOAA12158 is shownin Figure 6. It can be seen that the distribution of B SMF TL and HMI longitudinal magnetic field B HMIL is very similar (Figure 6(a) and (b)). The scatter plots of B SMF TL and B HMIL before and after correctingsaturation effect are shown in Figure 6(c) and (d), repectively. The B SMF TL starts to decrease when B HMIL is larger than 1300 G before correcting saturation effect. The linear correlation coefficient is 0.86. Aftercorrecting saturation effect in B SMF TL , the relationship of B SMF TL and B HMIL is closer to linear. Thelinear correlation coefficient increases to 0.96. Such good correlation indicates that the proposed correctionmethod for saturation in B SMF TL is effective for this active region.Figure 7 shows the comparison of B SMF TL and B HMIL for NOAA12305. It is also found that the B SMF TL starts to decrease when B HMIL is larger than 1300 G before correcting saturation effect. Thelinear correlation coefficient is 0.88. After correcting saturation effect in B SMF TL , a good correlation be-tween B SMF TL and B HMIL is found. The linear correlation coefficient increases to 0.96, which indicates
H.-Q. Xu et al.
Fig. 4: Scatter plots of | V /I | vs I/I m for NOAA12158 (panels (a)–(c)) and NOAA12305 (panels (d)–(f)).The green asterisks in panels (a) and (d) present the apex of the second-order polynomial fit (red line). Thecyan and blue asterisks present the ± σ uncertainty of apex. The corresponding I/I m values are markedusing the same color as those asterisks. The green and blue (yellow) lines are the linear (the second-orderpolynomial) fit to the data in panels (b) and (e). Panels (c) and (f) show the scatter plots of | V /I | vs I/I m after correcting magnetic saturation.that the proposed correction method for saturation in B SMF TL is also effective for activity region includesmulti-sunspots.We performed such pixel by pixel comparison for 9 ARs and listed the correlation coefficients in Table2. The correlation of B SMF TL and B HMIL is much better after eliminating magnetic saturation in B
SMF TL .So the detection and correction algorithms can be used to re-calibrate the longitudinal magnetic field instrong field region observed by SMFT .
The algorithm was proved to be completely effective by comparing the results with HMI data for individualactive region. To check the applicability of the algorithm for large sample, we tested it with
175 longitudinalmagnetograms of 175 ARs observed in 2013 by SMFT. The magnetic saturation generally occurs in strongfield region. Considering this actual situation, we set the following restrictions:(1) Only pixels where | V /I | > I/I m ≤ I min < I < I min is the minimum value of I/I m . If considering 1 σ error range, we can set I min < I − σ < I < I + σ < manual testing , the detected ARs are all correct.Only 1 AR with magnetic saturation was not detected by our method. So the accurate rate of detection is etection and Correction for Magnetic Saturation 9 Fig. 5: Maps of Stokes
V /I after correcting magnetic saturation for NOAA12158 (panels (a) and (b)) andNOAA12305 (panels (c) and (d)). The data in panels (b) and (d) are precessed with Gauss-smooth.Table 2: Correlation coefficient of B L for 9 ARs observed by SMFT and HMI NOAA Date Position C.C (before) C.C (after)11658 2013.11.19 S11W10 0.79 0.9411899 2013.11.16 N06W03 0.84 0.9311960 2014.01.25 S14E01 0.89 0.9512027 2014.06.07 N13W01 0.85 0.9512055 2014.05.12 N10W02 0.79 0.9312149 2014.08.27 N10E11 0.88 0.9412158 2014.09.10 N15E10 0.86 0.9612305 2015.03.27 S08W04 0.88 0.9612325 2015.04.19 N04E02 0.88 0.94
Notes: C.C (before) and C.C (after) represent the linear correlation coefficient before andafter correcting magnetic saturation in B
SMF TL , respectively. ∼ I/I m range, the above undetected AR can also be detected. It is found that magne-tograms of 5 ARs (the total is 42) were wrong after correcting saturation effect, which indicates that theaccurate rate of correction is ∼ either relatively small or with projection effect. Theprojection effect is complex, and we need to do a further analysis of its influence on saturation. Fig. 6: Panels (a) and (b) are B L maps of NOAA 12158 observed by SMFT and HMI , respectively . Panels(c) and (d) present the scatter plots of B L observed by SMFT and HMI. The data taken by SMFThave saturation in (c) and no saturation in (d). C.C is the linear correlation coefficient.
We developed an automatic detection and correction algorithms for saturation in longitudinal magnetic fieldobserved by SMFT based on the relationship between Stokes
V /I and I . It works well found in comparisonwith HMI data in case and sample study. The correlation of longitudinal magnetic fields between SMFTand HMI increased significantly after correcting for saturation effect. The accurate rate of detection andcorrection is ∼ ∼
88% respectively. There are total 43 out of 175 ARs with saturation effect. It means 75.4% ARs don’t need to correct saturation effect. We didn’t correct the scatter light when built the I - V /I relationship. The measured polarizationsignals are contaminated by scatter light ( I s ). E.g., if we consider the scatter light, the equation 2 willbe written as follow: VI = V l − V r V l + V r − I s I = V l + V r − I s (5) etection and Correction for Magnetic Saturation 11 Fig. 7: Similar as Figure 6, but for NOAA12305.
Generally, I s is determined at the solar limb. Here, we estimate I s using the intensity in quiet sun (acertain percent of I c ). We took NOAA 12158 as an example to estimate the effect of the scatter lighton the method. The result is shown in Figure 8. V /I in the umbrae increasing with larger scatter lightis subtracted from the observed data. When the contamination level is lower than 8% (Figure 8(a)-(d)), the Stokes
V /I vs I/I m curves are very similar and the areas of saturation are almost the samealthough the threshold of saturation is different. It may be due to the normalized I/I m being used.The saturation area decreases and the V /I vs I/I m curve is close to linear when the contaminationlevel is around 8% (Figure 8(e) and (f)), which shows that the measured polarized signals are likelyaffected more serious by scatter light than magnetic saturation. The above estimations indicate thatthe proposed method will not be affected by scatter light when the contamination level is lower, andthe scatter light can be corrected as a magnetic saturation effect. One advantage of this method is that it can calculate the threshold of saturation and correct it au-tomatically. Therefore, this method can be used for the routine longitudinal field observations. Anotheradvantage is that the used data acquired by one instrument which avoids a systematic error caused bycross-comparison. Especially, it can be used to correct the saturation effect in longitudinal magnetic fieldsin past 30 years taken by SMFT . This method can be used for FMG. The disadvantage of these method is
Fig. 8:
Scatter plots of | V /I | vs I/I m and V /I maps. (a) and (b): I s = I c × . (c) and (d): I s = I c × .(e) and (f): I s = I c × . In panels (a), (c) and (e), the red line is the second-order polynomial fit andthe marked I/I m corresponds to apex. The blue contours in V /I maps represent the
I/I m apexesmarked in panels (a), (c) and (e). that the correction for saturation is not very accurate when the active regions are far from disk center. Thismay caused by the projection effect. We will improve the method by considering the projection effect infuture. Acknowledgements
This work is supported by National Natural Science Foundation of China (NSFC)under No. 11703042, 11911530089, U1731241, 11773038, 11427901, 11427803, 11673033, U1831107,11873062, the Strategic Priority Research Program on Space Science, the Chinese Academy of Sciences etection and Correction for Magnetic Saturation 13 under No. XDA15320302, XDA15052200, XDA15320102, the 13th Five-year Informatization Plan ofChinese Academy of Sciences (Grant No. XXH13505-04). We acknowledge the use of data of SMFT/HSOSand HMI/SDO.