Long-Term Variations in Background Bias and Magnetic Field Noise in HSOS/SMFT Observations

Abstract

The Solar Magnetic Field Telescope (SMFT) at Huairou Solar Observing Station (HSOS) has conducted continuous observations of solar vector magnetic fields for nearly four decades, and while the primary optical system remains unchanged, critical components—including filters, polarizers, and detectors—have undergone multiple upgrades and replacements. Maintaining data consistency is essential for reliable long-term studies of magnetic field evolution and solar activity, as well as current helicity. In this study, we systematically analyze background bias and noise levels in SMFT observations from 1988 to 2019. Our dataset comprises 12,281 vector magnetograms of 1484 active regions. To quantify background bias, we computed mean values of Stokes $Q/I$, $U/I$ and $V/I$ over each entire magnetogram. The background bias of Stokes $V/I$ is small for the whole dataset. The background biases of Stokes $Q/I$ and $U/I$ fluctuate around zero during 1988–2000. From 2001 to 2011, however, the fluctuations in the background bias of both $Q/I$ and $U/I$ become significantly larger, exhibiting mixed positive and negative values. Between 2012 and 2019, the background biases shift to predominantly positive values for both Stokes $Q/I$ and $U/I$ parameters. To address this issue, we propose a potential method for removing the background bias and further discuss its impact on the estimation of current helicity. For each magnetogram, we quantify measurement noise by calculating the standard deviation ($\sigma$) of the longitudinal ($B_{\mathrm{l}}$) and transverse ($B_{\mathrm{t}}$) magnetic field components within a quiet-Sun region. The noise levels for $B_l$ and $B_{\mathrm{t}}$ components were approximately 15 Gauss (G) and 87 G, respectively, during 1988–2011. Since 2012, these values decreased significantly to ∼6 G for $B_{\mathrm{l}}$ and ∼55 G for $B_{\mathrm{t}}$, likely due to the installation of a new filter.

1. Introduction

The Solar Magnetic Field Telescope (SMFT) at Huairou Solar Observing Station (HSOS) is a 35 cm vacuum telescope equipped with a birefringent filter for wavelength selection and KD*P crystals to modulate polarization signals. The Fe I 5324.19 Å line is used for measurements. A vector magnetogram is built using four narrow-band (0.125 Å) filtergrams of Stokes $I,Q,U$ and V parameters. The center wavelength of the filter can be shifted and is normally at −0.075 Å for the measurements of longitudinal magnetic field and at the line center for the transverse magnetic fields [1]. The spatial resolution is approximately $2^{″}$, resulting from the local seeing effect. The measurement accuracy for the solar longitudinal magnetic field is about 10 Gauss (G), and the accuracy for the transverse magnetic field is about 150 G [2]. From 1988 to November 2001, observations were taken with a 512 × 512 pixel CCD. During December 2001 and 2008, the detector was a 640 × 480 pixel CCD. Since January 2009, observations have been taken with a 992 × 992 pixels CCD. From May 2011 to October 2012, SMFT underwent significant repairs and there were no regular observations during this period. Observations resumed in November 2012, with a new filter. A more detailed description of the instruments and observational techniques can be found elsewhere (e.g., Bao et al. [3]; Zhang et al. [4]).

Theoretical calibration of SMFT vector magnetograms was first made by Ai et al. [5]. Since then, several different methods of magnetic field calibration have been carried out. Wang et al. [6] used an empirical calibration and a velocity calibration method to calibrate the longitudinal magnetograms. Su and Zhang [7] used 31 points of the Fe i 5324.19 Å spectral line profile to derive the vector magnetic field by the non-linear least squares fitting technique. Bai et al. [8] improved the calibration process by fitting the observed full Stokes parameters using six points on the profile of the Fe I 5324.19 Å line and the analytical Stokes profiles under the Milne-Eddington atmosphere model, adopting the Levenberg–Marquardt least-squares fitting algorithm. Su et al. [9] summarized past historical experiences and introduced the standardized calibration procedure for vector magnetic field processing in detail.

The SMFT routine measurements of Stokes $I,Q,U$ and V parameters are being performed at a single-wavelength point. The vector magnetic fields are reconstructed under the weak-field approximation [10,11] using the following equations [5,12]:

$$\begin{array}{c}\hfill \begin{array}{c}{B}_{\mathrm{l}}=C_{\mathrm{l}}\left(V/I\right),\hfill \\ B_{\mathrm{t}}=C_{\mathrm{t}}{[{(Q/I)}^2+{(U/I)}^2]}^{1/4},\hfill \\ \varphi =0.5{tan}^{-1}(U/Q),\hfill \end{array}\end{array}$$

(1)

where $B_{\mathrm{l}}$ and $B_{\mathrm{t}}$ are longitudinal and transverse field components, $C_{\mathrm{l}}$ and $C_{\mathrm{t}}$ are corresponding calibration coefficients, and $\varphi$ is the azimuth angle.

SMFT has made significant contributions to the study of many scientific problems in solar physics. Using SMFT, Zhang [13] presented the vector magnetic field observations of active region NOAA 5395 and showed that the flare sites occurred near the magnetic islands and bays of opposite polarity and were associated with the change in the vector magnetic field. The active region NOAA 5395, observed in March 1989, was a prolific flare producer. It was the source of a solar flare that triggered geomagnetic storms, whose induced ground currents caused the collapse of Canada’s entire Hydro-Québec power system. Wang et al. [14] discovered that the vertical current can serve as a parameter for quantifying nonpotentiality and exhibits a strong correlation with solar flares. A comprehensive series of investigations has been conducted using SMFT data to examine the nonpotential characteristics of magnetic fields associated with the historically significant ‘Bastille Day’ flare of 14 July 2000 [15]. Magnetic (current) helicity can provide some information on the chirality and nonpotentiality of the magnetic field in the solar atmosphere. Seehafer [16] first discovered that most active regions show a negative sign of mean current helicity in the northern hemisphere and a positive one in the southern hemisphere. Subsequent statistical analyses using multiple vector magnetographs have consistently confirmed this initial helicity finding, though with minor discrepancies [17,18,19,20,21,22,23,24]. This tendency is commonly called the ‘hemispheric sign rule (HSR)’ of helicity. SMFT provided powerful data support for these helicity studies.

A commonly accepted view is that the magnetic field observed on the solar surface originates from the solar dynamo mechanism operating within the Sun’s convective zone [25]. However, the detailed structure and evolution of the Sun’s internal magnetic fields remain largely unknown, as the opacity of the solar atmosphere severely limits direct observations. Therefore, sustained high-precision observations of the photospheric magnetic field are crucial for probing the Sun’s internal structure and understanding the mechanisms that are driving solar activity cycles. In this study, we systematically evaluate background bias and the noise characteristics in the photospheric magnetic field measurements obtained by SMFT over a 32-year period (1988–2019). This comprehensive analysis establishes a reliable data foundation for subsequent investigations of current helicity and solar activity. Section 2 describes the observational dataset and outlines the methodology used to quantify both systematic background bias and measurement noise in the SMFT magnetic field measurements. Section 3 examines the temporal variations in background bias and noise levels, providing a detailed analysis of their evolution over time. Section 4 presents some discussion. Finally, Section 5 gives the conclusions.

2. Observations and Methods

2.1. The Observation

For our statistical analysis, we utilize 12,281 vector magnetograms covering 1484 active regions observed by SMFT between 1988 and 2019. To minimize the projection effect, we selected only those regions within $\pm {30}^{\circ }$ in both latitude and longitude. The yearly distributions of magnetograms (green) and active regions (blue) are shown in Figure 1. As introduced in Section 1, there are several calibration methods, and the calibration coefficients are different between them. Zhang [26] presented a comprehensive analysis of solar magnetic field observations, including calibration techniques and key challenges. In this study, we performed calibration of the vector magnetic fields by applying Equation (1). The longitudinal and transverse field calibration coefficients were set to $C_{\mathrm{l}}=8381$ G and $C_{\mathrm{t}}=6790$ G, respectively, as established by Su and Zhang [7].

2.2. The Method for Calculating Background Bias

Figure 2a is an example of SMFT vector magnetograms, which reveals that the transverse magnetic fields (red arrows) exhibit systematically strong signals with nearly identical or opposite orientations in regions distant from the sunspot. This coherent pattern, maintaining consistent directional alignment across large areas, clearly represents background bias rather than random noise. This bias may originate from the polarization signal. To quantify the background bias, we computed the mean values of Stokes $V/I$, $Q/I$ and $U/I$ across the entire magnetogram and denoted them as $V_{\mathrm{m}}$, $Q_{\mathrm{m}}$, and $U_{\mathrm{m}}$, respectively. In principle, these mean values should be zero in the field of view (FoV) encompassing complete active regions. For this active region, $V_{\mathrm{m}}=0.15\times {10}^{-3}$, $Q_{\mathrm{m}}=0.85\times {10}^{-3}$, and $U_{\mathrm{m}}=1.91\times {10}^{-3}$, respectively. $V_{\mathrm{m}}$ is small, which can be neglected, while $Q_{\mathrm{m}}$ and $U_{\mathrm{m}}$ are relatively large. The Stokes parameters were recalculated using the following equations:

$$\begin{array}{c}\hfill \begin{array}{c}{V}^{\mathrm{c}}/I=V/I-V_{\mathrm{m}},\hfill \\ Q^{\mathrm{c}}/I=Q/I-Q_{\mathrm{m}},\hfill \\ U^{\mathrm{c}}/I=U/I-U_{\mathrm{m}},\hfill \end{array}\end{array}$$

(2)

where $V^{\mathrm{c}}/I$, $Q^{\mathrm{c}}/I$, and $U^{\mathrm{c}}/I$ represent the Stokes parameters after removing background bias. The corrected magnetic field components were then recalculated using Equation (1). The corrected transverse magnetic field components, indicated by red arrows in Figure 2b, demonstrate the effective removal of the background bias. The residual fields now show physically meaningful patterns consistent with solar active region morphology.

2.3. The Method for Calculating Noise Level

The noise level is typically quantified through the standard deviation ($\sigma$) measured in quiet-Sun regions. To facilitate systematic analysis, we extracted four square regions (each $70\times 70$ pixels, blue squares in Figure 2a) from the magnetogram corners. For each region, we computed the standard deviations for both longitudinal ($B_{\mathrm{l}}$) and transverse ($B_{\mathrm{t}}$) magnetic fields. To minimize contamination by strong magnetic fields, the region exhibiting the minimum longitudinal field standard deviation (${\sigma }_{\mathrm{l}}$) among four corner regions was selected as the reference area for calculating both longitudinal and transverse noise levels in each magnetogram.

3. Results

3.1. Temporal Variation in Background Bias

Using the method described in Section 2.2, we estimated the background bias for each magnetogram. To examine the temporal variation in the background bias, we calculated annual median values as the representative yearly bias estimate. As shown in Figure 3a, the Stokes $V/I$ bias remained generally small throughout the entire dataset, with the exception of 3 notable outliers in 1996, 2003 and 2007. Figure 3b presents the annual variations in Stokes $Q/I$ background bias, revealing significant temporal fluctuations. During 1988–2000, the bias remained relatively small (mean absolute value: $0.23\times {10}^{-3}$) and predominantly positive, with 1997 being the sole exception. However, from 2001 to 2003, the bias shifted to negative values with substantially larger amplitudes, peaking at $-2.36\times {10}^{-3}$. The bias transitioned back to positive values during 2004–2007, reaching a maximum amplitude of $1.60\times {10}^{-3}$. From 2008 to 2011, it remained at a level of small negative values. Starting from 2012, the bias kept varying in the range between $0.87\times {10}^{-3}$ and $1.74\times {10}^{-3}$. A similar trend was observed in $U/I$ (Figure 3c). The bias remained small during 1988–2000 (mean absolute value: $0.22\times {10}^{-3}$), then shifted to large positive values in 2001–2003, peaking at $2.81\times {10}^{-3}$. From 2004 to 2011, the bias remained as small values except for the year 2007. To investigate this anomalous phenomenon observed in 2007, we conducted a manual inspection of all magnetogram records. No significant measurement errors were identified during this examination. Furthermore, our review of the maintenance logs revealed no relevant entries that could account for the observed phenomenon. Starting from 2012, the bias fluctuated between $0.38\times {10}^{-3}$ and $1.94\times {10}^{-3}$.

These biases in Stokes $V/I$, $Q/I$ and $U/I$ will affect both longitudinal and transverse magnetic field components through the calibration process using Equation (1). As above, we studied the annual median values, but this time on the derived magnetic fields rather than the Stokes parameters. Figure 4a demonstrates that the longitudinal magnetic field maintains consistently low bias levels, characterized by a mean absolute value of 3.5 G across all observational epochs. For the transverse field (Figure 4b), the bias variations remained relatively stable during 1988–2000, averaging at 137.7 G. However, significant fluctuations emerged between 2001 and 2011, ranging from 92.4 G to 417.9 G. During 2012–2019, the variations stabilized, maintaining values between 244 G and 315 G with a mean value of 295.8 G.

3.2. Temporal Variation in Noise Level

Prior to noise level calculation, we first remove the background bias. The noise level for each magnetogram is then estimated following the method outlined in Section 2.3. To analyze the temporal variation in noise level, we computed the annual median values as the representative noise level for each year. Figure 5a presents the annual variation in the noise level in Stokes $V/I$. From 1988 to 2009, the noise level fluctuated in the range between $1.11\times {10}^{-3}$ and $2.11\times {10}^{-3}$. However, a noticeable increase occurs in 2010 and 2011, coinciding with a major repair of the SMFT that affected observation quality. Beginning in 2012, a significant reduction in noise levels is observed, decreasing to within the range of (0.50–0.81) $\times {10}^{-3}$, most probably resulting from the installation of a new filter. As shown in Figure 5b,c, the yearly variation in noise level in Stokes $Q/I$ and $U/I$ is very simillar. Throughout the 1988–2011 observation period, the majority of annual noise levels remained below $2.20\times {10}^{-3}$, with mean values of $1.72\times {10}^{-3}$ for $Q/I$ and $1.78\times {10}^{-3}$ for $U/I$. Notably, the noise levels increase during 2009–2011, consistent with the trend observed in Stokes $V/I$. Since 2012, the noise levels in Stokes $Q/I$ and $U/I$ have both decreased to a mean value of approximately $0.70\times {10}^{-3}$.

The noise in Stokes $V/I$ propagates into the longitudinal magnetic field after calibration, while the noise in Stokes $Q/I$ and $U/I$ propagates into the transverse magnetic field. Figure 6a presents the annual variation in the noise level in the longitudinal magnetic fields. From 1988 to 2009, the noise level fluctuated in the range between 9 G and 18 G, remaining below 15 G for most of the years. However, a noticeable increase occurs in 2010 and 2011. An evident improvement in the noise level can be seen starting from 2012, with the levels falling to the 4–7 G range. Figure 6b displays the annual variation in transverse magnetic field noise levels. Throughout the 1988–2011 observation period, most annual noise levels remain below 100 G, with a mean value of 87 G. Notably, the transverse field exhibits a noise increase during 2009–2011, consistent with the trend observed in the longitudinal component. Since 2012, the noise level has fluctuated within the range of 44 G to 62 G, with a mean value of 55 G.

4. Discussion

The fundamental noise level of a telescope is typically determined by instrumental factors and observational techniques, while as a ground-based observational instrument, SMFT has its measurement noise level also critically dependent on atmospheric seeing conditions and transparency at the observatory site. Additionally, we adopted the standard deviation of the magnetic field in the corner region of magnetograms as an indicator of the noise level. If in regions with mixed strong and weak fields, the increased standard deviation primarily represents true magnetic field fluctuations rather than instrumental noise. The minor noise fluctuations observed in most years suggest they originate from seeing conditions and calculating methods, whereas the pronounced increase post-2009 and subsequent decrease after 2012 are primarily attributed to instrumental factors. During this period, the SMFT showed signs of performance degradation and underwent a filter replacement. A common technique for enhancing the signal-to-noise ratio is the multi-frame stacking method. The SMFT employs a 256-frame overlay method for routine observaton [3]. During data reduction, smoothing or binning techniques are typically employed to reduce the noise level in the measurements [20,27].

The bias caused only by instrument effects would be a stable quantity. We detected a temporal variation in the bias, indicating that this background bias may be caused by multiple sources—such as contaminants in the optical path, non-uniform distributions in imaging and polarization caused by telescope alignment error, and crosstalk between Stokes polarization parameters. We have conducted a thorough investigation of potential causes through multiple approaches, including (1) examination of telescope hardware components, (2) examination of the data acquisition software system, and (3) evaluation of the data calibration procedures. However, no significant errors were detected. This bias cannot be reliably identified through visual inspection of magnetograms. Identifying the fundamental causes of this bias remains a significant challenge. This bias can be reduced during the data reduction procedure. Other types of bias may be present in different telescopes. The east–west component of the magnetic field, $B_{\varphi }$ or $B_{\mathrm{p}}$, in low polarization signal areas changes its sign after the regions cross the central meridian was found in both the Vector Spectromagnetograph instrument (VSM) on the Synoptic Optical Long-term Investigations of the Sun telescope (SOLIS) [28] and the Helioseismic and Magnetic Imager (HMI) [29,30] aboard the Solar Dynamics Observatory (SDO) [31] magnetograms [32,33,34,35]. Leka et al. [36] performed novel investigations to quantify the bias, fully understand its source(s), and provided mitigation strategies.

To demonstrate the potential impact of background bias on magnetic field parameters, we present a comparative analysis of current helicity distributions both before and after bias removal. Figure 7a shows the vector magnetogram of active region NOAA 11654 observed by SMFT on 16 January 2013. The green and red arrows indicate transverse magnetic fields before and after bias correction, respectively. The transverse magnetic field is much smaller in the quiet region after background bias removal. Using the photospheric vector magnegotrams, we calculate the vertical current helicity as h_c = B_z· (∇ × ${\mathbf{B})}_{\mathrm{z}}$ [18,19], where B_z denotes the longitudinal magnetic field after deprojection, (∇ × ${\mathbf{B})}_z$ is the vertical component of the curl of the magnetic field vector $\mathbf{B}$. Figure 7b,c show the current helicity distribution before and after bias correction, respectively. The mean current helicity values are measured at $-0.19\times {10}^{-3}$ G²m⁻¹ (h_c1, before correction) and $-0.28\times {10}^{-3}$ G²m⁻¹ (h_c2, after correction), respectively. Figure 7d displays the residual current helicity distribution (h_c2 − h_c1), with a root-mean-square (RMS) value of $3.54\times {10}^{-3}$ G²m⁻¹. The discrepancy is evident in both weak and strong field regions. We selected 166 active regions observed by SMFT in 2013 for statistical study. The latitudinal profile of current helicity is shown in Figure 8. Prior to the removal of background bias, the positive current helicity predominates in both the northern and southern hemispheres, indicating a violation of the HSR in the northern hemisphere. After removing the background bias, negative current helicity dominates in the northern hemisphere, while positive helicity prevails in the southern hemisphere, more conforming to the HSR. After removal of background bias, 39 out of 166 active regions (23.5%) exhibit helicity sign reversal. This systematic difference suggests that the background bias likely contributes a positive component to the current helicity. When the intrinsic current helicity is weak, the systematic bias may be sufficient to alter its sign, potentially reversing the apparent chirality of the magnetic field structure.

On the one hand, magnetic fields and helicity play an important role in solar flares and coronal mass ejections. Park et al. [37] conjectured that the occurrence of the X3.4 flare is involved with the positive helicity injection into an existing system of negative helicity. Zhang et al. [38] suggested that there may be an absolute upper bound on the total magnetic helicity of all bipolar axisymmetric force-free fields. Thalmann et al. [39] demonstrated that the ratio of current-carrying helicity to total magnetic helicity shows a strong ability to indicate the eruptive potential of a solar active region. On the other hand, the current helicity on the solar surface is considered to be related to the $\alpha$-effect in the solar dynamo theory [40,41]. Therefore, it is essential to characterize and mitigate observational uncertainties in helicity measurements before investigating its physical properties. We will conduct a comprehensive analysis of solar-cycle variations in current helicity using SMFT observations.

5. Conclusions

In this study, we present a detailed analysis of the background bias and noise level in the SMFT photospheric magnetic field measurement over a 32-year observation period (1988–2019). Our results demonstrate that the background bias in Stokes $V/I$ ($B_{\mathrm{l}}$) maintains consistently low magnitudes with little temporal variation throughout the 32-year study period. In contrast, Stokes $Q/I$ and $U/I$ ($B_{\mathrm{t}}$) exhibit significantly larger temporal variations, particularly after 2001, with fluctuations exceeding 300 G in $B_{\mathrm{t}}$. These transverse-field biases may systematically affect current helicity calculations, potentially compromising the accuracy of solar cycle studies and investigations of solar activity. Although the physical origin of this bias remains unclear, we have developed an effective method to significantly reduce background bias in our measurements. The noise levels of both $B_{\mathrm{l}}$ and $B_{\mathrm{t}}$ components displayed a distinct two-phase evolution: (1) During 1988–2011, noise levels showed increasing trends with fluctuations, averaging approximately 15 G for $B_{\mathrm{l}}$ and 87 G for $B_{\mathrm{t}}$; (2) Post-2012, these values decreased significantly to ∼6 G for $B_{\mathrm{l}}$ and ∼55 G for $B_{\mathrm{t}}$, likely resulting from the installation of a new filter system.