A NEW METHOD TO QUANTIFY AND REDUCE THE NET PROJECTION ERROR IN WHOLE-SOLAR-ACTIVE-REGION PARAMETERS MEASURED FROM VECTOR MAGNETOGRAMS

Full-disk, high-cadence observations of Helioseismic and Magnetic Imager (HMI: Schou et al. 2012) vector magnetograms (Bobra et al. 2014) allow us to consistently measure magnetic parameters of solar active regions (ARs) even when they are far from the disk center. However, vector magnetograms away from the disk center suffer from projection errors. Projection errors increase with distance from the solar disk center. Based on a limited data sample of 36 AR vector magnetograms, Falconer et al. (2006) concluded that the projection error in whole-AR magnetic flux measured from line-of-sight (LOS) magnetograms of ARs within 30° from the solar disk center can be corrected by dividing it by cos²θ, θ being the heliocentric angle of the AR's position. Thus, for measuring AR magnetic flux, a LOS magnetogram can be used for ARs observed within 30° of the disk center by applying the above correction, but additional projection errors become significant beyond 30° and deprojection of vector magnetograms to the disk center becomes necessary.

In Figure 1, we show LOS and vertical-field components of HMI vector magnetograms of AR 11944 at seven different positions/times during its disk passage. The LOS component of AR magnetograms far from the disk center shows fictitious neutral lines, which occur when there is a change of polarity in the LOS component of the magnetic field but not in the vertical component of the magnetic field. Fictitious polarities always occur on the limb side of sunspots far from the disk center and are typically crescent-shaped, as seen in Figure 1. These artifacts are a projection effect.

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Several deprojection problems degrade the measurement of total magnetic flux. These include ambiguity resolution, large transverse-field noise, and foreshortening.

The transverse magnetic field direction in a vector magnetogram has an 180° ambiguity. This comes from the transverse-field direction measured from the linear polarization of light, which has only a 180° range. Several techniques have been developed to resolve this ambiguity (Metcalf et al. 2006; Leka et al. 2009; Georgoulis 2012). We use the HMI definitive data, already disambiguated by the minimum energy method (Leka et al. 2009; Borrero et al. 2011). We expect that the disambiguation error increases with AR distance from the disk center.

The noise in the LOS field in HMI vector magnetograms is 5–10 G, while the transverse-field noise is ∼100 G (Liu et al. 2012; Hoeksema et al. 2014). The farther the AR is from the disk center, the greater the component of the observed transverse field in the vertical field of the deprojected magnetograms, increasing the noise in the (deprojected) vertical magnetic field. This produces an overestimate of the AR's total magnetic flux. This error increases with increasing radial distance of the AR. This transverse-field noise effect can be clearly seen in Figure 1, where deprojected HMI active-region patches (HARPs) from original HARPs far from the disk center show obvious light and dark gray noise in quiet regions, which does not occur when the original HARP is near the disk center, such as on 2014 January 8 in Figure 1.

There is also a projection error due to foreshortening, which causes an underestimate of the AR's total magnetic flux. This error results from flux of both polarities being in the pixels that are on the polarity inversion lines in the LOS field. The farther the AR is from the disk center, the larger the area of the Sun that is covered by these pixels, and the greater the amount of lost flux. By blurring these pixels, the point-spread function (PSF) further increases this flux-loss (Yeo et al. 2014). The flux-loss is further increased with distance from the disk center by the increasing occurrence of false polarity inversion lines in the LOS magnetograms.

Some methods developed in the past to remove projection errors by transforming the AR's vector magnetogram to the disk center (Venkatakrishnan & Gary 1989; Venkatakrishnan et al. 1988; Gary & Hagyard 1990) retain some significant projection errors for ARs beyond 45° from the disk center. An alternative way to cope with projection error is to quantify the radial-distance dependence of the projection error in a measured whole-AR magnetic parameter and then correct for it. This is what we do in this Letter, i.e., we establish a method to quantify and reduce the net projection error in the magnitude (absolute value) of any whole-AR magnetic parameter measured from deprojected vector magnetograms. To demonstrate this method, we have used large-flux ARs (flux >10²² Mx) and one of the simplest magnetic parameters: AR total magnetic flux. We have used HMI vector magnetograms in the present demonstration but the method should work for any vector magnetograph.

We use the MAG4 (short for Magnetogram forecast) tool to measure the total magnetic flux of ARs. MAG4 is a code that automatically downloads and processes HMI magnetograms to forecast major flares, coronal mass ejections (CMEs), and solar proton events (Falconer et al. 2011, 2012, 2014). In the summer of 2015, MAG4 was upgraded to process vector magnetograms. Before that MAG4 used only LOS magnetograms. Now MAG4 first converts a vector magnetogram into vertical and horizontal field-vector components. This magnetogram is still in plane-of-sky pixels. MAG4 then resamples the magnetogram and creates square uniform pixels that map the surface of the Sun at the disk center using the method of Venkatakrishnan et al. (1988). Due to the relatively small size of HARPs, MAG4 does not take the spherical curvature of the Sun into account. We calculate the potential field from B_z (Alissandrakis 1981). In this Letter, we use a sample of 30,845 definitive-data HARP vector magnetograms. Although MAG4 measures many whole-AR magnetic parameters, here we establish our new method by concentrating on one parameter, AR total magnetic flux, which is the the most straightforward and least erroneous parameter to measure. The total magnetic flux is also chosen because it is one of the slowest evolving parameters of a mid-life AR.

The total magnetic flux is given by

$$\begin{eqnarray*}&&{\rm{\Phi }}=\int | {B}_{{\rm{z}}}| {da},\end{eqnarray*}$$

where B_z is the vertical magnetic field and the integral is over all pixels of the deprojected HARP having ∣B_z∣ >100 G. The magnetic centroid of the deprojected HARP and the distance of that point from the disk center in the plane-of-the-sky HARP are also determined. Since MAG4 only analyzes strong-field ARs, we use two other parameters measured from the deprojected HARPs to select strong-field ARs. One is the magnetic area

$$\begin{eqnarray*}&&A=\int {da},\end{eqnarray*}$$

where the integral is over the same pixels used for Φ, and the other is the length of strong-field neutral line

$$\begin{eqnarray*}&&{L}_{{\rm{S}}}=\int {dl},\end{eqnarray*}$$

where the integral is along all neutral-line intervals on which the potential horizontal magnetic field is greater than 150 G and separates opposite-polarity vertical magnetic fields no weaker than 20 G.

For the present Letter, we use the MAG4 HMI database, which has only ARs that are observed by HMI and that qualify as so-called strong-field ARs. MAG4 uses only strong-field ARs in order for the horizontal field on much of the AR neutral line to be well above the noise level, for low-noise evaluation of free-energy proxies given by integrals of the horizontal field along the strong-field intervals of the neutral line. Following all of our previous work in developing MAG4 (Falconer et al. 2011, 2012, 2014, and our foundational earlier work cited therein), we define a strong-field AR to be any AR for which the length L_S of strong-field neutral line exceeds 75% of the square root of the area A of the vertical-field flux Φ of vertical field stronger than 100 G : L_S/A^0.5 > 0.75.

For this Letter, only ARs with Φ > 10²² Mx are used. The 10²² Mx threshold was chosen because such large-flux ARs are the primary drivers of severe space weather. The same method can be applied to smaller-flux ARs, using a large set of smaller-flux (Φ < 10²² Mx) ARs to obtain the center-to-limb correction curve for those ARs.

Note that we have measured whole HARPs for the analysis. Some of these HARPs contain more than one NOAA AR. We measure all of the strong-field regions in a HARP, treating them as a single AR. This does not affect the results presented in the current paper. Accordingly, in this paper we often refer to a HARP as an AR.

The plots of measured magnetic flux Φ of five ARs through their disk transits are shown in Figure 2, each of which shows an upside-down U profile and an artificial 24 hr oscillation (Hoeksema et al. 2014 and Couvidat et al. 2016). Note that the x-axis is the distance from the disk center. The ARs in the east are given a negative distance so as not to have the east and west tracks overlay. Since none of these ARs transited through the disk center, none of the ARs reached R = 0. For each AR, due to the projection error from foreshortening, the measured values of the total magnetic flux are much smaller near the limbs than those near the disk center. The selected five ARs showed little evolution in the total flux during central disk passage. In other words, their fluxes did not rapidly grow or decay. The upside-down U profile is common to all ARs, including those that do show a real trend of growing or shrinking in size. The upside-down U profile makes using the time derivatives of an AR's flux problematic for analysis. For example, for an AR that is far east of the central meridian, we would not know if the AR's flux were actually growing as the AR rotated farther from the limb.

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We quantify the average radial dependence of the projection error using a large set of deprojected HARP magnetograms that yield disk-passage measurements of the total magnetic flux Φ of each of 272 HARPs. For each HARP, we average the measured flux in the strong-field area of that HARP over the HARP's magnetograms taken during the time when the magnetic centroid of the HARP's strong-field area is within 100'' of the central meridian. Then, each measured Φ from that HARP during its disk passage is divided by that average flux. Figure 3 (left and middle panels) shows a 2D histogram of the distribution of this flux ratio from a sample of 30,845 individual HARP magnetograms, all with a strong-field flux greater than 10²² Mx. Unlike in Figure 2, in Figure 3 we leave the distance from the disk center unsigned.

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In Figure 3 (left panel), the yellow solid curve shows the average value of Φ in each of 20 equal radial-distance bins. It has a slight rise starting near 0.4 R_S or 25° (before the 30° leftmost vertical line) and then falls back to 1 by 45° (the middle vertical line). At 60° the solid yellow curve shows that the uncorrected total magnetic flux is 2/3 of the true value. The dashed yellow lines show the 1σ standard deviation, (roughly 2/3 of the data points are within the two dashed lines). This standard deviation should not be thought of as an error. As described later, nearly all of the spread is due to the growth or decay of the ARs. Most ARs are not as steady as those shown in Figure 2.

In Figure 3 (left panel), a Chebyshev fit is done to all the data points, using the first five even terms (Press et al. 1992, p. 190). This fit is shown by the green dotted curve. The Chebyshev fit matches the bin-average curve and provides a continuous function, which is used for applying the correction. The fit shows that Φ is slightly overvalued in the ∼30°–45° distance from disk center, and is significantly undervalued beyond 50°. The values of the Chebyshev coefficients of the fit are 0.0049, 0.00087, −0.0016, 0.00084, and −0.000081.

Figure 3 (middle panel), shows the corrected distribution with the standard deviation of the corrected distribution around the average curve. The average of the corrected AR flux values shows hardly any deviation from 1.

The standard deviation of the corrected distribution grows with the distance from the disk center. The predominant cause of the growth of the standard deviation is AR evolution (Figure 3, right panel). To estimate the average growth/decay of the magnetic flux of our ARs, we use the same sample during the inner 45° disk passage where the correction factor is negligible. We normalized these not by the central meridian passage, but by the furthest east as well as the furthest west measured AR Φ; so each HARP magnetogram between these two extreme positions is used twice. In effect, we are treating the easternmost point as the central meridian to simulate evolution in the western hemisphere, and treating the westernmost point as the central meridian to simulate evolution in the eastern hemisphere. These are then folded, in a similar way as the other two panels of Figure 3. The absolute time relative to the easternmost or westernmost magnetogram is plotted on the horizontal axis. This gives the flux evolution of our 272 ARs plotted in Figure 3 (right panel). In 3–4 days, the standard deviation grows to the size seen at 60 heliocentric degrees in the Figure 3 (middle panel). Comparison of the scatter in the right panel with that in the central panel of Figure 3 indicates that most of the scatter in the central panel is from AR evolution and that little (∼20% or less) in the central panel is from variance in the projection error among the 272 ARs.

For the five AR HARPs of Figure 2, we show the corrected flux in Figure 4. Artificial 24 hr oscillation remains in the corrected flux. We can see that within 60° the ARs show no inverted-U trend during the central disk passage. For tracking the evolution of an AR's total magnetic flux, this gives us the confidence to use the magnetic flux measured from deprojected vector magnetograms of the AR out to 60° degrees, from which we have removed the radial-distance average projection error shown in Figure 3. Very close to the limb (R > 0.95 R_S), all ARs show a rapid fall in Φ. This is due to extreme foreshortening near the limb and also due to the ARs taking a finite time to rotate on or off the disk, effects that are not corrected by our method.

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A deprojected vector magnetogram of an AR is not exactly the same as the vector magnetogram of the AR if it were observed on the solar disk center. This is due to foreshortening (a pixel nearer the limb covers a larger area), due to the PSF, due to large transverse-field noise that increasingly increases the noise in the vertical component of the deprojected magnetogram of ARs increasingly nearer the limb, and due to error in 180° ambiguity resolution for ARs near the limb. These combined effects result in the measured AR uncorrected total magnetic flux being 2/3 of the true value at 60° from disk center (Figure 3). Between heliocentric angles of 30° and 45° there is a slight overvaluation of the total magnetic flux of an AR. Our interpretation of this is that transverse-field noise (and perhaps disambiguation error) causes the average curve in Figure 3 to rise with distance from the disk center and that these projection effects are overcome and dominated by foreshortening error far enough from the disk center, resulting in the fall of the average curve with increasing distance from the disk center beyond about 40°. This interpretation was confirmed by projecting a deprojected vector magnetogram of AR 11944 to different longitudes, converting back to LOS and transverse magnetic field maps, and applying the PSF to these maps, then deprojecting the magnetogram, and measuring the total magnetic flux. The center-to-limb curve obtained was qualitatively the same as that in the left panel of Figure 3 (showed the same trends) but was quantitatively different. To do this more correctly would require constructing the Stokes-parameter maps for the projected magnetograms with PSF applied (i.e., construct what the HMI vector magnetograph would have recorded).

The projection error curve in Figure 3 was found by using a large sample of ARs, assuming that the average normalized true total magnetic flux of the ARs is not a function of AR position on the disk. From this curve, the average projection error in an AR's total magnetic flux can be removed as in Figure 4. This correction is good out to at least 60 heliocentric degrees from the disk center and is needed for the improvement of scientific studies, such as, for example, studies of evolutionary changes in AR magnetic flux.

Please remember that the correction curve we obtain from the method presented here is based on the average total flux of a large number of ARs during their disk-passages. The basic assumption is that the center-to-limb changes in Φ from the real evolution of ARs average out, e.g., that an equal number of ARs grow or decay during their disk-passages. Given the large number of ARs in our sample, this assumption seems appropriate. To the degree that this assumption is true, an increase or decrease in the corrected Φ out to 60° from the disk center is due solely to the AR's growth or decay. We have also ignored the small east–west asymmetry in the HMI magnetograph (Hoeksema et al. 2014).

After the correction is applied, there is still the error due to the 24 hr artificial oscillation shown in Figures 2 and 4, which is a few percent and is practically negligible for some applications. This can be corrected by a fitting (e.g., see Hoeksema et al. 2014 and Couvidat et al. 2016) that we do not apply here.

By correcting for the projection error and filtering out the 24 hr dependence, the magnitude of an AR's Φ can be well determined out to 60° from the disk center. Most of the standard deviation in Figure 3 (central panel) is due to AR evolution. The projection-error correction shown in Figure 3 is especially needed if the AR's time derivative of Φ is of interest. Figure 3 indicates that without the correction applied, only ARs within 45 heliocentric degrees can be used without much concern.

Although in this Letter we present the method by applying it to the simplest-to-measure magnetic parameter of an AR, the total magnetic flux, the method is suitable for correcting the magnitude (absolute value) of any whole-AR magnetic parameter, e.g., the various size, twist, and free-energy proxies studied in Tiwari et al. (2015) and Bobra & Couvidat (2015). Correcting free-energy proxies out to 60° from the disk center is needed to improve forecasts of an AR's CME/flare productivity.

In conclusion, we have presented a new method to quantify and reduce the projection error in whole-AR parameters that are measured from deprojected vector magnetograms. In this Letter, we have established this method by applying it to the total strong-field magnetic flux of 30,845 HARPs of 272 large-flux (Φ > 10²² Mx) ARs. We show that, with the correction applied, this parameter can be reliably used for scientific analysis of large-flux ARs observed up to 60° away from the solar disk center. We have found that the method works well for finding the center-to-limb correction curves for the absolute value of many other whole-AR magnetic parameters, including several free-energy proxies and signed parameters such as whole-AR magnetic twist. These curves will be presented in a more extensive future publication.

We thank the referee for leading us to make several improvements in the paper. Support for MAG4 development came from NASA's Game Changing Development Program, and the Johnson Space Center's Space Radiation Analysis Group (SRAG). In particular, the authors want to gratefully acknowledge the continued support and guidance of Dan Fry (NASA-JSC) and David Moore (NASA-LaRC). D.F. acknowledges support from NextGen Federal Systems for the completion of this work. S.K.T. is supported by an appointment to the NASA Postdoctoral Program at the NASA/MSFC, administered by USRA through a contract with NASA.