Data

We use a sequence of 260 full disk line-of-sight magnetograms from the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO) taken at midnight on the 1st and 16th day of each month from 01 May 2010 to 16 March 2021. The dates were chosen arbitrarily but they provide at least half a Carrington rotation (27.27 days) between subsequent observations meaning that for each month we get two completely independent views of the photosphere. Three exceptions to these dates are 16 September 2010, 16 December 2011 and 16 March 2021 where we take the map at 00:12:00 the same day, 23:12:00 the day before and 00:12:00 the same day, respectively. This is due to a lack of good data at midnight of those days. This gives us enough data to span almost a full solar cycle. Magnetograms measure the magnetic flux density (in Mx cm−2) on the solar photosphere, however we use the magnetic flux density as a proxy for the average magnetic field strength at each pixel (measured in Gauss, G). Each magnetogram is a 4096 × 4096 pixel image with a pixel area of 0.5 × 0.5 arcsec2. The data is corrected from line-of-sight to radial magnetic field strength using a radial cosine correction. Each pixel is multiplied by the area of the Sun which it represents in order to convert the data into a measurement of radial magnetic flux. To avoid inaccuracies in radial magnetic flux near the limb we ignore all pixels further than 60∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$60^{\circ}$\end{document} in latitude or longitude from the center of the Sun.

Feature Identification Method

Comparison of Different Methods

In order to identify and isolate individual magnetic features in a magnetogram we use a modified version of the clumping algorithm developed by Parnell (2002). The clumping algorithm defines a magnetic feature as a “contiguous group of same-signed pixels with absolute values greater than a lower cutoff”. These groups of pixels are known as ‘flux massifs’ as they are analogous to a mountain massif. There are two other common feature detection algorithms (DeForest et al. 2007), both of which have different definitions of magnetic features. The first of which, known as the downhill method (Welsch and Longcope 2003), finds flux peaks rather than massifs. The algorithm divides massifs into their individual ‘summits’ along saddle lines. The last method, the curvature method (Strous 1994; Hagenaar et al. 1999), finds flux cores by identifying local extrema and grouping together contiguous pixels that form a convex surface around the extrema. Figure 1 (a) shows a cartoon illustration of the differences in identification of features by the three different methods. We decided to employ the clumping algorithm as we feel it is the most robust to changes in the resolution of data being analysed. We decided against the downhill and curvature methods as they are sensitive to the number of local extrema. Increasing resolution tends to increase the number of individual ‘peaks’ within a flux massif and, hence, the downhill and curvature methods will identify many smaller features rather than the flux massif itself. We believe the flux massifs are the best representation of individual magnetic features. The clumping algorithm holds up well to increases in resolution as it does not split up the features present in lower resolution data. Therefore, any newly identified features would be a result of the increased resolution only. In addition, the curvature method is disregarded as it also fails to identify any features that have a ‘flat’ top due to their lack of curvature as is shown in Figure 1. This may be an uncommon occurrence but is worth taking into account.

Figure 1: (a) A cartoon illustration of three common feature identification algorithms. The clumping method, as originally devised by Parnell (2002), uses a single lower cut-off and identifies two separate features. The downhill method splits the larger feature into two separate features at the local minimum (in 3D this would be a saddle point). The curvature method only identifies two small features; the peak at the right hand side is too flat for the method to identify. (b) A cartoon illustration of the filtering process carried out by the separate thresholds. The left hand peak is created by ‘noisy’ pixels and should therefore be neglected. The clump cutoff, which is the same as the lower cut-off in panel (a), achieves this, however, it removes some pixels which we believe are genuine constituents of the feature at the right hand side. Expanding the feature out to the growth cutoff allows us to obtain better estimates of what we believe to be the feature’s true size.

Modification of Clumping Algorithm

We have modified the clumping algorithm by allowing for a second threshold (known as the growth cutoff) which is lower than the original threshold (known as the clump cutoff). The growth cutoff allows us to expand the identified features by including any bordering pixels that have an absolute value greater than the growth cutoff. We do this so we can identify features more accurately by including all pixels which contribute to the feature and avoid any ‘false positive’ features. We set the clump cutoff high enough to avoid identifying ‘noisy’ pixels as features but in doing so we omit any contributing pixels which are below the threshold. The growth cutoff allows us to incorporate these pixels into their respective features. The two threshold modification can alternatively be viewed as the growth cutoff acting as the original threshold which needs to be satisfied and the clump cutoff acting as a filter which tries to remove any ‘false positives’. Figure 1 (b) shows a cartoon illustration of this filtering in practise. As a final step we neglect all identified features which do not consist of at least five contiguous pixels.

Choice of Thresholds

The two cutoff values are applied as a means to neglect noisy pixels; therefore, we use the noise levels of the magnetograms themselves to assign the cutoff values. We estimate the noise level of the magnetograms following the technique of Parnell (2002). We fit a Gaussian curve to the core of a histogram of the pixel values assuming that the core of the histogram is associated with the noisy pixels. The range of the noisy pixels is then given as the full-width at half-maximum (FWHM) of the Gaussian fit and the noise level assumed to be the half-width at half-maximum (this accounts for the sign of the individual pixels). To give an illustration of the method, Figure 2 (a) shows the histogram of the magnetogram taken on 01 May 2014 with the Gaussian fit superimposed. Also shown is the FWHM (dotted). In this case the noise level is approximately 7.6 G. To determine the thresholds we perform a survival analysis; a statistical technique which can be used to determine the proportion of a population which will survive past a certain time. In our case we can use survival analysis to determine the fraction of pixels which are associated with noise. Figure 2 (b) shows the survival function for both the histogram and the fitted Gaussian (dashed and dash-dotted, respectively). The survival function is defined as 1−F(x)1 - F(x), where F(x)F(x) is the cumulative distribution function for the observed distribution. The ratio of the survival functions of both curves (solid) tells us what fraction of pixels with field strength above a certain value are associated with noise. From the figure we see that for pixels with magnetic field strength greater than 3 times the noise of the magnetogram (22.8 G) less than 1% of these are associated with noise. We therefore choose the clump threshold to be 3 times the noise level and allow for growth of features back to a growth threshold set at 2 times the noise level. However, the noise level is different for every magnetogram and we want to ensure uniformity of feature identification across the full solar cycle.

Figure 2: (a) A histogram (black solid line) of the magnetic field strength of the pixels is created. A Gaussian curve (blue solid) is fitted to the core of the histogram (any bins with count greater than or equal to 0.5). The full width at half maximum (FWHM) is also plotted (dotted). The noise of the magnetogram is estimated as the half width at half maximum (HWHM). For this magnetogram the noise level is approximately 7.6 G. (b) The survival function of the histogram (dashed) and the Gaussian curve (dash-dotted) are plotted. The ratio of the two curves (solid) indicates the fraction of the pixels above a certain magnetic field strength that are associated with noise. From the graph we observe that for a threshold of 3 times the noise level (22.8 G), less than 1% of the pixels above the threshold are associated with noise. This allows us to comfortably set a clump threshold of 3 times the noise level and a growth threshold of 2 times the noise level to better estimate the true size of the features.

Figure 3 shows the value of noise for each magnetogram in our data set. The first thing to note is the significant drop in noise level during April 2016. This is caused by a change in HMI’s observing scheme, known as “Mod-L”, to now combine measurements from two separate cameras. The result of this change is that the noise of magnetograms from HMI is significantly reduced (see Liu et al. 2016, for more details). The average values of the noise before and after the switch are approximately 7.993 G and 5.974 G, respectively. There is no significant change in the noise levels over time, therefore we can assume that using the average noise value before and after the drop as a proxy for the true noise is acceptable and provides uniformity across all maps. We then calculate the two thresholds, clump and growth, by multiplying the average noise level by 3 and 2, respectively. We apply these thresholds to the modified clumping method outlined in Section 2.1.2 and determine all magnetic features present on each magnetogram.

Figure 3: The noise level across the full data set. The drop in noise level is caused by a change in the way HMI takes measurements (Liu et al. 2016) which reduced the noise associated with magnetograms. The dashed lines show the average value of the noise level before and after the drop; approximately 7.993 G and 5.974 G, respectively. There is no obvious correlation between noise level and solar cycle so we use the average noise values as the thresholds for all maps to ensure uniformity in the feature detection.