Results

Following the example of previous work (e.g. Parnell et al. 2009), we use histograms to analyse the observed distribution of magnetic flux features. Figure 4 is an example for such a histogram. It shows the absolute frequency density of magnetic flux for magnetogram data taken on 16 March 2011, plotted on double-logarithmic axes. This example histogram highlights some general properties of magnetic flux histograms.

Figure 4: Histogram of frequency density of magnetic flux features detected in SDO HMI magnetogram taken at midnight on 16 March 2011 (red) and detected in a SOHO MDI magnetogram taken at approximately the same time (blue).

Firstly, over a considerable range of magnetic flux (1018–1020Mx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$10^{18},\mathrm{--},10^{20}~\mathrm{Mx}$\end{document}) the distribution seems to be well approximated by a straight line. In a double-logarithmic plot this is interpreted as an indicator of a power law distribution (e.g. Newman 2005; Parnell et al. 2009, see the Appendix for mathematical details).

Secondly, however, one notices that above 1020Mx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$10^{20}~\mathrm{Mx}$\end{document} the data becomes sparser and the straight-line behaviour is less evident. It is one of the aims of this paper to investigate the nature of the magnetic flux distribution over the full range of flux values; another aim is to find out whether and how the distribution varies over a solar cycle.

Thirdly, we notice a turnover in the lower flux tail (<1018Mx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$< 10^{18}~\mathrm{Mx}$\end{document}). For context we show in Figure 4 a comparison between the SDO HMI data and a SOHO MDI observation taken at the same time (for details how to account for the difference in measurement between MDI and HMI, see Liu et al. 2016). As one can see in Figure 4 the MDI based histogram has a turnover at a higher magnetic flux value due to the lower resolution of MDI compared to HMI, while otherwise the histograms seem to match closely. Although this is not a proof, we take this as an indication that the turnover in the HMI histograms is also caused by the limitation in the observational resolution of the instrument.

As the drop-off seems to be caused by instrumental effects, we have to exclude it if we are to obtain the true underlying distribution of magnetic flux features. To account for this we truncate the dataset to only include features with a magnetic flux greater than 1018Mx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$10^{18}~\mathrm{Mx}$\end{document} and re-bin the histogram. This limit is chosen somewhat arbitrarily but setting a hard limit is a more sensible approach than trying to determine separate limits on a case-by-case basis.

As already mentioned before, one notices that the truncated histogram data resemble a straight line in a double-logarithmic plot. This is usually interpreted as being indicative of an underlying power distribution function (e.g. Newman 2005; Parnell et al. 2009). Figure 5 shows histograms of the observed distribution of magnetic flux at different phases of the solar cycle, together with a power law model that has been fitted to the data using the maximum likelihood method (e.g. Scholz 2014).

Figure 5: Histogram of magnetic flux features and a fitted power law model at different times in the solar cycle. (a) 16 March 2011 representing rising activity. (b) 01 July 2014 representing solar maximum. (c) 16 August 2017 representing declining activity. (d) 16 August 2020 representing solar minimum.

One can see that the power law fits the data reasonably well in each case. However, it is also clear that in the higher flux tail the power law model starts to deviate from the observed distribution. The histogram values tend to be lower than the power law model; this is especially true during the declining phase and solar minimum. Therefore the question arises whether a power law is really the optimal mathematical model to represent the data or whether other distribution functions might represent the data better.

We have picked a (single) power law based on a purely subjective interpretation of the data. Clearly, more sophisticated means are needed to make an objective decision on whether or not a mathematical model is an accurate representation of the true distribution that the data come from. In this paper we use a number of different methods to achieve this.

We also want to investigate whether there might be alternative models which perform better than the single power law over the full solar cycle. Therefore, in addition to the single power law, we will test eight different probability density functions (PDFs) against the data: Exponential PDFLognormal PDFWeibull PDFTruncated Weibull PDF (abbreviated as Tr. Weibull from now on)Sharp Double Power Law PDF (abbreviated as Sharp-DPL from now on)Smooth Double Power Law PDF (abbreviated as Smooth-DPL from now on)Weibull-Lognormal PDFTruncated Weibull-Lognormal PDF (abbreviated as Tr. Weibull-Lognormal from now on) The mathematical details for each of these distributions can be found in the Appendix. We remark that apart from the single power law a few more of these distribution functions (exponential, lognormal, and Weibull) have been discussed in the context of magnetic flux distributions before (see e.g. Muñoz-Jaramillo et al. 2015).

We start our investigation of these model PDFs by first using a better graphical representation to assess how well a model distribution function fits the data called probability-probability plots (P-P plot). A P-P plot displays a model’s cumulative distribution function (CDF) against the empirical cumulative distribution function (e.g. Gibbons and Chakraborti 2020) on the unit square. A good fit between the model CDF and the empirical CDF would lead to a curve very close to the main diagonal. A typical example of P-P plots for all nine model distribution functions is shown in Figure 6 for a date around solar maximum (1 July 2014). From Figure 6 it is clear that the exponential, the lognormal, and the Weibull distribution do not seem to represent the data as well as the other six distribution functions. However, although P-P plots give a better impression of the quality of fit for a model probability density function, the results are still subjective. Furthermore, applying this method to all datasets in our sample would not be practical.

Figure 6: Example P-P plots for the nine model PDFs considered in the paper. These plots use data from 1 July 2014, which is around solar maximum. The solid line in each plot is the main diagonal and the dashed line is the curve generated by plotting the theoretical CDF against the empiral distribution function. See main text for more details.

Clauset, Shalizi, and Newman (2009) suggest that one way to check if a model is suitable or not is to determine the model’s ‘plausibility’ by generating a p_p_-value based on some goodness-of-fit statistic. We follow their approach by generating the so-called p_p_-value based on the Kolmogorov-Smirnov statistic (KS Stat). The KS Stat is a measure of the maximum difference between a model’s cumulative distribution function and the observed data’s empirical distribution function (e.g. Chakravarti, Laha, and Roy 1967).

The p_p_-value is determined by fitting a model to the data and calculating the KS Stat for the model. Synthetic datasets are then generated based upon the fitted model’s parameters, before refitting the model to the synthetic dataset and calculating a new KS Stat. The p_p_-value is the fraction of data sets that have a larger KS Stat than the original data. Again, following the example of Clauset, Shalizi, and Newman (2009), a model is rejected as implausible if the p_p_-value is below 0.1.

In Table 1 we provide the p_p_-values for all nine model probability density functions for the same data used in the P-P plots in Figure 6. One can see that the p_p_-values for the Exponential, Lognormal, and Weibull distributions are zero and hence below the threshold values of 0.1, which corroborates the conclusion that these three PDFs do not represent the data well and should be rejected.

Distribution | p-value
Single Power Law | 0.4
Exponential | 0
Lognormal | 0
Weibull | 0
Tr. Weibull | 1
Sharp-DPL | 1
Smooth-DPL | 1
Weibull-Lognormal | 0.2
Tr. Weibull-Lognormal | 1

Obviously, one should not base the rejection of a probability density function on a comparison with a single data set. Therefore we have calculated the p_p_-values for all models over the complete data set covering the full solar cycle.

It turns out that the p_p_-values of the Exponential, Lognormal and Weibull PDFs are very close to zero throughout the complete data set. Plots of the p_p_-values for these PDF models against time do not convey any additional information and are hence omitted. On the basis of this result we will discard these three PDF models from the analysis.

Figure 7 shows the temporal variation over the solar cycle of the p_p_-value for each of the remaining six models. In each plot, the 0.1 significance level is shown as a dashed line. For plausibility, a model’s p_p_-value must exceed 0.1.

Figure 7: Variation of pp-values for the remaining six PDF models over the full solar cycle. The dashed line shows the 0.1 significance level which plausible models must exceed.

For the Truncated Weibull, Sharp Double Power Law, Smooth Double Power Law and Truncated Weibull-Lognormal model PDFs, the p_p_-value is safely above the 0.1 threshold for all magnetograms in the sequence. We conclude that each of these models are plausible fits to the magnetic flux distribution at all times in the cycle.

However, both the Single Power Law and Weibull-Lognormal models drop below 0.1 on occasion. Interestingly, the Single Power Law’s p_p_-value shows correlation with the solar cycle, increasing as magnetic activity increases, and the Weibull-Lognormal model appears to be anti-correlated, decreasing as magnetic activity increases. This suggests that the Single Power Law may be a good fit at solar maximum but fails at solar minimum and likewise, the Weibull-Lognormal may be a good fit at solar maximum but may fail at solar minimum.

We therefore omit both the single power law and the Weibull-Lognormal PDFs from the further analysis and accept the other four PDFs as plausible fits to the data.

To be able to make a choice which of the remaining four PDF models is the most suitable, we compare a number of different goodness-of-fit tests across the full solar cycle. We have based our choice of goodness-of-fit test statistics on those employed by Muñoz-Jaramillo et al. (2015); namely the KS Stat, the relative Akaike Information Criterion (Rel. AIC), and the Akaike weights (Aw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{w}$\end{document}). However, we have added a fourth test which measures the relative log-likelihood of each model (Rel. Log-Lik).

The goodness-of-fit statistics are summarised in Tables 2 and 3 which show the average value, over the full solar cycle, of each test statistic for all four models, and the percentage of time that each model wins a particular goodness-of-fit comparison, respectively. It is clear from these tables that the smooth double power law is the best performing model for most of the solar cycle. It is difficult to decide whether the sharp double power law PDF or the truncated Weibull-lognormal PDF follow in second place based on the win percentages alone, but the truncated Weibull-lognormal PDF comes out ahead on average values. Finally, it is clear that the truncated Weibull PDF is the worst performer over the full cycle.

Model | Average value
KS Stat | Rel. AIC | Aw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{w}$\end{document} | Rel. Log-Lik.
Tr. Weibull | 0.0073 | 19.532 | 0.036 | −11.804
Tr. Weibull-Lognormal | 0.0062 | 7.469 | 0.176 | −2.772
Sharp-DPL | 0.0082 | 10.497 | 0.211 | −6.286
Smooth-DPL | 0.0055 | 1.966 | 0.577 | −1.020

Model | Win percentage (%)
KS Stat | Rel. AIC | Aw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{w}$\end{document} | Rel. Log-Lik.
Tr. Weibull | 20.00 | 2.69 | 2.69 | 0.00
Tr. Weibull-Lognormal | 25.77 | 13.07 | 13.07 | 26.54
Sharp-DPL | 6.54 | 21.92 | 21.92 | 6.92
Smooth-DPL | 47.69 | 62.31 | 62.31 | 66.54

Although the smooth double power law does not win all goodness-of-fit comparisons 100% of the time, it seems sensible to select the model which performs best on average. All four models are plausible fits to the data, but we wish to make a definitive choice in order to perform statistical inference. Inferences about the data may come in a number of different forms, for example, using the model to make future predictions of the long-term trends in the data; to provide a physical constraint in numerical models of the sub-surface processes; or to achieve a better theoretical understanding of the underlying solar dynamo processes.

Therefore we choose the smooth double power law as the best representation of the true distribution of magnetic flux features over the course of a full solar cycle. Figure 8 shows the histograms of magnetic flux distribution representing the same four phases of the solar cycle as in Figure 5 but with a smooth double power law model overplotted on the histogram rather than a single power law model. The smooth double power law’s flexibility allows it to capture the behaviour at the high flux tail to a much higher accuracy than the single power law. We remark that this result is largely in accordance with the “two-segment” power law distribution put forward by Song et al. (2024), although their distribution seems to correspond to what we call sharp double power law in this paper.

Figure 8: Histograms of the same data as in Figure 5 but with a smooth double power law model fitted to the data rather than single power law model.