STATISTICAL PROPERTIES OF SOLAR ACTIVE REGIONS OBTAINED FROM AN AUTOMATIC DETECTION SYSTEM AND THE COMPUTATIONAL BIASES

The four-step identification method discussed above involves a set of four and only four controlling or thresholding parameters. We have used the set of I_K = 250 G, S_O = 10 Mm, I_A = 50 G, and S_C = 10 Mm to illustrate the processes as shown in Figure 1. Apparently, a different set of controlling parameters would produce a different detection result, in terms of not only the number of ARs, but also the size of the regions detected. While an automated method is free from the subjectivity of human operators in identifying ARs, there is certainly a computational bias associated with the selection of controlling parameters. Here we vigorously investigate such computational bias.

In this paper, we use the standard contingency table to validate our auto-detection system, assuming that the NOAA catalog provides the "ground truth." A contingency table contains four parameters: (1) the number of true positives (TPs) or hits, (2) the number of false positives (FPs), or false alarmings, also called Type-I errors, (3) the number of false negatives (FNs), or missing detections, also called Type-II errors, and (4) the number of true negatives (TNs). Using CR 2000 as an example (Figure 2), the automated method identifies 14 ARs (N_AUTO, indicated by the black rectangular boxes), while the NOAA catalog reports 19 ARs (N_NOAA, indicated by the red circular symbols, while the sole yellow symbol is for a non-spot AR). For this rotation, the number of TP prediction is 15 (N_TP), meaning that 15 NOAA ARs have their reported center locations within the boxes of automated ARs. Correspondingly, the number of FNs is 4 (N_FN), the NOAA ARs that are not caught by the auto method. The number of FPs is two (N_FP), meaning that two auto ARs do not contain any NOAA ARs. While the above three parameters are well defined, the number of TNs (N_TN) cannot be determined by the system, since there is no negative event identified in the context of AR finding. A negative AR in a solar image would correspond to any surface area outside positively identified ARs, and thus is not a constrained entity. Nevertheless, for the convenience of this discussion, we arbitrarily assume that N_TN equals the number of observed positive events, or N_NOAA; thus, it is treated as a constant independent of the selection of controlling parameters.

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Based on the contingency table, several validation metrics can be constructed. One popular metric is the true positive rate R_TP, which measures the success of finding NOAA ARs, defined as

$$\begin{equation} R_{{\rm TP}} = \frac{N_{{\rm TP}}}{N_{{\rm TP}}+{N_{{\rm FN}}}} = \frac{N_{{\rm TP}}}{N_{\rm NOAA}}. \end{equation} \tag{ 1 }$$

This parameter also denotes the rate of missing events, which equals 1 − R_TP. Another popular metric is the false positive rate, which shows the rate of overdetection, or false alarming of the auto system. It is defined as

$$\begin{equation} R_{{\rm FP}} = \frac{N_{{\rm FP}}}{N_{{\rm FP}}+N_{{\rm TN}}} = \frac{N_{{\rm FP}}}{N_{{\rm FP}}+N_{\rm NOAA}}. \end{equation} \tag{ 2 }$$

Therefore, for the set of controlling parameters chosen above and for CR 2000, our auto-detection system yields a true positive rate of 78.9% and a false positive rate of 9.5%. A desirable system is to maximize the true positive rate and at the same time minimize the false negative rate.

We further introduce one more metric, the rate of compound AR (CAR; R_CAR), to address the issue of multiple-to-one mapping between NOAA ARs and auto ARs. For example, as shown in Figure 2, the largest AR (region number "1" in Figure 1(f)) contains three individual NOAA ARs. This is caused by the fact that a magnetic region seen in the magnetogram is much more extensive in size than the corresponding sunspot seen in white light (the size ratio is about 15, as discussed in the following section). As a result, while ARs appear as discrete sunspot groups in white light images, they may appear morphologically connected in magnetogram images. In particular, when the magnetic fields expand out into the corona, they may be interconnected and collectively determine solar activities. Therefore, for both computational vigor and physical justification, the multiple-to-one mapping is necessary in our system. In CR 2000, there are two ARs having such mapping. We may call these regions as CARs. Therefore, we introduce the rate of CAR as

$$\begin{equation} R_{\rm CAR} = \frac{N_{\rm CAR}}{N_{\rm AUTO}}. \end{equation} \tag{ 3 }$$

Thus, the rate is 14.3% for the example discussed.

We here present the variation of these metrical parameters with the solar cycle from CR 1909 (starting on 1996 May 5) to CR 2077 (ending on 2008 December 17) for the set of controlling parameters chosen above (Figure 3). The period after 2008 December is largely within the extended solar minimum, during which there is almost no AR. The thirteen-year-long period from 1996 to 2008 encompasses the entire 23rd solar cycle starting from its minimum and extending into the beginning minimum of the 24th solar cycle. Apparently, the AR solar cycle variation from our auto detection closely follows that reported by NOAA (Figure 3, top panel). Note that there are several missing data points from the auto detection (CRs 1937, 1938, 1939, and 1940), which are caused by the malfunction of the SOHO spacecraft in 1998. Further, the following CRs, 1941, 1944, 1945, 1956, 2011, and 2015, are not included in the plot because of incomplete MDI data in these rotations. We simply replace these missing data points with the value of previous effective data points. Further, in order to better view the solar cycle trend, we have applied a six-point running average on all the profiles shown in Figure 3.

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Throughout the solar cycle, the true positive rate R_TP varies between ∼40% and 90%, with an average value at 73.8%; the average is weighted by the number of auto ARs per Carrington rotation. On the other hand, the false positive rate R_FP varies between ∼0% and ∼25%, with an average value at 15.3% (also weighted by the number of ARs). The rate of compound ARs varies between zero and ∼20% with an average value at 13.0% (also weighted by the number of ARs). Apparently, this rate is very low during the solar minima when ARs are sparse on the Sun, and becomes relatively large during the solar maximum when a large number of ARs are simultaneously present on the Sun.

With the validation metrics defined above, we now consider the computational bias of our auto system, when different sets of controlling parameters are used (Figure 4). Each metric parameter shown in the figure is the average value obtained over the whole solar cycle from 1996 to 2008. In the first column, we show the variation of the metric parameters by varying the AR kernel pixel intensity threshold I_K from 100 G to 400 G, while the other controlling parameters are fixed (the size of the morphological opening operation S_O = 10 Mm, the AR pixel intensity threshold I_A = 50 G, and the size of the morphological closing operation S_C = 10 Mm). The four panels from top to bottom show the following parameters, respectively: (1) the ratio (R_AR) between the auto AR number (N_AUTO) and the NOAA AR number (N_NOAA), (2) the true positive rate (R_TP), (3) the false negative rate (R_FN), and (4) the rate of compound ARs (R_CAR).

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As seen in the figure, when I_K increases from 100 G to 400 G, the number of auto-detected ARs decreases from 4320 to 1060 (N_AUTO), corresponding to R_AR from ∼2.1 to 0.5. Note that during the period from 1996 May 5 to 2008 December 17, there were in total 3048 ARs reported in the NOAA catalog. Among these regions, 2286 ARs had crossed the central meridian, while the others either disappeared before or emerged after the central meridian. Since we use only Carrington synoptic maps in this study, the non-central-meridian-crossing ARs should not be used in the comparison. Further, excluding those CRs with no or incomplete MDI observations, the number of NOAA ARs used in our comparative study stands at 2085 (N_NOAA). Apparently, the number of ARs auto-detected is rather sensitive to the kernel pixel intensity threshold. As the threshold decreases, regions with weaker magnetic fields will be included. These regions may not correspond to any NOAA ARs. When I_K = 100 G, the resulted N_AUTO = 4320, and the false positive rate R_FP is as high as 55%, meaning that about half of auto ARs are not co-spaced with any NOAA AR. Nevertheless, the true positive rate R_FP reaches almost 90%. In contrast, when the kernel intensity threshold is chosen to be 400 G, almost all small regions including small NOAA ARs are rejected; only large NOAA ARs survive the auto-detection system. As a result, the rate of true positive is 45%, meaning that about 55% of NOAA ARs are missed in the detection. However, the rate of false positive is extremely low (close to zero), meaning that almost every auto AR has corresponding NOAA ARs.

In the middle column of Figure 4, similar variations are shown for controlling parameter S_O, when it increases from 6 Mm to 18 Mm, while the other parameters are fixed (I_K = 250 G, I_A = 50 G, and S_C = 10 Mm). As the morphological opening size increases, the number of ARs auto-detected decreases from 5053 to 649, corresponding to the ratio R_AR decreasing from 2.4 to 0.3. In the mean time, the true positive rate decreases from ∼95% to 35%, and the false positive rate decreases from ∼60% to almost zero. It demonstrates that the auto-detection system is sensitive to the size of the morphological opening operation.

As seen in the right column of Figure 4, our detection system only weakly depends on S_C, the structural size of the morphological closing operation. As the size increases from 5 Mm to 50 Mm, the number of ARs detected decreases from 1851 to 1309, corresponding to the AR ratio R_AR from 0.89 to 0.66. The true positive rate remains ∼73%, and the false positive rate decreases from ∼17% to 10%.

In all cases we have investigated, we choose the minimal AR pixel intensity threshold I_A at 50 G. This is an arbitrary but reasonable election. It is about three times as large as the standard deviation of the magnetic fields in MDI magnetogram images. Since this parameter is only used on the region-growing operation, it has minimal impact on the number of ARs detected. Nevertheless, it may affect the characterization of an AR, e.g., its area size and total magnetic flux. A lower value would make an AR grow larger, and a larger value would make an AR appear smaller.

Now, the essential question is what controlling parameters we should adopt. There is no simple answer to this. The auto-detected regions which are not in the NOAA AR catalog are also magnetic features with true physical meaning (e.g., ephemeral regions); they are not noisy features. These regions should be included if the purpose is to study any sizable magnetic feature on the Sun. Apparently, the NOAA AR catalog contains only large magnetic regions on the Sun, further biased toward those having obvious sunspots in white light. If we assume that the NOAA AR catalog is a "perfect" catalog or the ground truth, then a "perfect" auto-detection system would have the true positive rate at 100% and the false positive rate at 0%.

One popular method to optimize controlling parameters is to analyze the so-called Receiver Operating Characteristic (ROC) plot, or simply, the ROC curve. We show one such plot in Figure 5, which shows the true positive rate (Y-axis) versus the false positive rate (X-axis) by varying the AR kernel intensity threshold I_K (while fixing other parameters, as for the plots in the right column of Figure 4). The seven data points (indicated by the cross symbols) in the figure correspond to I_K from 100 G to 400 G, with an increment of 50 G for two neighboring points. The optimization is to find the data point that is farthest from the diagonal line (dotted line). The diagonal line is called the no-discrimination line, since a data point on the line would have the same true positive rate as the false positive rate; in other words, the benefit is canceled out by the cost.

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Apparently, the seven data points are almost in parallel with the no-discrimination line, with the point in the middle (I_K = 250 G) slightly further from the line. Therefore, from the benefit–cost point of view, there is no strong preference for any of the seven points. The decision is largely based on physical justification, depending on who prefers finding only large ARs (large threshold), or finding all ARs including smaller magnetic features (small threshold) at the cost of significant overdetection. In this study (including AR characterization discussed in the following section), we choose controlling parameters that make this compromise, i.e., intermediately high true positive rate, intermediately low false positive rate, and similar number of auto ARs as that of NOAA ARs. Because of this consideration, the "optimized" set of controlling parameters is found to be (I_K = 250 G, S_O = 10 Mm, I_A = 50 G, and S_C = 10 Mm) and the resulting validation parameters are (R_AR = 0.86, R_TP = 73.8%, R_FP = 15.3%, and R_CAR = 13.0%).

For each AR identified by the automated system, we calculate three sets of parameters, which characterize the basic physical properties of the AR: (1) location, (2) geometric size, and (3) magnetic flux. In Figure 6, we show in detail one particular AR as a sample before we discuss the statistical properties of all ARs. This region is the largest AR in CR 2000, denoted as region 1 in Figure 1(f).

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The geometric center of the AR (green plus symbol) is located at (1651, 120) (the two numbers are the Carrington longitude and latitude, respectively). On the other hand, the flux-weighted geometric center (green cross symbol) is at (1664, 122) or 16.0 Mm apart from the plain geometric center. The offset to the right is apparently caused by the strong flux concentration within the leading positive polarity. The geometric center for the leading positive polarity (blue plus symbol) is located at (1701, 110), while for the trailing negative polarity (red plus symbol) is (1620, 126). Thus, the distance between the leading and trailing polarities, as defined by the geometric centers, is 98.1 Mm. When we define the distance using the flux-weighted geometric centers, the value is as large as 119.2 Mm.

It is also straightforward to calculate the geometric area sizes and the magnetic fluxes threading through the area. The geometric area sizes for the total unsigned, positive, and negative fluxes for this AR are 22489.7 Mm², 8507.7 Mm², and 13981.9 Mm², respectively. The corresponding magnetic fluxes are 6.59 ×10²² Mx, 3.35 ×10²² Mx, and 3.23 ×10²² Mx, respectively. It is interesting to note that, while the ratio between the positive and negative magnetic fluxes is nearly unity (1.04 to be exact), the ratio of the areas is as large as 1.64. Apparently, the flux of the leading (positive) polarity is concentrated in a relatively smaller area, while that of the trailing polarity disperses more. This kind of leading–trailing asymmetry is a well-known fact of bipolar ARs.

Further, we have applied morphological analysis on individual ARs (Figure 7). The process is almost identical to the one used for the entire synoptic chart as discussed in Section 2, except for the difference of the controlling thresholding parameters. While the two intensity controlling parameters are the same: 250 G for the kernel and 50 G for the nominal AR pixel, the two size parameters are much smaller, both of which are 2 Mm (instead of 10 Mm for AR identification). As shown in Figure 7, we are now able to identify small fragments in the AR. There are 8 fragments of positive polarity (indicated by the white patches and blue symbols at the center of each patch) and 15 fragments of negative polarity (dark patches and red symbols). Therefore, there are in total 23 fragments in this AR. It illustrates that, even in an AR which appears as a single big clump of magnetic flux over the surface of the Sun, the magnetic flux is far from an even distribution in space. The magnetic flux in an AR is fragmented and highly clumpy, concentrated in small patches. This kind of fractal property of ARs has been studied by several authors (Lawrence et al. 1993; Meunier 1999; McAteer et al. 2005; Conlon et al. 2008). In white light images, it is well known that an AR contains multiple sunspots. In a sense, these fragments are analogous to individual sunspots in an AR. To understand the relation between these magnetic fragments and individual sunspots require a study of direct comparison between magnetogram images and white light images, which is beyond the scope of this paper.

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With the basic parameters obtained for all ARs, we now investigate their global and statistical properties. In Figure 8, we show the variation of AR number, number of fragments, area size, and magnetic flux per Carrington rotation with respect to the phase of solar cycle from 1995 to 2008. In Table 1, we further list the statistical measures of these basic parameters averaged over the solar cycle. Corresponding data from the NOAA AR catalog are also shown as a comparison.

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As shown in Figure 8 (top panel), there is apparently a good correlation between the auto (black line) and NOAA (red line) AR numbers, since they both show very similar solar cycle variation. Both numbers indicate that there are double peaks during the solar maximum, one in early 2000 and the other in late 2001; the separation of the two peaks is about one and a half years. In the NOAA AR numbers, the two peaks have almost equal strength, while in the auto AR numbers, the first peak is stronger. There are typically about 30 ARs per Carrington rotation during the solar maximum, while there are only a few or zero during the solar minima at 1996 and 2007/2008. The highest AR number per Carrington rotation is 39 from the NOAA catalog and 37 from our automated method. The AR number averaged over the whole solar cycle is 13.5 from the NOAA catalog and 10.9 from the automated method.

In the second panel of the figure, we show the number of AR fragments per rotation from the automated method (black line) and the number of sunspots (red line) per rotation from the NOAA catalog. There are about 300 sunspots per rotation during the solar maximum, while the number of fragments is slighter smaller. The highest sunspot number per rotation is 413, while the highest fragment number per rotation is 327. In terms of mean value over the solar cycle, the sunspot number per rotation is 126.2, and the fragment number is 85.3. We believe that counting small fragments within an AR is probably more accurate with the automated method. In the NOAA catalog, the following formula, the so-called Zürich method (e.g., Hathaway 2010), is used to define the sunspot number:

$$\begin{equation} R = k(10g+n), \end{equation} \tag{ 4 }$$

where g is the number of sunspot groups, n is the number of individual sunspots, and k is a correction factor for the observer. It is recognized that, when solar images are inspected by human eyes, it is far easier to identify sunspot groups than to identify each individual sunspot. However, the automated method is able to identify individual fragments efficiently and objectively, free of the rather arbitrary assignment of sunspot groups. On the basis of individual ARs, Table 1 shows that an AR has 8.0 sunspots on average, similar to the 7.8 fragments per AR obtained using our automated method. The maximum number of sunspots an AR can contain is 90, while the maximum number of fragments of an AR is 69.

In the third panel of Figure 8, we show the solar cycle profile of the total geometric area of ARs (black line) per Carrington rotation; the unit is in milli, or one-thousandth of the total solar surface (TSS). Magnetic ARs occupy about 30 thousandth or 3% of the total solar surface during the solar maximum. This number is about 15 times larger than the area occupied by the sunspots (red line) in white light as reported by NOAA. Throughout the solar cycle, there is obviously a similar ratio between the AR magnetic area and the sunspot area. The mean area per Carrington rotation over the solar cycle is 10.8 thousandth of the TSS using our automated method, and is only 0.73 thousandth from the NOAA catalog. The maximum value of area per rotation is 43.3 and 2.95 thousandth for the automated and NOAA ones, respectively. All these numbers consistently show that an AR defined by extended magnetic fields in a magnetogram image is about 15 times as large as that defined by the dark region in a corresponding white light solar image.

In the last panel of the figure, we show the solar cycle variation of AR magnetic fluxes derived from the automated method. The three lines represent the total unsigned flux (black), positive flux (blue line), and negative flux (red line) per Carrington rotation, respectively. Note that there is no report on magnetic flux from the NOAA catalog. During the solar maximum, there are about 5 ×10²³ Mx of magnetic flux concentrated in ARs per Carrington rotation. The mean magnetic flux per rotation over the solar cycle is 1.83 ×  10²³ Mx and the maximum magnetic flux is 6.96 × 10²³ Mx. For individual ARs, the mean magnetic flux is 1.67 × 10²² Mx and the maximum magnetic flux is 1.97 × 10²³ Mx.

It is interesting to point out that, while there are double peaks during the solar maximum with similar strength in the NOAA AR number, the other parameters, including sunspot number, AR area, and flux, show an outstanding peak in late 2001. The strength of this peak is significantly larger than that in early 2000. It has been noted that the dates of solar cycle maxima, when they are determined by difference indexes, i.e., sunspot numbers versus sunspot areas, could be significant different, from a few months to a few years (Hathaway 2010). Further, we note that the peak magnetic flux in late 2001 is mainly caused by the large area size of the emerged ARs, but not the mean intensity of the magnetic field. In terms of the amount of magnetic flux emerged onto the surface of the Sun, it is fair to say that the 23rd solar cycle peaked in late 2001 but not in early 2000.

Statistical frequency distributions of AR sizes are studied in this subsection. There are 1730 ARs identified by the automated system based on MDI high-resolution synoptic maps from 1996 to 2008, using the "optimized" set of controlling parameters. We show the area size distribution and magnetic flux distribution of these ARs in Figures 9 and 10, respectively, plotted in a log–log scale. The two distributions appear similar and to be Gaussian-like. Therefore, we fit the distribution to a log-normal function. For the area size, the fitted function is

$$\begin{equation} \frac{dN(A)}{d\log A} = 76.2\times \exp \left\lbrace -\frac{(\log A - 3.52)^2}{0.454}\right\rbrace, \end{equation} \tag{ 5 }$$

where A denotes the AR area in units of Mm². The bin size of the distribution with respect to log A is 0.05. The reduced χ² test of the goodness of fit is 25.2. For the magnetic flux, the fitted function is

$$\begin{equation} \frac{dN(F)}{d\log F} = 70.8\times \exp \left\lbrace -\frac{(\log F - 1.89)^2}{0.541}\right\rbrace, \end{equation} \tag{ 6 }$$

where F denotes the unsigned AR magnetic flux in units of 10²⁰ Mx. The bin size of the distribution with respect to log F is 0.05. The reduced χ² test of the goodness of fit is 57.2.

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The log-normal distribution is consistent with that found by Bogdan et al. (1988). However, it is different from the functions suggested by other workers. The exponential distribution of ARs was found by Tang et al. (1984), Schrijver et al. (1997), and Zharkov et al. (2005). The polynomial function was used by Harvey & Zwaan (1993) to fit their data. The power-law distribution was found by Parnell et al. (2009). The functional form of AR distribution may reveal the physical mechanism that generates these ARs. Bogdan et al. (1988) argued that the log-normal distribution is a result of magnetic fragmentation in the convection zone. On the other hand, Schrijver et al. (1997) demonstrated that frequent fragmentation and collision (or merging) of magnetic features lead to the exponential distribution, favoring the cause of near-surface dynamo. Our data indicate that large ARs tend to be caused dominantly by the fragmentation process, thus favoring the global dynamo model occurring through the convection zone (Bogdan et al. 1988).

Nevertheless, our fitting has a large χ² value, indicating that the log-normal function is a rather poor fitting to the observed data. Careful inspection of Figures 9 and 10 reveals that the fitting is better in the higher end of the distribution than in the lower end. Toward the lower end, the frequency distribution drops much faster than the normal distribution. We believe that the fast drop is caused by the computational bias. For this instance of automatic detection, the set of controlling parameters is chosen to be (250 G, 10 Mm, 50 G, and 10 Mm), which favors the detection of large magnetic features, while selectively remove features of small size and weak intensity.

To further illustrate the influence of computational bias, we have made one instance of automatic detection with significantly reduced thresholds. We choose the set of controlling parameters to be (150 G, 6 Mm, 50 G, and 6 Mm), thus favoring the detection of weaker and smaller magnetic features. The smaller structural size for the morphological opening operation makes the system not only detect small non-AR features, but also tends to fragment an AR into multiple pieces. The smaller structural size for the morphological closing operation also reduces the possibility of grouping neighboring magnetic features. As a result, the automated system now finds 14,431 magnetic features, about 8.3 times as many as the number of ARs found at the instance of (250 G, 10 Mm, 50 G, and 10 Mm). Most of the detected magnetic features are not nominal ARs; they could include fragments of decayed ARs, ephemeral regions, and even large network features in the quiet Sun.

In Figure 11, the frequency distribution of these features with respect to magnetic flux is shown. Apparently, the system can detect magnetic features with flux as small as 1.0 × 10²⁰ Mx, which is about eight times smaller than that from the instance for ARs. Interestingly, with this instance of detection, the AR distribution now becomes largely a power law but with significant deviation at the two extreme ends. In the high end, which is of the largest ARs, the distribution decreases much faster than the power law, which could be fit by a softer power law, or exponential, or even Gaussian. In the lower end, the precipitous drop is likely caused by the detection cutoff of the auto system. The vast majority of detected features, starting from 2.8 × 10²⁰ Mx (point A in the figure) and ending at 6.3 × 10²² Mx (point B), can be well fitted by a power-law function, described as

$$\begin{equation} \frac{dN(F)}{d\log F} = 1.46\times 10^3 \times F^{-0.630}, \end{equation} \tag{ 7 }$$

where F again denotes the unsigned AR magnetic flux in units of 10²⁰ Mx. The bin size of the distribution with respect to log F is again 0.05. In this case, the unreduced χ² test of the goodness of fit is only 0.146, a value that strongly supports the null hypothesis of the power-law function.

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This result is consistent with that of recent work by Parnell et al. (2009), who found a power-law distribution of solar magnetic features over more than five decades in flux, expanding from large ARs to small bipolar regions in the quiet Sun. The only exception is that our data show a strong deviation from the dominant power law for those largest ARs, i.e., with flux larger than 6.3 ×10²² Mx. The change of AR distributions, from log-normal to power law when the controlling parameters are lowered, suggests important effects of computational bias on analysis results. One has to consider the bias when the results are discussed in scientific context.

In this section, we investigate the distribution of AR locations on the Sun and the variation with the solar cycle. Figure 12 shows the heliographic latitude (Y-axis) of the 1730 ARs obtained by the automated method with respect to their time crossing the central meridian. The figure reproduces the classical Butterfly Diagram of solar ARs (Spörer's Law), which is caused by the emerging locations of ARs progressively drifting toward the equator, the sign of deep meridional flow of solar dynamo (Hathaway et al. 2003). The equatorward drifting motion from the two hemispheres is approximately symmetric to the equator, but not exact (more discussion later). As shown in the figure, almost all ARs are below 40° latitude, with a few exceptions in the southern hemisphere. The AR maximum latitudes in the northern and southern hemispheres are 40° and 48°, respectively.

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ARs from the 23rd solar cycle started to emerge at high latitudes (∼30°) in late 1996. As the cycle progressed, not only did the emerging rate of ARs (as indicated by the number of ARs per Carrington rotation) increased, but also the latitudinal zones or bands of emerging ARs grew wider. Further, the centroids of the AR bands (indicated by the red lines) were drifting continuously toward the equator. AR bands were as wide as 30° during the rising phase of the solar cycle. In the middle of 2002, AR bands started to narrow down, and the number of emerging ARs also started to decrease. On the other hand, the centroids of the bands continued to move toward the equator. ARs completely disappeared in both hemispheres in the middle of 2008, and the absence of ARs was even earlier (by more than half year) in the northern hemisphere.

To quantify the drifting motion of the AR bands, we plot and analyze the centroid locations with respect to the solar cycle in Figure 13. The centroid location is derived from the mean latitude of all ARs per Carrington rotation weighted by magnetic flux. In order to have a clean representation of the drifting, we made an effort to use only ARs that emerged in Cycle 23. In the beginning of the cycle, we have removed the low-latitude ARs from Cycle 22, and in the end of the cycle, removed the high-latitude ARs from Cycle 24. Further, we consider only the folded latitudes from the equator, without differentiating between the northern and southern hemispheres. To the first order of approximation, the overall drifting can be described by the straight line shown in the figure. The line is the linear fit to the data points. The linear fitting formula is

$$\begin{equation} \delta = 24\mbox{$.\!\!^\circ $}4-0\mbox{$.\!\!^\circ $}137N_{{\rm CR}}, \end{equation} \tag{ 8 }$$

where δ is the latitude in units of degree and N_CR is the number of Carrington rotation elapsed from the first data point in the line. The first cycle-23 AR appeared in CR 1919 and the last cycle-23 AR in CR 2070. The fitting line shows that the AR band centroid started at about 25° and ended at about 5°. The average drifting rate is 0137/CR, with a standard deviation of 0003/CR. Converting to the drifting speed measured in years, since one Carrington rotation is 27.2753 days, and one year is 365.25 days, we obtain

$$\begin{equation} V_d = 1\mbox{$.\!\!^\circ $}83\pm 0\mbox{$.\!\!^\circ $}04\, \rm yr^{-1}. \end{equation} \tag{ 9 }$$

In linear terms, the drifting speed is

$$\begin{equation} V_d = 0.708\pm 0.015\, \mbox{m s}^{-1}. \end{equation} \tag{ 10 }$$

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We emphasize that this drifting speed or rate is an average value over the whole solar cycle. As seen from Figure 13, there are certain deviations from the oversimplified straight line. There seems a plateau for two years from 1997 to 1999. This plateau is followed by a speedy drifting much faster than the average speed. In at least four occasions, there appears a short period of "reverse" drifting, i.e., the centroid is moving away from the equator rather than toward it. The short period of "reverse" drifting is probably caused by short bursts of ARs emerging in usually high latitudes.

Earlier studies suggested that the drifting motion be fitted by a quadratic motion, based on historic sunspot data (Li et al. 2001; Hathaway et al. 2003). The quadratic fitting indicates that the drifting motion has a large velocity in the beginning of the solar cycle, gradually slow down, and eventually goes to zero at the end of the cycle. However, our data do not show such a gradually slowing trend. Instead, it seems to be a linear drifting on average, superposed with intermittent bursts of reverse drifting.

Finally, we want to mention the observation of the north–south asymmetry of the AR distribution. While the Butterfly Diagram and the drifting motion appear largely symmetric, there is also a noticeable asymmetry. Among the 1730 ARs obtained from our automated method, 938 of them are from southern hemisphere, while 792 are from the northern hemisphere. Therefore, over the whole solar cycle, there are about 18% more ARs appearing in the southern hemisphere than in the northern hemisphere. This south-over-north asymmetry mostly occurred during the declining phase of the solar cycle. In terms of magnetic flux, there is in total 1.55 × 10²⁵ Mx that emerged in the southern hemisphere, while 1.33 × 10²⁵ Mx emerged in the northern hemisphere; these numbers correspond to an asymmetry of about 17%. Similar north–south asymmetry has been reported earlier (Temmer et al. 2002; Zharkov & Zharkova 2006). The cause of this asymmetry is not understood. An investigation of the total budget of magnetic flux in the hemispheres requires knowledge of smaller magnetic regions, such as ephemeral regions. We will investigate the asymmetry issue in more detail in a separate work.