# Solar Anti-Hale Bipolar Magnetic Regions:
A Distinct Population with Systematic Properties

Andrés Muñoz-Jaramillo  Southwest Research Institute
1050 Walnut Street, #300
Boulder, CO 80302, USA  Benjamín Navarrete  Departamento de Astronomía,
Universidad de Chile, Casilla 36-D, Santiago, Chile  Luis E. Campusano
Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Santiago,
Chile

###### Abstract

Besides their causal connection with long and short-term magnetic variability,
solar bipolar magnetic regions are our chief source of insight into the
location, size, and properties of large-scale toroidal magnetic structures in
the solar interior. The great majority of these regions ($\approx$95%) follow
a systematic east-west polarity orientation (Hale’s law) that reverses in
opposite hemispheres and across even and odd cycles. These regions also
present a systematic north-south polarity orientation (Joy’s law) that helps
build the poloidal field that seeds the new cycle. Exceptions to Hale’s law
are rare and difficult to study due to their low numbers. Here we present a
statistical analysis of the inclination (tilt) with respect to the equator of
Hale vs. Anti-hale regions spanning four solar cycles, considering two
complementary tilt definitions adopted in previous studies. Our results show
that Anti-Hale regions belong to a separate population than Hale regions,
suggesting a different originating mechanism. However, we find that Anti-Hale
region tilts present similar systematic tilt properties and similar
latitudinal distributions as Hale regions, implying a strong connection
between the two. We see this as evidence that they belong to a common toroidal
flux system. We speculate that Anti-hale regions originate from poloidal field
sheared and strengthened ”on-the spot” after the emergence of hale regions
with very strong poloidal contribution. Thus, they are not in contradiction
with the idea of largely coherent toroidal flux systems inside the solar
interior.

Solar physics — Sun: magnetic field — sunspots — Hale’s law

##  1 Introduction

Bipolar magnetic regions (BMRs) are regions of strong magnetic fields that
emerge to the solar photosphere the solar photosphere, acting as the main
source of space weather events. They typically form sunspots, where strong
magnetic field bundles cross the photosphere (van Driel-Gesztelyi & Green,
2015), appearing as dark contrast regions on the solar photosphere in visible
light. The number of BMRs present at any given moment on the solar surface
waxes and wanes during the solar cycle. Although we cannot observe the solar
interior where BMRs originate, their observation give us insight into the
dynamical behavior of the large-scale internal solar magnetic field.

Figure 1: Schematic description of the different combinations of tilt and hale
orientations and how they translate to angles depending on the reference frame
used. Blue angles use the negative polarity as the reference frame and go
between $-180^{o}$ and $180^{o}$. This definition is better suited to
highlight the difference between Hale (HH) and Anti-Hale (AH) regions. Red
angles use the leading polarity as the reference frame and go between
$-90^{o}$ and $90^{o}$. This definition is better suited to highlight the
difference between Joy and anti-Joy regions.

Three main empirical laws govern populations of active regions: Spörer’s Law
(Carrington, 1858) states that the latitude of emerging sunspots decreases
with time as the solar cycle goes forward (from mid-latitudes towards the
equator), giving rise to the time vs. latitude plot called the butterfly
diagram (see Fig. 2). Hale’s law (Hale & Nicholson, 1925), which establishes
that BMRs present a systematic East-West magnetic polarity orientation for a
given solar hemisphere – for example, if the positive pole of the BMR trails
behind the negative pole at the northern solar hemisphere in the east-west
direction, then the negative pole will be behind the positive in the south in
the same direction. The leading polarity of each hemisphere is reversed for
odd and even cycles. Finally, Joy’s law (Hale et al., 1919) expresses that
BMRs tend to have a systematic North-South orientation, with the leading
(trailing) polarity being closer to the equator (pole). This law is typically
quantified through the measurement of a BMR’s tilt (i.e. the angle between a
line that passes between the center off a BMR’s polarities and a line parallel
to the solar equator). Regions that follow this law are typically refered to
as Hale (HH) regions.

There is a small fraction of BMRs that violate Hale’s law and its hemispheric
rule (Richardson, 1948): these “anti-Hale (AH)” regions have an opposite East-
West polarity orientation than most of the BMRs in the respective hemisphere.
These regions comprise between 5-10% of all BMRs (Richardson, 1948; Wang &
Sheeley, 1989; Khlystova & Sokoloff, 2009; McClintock et al., 2014; Stenflo &
Kosovichev, 2012; Li, 2018). In spite of their relatively small numbers, AH
regions provide important clues to the structure of the internal magnetic
field.

Previously, Stenflo & Kosovichev (2012) argued that AH regions are a distinct
population, rather than anomalies of the HH groups. They suggest that the
appearance of AH regions in similar latitudes that HH regions rules out the
possibility of well-defined, coherent toroidal flux systems as a source of all
active regions. On the other hand, McClintock et al. (2014) hypothesized that
AH regions are part of a continuous distribution of regions that stems from
the HH population, a picture that more consistent with the existence of a
largely coherent toroidal flux system. Here we provide further evidence that
AH regions comprise a different population than HH regions, implying that
their origin is different than that of HH. However, we still find a strong
connection between some of their statistical properties, which we take as
evidence of a causal connection between them.

In Section 2 we introduce the tilt definitions used in this work. Sections 3
and 4 describe the data and the determination of flux cut-offs and calibration
factor respectively. In Section 5 we determine the Hale and anti-Hale
fractions by hemisphere. Section 6 describes the combination of BMRs from
different hemispheres and Cycles. In section 7 we analyze to what extent do
Hale and Anti-Hale BMRs follow Joy’s Law. In section 8 we report the optimal
analytic fit to the distribution of Hale and anti-Hale region tilts. In
section 9 we demonstrate that Hale and anti-Hale region tilts cannot be
described using a unique distribution. Finally, in Section 10 we discuss the
possible origin of the Anti-Hale regions and in Section 11 we provide a
summary with the main conclusions.

##  2 Absolute vs. Relative BMR Magnetic tilt definitions

There are two main ways of defining the tilt of a BMR that showcase different
aspects of BMR inclination (see Fig. 1). They differ on the way BMR poles are
chosen as references. They have different flavors in the way they represent
the combination of HH vs.AH and Joy (leading polarity closer to the equator)
vs. anti-Joy (leading polarity closer to the pole).

One possibility is to always use the same polarity (typically the negative
polarity) as the location of the reference frame and the other polarity
(typically the positive) to determine the tilt angle (see for example Li &
Ulrich, 2012; McClintock et al., 2014). Under this definition, tilt angles
vary between $-180^{o}$ and $180^{o}$. This reference frame (and associated
angles) are shown using blue ink in Fig. 1. Under this definition, HH vs. AH
regions will separate into angles with a magnitude larger vs. smaller than
$90^{o}$ (depending on the hemisphere and cycle). Joy vs. anti-Joy are in turn
differentiated by a change of angle sign for a given hemisphere and cycle.
This approach, which we refer to from now on as the “absolute tilt”, enables
the simultaneous analysis of Hale orientation and Joy tilt angle, while
treating all BMRs as part of a continuous distribution of angles. It contains
the implicit assumption that HH and AH regions belong to a continuously
distributed population.

The second possibility is to always use the leading polarity (regardless of
its polarity) as the location of the origin and the other polarity (the
trailing polarity) to determine the tilt angle (see for example Wang &
Sheeley, 1989; Howard, 1991; Stenflo & Kosovichev, 2012). Under this
definition, tilt angles vary between $-90^{o}$ and $90^{o}$. This reference
frame (and associated angles) are shown using red ink in Fig. 1. Under this
definition, HH vs. AH regions have to be explicitly separated since there is
no information in the tilt angle that conveys Hale orientation. Joy vs. anti-
Joy BMRs are very easily differentiated because Joy (anti-Joy) regions always
have positive (negative) angles in the Northern hemisphere (regardless of the
cycle) – signs reverse for the Southern hemisphere. This approach, which we
refer to from now on as the “relative tilt”, focuses exclusively on the
relative position of BMR leading and trailing polarities. It contains the
implicit assumption that HH and AH regions belong to different populations.

To this date, analyses of tilt and Hale-orientation have chosen one approach
or the other. We speculate that it is not accidental that studies that use the
absolute tilt definition conclude that BMRs arise from a continuous
distribution of angles, whereas studies that use the relative tilt definition
conclude that HH and AH populations are different. The aim of this study is to
determine the significance limit at which we can reject the null hypothesis
that the HH and AH regions constitute a single population and this is only
possible to do if both definitions of tilt are studied simultaneously.

As done by Howard (1991), we compute the magnetic tilt angles as:

$\tan(\gamma)=\Delta\lambda/(\Delta\phi\cos(\bar{\lambda}))$ (1)

Where $\bar{\lambda}$ is the midpoint latitude of the polarity pair,
$\Delta\lambda$ and $\Delta\phi$ are the differences between latitude and
longitude respectively. For the absolute tilt calculation we set the negative
polarity as the reference polarity and use the arc-tangent function with two
arguments so that the resulting angles go between $-180^{o}$ and $180^{o}$.
For the relative tilt calculation we use the leading polarity as reference and
use the single argument arc-tangent so that the resulting angles go between
$-90^{o}$ and $90^{o}$. We ensure that tilt angles are positive (negative) for
Joy regions in the Northern (Southern) hemispheres. For both relative and
absolute tilts we use the division shown in Figure 1 to separate our data into
Hale and AH regions depending on cycle and hemisphere.

Figure 2: Solar Latitude of active regions as a function of time (Butterfly
diagram) for the last four solar cycles. Blue data is from KPVT/512, purple
for KPVT/SPMG, orange for SOHO/MDI and green color for SDO/HMI measurements.
The overlap regions between instruments are plotted in black color. Anti-Hale
regions are shown in red.

##  3 Data

We use data from two ground-based and two spaced-based instruments. Our
ground-based instruments are part of the Kitt Peak Vacuum Telescope survey.
The first instrument is the 512 Diode Array Magnetograph (from 1976 to 1993
Livingston et al., 1976) and the second the solar Spectromagnetograph (SPMG,
from 1992 to 2001 Jones, 1992). Both instruments obtained images at a cadence
of one image per day. The 512 instrument has a resolution of 1” per pixel;
SPMG has a resolution of 1.15” per pixel. Our space-based instruments are the
Michelson Doppler Imager (MDI; Scherrer et al., 1995, 1996-2010), on board the
Solar and Heliospheric Observatory (SOHO) collected 136,839 magnetograms from
1996 to 2011 at a 96 minutes cadence and a resolution of 2” per pixel. We also
use the data from the Helioseismic and Magnetic Imager (HMI; Scherrer et al.,
2012, 2010-present), on board the Solar Dynamics Observatory (SDO), which
produces magnetograms with a resolution of 0.5” per pixel at a 45 second
cadence.

Figure 3: Empirical BMR flux distributions for the overlapping periods between
instruments. Between KPVT/512 and KPVT/SPMG from April 1992 to April 1993,
SPMG-MDI from May 1996 to December 1999, and MDI-HMI from April 2010 to April
2011. Blue line represents the distribution of KPVT/512, KPVT/SPMG in orange,
SOHO/MDI in green, and SDO/HMI in red.

Our BMR dataset consists of 9,243 individually tracked BMRs, obtained using
the Bipolar Active Region Detection (BARD) code described by Muñoz-Jaramillo
et al. (2016). Here we use a single tilt measurement per BMR, at the moment of
maximum flux, to avoid folding the time evolution of a BMR tilt during a BMR’s
lifetime into our analysis. For each BMR we measure the flux, area, and flux-
weighted latitude and longitude centroids of the positive and negative
polarities. Data is shown in Figure 2. Our data covers four solar cycles
(21-24) as seen by the four different instruments (KPVT/512, KPVT/SPMG,
SOHO/MDI, and SDO/HMI).

###  3.1 Combination of data from different instruments

Figure 4: Two dimensional optimization of calibration factor and flux cut for
the overlapping periods between instruments. Left: Brightness indicates the KS
distance between distributions. Blue vertical hatches (diagonal red hatches)
indicate the region where the Kolmogorov-Smirnov (Anderson-Darling) statistics
rejects the hypothesis of the two populations belonging to the same
distribution. White dashed lines show the optimal calibration factor
(horizontal) and universal magnetic flux cutoff (vertical) that maximize the
similarity between distributions. The only distribution that needs to be
calibrated is SOHO/HMI, with a factor of 0.8, which makes it homogeneous both
with KPVT/SPMG and SDO/HMI. We find that a universal flux cutoff of $3\times
10^{21}$Mx is an optimal compromise between maximizing distribution
homogeneity and minimizing the number of BMRs that have to be rejected. Right:
Homogenized distributions showing the universal cutoff of $3\times 10^{21}$Mx
with a black dashed line. MDI flux has been applied a 0.8 correction factor.

Figure 3 shows the distributions of net BMR magnetic flux during overlap
intervals (512-SPMG, SPMG-MDI and MDI-HMI). In each of these histograms we
observe that there is general agreement for strong BMRs, with the
discrepancies arising for small objects. This discrepancy arises from an
artificial cut-off introduced by the difference in sensitivity and resolution
across magnetographs (very clear in the case of the MDI vs. HMI comparison).
Ensuring that the populations across instruments sample a similar range of
fluxes is important because tilt scatter is dependent on flux (Stenflo &
Kosovichev, 2012; Jiang et al., 2014). Furthermore, due to the fact that small
objects are more numerous, they have a sizable effect on tilt statistics.

Given that our tilt angles are calculated using the flux weighted centroid of
the positive and negative regions in a BMR, we believe our tilt measurements
are largely independent of instrument cross-calibration issues and are
internally consistent. Because of this a rough calibration factor between
instruments, coupled with a universal flux cut-off across instruments, are
sufficient to ensure that we are comparing populations with similar flux
statistics. We demonstrate quantitatively that this is the case in Section 6.

##  4 Determination of flux cut-offs and approximate calibration factors

We take advantage of the time-overlap between instruments to derive an
approximate flux calibration, as each of the instruments has its own
resolution and observational systematic issues. To find the optimal
calibration factor, we minimize the two-sample Kolmogorov-Smirnov (K-S)
distance between the flux distribution of the overlapping period for each pair
of instruments. We do this after applying a calibration factor to one of the
instruments and a common flux cutoff for both the target and calibrated
instrument. This form of calibration was used by Muñoz-Jaramillo et al.
(2015b) to calibrate sunspot group areas. Calibrating the strong end of the
distribution (i.e. the big BMRs that all instruments see well) is the best way
of homogenizing our data because small BMRs are systematically under-observed
due to cadence and sensitivity issues. Furthermore, these systematic issues
are different for each instrument. Introducing a cut-off allows the fit to
focus on the large BMRs that all instruments can see equally well.

Figure 4 shows the result of the 2-dimensional optimization (calibration
factor plus a common cutoff after calibration). Dark (light) regions indicate
a large (small) K-S distance. Hatched areas indicate parameters for which the
K-S test and the Anderson-Darling (A-D) tests reject the hypothesis that BMRs
observed by different instruments belong to the same population. There are
different combinations of factors and flux cuts that optimize the match
between distributions. However, there is a trade off between K-S distance and
cutoff: the more regions we leave out, the easier it is to find a fit.
However, leaving out too many regions is counter productive because it will
reduce the number of regions we can analyse. We find two main optimal regions:
a region around a cutoff of $3\times 10^{21}$ and another around $10^{22}$. We
pick the cutoff value of $3\times 10^{21}$ in order to retain as many BMRs as
possible for our analysis. This translates into a calibration factor around 1
for KPVT/512 vs KPVT/SPMG, 0.8 for SOHO/MDI vs for KPVT/SPMG, and 0.8 for
SOHO/MDI vs SOHO/HMI. Thus a 0.8 factor applied to MDI is sufficient to get
approximate consistency in the flux distribution for all instruments and all
cycles. This factor in good agreement ($1/0.8=1.25$) with the 1.3 factor found
by Liu et al. (2012) to calibrate HMI to MDI for strong magnetic fields,
considering that the focus of our analysis are BMRs containing the strongest
magnetic fields in the solar photosphere.

Figure 5: Anti-Hale fraction of all BMRs. As a function of solar cycle number
(Left) and solar cycle amplitude (Right). The Northern (Southern) hemisphere
is shown using blue (orange) markers.

Once the calibrated factor is applied, we can see a better agreement between
the flux distributions of the different instruments for strong BMRs (see right
panels of Fig. 4 vs. panels in Fig. 3). We use $3\times 10^{21}$Mx as a
universal cutoff for all instruments (after applying the 0.8 factor to MDI),
shown in the right column of Fig. 4 as a vertical black dashed line. Any BMR
with less flux than this cutoff is removed from our analysis. We discard
$1,232$ BMRs leaving $\sim 87\%$ from the initial $9,243$ BMRs in the
database. Additionally, we only include BMRs from a single instrument in each
overlap. This ensures that we do not count each BMR more than once, leaving
$7,511$ flux-normalized BMRs for use in our analysis of tilts.

Table 1: Numbers and fractions of Hale and anti-Hale BMRs per cycle and hemisphere. Cycle | Hem. | # | Hale | anti-Hale
---|---|---|---|---
# | % | # | %
21 | N | 1001 | 933 | 93 | 68 | 7
S | 1065 | 979 | 92 | 86 | 8
22 | N | 1022 | 959 | 94 | 63 | 6
S | 1194 | 1121 | 94 | 73 | 6
23 | N | 938 | 900 | 96 | 38 | 4
S | 1133 | 1084 | 96 | 49 | 4
24 | N | 570 | 552 | 97 | 18 | 3
S | 588 | 562 | 96 | 26 | 4
Total: |  | 7511 | 7090 | 94 | 421 | 6

##  5 Hale and anti-Hale fractions by cycle and hemisphere

Table 1 and Figure 5 show the numbers and fractions of the Hale and AH
sunspots for each hemisphere and cycle. Overall, the fraction of AH sunspots
in our data is $5.61\%$ which is consistent with previous estimates
(Richardson, 1948; Wang & Sheeley, 1989; Khlystova & Sokoloff, 2009; Stenflo &
Kosovichev, 2012). Other studies (McClintock et al., 2014; Li, 2018) have
reported a larger fraction of AH BMRs ($~{}8\%$). This discrepancy is likely
driven by the inclusion of small objects in other surveys, which invariably
dominate BMR statistics due the fact that smaller regions are more numerous.
Li (2018) reports an AH fraction of $~{}8\%$, but applying a flux cutoff of
$3\times 10^{21}$ to their Figure 10, like the one we use in this work,
reduces the fraction to $~{}6\%$. This does not invalidate in any way their
results. It simply highlights the importance of understanding the consequences
of working with objects near a survey’s threshold of detection. It also
highlights the difficulty of comparing surveys with different (and
unspecified) detection thresholds.

There seems to be a downward trend in the proportion of AH BMRs from cycle to
cycle in our observed cycles (21-24; see left panel of Fig. 5). This trend is
also visible in the results of McClintock et al. (2014) and Li (2018). It is
possible that there is a relationship between cycle amplitude and how many AH
regions it produces (see the right panel of Fig. 5): the stronger the cycle,
the higher the proportion of AH regions. This relationship does not seem to be
very strong, but it is still significant after estimating the Wilson
uncertainty interval for a binomial distribution (Brown et al., 2001). As will
be explained in Section 10, we speculate that this hints to connection between
AH regions and preceding HH regions with a strong poloidal field component.

Table 2: Kolmogorov-Smirnov test of the progressive combination of cycle and
hemispheric BMRs into a unified set. Absolute Tilt
---
Population | Hemisphere | Cycle | D-statistic | p-value |  | D-statistic | p-value |  | D-statistic | p-value
Hale | N | 21+23 | 0.024 | 0.94 | $>$ | 0.023 | 0.78 | $>$ | 0.018 | 0.54
22+24 | 0.089 | 0.01
S | 21+23 | 0.049 | 0.16 | $>$ | 0.039 | 0.11
22+24 | 0.049 | 0.16
Anti-Hale | N | 21+23 | 0.100 | 0.94 | $>$ | 0.107 | 0.61 | $>$ | 0.077 | 0.53
22+24 | 0.100 | 0.94
S | 21+23 | 0.112 | 0.77 | $>$ | 0.077 | 0.745
22+24 | 0.114 | 0.41
Relative Tilt
Population | Hemisphere | Cycle | D-statistic | p-value |  | D-statistic | p-value |  | D-statistic | p-value
Hale | N | 21+23 | 0.024 | 0.94 | $>$ | 0.022 | 0.78 | $>$ | 0.020 | 0.54
22+24 | 0.07 | 0.04
S | 21+23 | 0.041 | 0.35 | $>$ | 0.039 | 0.11
22+24 | 0.054 | 0.22
Anti-Hale | N | 21+23 | 0.166 | 0.46 | $>$ | 0.108 | 0.61 | $>$ | 0.077 | 0.53
22+24 | 0.230 | 0.40
S | 21+23 | 0.112 | 0.77 | $>$ | 0.114 | 0.41
22+24 | 0.311 | 0.04

##  6 Combination of BMR’s from different hemispheres and Cycles

Polarity orientation is opposite for the Northern and Southern hemispheres and
reverses with every new cycle. This means that the absolute (relative) tilt
angles that represent the different combinations of Joy vs. anti-Joy and Hale
vs. AH orientations can adopt different sets of values for different
hemispheres and odd/even cycles; 4 in the case of the absolute tilt and 2 in
the case of the relative tilt (see Figure 1). In order to take full advantage
of four cycle’s worth of BMRs, we first verify that BMR tilts in different
cycles can be considered to belong to the same population and then proceed to
fold them into a single population by modifying their tilt angles to become
what they would be if hemispheric and cyclic reversals would not take place.

Table 2 shows the results of the Kolmogorov-Smirnov (KS) test for a
hierarchical merging of our BMRs into a single population. We perform this
merging separately for Hale and anti-Hale BMRs. Each pair of columns named
D-Statistic and p-value indicate the maximum difference between the empirical
cumulative distribution functions of both populations D-Statistic and the
significance with we can reject the null hypothesis that they belong to the
same population (1 - p-value)*100%.

Merging BMR tilts belonging to different cycles and hemispheres requires the
following manipulations (see Fig. 1):

  * •

In the case of relative tilt, all BMRs from the same hemisphere can be
combined without manipulation and BMRs of different hemispheres can be
combined after a sign change.

  * •

In the case of the absolute tilt, tilt angles belonging to opposite
hemispheres and subsequent cycles are consistent (i.e. see bottom-left and
top-right of Fig. 1). The other combination of cycle parity and hemisphere
needs to be adjusted by 180o in order to make all angles consistent.

We first start by combining BMRs from the same hemisphere for odd (21+23) or
even (22+24) cycles. Overall, the null hypothesis that BMRs from the same
hemisphere and odd/even cycles belong to the same population holds in almost
all cases. The only exception are 22N+24N for Hale regions and 22S+24S for AH
regions. We suspect that in these cases the null hypothesis is rejected due to
the fact that our observations for cycle 24 are incomplete, so we may not have
sufficient statistics to fully sample the tilt population (i.e. randomly
drawing BMRs from the common distribution will, in some cases, result in
different populations). However, as we progressively merge all regions of the
same hemisphere for all cycles (6th and 7th columns of Table 2) and all Hale
or AH regions (last two columns of Table 2) the null hypothesis that BMRs from
all cycles and hemispheres belong to the same population holds with a very
high degree of significance.

Figure 6: Top: Relative abundance of Hale (dark purple) and Anti-Hale (light
blue) regions as a function of latitude. The numbers on each column indicate
the number of regions in a given bin. Bottom: Boxplot illustrating the
distribution of tilt angles for each latitude. Bold horizontal lines mark the
median of the distribution. The sloping notches indicate the uncertainty on
the median estimation calculated using 1000 random subsets of the population.
The box indicates the extent of the 2nd and 3rd quartiles in the distribution
and the whiskers mark the 5% and 95% percentiles. Light-red denotes bins with
so few points that no meaningful estimation of the median can be made.

##  7 Joy’s Law for Hale and Anti-Hale BMRs

The combination of all cycles and hemispheres gives us enough statistics to
look at the dependence of tilt angle on latitude for AH BMRs in comparison to
HH BMRs. Figure 6 shows the relative abundance of HH and AH regions for
different latitude bins, as well as box plots of HH and AH relative tilt angle
distributions as a function of latitude. Overall, the relative abundance of HH
and AH regions as a function of latitude (top panel of Fig. 6) is very
similar. Performing a K-S test between the latitudes of HH and AH regions we
find that we cannot reject the null-hypothesis that they are the same
population with full confidence (p-value of 0.07). We interpret this as
evidence that the majority of AH regions are likely connected in some way with
the flux that gives rise to HH regions. The two distributions peak slightly
differently, with the AH distribution peaking at a higher latitude than the HH
distribution. Additionally, there is a significant difference between the two
distributions near the equator (0-5 bin). We interpret this as evidence that
some of the AH regions near the equator are likely normal HH regions belonging
to the opposite hemisphere.

In terms of the latitudinal dependence of the distribution of tilts (bottom
panel of Fig. 6). The first clear difference between HH and AH regions is the
spread in the distribution of tilt angles: for Hale regions 50% of all tilt
angles within each latitude bin are within 10o of the median, whereas for AH
regions they are within 20o. HH tilts systematically increase with latitude
and, thanks to the very good statistics, the median tilt can be determined
with good certainty (the uncertainty interval indicated by the notches in the
boxes is $\sim\pm 3^{o}$). AH regions, on the other hand, present
significantly more uncertainty in the position of their median ($\sim\pm
10^{o}$) due to their larger scatter and poorer statistics. In the case of the
30-55o and 35-40o bins (denoted with a ligh-red color) the uncertainty is so
large, that any measurement of the median is meaningless. The uncertainty in
AH medians makes it hard for us to place strong constraints on the dependence
of AH tilt angle on latitude. Nevertheless, it is very clear that there is a
systematic hemispheric tendency of the same sign as the one presented by HH
regions: i.e. the leading polarity of AH regions also tends to be closer to
the equator than the trailing polarity. It also seems that the median tilt
does increase with latitude starting int the 10-15o bin. It is possible that
the two first bins (0-5o & 5-10o) reflect the impact of regions belonging to
the opposite hemisphere that emerge across the 0o line.

Table 3: Akaike weights and KS test of 7 analytic distributions for HH and AH
regions. Distributions are ordered according to the relative AIC for HH
regions. Out of these distributions, the best fit to HH (AH) regions is the
T-student (von Mises) distributions (denoted using bold letters). Model
Selection and Goodness of Fit
---
Fitted |  | Hale - Best: Student’s T |  | Anti-Hale - Best: von Mises
Distribution |  |  $\Delta$AIC | AICw | K-S distance | K-S p |  |  $\Delta$AIC | AICw | K-S distance | K-S p
Student’s T (HH) |  | 0 |  $\sim$1.00 | 0.015 | 0.07 |  | 51 |  $\sim$0.00 | 0.033 | 0.73
Laplace |  | 41 |  $\sim$0.00 | 0.019 | 0.01 |  | 79 |  $\sim$0.00 | 0.042 | 0.44
Logistic |  | 181 |  $\sim$0.00 | 0.026 |  $\sim$0.00 |  | 59 |  $\sim$0.00 | 0.023 | 0.98
von Mises (AH) |  | 255 |  $\sim$0.00 | 0.043 |  $\sim$0.00 |  | 0 |  $\sim$1.00 | 0.047 | 0.29
Skew-Normal |  | 687 |  $\sim$0.00 | 0.059 |  $\sim$0.00 |  | 51 |  $\sim$0.00 | 0.034 | 0.71
Normal |  | 909 |  $\sim$0.00 | 0.071 |  $\sim$0.00 |  | 49 |  $\sim$0.00 | 0.033 | 0.73
Cauchy |  | 1265 |  $\sim$0.00 | 0.057 |  $\sim$0.00 |  | 190 |  $\sim$0.00 | 0.084 | 0.01
Best fitting parameters
|  | Hale - Best: Student’s T |  | Anti-Hale - Best: von Mises
Fitted |  |  |  |  | Standard |  |  |  | Standard |
Distribution |  | n | Location $\mu$ | Scale $s$ | Deviation |  | $\kappa$ | Location $\mu$ | Deviation |
Student’s T (HH) |  | 3.45 | 7.5o | 14.46o | 22.31o |  |  |  |  |
von Mises (AH) |  |  |  |  |  |  | 2.62 | 2.85o | 41.55o |

##  8 Optimal analytic fit to tilt angle populations

We now identify the analytic distributions that best capture the populations
of HH and AH tilt. We use two criteria to make model discrimination The first
one is Akaike’s information criterion (AIC; Akaike 1983). The AIC is a
powerful tool for discriminating between different non-nested models by making
an estimate of the expected relative distance between the fitted model and the
unknown true mechanism that generated the observed data. The AIC for a model
$M_{j}$ is defined as:

$\mathrm{AIC_{j}}=-2\mathcal{L}(M_{j})-2n_{j},$ (2)

where $\mathcal{L}(M_{j})$ is the log-likelihood of model $M_{j}$:

$\mathcal{L}(M)=\sum_{i=1}^{n}\log(\mathrm{P}(D_{i}|M)).$ (3)

and $n_{j}$ the number of parameters of model $j$. The model with the smallest
AIC is chosen as the best. Minimizing AIC looks for the model with the largest
log-likelihood. However, log-likelihood alone is not sufficient to
discriminate between models because it is biased as an estimation of the model
selection target. This bias was found by Akaike (1983) to be approximately
equal to each model’s number of parameters ($n$), and thus the presence of the
second term in Eq. 2. Together, log-likelihood and $n$ are used to strike a
balance between bias and variance (or the trade-off between under-fitting and
over-fitting).

The relative nature of the AIC is better represented by calculating the
relative AIC differences:

$\mathrm{\Delta^{AIC}_{j}}=\mathrm{AIC_{j}}-\min(\mathrm{AIC}).$ (4)

This in turn can be used to estimate the likelihood of a model given the data:

$\mathcal{L}(M_{j}|D)\propto\exp\left(-\frac{\mathrm{\Delta^{AIC}_{j}}}{2}\right),$
(5)

and use it to calculate the Akaike weights:

$\mathrm{Aw_{j}}=\frac{\exp\left(-\frac{\mathrm{\Delta^{AIC}_{j}}}{2}\right)}{\sum_{k=1}^{K}\exp\left(-\frac{\mathrm{\Delta^{AIC}_{k}}}{2}\right)},$
(6)

which are a measure of the probability that the model $M_{j}$ is the best
model given the data. For more information about AIC we recommend the
excellent book by Burnham & Anderson (2002).

It is very important to highlight that the significance of AIC is strongly
dependent on an appropriate choice of models. Applying AIC to a set of very
poor models will always select one estimated to be the best (even though that
model may still be poor in an absolute sense). To account for that, we also
use the one sided K-S test under the null hypothesis that the HH and AH
populations come from a given analytical distribution. Table 3 shows the
result of our fits. We test all distributions with infinite or circular
support in the SciPy stats package (Virtanen et al., 2020).

The best fit, by far, to the HH tilt angle population is a Student’s T
distribution. However, it is also the only one for which the null hypothesis
of the K-S test cannot be rejected with confidence. We find this a difficult
result to interpret. The Student’s T distribution is overwhelmingly used to
estimate the uncertainty in the estimation of the mean of a population
governed by a normal distribution, given a small sample of observations, when
the true variance of the distribution is unknown. It is defined as:

$f(x|n,\mu,s)=\frac{\Gamma\left(\frac{n+1}{2}\right)}{s\sqrt{n\pi}\Gamma\left(\frac{n}{2}\right)}\left(1+\frac{(x-\mu)^{2}}{ns^{2}}\right)^{-\frac{n+1}{2}},$
(7)

where $\Gamma$ is the gamma function, $n$ is a shape parameter typically
associated with the number of degrees of freedom of a T distribution, $\mu$ is
the location parameter, and $s$ the scale parameter. We could not find an
example where the T distribution is used to describe a physical population nor
a physical generative model for this distribution. We thought perhaps that
this outcome stems from the fact that we were folding BMRs from all latitudes,
each maybe with a normal distribution, but found that the T distribution is
also the best fit, by far, for every 5o latitudinal bin shown in Figure 6. One
possible explanation could be that the interaction of the rising flux-tube
that gives rise to BMRs is buffeted by a limited amount of effective
interactions with helical convective turbulence. These would act as the small
sample that gives rise to a mean imparted tilt described by a distribution
with heavier tails than a normal distribution; even if the buffeting itself
would be normally distributed. The optimal fitted parameters (see Table 3) are
3.45 degrees of freedom ($n$), which we speculate could be related to the
effective number of interactions with turbulence during BMR rise; a location
($\mu$) of 7.5o, which quantifies the systematic tilt presented by HH BMRs;
and a scale ($s$) of 14.46o which translates to a standard deviation of
22.31o.

In the case of AH regions, the best fit (by far) is a von Mises distribution
(also known as the circular normal distribution), defined as:

$f(x|\kappa,\mu)=\frac{e^{\kappa\cos(x-\mu)}}{2\pi\mathrm{I_{0}}(\kappa)},$
(8)

where $\kappa$ is a scale parameter analogous to $\sigma^{2}$ in a normal
distribution, $\mu$ is the location parameter, and $\mathrm{I_{0}}(\kappa)$ is
the modified Bessel function of order 0. The von Mises distribution has
circular support ($[-\pi,\pi]$) so its better suited for angular random
variables. The optimal fitted parameters (see Table 3) are a location ($\mu$)
of 2.85o, which demonstrates that AH BMRs also have a hemispheric systematic
inclination (although not as marked as HH regions); and a scale factor
($\kappa$) of 2.62 which translates to a standard deviation of 41.55o.
Although AIC indisputably shows a von Mises distribution as the best fit to
our AH data, the K-S test can only confidently reject the Cauchy distribution
as a possible analytical fit to AH tilts. All other distributions are still
consistent enough with the data given the K-S test. Having more cycles worth
of AH observations would help give certainty to our results. However,
repeating the exercise for the individual 5o bins shown in Figure 6 has the
same result as with the HH distribution. In every single case the AIC finds
the von Mises distribution to be the best distribution by far.

Figure 7: Analytic fit to the relative tilt angle distribution for HH (dashed
purple) and AH (dashed blue) populations. The empirical distributions
associated with each population are shown using the same colors, but a solid
line .

Figure 7 shows the relative tilt angle distribution for HH and AH regions once
all cycles and hemispheres have been combined, as well as the optimal fits to
HH and AH BMR tilts. The AH distribution is clearly more spread than the HH
distribution, which is also evident in Figure 6 and Table 3. It also shows how
both distributions are shifted towards positive relative tilts, exhibiting the
same hemispheric systematic tendency for HH and AH regions.

Figure 8: Left: Analytic fits of the von Mises (solid blue) plus Student’s t
(dashed purple) distributions to the absolute tilt of our entire database
(solid gray). Tilt angles have been shifted by 1800 to place the focus of the
plot on the AH regions. Right: Zoom in on the AH peak. The black dotted line
is the same of both plots, as are the orange vertical dashed lines at
$\pm$90o.

##  9 The Anti-Hale population is not explained by Hale’s tilt distribution
tails

Figure 8 demonstrates that the tails of the HH distribution (dashed purple)
cannot explain the relative frequency of AH regions, which needs to be
explained by a second distribution (solid blue). The standard deviation
associated with the HH and AH distributions indicate that in the vicinity of a
tilt angle of $\pm$90o (dotted orange vertical lines), the majority of AH
regions are likely to belong to the HH population. However, the further from
$\pm$90o the more likely that AH regions belong to the AH distribution. Around
14o past $\pm$90o, AH regions are equally likely to belong to the HH and AH
distributions, after which they are significantly more likely to belong to the
AH distribution.

Given that the number of AH regions in our data is large enough (421), the
results from both AIC and K-S rule out with a high degree of certainty that AH
regions belong to the HH distribution. We take the latter as evidence that AH
regions constitute a distinct population and therefore their origin is
different than the one of HH regions.

##  10 What is the origin of Anti-Hale regions?

The excess frequency of AH regions in the most equatorial bin (0-5o), relative
to the HH distribution (see Fig. 6), indicates that these equatorial AH
regions are likely associated with the dominant toroidal flux system of the
opposite hemisphere (as reported by McClintock et al., 2014).

For all other bins (5-35o) the latitudinal distribution of HH and AH regions
are statistically indistinguishable. We see this as evidence of a strong
connection between them. The systematic properties of AH regions suggest that
they also originate from magnetic fields that are primarily toroidal (hence
their systematic East-west orientation and systematically positive relative
tilt. We speculate that AH regions are connected to the prior emergence of HH
regions with a very strong associated poloidal field contribution. This
poloidal field could be sheared on the spot by differential rotation into a
toroidal field (of the opposite polarity) that is strong enough to produce an
AH region. This hypothesis may also explain why AH regions seem to be more
frequent in stronger cycles (see Fig. 5), given that there is an increased
likelihood of seeing large sunspot groups in stronger cycles Muñoz-Jaramillo
et al. (2015a). Our hypothesis could be validated if the emergence of AH
regions is preceded consistently by HH regions with strong poloidal field
contributions at a similar latitude and longitude. If our hypothesis is true,
AH regions would still be consistent with the existence of largely coherent
toroidal flux systems inside the solar interior and may be playing an even
more important role on the solar dynamo that is currently believed.

##  11 Summary

We use a database consisting of 9,243 unique bipolar magnetic regions
belonging to the last 4 solar Cycles (21, 22, 23, 24) measured by two ground-
based instruments, KPVT/512 and KPVT/SPMG and two space-based instruments,
SOHO/MDI and SDO/HMI. Bipolar magnetic regions belonging to different
instruments are calibrated by homogenizing BMR flux distributions and applying
a flux threshold below which measurements of bipolar magnetic regions are
likely to be strongly affected by instrument discrepancies. We find that the
only instrument that needs a calibration factor (0.8) is SOHO/MDI, which is in
good agreement ($1/0.8=1.25$) with the 1.3 factor found by Liu et al. (2012)
to calibrate SDO/HMI to SOHO/MDI for strong magnetic fields, and adopt a flux
threshold of $3\times 10^{21}$Mx. This leaves 7,511 bipolar magnetic regions
that we consider to be homogeneous enough to be studied together. Out of this
revised database, 421 (5.61%) bipolar magnetic regions have anti-hale polarity
orientation, which enables a robust statistical characterization that is not
possible without the combination of observations from multiple solar cycles.

Our results show that we can reject, with a very high degree of certainty,
that anti-Hale bipolar magnetic regions belong to the same population as Hale
regions. We also rule out, with a very high degree of certainty, that anti-
Hale bipolar magnetic regions can be explained as Hale regions whose polarity
orientation is an extreme realization of a unique physical mechanism. Only
anti-Hale regions that are oriented almost completely in the north-south
direction (within 14o of a $\pm$90o fully north-south tilt) are likely to be
flipped Hale regions.

Our results support the claim by Stenflo & Kosovichev (2012) that anti-Hale
and Hale regions do not have the same origin, and reject the suggestion by
McClintock et al. (2014) that the anti-Hale regions can be accounted by the
tails of the tilt angle distribution of the Hale bipolar magnetic regions.
However, we disagree with the interpretation of Stenflo & Kosovichev (2012)
that the existence of anti-Hale regions is in contradiction with the existence
of a largely coherent toroidal flux system.

Our results show that Anti-Hale regions have a preferred East-West orientation
and follow the same Joy hemispheric tendency as Hale regions: i.e. anti-Hale
regions also have a tendency to have leading (trailing) polarities that are
closer to the equator (pole). This means that they originate from largely
toroidal fields with a polarity orientation opposite to each hemisphere’s
dominant flux system. We speculate that these toroidal fields are sheared
locally (”on the spot”) by differential rotation from the poloidal field
associated with preceding Hale regions possessing a strong poloidal signature.
This would explain why the relative frequency of anti-Hale regions seems to be
larger for stronger cycles. It also explains why they are observed coexisting
with Hale regions at similar latitudes. Finally, it also explains why anti-
Hale regions have a latitudinal distribution that is statistically
indistinguishable from that of Hale regions. This hypothesis can be validated
(falsified) in future work by studying if there is a consistent presence of
Hale regions with strong poloidal field contributions at similar latitude and
longitude preceding anti-Hale emergence. If our hypothesis is true, the
generation of anti-Hale regions would be acting as a strong form of non-linear
quenching mechanism for the solar cycle because they would counteract the
strong positive contribution of the originating Hale regions. It would also
reconcile the existence of anti-Hale regions with the commonly held picture of
a largely coherent toroidal flux system

The main results of this work are the following:

  1. 1.

The tilts of Bipolar Magnetic Regions from different cycles and hemispheres
are consistent enough with each other that they can be considered to be from
the same population and combined together.

  2. 2.

There seems to be monotonic relationship between cycle amplitude and the
relative abundance of anti-Hale region emergence.

  3. 3.

The latitudinal distribution of Hale and anti-Hale regions can be considered
to be the same, with the exception of the equatorial region (0-5o) where anti-
Hale regions are comparatively more frequent.

  4. 4.

Anti-Hale regions present the same systematic relative tilt angle inclination
than Hale regions. i.e. anti-Hale regions’ leading polarity tends to be closer
to the equator than their trailing polarity.

  5. 5.

Anti-Hale region relative tilt also seems to increase with latitude, but this
result may not be significant.

  6. 6.

Hale region relative tilt distribution is best fitted by a Student’s T
distribution centered on 7.5o and with a standard deviation of 22.31o.

  7. 7.

Anti-Hale region relative tilt distribution is best fitted by a von Mises
distribution centered on 2.85o and with a standard deviation of 41.55o.

  8. 8.

Hale and anti-Hale regions cannot be considered to belong to the same
population.

  9. 9.

Anti-Hale regions cannot be explained solely as Hale regions whose polarity
has been flipped. i.e. as exceptional Hale regions.

##  12 Data Repository

The database used for our analysis has been published by Muñoz-Jaramillo et
al. (2021) in the Harvard Dataverse and can be accessed at
https://doi.org/10.7910/DVN/QEMSZ2.

##  13 acknowledgments

LEC acknowledges partial support from the Center of Excellence in Astrophysics
and Associated Technologies (PFB06, CONICYT), and posthumously thanks Dr.
Adelina Gutiérrez for her pioneering Solar Physics course (1971) at the
Universidad de Chile. This research was funded by NASA Living With a Star
grant NNH15ZDA001N. All our analysis was performed using the open-source NumPy
(Harris et al., 2020) and SciPy (Virtanen et al., 2020) packages. We thank
Aimee Norton, an anonymous referee, and ApJ’s statistics editor for very
useful feedback and suggestions.

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