We compare the characterization of the SAA based on previous studies vs. our proposed measures. This characterization includes both the area and the coordinates of the SAA center.

Previous studies defined the SAA area as that where the geomagnetic field intensity \(|{\vec {B}}|\) at Earth’s surface is lower than 32,000 nT (De Santis et al. 2013; Pavón-Carrasco and De Santis 2016):

$$\begin{aligned} |{\vec {B}}|<32,000\, \mathrm{nT}. \end{aligned}$$

(1)

From hereafter we term the area based on (1) as *S*0. This definition is practical for space safety purposes. However, from a more fundamental point of view, (1) is affected not only by regional spatio-temporal field variations, but also by global changes. Fig. 1 illustrates this point. Under a hypothetical scenario of entire field magnitude decrease with no pattern change, a fixed critical threshold (such as 32,000 nT) for the SAA would suggest that its area increases despite no regional variation.

To overcome this possible problem, alternatively, we factor the critical intensity value by the instantaneous mean surface intensity outside the SAA normalized by its value at the middle of the investigated period, which we term \(F_{\mathrm{out}}\):

$$\begin{aligned} F_{\mathrm{out}}={{\int _0^{2\pi } \int _0^{\pi /2} |{\vec {B}}|(t) \sin \theta d\theta d\phi + \int _{\pi/2}^{3\pi /2} \int _{\pi /2}^\pi |{\vec {B}}|(t) \sin \theta d\theta d\phi } \over {\int _0^{2\pi } \int _0^{\pi /2} |{\vec {B}}|(1930) \sin \theta d\theta d\phi + \int _{\pi /2}^{3\pi /2} \int _{\pi /2}^\pi |{\vec {B}}|(1930) \sin \theta d\theta d\phi } }. \end{aligned}$$

(2)

For this purpose, for the area outside the SAA, we consider the northern hemisphere plus the Pacific (i.e. between \(90^\circ \hbox {E}\) and \(270^\circ \hbox {E}\)) southern hemisphere:

$$\begin{aligned} |{\vec {B}}|<32000 F_{\mathrm{out}} \, {\mathrm{nT}}, \end{aligned}$$

(3)

where \(\phi\) and \(\theta\) are longitude and co-latitude, respectively. Similar planetary-scale averages were recently invoked to quantify the Pacific/Atlantic geomagnetic SV dichotomy (Dumberry and More 2020) or the northern/southern differences in the SV-induced neutral density of the thermosphere (Cnossen and Maute 2020). Note that while the choice of the mid-term year 1930 in the denominator of (3) is completely arbitrary, this has no consequence on the resulting rate of change of the SAA area. From hereafter, we term the area based on (3) as *S*1. With this definition, the critical value varies with the mean surface intensity away from the SAA. As such, it captures the regional variation of the SAA area, independent of the change in the field magnitude elsewhere.

Next, previous studies tracked the SAA center based on the point of minimum field intensity, both at Earth’s surface (Hartmann and Pacca 2009; Finlay et al. 2010; Aubert 2015; Terra-Nova et al. 2017, 2019) and at higher altitudes (Anderson et al. 2018). Following Terra-Nova et al. (2017), we reproduce this result by first searching for the grid point with lowest intensity and then applying second-order polynomial interpolations using two neighboring points in each direction to resolve off-grid values. From hereafter, we term these coordinates as Min. Note that this definition of a center is advantageous in some useful applications, e.g. in determining the maximum cutoff of radiation vs. duration at a certain radiation level for a spacecraft traversing the SAA. In the context of a core origin, if the shape of the SAA is significantly anisotropic, the minimum point might not well represent the center of the structure (for an illustration see Fig. 2).

Alternatively, centers of mass were invoked to identify and track centers of intense flux patches on the CMB in numerical dynamos (Amit et al. 2010) and geomagnetic field models (Amit et al. 2011). Centers of mass were also used to identify the SAA at \(\sim 800\) km altitude (Casadio and Arino 2011; Schaefer et al. 2016). Here, we calculate centers of mass to determine the longitude and co-latitude of the SAA center, \(\phi _{\mathrm{cm}}\) and \(\theta _{\mathrm{cm}}\), respectively, at Earth’s surface:

$$\begin{aligned} \phi _{\mathrm{cm}}= & {} {{\sum _S \phi _i w_i}\over {\sum _S w_i}}, \end{aligned}$$

(4)

$$\begin{aligned} \theta _{\mathrm{cm}}= & {} {{\sum _S \theta _i w_i}\over {\sum _S w_i}}. \end{aligned}$$

(5)

The summations in (4)–(5) are over the SAA area, either *S*0 or *S*1, which we term CM0 and CM1, respectively. The weight *w* is given by the inverse of the intensity

$$\begin{aligned} w_i={{1}\over {|{\vec {B}}_i|}}. \end{aligned}$$

(6)

Determining the center of the SAA based on the center of mass of its area well represents the center even if its shape is significantly anisotropic.

Finally, we monitor the time dependence of the value of minimum surface intensity \(|{\vec {B}}|_{\rm min}\). In addition, we define a relative minimum surface intensity \(|{\vec {B}}|_{\mathrm{min}}^{\mathrm{rel}}\) with respect to the instantaneous field intensity outside the SAA. Similar to (3), we calculate \(|{\vec {B}}|_{\mathrm{min}}^{\mathrm{rel}}\) using \(F_{\mathrm{out}}\) (2) as follows:

$$\begin{aligned} |{\vec {B}}|_{\mathrm{min}}^{\mathrm{rel}}=|{\vec {B}}|_{\mathrm{min}}/F_{\mathrm{out}}. \end{aligned}$$

(7)

Both areas *S*0 and *S*1 were calculated using a simple trapezoid numerical scheme. Tests of the dependence of the results on the grid size show very weak sensitivity and fast convergence with increasing resolution. For all calculations we used a \(1^\circ \times 1^\circ\) grid in longitude and co-latitude. With this grid size, the computed properties (i.e. the area and coordinates of the center) practically reach asymptotic values with decreasing grid size.