Abstract
Geomagnetically induced currents (GICs) pose a significant space weather hazard, driven by geomagnetic field variation due to the coupling of the solar wind to the magnetosphere-ionosphere system. Extensive research has been dedicated to understanding ground-level geomagnetic field perturbations as a GIC proxy. Still, the non-uniform aspect of geomagnetic fluctuations make it difficult to fully characterize the ground-level magnetic field across large regions of the globe. Here, we focus on localized geomagnetic disturbances (LGMDs) in the North American region and specify the degree to which these disturbances are localized. Employing the electrodynamics-informed Spherical Elementary Current Systems (SECS) method, we spatially interpolate magnetic field perturbations between ground-based magnetometer stations. In this way, we represent the ground magnetic field as a series of heatmaps at high temporal and spatial resolution. We leverage heatmaps from storm time during solar cycle 24 to automatically identify LGMDs. We build a statistical picture of the frequency with which LGMDs occur, their scale sizes, and their latitude-longitude aspect ratios. Additionally, we use an information theory approach to quantify the dependence of these three attributes on the phase of the solar cycle. We find no clear influence of the solar cycle on any of the three attributes. We offer some avenues toward explaining why LGMDs might behave broadly the same whether they arise during solar maximum or solar minimum.
1 Introduction
Detailed descriptions of geomagnetic field activity near the surface of the Earth are of acute interest to a variety of communities. Utility providers (including, but not limited to, electric power transmission companies, railway companies, and fluid pipeline operators) can wield information about ground-level geomagnetic activity as a means to mitigate disruption to their infrastructure by geomagnetically induced currents (GICs) (; Patterson et al., 2023; ). It is common to derive an induced geoelectric field, and the corresponding GICs, based on the measured geomagnetic field where no direct GIC measurements are available (). Additionally, scientists value the ground-level magnetic field as an indicator for phenomena in the atmosphere, the inner magnetosphere, and beyond (e.g., Russell et al. (2008), ). The United States government maintains a National Space Weather Strategy and Action Plan to safeguard economic interests and national security against the threat of geomagnetic storms among other space weather effects (; ).
Of particular importance to these communities is knowledge about exactly where on Earth the horizontal component of the geomagnetic field, , is experiencing (or will experience) a perturbation away from its baseline value. After all, disturbances to the geomagnetic field are often spatially localized, especially during periods of intense temporal variability (), which is when the threat from GICs is most severe (Pirjola, 2000). Areas in which is markedly more perturbed from baseline than in the surroundings can be viewed as instances of this localization effect, and will hereafter be referred to as localized geomagnetic disturbances, or LGMDs, following the terminology used by Vandegriff et al. (2024). In this study, we focus on LGMDs in , the north-south vector component of . A precise characterization of an LGMD’s spatial extent could help explain, for example, why GICs might have occurred in one region’s power network but not in that of another.
Several explanations for the localization effect have been proposed. It is known that the nonuniform electric conductivity of Earth’s crust plays a role in determining the deflection of all components of the geomagnetic field, including , implying that localized magnetotelluric structures can help produce LGMDs (). Substorm activity also contributes to localized magnetic and electric field enhancements by way of intensification of the westward auroral electrojet and current structures therein (Viljanen et al., 2001; Ngwira et al., 2015; 2018). Additionally, geomagnetic pulsations, such as those caused by Kelvin-Helmholtz instabilities in outer-magnetosphere plasma flow, have been shown to produce LGMDs during geomagnetic storms even in mid-latitude regions (). Depending on the space weather conditions and the geographic location on Earth, these phenomena and others may vary in the magnitude of their contribution to the generation of LGMDs. Though the full suite of LGMD drivers has yet to be totally and precisely characterized, the contributing factors mentioned above represent rapid progress in understanding this complicated localization effect.
Some studies have investigated LGMD spatial scale by comparing readings at multiple stations, with attention paid to the geographic distance between those stations (; Pulkkinen et al., 2015). To give some recent examples, found that the time derivative of the horizontal component of the magnetic field could be as much as three times higher at a single magnetometer station compared to the average of all magnetometers situated across a relatively small (500 km) region of Fennoscandia. They also found that nonuniformities in the geomagnetic field at the 200 km scale can be responsible for a large percentage of the strength of GICs. Similarly, examined the BEAR Fennoscandian magnetometer network and the CARISMA North American network to determine that the correlation scale is only about 500 km between magnetometers located at similar latitudes, and this scale drops to a mere 100 km in the vicinity of the auroral oval.
However, single-point measurements from magnetometers are insufficient to fully describe the size and shape of LGMDs because the distance between magnetometer stations may be larger than the scale sizes of the LGMDs themselves. That is to say, too sparse a distribution of magnetometer stations will not allow size scales to be precisely measured. There may be a few points of measurement, but the magnetic field’s behavior between those points is unknown. To overcome this deficiency, one requires a method by which to estimate (or its time derivative ) at any point of interest regardless of the presence of a nearby magnetometer station or of an LGMD; in other words, a robust interpolation method is needed. Simply using the measurement of from the nearest magnetometer is generally not sufficient because it could differ greatly from the actual value at the point of interest, depending on the characteristics of the surrounding geology () and magnetospheric conditions (Viljanen, 1998). Even using a proximity-weighted average from multiple nearby magnetometers’ readings may result in notable estimation errors in regions where magnetometers are far apart from one another (). Applying the Spherical Elementary Current Systems (SECS) interpolation method addresses this need effectively. While SECS is complicated compared to several geometric methods, its electrodynamics-based approach delivers enhanced accuracy, particularly in magnetically disturbed conditions (). Interpolation using spherical harmonic analysis – for instance, as in Upendran et al. (2022) – is a competitive alternative to SECS; however, spherical harmonic analysis is more susceptible to interpolation artifacts (Rigler et al., 2019).
The goal of this investigation is to create a dataset of LGMDs on the North American region and to characterize their sizes and shapes during intervals of enhanced activity caused by geomagnetic storms. We use SECS to generate maps of geomagnetic field perturbations with high spatial resolution from a large cache of historical magnetometer data spanning storm times from 2009 through 2019. We then analyze the LGMDs that are displayed in these maps to reveal details about the frequency with which they occur, their spatial extent, and their longitude-latitude aspect ratios.
Finally, we quantify the solar cycle dependence of our LGMD dataset based on the Kullback-Leibler divergence, an important concept in information theory. Information theoretical approaches can be powerful tools when applied to space physics and space weather. used the Kullback-Leibler divergence to quantify the relative importance of various solar wind parameters in the generation of chorus waves in the magnetosphere. Wing et al. (2018) studied the memory of the solar cycle and Snelling et al. (2020) studied the causal relationship between solar flares using transfer entropy and mutual information, respectively, two quantities which are closely related to the Kullback-Leibler divergence. Readers interested in further uses of information theory in space science should refer to , , and references therein.
2 Data and data preparation
Obtaining the most complete picture of LGMD characteristics requires simultaneous data from multiple magnetometers. These data must describe long enough time scales so as to capture any variation due to the solar cycle. They must also exist at a fine enough temporal resolution to make clear how LGMD attributes evolve during periods of high geomagnetic activity. To address these requirements, we use 1-min cadence baseline-subtracted ground magnetic field data from the SuperMAG collaboration (). Our analysis in this work specifically examines the northward component of the perturbation vector of the interpolated magnetic field. We choose a single vector component because the northward () and eastward components do not necessarily behave the same (Viljanen and Tanskanen, 2011), and a thorough comparison between components is best reserved for future studies.
This study examines LGMD scale size over a particular region of interest (though it could, in principle, be replicated to study LGMDs across any given section of Earth’s surface). We choose the region of interest to be within the quadrangle which is aligned with the standard geographic coordinate system, and which extends from 45N to 55N and from 80W to 130W, as indicated by the orange region in Figure 1. Geographic coordinates are used here to be consistent with the use of geographic coordinates in Section 3, and offer no substantial change to the region size, shape, or data availability compared to geomagnetic coordinates. This particular region was chosen because it is of interest to contemporary GIC research: it lies at middle latitudes, is a geographically large region with good magnetometer coverage, and constitutes much of North America, which hosts a wide variety of subterranean magnetotelluric structures that influence the surface geomagnetic field. Such structures have been studied extensively, such as by , and mapped as part of the National Science Foundation’s EarthScope project (Schultz, 2009). Geomagnetic storms have created GIC-related disruptions in mid-latitude North America, including the 2003 “Halloween storm” (minimum SYM-H index 432 nT) which markedly affected power systems (; Piccinelli and Krausmann, 2014) and the March 1989 storm (minimum SYM-H 710 nT) which caused widespread blackouts in the northeastern United States and eastern Canada (; ; ). To the west of the Atlantic coast of North America, where the conductivity of the crust tends to be higher, the March 1989 storm still created hazardous ground-level geoelectric fields (Wei et al., 2013). Even storms weaker than the March 1989 storm have caused GICs in this western part of the region (e.g., ; , the latter analyzing an event which occurred as recently as 2023).
FIGURE 1
We use only data from magnetometers that are within the larger region extending from 35N to 70N and from 60W to 150W (also in geographic coordinates), shown as the blue quadrangle in Figure 1. This allows the region of interest to be surrounded by magnetometer stations, a constraint that benefits the SECS method by reducing the magnitude of interpolation error at the edges. We further restrict the magnetometers we use based on data availability. We only use those with more than 25% of the data available each year, and less than 35% missing data averaged across all years, from 2009 through 2019. (The final selection of magnetometers is listed in Supplementary Table S1). After this step, our dataset consists of measurements from 25 magnetometers whose locations are marked in Figure 1.
We then discard timesteps wherein any magnetometer is missing a data point. Our SECS calculations described in Section 3 are only carried out when data from every magnetometer is available, which ensures consistency across timesteps with regard to SECS inversion. Therefore, we make this data availability restriction in order to preserve as many data points as possible while maximizing the number of magnetometers available for analysis.
Our choice of years across which our dataset spans – 2009 through 2019 – represents one full solar cycle (Solar Cycle 24). This allows all phases of the solar cycle to be captured, which is important because geomagnetic activity, as well as the magnetospheric processes that cause it, depend on the current solar cycle phase. We have chosen Solar Cycle 24 in particular because it is the most recent complete cycle and has the most SuperMAG stations with 1-min data available.
Finally, we examine only times during which geomagnetic storms occur. We use the same list of storms as Pinto et al. (2022), who define a storm as the period where the SYM-H index decreases below −50 nT and remains below this threshold for at least 2 h, plus 12 h of lead time before the SYM-H minimum and 24 h of recovery time afterward. Thus our final magnetometer dataset is composed of 183 storms and a total of 395,463 data points (equivalent to a total of 274.6 days). This is about 7% of the 5,784,480 data points that are in the full 2009–2019 period. We reduce our dataset in this way because we wish to study LGMDs during periods of enhanced geomagnetic activity.
3 Methods
3.1 Spherical elementary current systems
As mentioned before, extending measurements to regions without magnetometers can be a difficult task due to the influence of localized phenomena throughout the magnetosphere and the lithosphere. This study seeks to overcome such difficulties and precisely reconstruct geomagnetic field variations by interpolating with the Spherical Elementary Current Systems (SECS) technique (). SECS is commonly used to estimate geoelectric fields for the purpose of calculating GICs over regional-scale infrastructure (e.g., Wik et al. (2008), ). A summary of the technique is presented in this section; readers looking for a more in-depth explanation are encouraged to refer to Vanhamäki and Juusola (2020).
SECS interpolation operates by representing the ionospheric current distribution with a parametric model. Empirical determination of the model’s parameters at each moment in time is carried out based on magnetometer observations, which allows for the assumption that the model current distribution causes all observed ground-level geomagnetic activity. Once the model is fitted using this approach, one can apply the Biot-Savart law to the model current distribution in order to calculate the magnetic field deflection at any point within a region of interest.
The model current distribution is a superposition of curl-free and divergence-free vector functions defined on a common spherical surface, each with a different location for its pole. These basis functions are called elementary current systems. One is free to choose the placement of the poles of elementary current systems, but typically they are arranged in a latitude-longitude grid for convenience. By Helmholtz’s Theorem, any smooth, differentiable vector function can be exactly decomposed into such basis functions; using a finite number of basis functions at worst only approximates the original vector field. The divergence-free part () and curl-free part () of each elementary current system are defined as (Equation 1):
One would typically consider the coordinate system in which is the radial distance from the center of Earth, and and are geographic longitude and colatitude, respectively. The “primed” spherical coordinate system differs from the standard “unprimed” system in that its north pole, where , is defined to be at the location of that current system. The azimuthal angle is defined accordingly. As opposed to the geographic unprimed coordinate system, we will hereafter call a primed coordinate system an egocentric coordinate system, since the pole of each current system has its own unique primed coordinate system centered on itself.
Here the currents are assumed to lie in an infinitely thin current shell of radius (which typically takes the value + 110 km, where is the radius of Earth, so as to approximately correspond with the heights of the maximum Pedersen and Hall conductivities). The factors and are empirically determined at every moment in time in order to fit this model ionosphere to magnetometer data. The total vector field representing ionospheric currents may not actually equal the currents in the real ionosphere, especially if the associated field-aligned currents are not approximately radial. However, the elementary currents still form a basis from which the magnetic field can be later reconstructed.
Under the assumptions of this current-shell model, it can be shown that the ground magnetic effect of the curl-free current systems vanishes within the current surface (). This allows to be rewritten more simply as . Thus, the horizontal component of the magnetic field due to a single current system at a point is solely due to and can be shown to be (Equation 2):
The constant in this equation denotes the magnetic permeability of free space. There is also a radial component to this magnetic field, but it will not be used in this analysis because we are only concerned with horizontal field components (specifically, this study eventually only examines the northward component). Then, the total horizontal vector magnetic field is the sum of the magnetic contributions from each current system.
To determine the contributions from each current system, one can define a vector containing the ground-truth magnetometer measurements at stations 1 through , a vector of scaling factors for current systems 1 through , and a transfer matrix (Equation 3):
Each element represents the or component of the magnetic effect that current system has on magnetometer . One must then solve the matrix equation for and plug the resulting scaling factors into Equation 2 to obtain interpolated magnetic field values. This is typically done using singular value decomposition and setting small singular values to 0 to properly condition the matrix. One usually considers “small singular values” to mean those that are less than the maximum singular value multiplied by some cutoff coefficient . The best value of depends on the specifics of the SECS model in question; refer to Section 3.2 for information on how this value was determined for the present case.
When using Equation 2 to find the contribution of each current system to the magnetic field, it is important to notice again the difference between the geographic and the egocentric coordinate systems. These contributions are most easily added together when recognizing that a contribution vector at geographic colatitude and longitude due to a current system at geographic colatitude and longitude may be transformed from the egocentric to the geographic coordinate system with Equations 4, 5:
The total instantaneous magnetic field in the geographic coordinate system is then the sum of each contribution vector , which we calculate component-wise to find the northward component, , and the eastward component, . Since is this study’s variable of interest, we focus solely on without using .
3.2 Model details
We define our model current distribution as the superposition of a set of current systems whose poles are placed on a quadrangular latitude-longitude grid extending from 40.5N to 60.5N and from 70.5W to 140.5W (the green quadrangle in Figure 1) in geographic coordinates. The current system array is offset half a degree from whole-number placement to avoid colocating the current systems with points at which the magnetic field is evaluated, as this would cause a singularity to arise in Equation 2. There is also the danger of a singularity if a current system is colocated with a magnetometer; however, this becomes obvious with a cursory inspection of the interpolation outputs. If one places the current systems such that this is not a problem, one can rest assured that no singularities will arise later because the locations of current systems and magnetometers do not change over time.
Poles are placed every 2°, making this a 10-by-35 array of current systems. The performance of SECS interpolation is not especially sensitive to small changes to the number and spacing of the current systems, so the 2-degree separation was chosen based on its use in . We further validated this choice by performing joint Bayesian optimization of the number of rows and columns of current systems (while keeping the boundaries of the array fixed) using the Optuna package for Python (). The objective of this optimization was to minimize the absolute error between the model outputs at the location of each magnetometer and the measurements of the magnetometers themselves, averaged across magnetometers in each timestep, and then averaged across all timesteps in the dataset. The errors at each magnetometer were calculated using a leave-one-out scheme in which the model was fit using data from all magnetometers except the one in question, meaning a new model was fit for each magnetometer, all identical except for which magnetometer was omitted. The optimization found that the error was minimized for an 8-by-39 array of current systems. This agrees with our similar choice of array dimensions, whose error is less than 1% higher than the optimum. As such, we kept the 10-by-35 array so as to maintain a rounded 2-degree separation, as well as for possible future comparison to studies such as .
Together with the number of rows and columns of current systems, the above procedure also jointly optimized a third quantity: the cutoff coefficient . We use the best value determined through this optimization, . This value is comparable in magnitude to what has been used in previous related studies (; Weygand et al., 2011).
When creating a SECS-based model, it is sometimes helpful to introduce an additional current sheet located at a negative altitude (that is, internal to the planet). Doing so may allow one to delineate which part of geomagnetic perturbations arise from ionosphere-associated phenomena and which perturbations arise from currents induced in the subsurface (Pulkkinen et al., 2003). However, the goal of the SECS model as it is used here is only to reproduce geomagnetic perturbations, not to explain them in terms of their contributing equivalent currents. As a result, the additional current sheet has not been employed in this study.
3.3 Identification and characterization of LGMDs
Measurements of are interpolated to give estimates of this field component at a set of points within the region of interest. The points could have any spatial density since they are sampled from the output of the SECS method, which is a continuous function. We limit ourselves to making estimations on a regularly spaced 14-by-32 grid of points to keep computation time under control. Each of these “heatmaps” (e.g., Figure 2) is associated with a set of simultaneous magnetometer measurements, so exactly one heatmap is created per timestep in our dataset.
FIGURE 2
Once the heatmaps have been created, one can visually inspect them to view the estimated extent of LGMDs. However, we need to define a method to locate LGMD in a given heatmap. We do so here by granting a quantitative aspect to the definition of LGMD given in Section 1; that is, by defining a threshold value . Any point in a heatmap whose absolute value exceeds that of the threshold (i.e., ), is considered part of an LGMD. Points identified this way that are adjacent, including diagonally, are considered to be part of the same LGMD. We then draw a contour line around each LGMD in a heatmap using the marching squares contour-finding algorithm implemented in the measure.find_contours function of the scikit-image Python package (Van Der Walt et al., 2014).
To identify LGMDs in this study, we define our perturbation threshold as = 25.95 nT, which corresponds to the absolute value of the median of the distribution of the minimum values from each heatmap. It is not a particularly strict threshold, but that is because we are interested in all LGMDs, not only anomalously intense ones. LGMDs that extend past the edge of the region of interest, and thus are not associated with a closed contour, cannot be characterized with confidence, and are thus completely omitted from our analysis. Examples of both closed and open contours drawn with our method can be seen in Figure 2.
All our results in Section 4 are based on the heatmaps created using the procedure outlined in this section. Some are, more specifically, based on the perimeters of LGMDs identified in the heatmaps, each of which is calculated by adding up the distances between successive vertices of the contour that surrounds it. One must respect the spheroidal shape of Earth when computing these distances. Given two contour vertices with (latitude, longitude) coordinates () and () in radians and the radius of Earth = 6,378 km, we find the haversine distance between vertices (Equation 6) rather than the Euclidean distance as
In this work, we focus on perimeter as a measure of LGMD size. While there is no fundamental problem with investigating the behavior of LGMDs with different measures – for example, diameter – concentrating on the perimeter of LGMDs removes a degree of arbitrariness from the study. In the example of studying LGMD diameter, one would be forced to confront the question of along which direction to calculate that diameter, especially considering that LGMDs can vary widely in shape (an effect which is displayed in Figure 4). This is not an insurmountable issue; we hope that future work will extend the simpler baseline focus of this study to include other size metrics.
We also wish to study the shape of LGMDs, and pursue analysis of their aspect ratios as a way of doing so. We define the aspect ratio for a given LGMD as (Equation 7):
with denoting the latitude and denoting the longitude of the easternmost, westernmost, northernmost, and southernmost points (subscripts E, W, N, and S, respectively) on the LGMD’s perimeter. This quantity describes the ratio of the LGMD’s longitudinal extent to its latitudinal extent, with the longitudinal extent being measured along the line of latitude halfway between the easternmost and westernmost points. Section 4 contains the results of the application of this definition to our set of computed contours.
4 Results and discussion
4.1 Full-solar-cycle distributions
After applying Spherical Elementary Current Systems and our contour-finding algorithm, we can study the behavior of LGMDs by examining relevant statistics of the resultant LGMD perimeter dataset. For that, we first consider the distribution of the number of LGMDs identified in each of our 395,463 heatmaps shown in Figure 3a. The distribution has a significant skew to the right, and no heatmap was ever observed to contain more than six LGMDs. A large proportion – about 58% – of heatmaps exhibit no LGMDs, though the exact amount is somewhat sensitive to the threshold used to discriminate LGMDs from background geomagnetic activity.
FIGURE 3
We can also consider the perimeters of the LGMDs, as shown in Figure 3b. This figure describes the overall distribution of perimeter values. It is agnostic toward simultaneity and simply includes all LGMDs; that is, any two LGMDs contribute their perimeters toward this distribution even if they reside in the same heatmap. This distribution has a significant positive skew and shows that the mean, median, and mode LGMD perimeters are 1,620 km, 1,372 km, and 550 km respectively. Also of note is that the largest LGMD was observed to be 12,024 km in perimeter. As a point of comparison, the region of interest measures 9,344 km in perimeter, but it is possible for an LGMD perimeter to exceed this value if its associated contour forms a concave shape. However, most LGMDs are smaller than 9,000 km in perimeter, with only a handful of cases in the entire dataset close to the size of the region of interest.
Importantly, the typical perimeter of LGMDs as per these results is of the order that one would expect from estimates of the few-hundred-kilometer spatial scale derived using other methods, such as those mentioned in Section 1 (e.g., ; ). This approximate correspondence between perimeter and diameter holds unless the LGMD in question is extremely oblong, which is quite rare according to the results contained in Figure 4.
FIGURE 4
Next, we examine the distribution of the aspect ratios of observed LGMDs. The distribution shown in Figure 4 is roughly symmetric about its central value at = 0.29 ( = 1.95). This value is markedly positive, meaning that we observe LGMDs with a longitudinal extent greater than their latitudinal extent more frequently than those with the opposite orientation. In fact, only 5.7% of the mass of the log aspect ratio distribution lies below zero. This means that over 94% of observed LGMDs are at least slightly oblong along the east-west direction. The cause of this asymmetry around = 0 is not entirely clear, but it may be related to the fact that current systems which induce perturbations to , such as the auroral electrojets, are also extended longitudinally.
One must note that our choice of the region of interest could be biasing this distribution in the positive direction by “cutting off” some data points on the negative end of the distribution. This is because the region of interest is wider in the longitudinal direction than it is in the latitudinal direction. To illustrate this, an LGMD which has been found to have a longitudinal extent of 1,400 km might be counted among those in Figure 4, but one with a latitudinal extent of 1,400 km would surpass the boundaries of the region of interest; its perimeter contours would not be closed, and thus it would not be represented in Figure 4. If this bias does exist, it could be corrected by extending the region of interest to have the same latitudinal extent as its longitudinal extent. However, for our purposes, this is not practical, because there are not sufficient magnetometers with available data to support such an expansion without reducing interpolation quality. Instead, we examined the distribution of aspect ratios of only LGMDs less than 1,000 km in perimeter. The LGMDs in this subset are too small to be cut off by the boundaries of the region of interest, even if they are oblong in the north-south direction. When normalized, the distribution of this subset is almost identical to the full distribution. We take this to be sufficient evidence that the dimensions of the chosen region of interest do not significantly impact the distribution of LGMD aspect ratios, at least for statistical purposes.
We have inspected the distributions of three attributes: the number of LGMDs per heatmap, the perimeter of LGMDs, and the aspect ratio of LGMDs. For readers interested in additional details concerning these LGMD attributes, plots of the distributions of perimeter and aspect ratio, separated by the number of simultaneous LGMDs, are available in the Supplementary Material. Additional statistics, whose significance is not analyzed in this work, are also available in the Supplementary Figure S3–S8.
4.2 Impact of solar cycle phase
We are also interested in whether the phase of the solar cycle has any relationship to the statistics explored in the previous sections. Therefore, we now proceed with an analysis of two different subsets of our dataset: the set of timesteps surrounding solar maximum (14 October 2012 through 31 January 2015) and those surrounding solar minimum (1 January 2009 through 31 October 2009 and 4 March 2018 through 31 December 2019) for Solar Cycle 24. The dates defining these intervals are set following Reyes et al. (2021), who precisely defined the phase boundaries for that solar cycle.
In this section, we reproduce each histogram from the previous section to include the solar maximum and solar minimum data subsets alongside the full dataset. These subsets each contain different numbers of data points because each represents a different range of instances in time. Therefore, the distributions shown have been normalized such that the area under each is 1. This is in contrast to the distributions in Section 4.1; those essentially represent the relevant attributes’ probability distributions, whereas the plots in this section show probability density functions (PDFs) because they have been normalized in this way. We show PDFs so that the data subsets may be directly compared. The distribution of the number of LGMDs per heatmap, though it represents a discrete variable as opposed to a continuous variable, can be represented by this normalization as a PDF because the “bin” width is 1.
The effects of separating solar minimum and solar maximum from the overall dataset are shown on the number of LGMDs (Figure 5), their perimeter (Figure 6), and their longitudinal-latitudinal aspect ratios (Figure 7). For each of these three statistics, the trends of the solar maximum and solar minimum subsets follow each other very closely. This motivates the question: are the differences that do exist between the subsets simply due to aleatoric variability (that is, do they only exist by chance?), or are they caused by some higher-order physical phenomena? To answer this question, we can compute a statistical divergence between the distributions corresponding to the solar maximum and solar minimum subsets. If this divergence is sufficiently large, that will be evidence against the hypothesis that the distributions differ only because of aleatoric effects.
FIGURE 5
FIGURE 6
FIGURE 7
We first quantify the distance between a PDF associated with the solar maximum dataset, , and that associated with the solar minimum dataset, , using , the symmetrized form of the Kullback-Leibler divergence () (Equation 8):
Here, the sums are over the “bins” of the discretized PDFs. The width of each bin is , which is written without subscript because, for this study, each bin within the same PDF has the same width. We use the symmetrized form of the divergence to avoid confusion that may arise due to generally not being equal to .
From an information theory perspective, measures how much information would be absent if one PDF were to be used to approximate the other. As the distance between and increases, will also increase. However, there is some ambiguity as to the threshold above which can be said to indicate that the differences between and are due to more than just aleatoric variability. Some clarification appears as one takes to various limits. First, the symmetrized divergence between a distribution and itself is zero. Second, while there is no upper bound on even for PDFs, one can create a “benchmark” PDF known a priori to be sufficiently distant from . If is much closer to than it is to 0, that may point to a phenomenological difference between and .
For this comparison, we simply define as the PDF describing a normalized uniform distribution, which is fundamentally different from any of the PDFs shown in Figures 5–7. The values for each of the statistics presented in this section are shown in Table 1. Also shown for comparison are the symmetrized divergences between and and between and for LGMD number, perimeter, and the base-10 logarithm of aspect ratio, . Because any conclusion drawn about has a linear equivalent that can be drawn about , we will refer to the “base-10 log aspect ratio” just as the “aspect ratio” for the sake of simplicity.
TABLE 1
| Symmetrized Kullback-Leibler divergence | |||
|---|---|---|---|
| Number | Perimeter | ||
| 3.6318 | 2.6811 | 3.3926 | |
| 3.1420 | 2.8600 | 3.3926 | |
| 0.0121 | 0.0522 | 0.0601 | |
Values for the symmetrized Kullback-Leibler divergence between various probability density functions (PDFs) formed from data from solar maximum () and solar minimum (). Divergences from , the PDF of a uniform distribution, are shown for comparison.
It is clear from the reported values that the PDFs of each statistic shows much less divergence between solar maximum and solar minimum than they do between a uniform PDF and either solar maximum and solar minimum. The values of are considerably larger for perimeter and aspect ratio than they are for LGMD number, but this likely due to the larger number of bins in those two quantities’ PDFs accentuating the divergences between PDFs; for perimeter and aspect ratio is in each case more than an order of magnitude smaller than either or . The fact that is so small in comparison does not suggest than these three LGMD attributes exhibit any dependence on the phase of the solar cycle.
The information theory perspective makes the criterion interpretable. However, there are some disadvantages to this approach. The effect of the logarithms in the definition of (Equation 8) emphasizes the relative importance of the tails of the PDF to the symmetrized divergence. This could raise the concern that outliers in each of the attributes’ distributions may be influencing the conclusions drawn from Table 1. Additionally, bins in which one of the PDFs is equal to zero must be ignored to keep well defined. This means that some small number of bins in the perimeter and aspect ratio PDFs are not taken into account by this criterion. These drawbacks do not invalidate the use of in this context, but they do suggest another criterion may be needed to verify the preliminary conclusions about the effect of the solar cycle. We have used the total variation distance (Equation 9) to supplement the use of :
The total variation is computed outside of “logarithm space”, which avoids both drawbacks mentioned regarding . The values for each of the statistics presented in this section are shown in Table 2. The divergence between and remains the smallest divergence by over one order of magnitude for the number of LGMDs. For their perimeter and their aspect ratio, it is smaller by about a factor of eight. Using the total variation distance as a criterion reinforces the results obtained with the initial Kullback-Leibler analysis, which indicate insufficient evidence to suggest a phenomenological difference between solar maximum and solar minimum LGMD aspect ratios.
TABLE 2
| Total variation distance | |||
|---|---|---|---|
| Number | Perimeter | ||
| 0.6133 | 0.7075 | 0.6763 | |
| 0.6347 | 0.7074 | 0.6609 | |
| 0.0318 | 0.0868 | 0.0842 | |
Values for the total variation distance between various probability distributions formed from data from solar maximum () and solar minimum (). Distances from , the PDF of a uniform distribution, are shown for comparison.
One might expect the solar cycle to have an effect on the PDFs of these three LGMD attributes. After all, the solar cycle has an effect on the general geoeffectiveness of the solar wind–strong geomagnetic storms, storms due to coronal mass ejections, and strong ground-level geomagnetic activity are all more common during solar maximum, for instance (Webb, 1991; Richardson and Cane, 2012; ). However, as we move away from global size scales toward the smaller scales of LGMDs, we can not reject the hypothesis that the solar maximum and solar minimum distributions differ only because of aleatoric effects.
This does not necessarily mean that the solar cycle has no influence on the properties of LGMDs, only that any influence that does exist could not be detected with the methods used here. This is perhaps due to the fairly strict choice of the uniform distribution’s PDF as a benchmark PDF. One distribution being more similar to a uniform distribution than to the other solar cycle phase’s distribution – the criterion necessary to reject the baseline hypothesis – is a high bar to meet. There still may be some much smaller solar cycle effect that could require a more sensitive test to extract. We have used the uniform distribution’s PDF as a benchmark even still because other tests would include further amounts of arbitrariness. For instance, one could repeat this analysis on LGMD aspect ratio, replacing with the PDF of a log-normal distribution, but it would be difficult to justify a choice of mean and variance for this new benchmark. Because of this, first investigating the influence that other phenomena have on LGMD attributes would likely be more fruitful than identifying a more sensitive test.
Though this numerical analysis suggests that the distribution of aspect ratios does not strongly depend on the phase of the solar cycle, it does not reveal the physical processes that do determine the shape of the PDFs shown. This motivates a cause-and-effect analysis of LGMD aspect ratio, which likely deserves a study all to itself. Such a study could explore phenomena inside the magnetosphere, where the effect of the solar wind is modulated by a multitude of current systems, and it could connect these phenomena to the LGMD attributes lain out here. For instance, it could investigate the shape of localized structures embedded within the auroral electrojet and how they correspond to LGMDs. Similarly, heeding the results of Vandegriff et al. (2024) wherein points on Earth’s surface were magnetically mapped to spatially dispersed locations in the magnetotail, it may be of interest to examine the influence of large-scale features in the magnetosphere on LGMD aspect ratio.
Better establishing the empirical relationship between the solar wind, magnetospheric and ionospheric processes, and the characteristics of LGMDs will provide knowledge about the Sun-to-Earth phenomenological chain, and help to describe the link between LGMDs’ attributes and their drivers.
5 Conclusion
Understanding the complete spatial picture of ground-level geomagnetic disturbances is key when it comes to protecting GIC-susceptible infrastructure and developing our knowledge of magnetospheric physics. We use the Spherical Elementary Current Systems method to interpolate readings from 25 magnetometer stations in a way consistent with physics that drives activity in the magnetosphere. We densely sample the interpolations to create heatmaps showing the behavior of within a wide region of mid-latitude North America.
From hundreds of thousands of these heatmaps, representing nearly all storm time in Solar Cycle 24, we analyze three statistics: the number of LGMDs per heatmap, the perimeters of LGMDs, and the aspect ratios of LGMDs. We develop frequency distributions of these statistics, finding that the typical scale size of LGMDs reconstructed via our methods agrees with scale sizes presented in related studies.
Finally, we subsample these frequency distributions to obtain ones that just represent data collected near solar maximum and ones that represent the same for solar minimum. For each statistic, the distributions from different phases of the solar cycle follow much the same trends; however, there are small differences between LGMDs that occur near solar maximum and ones that occur near solar minimum. These differences are determined to be insignificant by calculating the respective distributions’ probability density functions (PDFs), then comparing their Kullback-Leibler divergences and their total variation distances. In other words, the phase of the solar cycle does not strongly direct the attributes of LGMDs during storm time.
This study opens the path toward continued investigation of LGMDs and their drivers. Some possible areas of future study include investigating any correlation between LGMD attributes and the intensity of the storm during which those LGMDs occur. On the other hand, since we found no influence from the solar cycle, perhaps magnetospheric and ionospheric drivers are considerably more important than solar wind effects in determining the characteristics of LGMDs, and perhaps substorm activity in particular is more relevant than a focus on geomagnetic storms. This would agree with the findings of Ngwira et al. (2018), Ngwira et al. (2025) (among others), which emphasize the behavior of ionospheric currents and the mesoscale magnetospheric structures to which those currents systems are magnetically connected. It may, therefore, be constructive to correlate LGMD attributes with measurements of the ionosphere and aurorae above them, or, taking inspiration from Vandegriff et al. (2024), to correlate LGMD attributes with structures in the magnetotail in physics-based simulations.
Though we have investigated only the northward perturbation component in the present work, it should be noted that our methods could be applied to the eastward () or horizontal () components of perturbation. We acknowledge, however, that these quantities do not necessarily behave the same, and the dominant east-west flow of the auroral electrojets may have implications for small current structures driving LGMDs which might not affect current structures that drive LGMDs. Comparing the components’ behaviors as they pertain to this work would make a good topic for future study.
Similarly, our methods have applicability to the time derivative of magnetic field perturbations , though in this study we have investigated only the deflection from the magnetic field’s baseline value (, in this case). For comparison to the statistics on LGMDs presented in Section 4.2, we show corresponding distributions for LGMDs in the Supplementary Material. There, LGMD identification is based on intense rate of change instead of on intense perturbation magnitudes. A more thorough analysis of LGMDs has not been performed because of the noise which is introduced by taking a discrete derivative over to obtain .
There is no perfect interpolation method since all such methods are mere models of the physics governing the geomagnetic field. However, one cannot ignore the utility of interpolation in enabling advanced analysis techniques that could not be applied to single-point measurements. When its limitations are understood and accounted for, spatial interpolation has the power to facilitate geospace dynamics research and to enable the development of more effective and actionable predictive models.
Statements
Data availability statement
The magnetometer data which was used for this work is from the SuperMAG collaboration, and can be found at https://supermag.jhuapl.edu/. Software created for this work can be found in .
Author contributions
RM: Formal Analysis, Writing – review and editing, Methodology, Writing – original draft, Data curation, Investigation, Conceptualization. AK: Supervision, Investigation, Writing – review and editing, Funding acquisition. JM: Investigation, Writing – review and editing, Formal Analysis. VP: Investigation, Formal Analysis, Writing – review and editing. MC: Formal Analysis, Investigation, Writing – review and editing. DH: Writing – review and editing, Funding acquisition.
Funding
The author(s) declare that financial support was received for the research and/or publication of this article. This work was supported by NSF EPSCoR Award OIA-1920965 and NSF Award 2331527. VP thanks ANID grant SA772100112 and DICYT Regular grant 042431PA.
Acknowledgments
We thank all members of the MAGICIAN team at UNH and UAF who participated in the discussions leading to this article. We also thank the SuperMAG collaboration for providing the data used in this study.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declare that no Generative AI was used in the creation of this manuscript.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fspas.2025.1610276/full#supplementary-material
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Summary
Keywords
space weather, geomagnetically induced currents, localized geomagnetic disturbance, ground based magnetometer, spherical elementary current systems
Citation
Mukundan R, Keesee AM, Marchezi JP, Pinto VA, Coughlan M and Hampton D (2025) Localized geomagnetic disturbances: a statistical analysis of spatial scale. Front. Astron. Space Sci. 12:1610276. doi: 10.3389/fspas.2025.1610276
Received
11 April 2025
Accepted
21 July 2025
Published
13 August 2025
Volume
12 - 2025
Edited by
Georgios Balasis, National Observatory of Athens, Greece
Reviewed by
Stavros Dimitrakoudis, National and Kapodistrian University of Athens, Greece
Adamantia Zoe Boutsi, National Observatory of Athens, Greece
Updates
Copyright
© 2025 Mukundan, Keesee, Marchezi, Pinto, Coughlan and Hampton.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Raman Mukundan, raman.mukundan@unh.edu
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.