**Abstract**—The paper addresses the variability properties of the intensity and direction of the magnetic field during a substorm as well as magnetically quiet periods. The main focus is on the properties of the variations in the time derivative of the magnetic field *d***B**/*dt* which are a factor of particular importance for the problem of the geomagnetically induced currents. A method of two- (2D) and three-dimensional (3D) diagrams is proposed for visual representation of the directional variations of a vector field. As an example, the geomagnetic field variations during the isolated substorm of October 17, 2015 are analyzed with the use of the chain of the IMAGE network magnetic stations. It is confirmed that *d***B**/*dt* in the horizontal plane has a much stronger variability than the geomagnetic field perturbation Δ**В**. The proposed 3D diagrams show that the *d***B**/*dt* polarization plane is inclined towards the Earth’s surface, which is probably due to the irregularity of the field of geomagnetic fluctuations.

## INTRODUCTION

Among the most significant space weather manifestations that can affect normal operation of the ground-based technological systems are geomagnetically-induced electric currents (GIC) arising in these systems during the abrupt changes in the geomagnetic field (Pulkkinen et al., 2006). To date, the planetary geospatial structure and the characteristic morphological features of the magnetic disturbances Δ**В** have been fairly well described by a variety of the empirical and numerical models, e.g., (Lukianova and Christiansen, 2006). The strongest magnetic disturbances on the ground level are generated in the high latitudes by the auroral electrojet which sharply intensifies during the substorms and in the midlatitudes by the ionospheric electrojet currents. At the same time, the time variations in the magnetic field *d***B**/*dt* relevant for GIC excitation need attention and careful analysis. Important role in the generation of rapid changes in the magnetic field can be played by the small-scale ionospheric current systems with a spatial extent on the order of hundreds of km (Viljanen et al., 2001; Belakhovskii et al., 2018). The physics of the origin of these current systems responsible for rapid geomagnetic variations is still unclear (Engebretson et al., 2019).

Besides, detection of the small-scale variations in the magnetic field from the network station data may substantially promote the solution of one of the most challenging problems of modern space geophysics—localizing the nucleation region and identifying the mechanism of magnetospheric substorms. Presumably, the conditional “epicenter” of the magnetic fluctuations intensity in the range of periods of 10–30 s pinpoints the geomagnetic projection of the region of the future onset of the expansion phase of a substorm in the nighttime magnetosphere (Murphy et al., 2009). However, understanding the generation mechanism of these fluctuations calls for further research.

For the natural systems that are close to the stability threshold, the dynamics of fluctuations is particularly important. The physics of magnetospheric substorms still lacks a conclusive understanding, whether the small-scale fluctuations of the field during the substorms and in the pre-substorm periods are caused by the same mechanism which merely sharply intensifies at the explosive phase. The presence of the small-scale current systems superimposed on the ionospheric electrojet is reflected in vigorous variations of the field *d***B**/*dt*; therefore, vector properties of the time-derivative fluctuations of the field can provide additional information about their origin.

For describing the variability of the geomagnetic field we need methods for identifying the fine structural characteristics of the fluctuation pattern of the field. This paper addresses the search for new characteristics of variations of the field and the attempt to use these characteristics for describing the variability of the geomagnetic field during the substorms and quiet periods.

## METHODS FOR ANALYZING THE VARIABILITY OF THE FIELD

The initial data from a magnetometer at the geomagnetic station can be represented in the form of a vector time series with the elements consisting of three (*X*, *Y*, *Z*) components of **B**(*t*_{1}), **B**(*t*_{2}), … **B**(*t*_{k}), **B**(*t*_{k + 1}), …, where (*t*_{k + 1} − *t*_{k}) = Δ*t* is the sampling interval. The data of a standard three-component fluxgate magnetometer contain the values of the field accurate to some fixed value. The local variation of the magnetic field is defined by Δ**B**(*t*) = **B**(*t*) − **B**_{0} where the undisturbed background field **B**_{0} is either a median over the analyzed time interval or a field value before the onset of the disturbance.

The perturbation of the horizontal component of the magnetic field Δ**B**_{t} = {ΔB_{X}, ΔB_{Y}} at the observation point is associated with the equivalent ionospheric current **J** above it: Δ**B** = (2π/c)[**J** × **n**], where **n** is the normal to the ionospheric plane, i.e. vector **J** makes an angle of π/2 with Δ**B**_{t} measured in the clockwise direction. The equivalent ionospheric current can differ from the real horizontal current in the ionosphere in the regions with sharply inhomogeneous ionospheric conductivity.

We can also link this time series with the series of time derivatives *d***B**/*dt*. Provided horizontal homogeneity of geoelectric properties of the underlying medium, the orientation of vector *d***B**/*dt* corresponds to the direction of the excited telluric field **Å** and currents in the surface layers of the Earth.

We consider the local variability properties of the magnitude and direction of the magnetic field. The variability of horizontal components of vectors Δ**B**_{t} and *d***B**_{t}/*dt* can be inferred from two-dimensional (2D) polar diagrams of these vectors. In the three-dimensional (3D) space, i.e. with the consideration of the vertical *B*_{Z} component, the vector orientation of the disturbed geomagnetic field is characterized by the normalized vectors of unit length Δ**B**/|Δ**B**| and *d***B**/*dt*/|*d***B**/*dt*| which are defined by the points on the unit sphere. A point can correspond to both a large and a small value of the derivative; therefore, this approach raises certain difficulties associated with the accuracy of the magnetic data (about 0.1 nT) and the magnitude of the derivative per se. Therefore, in the 3D diagrams, only the directions of the derivatives absolute values of which exceed the 80% quantile will be shown. Thus, the field direction is characterized by a vector of unit length in a 3D space, and the variations in the direction of the field are identified with the points lying on the unit sphere. This method for representation of data from magnetic stations and observatories is more thoroughly described in (Khokhlov et al., 2019).

This representation elucidates the geometric structure of the variations but impedes calculation of the traditional statistical characteristics—mean values, variances, and correlations, because the points on the sphere do not allow arithmetic operations. However, there are methods for describing and analyzing the data on a sphere (e.g., (Watson, 1983; Mardia and Jupp, 2008)), which differ from the traditional statistical approaches. The condition for intelligent applicability of the statistical methods is that the observed quantities possess a certain stationarity, i.e., the data should correspond to an independent sample of a general population. Evidently, a combination of observations for the disturbed and quiet periods does not meet this criterion. However, a complexly organized random process can be analyzed separately on the different time segments under the assumption that within each segment the parameters of the process remain more or less constant, although in this approach the volume of the considered data decreases.

## SUBSTORM OF OCTOBER 17, 2015

We present the example of applying the proposed method to the analysis of the variations in the geomagnetic field during the isolated substorm of October 17, 2015, based on the 10-s data from the longitudinal chain of IMAGE magnetometer stations (Table 1) (Tanskanen, 2009). Figure 1 shows the example of the magnetograms (*X*-component) and the graphs of calculated magnetic field variability (*dX*/*dt*) for four IMAGE stations Hopen Island (HOP), Nordkapp (NOR), Ivalo (IVA), and Oulujärvi (OUJ). The substorm onset was detected at ~18:10 UT. The substorm begins at the geomagnetic latitude Ф ~ 65° (IVA), and this activation (including the sharply intensified westward electrojet) migrates then to the higher latitudes Ф ~ 70° (HOP). At the substorm maximum (NOR), intense irregular quasi periodic fluctuations classified as *Pi*3 pulsations are observed. Within the time fragment selected for the analysis (1700–2000 UT, duration 3 h), two characteristic intervals (separated in Fig. 2 by a vertical red line) are distinguished:

(1) substorm growth phase;

(2) substorm expansion phase and subsequent *Pi*3 pulsations.

Let us consider the data of the magnetic stations that were located in the characteristic regions during the substorm development:

—at the maximum of the westward auroral electrojet, at the latitude of the maximum negative bay variation in *X*-component (HOP);

—in the region of the most intense *Pi*3 pulsations (NOR);

—within the ionospheric projection of the supposed substorm nucleation zone (IVA); and

—in the region of the ionospheric electrojet currents (OUJ).

We select the time moment 70 min from the beginning of the analyzed interval for all stations as a characteristic moment separating (1) the growth phase of the substorm and (2) the expansion phase and subsequent recovery. The duration of intervals (1) and (2) is 70 and 110 min, respectively, which is sufficient for statistical estimation.

This division into a weakly disturbed field (growth phase) and strongly disturbed field (expansion phase) is approximate because the onset of the substorm expansion phase (the breakup) is observed at different time in the different latitudes, as can be seen from the comparison of the *X*-component magnetograms, due to the migration of the epicenter of substorm activation during its polarward expansion (Fig. 2). However, the burst in |*dХ*/*dt*| at the onset of the substorm growth phase occurs almost simultaneously (at the 48th and 72th minutes) in the different latitudes.

## ANALYSIS OF GEOMAGNETIC FIELD VARIABILITY AT DIFFERENT PHASES OF THE SUBSTORM BASED ON A 2D POLAR DIAGRAM

Let us separately consider the orientation of Δ**B** and *d***B**/*dt* at the growth phase and at the expansion phase of the substorm sequence at different latitudes. The fluctuations of the field during these periods may have different characteristics because their physical mechanisms may differ. Let us begin with considering the polar 2D diagrams of the horizontal components Δ**B**_{t} and *d***B**_{t} /*dt* on the ground (*XY* plane). The polar diagrams of vector Δ**B**_{t} at all stations (not shown in this paper) indicate predominantly southerly orientation of the disturbance (i.e., |Δ*X*| \( \gg \) |Δ*Y*|), consistent with the enhancement of the westward electrojet during the substorm development. This suggests that the disturbance of the field mainly varies in magnitude and is created by the ionospheric current flowing in the westerly direction. The polar 2D diagrams of the horizontal *d***B**_{t} /*dt* components on the ground (*XY* plane, unit circle) for HOP, NOR, IVA, and OUJ stations are shown in Figs. 3 and 4 for the growth phase and expansion phases of the substorm, respectively.

At the same time, the polar diagrams of *d***B**_{t} /*dt* (Figs. 3 and 4) indicate that the variations in this vector have a significantly wider scatter than variations in Δ**В**_{t}. At the substorm growth phase (interval 1), at the stations located at latitudes 65°–67° (NOR, IVA), |*dB*_{X}/*dt*| is predominantly larger than |*dB*_{Y}/*dt*| (Fig. 3). At the middle latitudes (OUJ) and polar latitudes (HOP), variations in *d***B**_{t}/*dt* become almost isotropic. Thus, the time derivative of the field *d***B**_{t}/*dt* (and, hence, the telluric current induced by it) strongly varies not only in magnitude but also directionally. At the substorm expansion phase (interval 2), polar 2D diagrams have a qualitatively similar shape (Fig. 4).

Thus, although at the growth (preparatory) phase and at the expansion (main) phases of the substorm, fluctuations become more intense in magnitude, the directional properties of these variations remain qualitatively unchanged.

## ANALYSIS OF GEOMAGNETIC FIELD VARIABILITY BASED ON SPHERICAL 3D DIAGRAMS

Using the method (Khokhlov et al., 2019), we plot the *d***B**/*dt* directions on a 3D sphere for all the substorm phases at different latitudes (Fig. 5). Because of the limited accuracy of magnetometers (~0.1 nT), the distributions of the calculated directions on the sphere are strongly discrete. Therefore, for identifying the structure of the variations we need the data for a fairly long observation period which would allow us to select a sample for the analysis without including the values that are at the limit of measurement accuracy. Hence, when constructing the 3D diagram, we combined intervals 1 and 2 considering the fact that the 2D diagrams did reveal a qualitative difference in the properties of the fluctuations at different phases of the substorm.

The distributions of the directions for *d***B**/*dt* reveal a certain structural pattern. The spherical 3D diagrams show that the directions are polarized along some inclined plane and the location of this plane depends on the location of a particular magnetometer. This structure becomes completely evident when we draw the distribution of 20% of the largest (in magnitude) values (Fig. 5, bottom).

Discerning this structure in the behavior of the Δ**B** directions is more difficult because the derivatives and their contribution in Δ**B** fairly differ in their magnitudes. The visualization of the directions of the derivatives included a highly nonlinear procedure of normalizing the vector of the derivative, and it is after this processing that the discussed structure became noticeable.

Actually, for any configuration of the points on the sphere, we may consider the known orientation tensor and find its eigenvectors and eigenvalues. The smallest eigenvalue corresponds to the vector perpendicular directions of which define the plane passing through the center of the sphere. If the points on the sphere tend to cluster in the vicinity of the great circle formed by the intersection of this plane with the sphere, the smallest eigenvalue noticeably differs from the other eigenvalues by its magnitude. The opposite is also true: based on the relative magnitude of eigenvalues, we may conclude about the degree of concentration of the points; in particular, when all the eigenvalues are approximately equal, the distribution of the points is close to uniform.

Unfortunately, the standard statistical estimates are inapplicable here: in the considered case, we are dealing with a clearly non-stationary process, and, therefore, for elucidating the clustering of the directions we need the data on their structure for a substantially longer period of observations—from one year and longer.

Examining the real data and 3D-representation of the directions, we can see that (a) in most cases, the smallest eigenvalue proves to be significantly smaller than the other two eigenvalues, which corresponds to the pronounced clustering of the directions of the derivative along a certain plane; (b) the degree of this clustering is the higher, the greater the magnitudes of the derivatives directions of which are considered. For the set of the magnetic stations HOP, NOR, OUJ, IVA used in this work, the clustering of the strongest directions is rather poorly pronounced even in the case of a long observation period. For example, the eigenvalues for the NOR station are (1.94; 3.01; 5.14). The sets of eigenvalues for the other stations also have a small relative difference. At the same time, this clustering is frequently fairly manifest, e.g., as in the ace for the strongest variations (selected by the quantile 97% criterion) recorded by the geomagnetic observatory St. Petersburg (SPG) (Sidorov et al., 2017), where the set of eigenvalues is (0.25; 1.94; 2.30) (Fig. 6).

In particular, clustering is most apparent during the magnetically disturbed periods. At the same time, in the pre-substorm period, clustering about the plane is much less evident. Mathematically, this means that the smallest eigenvalue of the orientation tensor for this period does not strongly differ from the other eigenvalues.

## PROBABLE CAUSES OF *dB*/*dt* POLARIATION PLANE INCLINATION IN SPACE

The emergence of the vertical *B*_{Z} component of geomagnetic variations in the vicinity of the Earth’s surface reflects the presence of horizontal inhomogeneities in the geoelectrical structure of the Earth’s crust or in the primary field **B**_{t} of the pulsations. The geoelectrical properties of the surface layer of the Earth are described by surface impedance *Z*_{g} (the impedance for homogeneous conductivity σ is \({{Z}_{g}} = \exp ({{ - i\pi } \mathord{\left/ {\vphantom {{ - i\pi } 4}} \right. \kern-0em} 4})\sqrt {{{\omega \mu } \mathord{\left/ {\vphantom {{\omega \mu } \sigma }} \right. \kern-0em} \sigma }} \)). In the case of a strong skin effect (which condition holds very well for the typical variations with a period longer than 1 min above low-resistive sections), these two mechanisms responsible for the emergence of a vertical component in the magnetic field of pulsations are clearly evident from the formula (Wait, 1954):

The time derivatives of the magnetic field are linked by the same relation. From formula (1) it follows that the emergence of the *B*_{Z} component can be caused by the inhomogeneity of the horizontal component of the primary field (\(\nabla {{{\mathbf{B}}}_{t}} \ne 0\)) or by the horizontal inhomogeneity of surface impedance (\(\nabla {{Z}_{g}} \ne 0\)). For a homogeneous plane wave above a geoelectrically horizontally homogeneous surface, *B*_{Z} = 0.

In the case of typical disturbances with *B*_{X} \( \gg \) *B*_{Y}, if we neglect the contributions of *B*_{Y} component and the inhomogeneity of *Z*_{g}, the relationship between the vertical and horizontal components will be described by the following formula:

For a perturbation that decreases towards lower latitudes, e.g., *B*_{X} *~* exp(−*x*/*a*), the emergence of a small *B*_{Z} component, on the order of the ratio of the skin length \({{\delta }_{g}} = \sqrt {{2 \mathord{\left/ {\vphantom {2 {\mu \omega \sigma }}} \right. \kern-0em} {\mu \omega \sigma }}} \) to the scale size *a* of the transversal inhomogeneity of the field:

Since the variations in *B*_{Z} and *B*_{X} are coupled according to formulas (1) and (2), this will lead to the inclination of the polarization plane of the field of geomagnetic variations. The inclination angle can be estimated as \({\text{tan}}\alpha = \left| {{{{{B}_{Z}}} \mathord{\left/ {\vphantom {{{{B}_{Z}}} {{{B}_{X}}}}} \right. \kern-0em} {{{B}_{X}}}}} \right|\).

Besides, in the presence of a horizontal inhomogeneity of surface impedance *Z*_{g}, the *B*_{Z} value is affected by the contribution of the term \({{B}_{t}}\nabla {{Z}_{g}}\). Its magnitude depends on the polarization direction of the primary field and on the orientation of the impedance gradient.

## DISCUSSION

Since the stationarity of the time series of geomagnetic variations during storms and substorms cannot be taken for granted, any numerical characteristics of statistical origin should be regarded with caution and only as the indirect arguments suggesting the differences in the behavior. On the contrary, simple visual comparisons of the presented 2D and 3D diagrams provide more information, similar to how easily the period of the substorm itself is distinguished from the period of its preparation. For numerous observations, the time series of the first differences (derivatives) proves to be closer to stationary, although in the case of the strong magnetic disturbances, this closeness to stationarity, if any, is only observed on some observations segments of limited duration.

The proposed method of 2D and 3D diagrams makes it possible to visualize the directional variations of the vector field. The analysis of the magnetic data during the development of a substorm revealed the following characteristics:

—The polar 2D diagrams for *d***B**_{t}/*dt* show that variations of this vector have a significantly wider directional scatter than variations in Δ**В**_{t} during all phases of a substorm. In the middle and polar latitudes, variations in *d***B**_{t}/*dt* are almost isotropic.

—During the undisturbed periods, the directions of the derivatives on the 3D diagrams group around the characteristic planes of each observation site. The larger the magnitude of the derivative, the better it is approximated by the corresponding plane. Thus, we may speak of the polarization effect of the directions of the derivative which is observed on the large derivatives and becomes vague on the small derivatives.

It is not possible to associate the orientation of these planes with the latitude of a magnetic observatory alone. The polarization plane is defined by both geographic coordinates and its inclination quite strongly varies in 3D space. The region of Fennoscandia is characterized by a complex geoelectrical structure of the crust with surface impedances differing by two orders of magnitude between different regions. In our opinion, the discussed spherical 3D diagrams can, on one hand, characterize the directional properties of the fluctuations of auroral electrojet in a given region, whereas the deviations of the orientation of the planes of variability of geomagnetic fluctuations from the horizontal plane can, in turn, be associated with local inhomogeneity of crustal impedance.

## CONCLUSIONS

This paper presents the examples of using 2D (polar) and 3D (on unit sphere) diagrams for visualization of the directional variations of the magnetic field vector. The variability properties of the magnitude and directions of the magnetic field were observed during a substorm and during quiet periods for four magnetometer stations of the IMAGE network (HOP, NOR, IVA, OUJ). As a separate example, we considered the data from the magnetic observatory St. Petersburg (SPG) for the isolated substorm of October 17, 2015, which was selected as a focus event.

The analysis of the magnetic data based on polar 2D diagrams during the growth phase of the substorm indicates that variations in the *d***B**_{t}/*dt* vector have a significantly wider directional scatter than variations in Δ**В**_{t} at all substorm phases. It is also established that variations in *d***B**_{t}/*dt* at the mid- and high latitudes become almost isotropic. The observations of the variability of the geomagnetic field based on the 3D diagrams revealed the characteristic clustering of the directions of this variability. During the undisturbed periods, the *d***B**_{t}/*dt* vector directions are grouped around the characteristic planes of each magnetic station. The 3D diagrams based on the considered data show a complex polarization pattern which is likely to result from the inhomogeneous character of the field of geomagnetic fluctuations.

The methods described in this paper can be useful in the general problem of describing the variability structure of the geomagnetic field and the related GIC.

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## ACKNOWLEDGMENTS

We are grateful to the scientific institutions and teams maintaining and operating the international network of IMAGE magnetometers (http://space.fmi.fi/image/www/). We acknowledge the use of data provided by the Shared Research Facility “Analytical Geomagnetic Data Center” of the Geophysical Center of the Russian Academy of Sciences (http://ckp.gcras.ru/).

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This work was supported by the Russian Science Foundation (project no. 17-77-20034 “Creation of maps of geomagnetic activity characteristics zoning for the territory of the Russian Federation”).

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Translated by M. Nazarenko

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Khokhlov, A.V., Pilipenko, V.A., Krasnoperov, R.I. *et al.* Geomagnetic Field Variability Analysis Based on Polar Diagrams.
*Izv., Phys. Solid Earth * **56, **854–863 (2020). https://doi.org/10.1134/S1069351320060038

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### Keywords:

- substorms
- geomagnetically induced current
- geomagnetic fluctuations