Introduction
The Earth has its own magnetic field, which is also widely known as the
geomagnetic field (also called “the Earth's geomagnetic field” or “the Earth's magnetic
field”) (Bartels et al., 1939). The Earth's magnetic field varies
continuously with both time and ambient space and can be represented as a
superposition of the main field, the local field, and the variable field
(Zaitsev et al., 2002). The above elements of the Earth's magnetic field are
typically described using the rectangular coordinate system, where the axes
are directed towards north, east, and downwards (Fig. 1).
The Earth's magnetic field vector can be represented either by components X,
Y
and Z in the Cartesian reference system with axes to geographical north,
east, and vertically downwards, respectively, or by components H
(horizontal), D (declination) and Z in the cylindrical system. Another
alternative is using components F (total intensity), D and I (inclinations) in
the spherical system. H points to magnetic north, D is the angle between the
geographical and magnetic meridians and is positive to the east, while I is
the angle between the horizontal component and the intensity vector.
Geomagnetic field components.
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As a rule, the horizontal component of the field (H component) is used for
calculating the indices of geomagnetic activity (Menvielle et al., 1995;
Nowożyński et al., 1991) and analyzing variations in the Earth's
magnetic field during magnetic storms. The observed temporal variations in
the Earth's magnetic field vector components are referred to here as the
geomagnetic signals.
This paper is particularly focused on the development of analysis
techniques for geomagnetic signal fluctuations characterizing the complex
spatiotemporal structure and dynamics of the variable geomagnetic field. The
variable part of the field is induced by the corpuscular flows of the
magnetized plasma emanating from the Sun along with solar wind.
Detailed characterization of the fluctuation phenomena in observational
geomagnetic signals at multiple scales from global effects to local
perturbations is essential for the understanding of the intensity, type, and
development of a magnetic storm.
The complex structure of geomagnetic signals and an insufficient number of
adequate mathematical models make these data difficult to analyze using manual
techniques. Conventional approaches mainly employ basic time-series analysis
models and methods that include various smoothing operations (smoothing and
trend extraction, Chen, 2007; Joselyn, 1979; Rangarajan, 1989; Sucksdorff et
al., 1991). Periodic changes and patterns in the data are typically
analyzed using traditional Fourier techniques (Berryman, 1978; Golovkov et al.,
1989). However, observational geomagnetic signals are often nonstationary
and exhibit a heterogeneous multiscale structure (e.g., Consolini et al.,
2013; Klausner et al., 2013). Therefore conventional analysis techniques
(smoothing and trend extraction, Fourier techniques), while being able to
provide a rather general picture, result in the smoothing of the local
perturbations that often contain important information about geomagnetic
field activity and are explicitly associated with the development of
magnetic storms.
To overcome the above limitations, we suggest here a specialized nonlinear
approach to the analysis of the geomagnetic signals that is based on the
wavelet transform (Mallat, 1999; Holschneider, 1995). In this paper, we
studied variations in the geomagnetic field and estimated their
characteristics using the approach based on wavelet packets and first
suggested in the papers from Mandrikova et al. (2012, 2013b).
Nowadays wavelets and wavelet packets are among the most frequently applied
mathematical tools in signal processing (e.g., Hafez et al., 2010; Jach et
al., 2006). Regarding applications in geophysics and, in particular,
the Earth's magnetic field studies, we would like to emphasize some of the
most significant advantages of wavelet-based approaches (Jach et al., 2006;
Kumar and Foufoula-Georgiou, 1997; Mandrikova et al., 2011; Nayar et al.,
2006; Rotanova et al., 2006; Xu et al., 2008):
Wavelets and wavelet packets are capable of tracking the multicomponent
structure of the observational geomagnetic data, considering that
geomagnetic signals exhibit multiscale features. The local multiscale
components are largely hindered by trends, thus altogether constituting a
complex signal structure.
Unlike wavelets, wavelet packets are a more flexible signal processing tool.
Wavelets work only with a low-frequency component at each decomposition
level and leave a high-frequency one unchanged. In contrast, wavelet packets
act to decompose both the high-frequency and the low-frequency components at
each decomposition level, providing better resolution and finer splitting of
the time–frequency domain.
Wavelets and wavelet packets provide fast computational techniques for
finding wavelet coefficients. These techniques are very important for
processing long and/or high-resolution data sets.
Currently, the wavelet transform is steadily becoming more and more popular
in the area of geomagnetic data analysis. Wavelet applications focus on
studying nonstationary processes in the magnetosphere preceding and
accompanying magnetic storms (Balasis et al., 2006), analyzing the
dynamics of geomagnetic activity and detecting singularities (Zaourar et
al., 2013), removing noise (during data preprocessing) (Kumar and
Foufoula-Georgiou, 1997), studying the dynamics of the processes in the
magnetosphere–ionosphere system (Kovacs et al., 2001), extracting the periodic components caused by the Earth's rotation (Jach et al., 2006; Xu et
al., 2008), extracting low-frequency signals in the external magnetic
field and specifying models of the magnetic field (Kunagu et al., 2013),
finding precursors of intense solar flares (Barkhatov et al., 2016),
automatically detecting magnetic storm development (Hafez et al., 2010),
studying properties and characteristics of the waves of ultra-low frequency
(ULF) of the magnetosphere (Balasis et al., 2012, 2013, 2015), and studying characteristics of solar daily
variations based on data from ground-based magnetic stations (Klausner et
al., 2013), as well as several other issues. Additional applications of
wavelets include the estimation of different geomagnetic activity indices
such as the K index (Mandrikova et al., 2012, 2013b),
Dst index and the wavelet-based index of storm activity “WISA” (Jach et al., 2006; Xu et
al., 2008).
Recently, a multiscale analysis of geomagnetic data has been applied to reveal
the anisotropic and nonintermittent character of geofields, helping to
distinguish between their strong and wide-range variability (Lovejoy et al.,
2001). Furthermore, nonlinear effects have been described in the framework
of multifractal models with particular applications to multifractal and
magnetization fields (Pecknold et al., 2001). Additional applications of
multiscale analysis to the geomagnetic field include descriptions of its
long-term horizontal intensity variation, which are capable of tracking its
intermittency and representing a more complex nature of geomagnetic response
to solar wind changes than previously thought (Consolini et al., 2013).
The model proposed here is based on multiscale wavelet analysis and allows us
to study characteristic variations in the geomagnetic field and
nonstationary short-term changes characterizing fast-flowing processes in
the magnetosphere. We also discuss how this model can facilitate an in-depth
analysis of geomagnetic field variations. Previously we have shown that the
wavelet-based multiscale model allows us to automate the procedure of
determining the “quietest” days for calculating the Sq variation and
K index by using the Bartels technique (Bartels et al., 1939) and automatic
extraction and estimation of perturbations in the geomagnetic field
(Mandrikova et al., 2013a, 2014). In this paper, in order
to perform a more detailed analysis of the geomagnetic data and study
nonstationary short-term variations in the geomagnetic field, we suggest an
enhanced version of this model. We discuss the potential area of
application of the suggested model and some practical techniques based on
this model. Using a prominent example of the analysis of geomagnetic data
from a network of ground-based stations, we demonstrate the potential of the
suggested approach for studying variations in the field and extracting subtle
features during periods of increased geomagnetic activity.
The paper is organized as follows. In the “Data used in the study” section,
we provide data used in the research and information about the observatories registering
these data. The section also contains information on the analyzed magnetic
storms. In the “Material and methods” section, we provide a brief
theoretical outline of our wavelet-based approach, including the suggested
multiscale model and associated algorithms to assess characteristic
variations and local perturbations in the geomagnetic field during
periods of increased geomagnetic activity. In the “Experimental results and
discussion” section, we validate our model and algorithms using the
observational data obtained at magnetic observatories of Institute of
Cosmophysical Research and Radio Wave Propagation (IKIR) and Y. G. Shafer
Institute of Cosmophysical Research and Aeronomy (IKFIA) of the Russian
Academy of Sciences and Guam observatory (United States Geological Survey).
In the “Conclusions” section, we have provided the main results of our
research.
Geographical position of observatories that provided data used in
this study.
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Materials and methods
Model of geomagnetic field variation
In the wavelet domain, the
geomagnetic field variations can be represented as (Mandrikova et al., 2012,
2013b):
f0(t)=fchar(t)+∑jpertgjpert(t)+e(t)=∑nc-m,nφ-m,n(t)+∑j∈I∑ndj,nΨj,n(t)+∑j∈I∑ndj,nΨj,n(t),
where
fchar(t)=∑nc-m,nφ-m,n(t) is the
characteristic signal component that characterizes typical variations in
the geomagnetic field;
gjpert(t)=∑ndjpert,nΨjpert,n(t) is the perturbed component (here and in the following j∈I is denoted
as jpert), which characterizes geomagnetic perturbations arising during
the periods of increased geomagnetic activity (during the quiet periods
gjpert=0);
e(t)=∑j∈I∑ndj,nΨj,n(t) is
the noise;
Ψj=Ψj,nn∈Z is the wavelet basis;
φ-m=φ-m,nn∈Z is the basis
generated by a particular scaling function;
c-m,n and dj,n are the coefficients calculated as c-m,n=f,φ-m,n and dj,n=f,Ψj,n, where the symbol
… implies the scalar product;
I is the index set for the perturbed components;
j is the scale;
m is the wavelet packet decomposition level;
n is the sample number.
Identification of the characteristic component of geomagnetic field
variation
To estimate the characteristic component fchar(t) for a
given f0(t) we introduce the operator D such that f^char=Df0. This particular definition of D depends on the given a priori information. Since
the a priori probability distribution is unknown, we consider its minimax estimate
as recommended by Levin (1963) and Mallat (1999).
Then the purpose is to minimize the maximum risk for the set Θ with
fchar. In order to control the risk we next calculate the maximum risk
r(D,Θ)=supfchar∈Θr(D,fchar),
where r(D,fchar)=Efchar-Df.
The minimal risk is the lower boundary calculated for all operators D:
r̃(Θ)=infDr(D,Θ),
and the task is to find the operator D satisfying Eq. (2).
The component reflecting the characteristic changes in diurnal variations in
the Earth's magnetic field is called the quiet-day diurnal variation (Sq variation) (Bartels et al., 1939; Chen et al., 2007; Klausner et
al., 2013; Mandrikova et al., 2012, 2013b). The Sq variation is
characterized by the Sq curve, which is calculated as an average
smoothed curve over several quiet diurnal variations in the geomagnetic field
observed in the neighboring days (typically, variations over 3–5 days are
averaged). This averaging is required because the Sq variation does
not remain constant and its day-to-day reproducibility is quite limited.
Since no functional description of the probability distribution of the
Sq variation universal for all locations is available, it is
advisable to use the minimax approach for finding the best solution (Levin,
1963; Mallat, 1999).
Wavelet packet transform will be employed as a solution operator. In this
case the characteristic variation is introduced as the approximation
component determined in the wavelet domain by the coefficients
c¯-m,n=c-m,nn=1T. We will take the
Sq curve as a reference function for the error estimation since it
reflects the quiet-day diurnal variation in the given geomagnetic signal
(Bartels et al., 1939; Mandrikova et al., 2012, 2013b). The approximation
error in the wavelet domain can be expressed as
Um=1T∑n=1Tc-m,n-c-m,nSq2,
where
c¯-m,n=c-m,nn=1T is the coefficient
vector of the approximating signal component;
c¯-m,nSq=c-m,nSqn=1T is the
coefficient vector of the approximating component of the Sq curve;
j is the scale;
m is the wavelet packet decomposition level;
n is the sample number;
T is the total number of samples per day.
According to Eq. (3) the estimation error depends on the decomposition level m
and therefore there is an obvious need to find the decomposition
level m∗, which provides the smallest approximation error for
fchar(t).
Here we suggest a numerical stepwise algorithm for identifying the
characteristic component of the geomagnetic signal model as outlined below.
The geomagnetic signal f0(t) is divided into segments of duration T,
where T is equal to 1 day (N is the total number of discrete samples in
the entire signal):f0tnn=1N=f0tnn=1T,f0tnn=T+12T,…,f0tnn=N-T+1N.
The Sq curve and each segment of the geomagnetic signal are transformed into
the wavelet domain using wavelet packets. The wavelet-packet transform is
performed for m=-1,-2,…,-J, where J is determined by the segment
length T: J≤log2T. Finally, we obtain the Sq curve
components and each segment in the following form:f-mSq(t)=∑nc-m,nSqφ-m,n(t),f-ml(t)=∑nc-m,nlφ-m,n(t),where l is the segment number.
The reconstruction of the components f-ml and f-mSq is
performed at each level m. Then the components are expressed as
f0(-m),l(t)=∑nc0,n(-m),lφ0,n(t) and
f0(-m),Sq(t)=∑nc0,n(-m),Sqφ0,n(t), and U(-m),l is estimated by applying expression
Eq. (3):U(-m),l=1T∑n=1Tco,n(-m),l-c0,n(-m),Sq2.
The decomposition level m∗, which provides the lowest risk, is calculated asr̃(-m∗)=min-mmaxlU(-m),l.
The characteristic component of the geomagnetic signal is finally
expressed asf-m∗(t)=∑nc-m∗,nφ-m∗,n(t).
The resulting estimate can be improved by choosing the value for φ that provides
the lowest approximation error.
Using the data from the Paratunka observatory for 2002–2008 and the
algorithm above (steps 1–5), we calculated the estimation error of the
characteristic field variation for various wavelet bases and decomposition
levels. The goal was to find the optimal scale m∗ that provided
with the smallest approximation error for fchar(t) (see Eq. 3). Figure 3
shows the error of the characteristic variation estimation Um versus
the decomposition level m. The figure indicates that for the chosen
example, the smallest approximation error is obtained for the sixth
decomposition level using the Daubechies wavelets (Daubechies, 2001) of the
third order (see Fig. 3). Thus the reconstruction of the geomagnetic field
variation component in the wavelet decomposition basis can be expressed as
(see Eq. 1)
f(t)=fchar(t)+∑jpertgjpert(t)+e(t)=∑n∈Zc-6,nφ-6,n(t)+∑jpertgjpert(t)+e(t),
where c-6,n are the approximating coefficients of the sixth
decomposition level for the wavelet packet decomposition, φ-6,n is the basis function, and the component gjpert determines detailed
coefficients containing perturbations. The index set I includes the second,
fourth, fifth, and sixth scale levels.
Error of the characteristic variation estimation Um versus the
wavelet decomposition level m.
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Extraction of the perturbed components of geomagnetic field
variation
The degree of the geomagnetic signal disturbance is the so-called
perturbation magnitude (Bartels et al., 1939), which can be assessed
by calculating the difference between the greatest and the smallest
deviations of the current field variation from the characteristic diurnal
variation, namely the Sq curve. In the suggested model (Eq. 1) the
geomagnetic perturbations are characterized by the component
fpert(t)=∑jpertgjpert(t),
where gjpert(t)=∑ndjpert,nΨjpert,n(t), j∈I are denoted as jpert.
Decomposition of the geomagnetic signal and its components in the
period of a magnetic storm on 22–24 October 2016: (a) signal
decomposition scheme, with perturbed components marked by the grey color;
(b) H component of the Earth's magnetic field, Paratunka
observatory; (c) perturbed components of the geomagnetic field
variations.
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In order to identify the perturbed component of the geomagnetic signal model,
we next employ the wavelet-packet tree components gjpert(t), which
characterize the respective perturbations. According to the results
published earlier in Mandrikova et al. (2012, 2013b),
the geomagnetic disturbance Aj of the wavelet-packet tree component
gj(t)=∑ndj,nΨj,n(t) can be determined as
Aj=maxndj,n.
Then the identification of these components is performed following Rule 1:
j∈I,ifmAjv>mAjk+εj,
where mAjv is the sample average of the greatest
wavelet coefficients (for scale j) for perturbed days, mAjk is the average of the greatest wavelet coefficients (for scale
j) for quiet days, v is the index of the perturbed field variation, k
is the index of the quiet field variation, and εj determines a
systematic shift between the perturbed and the quiet days.
Assuming Ajk is normally distributed with mean μjk and
variance σjk, it is possible to estimate εj
as
εj^=x1-α/2σjknk,
where σjk is the variance of the greatest wavelet coefficients
(for scale j) for quiet days (this variance is determined as a result of
multiple measurements); x1-α/2 is the 1-α/2 quantile of
the standard normal distribution; nk is the number of analyzed
quiet-field variations. For α=0.1 the confidence probability is Pr=1-α/2=0.95, the quantile is x1-α/2=1.96, and
εj=1.96σjn.
The scales jpert are obtained from Eq. (7), correspond to the
perturbed components gjpert of the model, and characterize the
storminess of the magnetic field.
Figure 4 exemplifies the geomagnetic signal decomposition for the observatory
PET (Kamchatka) data, including the results of the extraction of perturbed
components of the field variation by applying Rule 1 for the perturbed period
during 22–24 October 2016. All decompositions included here and below were
performed based on a third-order Daubechies wavelet determined by minimizing
the approximation error (Mandrikova et al., 2012, 2013b). Signal components
with perturbations are shown in the diagram in grey (Fig. 4a). The analysis
of the results in Fig. 4b confirms the complex and nonstationary structure of
a geomagnetic signal, which includes multiscale components of the wide
frequency band arising at random time points and characterizing periods of
increased geomagnetic activity. One can see that, particularly prior to the
magnetic storm, on 20–22 October the component g42 already
contains short-term (instantaneous) increases in the magnitude of the wavelet
coefficients (Fig. 4c, indicated by dashed ellipses), which are possibly
connected with instantaneous changes in the parameters of the interplanetary
environment (currents at magnetopause) (Gonzalez et al., 1999; Yermolaev and
Yermolaev, 2010; Zaitsev et al., 2002). During the event, geomagnetic perturbations exhibit a wider
spectrum and the magnitude of the wavelet coefficients increases drastically.
Remarkably, a considerable growth in the geomagnetic perturbations on
22–24 October could also be observed, especially during the evening and
night (18:00–06:00 LT), which appeared to be more pronounced in the components g61 and gpert (see Fig. 4c) and is probably associated with
current intensification in the tail of the magnetosphere (Gonzalez et al.,
1999; Yermolaev and Yermolaev, 2010; Zaitsev et al., 2002).
Quiet-day variations obtained in nonautomatic mode (red curve)
and in the automatic mode (black curve): (a) Paratunka observatory;
(b) Yakutsk observatory.
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Determining the quietest days and calculation of the Sq variation
In order to determine the characteristic diurnal variation, one has to identify the quietest diurnal
variations for the analyzed time period and then calculate the average
smoothed curve by averaging the variations over these days (usually the 5 quietest
diurnal field variations within a period of 1 month are considered). The resulting
curve determines the quiet-day diurnal variation in the geomagnetic field.
Identification of the quietest diurnal variations can be performed
automatically by another suggested Rule 2:
if
Ajpert(1)=1L∑n=1Ldjpert,n(1)>1L∑n=1Ldjpert,n(2)=Ajpert(2),
where L is the component length, then gjpert(1)(t)=∑n=1Ldjpert,n(1)Ψjpert,n(t) for
the scale jpert is more perturbed than gjpert(2)(t)=∑n=1Ldjpert,n(2)Ψjpert,n(t). Then Ajpert=1L∑n=1Ldjpert,n characterizes the degree of disturbance of the signal
component for the scale jpert.
Thus, by applying Rule 2 we can automatically detect the quietest
diurnal variations in the magnetic field for the current month (normally
the 5 quietest days are used) and then construct the average smoothed curve,
namely the Sq curve, which is the zero baseline of the K index
values (Bartels et al., 1939; Mandrikova et al., 2012, 2013b). Figure 5
indicates that by using Rule 2 the characteristic variations in the
geomagnetic field and the quiet-day diurnal variations can be reconstructed
using the suggested wavelet-based technique in a fully automatic mode. In
contrast to the suggested approach, existing techniques do not allow for the
automatic performance of this operation. At present, we have performed the
software implementation of this technique for Kamchatka (PET, IKIR RAS) and
Yakutsk (YAK, IKFIA SD RAS), and the results of K index during its online
calculation are presented at http://www.ikir.ru/en/Data/datalfg.html
(last access: 30 August 2018) and
http://ysn.ru/intermagnet/kindex (last access: 30 August 2018).
Extraction of weak and strong perturbations in the geomagnetic
field
Let us consider three possible geomagnetic field activity levels:
activity level h0 – the field is quiet (magnetic field is
quiet);
activity level h1 – the field is weakly disturbed (magnetic
field is weakly disturbed);
activity level h2 – the field is disturbed (magnetic field
is disturbed).
According to these activity levels we can convert the mathematical model Eq. (1)
to the following form:
f(t)=fchar(t)+∑(jpert,n)∈I1djpert,nΨjpert,n(t)+∑(jpert,n)∈I2djpert,nΨjpert,n(t)+e(t),
where
fchar(t) is the characteristic component,
gpert,1(t)=∑jpert,n∈I1djpert,nΨjpert,n(t) is the component characterizing
weak geomagnetic perturbations,
gpert,2(t)=∑jpert,n∈I2djpert,nΨjpert,n(t) is the component characterizing
strong geomagnetic perturbations,
Ψjpert=Ψjpert,nn∈Z is the
wavelet basis,
djpert,n=f,Ψjpert,n are
the wavelet coefficients,
jpert is the scale,
n is the sample number,
I1,I2 are the index sets,
e(t) is the noise.
Consider the following conditions of djpert,n for the introduced geomagnetic field activity levels:
hjpert,n0 – the coefficient is quiet;
hjpert,n1 – the coefficient is weakly disturbed;
hjpert,n2 – the coefficient is disturbed.
The degree of the magnetic disturbance is determined for its given magnitude
by Eq. (6). To estimate djpert,n(jpert,n)∈I1 and djpert,n(jpert,n)∈I2 the threshold functions F1 and F2 are applied as follows:
f(t)=fchar(t)+∑jpert,nF1(djpert,n)Ψjpert,n(t)+∑jpert,nF2(djpert,n)Ψjpert,n(t)+e(t),F1x=0,ifx≤Tjpert,1orx>Tjpert,2x,ifTjpert,1<x≤Tjpert,2,F2x=0,ifx≤Tjpert,2x,ifx>Tjpert,2.
The coefficients with the quiet condition hjpert,n0 are considered noise (they are equal to zero).
Both F1x and F2x determine the
decision rules (Levin, 1963) for the condition of wavelet coefficients. Thresholds
Tjpert,1 and Tjpert,2 split the coefficient space X
into three nonintersecting areas: X0,X1,X2.
In our case the decision rule is deterministic: if the given data set falls
in Xi, the hypothesis that a coefficient has condition hjpert,ni is true. When a particular decision rule is used for the condition
hjpert,ni, the average losses are
Ji(x)=∑l=02ΠilPx∈Xlhjpert,ni,
where Πil is the loss function for erroneous decisions (each
erroneous decision has its own cost), Px∈Xlhjpert,ni is the conditional probability of
a data set falling in the area Xl, if condition hjpert,ni
has occurred and i≠l, i,l are the condition indices.
The conditional average of the losses for the given condition hjpert,ni is known as the conditional risk. Averaging the conditional risk function
for each of the conditions hjpert,ni,i=0,1,2 provides the average risk:
J∗=∑l=02piJi,
where pi is the a priori probability of the condition hjpert,ni.
The value J∗ is the quality criterion for finding the decision rule.
The best rule is the one providing the lowest average risk (known as the
Bayesian risk; Levin, 1963).
Since the a priori distribution of the conditions is unavailable, we will use the a posteriori risk
to obtain the best rule:
Jl(x)=∑i=02ΠilPhjpert,nix∈Xl,
where the a posteriori probabilities Phjpert,nix,i=0,1,2 provide the most complete characteristic of the
conditions hjpert,ni for the given observational data. For the
simple loss function
Πil=Π,i≠l,0,i=l,
the a posteriori risk Jl(x) equals
Jl(x)=Π∑i≠lPhjpert,nix∈Xl.
In this case the quality criterion for the decision rule is the smallest
number of errors. The thresholds Tjpert,1 and Tjpert,2
are determined by the best decision rule, in particular the rule that
provides the lowest value of Jl(x).
By minimizing Jl(x) we estimated the thresholds Tj,1 and Tj,2j∈I for the region of Kamchatka. The estimates were based on the
geomagnetic data from the Paratunka station for the period between 2002 and
2008. The disturbance degree of the geomagnetic field was characterized by
the K index:
The coefficients belong to the area X0 (have the quiet condition hjpert,n0), if the current value of the K index is equal to 0 or 1.
The coefficients belong to the area X1 (have the weakly perturbed condition hjpert,n1), if the current value of the K index is equal to 2, 3 or 4.
The coefficients belong to the area X2 (have the perturbed condition hjpert,n2), if the current value of the K index is greater than 4.
Application of operations (9) and (10) allows one to automatically extract
weak and strong perturbations characterizing the activity level of the
studied geomagnetic signal and thus to extract the information about the
activity level of the geomagnetic field in the place of observation. The
estimates have minute-scale time resolution, which allows one to obtain more
detailed and prompt information about the activity of the geomagnetic field.
It is also important that these transformations can be performed fully
automatically.
Data processing results for the period from 26 February 2011 to
2 March 2011: (a) speed of solar wind (b)
Bz component of the interplanetary magnetic field; (c)
AE index; (d) Dst index; (e) magnetic
field variation for the Paratunka observatory; (f) magnetic field
variation for the Yakutsk observatory; (g) identified perturbed
components of the field variations (blue line – Yakutsk observatory, red line
– Paratunka observatory); (h) results of applying operation 9 (above the plots we can see K indices of the stations YAK and PET);
(i) results of applying operation 10. The vertical dashed line
indicates the onset of a magnetic storm.
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Figure 6 presents the event on 1 March 2011 caused by the high-speed flow of
solar wind from the coronal hole (Space Weather Prediction Center,
ftp://ftp.ngdc.noaa.gov/STP/swpc_products/daily_reports). The figure
exemplifies the results of extracting weak (operation 9, Fig. 6h) and strong
(operation 10, Fig. 6i) geomagnetic perturbations. Figure 6 also shows
perturbed components of the geomagnetic field variations extracted using
Rule 1 (Fig. 6g).
Prior to a magnetic storm the speed of solar wind did not exceed
400 km s-1, Bz component of the interplanetary magnetic field (IMF)
varied in the range of ±5 nT. The structure of the obtained data
components (Fig. 6g) indicates some general regularity of the geomagnetic
field variations at the analyzed stations YAK and PET. Prior to the event one
can observe periods of moderate increase in the geomagnetic activity
(indicates in Fig. 6g by the dashed ellipses), which correlate with the
periods of moderate increase in the AE index (Fig. 6c, 26 February from 09:30
to 15:00 UT, 27 February from 12:15 to 13:20 UT, from 18:10 to 19:35 UT,
28 February from 19:30 to 20:25 UT; 1 March from 00:50 to 01:30 UT) that
are probably connected with the field of the current system of polar
perturbations. By applying threshold functions (operations 9 and 10, Fig. 6h,
i) one can confirm the appearance of weak perturbations in the geomagnetic
field prior to the event at the high-latitude YAK station, while at the PET
station (midlatitude), the geomagnetic activity did not exceed the
corresponding threshold (Eq. 9). The values of K indices at the PET and YAK
stations (Fig. 6h) also confirm a moderate increase in the geomagnetic
activity at high latitudes. Furthermore, at the beginning of the day on
1 March (from 05:00 UT) the speed of solar wind started increasing and the
component IMFBz contained oscillations
±10 nT. Between 07:00 and 10:00 UT on 1 March the Dst index
increased up to 20 nT, which confirms the outbreak of a magnetic storm
(Gonzalez et al., 1999; Yermolaev and Yermolaev, 2010; Zaitsev et al., 2002).
At the analyzed stations YAK and PET, one could observe weak perturbations
(up to 10 nT, Fig. 6h). After 10:00 UT one could observe the onset of the
main phase of a magnetic storm, which is characterized by a dramatic decrease
in the Dst index (to -88 nT). During the main phase of the storm on
1 March from 09:10 to 15:45 UT, from 17:00 to 18:45 UT, from 19:45 to
20:45 UT one could see a dramatic rise of AE indices (to 1350 nT), which
confirms strong substorms in the auroral area. An analysis of the perturbed
components of the geomagnetic field variations (Fig. 6g) shows clear
correlations between the periods of increase in AE indices and significant
short-term increases in the geomagnetic activity at the YAK station
(characterized by the abrupt peaks of high magnitude in the perturbed
component) mostly during nighttime (from 21:00 to 06:00 LT) that could
probably be associated with auroral processes and the intensification of
currents in the magnetosphere's tail. The results of applying threshold
functions (operation 10, Fig. 6i) confirm a significant increase in the
activity at the YAK station during the main phase of the storm (1 March from
22:10 to 02:50 LT). At the midlatitude station PET, perturbations were
rather moderate (did not exceed the thresholds Tjpert,2,
Fig. 6i) and exhibited activity on the low-frequency spectrum, which allows
us to attribute them to the intensification of the ring current during the
main phase of the storm (Gonzalez et al., 1999; Yermolaev and Yermolaev,
2010; Zaitsev et al., 2002). The recovery phase lasted for several days and
was followed by continuous auroral activity (Fig. 6c) and weak perturbations
in the field at the PET and YAK stations, which is typical (Gonzalez et al.,
1999) of a storm caused by the high-speed flow of a coronal hole.
Data processing results for the period from 26 February 2011 to
4 March 2011: (a) AE index; (b) magnetic field
variation for the Paratunka observatory; (c) – magnetic field
variation for the Yakutsk observatory; (d) results of applying
adaptive thresholds Eq. (12), the red color indicates positive perturbations
(increases relative to trend), the blue color shows negative ones (decreases
relative to trend); (e) results of applying operation (13), the red
color indicates positive perturbations (increases relative to trend), the
blue color shows negative ones (decreases relative to trend). The vertical
line indicates the onset of a magnetic storm.
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Extraction and estimation of nonstationary short-term variations in
the geomagnetic field
Due to the continuous variability of magnetospheric processes, especially
during perturbed periods, we can introduce the adaptive thresholds
Tjpertad and the coefficients djpert,njpert,n∈I, determining the component gjpert in Eq. (1):
djpert,n=djpert,n+,ifdjpert,n≥Tjpertaddjpert,n-,ifdjpert,n≤-Tjpertad,
where Tjpertad=U∗Stjpert,
Stjpert=1l-1∑k=1ldjpert,n-djpert,n‾2, djpert,n‾ is the average value calculated in
the gliding window of duration l, and U is the threshold coefficient.
Then following Eq. (6) the intensity of positive (I+) and negative
(I-) perturbations in the geomagnetic field at the time point t=n can
be determined as
In+-=∑jpertdjpert,n+-.
Figure 7 shows the results of applying operations (12) and (13) with the
following parameters: coefficient U=2 and window length
l=720samples (corresponding to 12 h), Fig. 7d, e during the
event on 1 March 2011 (the event is described above, see Fig. 6 and the
description in Sect. 3.5). The analysis of the results in Fig. 7 confirms the
efficacy of the adaptive thresholding Eq. (12) and shows that this allowed
for the extraction of nonstationary short-term changes in data characterizing the
appearance of weak increases in geomagnetic activity at the YAK and PET stations that preceded a major magnetic storm. The extracted perturbations could
be observed nearly synchronously at the PET and YAK stations, correlated
with the increase in the AE index, and were probably associated with
short-term (instantaneous) changes in the parameters of the interplanetary
environment (Gonzalez et al., 1999; Yermolaev and Yermolaev, 2010; Zaitsev et
al., 2002). During the initial phase of the storm the intensity of the
geomagnetic perturbations at the analyzed stations increased drastically (see
Fig. 7e). During the main phase of the storm we also observed short-term
dramatic increases in the intensity of the geomagnetic perturbations (see
Fig. 7e). Thus, the application of Eqs. (12) and (13) allowed us to extract and
estimate nonstationary (within the analyzed window of duration l)
short-term increases in the geomagnetic activity, which provide a more
in-depth view of the dynamics of geomagnetic processes.
Processing results of the data for 6–8 January 2015; (a)
Bz component of the interplanetary magnetic field; (b) speed of
solar wind; (c) AE index; (d) Dst index;
(e) H component of the magnetic field; (f) identified
perturbed components of the geomagnetic field variations; (g)
results of applying threshold function (9); (h) results of applying
threshold function (10). The vertical dashed line indicates the onset of a
magnetic storm.
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Experimental results and discussion
The suggested approach has been further validated for the events that
occurred on 7 January 2015 and on 17 March 2015. These events have
been studied by the authors in the works of Madrikova et al. (2017a, b). The
results of corresponding tests are provided in Figs. 8–11. The first
analyzed event, which happened on 7 January 2015 (see Figs. 8, 9), was
associated with the coronal mass ejection (CME; the catalogue of ICMES by
Ian Richardson and Hilary Cane,
http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm) that
occurred 3 days before exhibiting the typical phases of the
Dst variation (Gonzalez et al., 1999; Yermolaev and Yermolaev, 2010;
Zaitsev et al., 2002). Prior to the storm the speed of solar wind was
greater than average (> 400 km s-1) (Gonzalez et al.,
1999; Yermolaev and Yermolaev, 2010; Zaitsev et al., 2002) and the
Bz component experienced a change of ±11 nT. Figure 8 shows that on
6 January, prior to the event, the increase on the AE index (Fig. 8c) at
the analyzed stations was accompanied by weak perturbations in the
geomagnetic field (see Fig. 8g, calculated by Eq. 9): at 07:00–11:00 UT,
16:00–18:30 and 19:20–21:10 UT at the YAK stations, at 8:00–11:00 and
17:00–21:10 UT at the PET stations. These results are in accordance with
those of Davis (1997) and Zhang and Moldwin (2015), where prior to magnetic storms, one can
observe characteristic increases in solar wind parameters and the power of IMF
followed by increases in the geomagnetic activity indices (AE,
Kp). The coincidence of the periods of increased geomagnetic
activity at the analyzed stations with the periods in the AE index
increases following fluctuations in the Bz component (Fig. 8a),
allowing
us to suggest the connection of the extracted geomagnetic perturbations with
the nonstationary changes in the parameters of the interplanetary environment
and the intensification of auroral activity (Gonzalez et al., 1999; Yermolaev and
Yermolaev, 2010; Zaitsev et al., 2002). At the beginning of the day on
7 January, the Bz component turned to the south (at 00:20 UT) and in this
period decreased to the value of -5 nT at both the YAK and PET stations. At
the same time, short-term perturbations (from 01:15 to 01:30 UT, Fig. 8g)
could be observed. Also, at the initial phase of the storm,
increases in the Dst index (from 06:00 UT) and in auroral
activity (see Fig. 8c) could be observed, accompanied by weak perturbations
in the geomagnetic field at both the YAK and PET stations (Fig. 8g).
Processing results of the data for 6–8 January 2015; (a)
Bz component of the interplanetary magnetic field; (b) speed of
solar wind; (c) AE index; (d) Dst index;
(e) calculations following Eq. (13). Red color indicates positive
perturbations (relative to trend), blue indicates negative (relative to trend). The
vertical dashed line indicates the onset of a magnetic storm.
![]()
During the main phase of the storm the variations in the geomagnetic field at
the analyzed stations exhibited a considerably different structure (see
Fig. 8e, f) due to the location of these stations: YAK (52∘ of the
geomagnetic latitude, 163∘ of the geomagnetic longitude) is located
in the auroral area, while PET (45∘ of the geomagnetic latitude,
137∘ of the geomagnetic longitude) is located in the midlatitude area.
Figure 8f, g, h show that the increase in the geomagnetic perturbations and
the moments of extrema (where perturbations reached 175 nT at the YAK
station
while they reached 50 nT at the PET station) occurred at the stations at the
same time, mostly during nighttime or evening hours.
An application of Eqs. (12) and (13) to the data from a network of meridionally
located stations (from high latitudes to the Equator) shows the distribution of
the perturbations along the meridian of observations and confirms the general
dynamics of nonstationary short-term perturbations in the geomagnetic field
prior to a magnetic storm and during the event (see Fig. 9e). Quantitative
estimates (by Eq. 13, Fig. 9e) show significant correlations of the
extracted geomagnetic perturbations with the AE index, not only in
their occurrence times, but also in their intensities.
One can see that several hours prior to the onset of the magnetic storm
(indicated by vertical dashed line in Figs. 8–11), weak variations in the
interplanetary magnetic field (±5 nT), a moderate increase in the
AE index (up to 150 nT), and a short-term moderate increase in the
geomagnetic activity at the equatorial station GUA (shown in Fig. 9e by
dashed circle: 7 January from 00:50 to 01:45 UT) were visible. This confirms the
connection of the extracted perturbations with the auroral processes and also
with the increase in the magnetosphere's tail currents during the main phase
of a magnetic storm. Possible connections of the ring current with the processes
in the auroral area are provided in Mendes et al. (2005). The reconstruction
phase was short, at 20:00 UT the Dst index increased to -35 nT,
which is common for the events from a CME (Gonzalez et al., 1999). At the end
of the day on 7 January, fluctuations in the Bz component of the
interplanetary magnetic field (±10 nT, Fig. 8a) accompanied by the
fluctuations of the speed of solar wind (Fig. 8b) and followed by weak
perturbations in the geomagnetic field at both YAK and PET stations (see
Fig. 8g) as well as at the equatorial station GUA (see Fig. 9e) could be
observed.
Processing results for observations on 15–18 March 2015;
(a) Bz component of the interplanetary magnetic field;
(b) speed of solar wind; (c) AE index;
(d) Dst index; (e) H component of the magnetic
field; (f) identified perturbed components of the geomagnetic field
variations; (g) results of applying the threshold function (9);
(h) results of applying the threshold function (10). The vertical
dashed line indicates the onset of a magnetic storm.
![]()
Figure 10 shows similar results obtained during the magnetic storm on
17 March 2015. This event is characterized as a “double storm” (magnetic
storm with two main phases) and is caused by two separate emissions of the
solar substance (the catalogue of ICMES by Ian Richardson and Hilary Cane,
http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm). Prior
to a magnetic storm the speed of solar wind gradually increased from 330
to 430 km s-1. Figures 10f–h and 11e, f show similar dynamics in the
process preceding a storm. With fluctuations of the Bz component
(±12 nT) and an increase in the AE index (up to 540 nT on
16 March from 02:50 to 10:00 UT), we can observe short-term weak
geomagnetic perturbations at the analyzed stations. Synchronous perturbations
at the analyzed stations (from those located at high latitudes to the Equator), their
nonstationarity, and correlation with the AE index indicate a possible
connection between the extracted perturbations and the variability of the
interplanetary environmental parameters and an intensification of the auroral
currents. Weak variations in the interplanetary magnetic field (±6 nT)
accompanied by an increase in the AE index (up to 117 nT) and
a short-term moderate increase in the geomagnetic activity at the equatorial
station GUA (time period in Fig. 11f is indicated by the dashed line:
17 March from 00:00 to 03:10) several hours prior to the onset of the storm
could be observed as well. The observed dynamics in interplanetary
environmental parameters and geomagnetic activity variations is similar to the event
considered above and accords with the results of Davis (1997) and Zhang and
Moldwin (2015). At 04:00 UT on 17 March, due to the arrival of solar mass
from CME (the catalogue of ICMES by Ian Richardson and Hilary Cane,
http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm), the
speed of solar wind reached 510 km s-1 while the
Bz component of the interplanetary magnetic field reached 26 nT. In
45 min (at 04:00 UT) at the PET station, the onset of a magnetic storm was
registered. At the YAK station the initial phase of the storm was less
noticeable (Fig. 10e–g). The strongest geomagnetic perturbations at this
station began occurring 08:50 UT (Fig. 10h), with their magnitude reaching
307 nT, (Fig. 10f). This was accompanied by the reduction in the Dst index in this
period to -77 nT, and the AE index reaching 1055 nT.
Then the speed of solar wind reached 640 km s-1 and due to high-speed
flows of solar mass from the second CME (reminiscent of the scenario considered
in Gonzalez et al., 1999) beginning at 13:30 UT, one could observe the second main
phase of the storm followed by strong perturbations in the geomagnetic field
(at the YAK station the magnitude of perturbations reached 370 nT, while it reached 76 nT at the Paratunka station; see Fig. 10f, h, respectively) accompanied by a
strong decrease in the Dst index (to -224 nT). During this period
there were strong substorms in the auroral area, where the AE index
reached a maximal value of -2250 nT (see Fig. 10c). A detailed analysis of the event
based on the application of Eqs. (12) and (13) (see Fig. 11f) indicates that at the
beginning of a magnetic storm, at all analyzed stations (from those located at high latitudes to
the Equator), one could notice a short-term increase in the geomagnetic
activity. During the fluctuations of the Bz component of the
interplanetary magnetic field (to ±23 nT, Fig. 11a) and during an increase in
the AE index (from 06:00 to 09:00 UT, Fig. 11c), strong short-term
perturbations were observed, mostly at the stations closer to the north,
particularly YAK, PET, and MGD (see Fig. 11f). After the arrival of
high-speed flows of solar mass from the second CME on 17 March from 12:35 to
15:15 UT, one could observe a significant increase in the AE index
(Fig. 11c), a decrease in the Dst index (Fig. 11d), and strong
short-term perturbations in the geomagnetic field at all analyzed stations
(Fig. 11f). An analysis of perturbed components of the field variations
(Fig. 11e) and a comparison of the results with the results of Eqs. (12) and
(13) (Fig. 11f) show that during the time of the greatest decrease in the
Dst index (Fig. 11d) at low-latitude stations KHB and GUA, there were
strong geomagnetic low-frequency spectrum perturbations (fluctuations with
the period from 20 to 50 min, see Fig. 11e), which likely indicate their
connection with the strong intensification of the ring current during the second main
phase of a magnetic storm.
Processing results for observations on 15–18 March 2015;
(a) Bz component of the interplanetary magnetic field;
(b) speed of solar wind; (c) AE index;
(d) Dst index; (e) identified perturbed
components of the field variations; (f) results of applying
Eq. (13), red color indicates positive perturbations (increases relative to
trend), blue indicates negative (decreases relative to trend). The vertical dashed
line indicates the onset of a magnetic storm.
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Our results indicate the complex dynamics of the spatiotemporal distribution
of geomagnetic perturbations during the periods of increased solar activity
and magnetic storms. A detailed analysis of the events on 7 January and
17 March 2015 confirmed the occurrence of weak short-term perturbations in
the geomagnetic field prior to magnetic storms. The extracted perturbations
were observed at all analyzed stations (from those located at high latitudes to the Equator),
exhibited nonstationary behavior, and were accompanied by the fluctuations of
the Bz component of the interplanetary magnetic field and increase in the AE index. These results are in accordance with those of Davis (1997) and
Zhang and Moldwin (2015), which allows us to suggest their external nature
and connection with the nonstationary impact of solar wind on the Earth's
magnetosphere. In Davis (1997) and Zhang and Moldwin (2015), it has been shown
that increases in solar wind parameters and the subsequent increases in
geomagnetic activity (AE, Kp indices) can be observed prior
to the abrupt turns of the IMF towards the south, then leading to magnetic
storms (Lockwood, 2016).
The analysis of the results of this work also showed correlations of the
occurring geomagnetic perturbations with the AE index not only in
their occurrence times but also in their intensities. One possibility of
extracting such abnormal effects as a result of processing ground-based
geomagnetic data has also been suggested in Barkhatovetal (2016) and Sheiner and
Fridman (2012) and was mentioned briefly in Mandrikova et al. (2013a). The
analyses of the
authors Barkhatov et al. (2016) and Sheiner and Fridman (2012), based on
observational data and the joint analysis of the oscillations of
the H component of the geomagnetic field with the oscillating processes on
the Sun, have shown that the probability of these abnormal effects is high and
reaches nearly 90 %. Here we have confirmed this effect using a very
different approach and have shown explicitly that the suggested technique can
successfully extract corresponding events. An analysis of the variations in the
Dst index in the periods preceding magnetic storms can be found in Balasis
et al. (2006), where we can also find the assumption that the critical
feature of persistence in the magnetosphere is the result of combining
solar wind with the internal magnetosphere activity (the magnetosphere is affected
by solar wind).
Accordingly, an important aspect of this approach is the possibility of
extracting prestorm anomalies based on the analysis of the ground-based
data and the possibility of the automatic implementation of the technique,
with online performance exhibiting only minor delays. Several hours prior to
the analyzed magnetic storms, weak variations in the interplanetary magnetic
field (±5 nT for 7 January and ±6 nT for 17 March) were
accompanied by a moderate increase in the AE index (to 150 nT on
7 January and 117 nT on 17 March) and a moderate increase in the
geomagnetic activity at the equatorial station GUA. During the main phases of
the analyzed magnetic storms the geomagnetic perturbations increased
drastically, exhibiting a nonstationary spectrum depending on the station
where the data were measured, which could be attributed to the complex dynamics of
the current system during magnetic storms (Gonzalez et al., 1999; Yermolaev
and Yermolaev, 2010; Zaitsev et al., 2002).