We suggest a wavelet-based multiscale mathematical model of geomagnetic field variations. The model is particularly capable of reflecting the characteristic variation and local perturbations in the geomagnetic field during the periods of increased geomagnetic activity. Based on the model, we have designed numerical algorithms to identify the characteristic variation component as well as other components that represent different geomagnetic field activity. The substantial advantage of the designed algorithms is their fully automatic performance without any manual control. The algorithms are also suited for estimating and monitoring the activity level of the geomagnetic field at different magnetic observatories without any specific adjustment to their particular locations. The suggested approach has high temporal resolution reaching 1 min. This allows us to study the dynamics and spatiotemporal distribution of geomagnetic perturbations using data from ground-based observatories. Moreover, the suggested approach is particularly capable of discovering weak perturbations in the geomagnetic field, likely linked to the nonstationary impact of the solar wind plasma on the magnetosphere. The algorithms have been validated using the experimental data collected at the IKIR FEB RAS observatory network.

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

Geomagnetic field components.

As a rule, the horizontal component of the field (

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

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

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.

Experiments were carried out using the geomagnetic data (horizontal
component of the magnetic field) obtained at the IKIR observatories
Paratunka (PET), Magadan (MGD) and Khabarovsk (KHB) (in the eastern region of
Russia). Additional data sets for the analysis were kindly offered by the
Yakutsk (YAK) observatory of the IKFIA (Siberian region of Russia). Magnetic
data from the Guam observatory were obtained from INTERMAGNET (GUA,

Observatories whose data were used.

We use only magnetic data at minute-scale resolution obtained at
observatories in accordance with INTERMAGNET Standards
(

The characteristics of the magnetic storms analyzed in the paper.

The results of our analysis were compared with data from the interplanetary
magnetic field and the parameters of solar wind
(

In the wavelet domain, the
geomagnetic field variations can be represented as (Mandrikova et al., 2012,
2013b):

To estimate the characteristic component

Then the purpose is to minimize the maximum risk for the set

The minimal risk is the lower boundary calculated for all operators

The component reflecting the characteristic changes in diurnal variations in
the Earth's magnetic field is called the

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

According to Eq. (3) the estimation error depends on the decomposition level

Here we suggest a numerical stepwise algorithm for identifying the
characteristic component of the geomagnetic signal model as outlined below.

The geomagnetic signal

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

The reconstruction of the components

The decomposition level

The characteristic component of the geomagnetic signal is finally
expressed as

The resulting estimate can be improved by choosing the value for

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

Error of the characteristic variation estimation

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

Decomposition of the geomagnetic signal and its components in the
period of a magnetic storm on 22–24 October 2016:

In order to identify the perturbed component of the geomagnetic signal model,
we next employ the wavelet-packet tree components

Assuming

The scales

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

Quiet-day variations obtained in nonautomatic mode (red curve)
and in the automatic mode (black curve):

In order to determine the

Identification of the quietest diurnal variations can be performed automatically by another suggested Rule 2:

if

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

Let us consider three possible geomagnetic field activity levels:

activity level

activity level

activity level

According to these activity levels we can convert the mathematical model Eq. (1)
to the following form:

Consider the following conditions of

The degree of the magnetic disturbance is determined for its given magnitude
by Eq. (6). To estimate

Both

In our case the decision rule is deterministic: if the given data set falls
in

The conditional average of the losses for the given condition

The value

Since the a priori distribution of the conditions is unavailable, we will use the a posteriori risk
to obtain the

By minimizing

The coefficients belong to the area

The coefficients belong to the area

The coefficients belong to the area

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:

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,

Prior to a magnetic storm the speed of solar wind did not exceed
400 km s

Data processing results for the period from 26 February 2011 to
4 March 2011:

Due to the continuous variability of magnetospheric processes, especially
during perturbed periods, we can introduce the adaptive thresholds

Then following Eq. (6) the intensity of positive (

Figure 7 shows the results of applying operations (12) and (13) with the
following parameters: coefficient

Processing results of the data for 6–8 January 2015;

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,

Processing results of the data for 6–8 January 2015;

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

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 (

Processing results for observations on 15–18 March 2015;

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,

Then the speed of solar wind reached 640 km s

Processing results for observations on 15–18 March 2015;

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

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 (

To summarize, we have suggested, implemented, and validated a mathematical model and automated algorithms to analyze and describe the geomagnetic field variations based on the wavelet-based multiscale approach. Our results indicate that the model is particularly capable of reflecting the characteristic variation and local perturbations in the geomagnetic field during periods of increased geomagnetic activity. The efficiency of applying the wavelet transform in the analysis of geomagnetic data and the study of nonstationary processes in the magnetosphere can also be found in the works of other authors (e.g., Mendes et al., 2005; Hafez et al., 2013). In our research, we have suggested Rule 1 (operation 7) for identifying components containing geomagnetic perturbations. The magnitudes of the components extracted using Eq. (7) allow us to estimate the degree of geomagnetic activity.

In most cases when we need to extract nonstationary changes in the geomagnetic field we use the threshold (Balasis et al., 2013; Jach et al., 2006; Hafez et al., 2013; Mendes et al., 2005) together with the wavelet transform. In our work, we have suggested the technique of threshold estimation, and these thresholds allow us to extract geomagnetic perturbations in varying intensities (see Eqs. 9 and 10). We have also considered the adaptive threshold (see Eq. 12) for a more detailed analysis of the dynamics of the process. Analyses of moderate magnetic storms on 1 March 2011 and 7 January 2015 and of strong magnetic storms on 17 March 2015 have shown the efficiency of the suggested solutions.

Our experimental results clearly indicate the high sensitivity of the suggested technique and the possibility of its application in the in-depth study of the dynamics and spatiotemporal distribution of the geomagnetic perturbations (based on processing data from a network of geomagnetic observatories) for different levels of activity in the geomagnetic field. The suggested algorithms can be fully automated and allow one to find the moments of the increased geomagnetic activity and estimate quantitative characteristics of the degree of field perturbation. The suggested approach has high temporal resolution (up to 1 min), which helps us obtain detailed information on the activity level of the geomagnetic field in the case of nonstationary events. Moreover, the suggested approach is particularly capable of discovering weak perturbations that can occur prior to strong magnetic storms and could be associated with the nonstationary impact of the solar wind plasma on the magnetosphere.

Geomagnetic data from the stations of the Institute of
Cosmophysical Research and Radio Wave Propagation are available at the
following link:

OVM has developed the model and helped develop the algorithm, performed a detailed analysis of the processing results and prepared the manuscript. ISS has performed the estimation of model parameters, developed and implemented the algorithms, performed data processing and prepared the manuscript. SYK has performed primary data processing and helped analyze and edit the manuscript. VVG helped develop the model. DMK helped to analyze and edit the manuscript. MIB helped to analyze and edit the manuscript. All the authors took part in analyzing and discussing the results.

The authors declare that they have no conflict of interest.

The research is supported by the grant of the Russian Science Foundation
(project no. 14-11-00194). We would like to thank the staff of the geomagnetic
observatories at IKIR FEB RAS and at IKFIA SB RAS for providing high-quality
experimental data. The results presented in this paper rely on the data
collected at the Guam observatory. We thank USGS for supporting
its operation and INTERMAGNET for promoting high standards of magnetic
observatory practice (