Wavelet Analysis of Big Data in the Global Investigation of Magnetic Field Variations in Solar-Terrestrial Physics

TL;DR

This paper applies wavelet analysis to large-scale solar-terrestrial data to uncover correlations and understand geomagnetic phenomena, contributing to the emerging field of AstroGeoInformatics.

Contribution

It introduces a wavelet-based approach to analyze diverse big data sources in Solar Terrestrial Physics for the first time.

Findings

Identified correlations in wavelet coefficients across datasets.

Enhanced understanding of geomagnetic dynamics.

Demonstrated the utility of wavelet analysis in AstroGeoInformatics.

Abstract

We provide a Wavelet analysis of Big Data in Solar Terrestrial Physics. In order to explain and predict the dynamics of the geomagnetic phenomena we analyze high frequency time series data from different sources: 1. The Interplanetary Magnetic Field (from the ACE satellite). 2. The Ionospheric parameters - TEC (from ionospheric sounding stations). 3. The ground Geomagnetic data (from ground geomagnetic observatories, located in middle geographic latitudes). We seek for correlations in the wavelet coefficients which explain the dynamics of different magnetic phenomena in the Solar Terrestrial Physics. The large variety of data used in our research from both Solar Astronomy and Earth Observations makes it a contribution to the newly developing area of AstroGeoInformatics.

Equations

$$D\left(\theta ,\phi ,t\right)=c_0\left(\theta ,t\right)+\sum _{n=-\infty }^{\infty }{c}_0\left(\theta ,t\right)\sin \left(n\phi +{\alpha }_n\left(\theta ,t\right)\right);$$

$$D=D_{st}+DS;$$

$$DS(H)=D\left(H\right)-D_{st}(H)$$

$$0<C_{\psi }:={\int }_{-\infty }^{\infty }\frac{{\left|\hat{\psi }\left(\omega \right)\right|}^2}{\left|\omega \right|}d\omega <\infty$$

$$\hat{\psi }\left(0\right)=0$$

$$\int \psi \left(\omega \right)d\omega =0.$$

$$W_{\psi }f\left(a,b\right):=\frac{1}{\sqrt{a}}{\int }_{-\infty }^{\infty }f\left(t\right)\psi \left(\frac{t-b}{a}\right)dt$$

$$k=\frac{1}{a}.$$

$$f\left(t\right)=\frac{1}{C_{\psi }}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{1}{\sqrt{a}}{W}_{\psi }f\left(a,b\right)\psi \left(\frac{t-b}{a}\right)\frac{da}{a^2}db$$

$$D_{\epsilon }=\left\{\left(a,b\right):\left|W_{\psi }f\left(a,b\right)\right|<\epsilon \right\}$$

$$I_{\epsilon }:=\frac{1}{C_{\psi }}\underset{D_{\epsilon }}{\int \int }\frac{1}{\sqrt{a}}{W}_{\psi }f\left(a,b\right)\psi \left(\frac{t-b}{a}\right)\frac{da}{a^2}db.$$

$$\left|f\left(t\right)-I_{\epsilon }\left(t\right)\right|\le \delta \text{for all }t\in \mathbb{R}$$

Full text

We seek for correlations in the wavelet coefficients which explain the dynamics of different magnetic phenomena in the Solar Terrestrial Physics.

The large variety of data used in our research from both Solar Astronomy and Earth Observations makes it a contribution to the newly developing area of AstroGeoInformatics.

1 Introduction to Big magnetic Data in Solar-Terrestrial Physics

Following the present concepts, Big data in Solar-Terrestrial Physics would refer to large amounts of measured data

whose source is not homogeneous 2. 2.

having considerable dimension 3. 3.

the size and the format excess the capacity of the conventional tools to effectively capture, store, manage, analyze, and exploit them, and finally, 4. 4.

having a complex and dynamic relationship.

Institutions are increasingly facing more and more Big data challenges, and a wide variety of techniques have been developed and adapted to aggregate, manipulate, organize, analyze, and visualize them. The techniques currently applied in Big Astronomical and Earth Observation data usually draw from several fields, including statistics, applied mathematics and computer science, and institutions that intend to derive value from the data should adopt a flexible, reliable, and multidisciplinary approach. In particular, utilizing Big data in Solar-Terrestrial Physics and its analytics will improve the performance of prediction mechanisms for unusual geomagnetic events as for example geomagnetic storms.

Normally, Big data in Solar-Terrestrial Physics are accessible, but there are fewer tools to get value out of them as the data are immediately available only in their most coarsest form or in a semi-structured or unstructured format.

One broad way of using Big magnetic data in Solar-Terrestrial Physics to unlock significant value is to collect the data at a tick-by-tick level, i.e. at higher frequency. Let us remark that the standard registrations of the geomagnetic field variations has the order of about $0.1-10$ mHz (i.e. of periods $1,66$ min till $2,77$ hours). When considering geomagnetic data at higher frequency (here, high frequency is understood from the point of view of the standards in Geomagnetism), usually such data illustrate the complex structure of irregularities and roughness (i.e., multifractal phenomena) due to huge amounts of microstructure noise. The non-homogeneity characterized by multifractal phenomena is caused by a large number of instantaneous changes in the geomagnetic field due to geomagnetic storms and various sources of noises as, for example, the low frequency plasma instability modes. Therefore, mining big geomagnetic data needs to intelligently extract information conveyed at different frequencies. At the present moment the registration of geomagnetic signals has the maximal frequency of seconds, however majority of the geomagnetic or ionosound data are still collected at maximal frequency $4$ or $5$ min. as we will see below.

With the classic assumption of Data Mining, data are generated by certain unknown function representing signals plus random noise (see e.g. one of the bibles of modern pattern recognition [5] and the references therein). Decomposing big magnetic data in Solar-Terrestrial Physics is equivalent to extracting the systematic patterns (i.e., approximate the unknown function) conveyed in the data from noise, which is the standard approach of the classic signal processing theory brilliantly presented in the famous handbook [29].

The situation in the modeling of Geomagnetic fields and in particular geomagnetic storms falls in the framework of analyzing jump events in Big Data, which has been recently thoroughly studied by various researchers. In particular, it has been studied in the context of financial time series, in the nice research of Sun and Meinl, [34], [35]. They point out, that a specific problem arises when the trend component exhibits occasional jumps that are in contrast to the slow evolving long term trend and one needs to apply appropriate tools to analyze these singularity events.

In Geomagnetism, jumps are often caused by some unexpected large Geomagnetic storms or by predictable changes in the sectorial structure of the Interplanetary magnetic field. Traditional linear denoising methods (e.g., moving average) usually fail to capture these jumps accurately as these linear methods tend to blur them. On the other hand, nonlinear filters are not appropriate to smooth out these jumps sufficiently, because the patterns extracted by nonlinear filters are not stationary to present long run dynamic information. The situation is rather similar to that in the case of financial time series, as noted in [34] and in the references therein.

The present Chapter puts the foundations of a research in the Global structure of the magnetic phenomena in Solar-Terrestrial Physics, by systematically applying Wavelet Analysis (WA) to the available Big Data. We use Continuous Wavelet Transform (CWT) to analyze the structure and the dynamics of Geomagnetic storms. The wavelet method has been shown to be one of a number of multifractal spectrum computing methods and proven to be a reliable in signal processing as established in the classical monograph [24]. It has proven to be very suitable for analyzing time series analysis – for example, smoothing, denoising, and jump detection very diverse areas of science, finance, economics, see e.g., [30], [9], [12], [26], [22].

An important advantage of the wavelet method in analyzing magnetic phenomena in Solar-Terrestrial Physics, where factors of different scales interfere, is that it performs a Multiresolution analysi (MRA), in other words, it allows us to analyze the data at different scales (each one associated with a particular frequency passband) at the same time. In this way, wavelets enable us to identify single events truncated in one frequency range as well as coherent structures across different scales. Many recent studies have applied wavelet methods in mining geophysical and geomagnetic data, some very recent references are [37], [38], [19], [21], [20], [31].

Let us recall that there are in fact two versions of Wavelet Analysis: Continuous Wavelet Transform (CWT) and the Discrete Wavelet Transform (DWT). In both of them, there is a large variety of wavelet functions by which one may perform the signal decomposition. A common approach in choosing the wavelet function is to use the shortest wavelet filter that can provide reasonable results, see e.g. [30]. The main challenge in performing WA is how to determine the combination of wavelet function, level of decomposition, and threshold rule to reach an optimal smoothness that generally improves the performance of classic models after denoising the data. We have provided at the end a short Appendix which explains briefly the technical details of WA and our choice of wavelet function.

Another major advantage of Wavelet Analysis (either discrete or continuous), which makes it well adapted to the purposes of Big Data is that, similar to the classical Fourier analysis, there exist very fast algorithms for processing large amounts of data, [24].

Due to space restriction, in the present Chapter we have limited ourselves with just preliminary research of the correlations which exist among the data in the frequency domain (the coefficients of the CWT of the time series). Although the results obtained are very interesting and promising, we have not provided a more detailed statistical analysis (as e.g. in [34], [35]) which would uncover much deeper and interesting connections between the different factors playing role in the Geomagnetic phenomena. We leave such analysis for follow up research which is in progress.

The structure of the Chapter is as follows: in section 2 we recall the Big Picture of the Solar-Terrestrial Mechanism – in quiet and in stormy periods. This has to give an idea to the unexperienced reader about the complexity of the manifestation of a geomagnetic storm and the necessity to apply modern methods of Machine learning to Big Data, for a deeper understanding of the phenomena. In section 2.3 we provide a short description of the different components of the ground geomagnetic field provided by Chapman’s analysis – the global index $D_{st}$ and local disturbance index $DS$ which show how complicated the structure of the geomagnetic field on Earth is. In section 2.6 we provide basic information about two famous geomagnetic storms. In section 2.7 we provide basic information about the different types of data used for analyzing the Big picture of the magnetic phenomena in the Solar-Terrestrial Physics. In section 3 we provide the results of the application of CWT to the $3$ main types of data in the form of Simultaneous Time Series (Interplanetary Magnetic Field data, Ionospheric TEC data, Geomagnetic data), in some quiet days and in days of Geomagnetic storms. For every experiment carried out, we provide some empirical observations on the correlations between the CWT coefficients (the frequency domain) for simultaneous time series. In a forthcoming research we will apply the methods of Statistical Analysis for a more rigorous analysis of these observed correlations between the different types of data.

The large variety of data used, from both Solar Astronomy and Earth Observations, makes our research a contribution to the newly developing area of AstroGeoInformatics.

2 Mechanism of generating strong geomagnetic storms (long period

geomagnetic field variations)

The main purpose of the present study is to analyze different sources of data, and to discover correlations between the (relatively) high frequency geomagnetic variations (wave packages with short period about $0.1$ mHz till $10$ mHz) which happen not only during Geomagnetic storms but also in more quiet periods.

Before carrying out such an analysis we will provide the global context of the Geomagnetic picture in the Solar-Terrestrial interactions.

2.1 The Big picture of the Solar-Terrestrial Physics - quiet and

disturbed Geomagnetic phenomena

In the present section we will outline the Big picture of the influence of the solar activity on the Interplanetary Magnetic field, the Ionospheric parameters, and the (ground) Geomagnetic field.

First of all, we speak about events happening in the Magnetosphere, i.e. in the region of space surrounding Earth where the dominant magnetic field is the magnetic field of Earth, rather than the magnetic field of interplanetary space. The Magnetosphere is formed by the interaction of the solar wind with Earth’s magnetic field. The Earth’s magnetic field is continually changing as it is buffeted by the solar wind.

In quiet periods the Sun emits a flow of particles (solar wind) having a relatively constant intensity and speed, which start with appr. $50$ km/sec and is accelerated to about appr. $360$ km/sec close to Earth (which is obtained by the Parker model). The speed is accelerated by a mechanism which is still not known but most probably due to energy transfer in the solar wind. 2. 2.

In a disturbed state, the hyper-activity of the Sun causes Coronal Mass Ejection (CME) which increases the amount of charged particles; they have the macro-speed about $1500-2500$ km/sec when they leave the Sun; this speed decreases to about $500-700$ km/sec when they approach the Earth. 3. 3.

In the Interplanetary medium there is a collisionless plasma where the particles are electrons and ions (protons); the satellite ACE collects the data of the Interplanetary Magnetic Field (IMF) at the distance $1.5$ Million km from Earth in this plasma (see Figure 1); the most important is the $Z-$component of IMF called $B_{z}$ (which is perpendicular to the ecliptic) which influences the formation of the storm; during the strong storms the component $B_{z}$ is negative. 4. 4.

After that, the flow of particles approaches the Magnetosphere (about $12000-25000$ km) which is the belt of Van Allen (mathematical figure called torus), where they are caught by the Earth’s magnetic field (the Earth’s dipole); this looks like a cavern but they are mainly concentrated in the equatorial domain. The so-called Ring Current is formed in the Van Allen belts. In the Van Allen belts one does not have plasma, but there is a kinetic movement of the different particles. 5. 5.

Then at about 1000 km from Earth they enter the Plasmosphere and the Ionosphere which is a plasma (partially ionized gas), and the laws describing the plasma state start to work. There are collisions among the particles. Hence, more wave effects may arise compared to the collisionless plasma. The ionosound stations acquire the values of the Ionospheric parameters at these heights. For more details see the excellent exposition in the classical monograph [27]. 6. 6.

In the quiet days a regular source of disturbances in the earth’s ionosphere is also the solar terminator (at sunrise and sunset).

On Figure 1 below we provide the overall picture of different sources of measurements which we have used in our study.

On Figure 2 we provide the Magnetic field structure.

2.2 The geomagnetic storms

The geomagnetic storms are by their nature long period geomagnetic field variations.

Geomagnetic storms are phenomena directly related to solar activity. They result from the interaction of the magnetosphere and the ionosphere with changes in the interplanetary conditions caused by closed interplanetary magnetic structures. These structures are formed in the active centers of the Sun’s chromosome as a result of a sudden impulse ejection of the substance in the quiet solar wind called Coronal Mass Ejection (CME).

The storm is mainly characterized by a decrease in the horizontal component $H$ of the geomagnetic field, which encompasses the entire planet during a geomagnetic storm. At low latitudes, for a long time, a current system called the Ring Current is formed around the Earth at the distance approximately $2$ to $4$ Earth radii. The nowadays idea of the size and strength of the Ring current system gives us reason to view it as a toroid inside the magnetosphere (in the area of radiation belts, called Van Allen belts) and formed by particles of the solar wind. The geomagnetic field in the magnetosphere captures particles from the interplanetary plasma by creating an axisymmetric distribution of these particles in space. This leads to an amplification of the Van Allen radiation belt. The perturbed geomagnetic field associated with the so-called main storm phase in many cases is not axisymmetric. The analysis of the perturbed field shows that it can be represented as a sum of an axisymmetric part and an asymmetric component. This indicates that the proton belt is substantially asymmetric, especially in the early part of the main phase of the storm. Thus, asymmetry appears as an essential feature of Ring Current formation.

The intensive proton belt during a storm significantly deforms the magnetic field of the magnetosphere and changes its structure. In particular, we see that the domain of particle trapping approaches the Earth, in the interplanetary environment, and the polar ray oval shifts in the direction of the equator.

The intensity of the storm strongly depends on the geographic latitude. Thus, the decreasing of the horizontal component of the field during the storm in the different ground magnetic observatories is similar in shape but different in amplitude (see e.g. Figure 13 with the $H-$component of the geomagnetic field from observatory PAG in Bulgaria, and Figure 14 with the $DS$ index of the geomagnetic field data from observatory SUA in Romania). At low latitudes, this amplitude is larger and decreases as the latitude increases. There is a difference of the amplitude of the $H-$component of the magnetic field for the different longitudes. This asymmetry of the field is related to the asymmetry of the Ring Current considered above.

Geomagnetic storms are very diverse, but they are subdivided into two main types - ”standard” and ”with sudden start”. Geomagnetic storms of the second type are characterized by the absence of a pronounced sudden start. But in practice the main features of these storms during the main phase are like those of the standard type. Therefore, the initial contraction of the magnetosphere is not a prerequisite for the occurrence of the main phase of the storm.

For our purposes it is enough to mention that the type of Sun activity (solar chromospheric disturbances near or far from the solar equator) may cause different types of storms. The formation and spread of the interplanetary disturbed structure of plasma and the interplanetary magnetic field is a complex magnetohydrodynamic process that is essentially three-dimensional. The statistical analysis shows that more than $80\%$ of the strong geomagnetic storms are associated with intensive processes in the active centers of the Sun. For average storms, this percentage is smaller and is between $60-80$. Briefly, the Solar astronomy may provide an important information which has to be taken into account if predictions are needed.

It happens that important Sun processes do not have any impact on the geomagnetic field. This can occur in a relatively small number of cases when the disturbed interplanetary structure does not affect the point of the Earth’s orbit in which it is at that moment. The precise understanding of these phenomena requires the joint efforts of astronomers and geophysicists. All this indicates that knowing and predicting geomagnetic storms depends on knowing the propagation of interplanetary disturbances as a hydrodynamic process in three dimensions, [32]. For example, it has been found that the direction of the interplanetary magnetic field (IMF) vector $B$ is essential to unlock the geomagnetic storm mechanism. In particular, if the $Z-$component (denoted by $B_{z}$ in the coordinate system where the axis $Z$ is perpendicular to the ecliptic) is negative, then this indicates a possibility for a very strong geomagnetic storm.

It would be interesting for the unexperienced reader to hear about the parameters of a simulated Coronal Mass Ejection (CME), see details in [32], where a magnetohydrodynamic model is numerically implemented: the release energy is E, = $6.0\times 10^{22}$ J; the release mass is M, = $2.5\times 10^{10}$ kg; the initial velocities of the ejected flow are: the radial $u_{d}=1500$ km / sec. and the tangential is $v_{d}=0$; the duration of CME is $180$ seconds; the angle of the small conic area associated with the CME is $27^{0}$, see a model with these realistic parameters simulated in [32]. To understand how the southern direction of the interplanetary magnetic field is formed, we will look at the results of the computer modeling of the disturbance propagation in the interplanetary environment caused by the CME. As more details are provided in [32], let us shortly describe the dynamics of the simulated disturbance propagation in the interplanetary medium: After the CME happens at $t=0$ and has duration $3$ min., about $40$ hours later the disturbance reaches Earth’s orbit and (see Figures 3, 4 below and the Figures in [32]).

Remark 1

REMOVE = Remove: Figure 3 and 4) the resulting tangential velocity $v$ of the disturbed solar wind is shown on Figure 3,

from which it is seen that $v$ has positive and negative values.

Remark 2

REMOVE = This picture is in a meridional plane, defined by Sun-Earth as $x$ axis, and $Z$ vertical to the ecliptic; in it the tangential velocity $v$ is the component of the velocity along the $Z$ axis. The magnetic field components are shown in Figure 4, which is again in the meridional plane.

We put the following figures here: Below on Figure 5, 6, 7 we provide the radial solar wind velocity $U$ at different times: $10,$ $30,$ $50$ hours after the CME; note that the value of $U$ on the Figures is dimensionless:

Finally, on Figure 8 we provide the Contour plot of the radial velocity at $40$ hours after the CME:

On Figure 9 we provide the tangential velocity $V$ of the disturbed solar wind at $40$ hours after the CME; the values of $V$ are dimensionless:

and on Figure 10 below we provide the contour plot of the tangential velocity $40$ hours after the CME:

What concerns the magnetic field, we refer to the paper [32], where a detailed figures of the disturbed tangential IMF $B_{z}$ are provided. In [32] it is seen that the tangential component of the magnetic field vector $B_{t}$ (which in fact coincides with the $B_{z}$ component in the geocentric Cartesian coordinate system) also has positive and negative values. Thus a structure is formed in which the direction of the disturbed magnetic field vector is changing to the north or to the south (in the same coordinate system mentioned above). As we know these changes are the ones found to be the main cause of the geomagnetic storms at coupling of the disturbed interplanetary magnetic field with the Earth’s magnetosphere magnetic field. From the results of the simulations it is visible that the initial relatively closely disturbed conical area, caused by the CME, also expands in the tangential direction during the propagation and has values, which are comparable with the data gathered for example from the observations near the Earth orbit.

The above explanation justifies why is the component $B_{z}$ of the IMF so important for the understanding of the Geomagnetic storm.

2.3 Ground Geomagnetic Field, and geomagnetic activity index during a

storm

One of the classical models of geomagnetic storm is the Chapman model, [2]. It describes the disturbance of the (ground) geomagnetic field variation during a geomagnetic storm.

According to Chapman’s analysis, [2], if the time $t=0$ denotes the sudden start of the storm, then at a certain time point $t$ the disturbed magnetic field vector $D$ measured at a point on Earth’s surface (with components denoted by $D(H),$ $D(D)$ and $D(Z)$ ) can be expressed in a Fourier series expansion as follows:

[TABLE]

here $\theta$ is the complement of the geomagnetic latitude to $90^{0}$ , $\varphi$ is the geomagnetic longitude, and $\alpha_{n}$ is the phase angle. The first and the second terms in this expression represent the axially symmetrical component of the dipole axis and the asymmetric part of the disturbed field that varies with the longitude $\varphi.$ These components are referred to respectively storm-time variation ($D_{st}$) and local time-dependent disturbance ($DS$), which contain daily regular variation $S_{r}$ of type $S_{q}$ and the variance from the asymmetric part of the Ring current. So we can write:

[TABLE]

here $DS$ is a geomagnetic index that characterizes local storms. The variation of the horizontal component $D_{st}\left(H\right)$ of the $D_{st}$ field is a function of the variable $\theta.$ It is larger at low latitudes. The declination $D_{st}\left(D\right)$ has little change during the geomagnetic storm. The vertical component $D_{st}\left(Z\right)$ also changes slightly compared to the horizontal component. Thus, the $D_{st}$ variation is practically parallel to the Earth’s surface except in the areas of the polar cap, where the vertical component has larger positive changes. For that reason the main interest represents the horizontal component $D_{st}\left(H\right).$ Hence, we have the formula for the horizontal components

[TABLE]

where usually $D\left(H\right)$ is denoted simply by $H\left(t\right),$ and called the horizontal component of the field.

By neglecting $D_{st}\left(D\right)$ and $D_{st}\left(Z\right)$ as small quantities, it is apparent from the above analysis that the magnitude of the geomagnetic storm is determined mainly by the horizontal component $D_{st}\left(H\right)$, and this is the geomagnetic activity index during the storm. It describes the intensity of the symmetrical part of the circular current that occurs in the equatorial area of the magnetosphere during a magnetospheric storm. This is the global part of the geomagnetic storm index. It represents the mean value of the disturbed horizontal component of the geomagnetic field determined by data from several low-level observatories, distributed by geographic lengths. On quiet days, this variation may be around $\pm 20$ nT, but during a geomagnetic storm it reaches large negative values of the order of hundreds of nT.

Remark 3

The determination of the $D_{st}$ index is provided every hour, and is practically determined by taking an average of the data through a consortium of geomagnetic observatories. The process of practical determination of this geomagnetic index is described in detail in the IAGA bulletin.

Remark 4

Geomagnetic field variation data in the past decades contained mean hour values, however, the present data contain mean minute values, and in all INTERMAGNET observatories, the registration is done with mean-second values. This shows an increase in the information about the geomagnetic field and one may speak about ”big data”-shift of the measurement paradigm.

The $D_{st}$ index during the 2003 storm

In order to give an idea about the general form of the $D_{st}$ index during the storm we provide the $D_{st}$ data during the storm in October $29,$ $2003$, downloaded from the WDC (World Data Center) for Geomagnetism in Kyoto; the data available are for $3$ days, every hour, see Figure 11. On Figure 11 we see that the $D_{st}$ variation has two big decreases due to two different CMEs.

2.4 Ionospheric parameters from ionospheric sounding stations

Let us remind that the ionosphere of the Earth can be seen as a conductive layer. Its motion induces an electromotive force $v\times B$. Here $v$ is the vector of the drift velocity of the charged particles (not to be mixed with the tangential macro-velocity of the solar wind!), and $B$ is the geomagnetic field vector in the ionosphere.

Let us mention that the ionosphere is influenced by different factors which cause some long-period (low frequency) ionospheric parameter changes:

Solar tidal movements and movements, caused by the periodic warming of the atmosphere by the Sun, cause changes in the geomagnetic field, called quiet-solar variations. The latter, as noted above, are referred to as $S_{q}$. In turbulent conditions, we have a regular variation $S_{r}$ associated with the local time, which differs from $S_{q}$ due to the influence on the magnetospheric storm ionosphere. As with $S_{q}$, the conditions in the high atmospheric layers are particularly important and essential. For example, the concentration of the charged particles in the ionosphere, which depends on UV ionization, as well as the degree of invasion of charged particles in the ionosphere from the above areas, such as the plasosphere and the magnetosphere. 2. 2.

The distribution in the ionosphere of the electromotive forces, determined by the lunar tidal movements of the atmosphere, is approximately fixed in relation to the moon. Thus, the induced current and the resulting magnetic field is also fixed for an observer on the moon. Each magnetic observatory performs one turn per day about this distribution, turning in a circle defined by the latitude. Therefore, the stations register a variation of the geomagnetic field over time. This variation is called lunar-day and is denoted by $L$.

2.4.1 Data about the parameters of the ionospheric plasma

In the present work we have used data for different parameters of ionospheric plasma in a specific local area, as TEC, F2, etc. In the experiments presented we have limited ourselves with the TEC data since it is considered as the most important of all parameters. For example, we use the ionosphere sounding data from ground ionospheric stations, which are chosen to be located near the (ground) geomagnetic field registration points. This allows by comparison of the two ”signals” to seek for the presence or absence of a possible correlation between them. Thus, we have the possibility of identifying the origin of individual modes (wave packages) and groups of modes.

The ionospheric and geomagnetic data are synchronized in universal time so that we can monitor for the presence or absence of simultaneity of the two signals. We have used data on ionospheric parameters with a sampling frequency comparable to the geomagnetic data, namely, every $5$ minutes. This is very interesting in terms of decomposition of geomagnetic variations. Of course, the modes associated with extra-ionospheric origin can also be identified and this has already been commented (as for example the geomagnetic pulsations).

Let us note that in recent years, the ionospheric data registration also tends to increase the sampling frequency and the sampling has now reached in some stations a period of $5$ minutes between two samples. Given that a large number of ionospheric plasma parameters are measured at these stations, we can assume that they may also be referred to as ”Big Data” paradigm.

For example, on Figure 12 below we provide a graph of the TEC data from the ionosound station in Athens, on a quiet day, January 4, 2018. The variation of the TEC data is shown. We see that the main trend is given by the daily variation of the TEC. The fast oscillations with periods less than $3$ hours are observed during the whole day. At midday time we observe modes (wave packages) with a larger amplitude which are apparently of soliton type. This phenomenon has been studied in [4].

Remark 5

In an interesting research of the short period modes (wave packages) in the Ionosphere, V. Belashov has created soliton model for their explanation, see [4] and references therein; matching of the model to the TEC values is presented on p. $338.$ See also [33].

2.5 Emergence of higher-frequency modes in the Ionospheric parameters

and in the IMF which are related to the ground Geomagnetic field variations

Up to here we have considered the long-periodic Geomagnetic variation phenomena were considered above. Below we describe shortly the generation of short-period (high-frequency) Geomagnetic variations caused by low-frequency plasma instabilities.

As already mentioned above, the plasma in the ionosphere, plasmospher and the interplanetary colisionless plasma medium have considerable instability. Instability is also occurring in the radiation belts of the Earth, i.e. in the Magnetosphere although most of the latter do not realize the conditions characterizing the medium as plasma. Some examples of cosmic plasma show the presence of instabilities that produce non-thermal waves and various distribution functions of kinetic particles parameters, especially in the ionosphere and the plasmosphere.

Instabilities can also cause a non-linear effect of wave propagation in the ionosphere, plasmosphere and interplanetary medium, and they also cause collisions and acceleration of fast particles in astrophysical plasma.

For the current work, it is important to note the low-frequency instabilities in terms of natural plasma frequencies (MHD instabilities, fluid instabilities and drift instabilities). They create the low-frequency modes in the Geomagnetic field and in the Interplanetary magnetic field. The properities of the wave modes are strong functions of frequency. For low-frequency plasma modes, the circular frequency of the waves is much smaller than the natural frequencies as the plasma frequency and the cyclotron frequency of the plasma.

Practically, very often these low-frequency plasma modes coincide with the considered by us higher-frequency geomagnetic field variations. In this work, we use geomagnetic field registration data on the ground with periods more than $1$ minute, that are associated with low-frequency modes in the Earth’s plasma cosmic environment, and also with Ring current fluctuations in the Magnetosphere.

In plasma, the macroinstabilities occur in the low-frequency mode and usually involve the magnetic field (not just the ground geomagnetic field!). Therefore, in the present research we use data about the variations of the** **magnetic field in different areas of the near Earth space and those generated in the interplanetary space.

What concerns the short period (higher-frequency) ionospheric parameter variations, let us recall that the variation can be decomposed, as a superposition of sources located in the magnetosphere and the ionosphere and also associated with the various tidal movements of the high Atmosphere. The high frequency (ground) geomagnetic field variations may be a result of short period Ionospheric parameters variations.

It is clear from the above that it is **necessary **to use data on the ionosphere status as well as data of the **interplanetary magnetic field (IMF), **and the ground Geomagnetic field during the storm.

In the present work, in each case, we choose to study the behavior of the various magnetic field components that are associated with certain processes in the respective area of observation.

In the Interplanetary medium disturbances of the IMF and of the solar wind propagate, as we have already mentioned above. Beyond that, in the Interplanetary medium different low-frequency plasma modes are generated, as a result of the plasma instabilities. Data on IMF variations induced by macro-instability of the interplanetary plasma medium are recorded by satellites located outside the Earth’s magnetosphere, as for example the satellite ACE whose data we use. These data contain the values of the components of the magnetic vector as well as the values of its magnitude, measured by different instruments. In order to study these instabilities and related waves in the interplanetary medium, we use data for the IMF with data sampling $4$ minutes. Magnetic modes in these media and in this frequency range are associated with the so-called macroinstability.

Finally, one of the main objectives of our study is to identify modes with **short periods **(with frequencies much larger than the natural plasma frequencies), which are generated by microinstabilities, and are caught by the ground geomagnetic field registration and in the plasma ionospheric parameters.

2.6 The strong geomagnetic storms in $2003$ and $2017$ to be analyzed

In the present research we apply Wavelet Analysis to analyze short-period Geomagnetic Field variations, Ionospheric parameter variations, and Interplanetary magentic field variations, during the manifestation of the two famous strong geomagnetic storms during the $23$rd solar cycle in $2003,$ and during the $24$th solar cycle in $2017.$

2.6.1 The storm in 2003

The data obtained from Solar Astronomy observations have provided the following report:

During the $23$rd solar cycle in October and November $2003$ there were two very strong storms. One started on October 29 and the other began on November $21$. In the last ten days of October 2003, the lean activity has gone to an extremely high level. On October 18, a large active region (AR), turning north of the solar equator, was designated by NOAA as AR 484. On October 28, AR 484 was located near the sub-Earth point of the solar disk $8^{0}$ on the east of the central meridian and $16^{0}$ north latitude. At 11:10 UTC AR 484 produced one of the largest solar flares for the current solar cycle. This flare was classified as X17 (peak X-ray flux $1.7\times 10^{-3}$ W/m2 ).

An extreme CME with a radial plasma velocity of $2500$ km/s was observed. The mass ejected from this CME was in the range of $1,4-2,1\times 10^{13}$ kg, and the kinetic energy released was $4,2-6,4\times 10^{25}$ J. The following day, October 29, AR 484 again produced a large eruption. This peak was X10 (X-ray flux $10^{-3}$ W/m2 ) at $20:49$ UTC. I was targeting the Earth halo at a speed of $2000$ km / s and kinetic energy of $5,7\times 10^{25}$ J. The interplanetary magnetic field (IMF) reached about -50 nT, its normal value, in calm conditions, is ten times lower. The shock wave of the event on Oct. 28 was determined by the Advanced Composition Explorer (ACE) spacecraft at $05:59$ UTC. At $06:13$ UT was registered an SSC pulse, marking the beginning of the sixth storm by the registration stamp (since 1932). On 29 and 30 October the planetary index $Kp$ reached value 9. The geomagnetic storm continued until November 1 and had a horizontal component down to around -400 nT. The highest value of the $D_{st}$ index was registered on October 30 at $23:00$ UT.

2.6.2 The storm on 7-8 September, 2017

The other storm considered in the present work is the one on September $7-8,$ $2017$. It was one of the most flare-productive periods of now-waning solar cycle $24$. Solar active regions (AR) $2673$ and $2674$ both matured to complex magnetic configurations as they transited the disk. AR2673 transformed from a simple sunspot on 2 September to a complex region with order-of-magnitude growth on 4 September, rapidly reaching beta-gamma-delta configuration. In subsequent days the region issued three X-class flares and multiple partial halo ejecta. Combined, the two active regions produced more than a dozen M-class flares. As a parting shot AR2673 produced: 1) an X-9 level flare; 2) an associated moderate solar energetic particle event ;and 3) a ground level event, as it arrived at the solar west limb on 10 September. From $4-16$ September the radiation environment at geosynchronous orbit was at minor storm level and $100$ MeV protons were episodically present in geostationary orbit during that time frame. The early arrival of the coronal mass ejection associated with the 6 September X-9 flare produced severe geomagnetic storming on 7 and 8 September. The full set of events was bracketed by high speed streams that produced their own minor-to-moderate geomagnetic storming.

2.7 Acquired Data for short period variations of the Geomagnetic

field, the Ionospheric parameters, and the IMF,

As we have explained in section 2, the global picture of the geomagnetic phenomena is very complicated and dynamic. For that reason, for the explanation as well as for the prediction of its dynamics one needs to attract as much as possible observable data, which form the basis of our Big Data analysis.

We analyze high frequency time series data from different sources. Let us remind that ”high frequency” registrations in Geomagnetism are of the order $0.1-10$ mHz (i.e. of periods $1,66$ min till $2,77$ hours). The following three high frequency time series were acquired:

$T_{G}:$ Time series for the ground Geomagnetic data (from ground geomagnetic observatories, $1$ min. sampling and $1$ sec. sampling), in nT. Our main objective is to seek for correlations in the wavelet coefficients of the CWT of the above time series which explain the dynamics of different geomagnetic phenomena. 2. 2.

$T_{IP}:$ Time series for the Ionospheric parameters - TEC (from ionospheric sounding stations, $5$ min. sampling), in TEC unit. 3. 3.

$T_{IMF}:$ Time series for the Interplanetary Magnetic Field (from the ACE satellite, $4$ sec. sampling), in nT.

2.8 Data about the strongly disturbed geomagnetic field in October and

November 2003 and September 2017

2.8.1 The H-component of the geomagnetic data from

**Panagiurishte (**PAG) observatory

On Figure 13 below we have the variation of the $H-$component (see formula (2)) registered at the geomagnetic observatory PAG during the geomagnetic storm on October $29,$ $2003.$ The data are mean-hour values, registered from [math] h on October $29$ till $24$ h on October $31.$

2.8.2 The $DS$ index from the Surlary (SUA) geomagnetic data

On Figure 14 below, we provide the data for the index $DS$ variation (defined in formula (1) during the storm $29-31$ October, $2003,$ registered at the Surlary (SUA) geomagnetic observatory in Romania. The sampling of the data is $1$ minute.

We see all details due to the fact that the data are provided every single minute.

Remark 6

It is questionable whether it is worth applying the CWT to the $DS$ index given by $H\left(t\right)-D_{st}$ (see formulas (1) and (2)), or directly to the rough data of the $H-$component, since we are in principle seeking for variations of $H,$ but $D_{st}$ index is provided on hourly basis. This might create artificial jump every hour.

3 EXPERIMENTS

In the experiments to follow, we are motivated by the interest to discover wave packages of short periods and their correlations in the three different sources of data which we have already discussed:

(ground) Geomagnetic data, from geomagnetic observatories 2. 2.

Ionospheric data (TEC values, from ionosound stations) 3. 3.

IMF data (from satellite ACE),

We would like to identify any kind of correlation and causality among them by applying the method of Wavelet Analysis.

3.1 References on applications of Wavelet Analysis to

Geomagnetism

Before presenting the results of our experiments, we provide some references which might be useful to the reader.

Let us note that in a number of works, [25], [36], [3], [6], [14], [16], [37], the variation of geomagnetic data (in particular of $D_{st}$) is analyzed by means of wavelet analysis. Recently, wavelet analysis of geomagnetic field perturbations was widely used in the study of tsunami waves [17], [18], [19], [21], [20], [31]. In [14], [38] wavelet analysis of the geomagnetic field is used to define a new index, alternative to $D_{st},$ but on a minute basis.

3.2 Experiments with data on a quiet day, 28 July, 2018

In order to have controls over the statistical behavior of the geomagnetic data during geomagnetic storm, we have taken data for quiet days from two geomagnetic observatories – in situ repeat station at Balchik (Bulgaria), and from another geomagnetic observatory Surlary (SUA), in Romania; the distance between them is about $190$ km. This means that they have almost identical conditions, and it is well known that there are no strong magnetic variations from natural or artificial character in the regions.

3.2.1 Visualization of the Wavelet Analysis

We have provided a brief summary on the Continuous Wavelet Transform in Appendix, section 6.

In the experiments below we perform CWT to time series $f\left(j\right),$ where $f\left(j\right)=F\left(\delta t_{j}\right)$ for some ”continuous time series” $F\left(t\right),$ and $t_{j}\delta$ are the sampling times on a uniform mesh, $t_{1},t_{2},...,t_{N}$. Here $\delta$ denotes the sampling interval, for example, we have $\delta=1$ second, $4$ seconds, $1$ minute, $1$ hour, etc. We visualize the absolute values $\left|W_{\psi}f\left(a,b\right)\right|$ of the CWT coefficients $W_{\psi}f\left(a,b\right)$ defined in formula (3). In all our experiments the shifts $b$ (visualized on the $X-$axis) run through the full set of indexes $1,2,...,N.$ On the other hand, the non-negative parameter $a$ in formula (3) which denotes the scaling parameter (and has the meaning of periodicity), is interesting for us mainly for shorter intervals. Hence, in some experiments we consider only periods $a\leq N_{1}$ for some maximal period $N_{1}<N.$ The parameter $a$ is visualized on the $Y-$axis. In order to get a better idea of the behaviour of the CWT coefficients $W_{\psi}f\left(a,b\right),$ we find it instructive to have the visualization of $\left|W_{\psi}f\left(a,b\right)\right|$ both as a heatmap and as a contour map, see for example, Figure 16 below.

3.2.2 Experiments with Balchik Geomagnetic data, $28$ July, $2018,$ 1

second data

The $H-$component geomagnetic data were collected every second, for 24 hours, at a repeat station in Balchik (Bulgaria), on $28$ July, $2018.$

We provide the graph of the time series of the data Figure 15 below. The daily variation of the field is clearly visible as the main trend, and also some rather permanent short-period variations.

Below, on Figure 16, we provide the experiments, namely, the heatmap (on top) and the contour map (on bottom) of the absolute values of the CWT coefficients $\left|W_{\psi}f\left(a,b\right)\right|$ of the $H-$component .

On Figure 16 we see that in the CWT of the time series of the Balchik geomagnetic data, some very interesting details are identified. Around midday, some wave packages with periods about $100$ min are clearly visible. They show the possibility for soliton like oscillations , related to the solar terminator, theoretically studied in [4], and which has been recently observed in the Wavelet Analysis experiments with geomagnetic data in the paper [33].

3.2.3 Experiment with SUA geomagnetic data on 28 July, $2018$

We have made experiments with the time series formed by the $H-$component of the geomagnetic data from SUA (Surlary, Romania), on whole day, July 28, $2018$; the sampling is every minute. The heatmap (on the top) and the contour map (on the bottom) of the CWT is provided on Figure 17.

Remark 7

On the Figures 17 we see wave packages of $100$ minutes during the whole day, and wave packages with periods below $180$ minutes about midday. At midday one observes an intensive process of generation with a period between $100$ to $150$ minutes. Right at the same time interval (between $600$ and $700$ minutes) there are wave packages with period $30-50$ minutes. One may suggest that the latter phenomenon may be generated by the solar terminator, and have a soliton structure, [4].

Remark 8

There is a lot of similarity at the scale of 20-60 minutes between the CWT of Balchik data and the CWT of SUA data:

We compare the CWT of the Balchik data in Figure 16, and the CWT of the SUA data in Figure 17. We see that the wave packages with periods about $80$ minutes are much more expressed in the Balchik second data, than on the SUA minute data. This shows that the oscillation phenomena carry persistent character. In particular, as was concluded in the paper [33], we may suggest the existence of soliton like patterns at the periods of 40 to 60 min.

Remark 9

CONCLUSIONS: Wave packages with periods from 10 to 100 min. exist during quiet geomagnetic days, which are however predominant at midday.

3.3 Experiments with data for the Geomagnetic storm on $7,$ $8$

September, $2017$

In the following we provide a Wavelet Analysis of the data from ground geomagnetic field, Ionospheric parameters, and IMF, collected during this strong geomagnetic storm.

First we provide the picture of the main trend which is determined by the $D_{st}$ index.

3.3.1 The $D_{st}$ for the period 7-10 Sept., 2017

This Figure shows the $D_{st}$ graph for the period, Figure 18:

As we have described the $D_{st}$ index in section 2.3, it shows a very unusual behaviour after the storm of 7-10 Sept., 2017.

Remark 10

On Figure 18 we see the variation of the $D_{st}$ index during the manifestation of the geomagnetic storm in September, $2017.$ On the Figure we see the two decreases of the magnetic field, caused by two event on the Sun surface, and also a very long recovery phase of the storm during the period $9-10$ September. This long recovery phase is most probably related to the lack of short-period variations in the geomagnetic records in the observatory on September $9$ and $10.$ The decay of the Ring current cannot create significant geomagnetic variations on the ground in the region of PAG observatory. However, as we have remarked above, the ionospheric macroinstabilities during these two days cannot create a variation in the ionospheric current system, which itself would create variations to be registered by this ground observatory.

3.3.2 Experiments with Geomagnetic data from PAG, 7-10 Sept., 2017, 1

minute data

We have acquired the $H-$component of the geomagnetic data from the Panagyurishte (PAG) geomagnetic observatory. These data are every 1 minute sampling period. We provide the heatmap (on the top) and the contour map (on the bottom) of the CWT, see Figure 19.

We see that during $7$ and $8$ September of the geomagnetic storm we may identify wave packages with periodicity $20-100$ minutes. However, on $9$ and $10$ September, one cannot identify wave packages with periods in the interval $20-100.$

3.3.3 Experiments with Ionosperic data from Athens, 7-10 Sept., 2017,

$5$ min. data

We have taken the ionosound TEC data from the ionosound station in Athens (ATN). The data are measured for full four days 7-10 Sept., 2017, every 5 minutes (this frequency is the modern standard for sampling of ionosounding data).

On Figure 20 we provide the heatmap (on the top) and the contour map (on the bottom) of the CWT of the TEC time series.

We see that the short period scales which are interesting for us really show regular patterns. For that reason we have restricted the scales only to 50, and we show the result below, Figure 21:

We see that during the geomagnetic storm we have a lot of wave-packages and regular patterns. However what is not less interesting, similar pattern appear during the two days $9$ and $10$ September, (during the recovery phase of the storm), at midday time, having periods $20-50$ minutes. This seems to be due to the solar terminator influence, as was suggested in [4], [31].

3.3.4 Experiments with IMF data from ACE satellite, 7-10 Sept., 2017,

$4$ min. data

We retrieved the $B_{z}$ component of the IMF from the ACE satellite on 7-10 Sept., 2017, every 4 min. data.

On Figure 22 we provide the CWT of the time series containing the values of $B_{z}$; again the heatmap is on the top and the contour map is on the bottom of the Figure.

An interesting observation is that in the ACE data and in the ground PAG data one cannot identify any wave-packages of periods $20-100$ min, after the storm, i.e. on Sept. $9,10,$ $2017.$ This makes us believe that there is a correlation between the two observable values. This is in a strong contrast to the Ionospheric observations provided above on Figure 20 and Figure 21, where such wave packages are available. This shows that in some cases the ionospheric plasma generates short-period modes with a relatively small amplitude. This would explain the lack of similar modes in the ground geomagnetic data of PAG. On the other hand, it is clear that on $9$ and $10$ September, these short-periodic modes are the result of eigen-oscillations of the ionospheric plasma, which are not caused by the influence of the Interplanetary Magnetic field.

3.4 Experiments with data for the 2003 strong geomagnetic storm

Since the geomagnetic storm in 2003 was unusually strong, it has become as a handbook example for the testing the analysis tools. However, in 2003 there were not so many data available. In particular, the ionosound data are not available with the present sampling, but only hour data.

We provide below the results for Wavelet Analysis for the following data

1.  In $2003$, for different (ground) geomagnetic ground observatories (SUA) we have 1 minute data for the  $H-$component.

2.  ACE satellite data for IMF are available at frequency $4$ min.

On the other hand, during this storm the Ionospheric data are only mean-hour which is insufficient for the present analysis.

First of all, we have the main trend of the magnetic fields provided by the $D_{st}$ data on Figure 11 above.

3.4.1 Experiments with geomagnetic data from SUA, on 28-29 Oct.,

2003

One may analyze the original source data $H-$component, or $DS$ obtained after subtracting the $D_{st}$ from the data, see formula (1). As we said above, since the $D_{st}$ data are given every hour this may create artefacts every whole hour. We provide the contour plot of the CWT for the $H-$ component of the geomagnetic field from SUA, 28-29 Oct., 2003, sampling 1 minute, on the following Figure 23.

On Figure 23, we may clearly identify short-period wave packages with periods below $3-4$ hours, as well as with periods $4-12$ hours, but also with periods about $24$ hours. The last are related with the main and recovery phase of the storm, i.e. with the Ring current. The wave packages with periods below $3$ hours may ber related to the fluctuations of the Ring current and the ionospheric instabilities generating such modes. Modes with periods below $100$ minutes are generated mainly at midday, and may be related to the solar terminatory and are eventually of soliton type, as was mentioned already above, see also [4], [33].

On Figure 24 we provide the CWT of the $DS$ data.

3.4.2 Experiments with spline smoothing of $D_{st}$

We have provided some interesting experiments which show the effect of subtracting of $D_{st}$ after smoothing the $D_{st}$ with splines, Figure 25.

On the bottom of Figure 25 we have the CWT of the $H-$component and on the top we have the CWT of the $DS=H-\widetilde{D_{st}}$ where $\widetilde{D_{st}}$ is the smoothed $D_{st}$ (which as mentioned is provided by the WDC of geomagnetism in Kyoto on hourly basis).

This shows that one has to be careful when subtracting the $D_{st}$ index (which is a step function) from the $H-$component since this creates non-smooth signal and the Fourier or Wavelet analyses generate artificial frequencies.

3.4.3 Experments with IMF data, on 28-29 Oct., 2003, $4$ min. data

We have provided the CWT for the $B_{z}$ component of the IMF on the following Figure 26. Again the heatmap of the CWT is on the top, while the contour map is on the bottom.

On the Figure 26 we see various families of wave packages, which may be separated into two types: those with periods less than $100$ minutes, and those with periods between $100-450$ minutes. Their explanation is related to the complex structure of the disturbance of the $B_{z}$ component during the strong geomagnetic storm.

4 CONCLUSIONS

The main objective of the present research is to apply Wavelet Analysis to Big data in the Solar-Terrestrial Physics, for the investigation of short period variations of the (ground) geomagnetic field/Ionospheric parameters in a region with mean geographic latitude. Thus, by applying Continuous Wavelet Transform to large amount of heterogeneous data (geomagnetic field, ionospheric parameters and IMF), we have identified modes (wave packages) with different periods, of the order of $20$ to few hundred minutes with a significant amplitude, which is enough to be registered by the equipment in the geomagnetic observatories.

As it is known, in the same range there exist the so-called geomagnetic pulsations, but they have a very low amplitude and exist for a short time only during the night hours for this geographic latitude. Unlike the geomagnetic pulsations, the short period variations of our interest, have significant amplitude, and are identified in the present research; they are discovered during the whole day and may be divided into modes with periods less than $3$ hours and modes which have a period greater than $3$ hours. 2. 2.

Our analysis of the variations of the geomagnetic field, the ionospheric plasma parameters, and the IMF, has shown persistent short term periodic events, as wave packages. The short period modes (wave packages) of the variations which we have identified have a clear explanation (e.g. from plasma physics) and are caused by macro-instabilities in different domains of the near Earth space environment. 3. 3.

We have identified the presence of modes with periods lower than $3$ hours, generated predominantly by the ionospheric plasma, but also similar modes which exist in the IMF. The ability of Wavelet Analysis to uncover Multiresolution structure of the data, gave us possibility to identify short-periodic wave packages in the geomagnetic field variations, in IMF and in the Ionospheric parameters. 4. 4.

The present research represents a contribution to the newly developing area of AstroGeoInformatics due to the large spectrum of the analyzed phenomena which belong to the Solar-Terrestrial Physics.

5 Thanks

The authors due thanks to the editors of this volume, Petr Skoda and Adam Fathalrahman, for the patience and their assistance. Thanks extend to Aleksandra Nina (Belgrade) for the discussions on the ionosphere. The two first-named authors thank the Project on Modern mathematical methods for Big Data, DH 02-13 with Bulgarian NSF, and also the Project SatWebMare with ESA (in the PECS framework). Last but not least, all owe thanks to the COST action BigSkyEarth with EU. OK thanks the Alexander von Humboldt Foundation.

The services of the World Data Center for Geomagnetism, at the Kyoto University, Japan, the INTERMAGNET network, the ACE Science Center at Caltech, and the GIRO center, University of Massachusettes at Lowell, are gratefully acknowledged.

6 APPENDIX on Wavelet Analysis and its applications to geomagnetic

data

In the present research we have decided for Continous Wavelet Transform (CWT). We provide the essentials of the CWT and some useful references for the applications of Wavelet Analysis.

6.1 Technical stuff

We will say that the integrable function $\psi$ is a wavelet function if it satisfies the following properties:

the admissibility condition holds

[TABLE] 2. 2.

the zero integral condition holds

[TABLE]

where $\widehat{\psi}\left(\omega\right)$ is the Fourier transform of $\psi.$ This condition is equivalent to

[TABLE]

We consider only real valued functions $\psi.$

Once the wavelet function $\psi$ is fixed, then for every integrable function $f$ (which is considered to represent the signal) which has a sufficient decay at $\infty,$ and for every two real numbers $a,b\in\mathbb{R}$ with $a>0,$ we may define the CWT $W_{\psi}f\left(a,b\right)$ by putting:

[TABLE]

The number $a$ is called scale, and $b$ is called translation (shift). Recall that the usual definition of the frequency $k$ is then given by putting

[TABLE]

Unlike the usual Fourier transform where the dimension of the variable of the signal $f\left(t\right)$ is transformed into the same dimensional frequency domain, here we see that the CWT $W_{\psi}f\left(a,b\right)$ depends on two variables. We are able to reconstruct the original signal $f$ from this representation, by means of the Calderon inversion formula ([15], [24]):

[TABLE]

As in the DWT it is always the question to find some reasonable approximation in the Calderon formula which takes into account only the larger values of $\left|W_{\psi}f\left(a,b\right)\right|$, which will result in an approximation of the double integral in the equality in formula (4). Thus, the question is, whether it is possible to use just a part of the integration domain? This may be achieved in different ways; one approach is to apply a a threshold on the absolute value of the CWT $\left|W_{\psi}f\left(a,b\right)\right|$, say $\varepsilon>0,$ and define the domain

[TABLE]

and then consider the approximation integral

[TABLE]

so that the remainder would satisfy

[TABLE]

Unlike the discrete wavelet transform (DWT) here we do not have a clearly defined Multi-resolution Analysis (MRA), and also there are no father wavelets (scaling functions). However, in general, one may use the wavelets in the Discrete Wavelet theory and apply them in CWT if they are smooth enough. The CWT is very convenient tool to detect and characterize singularities in functions, in order to distinguish between noise and signal (see [15], [24]). In particular, one may use CWT to study fractal behaviour of the signals.

In a wide area of applications, people use CWT with a wavelet function $\psi$ equal to the Mexican hat and the Morlet wavelet, although these two functions do not enjoy the usual scheme of Multiresolution Analysis as introduced in [24], [15]. In the present research, after numerous experimentations, we have decided for the symlet family of functions, which are a modified version of Daubechies wavelets family db, since they enjoy increased symmetry, [1]. We have applied the sym8 wavelet function, provided on Figure 27 below (see also the website http://wavelets.pybytes.com/wavelet/sym8/).

However it is important to remark that the experiments with many other wavelets $\psi,$ have shown that the singularities which we detect by the symlets may be analyzed with the same succes by applying the other wavelets; completely subjectively, we have found that sym8 gives in average one of the best possible visual picture. This fact shows that our observation is due to persistent physical events and may not be an artefact which is due to the particular wavelet which we choose.

6.2 CWT of some simple functions

An important control of the Wavelet Analysis method is to consider the CWT of the simple jump functions, as for example impulse trains which are sums of Dirac delta functions. It gives us idea about the behavior of the CWT of more complicated signals. A main reason to consider these impulse trains is the result of Belashov [4], who has very successfully modeled (better than the traditional IRI model) the ionospheric data by assuming that the disturbances of the Ionosphere are related (even in quiet days!) to wave packages having soliton character; we have already announced this resemblance of our analysis in [33].

The following Figure 28 shows the graph of a simple impulse train.

It has the CWT shown on Figure 29. The $y-$axis shows the lenght of the period (the scale $a$ in the CWT $W_{\psi}\left(a,b\right)$ and the $x-$axis shows the number $b$).

Remark 11

From the above Figure 29 we see that a package of periodic pulses has considerable CWT $\left|W_{\psi}\left(a,b\right)\right|$ for lower periods $0\leq a\leq 105,$ but it also shows an “integral effect” and shows a considerable CWT $\left|W_{\psi}\left(a,b\right)\right|$ for longer periods $a\geq 1030.$ The moral of this observation is that one has to be really careful when solving the Inverse problem, i.e. when making conclusion about the singularities of the original signal $f\left(t\right)$ judging by the large period behaviour of the CWT $W_{\psi}\left(a,b\right).$

Remark 12

Another interesting example is provided also in Wikipedia,

https://en.wikipedia.org/wiki/Continuous_wavelet_transform, where CWT is provided of a frequency breakdown signal by using the symlet as a wavelet function with 5 vanishing moments, see section 6.