Automatic Characterisation of Magnetic Indices with Artificial Intelligence

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Automatic Characterisation of Magnetic Indices

with Artificial Intelligence

Veronika Haberle

Space Engineering, master's level 2020

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

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with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence with Artificial Intelligence

written by

Veronika Haberle

^1,2

supervised by Aur´elie Marchaudon¹ Victoria Barabash²

In cooperation with

L’Institut de Recherche en Astrophysique et Plan´etologie in Toulouse, France

January - June 2020

This Master Thesis is submitted in fulfilment of the requirements for the Joint Master Program in Space Sciences and Technology

1 Universit´e de Toulouse, UPS-OMP, IRAP, Toulouse, France

2 Lule˚a University of Technology, Kiruna, Sweden

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Abstract

The complex interactions between the Sun and Earth are referred to as Space Weather. Key parameters include magnetic indices which quantitatively describe geomagnetic activity by determining a baseline that removes the background magnetic field and allows quantification of the remaining activity during geomagnetic events. However, most used indices have a low temporal resolution and rely on a sparse and frozen network of ground magnetic observatories. This thesis introduces a novel way of determining the baseline for future high temporal and spatial resolution magnetic indices.

Firstly, the main phenomena and effects of Space Weather are outlined, followed by a review of currently used magnetic indices and their derivation. The computation of a novel baseline introduced in this work relies on basic statistical methods which are applied on magnetic data from a dense and flexible network of ground observatories for the period 1991-2016. The focus is on the investigation of geomagnetic quiet periods for which average annual activity at each observatory is determined. A global latitudinal normalisation function with dependency on solar activity for quiet periods is found.

The analysis of the newly derived baseline shows that it provides the temporal, spatial and amplitudinal resolution needed to characterise geomagnetic disturbances adequately. The residual signal has the capability of being used as the basis for further quiet period studies. A first attempt of new indices based on the introduced derivation shows a good agreement with already existing high temporal and spatial resolution magnetic indices.

Future indices derived with this baseline lay a favourable fundament for the application of artificial intelligence methods.

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1 Introduction

Nowadays an increasing amount of our culture exists only in digital form (Hodson 2018). During the last few decades the digital revolution has brought humanity access to a wide range of global communication opportunities, navigation and online available knowledge, all of which are mainly powered by electricity.

We have started populating our near-Earth environment with technology to support these infrastructures and to explore our solar system in order to gain a better understanding of the world we live in. We have become dependent on these sensitive electrical systems and in order to protect this advancement, the need to understand not only its threats but also its risks is crucial. Space Weather is the domain which describes the influence of our Sun on our well-being and technology. Some socio-economical implications of Space Weather include power grid failures due to geomagnetically induced currents on ground, disruption of satellite communication and navigation due to the perturbation of the upper atmosphere and risks of radiation damages, both on space- and aircraft (Wolfert et al. (2017), Desmaris (2016)).

In the following, the physical phenomena relating to Space Weather are presented including the role of the Sun and its impact on Earth’s near-space environment. The complex range of interactions are summarized in order to give the reader an overview of the most important phenomena, as well as examples of associated effects.

As a start, the most important properties of plasma are presented, as they are the physical basis of Space Weather. Afterwards the Sun and its interaction with the Earth’s magnetic field and arising phenomena are described, followed by examples of Space Weather impacts on our technology. Finally, the role of magnetic indices and the objectives of this work are presented.

1.1 Basic Plasma Physics

We briefly present the basic concepts from plasma physics as they are useful for a thorough understanding of the following study. The interested reader may find a more detailed explanation in Baumjohann &

Treumann (2012).

Plasma is an ionised gas and is the most abundant state of matter in our Solar System. It makes up 99 % of the known matter. To be in an ideal state, a plasma has to meet three conditions:

1. Quasi-neutrality holds when the Debye length λD, depending upon the plasma’s temperature and density, is much smaller than the length dimension of the studied system.

2. The plasma parameter, which tells how many particles are in a sphere with radius λ_D, exceeds 1. Implying that, collective electrostatic interactions are dominating over binary collisions. Then particles can be treated as if they only interact with a smooth background field.

3. The plasma frequency, depending on the square root of the plasma density, has to be much smaller than the collision frequency with neutrals (for non fully-ionised plasma).

Single Particle Motion The most important forces acting on a plasma are the Lorentz force F_L = q (v × B) and the Coulomb force F_C= qB. The equation of motion for a charged particle derives as

mdv

dt = q (E + v × B) . (1)

Without an electric field, the particle exhibits a gyration motion

with gyrofrequency ωg = ^|q|B_m

which has opposite directions for electrons and protons. Taking the electric field into consideration results in a drift of the particle. The E × B drift is the motion of the particle perpendicular to both, the electric and magnetic fields. The direction of this drift does not depend upon the charge and thus no current is created. A magnetic field gradient results in a drift perpendicular to the magnetic field and its gradient, the direction depending on the charge of the particle. When the magnetic field is curved additionally, the particles experience a curvature drift which is proportional to the parallel particle energy and perpendicular to the magnetic field and its curvature, having opposite directions for ions and

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electrons. Together, the gradient and curvature drift are the source of one of the main magnetospheric currents, called the ring current (see chapter 1.3.1).

Collisions and Conductivity When a plasma population is observed, particle interactions may become important. The driving parameter for particle interaction within a plasma is the collision frequency and the mean free path length, as well as their plasma frequencies. In an unmagnetised plasma, when electrons move and their collision partners (ions and, if not fully ionised, neutrals) can be assumed to be at rest, they carry a current, generating an electric field. This is described by Ohm’s law, which resistivity depends upon the electron density and the collision frequency. In a magnetised plasma, the current density depends upon the electric field and the magnetic field

j = σ₀(E + v × B) . (2)

In a fully ionised plasma, the plasma conductivity σ₀ is a scalar. In most geophysical plasmas, the collision frequencies are extremely low and the conductivity can be approximated as infinite. However, in a partially ionised and magnetized plasma, such as the one existing in the upper part of the terrestrial atmosphere, called the ionosphere, collisions are high, introducing an anisotropic conductivity tensor of the form

j = σE =





σ_P −σ_H 0 σH σP 0

0 0 σ_||



E .

The Pedersen conductivity σP acts along E⊥. The Hall conductivity σH acts along the E⊥× B direction.

Both being perpendicular to the magnetic field. σ_kis the conductivity that arises on the parallel direction of the magnetic field. Pedersen and Hall conductivities depend upon the ratio of the collision frequency and the gyrofrequency. This becomes especially important in the ionosphere of Earth (c. f. chapter 1.3.2). As particles can move freely on magnetic field lines, σ_k is always higher than σ_P and σ_H. Parallel, Pedersen and Hall currents can be generated and coexist in such a system.

1.2 The Sun

In astrophysics, the Sun is classified as a G-class main sequence star with luminosity class V (Gray &

Corbally 2009). Its main constituent is Hydrogen, followed by around one fourth of Helium and traces of other, heavier elements. The immense power of the Sun is produced in its hot and high pressure core, burning Hydrogen to Helium. This is by far not the only interesting aspect of the Sun: as our closest star, it supplies our planet with life-sustaining energy through electromagnetic radiation. The solar spectrum can be described by a black-body spectrum of ∼5800 K with additional contributions in the extreme ultra-violet (EUV) and X-ray regime from its atmosphere. This contribution in the high energy part of the spectrum is the major source of heating and ionisation of the higher parts of Earth’s atmosphere. Furthermore, the Sun continuously emits charged particles from the highest layers of its atmosphere, the solar corona, into space, called the solar wind. It can be classified as slow solar wind having average velocities of around 400 km s⁻¹ and fast solar wind with velocities of up to 800 km s⁻¹. The expanding solar wind also carries the frozen-in magnetic field of the rotating Sun, moving outward in an Archimedian spiral, where it forms the Interplanetary Magnetic Field (IMF). The angle at which the IMF impacts the magnetic field of Earth depends upon the speed of the solar wind being on average 45° for the slow solar wind. The solar wind’s polarity (positive or negative) depends upon its source region on the Sun. The impacting polarity has an important role in Space Weather: a positive polarity, simply called the southward (−Bz) component, leads to magnetic reconnection between the IMF and the dayside part of Earth’s magnetic field. This reconnection process¹ leads to an increased injection of plasma and energy into the magnetosphere, providing energy for phenomena that trigger geomagnetic disturbances like storms on Earth (Bothmer & Zhukov 2007).

The Sun is not a static object, it rotates with a period of ∼27 days and exhibits transitions of maximum and minimum activity during the so-called 11-year solar cycle. The 11-year solar cycle is very well described by the F10.7 index, relating to the 10.7 cm solar radio flux (see Tapping 2013, for a review).

During this cycle the Sun’s magnetic field is changing, from an almost dipole configuration during

1 Reconnection occurs when opposite directed field lines merge to form new reconfigured field

lines, converting energy in large scales (e. g. in the dayside magnetopause and magnetotail). 2

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minimum to a highly structured morphology during maximum. These changes also affect the properties of the solar wind and influence the occurrence of sporadic high energetic events like coronal mass ejections (CMEs). CMEs are eruptive events that blast a high amount of mass (charged particles from the Sun’s atmosphere) with speeds of up to 2000 km s⁻¹ into interplanetary space. These extreme events, when reaching the Earth, may be highly geoeffective and consequently can produce major geomagnetic storms (Baumjohann & Treumann 2012).

1.3 The Earth

The geomagnetic field, or core field, is dominated by fields generated from a self-sustained dynamo in the Earth’s fluid outer core. This magnetic field acts as an obstacle for the incoming solar wind and, as a result, the Earth’s magnetosphere is created, appearing as a cavity in the solar wind. The Earth’s neutral atmosphere overlaps partially with a layer of ionised atoms that form the ionosphere.

Currents flow through this conducting layer and create a complex coupling between the magnetosphere and ionosphere. Hence, magnetic variations measured on the Earth’s surface are not solely due to the main field but have a significant contribution resulting from this coupling.

1.3.1 The Magnetosphere

When the supersonic solar wind hits the Earth’s magnetic field, the plasma is slowed down and a big portion of its kinetic energy is converted into thermal energy, forming a bow shock. As a result, the plasma behind the bow shock has higher temperatures, pressure and magnetic field strength with subsonic speeds. This region is called the magneto-sheath. The magnetopause is mainly characterised by the pressure balance between the external solar wind pressure and internal magnetic pressure and marks the boundary of the magnetosphere (Vasyliunas 1983). On the day-side, magnetic field lines are compressed. The subsolar point of the magnetopause has a typical distance of 10 R_E² . However, this distance is highly variable with varying solar wind properties. On the night-side, the field lines are elongated to form the magnetotail. There are theoretical and observational estimates that the magnetotail extends out up to 5000 R_E (Cowley 1991). There is a reversal in direction of the magnetic field lines between the northern and southern regions of the tail, resulting in two separate lobes. These northern and southern lobes are divided by a current sheet. The polar cusps are regions around the Earth’s magnetic poles where magnetic field lines merge with the IMF. On these funnels, solar wind particles are able to penetrate inside the magnetosphere, down to the upper atmosphere in dayside auroral zones.

The magnetosphere is populated with various distinct plasma populations (in terms of energy and density). For example, the toroidal-shaped radiation belts are located between 2 and 6 R_E, consisting of energetic particles that are trapped on magnetic field lines oscillating between the two hemispheres.

The plasmasphere, a torus-shaped volume located within the radiation belts, consists of cool and dense plasma. In general, some of the plasma comes from the injection by solar wind and some from the ionosphere. These plasma populations are under the influence of external forces. Ions and electrons may flow in opposite directions, creating currents. As figure 1 illustrates, the overall shape of the magnetosphere is accompanied by electrical currents. They play an important role as external sources for the geomagnetic field measured at ground or in space. In the following the most important current systems are briefly reviewed (Campbell 2003).

Magnetopause Currents

The magnetopause currents (also called Chapman-Ferraro currents) arise from the charge separation of the solar wind encountering the terrestrial magnetic field. Electrons are deflected westward and ions eastward which results in an eastward current.

Magnetotail Currents

The magnetotail is accompanied by tail currents which flow on its surface and a neutral sheet current that flows in the central plasma sheet of the tail.

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Figure 1: Three dimensional view of the magnetosphere illustrating currents, magnetic fields and plasma regions (adapted from Kivelson & Russell 1995)

Ring Current

The ring current encircles the Earth in the equatorial plane at a distance of about 2 - 9 R_E. Its sources are trapped high-energy particles which, apart from their bounce motion, exhibit a drift motion. The electrons drift eastward and the protons westward, resulting in a westward current.

During geomagnetic storms, the ring current can be significantly increased and recorded by ground magnetometers.

Field-Aligned Currents (FACs)

The field aligned currents (also referred to as Birkeland currents) are currents that flow on magnetic field lines connecting the magnetosphere with the high-latitude ionosphere. They arise directly from magnetospheric interactions with the solar wind and plasma flow through the magnetosphere. Two pairs of FACs are observed: Region 1 FACs are upwards directed on the duskside, and downward directed on the dawnside. Region 2 FACs are upwards on dawnside and downward on duskside. They connect via the Pedersen currents at ionospheric heights (see chapter 1.3.2). FACs also connect the divergence of the magnetopause, tail and ring currents to the ionosphere to get a closed circuit.

This short introduction of the magnetosphere already suggests the complex coupling between the solar wind, magnetosphere and ionosphere. In fact, in order to understand and predict phenomena, it is needed to consider all these subsystems as one unified system interacting constantly.

1.3.2 The Ionosphere

The neutral atmosphere of Earth can be classified by distinct temperature profiles. The troposphere, which reaches from the ground to approximately 12 km, is the layer we live in and temperatures fall with raising height. The stratosphere, located between 12-50 km is characterised by a raise in temperature due to the absorption of solar UV light by ozone. Between 50 and 85 km, The mesosphere exhibits fast falling temperatures with increasing height due to effective CO₂cooling. Above 85 km temperatures rise quickly in the thermosphere as solar UV and soft X-ray radiation can penetrate these heights easily, resulting in effective heating (de Pater & Lissauer 2015). This radiation does not only increase temperatures but is strong enough to invoke ionisation and thus is an important factor in the creation of the ionosphere. As

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the neutral atmosphere is characterised by its temperature profile, the ionosphere can be characterised by its electron density profile as illustrated in figure 2. The upper neutral atmosphere overlaps with the lower ionosphere, especially in the D- and E-region, implying a close interaction between the two regimes. Furthermore, the ionosphere exhibits a distinct day-night variation in plasma densities. In the following, the main ionisation and loss processes are briefly described. The interested reader may relate to a more detailed review in Baumjohann & Treumann (2012).

Solar UV Ionisation The ultraviolet part of the solar radiation is able to ionise atmospheric atoms and molecules, accounting for the main production process. The ion production rate has a pronounced maximum at a dedicated height. This maximum results from the decrease of the neutral atmosphere’s density and the increase of the solar radiation’s intensity with increasing altitude. This profile is called the Chapman profile (Chapman 1931).

Ionisation by Energetic Particles Energetic particles entering via magnetic field lines are able to ionise atmospheric atoms and molecules. This production process dominates during the night when there is no direct solar UV radiation. It is especially important at high latitudes due to dynamical processes at the magnetopause and in the tail. At the stopping height, where the particle is stopped and deposits the majority of its energy, the maximum in ionisation rate is found. This altitude is lower for electrons than for protons.

Recombination The process of a positive ion gaining an electron to form a neutral atom is called recombination. As this loss process depends upon the collision frequency of particles, it is dominant in the lower ionosphere, where collision frequencies are high due to the high density of neutral atoms and molecules.

Charge Exchange The process of exchanging electrons between interacting particles (e. g. neutral atoms with ions) is called charge exchange. This process changes the nature of the ions and is important in the high altitude F-region of the ionosphere, converting hydrogen ions into oxygen ions and vice-versa.

Figure 2: Typical electron density profiles for the mid- latitude ionosphere for day/night and solar cycle variations (from Pfaff 2012)

The combination of these processes form the characteristic density profile of the ionosphere as sketched for mid-latitudes in figure 2. At a height of about 60 to 90 km, the D-region is situated. Due to the relatively high density of neutral atoms from the atmosphere, high collision frequencies occur and the gas is only weakly ionised. During night- time, the production source of solar irradiation ceases and loss processes dominate, such that the D-region vanishes. At altitudes between 90 and 150 km, the E-layer forms due to the absorption of longer UV radiation and is partially ionised with a substantial contribution of neutral gas. Gener- ally, this layer is in balance between production, charge exchange and loss processes and thus, is in chemical equilibrium. Auroral particle precipitation is able to maintain the E-region during the night. Above the E-region, the F-region is classified into two subregions during daytime: The F1- region around 200 km forms during day-time by absorption of shorter UV radiation but disappears during the night as this production source ceases.

The F2-region at an altitude of about 300 km is

predominately populated by oxygen ions. It is created and maintained by the equilibrium of UV ionisation and diffusive transport along magnetic field lines. During the day, O⁺ is produced locally in the ionosphere and transported upward to the plasmasphere. During the night, the F-layer is sustained by

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the downward movement of the O⁺ from the plasmasphere to the ionosphere. As the F1-layer vanishes, it is common to refer to it simply as the F-region during night time. It is clear, that these layers have a major dependency on solar radiation which varies significantly over a solar cycle. Figure 2 also shows the electron density profiles for sunspot maximum, when solar activity is very high (relating to solar maximum) and for sunspot minimum when solar activity is very low (relating to solar minimum). The electron densities exhibit a variation of over one magnitude for maximum and minimum.

Figure 3: Conductivity profile of the E- and F-region in the ionosphere (from Yamazaki & Maute 2017)

Being populated with ionised plasma and neutral particles, the ionosphere supports electrical conductivity which depends upon electron- neutral and ion-neutral collisions, resulting in an anisotropic conductivity tensor (c. f. chapter 1.1).

An important implication of this conductivity property is the dynamo effect that arises within the E-region. The dynamo region is governed by the lighter electrons which electron cyclotron frequency being higher than the electron-neutral collision frequency and heavier ions which cyclotron frequency is lower than their collision frequency with neutrals. This leads to an E × B drift of the electrons, but the ions still move with the neutrals, creating a Hall current that is carried by the electron motion perpendicular to the electric and magnetic field. At an altitude where the cyclotron and collision frequency of ions becomes compara- ble, the ions start moving in the direction of the electric field and they carry a Pedersen current.

Based on these considerations, it is possible to model the three conductivities within the E-layer (Ya- mazaki & Maute 2017). Figure 3 illustrates the conductivity profile in the day-time ionosphere at mid- latitudes. The Hall conductivity (blue) peaks at lower altitudes with higher values than the Pedersen conductivity (in red). The parallel conductivity (in black) is always the highest.

These conductivities are the basis of ionospheric current systems. In the high-latitude ionosphere, the Pedersen currents close the FACs (c. f. chapter 1.3.1) in the auroral oval and are associated with auroral displays (see Baker 2019, for a review).

Figure 4: Global view of average S_q current vortices in both hemispheres (from Baumjohann & Treumann 2012) Solar Quiet Variations

In mid-latitudes, solar radiation heating creates tides in the neutral atmosphere. They affect the E- region dynamo and the resulting current system is called the solar quiet currents. This system forms two vortices, one in each hemisphere, touching at the geomagnetic equator, as illustrated in figure 4. Due to the dependency of the electron density and solar heating, the currents are concentrated on the dayside region (Baumjohann & Treumann 2012). Ground magnetometers³ are able to regis- ter the induced variability of this current system during geomagnetically quiet periods. The variations are a result of the disturbance of the solar quiet current system and are referred to as quiet fields Sq. (Campbell 1989). These variations exhibit a strong diurnal and seasonal dependency. In the following, the main temporal dependencies of

the Sq variations are outlined. For a detailed description, please relate to Yamazaki & Maute (2017).

3Basic principles of magnetometers are described in appendix A. 6

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Solar Cycle Dependency

The conductivity of the ionosphere is closely connected with the solar activity (the higher the activity, the easier it is to produce ions). This leads to variations in the S_q currents. They are typically two times larger during solar maximum than during solar minimum.

Seasonal Dependency

In general, the S_q amplitude is higher during the summer than during winter at mid-latitudes due to prolonged solar irradiation. Peak amplitudes can be found during both equinoxes close to the equator. An important factor that influences these seasonal changes can be attributed to upward propagating tides of the neutral atmosphere (Yamazaki & Kosch 2014).

Diurnal Dependency

Daily averaged variations are very similar considering one station, being higher during the day-time hours (when solar radiation is present) than during night. The S_q variations exhibit a significant location-wise and day-to-day dependency and can be significantly increased by other phenomena, e. g. for stations near the equator, the equatorial electrojet amplifies these signals significantly.

Electrojets At the equator, where the vortices of the Sq variations touch, an important current of the ionosphere is created: the equatorial electrojet. However, its strength is not solely due to the Sq

variations but depends also upon the geometry of the magnetic field. The nearly perpendicular incidence of solar radiation causes an enhancement of ionospheric conductivity, leading to the amplification of this jet current. Especially during equinoxes, the current is amplified significantly, most likely due to neutral winds.

Another important jet current system occurs at auroral latitudes, the so-called auroral electrojets, which are currents of Hall type. Due to particle precipitation in the auroral oval, ionisation is significant, leading to a much higher conductivity invoking these jets. They then can cause dramatic disturbances in the geomagnetic field at these latitudes, especially during magnetic storms (Baumjohann & Treumann 2012).

1.3.3 Earth’s Internal Magnetic Field

Figure 5: Magnetic elements in local geodetic coordinate system (from Olsen 2016)

Earth’s magnetic field, also called the core field or the main field, is the internal magnetic field generated by Earth’s con- vective liquid outer core. This creates around 93 % of the magnetic field strength at the Earth’s surface. The measured magnetic field averages at a strength in the order of 50µT, varying however between around 20 µT and 65 µT.

The strongest scalar magnetic fields are found at Earth’s polar regions and lowest fields at equatorial regions, sug- gesting a strong dependence on (magnetic) latitude. The core field exhibits large Earth-fixed spatial scales and varies on timescales of months to millennia (known as the secular variation), even millions of years when reversal processes are considered. For example, the magnetic pole in the northern hemisphere is currently slowly moving away from North America towards Siberia (Olsen & Mandea 2007).

Geomagnetic measurements are typically given in the local magnetic coordinate system as magnetic elements (components) X, Y and Z. It is an orthogonal right-handed coordinate system which axes point towards geographic North, geographic East and vertically down, respectively. These

components are illustrated in figure 5. Derived magnetic elements are the declination D which is the angle between geographic North and the magnetic North, the inclination I which is the angle between the local horizontal plane and the field vector, and the total intensity F which is the strength of the magnetic field. H depicts the horizontal intensity of the magnetic field vector and is sensitive to geomagnetic

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disturbances (Olsen 2016). In a first approach, the main field can be approximated by a dipolar field on ground. Its poles do not coincide with the rotation axis of the Earth but are tilted by approximately 11°. However, the dipolar representation remains a rough approximation. Indeed, the internal magnetic field contains shorter spatial wavelengths not only from its main field but also from its crustal field.

The measurements, acquired at the surface or at satellite altitudes, contain contributions from various internal and external sources spanning wide-ranges of spatial and temporal scales (Constable 2019).

Therefore, the geomagnetic field is represented in terms of a model in spherical harmonics, called the International Geomagnetic Reference Field (IGRF) model (Th´ebault et al. 2015). The model coefficients are determined every five years by the International Association of Geomagnetism and Aeronomy⁴ (IAGA) to account for temporal changes of secular variation. The spherical harmonic expansion shows that the contribution of the 1st order degree (the dipole part) accounts for 93 % of the total magnetic field. The remaining 7 % are attributed to external sources originating in the magnetosphere and ionosphere (Olsen 2016).

Reference Frame and Parameters for the Description of Magnetic Phenomena

Due to the tilt of the magnetic axis from the rotation axis, it is more convenient to describe magnetic phenomena in magnetic coordinates, rather than in geographic coordinates. A variety of magnetic coordinate systems exist, designed for different phenomena and regions. Within the scope of this work, we use the Eccentric Dipole Coordinate System (ED). Its Cartesian z-axis is aligned with the dipole axis, positively towards North, and is shifted about 500 km from the centre of Earth, thus eccentric dipole.

Its y-axis is perpendicular to the plane containing the dipole axis and the rotation axis of Earth. The x-axis completes the right-hand system. As a consequence of the slow drift of the magnetic poles, the coordinates of this system evolve over time. For an illustration please refer to the left lower panel of figure 16 which visualises the latitudinal drift for selected magnetic observatories.

Eccentric Dipole coordinates may be expressed in spherical coordinates (r, θ, φ) with r corresponding to the Earth’s radius, θ to the magnetic latitude and φ to the magnetic longitude. For the purpose of this work, we will use the spherical version of the ED (Laundal & Richmond 2017). Obviously, these also evolve over time. For a detailed description, the interested reader is referred to appendix B.

The angle µ, sometimes called the dipole tilt angle, is the angle between the geomagnetic dipole axis and the geocentric solar magnetospheric (GSM) z-axis (for more details, please, refer to appendix B).

This angle changes as a function of time of the day (due to the tilt of the dipole axis) and season (due to the inclination of Earth’s equatorial plane). In the northern hemisphere, µ has its maximum at the summer solstice and its minimum at the winter solstice, which is vice-versa for the southern hemisphere.

At the equinoxes, it is equal to zero (Cnossen et al. 2012). In the following tilt refers to the varying dipole tilt angle µ (not to be confused with the tilt between the dipole axis and the rotational axis of Earth with 11°, which is only varying on geological timescales).

Apart from the slow drift of the magnetic poles, varying on timescales of years, the ED is fixed with respect to Earth, having an absolute latitude and longitude. Many phenomena, however, vary with the zenith solar angle. It is suitable to organize data with respect to the position of the Sun. Therefore, we introduce the Magnetic Local Time (MLT). Its midnight magnetic meridian is defined as the meridian that is 180° magnetic longitude away from the subsolar point. An hour, where 1 h corresponds to 15°

magnetic longitude, is positive towards magnetic east. The MLT/magnetic latitude system rotates with respect to the Earth at the rate at which the subsolar point crosses magnetic meridians (Laundal &

Richmond 2017).

1.4 Geomagnetic Activity and Storm Enhancements

During quiet periods, the geomagnetic field shows smooth changes associated with the S_q variations.

However, the geomagnetic field can be highly disturbed by (sporadic) energetic events emitted from the Sun (like CMEs). These disturbances are called geomagnetic (sub-)storms and can cause changes in the measured magnetic field of up to 250 nT at mid-latitudes. A geomagnetic storm is a temporary (up to days) perturbation of the geomagnetic field and can typically be divided in three phases (see figure 6): the initial phase (which is due to compression effects of the solar event hitting Earth’s space environment), the

4http://www.iaga-aiga.org/ 8

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main phase (H component of the magnetic field decreases drastically) and the recovery phase (a gradual recovery of the field component, especially at equatorial latitudes below the ring current). Storms happen when the B_z part of the IMF is pointing South for a longer time, enhancing the coupling between the solar wind and the magnetosphere (see chapter 1.2 and Campbell (2003)). They are characterised by an enhancement of the ring current and the cause of the depression of the H component. A substorm, on the other hand, is a rather brief disruption of the magnetic field associated with an energy release coming from the magnetotail through the process of reconnection. Substorms, against the suggestion of the naming, are not necessarily connected to storms. They do appear during storm times, but can occur independently. Thus, substorms are more abundant than geomagnetic storms.

Associated with storms and substorms are rapid brightening and expansion of the entire auroral oval in both hemispheres and a significant amplification of ionospheric currents, especially for the auroral electrojets (Moldwin 2008).

Figure 6: Storm phases at different observatories close to the equator with varying H component.

1 gamma = 1 nT (adapted from Campbell 2003)

Figure 7: Different contributions and temporal ranges of phenomena altering the geomagnetic field (from Consta- ble & Constable 2004)

1.5 Variability of the Geomagnetic Field

The internal and external sources of the geomagnetic field described in the previous chapter are not static in space nor time. They span across a wide-range of temporal, spatial and amplitudinal spectra.

For example, the solar cycle induces a change of 10-20 nT with a period of 11 years. The mid-latitude Sq

variations feature diurnal and semi-diurnal changes with amplitudes of 20-50 nT. Substorms can alter the field easily by 100 nT in mid-latitudes and up to 1000 nT in polar regions, lasting hours or even days (Constable 2019). Figure 7 summarises the temporal variations and amplitudes associated with various phenomena. Through differing spatial and temporal extends, it is possible to distinguish them.

However, as can be seen, phenomena can also overlap and it is important to understand which signal contributes to what extend to the measurements.

1.6 Magnetic Data Acquisition

The sources of the geomagnetic field come in a high spatial and temporal variety. Sources which vary over geological timescales like pole reversals can be measured with palaeomagnetism techniques (Gilder

& Lhuillier 2019, and references therein). In the advent of the space age, it is not only possible to collect data from the ground, but to measure the magnetic field configuration at various locations at

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very high altitudes in space (Marchaudon 2019). There is no single instrument to measure the magnetic field (which is a vector field with magnitude and direction), but the combination of various instruments is needed to appropriately collect information. The most commonly used magnetometers in space and on ground are the Fluxgate Magnetometer, measuring field variations in one or more directions and the Scalar Magnetometer, measuring the magnitude of the magnetic field (see appendix A for further details). On ground, there are diverse facilities which are dedicated to measure magnetic fields. Magnetic observatories are location fixed facilities which provide the most accurate and long-term observations available. Magnetometer arrays are variable facilities which are build up to study magnetic phenomena occurring at distinct spatial scales. Marine and Airborne magnetic surveys are time and space limited surveys conducted on water and in airspace (Gilder & Lhuillier 2019).

1.6.1 Magnetic Observatories on the Ground

Magnetic observatories (also called stations) are characterised by the continues production of high-quality magnetic field data at stable locations. Nowadays, an increasing number of magnetic observatories are able to deliver quality magnetic field data with up to one second resolution. IAGA maintains a full list of observatories. The Real-time Magnetic Observatory Network, INTERMAGNET⁵ , ensures quality minute-resolution data of its member observatories and facilitates the distribution of it. Strict regulations are imposed, including one minute resolution vector data with a resolution of 0.1 nT. From the time of its founding in 1991 until now the amount of member observatories has not been static. Some magnetic observatories stopped producing data and dropped out completely, some became malfunctioning and dropped out for a limited amount of time only. Others started to provide data corresponding to the INTERMAGNET standards and dropped in (Gilder & Lhuillier (2019), Kerridge (2001)). Observatories can be identified by their three letter abbreviation (e. g. the French observatory Chambon la Foret has the abbreviation CLF). Appendix A gives details of the INTERMAGNET network from 1991 - 2016 and illustrates the location of observatories around the globe.

1.7 Geomagnetic Indices

The amount of geomagnetic data is vast and in order to characterise and quantify geomagnetic activity, magnetic indices have been introduced as early as 1939 (Bartels et al. 1939). Nowadays, they play a crucial role in the description of the Sun-Earth relationship and act as an important input for Space Weather forecast. In general, magnetic indices are concentrated information, delivered in a pertinent and reliable way, characterising geomagnetic activity. There is a series of officially acknowledged indices by IAGA, describing either the overall (planetary) geomagnetic activity or account for dedicated phenomena.

Each index, depending on their application, is taking into account a network of magnetic observatories at dedicated locations. Their derivation usually includes the determination of a quiet field baseline, which is then removed from the field measurements to receive the actual amount of disturbances in the remaining signal. Menvielle et al. (2011) gives a comprehensive overview of these indices, of which, some important ones are briefly reviewed in the following.

K-derived Indices

The K index describes geomagnetic activity on a scale from 0−9, where 0 indicates magnetic quietness and 9 very intense geomagnetic activity with a time resolution of three hours. The baseline removal process depends on subjective observation of the magnetometer curve. The only guideline to determine the baseline is ”The baseline is considered to be a smooth curve which is expected for magnetic quietness, taking into account the considered element and day” (Bartels et al. 1939). After applying the identified baseline, the resulting signal is classified upon its amplitude according to an observatory dependent scheme, ranging from 0 to 9.

Three IAGA endorsed geomagnetic activity indices are derived from these historical K indices: Kp, aa and am:

– The Kp index is derived from the K index in the form of conversion tables, taking into account 13 magnetic observatories. It is the result of the arithmetic mean value of the K indices of these

5https://intermagnet.github.io 10

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13 stations. Due to the historical context at the time of its creation, the Kp network is heavily weighted towards Europe and northern America. This index, as well as the K index, is limited to describe geomagnetic activity at mid-latitudinal regions in both hemispheres.

– The aa index was introduced to provide simple means of monitoring and determining global geomagnetic activity back to 1868. It is produced from the K indices of two antipodal magnetic observatories in England and in Australia. This index is a good measure to determine roughly magnetically quiet and magnetically disturbed days.

– The am index is based upon the aa index, with the difference of taking into account a wider longitudinal distribution of 24 sub-auroral observatories, which have an average magnetic latitude of around 50°.

Auroral Electrojet AE indices

AE indices (Nose et al. 2015a) were introduced to monitor the activity of the auroral electrojets. It is derived from the horizontal component H of 12 observatories in the northern hemispheric auroral zone. In order to find the quiet time baseline, the five quietest days of a month are averaged.

Disturbance storm time Dst index

The equatorial ring current is assumed to be the source of storm disturbance fields. Its activity is measured with the Dst index (Nose et al. 2015b), which derivation includes the H disturbance variations and the baseline value depending upon Sq variations. It is derived from four equatorial observatories.

During the time of introduction of these indices, especially the K-indices, no extended digital network was available, thus having a coarse temporal resolution and a limited number of stations. This makes it challenging to conduct detailed studies on the diverse magnetic phenomena. Especially for Space Weather forecast purposes, the need for a new generation of magnetic indices arises, asking for higher temporal and spatial resolution. Nowadays, high quality and high resolution data is available (for example from the INTERMAGNET network) and gives the possibility of satisfying this need. The first steps towards this goal have already been taken by Chambodut et al. (2015) who proposed the sub-auroral α15 index with a 15 minutes time resolution, taking into account 48 observatories over a large longitudinal range, covering the mid-latitudes.

1.8 Space Weather

Space Weather is defined by the US National Science Foundation as:

”Conditions on the Sun and in the solar wind, magnetosphere, ionosphere, and thermosphere that can influence the performance and reliability of space-borne and ground-based technological systems and can

endanger human life or health” (Wright et al. 1997)

It is important to understand the dangers it imposes and take protective measures accordingly. As humanity expands and fosters its technological reaches, Space Weather is able to impact a wide variety of areas covering spacecraft operations, radio communication, air traffic and power grid networks. Figure 8 visualises a broad overview of impacts. Koskinen et al. (2001) encompasses an exhaustive catalogue of Space Weather effects sorted into three categories: domain, phenomena and systems. In the following a few selected effects and impacted fields are outlined.

Scientific Payload on Spacecraft Missions During space-missions, satellites and their payload have to withstand the harsh environment of space while collecting scientific data, depending solely on electronics and sensible instruments. High energetic particles are able to penetrate the outer layers of satellites and harm these fragile components by corrupting data or, in the worst case, destroy essential electrical components.

Communication and Positioning Systems The ionosphere affects the propagation of radio frequency signals. Variable conditions in the electron density due to increased solar activity can lead to loss of signals, affecting crucial communication signals and positioning systems. This can lead, for example, to significant position errors in precision farming and decreased directional drilling accuracy.

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Power Grid Damage Magnetospheric and ionospheric currents are enhanced during geomagnetic disturbances and lead to geomagnetic variations accompanied by geoelectric fields, inducing geomagnetically induced currents (GICs) in power systems. GICs can lead to damages and malfunctioning of power grid systems causing major electricity outages on wide-area scales.

1.9 Objectives of this Work

Figure 8: Space Weather effects from https://www.

metoffice.gov.uk/

Space Weather poses an important risk to our technology dependent civilisation. And it is important to understand the physics behind the observed phenomena. One important measure of Space Weather are magnetic indices. The currently wide-used magnetic indices were introduced before the digital revolution and therefore use a slim network of observatories and data. Current models for the Earth’s magnetic activity and its reaction to disturbances still rely on these old- fashioned indices. Furthermore, as Space Weather is evolving, the need for an accurate and real- time forecast has risen which these indices are no longer able to provide. With the digital revolution, the amount of observatories providing high- quality data with minute resolution magnetic data has been steadily increasing and provides the basis for the next generation of indices with a high spatial and temporal resolution.

This work builds upon the α₁₅ indices introduced by Chambodut et al. (2015), which are based on a dense network of magnetic observato-

ries in both hemispheres and have a temporal resolution of 15 minutes. The objective of this Master Thesis is to rework their methodology by characterising and refining the baseline derivation. The first step comprises the computation of an improved baseline determination which can be calculated by a proposed automatic algorithm. In order to define a physically accurate index, the remaining signal after application of the newly found baseline is investigated with respect to the physics it is supposed to reflect. Such a novel baseline determination provides the foundation for future work including artificial intelligence methods.

2 Methodology

As we know from chapter 1, the magnetic activity on Earth is not homogeneous. An important parameter is the magnetic latitude, as magnetic activity is higher in the polar regions than at mid- and low-latitude regions (c. f. chapter 1.3.3). Therefore, to account for this variability, the data has to be corrected in latitude for each observatory. In a first step, this is done only for quiet periods, as defined by the aa index (c. f. chapter 1.7). After this normalisation, we seek a latitudinal correction to be applied over any period (independent of magnetic activity level). This normalisation may be seen as an improved baseline. With this, it is possible to define ”proto-indices”.

2.1 Data

Magnetometer data is obtained from the INTERMAGNET network between the years 1991 - 2016, covering the full solar cycle 23 and the maximum of solar cycle 24. In total, 146 stations delivered data for at least one year during this period. The vectorial data provided is described by the three magnetic components X, Y and Z (c. f. figure 5 and chapter 1.3.3), in minute resolution and theoretical precision

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Figure 9: Left: Amount of quiet days (as defined by the aa index) per year between 1991 and 2016. Right:

Amount of magnetic observatories in sub-auroral regions within the INTERMAGNET network per year between 1991 and 2016.

of 0.1 nT. For each magnetic observatory, we may calculate its magnetic location at a given point in time, taking advantage of the IGRF model. As a reference coordinate system, we have chosen to use Eccentric Dipole Coordinates as they are a reasonable balance between good approximation of the geomagnetic coordinate system (full spatial scales) and straightforward calculations for future operational purposes.

In the following, θ represents the ED latitude and φ the ED longitude. It is important to note here, that the implemented algorithm takes the geographic coordinates as an input and thus, is flexible to be used with any other magnetic coordinate system. As we would like to study the sub-auroral regions, we restrict our data to stations with |θ| ∈ [20, 60]. As the magnetic latitude evolves over time (c. f.

chapter 1.3.3), stations can enter or leave these boundaries and thus, the amount of considered stations is adapted dynamically. The right panel of figure 9 shows the amount of stations considered per year.

A detailed list of the used stations and their locations is presented in appendix A.

αH data A high-pass filter with cut-off frequency of 5.10 × 10⁻⁵Hz is applied to the minute resolution X(t) and Y (t) values. This strategy allows to filter out all non-relevant information (e. g. lower frequencies of diurnal and secular variations) and keep only the storm activity in the signal ∆X(t), ∆Y (t) after applying this baseline. From these filtered values, the horizontal intensity of the magnetic field vector is calculated by

αH(t) =p

∆X(t)²+ ∆Y (t)², (3)

which is, consequently, also in minute resolution with units of nT. This is in accordance with the experience gained with K-derived indices which show that magnetic activity at sub-auroral latitudes can be characterised using the variations measured in the horizontal component. The vertical component is more sensitive to internal induction effects and therefore is not considered (Bartels et al. (1939), Mayaud (1968), Chambodut et al. (2015)). The αH data serve as the basis for the upcoming methodology. Figure 10 illustrates the components X and Y (black) at the Chambon-la-Foret magnetic observatory (CLF), France, for several days at the end of 2001 and beginning of 2002 with superimposed baseline (red) determined by the filter. The remaining signals ∆X and ∆Y , as well as the resulting αH are plotted.

2.2 Latitudinal Normalisation for Quiet Periods

As a first step, we wish to investigate the behaviour of α_H solely for magnetically quiet periods between 1991 and 2016. The quiet day selection is based upon the aa index and its delivered products which allow to define periods of real magnetic quietness over 48 h centred on the considered UT day. The total amount of identified quiet days is 2572, with a minimum of 18 days in 2003 (maximum of solar cycle)

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and a maximum of 263 days during 2009 (minimum of solar cycle). The left panel of figure 9 summarises the amount of quiet days per year.

2.09 2.095 2.1 2.105

10⁴

-20 0 20

-620 -600 -580 -560

-20 0 20

Dec 26, 20010 Dec 27, 2001 Dec 28, 2001 Dec 29, 2001 Dec 30, 2001 Dec 31, 2001 Jan 01, 2002 Jan 02, 2002 Jan 03, 2002 20

40

Figure 10: The visualisation of the X and Y components as delivered by CLF (black) with superimposed baseline curve (red) and resulting ∆X, ∆Y and αH between the end of 2001 and beginning of 2002.

Bartels (1949) and Mayaud (1968) proposed that magnetic indices for sub-auroral latitudes should represent magnetic activity outside the equator and auroral zones. Therefore we will use a similar approach to represent activity at a magnetic latitude of ±50°. We will normalise the average annual quiet activity depending on the magnetic latitude θ by finding a normalisation function f_q. We start by finding a fit function yBF which describes the magnetic activity depending on the magnetic latitude per year. This implies that this annual fit function changes from year to year. We then pursue the goal of finding such a function that can be applied independently of the year (globally) by taking into account the solar activity.

2.2.1 Annual Representation of Geomagnetic Quiet Activity in Each Observatory

In order to characterise geomagnetic quiet activity (influenced mainly by the solar cycle) at one observatory in a specific year, we analyse the distribution of its α_H values for the quiet days of that year and look for a representative value for its average annual quiet activity. A natural way is to compute the probability distribution of those values and to estimate its probability density function for each observatory and year. Figure 11 visualises the probability distribution of the α_H on quiet days of the years 1997, 2000, 2003 and 2009, respectively, as blue stars for the French National Magnetic Observatory Chambon-la-Forˆet with magnetic latitude θ ≈ 46°. An empirically adequate model appears to be the log-normal distribution. Its characteristics can be extracted by fitting the distribution and computing its statistical properties. The log-normal fit is represented as a red line in figure 11 and the three properties mean, mode and standard deviation are indicated by a straight, a pointed-dashed and a dashed line, respectively.

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0 1 2 3 4 5 6 7 0

0.2 0.4 0.6

Probability

1997

0 1 2 3 4 5 6 7

0 0.2 0.4

0.6 2000

Mode StdDev Mean

0 1 2 3 4 5 6 7

0 0.2 0.4 0.6

Probability

2003

0 1 2 3 4 5 6 7

0 0.2 0.4

0.6 2009

Chambon la Foret

Figure 11: Probability density estimation of the α_H values in nT for the magnetic observatory Chambon la Foret on quiet days of the years 1997, 2000, 2003, 2009, respectively. The occurrence distributions are indicated with blue stars and the estimated log-normal distributions with a red line. The mean, mode and standard deviation are indicated as a straight, pointed-dashed and dashed line, respectively.

The differences between the distributions can be directly related to the solar activity. 2003 was a year with few quiet days and 2009 was a year with many. 1997 has more quiet days than 2000. Moreover, quiet days, as selected by aa, seem to have slightly higher values of activity during 2003 than during 2009. In general, the less quiet days are in a year, the higher the skewness (measure of asymmetry of the distribution) is and the higher its mean. These variations can also be observed for the mode and the standard deviation. The mean carries information about both, the mode and the standard deviation.

Therefore, it appears highly appropriate to use it as a representation for the average annual quiet activity α_{H year}(θ) for a certain year and station with magnetic latitude θ.

2.2.2 Annual Latitudinal Correction for Geomagnetic Quiet Activity

We now wish to describe the annual activity as a function of magnetic latitude for quiet geomagnetic periods only. Figure 12 presents examples of the latitudinal variations of the annual average αH year(θ) for the years 2003, 2005, 2007 and 2009. Each single marker corresponds to one observatory that delivered data during the considered year, i. e. the average annual quiet activity as a function of magnetic latitude⁶ θ, αH year(θ). Blue markers represent stations with |θ| ∈ [20, 60], while grey markers depict stations outside the sub-auroral range. A log-scale for the y-axis has been chosen to emphasise the variations in sub-auroral regions, as well as a dashed reference line at 1.6 nT is indicated. 2003 was the maximum of solar cycle 23, implying higher solar activity and thus, increased magnetic activity. 2009 was the minimum of solar cycle 23, implying lower solar activity and thus, less magnetic activity. As may be seen in figure 12, even considering solely quiet periods, the averaged magnetic quiet activity clearly depends upon the solar cycle as the curve is slightly flatter and lower during minimum (2009) than during maximum (2003). For stations outside of sub-auroral regions (grey stars), the effects of the electrojets are clearly noticeable. The equatorial electrojet intensifies the activity around θ = 0° and the auroral electrojet affects the activity around θ = 70°. At this point, it has to be noted that the latitudinal variation does not depend on the longitude sector when using an annual mean, as these variations will smooth out.

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-50 0 50 10⁰

10¹

2003

-50 0 50

10⁰ 10¹

2005

-50 0 50

10⁰ 10¹

2007

-50 0 50

10⁰ 10¹

2009

Figure 12: Magnetic latitudinal distribution of the average annual quiet activity α_H in nT for the years 2003, 2005, 2007 and 2009. Blue markers indicate stations with |θ| ∈ [20°, 60°]; grey markers indicate stations outside of the mid-latitudinal range. A reference line at 1.6 nT is indicated.

When only taking into account the sub-auroral dependency on latitude, one might suggest that it follows a quadratic trend. In fact, such curves have been fitted by quadratic functions in the past (Mayaud (1968), Chambodut et al. (2015)). However, when taking into account the activity between

|θ| ∈ [20, 80], the function of the inverse cosine seems a more appropriate fitting function. The inverse cosine is also related to the so-called L-shell which, in a dipole field, describes magnetic field lines that cross at the same distance from the Earth on the magnetic equator⁷ . The latitude λE, where a dipolar field line of given L-value intersects the Earth’s surface, is given by L = cos⁻²λ_E (McIlwain 1966). The inverse cosine has another important advantage: it is not sensitive to a reduced amount of data points, as the quadratic function (see appendix C) and, since it is a symmetric function, it doesn’t depend upon the sign of θ. The symmetry enables us to fit both hemispheres together in the following. We are now in the position of defining a fit function which, applied annually, provides coefficients c_1,y, c_2,y describing the activity depending on the latitude θ for each year ty,

y_BF(θ, t_y) = c_1,ycos θ^c^2,y. (4)

We call yBF the Best Fit function as it incorporates the minimum dispersion around 1 which we would get from such a fit for standardised values of α_H. We explain this procedure in the next subsection.

The best fit normalisation derivation needs a priori knowledge of the magnetic quiet activity of the full year. Additionally, its coefficients change annually. This does not appear convenient for the generation of operational real time values. One way to obtain a fit function independent of the year is to simply take the average of all the 26 annual coefficients. We obtain a global mean fit function in the form of

yM F(θ) = c1cos θ^c², (5)

with c_i = mean (c_i,y) , i = 1, 2; y = 1991 − 2016.

7Also called the apex of the field line. 16

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0 20 40 60 80 2

3 4 5

1997

0 20 40 60 80

2 3 4 5

2000

0 20 40 60 80

2 3 4

5 2003

0 20 40 60 80

2 3 4

5 2009

Figure 13: Illustration of the fit functions y_BF, y_{M F} and y_q as a red-dashed line, a red line and a green line, respectively, for the years 1997, 2000, 2003 and 2009. The blue stars correspond to the average annual quiet activity αH per magnetic observatory in nT, located at magnetic latitude θ in degree which reference value is taken on 30th June of the considered year. The vertical dashed lines mark the boundaries of the sub-auroral latitudes at 20° and 60°. Grey stars mark observatories outside this range.

For the years 1997, 2000, 2003 and 2009, figure 13 illustrates the two fits yBF and yM F with dashed- red and red lines, respectively (the third fit function yq in green will be described in the next section).

The blue stars correspond to the pair of magnetic latitude and average annual activity θ, α_{H year}(θ), the grey stars to pairs of stations outside the boundaries of the considered interval which is indicated by vertical grey dashed lines. It can be seen that yM F is the same for each year as its coefficients c1, c2

are static, whereas y_BF varies for each year to follow the annual dependency of average quiet magnetic activity. The question remains whether there is an improved global representation for the quiet activity.

2.2.3 Dependence on Solar Activity

From figure 11 and 12 we can suggest that the remaining activity over quiet periods depends upon solar activity which can be quantitatively described by the F10.7 index (see Tapping 2013, for a review). The mean quiet solar activity per year F_10.7,y is taken as the mean of F_10.7 of all quiet days in that year. In a first instance, we fit the inverse cosine with a linear dependency in F10.7,y annually

y_lin= (cd_1,y+dc_2,yF_10.7,y) cos θ^c^d^3,y⁺^c^d^4,y^F^10.7,y, (6) such that we receive 26 sets of coefficients cci,y, i = 1...4, y = 1991...2016. Figure 14 presents the correlation between the annual mean F_10.7,y on quiet days and the annual coefficients cd_1,y, dc_2,y, dc_3,y, cd_4,y. Coefficientsdc_1,y anddc_3,yshow a linear correlation, whereasdc_2,y andcd_4,yexhibit a logarithmic correlation.

Automatic Characterisation of Magnetic Indices with Artificial Intelligence