Developing a Phase Screen Model to analyse electron density measurements from the ICI-4 Rocket

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Developing a Phase Screen Model to analyse electron density

measurements from the ICI-4 Rocket

Saida Ramdani August 2021

Master’s Thesis

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Abstract

Global Navigation Satellite System (GNSS) signals can suffer from amplitude and phase fluctuations. These are referred to as scintillations and can strongly degrade the signal’s quality. If strong enough, scintillations can cause failure of the signal reception, as ground-based receivers can have troubles tracking the signal from satellites. S₄ (amplitude) andσ_φ (phase) describe the strength of scintillations and can be measured by ground-based receivers. In this study, a phase screen model is implemented in an effort to replicate the state of a small part of the ionosphere. This latter is a region of the Earth’s upper atmosphere subject to enhanced density irregularities. The phase screen model uses electron density in-situ measurements from multi-Needle Langmuir Probes mounted on the ICI-4 rocket. It was launched toward a region of enhanced electron densities the 19^th February 2015 at 22:06 UT. Before introducing the measured data, the model is first verified, and validated through multiple tests. The Super Dual Auroral Radar Network (superDARN) measured enhanced plasma velocity flow just 40 minutes before the launch. During the flight, the two cells convection pattern appear and plasma flow can be seen in the vicinity of the rocket’s path. However, PRN 31, the closest GPS satellite to the rocket’s trajectory, show very weak scintillations.

Meanwhile the implemented phase screen model calculates weak to intermediate scintillations. It was not possible to compare the scintillation data from the phase screen model, to the receiver’s measurements. However, the varying density structures measured by the rocket can allow intermediate scintillations, according to previous studies from in-situ measurements. Finally,S4 andσ_φ calculated by the model seem to be highly sensitive to the phase screen thickness L. A linear dependance is observed for amplitude scintillations, as increasing the phase screen thickness allow increased amplitude scintillations. However phase scintillations depict a totally chaotic pattern for L >30 km.

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Acknowledgements

I would like to express my sincere gratitude to my supervisors Lasse L. B. Clausen and Wojciech J. Miloch for their precious help and guidance throughout this thesis.

I am thankful to Professor Lasse Clausen for trusting me with this challenging and exciting project, and for his precious help and feedback during the meetings, and his support during stressful times. I would also like to thank Professor Wojcieck Miloch for his precious help during the meetings, his feedback on the different reports and for encouragements during stressful times.

A special thanks to Bjørn Lybekk who provided the receivers data from Bear Island, and, a thorough explanation and guidance on the way to use them. Also, the continuous help through E-mail with the Raw data extraction and use. Thank you to Yaqi Jin for his help on understanding some features from the receiver’s data.

I would like to thank everyone: Professors, Postdoc and PhD students from Plasma and Space Physics that took time to give me some tips on the project and for the various conversations on space and plasma Physics. Deepest and warmest thanks to my fellow Space Physics’s Master students for an amazing experience, and for all the fun we had in the office, and later, online. And, for their constant support throughout these stressful times far from home.

Finally, I am grateful for my family and friends, from home, for their constant support and encouragements throughout my stay in Norway, and my friends here in Norway for making these years memorable.

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List of Figures

2.1 Earth’s magnetosphere illustration . . . 9

2.2 Ionospheric density profile . . . 10

2.3 The Dungey cycle . . . 11

2.4 Electric field and convection cells in the polar cap . . . 13

2.5 The current systems in the ionosphere . . . 13

2.6 Polar cap patches propagation . . . 15

2.7 Trilateration . . . 17

2.8 GNSS frequency bands . . . 18

2.9 Fresnel zone . . . 22

3.1 Langmuir probe I-V curve . . . 28

3.2 SuperDARN radars layout . . . 30

4.1 Phase screen illustration . . . 34

4.2 Test 1 . . . 37

4.3 Angular spectrum geometric description . . . 39

4.4 Angle representation for two propagating waves . . . 40

4.5 Sinusoidal phase change using angular spectrum description . . . 41

4.6 Sinusoidal phase change propagation plot . . . 42

4.7 Test 2 . . . 42

4.8 Discrepancies zoom . . . 43

4.9 Propagating wave using Fresnel diffraction formula . . . 44

4.10 Illustration for different phase screen lengths . . . 44

4.11 Contour plot for phase screen lengthx∼2.4 . . . 45

4.12 Contour plot for phase screen lengthx∼2.4, from z = 3 km . . . 46

4.13 Sinusoidal phase change using Galileo E5a . . . 47

4.14 Sinusoidal phase change usingf = 300 MHz . . . 48

4.15 Sinusoidal phase change usingf = 10 GHz . . . 49

4.16 Sinusoidal phase change using f = 10 GHz and wave intensity at z = 800 km 49 5.1 Electron density measurements . . . 51

5.2 60 secondsS₄ and σ_φ calculated from the model . . . 54

5.3 1 secondS₄ and σ_φ calculated from the model . . . 55

5.4 IRI model ionospheric profile . . . 56

5.5 Receivers map and rocket trajectory . . . 57

5.6 Satellites SkyView from Bjørnøya . . . 57

5.7 Satellites SkyView from Hopen . . . 58

5.8 Satellites SkyView from Longyearbyen . . . 59

5.9 Satellites SkyView from Ny-Ålesund . . . 59

5.10 Satellites SkyView from Skibotn . . . 60

5.11 Scintillation representation for PRN 23, Skibotn . . . 61

5.12 Scintillation representation for PRN 29 . . . 62

5.13 Scintillations data for PRN 25 . . . 64

5.14 Scintillation representation for PRN 31 . . . 65

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5.15 Ionospheric convection cells att= 22 : 08 UT . . . 66

5.16 Convection patterns fort = 21 : 26 UT . . . 67

5.17 Interplanetary Magnetic Field . . . 68

5.18 Solar Wind bulk velocity . . . 68

5.19 Phase screen model varying thickness . . . 72

5.20 S₄ and σ_φ dependance to the phase screen thickness . . . 73

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CHAPTER 1 Introduction

1.1 Introduction

Nowadays, more and more users rely on Global Navigation Satellite System (GNSS) for different purposes, such as transportation, construction and off-road equipement tracking. In case of emergencies, for example during natural disasters, providing a good navigation and tracking system can reduce time response significantly. GNSS signals suffer from different alterations introduced by the various atmospheric layers they encouter during their propagation. It is without a doubt that a better understanding of the underlying phenomena and irregularities, that can affect satellite signals when travelling, is very important. More than 70 years of work and reseach have been put into the study of these perturbing phenomena. One important and well established fact is that the enhanced electron density regions, located in the ionosphere can have a direct negative effect on propagating signals. These irregularities have spatial scales that can range between few meters to hundreds of kilometers, and can diffract the signals travelling from satellites to ground-based receivers. Therefore, ionospheric irregulaties are of particular interest in order to allow a better understanding of the phenomena, mitigate the effect and consequences on GNSS signals, and one day, offer a forecasting space weather tool.

When a GNSS signal is sent from a satellite, located at around 20 000 km of altitude.

The electromagnetic radio wave propagates through the different layers, all the way to the ground where it is detected by receivers. Different information can be translated, depending on the satellites available in the vicinity of the receiver and the tracking purpose of the receiver. The main use of GNSS is for location, navigation and tracking, but it can also be used for mapping and time measurements. However, when an electromagnetic wave travels through the ionosphere, the main perturbing plasma medium, the different electron density fluctuations affect the signal by diffracting certain parts. This allows the different phase shifted parts of the signal to interfere when propagating to the ground. At the receiver’s level, the fluctuating signal’s amplitude and phase is known as scintillations and can drastically vary, causing partial loss of signal, to complete loss of lock. To better forecast these disturbance, some receivers allow to measure a phase and amplitude scintillation index that allow to assess the irregularities size and scale. However, the electron density enhancement that affect irregularities is highly dependant on the ionospheric dynamics, which in their turn are dependant on the solar winds given the Sun’s activity. Under certain conditions, reconnection occurs between the interplanetary magnetic field (IMF) and the Earth’s magnetic field lines. This starts a cycle, known as the Dungey cycle, where dayside and nightside reconnnection affect the ionospheric dynamics in the polar cap. During the dayside reconnection, the cusp is affected by the coupling between the IMF and the Earth’s magnetic field, and nightside reconnection allow energized particle precipitation. Finally, the plasma transport across the polar cap, allow for different structures of enhanced densities to appear which in their turn, as stated earlier, affect GNSS signals during their propagation to receivers. Therefore, by measuring the amplitude and phase scintillation

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on the ground, coupled with other observations, it is possible to monitor and keep track of the changes in the ionosphere.

Ground based measurements from receivers have a poor resolution of usually 50 Hz. This makes in depth investigation of the electron density fluctuations in the ionosphere difficult.

High-resolution in-situ measurements are then needed to make a full characterization of the irregularities. This was mentioned by Jøran Moen [Moen. et al., 2002] when large scale irregularity structures, observed by HF radar backscatter, was insufficient to explain and account for rapid growth of irregularities at decameter-scales. It was assumed that finer structures might be responsible for the observed perturbation. Therefore, higher resolution in-situ measurements were needed to assess these finer structures. The Investigation of Cusp Irregularities (ICI) is a project initiative from the University of Oslo, aimed to better understand the polar region’s smallest scales in the ionosphere using sounding rockets.

In this thesis, the ICI-4 rocket electron density measurements are used to study scintillations. The rocket was launched the 19^th February 2015, from Andøya Space Center, at 22:06 UT. The flight was 10 minutes long and the rocket reached an Altitude of 362 km. Ground based instruments diagnosed a patch of enhanced electron density directed toward the rocket’s trajectory. Therefore, in-situ measurements are intended to help assess and investigate these intercepted irregularities. For this specific study, the electron density data will be studied using a Phase Screen Model (PSM) to calculate amplitude and phase scintillation indices. These will then be compared with ground- based measurements from receivers, located close to the rocket’s path. The goal of this study is to offer a tool to measure scintillation’s strength from in-situ electron density measurements. Due to the constant varying electron density structures, the change in the ionospheric velocity flow and the location of the satellites, it is difficult to compare in-situ measurements to ground-based observations for small scale structures. Therefore, this phase screen model can be an additional tool to help understand the impact of small scale irregularity structures on propagating signals. Furthermore, this tool can be used in the future, onboard orbiting scientific satellites, carrying instruments for electron density measurements. Since these measurements are performed at a higher resolution, permanent monitoring accounts for high telemetry costs. Allowing the phase screen model, to directly calculate scintillation indices, onboard the satellite, will enable to send amplitude and phase scintillation indices at a much lower resolution, avoiding telemetry schemes. This process will allow an ongoing understanding of the irregularities, at much lower scales and with longer time period.

This study has two main parts. First, the phase screen sodel is implemented and validated using different tests. Then, the filtered electron density data measured by the rocket, are used in the model, and the phase and amplitude scintillations calculated are then compared with ground-based receivers, located at the vicinity of the rocket’s trajectory.

1.2 Outline

The rest of the thesis is organised as follows:

Chapter 2 presents the theoritical background, including latest understandings of scintillations, the different events involved in electron density fluctuations and the phase screen model theory.

Chapter 3 includes the different description of the instruments used in the study of the space weather conditions.

Chapter 4 introduces the numerical phase screen model implementation. This chapter also includes the different validation tests.

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1.2. Outline Chapter 5 presents the main results from the phase screen model and from ground based instruments. In addition, a discussion and analysis of the results is provided.

Chapter 6 presents a summary of the study and future works.

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CHAPTER 2 Theoretical background

An introduction to the theoritical background, as well as the basic mathematical tools for the Phase Screen Model implementation and the following data analysis, is presented in this section. First an introduction to plasma in space physics, the sun, the solar wind and the coupling process between the interplanetray magnetic field (IMF) and the earth’s magnetic field will be presented. Then the different consequences on the earth’s magnetosphere, and more particularly, in the ionosphere will be introduced. The mathematics behind the phase screen model will be developped and the consequences of a fluctuating electron density on a propagating signal will be described.

2.1 Plasma in space

Plasma physics is the study of the different dynamics that a plasma and its constituents can experience. A plasma is an ionised gas composed mostly of electrons and ions, but also neutrals. Due to the charged nature of its elements, a plasma can be studied through the motion of single particles. However, the collective behaviour and interaction of its constituents, giving rise to different current distributions, due to external electric and magnetic field, are what makes a plasma, a powerful medium to study and understand.

A plasma is described as a quasineutral gas that displays collective behaviors. Quasi- neutrality describes the overall charge neutrality of a plasma, while at certain small scales, depending on the plasma, the charged constituents can give rise to electric fields.

At a certain length, particles are able to shield out any electric field that can rise from local charge concentration, or through external potential, introduced into the system.

Therefore outside this length, the electron and ion densities are equal and the plasma is said to bequasi-neutral, and ’quasi’ is introduced to express that the plasma is neutral enough at certain scales, but not completely neutral to the point where one can exclude electromagnetic forces. The other important definition is the ’collective behavior’ that the plasma exhibits, which states that the collective effects of the particles is far more important than the Coulombian forces between its charged constituents [Chen, 2016].

Following these two important conditions, three main descriptions can be used to study plasmas. The fluid description of the plasma, where the charged particles are considered as one single fluid governed by fluid equations. Here the collective behaviors of the plasma, described previously, is important for this description, which can be referred to as Magneto-hydrodynamics (MHD). Following the same description, one can separate the plasma into two fluids from different charges, following its constituents. This second description is referred to as the two-fluid approximation of plasma with a mixture of charged gases [Pécseli, 2013]. Finally, certain phenomena cannot be described using the fluid description, therefore, the motions of each individual constituents, and their interactions, are considered using a kinetic description.

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2.1.1 The Debye length

As mentioned earlier, the quasi-neutrality dependance of a plasma comes from this ability that the plasma has to shield out rising electric fields. For example, if one inserts a positively charged ball inside a plasma, it will attract electrons and repel ions.

Therefore, a negative cloud is created around this ball, called a sheath. Consequently, at radial distances from this charged sphere, the potential falls very rapidly, to the point where it will equal to zero, at a certain distance [Chen, 2016]. This length at which the potential is screened out is called: The Debye length (λ_D) and is defined as:

λ_D =

s₀k_BT_x

nxq_x² (2.1)

Where^x refers to the different constituents, which in a plasma, are either the electrons e, or ions i. Thus,q_x refers to the charge, n_x the density, T_x is the temperature, ₀ is

the permittivity of free space andk_B the Boltzmann constant.

Hence, when a measuring device is introduced in a plasma, such as the Langmuir probes used in the ICI-rocket missions. The size and shape of the probe must be adapted to the plasma studied, since the length changes according to important plasma parameters, as described in equation 2.1.

2.1.2 The plasma frequency

The plasma is characterized by a natural and intrinsec oscillation called the plasma frequency. It is the natural oscillation of the electrons relative to the ions, this quantity is defined as:

ωpx=

sn_x(q_x)²

₀m_x (2.2)

Whereω_pxis the plasma frequency and can be described by the electron plasma frequency ωpe or the ion plasma frequency ωpi. In addition, mx refers to the mass.

2.1.3 Single particle motion and the drift velocity

When the particle density is very dilute, the electric and magnetic field that rises from the charged particles can be neglected compared to the externally imposed fields. Therefore, the plasma is descriped by the motion of its individual constituents. Depending on the relative direction of the electric field, in comparison to the magnetic field, the determination of the different trajectories of the charged particles, will give rise to different concepts, important to the study and understanding of the plasma behaviors in space. This is due to the electromagnetic force or Lorentz force. This latter affects charged particles moving with a certain velocity in an electric E and magnetic B field.

The most fundamental motion occurs when the electric field is parallel to the magnetic field, E k B, or when there is no electric field E = 0. In this case, the particle will gyrate in a circular orbit around the magnetic field with a radius known as the Larmor radius,rL, and a frequency, known as the cyclotron frequency orgyrofrequency (Ωc):

r_L = mxU⊥

q_xB (2.3)

Ωc= q_xB

m_x (2.4)

where m_x refers to the mass of the particle, U⊥ is the component of the velocity perpendicular to the magnetic field,q_x the charge and B the magnetic field’s magnitude.

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2.2. The Sun and the Solar winds Due to its dependence to the charge, the direction of gyration will differ whether the particle is an electron or an ion. When the velocity component of the particle is parallel to the magnetic field, the Lorentz force has no influence on the particle’s motion. Therefore, the particle will continue its initial motion following the electric field orientation. This is due to the fact that the Lorentz force is perpendicular to the magnetic field. This can be noticed in the Lorentz force equationF=q(E+u×B).

The second important quantity takes place when the electric and magnetic fields are perpendicular,E⊥B. The particle motion becomes a combination of a gyrating motion, due to B, and a drifting motion with a distinguished velocity called the E-cross-B drift, or mathematically E×B-drift.

u_D = E×B

B² (2.5)

The particle will drift in a direction both perpendicual to the electric, and magnetic field. This phenomena occurs often in the ionosphere and plays an important role in the plasma convection pattern. Other cases can occur, where the electric field can be replaced with an external force, such as gravity, or a non-uniform B field. However, these cases won’t be further studied here and a more complete description can be found in Pécseli, 2013.

2.1.4 MHD and the Frozen-in theorem

Another description used to study plasma, specifically the Earth’s upper ionosphere, the magnetosphere or the solar wind, is the single fluid description. The plasma is represented as one medium, interacting with magnetic and electric fields. This interpretation of the plasma is referred to as Magneto-hydrodynamic (MHD), and requires a set of equations from both fluid and electrodynamics. In this limit, very large time scales, are considered, thus, slow phenomena and low frequencies, where any charge separation is instantly short-circuited. The plasma is assumed to be a very good conductor. From the set of equations that can describe this model, the generalised Ohm’s law is used to describe the current density of the fluid:

J=σ(E+u×B) (2.6)

WhereE and B are the electric and magnetic fields,u is the plasma velocity and σ its conductivity. A limiting case of MHD, calledideal MHD is presented when the plasma is assumed to be an ideal conductor. The conductive term in equation 2.6 becomes σ→ ∞ therefore, to keep the current finite, equation 2.6 is reduced to:

E=−u×B (2.7)

Which implies that the electric field is induced by the motion of plasma with velocity u across the magnetic field. This concept in MHD is known as the Alfvén’s frozen-in theorem, where the magnetic field is frozen into the fluid and moves along with it. A more detailed derivation of the principle and its implications can be found in different articles and books Alfvén, 1950, Pécseli, 2013, Alfvén, 1942.

2.2 The Sun and the Solar winds

The sun is responsible for most of the space weather we encounter today. With a radius of approximately 109 times the earth’s radius, and a mass of 1,99·10³⁰ kg, this middle aged star contains a full plasma physics laboratory, inducing different processes from its core to the outermost layer,the corona. The solar magnetic field has an ≈11 years

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period cycle, where it goes from very low to very high magnetic activity. This cycle period is characterized by the number of sunspots, which are dark spots that are visible on the solar disk. A higher number of sunspots, reflects a disorganised and chaotic magnetic field with higher chances of increased outflow of particles [Russell et al., 2016, p.100].

During solar maxima the increased outflow of high energetic particles can reach speeds up to 800 km/s. These particles constitute the solar winds which propagates in the interplanteary medium. Fast solar winds are associated with open field lines that originates from coronal holes [Müller et al., 2012]. In the other hand, slow solar winds emanate from the coronal holes boundaries and have a velocity of 300−500 km/s. The solar winds are composed of protons (H) and alpha particles (He²⁺), but also other heavy ion species [Wimmer-Schweingruber, 2002, Müller et al., 2012]. Solar winds are a highly conductive plasma that can be described by theIdeal MHD model (Section 2.1.4).

Due to the Frozen-in theorem, the solar winds are able to drag the Sun’s magnetic field toward the Earth. This is referred to as the interplanetary magnetic field (IMF).

Using different spacecraft missions such as the Adavanced Composition Explorer (ACE), orWind spacecraft, situated at the L1 Lagrangian point. In-situ IMF measurements including solar wind speed, densities and the magnetic field total magnitude and vector quantities, are cross compared between different spacecrafts, and hourly averaged data are archived [NASA, 2021c]. The IMF vector components can be expressed in different coordinate systems, which one need to be aware of when using OMNI data. How- ever, the IMFB_z which depicts the northward direction is the most relevant in this study.

Other important solar coronal activity can change the composition and strength of the solar winds. Solar flares are rapid, impulsive brightening events that occur in the vicinity of sunspots. Massive amounts of stored energies can be released through flares. Coronal Mass Ejection (CME’s) are very strong events that allow the release of large amounts of matter as plasma. These eruptions are usually associated with geomagnetic storms on Earth [Veronig et al., 2018, Zurbuchen and Richardson, 2005]. A deeper description of the phenomena that can occur on the Sun can be found in different books and studies, for example Russell et al., 2016.

2.3 The Magnetosphere, the ionosphere and the Dungey cycle

The Earth magnetic field can be approximated to a dipole, with magnetic field lines directed from geographic south to north. The source of the magnetic field is located in the center core, and its axis is tilted compared to the Earth’s rotation axis. The magnetic field at an altitude of 100 km is approximately≈6.5·10⁴ nT [Pfaff Jr, 2012], and becomes weaker at increasing distances from the Earth’s center. The magnetosphere is this region of space surrounding the Earth which contains the magnetic field. At the boundary, the Earth’s magnetic field becomes so weak that an equilibrium region is created between its outward pressure and the solar wind’s pressure. This boundary is referred to as the magnetopause, and is located at typically 8−12 earth radius (R_E) in the sunward side.

Figure 2.1 illustrate the magnetosphere and the different main regions. First, when the solar wind encounters the outer boundary, it deflects around the magnetosphere, and a bow shock is formed on the dayside. This is due to the fact that the solar wind is supersonic [Pfaff Jr, 2012]. The magnetosheath, located below the bow shock, is a region of highly magnetic turbulence, where different complex currents can be found [Pécseli, 2013 and Russell et al., 2016]. Then, we can find the magnetopause, which is this equilibrium boundary between the earth’s magnetic field and the solar wind pressure. Due to the solar wind pressures exerted on the dayside magnetosphere, the

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2.3. The Magnetosphere, the ionosphere and the Dungey cycle

Figure 2.1: Illustration of the magnetosphere and its surroundings, with a representation of the dayside from left to right. The IMF is represented as white lines, and the Earth’s magnetic field lines are represented in red. Modified picture from Nasa media webpage NASA, 2021b.

Earth’s magnetic field is compressed, but streches out on the opposite direction. The nightside region is referred to as the magnetotail. Finally, one can notice two different kind of field lines, The ones that forms most of the magnetic field lines are ’closed’, where both ends of the line are attached to Earth. The closed magnetic field lines come out from the southward region and go back to northward Earth’s region, close to the polar cap. The other type of field lines are ’open’ field lines. They are situated at each poles, and characterised by only having one end of the field line attached to Earth. the other end extends into the interplanetary medium. The boundary between the open and closed field lines is referred to as the ’polar cap boundary’ or simply the ’open-closed field line boundary’ (OCB).

2.3.1 The ionosphere

The ionosphere is the most important Earth’s layer in this study, due to its content and its influence on radio communication. It stretches from 50 to 1000 km above Earth’s surface, and is the ionised region of Earth’s upper atmosphere. It contains electrons and charged ions and molecules. These different constituents are distributed within the ionosphere altitude. The electron densities strongly vary creating different density layers.

Two main processes allow for the creation of plasma. First, due to strong radiation coming from the sun, the extreme ultraviolets (EUV) are able of extracting electrons from neutral atoms. This process is called photoionisation. The second process is due to energetic particle precipitation, which will be introduced in later sections.

Photoionisation is responsible for the creation of the Earth’s main ionosphere, specifically at mid- and low-latitudes [Pfaff Jr, 2012].

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Photoionisation depends on the availability of the neutral constituents of the ionosphere [Chapman, 1930]. It also depends on the energy strength of the incoming solar radiation.

Therefore, the density profile of the ionosphere varies with increasing altitudes. Moreover, the density profile undergoes seasonal changes, but also varies from daytime to night time.

Figure 2.2: Ionosppheric electron density profile for both daytime and night time, with labeled layers. Implemented using the IRI-2016 model for Tromsø, the 17^th March 2015 for 12 : 00 and 23 : 00 UT Center, 2021

Figure 2.2 shows the electron density profile, throughout the altitudes. One can already notice the different layered regions. First, the lower layer of the ionosphere is named theD-layer and is located below 90 km. This layer depicts the lowest densities in the ionosphere. However this layer will not be discussed in this thesis and more details can be found in different litteratures, for example Pfaff Jr, 2012. Next, we can find theE-layer located at altitudes between 90 and 170 km, with the dominant ion species being N O⁺ and O₂⁺. At higher altitudes, between 170 and 500 km, we can find the F-layer. During the daytime, this layer is split into two sub-layers namely the F1-and F2-layers, characterised by enhanced peaks. These two layers can’t be distinguished in figure2.2.The F1 and F2 layers recombine and only form one layer during night time [Prölss, 2004, p.162]. The F-layer is characterised by the highest electron content in the ionosphere withO⁺ as dominating ion.

The ionisation rate describes the rate at which ionisation occur due to photoionisation.

This rate is balanced by the reverse process where the ionisation products, ions and electrons, recombine. This second process can be described by the ion or electron loss rate, where these constituents disappear by the mean of recombination. This depends on their local concentration [Friedrich et al., 2004]. Finally, From figure 2.2, we can notice that during night time, the overall density profile is reduced. This is due to the reduced solar radiation, which allows the recombination rate process to increase [Russell

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2.3. The Magnetosphere, the ionosphere and the Dungey cycle et al., 2016, Pfaff Jr, 2012 and Prölss, 2004].

2.3.2 The Dungey cycle

At the magnetopause,magnetic reconnection can take place on the dayside. The Earth’s magnetic field interacts with the IMF when itsBz component is in the opposite direction as the Earth’s field. After reconnection, the newly open field line gets pushed to the nightside, and onto the magnetotail. The two open field lines, from the Earth’s North and South region, might undergo another reconnetion if conditions allow it. However, this time the process go from two open field lines to a closed field line.

Figure 2.3: Illustration of the Dungey cycle and the path that takes the field lines when it undergoes a dayside, and a night side, reconnection. On the zoomed image (bottom-right), the footprint of the plasma, as it drags the magnetic field line, following

the frozen-in theorem. Figure from Russell et al., 2016.

Figure 2.3 shows the reconnection process through numbered steps from the dayside to the nightside. The travel of the open field line in the magnetosphere is included, as well as the plasma footprints on the polar cap. When reconnection occurs, a newly open field line appears (step 1 and 2 in figure 2.3). As the plasma is pulled antisunward across the polar cap, through processes that will be shortly described (steps 3, 4 and 5).

The newly open field line is pushed toward the nightside through the magnetosphere.

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If reconnection occurs on the nightside, a newly closed field line forms. The other reconnected side detaches and becomes interplanetary magnetic field as it is pushed away from magnetosphere (steps 6 and 7). Finally, the plasma carrying the magnetic field line is pushed back toward the dayside, through the lower latitudes of the polar cap (steps 8 and 9). This process then repeats itself, if the IMFBz remains negative.

This nightside reconnection allows a flow of particles to stream toward theauroral oval which is a region on the nightside of the polar cap. Due to these energetic particles, optical emissions can occur at lower regions of the ionosphere. These optical emissions are better known asauroras and will be adressed in the next section.

This magnetospheric convection is known as the Dungey cycle, named after James Dungey, who was the first to propose the theory in 1961 [Dungey, 1961]. In this study we are interested in the magnetic reconnection when the IMF B_z component points southward. However, northward reconnection when Bz >0 has been observed at tailward regions of open field lines [Crooke, 1992].

2.3.3 Electric fields, current systems and auroras

The MLT or Magnetic Local Time is customarily used to describe local phenomena closer to Earth. This system follows the Geocentric Solar Magnetospheric (GSM) coordinate system. The x-axis is based on the radially aligned Earth center to the sun and the y-axis points positively toward dusk. The description of the different electric fields and current systems in the ionosphere will follow the MLT coordinate system. There are different other coordinate systems that one need to be aware of when conducting data analysis from spacecraft in-situ measurements. A full description of the different systems can be found in Laundal and Richmond, 2017.

In the previous section, the Dungey cycle was introduced and figure 2.3 allowed a representation of the process in the magnetosphere, but also in the ionosphere. The magnetic field line path was followed and introduced as plasma footprints on the polar cap. These tracks are the result of the Frozen-in theorem described in Section 2.1.4.

Through the ideal MHD description, the plasma flows through the polar cap and creates an electric field given by equation 2.7. Where the velocity isu_sw the solar wind speed, and B is the Earth’s magnetic field directed perpendicular to the polar cap.

Therefore, an electric field perpendicular to B is imposed and is directed from dawn to dusk. In the same way, the plasma all the way to the ionosphere, follows this electric field through the E×B-drift in the antisunward direction. Therefore, the footprint seen in figure 2.3, for the first 6 steps goes from noon-to-midnight. When nightside reconnection occurs, the return flow from the tail into the night side region of the polar cap part of the auroral oval impose an electric field in the opposite direction. There- fore, at lower latitudes, on the flanks of the polar cap, the plasma flows sunward direction.

The different direction of the electric field in the polar cap and at lower latitudes creates a pair of convection cells following the E×B-drifts. These convection patterns are everywhere perpendicular to the electric field, and the convection velocity is proportional to the local electric field strength. Figure 2.4 represents the different electric fields, and the dual cell patterns. The electric field is directed from dawn-to-dusk within the polar cap, and points in different direction in the auroral oval. Whether it is the dawn side (toward the equator) or the dusk side (toward the pole), The convection patterns, resulting from the E×B configuration, rotate in different directions as well [Prölss, 2004, Pfaff Jr, 2012].

Figure 2.5 shows the different current systems that govern in the ionosphere. The current system that connects the ionosphere to the magnetopause, and vice-versa, is also illustrated in the figure. First, the field aligned currents (FAC), also known as Birekeland currents, are a system of currents that flow along the magnetic field lines.

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2.3. The Magnetosphere, the ionosphere and the Dungey cycle

Figure 2.4: Electric field and plasma convection pattern in the polar cap.

Figure 2.5: Illustration of the different current systems that govern in the ionosphere.

In figure 2.5, one can notice the FAC paths. The currents flowing from magnetopause toward the ionosphere (in blue) and the oppositely directed flow (in red) are presented.

These can be enhanced when the reconnection rate increases. These currents, created by electric field parallel to the magnetic field lines, are due to the shear in the magnetic field that forces the transport of the whole flux tube (the magnetic field line). This last is attached on one side to the electrically conducting Earth, and in the other to the magnetosphere. A more complete study can be found in Russell et al., 2016.

For this thesis , it is important to know that the field aligned currents are carried from Dawn to Dusk, across the polar cap. These are known as the Pedersen currents (in green in the figure) and follow the same direction as the electric field component (⊥B). The FAC form two distinguish regions namelyRegion 1 (R1) and Region 2 (R2) that are driven by two different phenomena. For R1 the current systems vary according to solar wind coupling, andR2is driven by magnetospheric stress [Russell et al., 2016]. Another set of currents, depicted in orange in the figure, are referred to as the Hall currents.

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These follow the opposite direction of the convection cells −E×B-drift, and are also referred to as auroral electrojets. During disturbed magnetic periods, all these current systems are enhanced. Strong field aligned currents cause enhanced Hall currents which in their turn, cause the expansion of the elctrojets. We can distinguish between two type of electrojets in the polar region: The Westward electrojet and Eatward electrojet. This expansion results from enhanced particle precipitation and enhanced ionospheric electric field [Russell et al., 2016, Pfaff Jr, 2012].

When these energized particles finally reaches the ionosphere, they collide with the plama constituents. The collision rate differs with the plasma density, and the outcome of the collisions differ also depending on the constituents availability. These excited particles are described in Section 2.3.1 as the second process by which the plasma density fluctuates, which is particle precipitation.

This process is responsible for the optical emissions observed at higher latitudes, close to polar regions. The physical explanation is rather straightforward. Since, the ionosphere also contains molecules and atoms, such as Oxygen (O) and Nitrogen (N₂), the precipitating particles, carrying different energies will excite the atomic and molecular electrons, to higher enery states. The de-excitation, or more commonly known as electronic shell relax allows the excited electron to return to the ’ground’ state. This will take a certain amount of time depending on the energy state level at which it got excited to. Therefore, allowing the release of a photon in a certain wavelength.

Depending on the atoms availability in the ionospheric layer, and the altitude at which this process occur, the photon released will carry a signature color that we can recognize as aurora. Oxygen atoms can emit in the yellow-green (λ= 557.7 nm) which are the most common auroras. The other emission from Oxygen is red and has (λ= 630.0 nm).

Nitrogen atoms emit in dark red (λ= 650 - 680 nm), and are usually located at higher altitude [Russell et al., 2016]. Other wavelength can also be emitted, depending on the altitudes, the ions and atoms availability, and also the energy at which the precipitating particle interact with the ionospheric constituents [Rönnmark, 1991].

2.4 Polar cap patches and ionospheric irregularities

As introduced previously, magnetic reconnection has a great influence on the different ionisation processes that can occur in the ionosphere. Due to particle precipitation and solar radiation, the ion/electron density content varies greatly. In its turn, this variation influences GNSS signals and radio communication, since electromagnetic (EM) waves passing through the ionosphere undergo different changes. This will be introduced in later sections. However, within the ionosphere itself, other phenomena can influence EM waves. These events are large scale patches disturbance, also referred to as blobs under certain conditions.

Polar cap patches are regions of high density plasma in the F-layer. They can reach densities that are 2 to 10 times larger than the background density and have scales of 100 to 1000 km [Livingston et al., 1984 and Moen, J. et al., 2013 and references therein]. They are generated in the dayside and propagate across the polar cap toward the nightside with drift speeds of approximately 500−1000 km/s [Livingston et al., 1984].

If they occur outside of the polar cap, in the auroral oval they are referred to as blobs [Crowley et al., 2000]. Polar cap patches have been studied and observed for decades.

However Livingston et al., 1984 was the first to point out that there is no evidence that particle precipitation are a source for the polar cap patches, nor polar rain fluxes. These latters are precipitating plasma from solar wind electrons that enters the magnetosphere through the cusp [Reidy et al., 2018 and reference therein]. Livingston et al., 1984 studied polar cap patches during moderatly geomagnetic conditions. Therefore, these enhanced density pockets can form independently of strong geomagnetic disturbance.

In the recent years, studies have been made to understand the origin and formation

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2.4. Polar cap patches and ionospheric irregularities of polar cap patches. However, till this day, polar cap patches mechanism formation are still not well-understood, and multiple processes have been proposed throughout the years. Despite not having a clear image of the mechanism behind the polar cap patches, most research agree on the strong correlation bwteen the IMF B_z directed southward and the increased amount of polar cap patches. In a more recent study, Kagawa et al., 2021 introduces a new D parameter which shows strong dependance with polar cap patches. The D parameter refers to the distance between the Altitude Adjusted Corrected GeoMagnetic (AACGM) magnetic pole and the ’solar terminator’.

This latter is the determining line between the dayside and night side across the polar cap. Another indeniable statement is the strong correlation between, polar cap patches and blobs, with scintillations observed at higher latitudes. [Moen et al., 2012; Moen, J.

et al., 2013, Jin, Y. et al., 2017, Thayyil et al., 2021].

Figure 2.6: Illustration of the events occuring at magnetic disturbance on the polar cap.

On the nightside, in the auroral oval region, one can notice active auroral enhancements.

The Tongue of Ionization, appreas in the dayside cusp c), however, fragments of TOI propagates toward the nightside, as polar cap patches. Picture from Moen, J. et al., 2013.

Figure 2.6 shows a polar cap patch, as it moves from dayside to the night side, the pink region is called Tongue of Ionisation (TOI). It refers to a region of enhanced density in the cusp polar cap, acquiring the flow from ionised plasma entering the ionosphere due to dayside reconnection [Moen, J. et al., 2013, Foster et al., 2005]. Polar cap Patches are believed to be a fragment of the TOI, that propagates through the convection line from noon to midnight. In the figure, we can notice that the the TOI (pink oval shape) is chopped into smaller scale patches, that are dragged toward the night side following convection cells pattern [Moen, J. et al., 2013].

Finally, polar cap patches afflicts the most intense phase scintillations when they exit the polar cap toward the auroral oval. This type of polar cap patch is called blob type 1. Like the first one, blob type 2 is an enhanced density structure in the F-layer, however formed locally in the auroral oval [Jin, Yaqi et al., 2016 and Y. Jin et al., 2014].

Many studies have been made, to understand the influence of polar cap patches on scintillations. It has been found that blob type 1 has a greater influence on scintillations than the second type [Jin, Yaqi et al., 2016]. Moreover, in a study on GPS scintillation associated with polar cap patches and cusp dynamics Jin, Y. et al., 2017. It has been found that phase scintillations seem to be moderate when only polar cap patches are involved. However, phase scintillations greatly increase when polar cap patches are associated with cusp auroral dynamics.

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In this study, the ICI-4 rocket was launched to intercept a polar cap patch that was directed toward a region of enhanced auroral activity [Lynnebakken, 2015]. The aim was to get in-situ measurements of electron densities as the polar cap patch combines with the auroral dynamics. As opposed to the previous cited study Jin, Y. et al., 2017 the rocket was launch toward the auroral oval in the nightside.

2.5 Ionospheric Instabilities

Different instabilities can be driven in the ionosphere due to the enhanced plasma flows initiated by the reconnection process. the Gradient-drift instability, also referred to as theE×B instability, takes place when an external force acts on a certain volume of plasma. This creates a charge separation and an electric field appear. Due to the presence of magnetic field, the electric field will create a disturbance that will grow into an instability [M. J. Keskinen and Ossakow, 1983]. This instability takes place when a gradient density is present in the plasma and usually breaks down large irregular structures into smaller scales. Spicher, A et al., 2015 Studied the potential role of GD instability on the growth of polar cap patches, using Swarm satellites in-situ measurements. This study concluded that scintillations observed on radio waves, may be the result of the influence of GD instabilities on polar cap patches.

Another instability that can occur is known as the Kelvin-Helmoltz instability (KHI).

It rises at a boundary of shear flow plasma, when there is different velocities at the interface between two fluids, and both are perpendicular to the magnetic field. A common example is when winds blow on the surface of the ocean, one can notice the formation of small waves on the surface. A more complete description of the theory behind KHI can be found in M. Keskinen et al., 1988. In the ionosphere, these little waves created due to this shear flow, can lead to density fluctuations. Therefore, this can give rise to scintillations on GNSS signals. Recently Spicher, A. et al., 2020 studied the impact of KHI on scintillations, through a quantitative nonlinear analysis.

It was concluded that, not only KHI can be a suggested process involved in the occurence of scintillation. Contrary to GDI, and under certain conditions in place, KHI exhibits a particular pattern in scintillations occurence. More details on the specific outcomes and the methods used can be found in the paper Spicher, A. et al., 2020.

One important shear flow instabilities, that have been found to cause important ionospheric irregularities [Moen, J. et al., 2013, Spicher et al., 2016; Spicher, A. et al., 2020] are the Reverse Flow Events (RFE). These are ∼ 100-200 km wide east-west elongated channels, that move in the opposite direction to the background plasma flow.

They are a type of KHI, however, covering a larger scale and flowing sunward. A more detailed study on RFEs can be found in Rinne et al., 2007.

2.6 Global Navigation Satellite System

GNSS stands for Global Navigation Satellite System, which describes the use of satellite for geo-spatial positioning and navigation. The most known of these satellite systems is GPS which uses 31 satellites for positioning and tracking. As of the begining of 2021, 31 constelation were in operation Portal, 2021.

GNSS satellites are located at medium Earth orbits (MEO) with an altitude of approximatly 20200 km. They have a revolution period of about 12 hours that may change depending on the satellite system. As of today, there exists six different global Navigation satellite systems. The most commonly known is GPS owned by the United States. We can also count: GLONASS owned by Russia, the European Union GNSS namedGalileo,BeiDou(BDS) from China,IRNSSfrom India, and finally, the Quasi- Zenith Satellite System (QZSS) owned by Japan [Space-Based PNT, 2021]. The main

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2.7. Signal structure

Figure 2.7: Two dimensional representation of trilateration for positionning.

task of these systems is to provide the user with a convenient way to constantly measure its distance to the satellite, and this is executed using a method known as: Trilateration.

This technique is defined by the process of using measurements of distances to determine a location. For this method, we need a minimum of four satellites to establish a location.

Figure 2.7 introduces a two-dimentional representation of this method, this is adapted from [Petrovski and Tsujii, 2012]. If we have only one satellite, the distance to this satellite can be anywhere within a certain radius of this one. Therefore, if we draw a circle around the satellite, by using its position as center, the receiver can be anywhere on that diameter. By introducing a second satellite, the positioning precision will highly increase since the two cirle will intersect at two different position. By knowing the distance to the satellites and their coordinates in ^xand ^y (in two dimension). One can retrieve the exact positions of the two intersecting points. By adding a third satellite, all circles will only intersect at one point. This allows to retrieve one only exact location (see figure 2.7). This works the same in three dimensions, where we have a sphere around the satellites that intersect with the earth at only one position corresponding to the location.

To measure an exact position, satellites broadcast their location and their clock time through a carrier electromagnetic wave. This latter travels at the speed of light. It is then possible to retrieve the distance from a location to the satellite using d = t∗v, wheret is the measured time of the received signal. However, all signals should carry the time in the same GNSS timeframe. This also accounts for the receivers and users epoch.

A fourth satellite is then used to accounts for the time error [Petrovski and Tsujii, 2012].

Finally, the range, or distance, to the satellite is measured by comparing the incoming signal, adjusted to the satellite timeframe, with a locally generated signal. This last is synchronized to the receivers timeframe. The clock bias calculated using the fourth satellite is also added to adjust for time errors.

2.7 Signal structure

As explained earlier, the range measurement is based on correlating two signals: The received satellite signal with a generated replica from the receivers. The satellite’s generated signal is composed of three layers:

First, a physical layer which is a waveform modified (modulated) for the purpose of

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Figure 2.8: GNSS frequency bands for GPS, GLONASS and Galileo. Figure from Subirana et al., 2021.

’carrying’ information. This layer is best known as the carrier wave and is usually a sine wave that does not carry any information. However, it contains the essential transmitted message. For example, GPS satellites transmits on five carrier frequencies, the most known are: The ^L1 carrier frequency (1575.42 MHz) and the L2 (1227.60 MHz) [Petrovski and Tsujii, 2012]. To limit intereference between carries frequencies from the other satellite systems, the International Telecommunication Union (ITU)strictly regulates the shared radio spectrum, by coordinating the assignment of frequencies to users. Figure 2.8 shows the different frequencies used by GPS, GLONASS and Galileo, and the limitation assigned by the ITU.

The second layer is the ranging code layer. It consists of a periodic sequence of pulses that randomly switches between 0 and 1. This is known aspseudorandom noise codes (PRNs) [Hofmann-Wellenhof and Herbert Lichtenegger, 2008].

Each satellite has its own unique PRN code, allowing proper identification at the receiver level. These codes also allow to determine the time shift, much needed to establich the distance to the satellites. However, in order to establish navigation points, the receiver needs to know the position of each satellites. The third and final layer is called the Navigation message containing information about the satellite’s orbit, the clock corrections, the week number and other system status.

2.8 Atmospheric effects

The carrier frequency propagates through space with a phase velocity

v_ph=λf (2.8)

Where λ is the wavelength of the signal and f its freqiency. The carried signal is propagated as a group with a slightly different frequencies and the velocity can be expressed as

v_gr =−df

dλλ² (2.9)

The phase and group velocity are related through the Rayleigh equation [Hofmann- Wellenhof and Herbert Lichtenegger, 2008, p.117]:

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2.8. Atmospheric effects

vgr =vph−λdv_ph

dλ (2.10)

The phase and group velocity become equal in nondispersive media, where the speed of the propagating wave is not depending on its frequency. By contrast, a dispersive medium is a medium in which waves of different frequencies propagate at different velocities, this is due to the dependence of the medium’s refractive index, on the frequency of each wave. The refractive index of any medium n is defined as the ratio between, the speed of light and the velocity of the wave:

n= c

v (2.11)

We can then define the group and phase velocity as:

v_gr = c

n_gr (2.12)

v_ph= c

nph (2.13)

We can replacing these two last equation into Rayleigh equation 2.10, which yields to :

c

n_gr = c

n_ph +λ c n²_ph

dn_ph

dλ (2.14)

By introducing the differential form of equationc= λf, and applying a series of algebra, equation 2.15 becomes:

n_gr =n_ph+fdn_ph

df (2.15)

where f is the signal frequency.

Seeber [Seeber, 2003, p-54] describes the refracting coefficient for phase propagation, in the ionosphere, as a power series

n_ph = 1 + c2

f² + c3

f³ + c4

f⁴ +... (2.16)

Where c_i coefficients are independent of the carrier frequency f, and depend on the elctron density N_e. By only taking the quadratic terms of the series 2.16

n_ph = 1 + c₂

f² (2.17)

Differentiating with respect tof, follows:

dn_ph=−2c₂

f³ df (2.18)

That we can substitute in equation 2.15, we get:

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n_gr = 1 + c₂

f² −f2c₂

f³ (2.19)

= 1− c₂

f² (2.20)

An estimate of c₂ is also presented in [Seeber, 2003] as: c₂ = −40.3N_e, which states thatc₂ is always a negative quantity, sinceN_e is always positive. We can replace this value in equations 2.17 and 2.20, andretrieve the relation n_ph <n_gr, and therefore: v_ph

> v_gr. Therefore, when travelling through the ionosphere (dispersive medium), the signal experiences a group delay and a phase advance, meaning that the carrier phase is advanced, while the ranging codes are delayed.

According to Hofmann [Hofmann-Wellenhof and Herbert Lichtenegger, 2008, p.119], the integral along the signal’s path, defines the measured range, and according to Fermat’s principle

s=^Z nds (2.21)

And the geometric range, LoS between the satellite and the user, where n= 1 is defined as:

s0 =^Z ds0 (2.22)

The ionospheric refraction is determined by calculating the difference between the measured and geometric range:

∆^iono =^Z nds−

Z

ds₀ (2.23)

Using the definition ofn_ph, andn_gr, the equation above can be written for a path length difference for phase propagation ∆ph and group propagation ∆gr:

∆^ionoph/gr =^Z (1± c2

f²)ds−

Z

ds₀ (2.24)

The integration is approximated for the first term, along the geometric range, according to Hofmann [Hofmann-Wellenhof and Herbert Lichtenegger, 2008, p.119], the formula becomes:

∆^ionoph/gr =±

Z c₂

f²ds₀ (2.25)

Recalling c₂ =−40.3N_e, are expressed as a function of the electron density

∆^ionoph =−40.3 f²

Z

dN_eds₀ (2.26)

∆^ionogr = 40.3 f²

Z

N_eds₀ (2.27)

Where, the ionospheric group delay and the ionospheric carrier phase advance can be measured by the receivers [Zhang et al., 2003], using two different known frequenciesf₁ andf₂, finally the Total Electron Content (TEC) from the satellite to the receiver is deduced

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2.9. Scintillations

T EC =^Z Neds0 (2.28)

Can then be calculated and is an important ionospheric index in GNSS, is is usually expressed usingTECU unit, where

1T ECU = 10¹⁶electrons/m² (2.29) The integral in equation 2.28 includes the electrons in a cylindrical column with a cross section of 1m², ranging from the receiver to the satellite.

2.9 Scintillations

Scintillations are a rapid fluctuation of a wave’s phase and amplitude. This phenomena is observed on the received signal indicating that the wave experiences fluctuations, throughout its propagation. Historically, the first known effect of scintillations was the observation of a fluctuating intensity from radio stars [Hey et al., 1946]. Multiple studies have been dedicated to understand scintillations [Kintner et al., 2007, Yeh and C.-H.

Liu, 1982, Aarons, 1997]. It has been found that if sufficiently intense, scintillations can strongly affect satellite based navigation systems (GNSS) and High-Frequency communication [Kintner et al., 2007, Moen, J. et al., 2013]. Due to the importance of these two systems, GPS scintillation studies have received a lot of interest. Two main reasons usually motivate the research. On one hand, the need to mitigate and understand the consequences of ionospheric scintillations on radio signals. On the other hand, the information carried by scintillation data allow to understand the dynamics of the ionosphere [Yeh and C.-H. Liu, 1982]. This second part is the main motivation for this thesis.

The strongest scintillations occur at the equatorial regions and at high latitudes [Kintner et al., 2007, Moen, J. et al., 2013, Jin, Y et al., 2015, and references therein]. As per previous sections, the description of solar wind magnetosphere coupling, as well as the following energy transfer through magnetosphere-ionosphere dynamics, allow for strong events to occur at high latitudes [Moen, J. et al., 2013 and references therin]. Several studies have been conducted on high latitude scintillations using the GPS L1 frequency.

It was then noticed that strong ampltitude scintillations at GPS frequency are more rare at high latitues [Spogli et al., 2009]. Therefore, most studies focused on phase scintillations for this region.

Scintillations in the polar cap are usually associated with polar cap patches due to the enhanced electron densities within [Moen, J. et al., 2013, Kintner et al., 2007]. The specific regions that presents the most scintillations occurence, in the polar cap are the cusp and auroral oval boundaries [Moen, J. et al., 2013].

The strength of scintillations are characterized by metrics calledS₄ index, for amplitude scintillations, andσφ for phase scintillations. S₄ is the normalized standard deviation of the signal’s intensity:

S₄ =

s< I² >−< I >²

< I² > (2.30)

Where I is the signal’s intensity.

Phase scintillations index is represented by the standard deviation of the received signal’s phase:

S₄ =^q< φ² >−< φ >² (2.31)

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These indices were introduced in 1978 by E. J. Fremouw, 1978 where <> is a temporal average over a certain time.

When severe scintillations occur, GPS receivers can experience strong amplitude scintillations (above 0.7). Amplitude scintillations between S₄ = 0.3 and S₄ = 0.5 are considered weak, and 0.5< S₄ <0.7 are considered moderate [Y. Liu et al., 2017].

Phase scintillation, on the other hand, are considered very weak or non existant when σ_φ < 0.1. Signals experience strong phase scintillations when σ_φ > 0.5. Weak and intermediate phase scintillations occur when 0.1 < σ_φ < 0.25 and 0.25 < σ_φ < 0.5, respectively [Y. Jin et al., 2014]. This will be used in the result sections when a thourough analysis of ground receivers scintillations will be presented.

2.10 Fresnel zone

The Fresnel zone, also referred to as Fresnel length or radius, is defined as a region of space between a transmitter and a receiver for wave propagation. When a radio signal propagates from a point to another, parts of the wave can follow a slightly different path from the primary one. This latter is also referred to as the Line-of-Sight (LoS).

At large scales, this can introduce a change in the direction of the wave. Therefore, if the change is large enough, the scattered wave, or non-LoS wave, can interact with the primary wave and introduce constructive or destructive interference. This is due to the phase shift between the two waves.

The change in the direction can be introduced by different events. An obstacle can cause the wave to reflect in a different direction therefore introducing an angle at which the wave is propagating. This is known as multipath [Braasch, n.d.].

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Figure 2.9: Illustration of the ellipsoidal Fresnel zones between a Transmitter and a receiver on ground.

We define different Fresnel Zones, where the impact of a scattered wave will differ depending on the zone at which it is scattered from. The first Fresnel zone is defined such as the difference between the direct path (wave following the LoS) and an indirect path (scattered wave) is below or equal to half the wavelength. This zone defines a low impact of the scattered wave on the received signal. Figure 2.9 shows the Fresnel zones for a transmitter and receiver on ground. Three different ellipsoidal Fresnel zones are represented. In addition, three Multipath environements by which signal’s can either be reflected, or diffracted are illustrated. More details on the Multipath environements can be found in Braasch, n.d.

In figure 2.9 the primary wave, following the LoS A to B is illustrated. The non-LoS is transmitted at the same time, however following a slightly different path. In the figure, the illustrated non-LoS wave travels fromA toC, then from C to the receiverB. The non-LoS wave will arrive at the receiver with a slightly different path, than the primary wave, therefore introducing a phase shift. Depending on the magnitude of this phase

Developing a Phase Screen Model to analyse electron density measurements from the ICI-4 Rocket