Automatic SLAMS detection and magnetospheric classification in MMS data

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Automatic SLAMS detection and magnetospheric classification in MMS data

CARL FOGHAMMAR NÖMTAK

KTH

SKOLAN FÖR ELEKTROTEKNIK OCH DATAVETENSKAP

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Automatic SLAMS detection and magnetospheric

classification in MMS data

Carl Foghammar Nömtak

Master’s Programme, Aerospace Engineering, 120 credits Date: August 21, 2020

Supervisor: Tomas Karlsson Examiner: Anita Kullen

School of Electrical Engineering and Computer Science Swedish title: Automatisk SLAMS-identifikation och magnetosfärsklassificering i MMS data

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c

2020 Carl Foghammar Nömtak

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Abstract | i

Abstract

Short Large-Amplitude Magnetic Structures (SLAMS) have been observed by spacecraft near Earth’s quasi-parallel bow shock. They are characterized by a short and sudden increase of the magnetic field, usually by a factor of 2 or more. SLAMS studies have previously been limited to small sample sizes because SLAMS were identified through manual inspection of the spacecraft data. This makes it difficult to draw general conclusions and the subjective element complicates collabora- tion between researchers. A solution is presented in this thesis; an automatic SLAMSdetection algorithm. We investigate several moving- window methods and measure their performance on a set of manually identified SLAMS. The best algorithm is then used to identify 98406 SLAMSin data from theMagnetospheric Multiscale(MMS) mission. Of those, 66210SLAMSwere detected when theFast Plasma Investigation (FPI) instrument was active. Additionally, we are interested in knowing whether a detected SLAMS is located in the foreshock or magnetosheath. Therefore, we implement a Gaussian mixture model classifier, based on hierarchical clustering of the FPI data, that can separate between the four distinct regions of the magnetosphere that MMS encounters; magnetosphere, magnetosheath, solar wind and (ion) foreshock. The identifiedSLAMSare compiled into a database which holds their start and stop dates, positional coordinates, B-field information and information from the magnetospheric classifier to allow for easy filtering to a specific SLAMS population. To showcase the potential of the database we use it to perform preliminary statistical analysis on how the properties of SLAMS are affected by its spatial and/or magnetospheric location. The database and Matlab implementation are available on github: https://github.com/cfognom/MMS_

SLAMS_detection_and_magnetospheric_classification.

Keywords

Short Large-Amplitude Magnetic Structures (SLAMS), Algorithm design,

Magnetospheric classification, Time series analysis,

Space plasma physics

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Sammanfattning | iii

Sammanfattning

Korta magnetiska strukturer med hög amplitud (SLAMS) har obser- verats av satelliter nära jordens kvasi-parallella bogchock. En kort och plötslig höjning av magnetfältsstyrkan är ett typiskt drag för SLAMS, vanligtvis med en faktor 2 eller mer. Forskning om SLAMS har tidigare varit begränsad till mindre fallstudier eftersom SLAMS identifierats genom manuell inspektion av satellitdata. Detta gör det svårt att dra generella slutsatser och det subjektiva elementet försvårar samarbetet mellan forskare. En lösning till detta problem presenteras i denna avhandling; en automatisk identifieringsalgoritm för SLAMS. Vi undersöker flera metoder och mäter deras prestanda på en uppsättning manuellt identifieradeSLAMS. Den bästa algoritmen används sedan för att identifiera 98406SLAMSi data frånMMS-uppdraget. Av dessa upp- täcktes 66210 SLAMS när FPI-instrumentet var aktivt. Vi är dessutom intresserade av att veta om en upptäcktSLAMSfinns i förshocken eller magnetoskiktet. Därför implementerar vi en Gaussisk klassificerare som bygger på hierarkisk klustring av FPI-data. Den kan separera de fyra distinkta regionerna av magnetosfären som MMS observerar;

magnetosfär, magnetoskikt, solvind och (jon) förchock. De identifierade SLAMS:en sammanställs till en databas som innehåller deras start- och stoppdatum, positionskoordinater, B-fältsinformation och information från magnetosfärsklassificeraren för att möjliggöra enkel filtrering till en specifik SLAMS-population. För att visa potentialen av databasen utför vi en preliminär statistisk undersökning av hur egenskaperna av SLAMS påverkas av deras rumsliga och/eller magnetosfäriska position. Databasen och Matlab-implementationen är tillgängliga på Git- hub: https://github.com/cfognom/MMS_SLAMS_detection_

and_magnetospheric_classification.

Nyckelord

Korta magnetiska strukturer med hög amplitud (SLAMS), Algoritmdesign,

Magnetosfärsklassificering, Tidsseriesanalys,

Rymdplasmafysik

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Acknowledgments | v

Acknowledgments

I would like to specially thank my supervisor Tomas Karlsson for giving me this thesis opportunity and for guiding me continuously during the thesis work. His expertise in space plasma physics and knowledge about the magnetosphere has been very helpful. I would also like to thank my classmate Linn Mandell for this thesis period. We had similar thesis objectives but were assigned to different spacecraft; Cluster II for Linn and MMS for me. Although we worked independently, we could still learn much by sharing our successes and mistakes with each other.

Finally, I wish to acknowledge my examiner Anita Kullen for taking her time to evaluate my work.

Stockholm, August 2020 Carl Foghammar Nömtak

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CONTENTS | vii

List of Figures

1.1 Example of SLAMS in MMS data. . . 2 2.1 The magnetosphere. Source: electroverse.net . . . 8 2.2 The patchwork model. Source: Schwartz and Burgess

1991 . . . 10 2.3 MMS orbit for phase 1 and 2. Source: NASA . . . 12 2.4 The ground loop hierarchy. Source: Argall et al. 2020 . . 14 3.1 Different types of moving windows over a couple of

SLAMS. . . 17 3.2 Different types of moving windows over a bow shock

crossing. . . 18 3.3 The RANSAC algorithm. . . 18 3.4 The background field using a moving median and the

detection threshold using a detection constant of 2. . . . 19 3.5 Merged SLAMS using a merging constant of 1.5.. . . 20 3.6 Recall-precision plot for every parameter setting combi-

nation. The curves indicate F1 score. . . 22 3.7 Recall-precision plot for every parameter setting com-

bination while excluding SLAMS under 1 second. The curves indicate F1 score. . . 23 3.8 Raw data plotted on F₁, F₂ and F₃ axes. . . 27 3.9 Hierarchical clustering using 9 clusters. . . 28 3.10 Merged clusters. The magenta cluster was discarded. . . 28 3.11 GMM clusters.. . . 29 3.12 Portion of time in each class. The position of the bow

shock and Earth is marked in cyan. . . 31 4.1 Left panel: Position of primary SLAMS by class. Right

Panel: Position of primary and secondary SLAMS. . . 36

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4.2 Left panel: Primary search time. Right Panel: Union of primary and secondary search time . . . 37 4.3 SLAMS rate, strength and relative strength. . . 39 4.4 Magnetosheath and foreshock SLAMS rate, strength and

relative strength. . . 40 4.5 Histogram plots of various SLAMS properties. . . 41 4.6 SLAMS size plotted against duration. . . 42 4.7 SLAMS properties plotted against Alfvén mach number

of surrounding flow. . . 43 A.1 Magnetosheath with a foreshock excursion. SLAMS are

detected both in the foreshock and magnetosheath. . . . 52 A.2 MMS exiting the magnetosphere through the quasi-

perpendicular bow shock. Some magnetosheath SLAMS are detected but its hard to see them at this scale. . . 53 A.3 Magnetosheath missclassified as foreshock. The classi-

fier is not perfect. . . 54 A.4 Alternating magnetosheath and solar wind. Likely

caused by in-and-out movements of the bow shock.

Thanks to a short time window and use of median, these short excursions are not predicted as SLAMS. . . 55 A.5 Closeup of a cluster of SLAMS. The drawback of a short

window can be seen here, as some SLAMS are not detected. 56 A.6 Several SLAMS are detected in this turbulent foreshock-

/magnetosheath transition zone. . . 57 A.7 Three SLAMS are detected in the foreshock, but the

SLAMS themselves are classified as magnetosheath. . . . 58

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LIST OF TABLES | xi

List of Tables

3.1 Tested parameter settings for the SLAMS detection algo-

rithm. . . 21

3.2 Classes of the classifier. . . 24

3.3 FPI ion data used by the the classifier. . . 25

3.4 MMS1 data intervals. . . 32

3.5 SLAMS detection algorithm settings. . . 32

4.1 Number of SLAMS per class. . . 36

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List of acronyms and abbreviations | xiii

List of acronyms and abbreviations

FGM Fluxgate Magnetometer FPI Fast Plasma Investigation FS Foreshock

GMM Gaussian Mixture Model GSE Geocentric Solar Ecliptic IMF Interplanetary Magnetic Field

MEC Magnetic Ephemeris Coordinates MMS Magnetospheric Multiscale MSH Magnetosheath

MSP Magnetosphere

RANSAC Random Sample Consensus

SLAMS Short Large-Amplitude Magnetic Structures SW Solar Wind

ULF Ultra Low Frequency

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Introduction | 1

Chapter 1 Introduction

1.1 Background

The interactions between plasma and magnetic fields are complex and to better understand it multiple scientific space missions have been launched with the goal of providing in-situ measurements. For example, Cluster II (2000) and the Magnetospheric Multiscale (MMS) mission (2015) which are multi-spacecraft satellites orbiting in close formation. Among other regions of interest in the magnetosphere, they target the complex bow shock where the supersonic solar wind is slowed down to subsonic speeds and diverted due to Earth’s magnetosphere. By studying this local region we hope to gain new knowledge about plasma phenomena that can be generalized to the rest of the universe.

In the quasi-parallel bow shock, i.e. when the angle between the bow shock normal and the interplanetary magnetic field is <45^◦, events known as Short Large-Amplitude Magnetic Structures (SLAMS) have been observed. They are characterized by a large increase in the magnetic field strength, typically by a factor of 2 and a short duration for an observing spacecraft in the order of 5 to 20 seconds, typically 10 seconds (Schwartz et al. 1992). Figure1.1 shows how SLAMS can appear in B-field time-series data.

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Figure 1.1 – Example of SLAMSinMMSdata.

1.2 Objective

The initial objective of this thesis was to design an algorithm for the MMS mission that can detect SLAMS in the solar wind, including foreshock. However, as progress was being made into accurately detecting the solar wind, detection of the magnetosheath, foreshock and magnetosphere came as a bi-product. Therefore, the objective of the thesis was extended to also include the detection of magnetosheath SLAMS.

Objective

Design an algorithm that can automatically detect SLAMS in MMS data. Additionally, the algorithm should distinguish between magnetosheath and foreshockSLAMS.

1.3 Scientific and engineering challenges

Here we have identified the key challenges of the thesis.

1. How can the background magnetic field be determined?

2. Knowing the background field and the local field, what should be the criterion for aSLAMS?

3. How should indications ofSLAMSbe joined/merged temporally?

4. How can we automatically distinguish between magnetosphere, magnetosheath, solar wind or foreshock?

5. How can we validate that the algorithm is working?

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Introduction | 3

1.4 Purpose

Previous studies of SLAMS have been case studies where SLAMS are identified manually. Often the studies are limited to a smaller number ofSLAMS, typically in the order of tens. Sometimes the studies are even analysing the same SLAMS, for example (Schwartz et al. 1992) and (Mann and Lühr1992). Because of this, it is not clear if we are looking at the full spectrum of SLAMS phenomena, so it becomes difficult to draw general conclusions. Additionally, the subjective element in the identification process potentially means that every researcher has their own definition of what a SLAMS is and as such makes it inconvenient for researchers to collaborate. It would therefore be beneficial to have an algorithm that could identify SLAMS automatically with a set of objective rules, thus eliminating subjectiveness and enabling the creation of large SLAMS databases without much effort. These databases could then be used as the starting point for large statistical studies which might reveal new insights undiscoverable by case studies.

1.5 Goals

The end goal of this thesis is to create a useful SLAMS database. This has been divided into the following four sub-goals:

1. Design an objective method for detectingSLAMS

2. Design a classifier which can separate between magnetosheath SLAMSand foreshockSLAMS

3. Process allMMSdata and create a database

4. Visualize the database and perform initial analysis using the database

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1.6 Research Methodology

The algorithm consists of two parts; the SLAMS detector and the magnetospheric classifier. The SLAMS detector uses Fluxgate Mag- netometer (FGM) data from MMS and can detect SLAMS-like events anywhere. To complement it, the magnetospheric classifier uses Fast Plasma Investigation(FPI) data to classify where in the magnetosphere the SLAMS are detected; magnetosphere, solar wind, magnetosheath or (ion) foreshock. Events discovered in the magnetosphere should probably be discarded as outliers but the rest is of interest.

For theSLAMSdetector, several different moving window strategies were investigated. Even though moving window methods are simple, we found that they were sufficiently good for this task and we did not need to use a more advanced method. We then picked the best algorithm based on qualitative analysis and its performance on a validation dataset.

The magnetospheric classifier is a Gaussian mixture model that was created by first sampling random FPI data. This data could then be clustered into four distinct clusters using only three dimensions. Three was chosen so that the clusters easily could be visualized during the design process. On each dimension is a carefully engineered quantity.

The first one is related to the cone angle of the solar wind bulk velocity, the second and third were derived from the omni-directional differential energy flux spectrum data. Details are found in chapter 3.

Finally, the Gaussian mixture model could be constructed by simply calculating the covariance and mean of each cluster. The Gaussian mixture model was a good fit to the data since each cluster was roughly globular. Due to lack of time, the classifier could not be validated using a validation dataset. Instead qualitative methods were used.

A SLAMS database was then created by processing all available MMS1 data at the time. For each detectedSLAMSwe include; start and stop dates, positional coordinates, B-field information, and information from the magnetospheric classifier.

Matlab was used to implement the algorithm. The full implementation together with the resulting SLAMS database can be found on Github: https://github.com/cfognom/MMS_SLAMS_

detection_and_magnetospheric_classification

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Introduction | 5

1.7 Scope

Because of disk space limitations, the data used in this thesis is limited to MMS1. As such the resulting database only contains SLAMS that were observed by MMS1 and any cross-spacecraft analysis have been left out. We believe that in many cases theSLAMSobserved byMMS2-4 will be the same as for MMS1 because of the close formation of MMS.

Nonetheless, getting data on the sameSLAMSfrom multiple points and at different times is still invaluable information, but it is outside the scope of the work. Luckily though, databases for MMS2-4 can easily be created. It is just a matter of downloading the MMSdata and running the code.

1.8 Structure of the thesis

Chapter 1, Introduction: Introduces the reader to the thesis, the con- cept ofSLAMS, the problems of previousSLAMS studies and proposes a solution. Also gives an overview of the research methodology.

Chapter 2, Background: Aims to educate the reader on prerequisite knowledge that is required to understand the thesis. For example, the structure of the magnetosphere, an overview of theMMSmission or the quasi-parallel and quasi-perpendicular concepts. Also contains a related work section.

Chapter 3, Methods: Detailed explanation of the methods used to detectSLAMS and classify the magnetosphere and how these methods were validated. Also explains how the SLAMS database was created and how the properties of SLAMSwere calculated.

Chapter4, Results and analysis: Here, the database is visualized and we analyse various properties ofSLAMS on a large scale.

Chapter 5, Discussion and conclusions: Contains a discussion of the results, conclusion of the thesis and lists future work.

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Background | 7

Chapter 2 Background

2.1 The magnetosphere

The region of space where Earth’s magnetic field dominates is known as the magnetosphere, see Figure2.1. It is shaped almost like a paraboloid, extending to unknown extents on the night side. On the day side it is usually extends to 10 Earth radii, after which a boundary layer known as the magnetopause is located. In the magnetopause the dynamic pressure from the solar wind and the magnetic pressure from Earth is equal. The solar wind, which is the flow of supersonic charged particles from the Sun, is slowed to subsonic speeds and diverted because of the magnetosphere. A shock front, known as the bow shock, is formed where this happens and it is located a few Earth radii ahead of the magnetopause. Between the bow shock and the magnetopause is the turbulent magnetosheath where the now subsonic solar wind flows around the magnetosphere. (Fälthammar1992)

What is fascinating about the bow shock is that it is essentially collisionless (the mean free path is about 1 AU), but yet exists. Almost no particle collisions occur, instead the shock is formed by interactions between electro-magnetic fields and particles.

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Figure 2.1 – The magnetosphere. Source: electroverse.net

2.2 The foreshock

Outside the magnetosphere, the Interplanetary Magnetic Field (IMF) of the Sun is dominating. When the magnetic field lines of the IMF and the bow shock normal are roughly parallel (< 45^◦), solar wind particles can be reflected and gyrate upstream. This region is the quasi-parallel bow shock, elsewhere we have the quasi-perpendicular bow shock. The space that the reflected particles occupy is known as the foreshock. In the foreshock, particles are thus flowing both upstream and downstream, giving rise to non-linear instabilities. In turn, these instabilities result in Ultra Low Frequency (ULF) waves which propagate sunward in the plasma rest frame but are convected by the faster solar wind in the opposite direction (Schwartz1991). ULF waves and SLAMS show similar properties and it is likely thatSLAMS grow from the ULF wave field (Thomsen et al. 1990; Schwartz and Burgess1991).

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Background | 9

2.3 Short Large-Amplitude Magnetic Struc- tures (SLAMS)

Initial observations of the quasi-parallel shock included studies by Greenstadt (1974) and Greenstadt et al. (1977). It was observed that the shock consisted of magnetic pulsations with sizes of 1-2 Earth radii.

The plasma distributions of the pulsations were neither characteristic of the solar wind or magnetosheath. Thomsen et al. (1990) found that the pulsations exhibit both convected and nested signatures. A convected signature refers to a signature which is convected past the spacecraft along the solar wind whereas a nested signature is created by the in-and-out motions of the bow shock. Schwartz and Burgess (1991) named the pulsations;Short Large-Amplitude Magnetic Structures (SLAMS) and introduced a patchwork model for the quasi- parallel shock crossing. The model is visualized in Figure 2.2, where SLAMS(shaded ovals) are convected along the flow, grow in amplitude (darken) and then finally merge. Schwartz et al. (1992) found that the characteristic properties of SLAMSwere:

1. Short duration: 5-20 seconds 2. Large amplitude: _B^B

0 ≥ 2

3. Monolithic and well defined magnetic signature

They also found that ULF waves quickly grow intoSLAMS. In 5-20 seconds ULF waves can grow to 2-4 times their initial amplitude. This was later confirmed by Lucek et al. (2008) which estimated SLAMS growth rate to a few seconds, using Cluster II data. Lucek et al. (2008) also gave an upper limit of 2700 km for the size of the transition zone (see Figure2.2), i.e. the region of whereSLAMS first appear to where they merge. This short length suggests that only one or a few SLAMS can be present at once in the transition zone.

A recent overview of the subject is given by Gopal (2014).

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Figure 2.2 – The patchwork model. Source: Schwartz and Burgess 1991

2.4 Geocentric Solar Ecliptic (GSE) coordi- nates

In this thesis we useGSEcoordinates extensively because of their use- fulness when describing magnetospheric phenomena which is highly dependent on the Sun’s location. The coordinate system is explained fully by the following list:

• Origin: Earth

• X-axis: Points from Earth to Sun

• Y-axis: In the ecliptic plane pointing in the opposite direction of Earth’s velocity vector around the sun

• Z-axis: Perpendicular to the ecliptic plane, pointing to the ecliptic pole

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Background | 11

2.5 The magnetospheric multiscale (MMS) mission

MMS is a four-spacecraft constellation mission, similar to the Cluster II mission, which has the primary objective of investigating magnetic reconnection near Earth (Burch et al. 2016). Magnetic reconnection occurs when plasma populations with anti-parallel magnetic field lines meet and convert magnetic energy into thermal and kinetic energy, accelerating particles perpendicular to their initial velocity. This phenomena often occurs in the magnetopause and in the magnetotail which are also the regions which MMS targets during its two phases.

However, MMS is also useful for SLAMS research since it often passes through the bow shock, especially during phase 2, and thus observes magnetosheath, solar wind and foreshock frequently.

2.5.1 Orbit and phases

In year 1 and 2, during phase 1, the orbit apogee was placed on the magnetopause as shown in Figure 2.3. Phase 2 begun in year 3 when the apogee was raised to target the magnetotail. The orbits are highly elliptical to maximize the time spent in the regions of interest (Burch et al. 2016). Because of Earth’s rotation around the Sun, the orbits will appear to precess around the ecliptic plane normal with one revolution per year when viewed from a GSE frame. Each spacecraft has a very similar orbit, meaning they are in a very close formation.

The separation distance is only 10 km minimum and 400 km maximum.

This allows for very accurate spatial gradient measurements.

2.5.2 Instruments

MMS carries many instruments, so we will only present those which are relevant for the thesis here. Because of limited download rate, it is not always possible to get full time resolution measurements from all instruments. At regions of low interest some instruments are disabled.

At especially interesting regions, such as those in 2.3, the instruments operate in burst mode at full speed. At other times the instruments operate in either fast, slow or survey mode. In this work we have not considered burst data since it is not available in large quantities.

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Figure 2.3 –MMSorbit for phase 1 and 2. Source: NASA

Fluxgate Magnetometer (FGM) (Torbert et al. 2016): The FGM is responsible for accurately measuring the magnetic field strength and direction. In survey mode this instrument produces 8-16 measurements per second. For redundancy and improved calibration, each spacecraft carries two FGMs, one digital and one analog. Obviously, this is the instrument we will use to detectSLAMS.

Fast Plasma Investigation (FPI) (Pollock et al. 2016): The FPI observes fast-moving plasma at a rate of 13.33 measurements per minute in fast mode. It can detect the number of electrons and ions encountering the spacecraft with multiple directions and energies.

Each spacecraft carries 8 sensors, 4 for ions and 4 for electrons, which together can observe the entire sky. We use the ion component of this instrument for the magnetospheric classifier.

Magnetic Ephemeris Coordinates (MEC): MEC is not actually an instrument, but it is accessed in the same way as other instruments.

MEC contains the positional data of MMS at a rate of 2 times per minute. From this "instrument" we extract the GSE coordinates of MMS.

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Background | 13

2.6 Related work

1. The work of Schwartz et al. (1992) has been a very useful resource when designing the algorithm. Their work is a detailed analysis of a set ofSLAMS. They present a set of criteria (the list in section2.3) that all of their identifiedSLAMSfollow. By knowing this, SLAMS can effectively be detected. Although, Schwartz et al. (1992) also express their concerns of using this rigid set of criteria when identifying SLAMS, as they expect a large variation of signatures in such a turbulent environment. While this may be true, in our opinion it is still better than relying solely on subjective judgement. Also, even if these criteria do not describe the full spectrum of SLAMS phenomena, using them will both expand our currentSLAMSdatasets and serve as a stepping stone towards discovering more accurate criteria.

2. Recent work of Argall et al. (2020) uses a recurrent neural network to detect magnetopause crossings for the MMS mission.

Locating magnetopause crossings is a crucial part of magnetospheric research, which previously was done manually. They also discuss a broader goal of fully automating MMS science objectives, called the ground loop hierarchy, see Figure 2.4. The first level of the hierarchy is regional classifiers that distinguish between; solar wind, magnetosheath and magnetosphere. In the next level, classifiers that are specific to each region are employed to detect regional events such as SLAMS in the solar wind. The magnetospheric classifier and the SLAMS detector presented in this thesis could serve as potential candidates as regional classifier and SLAMS classifier respectively in the proposed ground loop hierarchy.

3. We are not aware of any previous SLAMS detection algorithms, however, there exists several previous magnetospheric classifiers:

(Silva et al. 2020; Nguyen et al. 2019; Piatt 2019; Olshevsky et al. 2019). Silva et al. (2020) used a support vector machine in the number density vs. ion temperature space to classify magnetosphere, magnetosheath and solar wind in MMS data.

Nguyen et al. (2019) developed a method for the Themis B spacecraft to classify magnetosphere, magnetosheath and solar wind in order to locate magnetopause and bow shock crossings.

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Their method was also adapted to the Cluster, Double Star and Artemis missions and utilized plasma moments and magnetic field data. Piatt (2019) created classification model for magnetosheath and magnetosphere in order to detect magnetosphere crossings.

The model uses a Bayesian mixture model and an auto regressive component to classifyFPIandFGMdata fromMMS. Olshevsky et al. (2019) used a convolutional autoencoder and neural network to compress and classify three-dimensional FPI data from MMS.

Their network can classify magnetosphere, magnetosheath, solar wind and ion foreshock. Comparing our classifier to the above, only the method of Olshevsky et al. (2019) has the same capabil- ities as ours, since it can classify the foreshock. Their method is definitely more complicated than ours and achieves almost perfect accuracy (99%) on a test set. The accuracy of our classifier is yet to be determined, however, there is no reason why it could not be similar or even greater, as demonstrated by Silva et al.

(2020) whose approach is arguably even simpler than ours, yet achieves an accuracy of 99.9%.

Figure 2.4 – The ground loop hierarchy. Source: Argall et al.2020

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Methods | 15

Chapter 3 Methods

In this chapter we describe the methods used by the algorithm to detect SLAMS and classify in which part of the magnetosphere they are detected. We also describe how the database was created and how we used it to investigate the properties ofSLAMS.

3.1 Detecting SLAMS

The main characteristic of a SLAMS, as mentioned previously, is a sudden increase in the magnetic field. A local region of space will have a stronger magnetic field compared to the field in the surrounding regions. We will refer to this field as the background field or B₀. A good estimation of the background field is the key to accurately detecting SLAMS, but it is also the most challenging part.

We split the process of detecting a SLAMSinto 3 parts:

1. Estimating the background field 2. Detection

3. Merging

3.1.1 Estimating the background field

The most straightforward method of estimating the background field is to smooth out the data using a moving window. We consider three types of functions which have such an effect: mean, harmonic mean

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and median. Figures 3.1 and 3.2 show them in action on a couple of SLAMSand a bow shock crossing respectively.

Mean: The mean, or arithmetic mean x, is of course calculated as;

x = Pn

i=1x_i

n (3.1)

where x_i are the items (the magnetic field data in the times series) and n is the number of items. It smooths the curve while keeping the area under the curve constant. This means that the peak of a SLAMS will be redistributed to the surrounding areas. This can be seen in Figure 3.1at 13:20:00 and 13:20:10 where the mean curve jumps up due to the presence of the first and second SLAMS respectively. At 13:20:25 and 13:20:40 the curve jumps down again. For this method to work well it is necessary that the time spent in the background is much greater than the time spent inside SLAMS, so that the contribution of the SLAMS will be small in comparison the the background. The estimation gets gradually worse as theSLAMS density increases in the window.

Harmonic mean: The harmonic mean H is a variation on the mean which is biased to lower values. It is calculated as;

H = n

Pn i=1

1 xi

(3.2) where x_iare the items and n is the number of items. Perhaps a more intuitive variant of the harmonic mean is:

H =

n

X

i=1

xiwi wi = s − x_i

s s =

n

X

i=1

xi (3.3)

In essence, the harmonic mean can be interpreted as the weighted sum of the items, where a weight w_i is proportional to how small x_i is in relation to the sum of the items s.

As can be seen in Figures 3.1 and 3.2 the harmonic mean will always be lower or equal to the mean. It too will experience the aforementioned jumps when aSLAMSenters the window but they will be greatly suppressed.

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Methods | 17

Median: The median first sorts the items then picks the center value, or takes the mean of the two center values if there is an even number of items. The advantage of the median is that extreme values, such as those in a SLAMS, have no influence at all. This works well as long the SLAMS do not occupy more than 50% of the window. If they do, the median value will be picked inside a SLAMS, which is undesired. It’s an "all or nothing" method. However, this property is also an advantage when the window moves over a boundary crossing such as the one in Figure 3.2 where the weaker magnetic field of the solar wind suddenly increases because of the magnetosheath. In this case the median is the method which most quickly adapts to the new conditions. The mean curve tends to rise before the actual crossing happens and the harmonic mean holds on to the solar wind values longer.

Figure 3.1 – Different types of moving windows over a couple of SLAMS.

During the early stages of the design process we investigated the Random Sample Consensus (RANSAC) algorithm (Fischler and Bolles 1981) for the purpose of B₀ estimation. RANSAC randomly samples the data and fits linear models to it. It then returns the model which had the best fit, as shown by Figure3.3, giving outliers such as SLAMS no influence. In theory this should work well, however there were several problems with this approach. Firstly, the background field is not necessarily linear on larger time scales, meaning that the data had to be split into smaller time intervals or RANSAC has to be used in a moving window. The first alternative means that the quality of the

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Figure 3.2 – Different types of moving windows over a bow shock crossing.

estimation will vary depending if you look in the middle of a time interval (good) or at the edges (bad). The second alternative has the problem of being very computationally expensive, sinceRANSACis an iterative algorithm. Ultimately, we left RANSAC in favor of the other moving window methods which were simpler and quicker, while still giving satisfying results.

Figure 3.3 – TheRANSACalgorithm.

3.1.2 Detection

After the background field has been established we can multiply it with a constant, which we call the detection constant C_d, to create a new curve which is elevated from the background field. Anything above this curve will be considered aSLAMS, as shown in Figure3.4. In previous studiesSLAMS have typically been identified by _|B^|B|

0| > 2, so we will set the detection constant to 2 as well.

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Methods | 19

Figure 3.4 – The background field using a moving median and the detection threshold using a detection constant of 2.

3.1.3 Merging

The final step is the merging step where intervals which likely belong to the same SLAMS are merged. For example, in Figure 3.4 the two leftmost intervals should probably be merged. This is done by first determining the merging constant C_m, which will be used in the same way as the detection constant to create a new curve. Anything above this curve will be inside a merging interval. If there is one or more SLAMS inside a merging interval, the whole merging interval will be considered one SLAMS. This means that the merging constant should be less than the detection constant. The effect of this operation can be seen in Figure 3.5 which is a continuation of Figure 3.4. From qualitative testing we found that a merging constant that follows the rule in Eq. (3.4) gives good results. In essence, the merging threshold should be right in-between the background field and the detection threshold.

C_m = C_d− 1

2 + 1 (3.4)

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Figure 3.5 – MergedSLAMSusing a merging constant of 1.5.

3.2 Validating the SLAMS detector

The proposed method forSLAMSdetection involves several parameters that have to be set by the user and it is still not clear which moving window method is best. Therefore, we wanted to investigate how the parameters affect the performance. We did this by creating a validation set of manually detectedSLAMS, which could be used with theSLAMS detected by the algorithm to calculate precision, recall and F1 score.

This was done for several parameter settings as shown in table 3.1.

Precision is a measure for how good the algorithm is at detecting only

"true"SLAMSand is calculated as;

precision = T P

T P + F P (3.5)

where T P is the number of true positives and F P is the number of false positives. In this case the "true"SLAMSare those in the validation set. ASLAMS detected by the algorithm is considered true positive if it intersects at least one SLAMS in the validation set, false positive if no SLAMSin the validation set intersects it.

Recall is a measure for how many of the "true"SLAMSthe algorithm detected and is calculated as;

recall = T P

T P + F N (3.6)

where F N is the number of false negatives. F N was determined by counting allSLAMSin the validation set which are not intersected by a SLAMSdetected by the algorithm.

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Methods | 21

The F1 score is simply the harmonic mean of precision and recall:

F 1 = 2

precision⁻¹+ recall⁻¹ (3.7) Ideally both precision and recall should be high. However, it is usually the case that increasing one, decreases the other. It can therefore be useful to instead maximize the F1 score. If the F1 score is high, it means that both precision and recall are high too. These three values will give insights into how well the algorithm is performing as well as what settings are best.

To create the validation set we searched the MMS Science Data Center (2020) for burst intervals which contained the string "SLAMS"

in the description. This yielded 301 short intervals spread out over the lifetime of MMS. Note: we do not use burst data, we just use the burst interval dates. We randomly picked 60 of those to be included in the validation set. In these intervals we manually identified 116 SLAMS.

We acknowledge that this is a rather small set of SLAMS so instead of letting the results of this analysis be the final say, we mainly use them as guideline in the design process and to give us a hint of the performance.

Table 3.1 shows the parameter settings we tested. Every com- bination of these 8 settings were tested and the resulting precision, recall and F1 values are found in Figure 3.6. When processing each interval we made sure to load additional time before and after so that any edge effects of the moving window would not be prevalent inside the validation interval. In every case the merging constant was set according to the rule in Eq. (3.4).

Table 3.1 – Tested parameter settings for the SLAMS detection algorithm.

Setting Value Indicator

Moving window function

Mean Square

Median Diamond

Harmonic mean Circle Window size

1 min Small

2 min Medium

3 min Large

Detection constant 2 Green

2.5 Red

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Figure 3.6 – Recall-precision plot for every parameter setting combina- tion. The curves indicate F1 score.

We noticed that the algorithm detected many short SLAMS under 1 second which had not been considered when manually identifying SLAMS. So, we reran the tests excluding any SLAMS under 1 second to see the performance on only largerSLAMS. The results are found in Figure3.7. As expected the precision increased for all settings.

The conclusions that can be drawn from studying the precision- recall plots are the following:

1. Increasing window size increases recall while slightly decreasing precision.

2. Increasing the detection constant trades recall for precision.

3. Going from left to right in {harmonic mean, median, mean}

trades recall for precision.

4. The F1 score is higher for a detection constant of 2 rather than 2.5 when excluding shortSLAMS.

5. The F1 score is in the 0.6 to 0.8 range for most settings.

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Methods | 23

Figure 3.7 – Recall-precision plot for every parameter setting combina- tion while excluding SLAMS under 1 second. The curves indicate F1 score.

Taking everything into account from this and the previous section, we conclude that the best and most well rounded algorithm is to use the median with 1 minute window and a detection threshold of 2. In this way high recall and F1 score is achieved. Then, if desired, some recall can be traded for precision by simply filtering outSLAMS with a B_max/B₀ under 2.5 as a post processing step. Having a short window decreases the chances of detecting other longer duration events which are not SLAMS, such as hot flow anomalies or short magnetosheath excursions.

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3.3 Magnetospheric classification

The initial task was to detect SLAMS in the solar wind (including foreshock). As such, a classifier was required that could separate solar wind from other data. Instead of designing the classifier solely based on expert knowledge, we let the data speak for itself by employing clustering techniques. During the iterative design process, as the classifier got better and better at isolating the solar wind, separate clusters for the magnetosphere, magnetosheath and foreshock appeared as a bi-product. Thus, the classifier could be extended for those regions as well. Table3.2lists each class with its corresponding color indicator.

Table 3.2 – Classes of the classifier.

Class Indicator

Magnetosphere(MSP) Red Solar Wind (SW) Green Magnetosheath(MSH) Blue

(Ion)Foreshock(FS) Cyan

The construction of the classifier can be separated into four steps:

1. Data collection 2. Feature extraction 3. Clustering

4. Creating the Gaussian mixture model

We will now explain these steps in further detail.

3.3.1 Data collection

A smaller dataset which is representative of typical MMSobservations was necessary to be able to quickly iterate during the design process.

Thus, we collected data in 1000 randomly chosen points when theFPI instrument was active. The type of data collected is shown in table3.3.

Initially, clustering was attempted using the ion number density, ion bulk velocity and the B-field, however, we found that to fully separate the clusters and to even get a foreshock cluster it was necessary to use the variables in table3.3.

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Methods | 25

Table 3.3 – FPI ion data used by the the classifier.

Variable Symbol Description

energy E~ Sky-map energy bin centers

energy_delta ∆ ~E Sky-map delta energies energyspectr_omni φ~ Omni-directional ion energy

spectrum

bulkv_gse ~v Bulk ion velocity vector in GSE

The energy variables are vectors of 32 dimensions or bins. When a particle of a specific energy enters the instrument, its energy is recorded to the bin with the nearest center. The bin centers and bin widths are found in the energy and energy_delta variables respectively. The energyspectr_omni variable is the differential energy flux of each bin. To get the total energy flux per bin, energyspectr_omni must be multiplied element-wise with energy_delta.

3.3.2 Feature extraction

From the collected data we extract three features. This number was chosen so that the data could be visualized easily during the design process and also because most clustering algorithms work best in lower dimensions since the data gets spread out in higher dimensions. The features were chosen based on how a human might identify the regions of the magnetosphere.

The solar wind is emitted radially from the sun and will in comparison to the other regions have a velocity vector which mostly is aligned with the Sun-Earth vector, i.e. its cone angle is small. The cone angle θ_conein GSEcoordinates of the bulk ion velocity is;

θ_cone = arccos−ˆi · ~v

k~vk (3.8)

and we define the first feature F₁ as;

F₁ = log θcone

π (3.9)

where the cone angle is in radians. This means that the range of F₁ is (−∞, 0], however, we cap it to [−7, 0] as further precision is unnecessary and will simply throw of the classifier in rare cases. The

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reason for taking the logarithm here is to increase the resolution of small cone angles which is prevalent in the solar wind.

The next two features are derived from the energy spectrum measurements. Typically, inside the magnetosphere, mainly energy flux from high energy particles are observed. In the solar wind, magnetosheath and foreshock the energy flux mainly comes from lower energy particles, although, some high energy particles are also present in the magnetosheath and foreshock. To capture these characteristics the second feature F₂ is a measure of the "mean log particle energy of the energy flux". To calculate it we first require the energy flux per bin

~Φ;

~Φ = ~φ ∆ ~E (3.10)

where denotes element-wise multiplication. We also require the logarithm of the energy centers ~Elog:

E_log,i = log E_i (3.11)

Log is used here to counteract the exponential spacing of the energy bin centers. Then, F₂ is calculated as:

F2 = P32

i=1Φ_iE_log,i P32

i=1Φ_i (3.12)

The final feature F₃ is related to F₂. It is the "mean absolute deviation of the log particle energy of the flux". It was chosen because in the magnetosheath and to some degree in the foreshock, energy flux from many particle energies can be observed. This is not usually the case in the solar wind and the magnetosphere. The absolute deviation δ ~E is calculated as:

δEi = |Elog,i− F2| (3.13)

Then the mean absolute deviation, or F3 can be calculated as;

F₃ = P32

i=1Φ_iδE_i P32

i=1Φ_i (3.14)

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Methods | 27

3.3.3 Clustering

Figure3.8shows the raw data on the F₁, F₂ and F₃ axes. Clearly three clusters can be distinguished. As we will see, they correspond to the magnetosphere, magnetosheath and solar wind. What can not be seen clearly in this figure is that there is a less dense cluster under the big rightmost cluster which is the foreshock.

Figure 3.8 – Raw data plotted on F₁, F₂ and F₃ axes.

Because of the varying densities of the clusters, it was hard to find a clustering algorithm that would do a satisfactory job. Most simple clustering algorithms, which were available in Matlab, perform well on equally dense clusters but fail otherwise. There are of course more sophisticated clustering algorithms that can handle such data but it was hard to find a working implementation for Matlab. However, after much tweaking we managed to make it work using hierarchical clustering with the ward linkage method. This is shown in Figure 3.9.

The only problem was that 9 clusters had to be used in order to fully separate the foreshock from the magnetosheath. To solve this we merge the clusters belonging to the same region, as shown in Figure 3.10. The magenta cluster partially belongs to both the magnetosphere and the magnetosheath so it was discarded.

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Figure 3.9 – Hierarchical clustering using 9 clusters.

Figure 3.10 – Merged clusters. The magenta cluster was discarded.

3.3.4 Creating the Gaussian mixture model

The final step in the construction of the classifier was to create a Gaussian Mixture Model (GMM), which can be used to classify new data. Aside from using a GMM there are of course many other ways this can be done. Initially we tried to use support vector machines, however they lacked a straightforward method to get the probability

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Methods | 29

of belonging to a certain class. Luckily, in this case the clusters ended up being roughly globular, making Gaussian distributions a very good fit. To create the GMM, we simply calculate the mean and covariance of each cluster and assign them to each component of the GMM. The mixing proportions (the scaling of each component) are set to the prior probability of each cluster. In other words, the number of points in a cluster divided by the total number of points.

The GMM can be used to infer, given a new data point, which component/distribution it is most likely to belong to. This can be done by calculating the posterior probability, i.e. the probability after new evidence (the GMM) is taken into account. For each component, it is calculated by dividing the component probability density function by the sum of all probability density functions. Aside from the posterior probability, it is also possible to extract other useful quantities from a GMM, such as the Mahalanobis distance (distance in standard deviations) to each cluster or as already mentioned, the probability density function.

In Figure 3.11 the existing points have been assigned to a component based on which one had the highest posterior probability. As can be seen, the magenta cluster is now split into the red and blue clusters.

Figure 3.11 – GMM clusters.

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3.4 Validating the magnetospheric classifier

Ideally, to validate the magnetospheric classifier a similar procedure to how theSLAMSdetection algorithm was validated should be employed.

That way, accuracy, recall and F1 score can be computed. However, this requires a validation dataset which due to time constraints could not be created. Instead, the magnetospheric classifier was validated mainly by looking at random intervals and qualitatively evaluating it. Some of these intervals can be found in appendix A. We acknowledge that the classifier is not perfect and can be improved in many ways, but we still believe that it is good enough to be a useful tool for separating magnetosheath and foreshockSLAMS. This is especially true for larger studies where the smaller number of times when the classifier is wrong get overshadowed by the vast majority of times the classifier is right. As an additional validation procedure, which indicates that the classifier is performing well, we did the following:

When processing theMMSdata, we record where and for how long a specific class was observed. The durations are then placed into bins based on their position. The bins are 2-dimensional where, in GSE coordinates, the first axis is the x-distance to the average position of the bow shock and the second axis is the distance from the x-axis. The size of each bin is 1x1 R_E. For future reference we will refer to these bins as sectors. The bow shock is modeled as an ellipse based on the formula from Burgess and Scholer (2015);

R = K

1 + cos θ (3.15)

where R is the distance to the bow shock from Earth, K = 25R_E, = 0.8and θ is the angle from the Earth-Sun vector. Here, we assume that the bow shock is axially symmetric around the x-axis. Transforming the equation to GSE coordinates gives us an expression for the x-position of the bow shock x_BS given the distance from the x-axis ρ:

x_BS(ρ) = K ±pK²+ (²− 1)ρ²

²− 1 (3.16)

Now, Figure3.12can be created which displays a map of where each class was observed. Unpainted patches are sectors that eitherMMShas not visited or where the FPI instrument was inactive. It is clear from the figure that each class is observed in the correct place, indicating

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Methods | 31

that the classifier is performing well. The only exception is the patch of foreshock that is located far behind the bow shock in the bottom left panel. Luckily, it can easily be discarded by a simple positional filter.

Figure 3.12 – Portion of time in each class. The position of the bow shock and Earth is marked in cyan.

3.5 Creating the SLAMS database

Here we will explain the how the SLAMS database was created and what it contains. To begin, FGM, FPI and MEC data in the intervals specified in table3.4 were downloaded fromMMS Science Data Center (2020). Only in these intervals doesMMSexit the magnetosphere and has a chance of observing SLAMS, according to the orbit plot browser onMMS Science Data Center(2020).

Then, two kinds of interval populations are determined, primary and secondary intervals. Primary intervals are when FPI and FGM is active and secondary is when only FGM is active. SLAMS can be detected in both types of intervals but only in primary intervals can the

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Table 3.4 – MMS1 data intervals.

Interval

2015-10-01 - 2016-01-30 2016-11-01 - 2017-05-09 2017-09-08 - 2018-05-23 2018-09-23 - 2019-06-16 2019-09-26 - 2020-06-08

magnetospheric classifier operate. 1001 primary and 915 secondary intervals were found. We discard any interval shorter than 8 minutes because that is the longest time window that will be in use. Each interval is then processed one at a time and we discard any SLAMS detected in the first and last 4 minutes since edge effects are prevalent there.

Following the conclusions in section 3.2 the SLAMS detection algorithm was initialized with the settings in table3.5.

Table 3.5 –SLAMSdetection algorithm settings.

Setting Value

Moving window function Median

Window size 1 min

Detection constant 2

Merging constant 1.5

For eachSLAMSdetected we store the following information in the database:

1. Start and stop time.

2. GSEcoordinates, interpolated to the middle of the SLAMS.

3. Mean of the B0-field inside theSLAMS.

4. Mean of the |B|-field inside theSLAMS.

5. Max of the |B|-field inside theSLAMS.

6. Mean of ^|B|_B

0 inside theSLAMS.

7. Max of ^|B|_B

0 inside theSLAMS.

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Methods | 33

Additionally, for primary intervals, when the magnetospheric classifier can operate, we store the following properties for eachSLAMS:

1. Posterior for each class.

2. Mahalanobis distance for each class.

3. Log of the probability density function.

The nearest value from the middle of eachSLAMSis taken, but to allow for flexibility when filtering we also allow for a user specified list of windows in which the mean of each property is calculated. Of course, for the third property, before taking the mean we exponentiate and then take the logarithm afterwards. For this database we use 6 windows:

{15, 30, 60, 120, 240, 580} seconds. The reason for doing this is that we noted that some signatures of SLAMS, especially when several SLAMS are closely clustered, were classified as magnetosheath even when in the middle of the foreshock. In such cases using a smoothed value can be a better indicator of where theSLAMSis located.

The SLAMS together with associated information is located in two files. One for primary intervals; SLAMS_primary.csv and one for secondary intervals; SLAMS_secondary.csv. We also store the seconds spent inside each sector for each class in search_durations_primary.txt using the format;

Format of primary search durations

= [<MSP seconds> <SW seconds> <MSH seconds> <FS seconds> <total seconds>]

and in search_durations_secondary.txt using the format:

Format of secondary search durations

= <total seconds>

Also located together with these files is database_info.txt which contains information that apply to the database as a whole. For example which spacecraft was used, which intervals was searched, the prior probabilities of the classes and the settings for the algorithm.

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3.6 Investigating SLAMS properties

In this section we describe how the database was used to investigate some SLAMS properties. We are interested in discovering relations between the Alfvén Mach number, size, duration, strength and relative strength. In this case we limit the sample size to 1000 magnetosheath SLAMS, 1000 foreshock SLAMS and 1000 control points. The SLAMS were randomly picked from the database and the control points were randomly picked when FPI was active. The Alfvén Mach number M_A for aSLAMSor control point was calculated as;

M_A= v0

v_A (3.17)

where v₀ is the surrounding flow velocity and v_A is the Alfvén velocity. We estimated v₀ as the 60 sec median of the bulk ion velocity.

v_A was calculated as;

v_A = B₀

√µ₀ρ₀ (3.18)

where µ₀ is the vacuum permeability (1.256 637 062 × 10⁻⁶H m⁻¹) and ρ₀is the density of the surrounding flow. We estimated it as the ion number density times the proton mass (1.672 621 9 × 10⁻²⁷kg). The ion number density and B₀ were, again, estimated with a 60 sec median.

The size of a SLAMS could only be estimated for a subset of the SLAMS, because only 483 magnetosheath SLAMS and 558 foreshock SLAMShad at least one FPI data point inside them. This was required since the size was estimated by multiplying the mean ion velocity inside theSLAMSby the duration of theSLAMS.

The other properties which we were interested in; duration, strength and relative strength, are already available in the database as is.

Results of this investigation can be found in the results and analysis chapter, section4.2.

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Results and Analysis | 35

Chapter 4 Results and Analysis

The main result of this thesis work is the SLAMS database which is available here: https://github.com/cfognom/MMS_SLAMS_

detection_and_magnetospheric_classification/tree/

master/SLAMS_database/identified_SLAMS_mms1_

median60. However, to get a better understanding of the database, we will in this chapter visualize it with several different plots that show where SLAMS are detected and how their properties vary by spatial and magnetosperic location. In addition, we will also show the results from section 3.6, which will give us a better understanding of SLAMS properties in general and how foreshock and magnetosphere SLAMS differ.

4.1 Visualizing the database

The database contains 98406SLAMS of which 66210 were detected in primary intervals, whenFPIinstruments were active. Table4.1contains a breakdown of how manySLAMSare detected in each class. ASLAMS membership to a class is determined by which posterior is greatest either for the nearest value or for the mean of the values inside a window. Please note: This should not be confused with the window of the SLAMS detection algorithm which always is 60 sec median.

Henceforth, when referring to SLAMS detected in a certain class, we assume a 30 sec window. We believe this is the shortest window which still captures the surrounding regions.

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Table 4.1 – Number ofSLAMSper class.

MSP SW MSH FS

Nearest 134 414 59000 6662

15 sec mean 108 414 59833 5855 30 sec mean 101 472 59862 5775 60 sec mean 75 509 59442 6184 120 sec mean 60 594 58985 6571 240 sec mean 65 844 58314 6987 580 sec mean 118 1071 57805 7216

Figure 4.1 shows the primary SLAMS colored by class in the left panel and primary and secondarySLAMS in the right panel. Indicated in the figure is also the bow shock using Eq. (3.15) from section 3.4.

It is clear that Most SLAMS are detected around the theoretical bow shock. Magnetosheath SLAMS mainly behind it and foreshockSLAMS seem to have their mean position right on it. There are also smaller amounts being detected about 5 R_E upstream of the bow shock and onward, where the FPI instrument is only active during short time periods.

Figure 4.1 – Left panel: Position of primary SLAMS by class. Right Panel: Position of primary and secondarySLAMS.

The next figure, Figure 4.2, shows how much time was spent in each sector. It is good to have this as a reference, because sectors with a low search time will be less reliable. In the right panel, the yearly revolutions of MMS orbit can be seen. The first two years are

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the bright red sectors close to Earth. During these years the apogee was low enough to never reach outside the theoretical bow shock. Year three is seen as a slight increase in search time along the bow shock when the apogee was raised. Year four and five correspond to the two outer regions.

Figure 4.2 – Left panel: Primary search time. Right Panel: Union of primary and secondary search time

In Figures 4.3 we have centered the x-axis on the bow shock, creating the normal incidence frame. The panels display the rate, strength and relative strength of both primary and secondary SLAMS.

We see that SLAMS in general are observed mostly right behind the theoretical bow shock, in rates of 20-30 per hour. Their strength increases as the distance to Earth decreases, however, the relative strength is almost constant with a slight increase near the bow shock.

There also seems to be a slight increase in relative strength closer to the x-axis. Further upstream the values get unreliable because of the small amount ofSLAMS that were detected there.

In the next set of panels, Figure 4.4, we see the same types of plots but for magnetosheath and foreshock SLAMS specifically.

Magnetosheath is shown in the left column and foreshock in the right column. The rate of magnetosheath SLAMS increases while the rate foreshock SLAMS decreases as we move upstream. It may look like magnetosheath SLAMS are observed further upstream of foreshock SLAMS, however we have to remember that it is plotted against magnetosheath/foreshock hours, not any hour. Thus, it is unlikely to observe magnetosheath 4 R_E upsteam of the theoretical bowshock, but when we do, manySLAMSare detected. We refer to Figure3.12which

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shows where each class is observed, to complement this understanding.

This indicates that SLAMS become more common as we approach the bow shock from the Sun, then once we have moved past the bow shock SLAMS become rarer as we move further into the magnetosheath.

The strength plot is similar to the previous plots which show that strength increases as we move closer to Earth. The relative strength for foreshock SLAMS seems to be rather constant, with perhaps a slight increase as we move downstream and for magnetosheathSLAMS decreases as we move downstream. MagnetosheathSLAMS also seem to be relatively stronger closer to the x-axis.

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Figure 4.3 –SLAMSrate, strength and relative strength.

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Figure 4.4 – Magnetosheath and foreshock SLAMS rate, strength and relative strength.

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4.2 Properties of SLAMS

Here we present the results from section 3.6. Figure4.5 displays his- tograms showing the Alfvén Mach number of surrounding flow, relative strength, size and duration. The total count for each class is 1000 except in the bottom left panel which only has 483 of each class due to complications explained in section 3.6. In the top left panel we see that compared to control points (random points when FPI instrument is active) which mostly are subsonic, magnetosheath and foreshock SLAMS occur mostly in supersonic flows of higher Mach numbers, especially foreshockSLAMS. The top right panel shows that the relative

Figure 4.5 – Histogram plots of variousSLAMS properties.

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strength distribution of magnetosheath and foreshock SLAMS is very similar, but higher values are more likely to occur in the foreshock.

In the bottom left we see that, although the highest count occur in the 500-1000 km bin for both types of SLAMS, foreshock SLAMS are slightly larger than magentosheath SLAMS. The final bottom right panel displays similar distributions for the durations. Similar durations but different sizes could indicate that the foreshock SLAMS are larger because they are faster. Then once they reach the magnetosheath they are slowed down and compressed, slightly decreasing their size.

To better understand the relation between size and duration we have plotted them against each other in Figure 4.6. We notice a linear relation for foreshock SLAMS and shorter magnetosheath SLAMS.

However this linearity breaks down for magnetosheath SLAMS longer than 6-7 seconds.

Figure 4.6 –SLAMSsize plotted against duration.

In Figure 4.7 we have plotted SLAMS strength, relative strength, size and duration against Alfvén Mach number. The top left panel shows that stronger SLAMS occur at lower Mach numbers, this might just be because strongerSLAMSare observed closer to Earth where the Mach number is lower. If we instead look at the relative strength, in the top right panel, we see an opposite trend. Higher Mach number means slightly strongerSLAMS relative to the background field. In the bottom left panel, we again see that foreshock SLAMS are larger, but only for Mach numbers above 4. In the bottom right panel we see that as the Mach number increases, the duration of magnetosheathSLAMS

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decreases, which perhaps is to be expected. However, that is not the case for foreshockSLAMSwhich actually increases to a peak around 12 M_A. We do not put too much effort into understanding the large peak for low Mach numbers as it might just be inflated outliers. As can be seen in the top left panel of the earlier histogram plots, Figure4.5, the sample size in the subsonic forshock bin is almost zero.

Figure 4.7 –SLAMS properties plotted against Alfvén mach number of surrounding flow.

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Automatic SLAMS detection and magnetospheric classification in MMS data

Automatic SLAMS detection and magnetospheric classification in MMS data

CARL FOGHAMMAR NÖMTAK

Automatic SLAMS detection and magnetospheric

classification in MMS data

Carl Foghammar Nömtak

Abstract

Keywords

Sammanfattning

Nyckelord

Acknowledgments

Contents

List of Figures

List of Tables

List of acronyms and abbreviations

Chapter 1 Introduction

1.1 Background

1.2 Objective

1.3 Scientific and engineering challenges

1.4 Purpose

1.5 Goals

1.6 Research Methodology

1.7 Scope

1.8 Structure of the thesis

Chapter 2 Background

2.1 The magnetosphere

2.2 The foreshock

2.3 Short Large-Amplitude Magnetic Struc- tures (SLAMS)

2.4 Geocentric Solar Ecliptic (GSE) coordi- nates

2.5 The magnetospheric multiscale (MMS) mission

2.5.1 Orbit and phases

2.5.2 Instruments

2.6 Related work

Chapter 3 Methods

3.1 Detecting SLAMS

3.1.1 Estimating the background field

3.1.2 Detection

3.1.3 Merging

3.2 Validating the SLAMS detector

3.3 Magnetospheric classification

3.3.1 Data collection

3.3.2 Feature extraction

3.3.3 Clustering

3.3.4 Creating the Gaussian mixture model

3.4 Validating the magnetospheric classifier

3.5 Creating the SLAMS database

3.6 Investigating SLAMS properties

Chapter 4

Results and Analysis

4.1 Visualizing the database

4.2 Properties of SLAMS