ISTTOK Plasma Tomography Using Minimum Fisher Regularization

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ISTTOK Plasma Tomography Using Minimum Fisher

Regularization

Daniel Hachmeister Ferreira da Costa

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisor(s):

Prof. Horácio João Matos Fernandes

Prof. Diogo Manuel Ribeiro Ferreira

Examination Committee

Chairperson: Prof. João Pedro Saraiva Bizzaro

Supervisor: Prof. Horácio João Matos Fernandes

Member of the Committee: Prof. Carlos Alberto Nogueira Garcia da Silva

Resumo

Foi desenvolvido um algoritmo de tomografia de plasma para o tokamak ISTTOK. O algoritmo ´e uma aplicac¸ ˜ao da Regularizac¸ ˜ao de M´ınimo de Fisher e foi implementado e distribu´ıdo como um pacote em python. A tomografia de plasma ´e um problema de invers ˜ao mal condicionado. Os algoritmos de reconstruc¸ ˜ao tentam superar esta adversidade introduzindo conhecimento a priori sobre a soluc¸ ˜ao, o que requer ajuste emp´ırico. Em geral, estes algoritmos s ˜ao validados com fantomas artificiais ou com informac¸ ˜ao proveniente de outros diagn ´osticos. Neste trabalho, desenvolveu-se um aparto ex-perimental que permite o uso de fantomas f´ısicos para calibrar e validar o algoritmo de reconstruc¸ ˜ao implementado, permitindo tamb ´em a comparac¸ ˜ao entre duas implementac¸ ˜oes diferentes no que toca `a descric¸ ˜ao matem ´atica da amostragem espacial. Uma poss´ıvel aplicac¸ ˜ao do diagn ´ostico tomogr ´afico ´e demonstrada calculando a posic¸ ˜ao do plasma e observando o desvio de Shafranov.

Palavras-chave:

Abstract

A plasma tomography algorithm was developed for the ISTTOK tokamak. The algorithm is an instance of the Minimum Fisher Regularization and was implemented and distributed as a python package. Plasma tomography is an ill-conditioned inversion problem. Reconstruction algorithms try to overcome this issue by introducing some form of a priori knowledge that requires empirical tuning. In general, to validate the implementation of these algorithms, either artificial phantoms are used, or one must rely on information provided by other diagnostics. In this work, an experimental setup was developed that allows the use of physical phantoms to tune and validate the reconstruction algorithm used. This also allowed the comparison of two different implementations of the algorithm regarding the mathematical description of the spatial sampling. A possible application of the tomographic diagnostic is demonstrated by computing the plasma position and observing the Shafranov shift.

Keywords:

Contents

1.2 Plasma Emissivity . . . 2

1.2.1 Cyclotron Radiation . . . 3

1.2.2 Bremsstrahlung . . . 3

1.2.3 Electron-Ion Recombination and Electronic Transitions . . . 4

1.3 Objectives . . . 4

1.4 Thesis Outline . . . 5

2 Plasma Tomography 7 2.1 The Tomography Problem . . . 7

2.2 Etendue . . . .´ 8

2.3 Reconstruction Algorithms . . . 9

2.3.1 Least Squares Fitting . . . 10

2.3.2 Tikhonov Regularization . . . 10

2.3.3 Minimum Fisher Regularization . . . 11

2.3.4 Alternative Methods . . . 12

3 Proposed Methods 15 3.1 Line of Sight and Volume of Sight . . . 15

3.2 Algorithm Flowchart . . . 17

3.3 Camera Calibration . . . 19

3.4 Validation with Physical Phantoms . . . 19

4 Application to ISTTOK 21 4.1 ISTTOK Pinhole Cameras . . . 21

4.2 Algorithm Implementation . . . 24

4.2.1 Geometry Matrix . . . 24

4.2.2 Regularization Strength . . . 25

4.2.3 Boundary Conditions . . . 27

4.3 Physical Phantom Reconstructions . . . 27

4.4 High Field Side Activity . . . 29

4.5 Plasma Position . . . 30 4.6 MHD Activity . . . 31 5 Conclusions 35 5.1 Contributions . . . 35 5.2 Future Work . . . 36 Bibliography 37 A Technical Datasheets 41 A.1 Photodiodes AXUV20ELGDS . . . 41

A.2 Camara Circuit Schematic . . . 46

List of Figures

1.1 Bremsstrahlung radiation profile computed using equation 1.2 with typical ISTTOK values (ne= 3 × 1018m−3, Te= 150eV, Zef f = 1.5, ¯gf f = 2). . . 4

2.1 Optical system formed by the pinhole and photodiode. . . 9

3.1 A simplified version of the LoS calculation. Each square is simultaneously a pixel on the reconstructed image and an entry of matrix Ti. The grayscale represents the weight of that pixel to the measurement of a given sensor. The dashed line represents the LoS of that sensor. . . 16 3.2 Pinhole and sensor system. The solid angle Ω subtended by surface S dictates what

fraction of light reaches the detector emitted from point P . . . 16 3.3 Graphical representation of matrix Ti. On the bottom is the line of sight. On top is the

volume of sight overlayed with the integration along yy which reduces to the line of sight approximation. . . 17 3.4 Flowchart of the outer loop of the Minimum Fisher Regularization. . . 18

4.1 (a) Uncovered camera showing the photodiode array mounted on top of the printed circuit board. (b) Camera with the pinhole plate in place. . . 22 4.2 Cross sectional view of the ISTTOK port used by the tomography diagnostic and the

available lines of sight (LoS) . . . 23 4.3 Voltage readings from the sensors versus the expected ´etendue for the outer and top

cameras. . . 23 4.4 Representation of matrices Tion a color scale. From left to right: matrices T1−16, T17−32

and T33−48. Each matrix Ti contains one single line of sight but multiple matrices are represented per figure. . . 24 4.5 VoS of the sixth sensor of the outer camera with (a) 0.8mm of pinhole diameter and (b)

0.5mm of pinhole diameter. The white lines represent the border of the VoS. . . 25 4.6 Representation of matrices Tion a color scale. From left to right: matrices T1−16, T17−32

and T33−48. Each matrix Ticontains one single volume of sight but multiple matrices are represented per figure. . . 26

4.7 Plasma reconstructions made with (a) volume of sight matrix and (b) line of sight matrix. Bar plots show, for each camera, the signals measured by the detectors and the retrofit signals computed with equation (2.9). . . 26 4.8 Reconstructed emissivity with different weights of the norm outside the vessel represented

by the white circle. . . 27 4.9 (a) Cold cathode lamp and supporting structure inside the vessel replica; (b) real vs.

ex-pected signals from the VoS and LoS geometries. . . 28 4.10 Single sensor illumination a) & b) vs. multiple sensor illumination c) & d) for the LoS and

the VoS approximations. Bar plots show, for each camera, the signals measured by the detectors and the retrofit signals computed with equation (2.9). . . 29 4.11 (a) Line-averaged emissivity for typical ISTTOK operation; (b) Lines of sight for the top

and bottom cameras (numbered clockwise). . . 30 4.12 Plasma displacement in the horizontal (top) and vertical (bottom) directions. . . 31 4.13 Fluctuations of the plasma toroidal current, responsible for generating the poloidal

mag-netic field. Taken from [28]. . . 31 4.14 Spectrogram of the signal from the first sensor in the outer tomography camera (top), and

for one of the magnetic probes (middle). Plasma current (bottom). Shot #47309. . . 32 4.15 . . . 32

Glossary

COMPASS COMPact ASSembly tokamak.

DEMO DEMOnstration Power Station.

HFS High Field Side.

HT-7 Hefei Tokamak-7.

IL Innermost Loop or just Inner Loop of the Mini-mum Fisher Regularization algorithm.

ISTTOK Instituto Superior T ´ecnico TOKamak.

ITER International Thermonuclear Experimental Re-actor.

JET Joint European Torus.

LoS Line of Sight or the plural, Lines of Sight.

MFR Minimum Fisher Regularization is a widespread method for plasma tomography that relies on minimizing the Fisher Information of the plasma emissivity profile.

MHD Magnetohydrodynamics.

ML Maximum Likelihood.

OL Outermost Loop or just Outer Loop of the Mini-mum Fisher Regularization algorithm.

SSIM Structural Similarity.

SXR Soft x-rays.

TCV Tokamak `a Configuration Variable.

UV Ultraviolet.

VoS Volume of Sight or the plural, Volumes of Sight.

WEST Tungsten (chemical symbol W) Environment in Steady-state Tokamak.

Chapter 1

Introduction

ISTTOK ”Instituto Superior T ´ecnico Tokamak” is a toroidal research reactor for nuclear fusion by mag-netic confinement (tokamak) based at Instituto Superior T ´ecnico [1].

In a tokamak, hot plasma is confined by applying powerful magnetic fields [2]. Magnetic confinement works on the principle that charged particles (ions and electrons) that compose the hot plasma describe circular trajectories around the magnetic field lines. By establishing a magnetic field inside the tokamak, particles get trapped in their circular orbits preventing them from leaving the tokamak. The plasma inside the tokamak is then heated until the point nuclear fusion starts to take place.

Because of the extreme temperature necessary for fusion, it is essential that the plasma does not come in contact with the walls of the tokamak vessel, otherwise the plasma will cool down and damage the vessel walls. Maintaining the plasma well confined and positioned has to be an active procedure because the plasma has instabilities that naturally tend to disrupt confinement. Because of this, a toka-mak requires a set of tools to correctly determine the plasma characteristics, e.g. position, temperature, density, etc. These tools are generally referred to as diagnostic tools.

One of the diagnostics available at ISTTOK is the magnetic positioning diagnostic. This diagnostic measures the magnetic field generated by the plasma to determine its position inside the tokamak. This information is then used to control the plasma position in real-time by adjusting the magnetic confine-ment fields. Nevertheless, the magnetic diagnostic is subject to uncertainties and can ultimately have systematic errors of calibration and lead to the misdiagnosis of plasma position. It makes sense then to have more than one diagnostic to determine the same information through different methods, for redundancy.

An alternative to measuring the plasma position magnetically is doing so optically by ”looking” directly at the radiation emitted by the plasma to infer about its position. In practice, this is a challenging task because the complexity of these machines places mechanical constraints in accessing its interior. A way of optically determining position is through tomography.

In medical procedures, tomography is used to reconstruct images of slices of the human body from multiple projection radiographs [3].

Inspired by this, plasma tomography reconstructs the distribution of emissivity1 on a slice of the tokamak by measuring the total radiation emitted along lines that cross the reconstructed plane, the so-called lines of sight (LoS). Contrary to medical tomography where an arbitrary number of radiographs can be taken to yield high-resolution images, access to the tokamak places physical restrictions on the number of LoS available. For this reason, plasma tomography has been forced to apply ingenious methods that compensate for the lack of information. These usually rely on a priori expected plasma behavior that is enforced on the reconstructed images.

1.1

Plasma Tomography in Tokamaks

Although determining plasma position is a common use of plasma tomography, it is not its only applica-tion. Plasma tomography applications are as varied as the types of radiation emitted. In this section, we take a look at the most common applications of plasma tomography and the types of radiation relevant in fusion plasma applications.

Plasma tomography branches into several different applications that share the mathematical formal-ism but differ in terms of the radiation being observed.

Soft x-ray (SXR) radiation comes mainly from the hot plasma core where the gas is totally ionized and is especially useful for studying magnetohydrodynamic (MHD) activity and measuring plasma posi-tion [4].

In bolometry, a larger portion of the electromagnetic spectrum is observed (typically from x-rays to infrared). This diagnostic is ideal for measuring the total energy lost by the plasma in radiative pro-cesses [5]. Bolometry can also be used for impurity transport studies since these impurities emit a detectable amount of radiation in the visible and UV spectrum [6].

In fusion-oriented reactors such as JET, ITER and DEMO, analysis of neutron emission can also be performed using tomography which is useful for measuring fusion power output and tritium transport [7]. Plasma tomography is a non-invasive tool that can probe the plasma morphology and composition and thus it is extremely relevant for fusion applications were invasive physical probing is often not an option.

For a broader overview of plasma tomography and its applications see reference [8].

1.2

Plasma Emissivity

Being composed of charged particles, a plasma will interact with electromagnetic fields as well as pro-duce electromagnetic radiation, which is generically referred to as plasma emission. Plasma emission itself can be subdivided and classified according to the physical interaction underlying the radiative pro-cesses. The following section discusses the processes involved in radiation emission in fusion relevant plasmas.

1.2.1

Cyclotron Radiation

Fusion plasmas are usually permeated by a background magnetic field. A charge moving in a magnetic field feels a force perpendicular to its velocity resulting in a circular motion. This accelerated charge will then emit what is usually referred to as cyclotron radiation. The frequency of this radiation is given by

ω = nqB

m (1.1)

with q and m being the charge and mass of the particle, B the magnetic field strength, and n the harmonic number [9].

In typical tokamaks, the magnetic field is between 2T < B < 8T meaning that cyclotron radiation is emitted in a frequency range of 60GHz − 600GHz. Cyclotron radiation emitted by electrons is commonly used to measure the electron temperature [10].

When it comes to tomography, one needs not worry about this source of radiation because: (i) the plasma is optically thick to cyclotron radiation, (ii) the fraction of power radiated by cyclotron emission is small when compared to the remaining radiative processes, (iii) detectors used in tomography are sensitive to the visible, ultraviolet and x-ray spectra and are not sensitive to the wavelengths produced by cyclotron emission which are in the microwave range.

1.2.2

Bremsstrahlung

While in motion, an electron experiences forces due to the presence of other charged particles. When the electron is accelerated due to these forces it emits radiation. This type of radiation is called bremsstrahlung, german for ”braking radiation”. In the context of magnetic fusion energy (MFE) plasmas, the power radi-ated by bremsstrahlung per unit volume per unit solid angle per unit photon energy is given by [11]

εν = 1.2 × 10−39 n2_eZef f¯gf f √ Te e−ETe _(Wm−3_sr−1_eV−1_{) (1.2)} with • ne=electron density (m−3) • Te=electron temperature (eV) • E = photon energy (eV)

• Zef f =plasma effective charge number

• ¯gf f = Maxwellian-averaged free-free Gaunt factor, (accounts for quantum-mechanical correc-tions). Typically between 1-5 from soft x-rays to the visible spectrum

Figure 1.1 shows the plasma bremsstrahlung profile under typical ISTTOK conditions. Integrating over the entire frequency spectrum of bremsstrahlung photons yields a total power of roughly 5Wm−3. This calculation is not accurate but gives us an order of magnitude for comparison with other radiative processes.

100 ₁₀1 ₁₀2 ₁₀3

photon energy (eV) 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 ra dia te d po we r ( W m 3sr 1eV 1)

Figure 1.1: Bremsstrahlung radiation profile computed using equation 1.2 with typical ISTTOK values (ne= 3 × 1018m−3, Te= 150eV, Zef f = 1.5, ¯gf f = 2).

1.2.3

Electron-Ion Recombination and Electronic Transitions

In a fusion plasma, inelastic collisions between electrons and ions are also an important source of radiation. The most relevant interactions in terms of radiative significance are:

• Radiative recombination: A free electron is captured to a bound state of an ion releasing the excess energy as quanta with a continuous spectrum with a lower threshold on the electron binding energy. The final state of the ion is usually an excited state that is followed by radiative decay, emitting quanta distributed on a discrete spectrum.

• Dielectronic recombination: The available energy from the free-electron capture is absorbed by a bound electron that is promoted to a higher energy state. The ion then undergoes radiative decay to the ground state [12].

• Ion excitation by electron collision followed by radiative deexcitation to the ground level.

The recombination processes give rise to both a continuous and discrete spectrum of emission. On the other hand, excitation by electron impact always gives rise to a discrete spectrum. The discrete part of the spectrum is often used to measure ion-related quantities such as density and temperature. Some of the most intense spectral lines at ISTTOK are listed in table 1.1.

Ionization State W I H I O II He I O II C III H I C II O III H I C II

λ(nm) 407.4 434 441.6 447.1 459.3 464.7 486.1 514.5 559.3 656.27 678.39

Table 1.1: Most intense spectral lines in tokamak ISTTOK.

1.3

Objectives

The goal of this work is to implement a tomography diagnostic for ISTTOK based on the currently avail-able cameras. The algorithm of choice for the reconstruction will be the Minimum Fisher Regularization (MFR) algorithm, which is perceived as one of the leading algorithms for plasma tomography in toka-maks [8].

A demanding part of implementing a tomography diagnostic is its calibration and testing. Part of this work will thus be the calibration and validation of the diagnostic.

The importance of implementing a tomography diagnostic is that it unlocks the possibility of studying different research areas, namely:

• Detection of MHD modes with tomography - MHD modes are magnetic perturbations that should manifest in a deviation from the circular shape of the plasma cross-section [13].

• Study of impurity transport - sometimes heavy impurities are released from the tokamak wall onto the plasma. These manifest as a very bright but small localized source of light over the plasma background.

• Study of the plasma current inversion - Being one of the few tokamaks to have alternate current (AC) discharges, it is the perfect test bench for studying how the plasma behaves during inversion, something that has only briefly been explored at the HT-7 tokamak [14].

1.4

Thesis Outline

Chapter 2 starts with a general introduction to plasma tomography narrowing down to the methods used for this work in particular. It discusses key concepts such as ´etendue, local basis decomposition, the geometry matrix, cost functionals, and Fisher information.

Chapter 3 proposes a methodology for implementing a tomography inversion routine. It details some of the more intricate parts of solving the tomography problem and also how to perform a good calibration and testing of the tomography diagnostic.

Chapter 4 explains how the theory and methods described in chapters 2 and 3 were applied to the ISTTOK tokamak. It reports on the hardware calibration, the algorithm implementation, testing, and finishes with a couple of example usages of the diagnostic.

Chapter 5 highlights the contributions of this work and finishes with some concluding remarks and suggestions for future work.

Chapter 2

Plasma Tomography

This chapter introduces some of the mathematical formalism used in plasma tomography. We start from the general statement of the tomography problem and then proceed to detail the algorithm used to solve it in the context of this work.

The algorithm of choice for this work, the Minimum Fisher Regularization (MFR), is described in section 2.3.3 and was first proposed by Anton et. al. in 1996 [15].

2.1

The Tomography Problem

Abstractly speaking, the tomography problem consists of determining the best approximation to a 2D scalar function, knowing only the value of a finite number of line integrals of this function, and taking some assumptions regarding the function, e.g. boundaries and smoothness. In practice, we want to determine the plasma emissivity profile on a cross-section of the tokamak having the measurements from multiple light sensors placed around its cross-section. These sensors are usually small and have a small viewing angle. We can think of the volume seen by each sensor as a thin cone which will be referred to as the volume of sight (VoS).

In most applications, optical sensors for tomography are implemented using a photo-sensitive ele-ment (a photo-diode or bolometer) covered by a pinhole to create the required narrow VoS1_.

If we assume that the plasma radiates isotropically, the local radiated power can be defined as a scalar function g(r, ν), which is a function of position r and frequency ν.

The output of sensor i will be a measurement of the total radiation that reaches it from inside its VoS. At each position r, light is directed towards the sensor along a solid angle Ωi(r). Because no light reaches the sensor from outside the VoS we have that Ωi(r) = 0 for these positions. Note also that sensors can have a frequency-dependent response η(ν). Taking all this into account, the output fi of each sensor i will be:

fi= Z VoSi Z ∞ ν=0 Ωi(r) 4π ηi(ν)g(r, ν)dνdr (2.1)

Where the spatial integral only needs to be taken inside the VoS of each sensor. Because of the

frequency-dependent sensitivity, each sensor will measure the convolution of the incident light with its frequency response. We define this quantity as the perceived emissivity G(r):

fi= Z VoSi Ωi(r) 4π G(r)dr , G(r) = Z ∞ ν=0 η(ν)g(r, ν)dν (2.2)

If the VoS of a detector is a cone narrow enough, we can assume that the emissivity g(r, ν) is constant in a surface (A) perpendicular to the axis (S) of the cone. The line S is referred to as ”line of sight” (LoS). The previous equation can be rewritten replacing the volume element dr with an area element times a line element dAds:

fi= Z Si G(r) Z A Ωi(r) 4π dAds , Z A Ωi(r) 4π dA = ci (2.3)

The second equality stems from the conservation of the ´etendue (see section 2.2). One way to think of this result is that, as the distance to the sensor increases, the solid angle subtended by the sensor decreases but the total area A increases. The conservation of ´etendue states that these effects cancel out exactly. We get the final expression:

fi= ci Z

Si

G(r)ds (2.4)

This simplified expression reveals how each sensor effectively measures a line integral of the (perceived) emissivity weight by the ´etendue of the sensor plus pinhole system.

2.2

Etendue

´

” ´Etendue” comes from the French vocabulary and translates into ”extent”. It is a measure of how spread out is light in terms of area and angle while traveling in an optical system [16]. Since most tomography systems employ pinhole cameras, this is the framework that will be used to explain the concepts of

´etendue and conservation of ´etendue, which are central in plasma tomography.

Figure 2.1 shows a typical pinhole set up where a surface A is defined at a certain distance from the sensor surface S. Both surfaces are separated by the pinhole, which is an opaque plane with a small hole.

Light that radiates from the area element dA and that passes through the pinhole casts a shadow P on the sensor plane. The infinitesimal ´etendue is given by the product of the effective area cos θdA with the solid angle dΩ.

dU = cos θdAdΩ, (2.5)

and in turn

dΩ = cos φdS/d2 (2.6)

where d is the distance between infinitesimal areas dA and dS.

A dA θ dΩ P dS S φ

Figure 2.1: Optical system formed by the pinhole and photodiode.

that go through the pinhole. Integrating over dΩ is equivalent to integrating over the intersection between surfaces S and P. U = Z A Z S∩P

cos θ cos φ dSdA

d2 (2.7)

It can be shown that for such a system, the ´etendue U is conserved throughout the light path [16]. This means that the choice of surface A is arbitrary, as long as it includes all points from where light can reach the sensor.

In terms of plasma tomography, the conservation of ´etendue means that if the plasma is closer to the camera the light reaching the sensors is more intense but comes from a smaller area. On the other hand, if the plasma is further away from the camera, its light is dimmer but the area seen by the camera is larger. These effects cancel out exactly, which motivates the simplification done in equation 2.3.

Since the choice of surface A is arbitrary, measuring the ´etendue of each sensor can be achieved by using a uniform and isotropic light-emitting surface. This will be explored later on when discussing the calibration of the tomography diagnostic.

2.3

Reconstruction Algorithms

The goal of tomography is to start from the line integrals fi(from equation 2.4) and determine the emis-sivity function G(r) restricted to a 2D cross-section of the tokamak. However, this problem is impossible to solve exactly unless one has access to an infinite set of such line integrals. The idea here is to find a good approximation of G(r) given the reduced number of line integrals we have, due to constraints in placing sensors on the tokamak.

The algorithms described next rely on decomposing the emissivity distribution G(r) into a local basis. More specifically, G(r) will be discretized into pixels in an image, and producing said image is the final goal of the reconstruction algorithm.

2.3.1

Least Squares Fitting

Let us start with discretizing the function G(r) into an r × c matrix G whose entries are pixels of the plasma cross-section we want to image. Then, for each sensor i, we define another r × c matrix Ti. Every element of matrix Ti contains the length of the intersection between a given pixel and the line of sight (LoS) of sensor i. The line integral in equation 2.4 is then approximated by summing over the element-wise product between the matrices G and Ti. If we reshape each matrix into a single row vector we can also write

fi= tTi · g (2.8)

where ti and g are the vector versions of Ti and G and (·) denotes the usual matrix multiplication. We now define the vector f composed of the sensor measurements fiand the matrix T whose rows are the vectors ti. With these definitions, we can write the previous equation for all sensors at once.

f = T · g (2.9)

Here, T is an s × p matrix where s is the number of sensors and p is the number of pixels of the reconstructed image. This matrix is an important concept in tomography and is often referred to as the geometric or contribution matrix. The details of computing matrix T are given in section 3.1. For now, one can think of it as a transformation that maps each possible emissivity distribution into a set of sensor measurements.

The tomography problem consists in determining vector g for a certain set of measurements f and given the geometry matrix T. Solving g for f cannot be done by inversion of the geometry matrix T because it is always non-invertible or ill-conditioned. If s is larger than p, the problem can be solved by least-squares fitting minimizing the χ2_{parameter [17].}

χ2= ( ˜T · g − ˜f )T· ( ˜T · g − ˜f ) (2.10)

Here, ˜Tij = Tij/σiand ˜fi = fij/σi where σiis the standard deviation of the measurement of sensor i. A rule of thumb for accuracy is having 1.5 times any many sensors as pixels (s/p = 1.5). On ISTTOK with a set up that consists of 48 LoS, this would lead to a minimum pixel width of at least 15 millimeters.

2.3.2

Tikhonov Regularization

If there are more pixels than 2/3 the number of sensors, a least-squares fitting should not be used. In fact, for p > s the least-squares solution will always be overfitted to the data. This overfitting can be

avoided if the function is required to have a specific shape or behavior. The requirement is introduced in the form of an additional term in the cost function that penalizes solutions that drift from the required behavior.

φ = χ2+ αR (2.11)

Here, R is the regularizing functional that is positive and large for ”bad” solutions and α is the regular-ization parameter that determines the relative importance of the regularregular-ization. To ensure positivity, R can be written as a quadratic form

R = ||Γ · g||2 _(2.12)

where Γ is a linear operation on vector g and || · || is the euclidean norm. Introducing matrix H = ΓT_Γ, equation 2.11 can be rewritten as

φ = χ2+ α gT· H · g (2.13)

where H is a symmetric matrix that defines the quadratic form R. The solution that minimizes φ is given by:

g = ( ˜TT· ˜T + αH)−1· ˜TT· ˜f (2.14) In general, the regularization is chosen as to impose some sort of smoothness that is expected in plasma behavior. The zeroth-order regularization corresponds to minimizing the norm of the solution.

R = ||g||2_{= g}T_{· g , H = I ≡}

Identity (2.15)

The first-order regularization accounts for the minimization of gradients.

R = ||∇xg||2+ ||∇yg||2 , H = ∇Tx· ∇x+ ∇Ty · ∇y (2.16)

Where ∇x(y)are matrices that represent the finite difference operators corresponding to the gradient in the x(y) directions. The second-order regularization minimizes the curvature of the solution.

R = ||∆g||2 , H = ∆T· ∆ (2.17)

Where ∆ is the appropriate matrix form of the Laplace operator.

2.3.3

Minimum Fisher Regularization

Minimum Fisher Regularization (MFR) works by imposing a strong smoothness where g is small (low plasma emissivity) but allowing g to be less smooth where it is large. It does so by minimizing the Fisher information defined in 1D as:

IF = Z ₁ g(x)  ∂g(x) ∂x 2 dx (2.18)

Contrary to the previous regularizations, MFR leads to a matrix H that depends on g. Because of this, it has to be solved iteratively.

g(n+1)= ( ˜TT· ˜T + αH(n))−1· ˜TT· ˜f (2.19) Matrix H is similar to the one defined in equation 2.16 for the first-order regularization, but now the gradient at each point has to be weighted by the corresponding value of g

H(n)= ∇T_x· W(n)_{· ∇}

x+ ∇Ty· W (n)_{· ∇}

y (2.20)

Where W is a diagonal matrix whose diagonal elements are the inverse of the elements in vector g. To avoid division by too small numbers the values in g should be clipped to a small positive number e.g. 10−10. The iterative process of solving the MFR method can be started with W(0) _{≡ I, which in} turn means that the first iteration is simply the result with the regularization introduced by equation 2.16. For a large enough regularization constant, the iterative process described by equation 2.19 will quickly converge to its fixed point (usually well-within 10 iterations)[8].

Research regarding MFR has provided multiple ways to improve its quality and performance among which are:

• Anisotropic smoothing [18]. This method applies a preferential smoothing of the emissivity profile along the magnetic flux surfaces. Since this is the expected behavior of the plasma, the recon-struction results are physically more likely to be correct. Anisotropic smoothing can introduce arti-facts if the plasma drifts from this behavior or if the magnetic flux surfaces are wrongly computed. Nevertheless, it is recommended to use this technique [8].

• Fast MFR. Improvement in performance is extremely relevant because: (i) MFR can take a long time to run, making it difficult to do full pulse reconstructions. (ii) A method that can work in real-time is desirable, e.g. to monitor fusion power output using neutron tomography or for real-real-time control applications. One of the developments in this area is the concept of rolling iteration in which the convergence criteria are relaxed and the initial guess for the iterative process at each time step is the reconstruction from the previous one [19].

2.3.4

Alternative Methods

In current tokamak tomography research [8] three methods stand out. The minimum Fisher regulariza-tion (MFR), the maximum likelihood method (ML) and the approach using neural networks.

The MFR has been widely tested [8] and has proved its robustness several times, e.g. TCV [15], COMPASS [4] and WEST (former Tore Supra) [20]. The quality reported in the literature, along with the relative simplicity and the continued development of this method, make MFR the natural choice for the initial implementation of a tomography diagnostic at ISTTOK.

An alternative to the MFR is the maximum likelihood tomography [5]. This is also an iterative non-linear method that tries to find the emissivity profile that is most probable given the measurements in the

sense of the maximum likelihood. One of the advantages of this method is that it allows the estimation of uncertainties in the reconstructed profile [8]. A disadvantage is that it is significantly slower than MFR. Yet another possible approach to this problem is the use of neural networks. Starting from recon-structions performed by tested algorithms such as MFR or ML, a neural network can be trained to output a reconstruction taking as input the measurement data. Such neural networks have successfully been developed and perform significantly faster than the two previous methods [6]. This is thus a good candidate for real-time applications but carries the drawback of needing a good training set to perform accurately.

Chapter 3

Proposed Methods

The last chapter taught us the fundamentals of plasma tomography algorithms but overlooked, for ex-ample, the details of computing the geometry matrix T from equation 2.9.

This chapter begins by explaining two different methods to compute the matrix T. We will then de-scribe a method to calibrate pinhole cameras and a method to test and validate a working reconstruction algorithm.

3.1

Line of Sight and Volume of Sight

The geometric matrix T from equation 2.9 can be computed in several ways. Equation 3.1 shows the shape of matrix T that is composed of elements tij. The goal of computing the geometry matrix is to assign a value to each element such that tijis the relative contribution of pixel j to the measurement of sensor i. T = . . . . .. . ... ... ... ... ... ... . . . tij . . . . .. . ... ... ... ... ... ... . . . .                   jpixels isensors (3.1)

The simplest approach is to assume that tij is the length of the intersection between the line of sight of sensor i and pixel j. This is illustrated in figure 3.1 that shows an example of the intersection between a line of sight and a pixel map. Figure 3.1 is thus a graphical representation of matrix Ti that was discussed in section 2.3. This matrix is then reshaped into a row vector to form one row of matrix T. Finally, each row of matrix T, which corresponds to a certain sensor, has to be multiplied by the corresponding ´etendue which can be computed from equation 2.3.

A more complete description needs to account for the finite dimensions of each sensor and the pinhole. Each sensor measures light coming from points inside a given volume of sight (VoS). Each point inside the VoS contributes differently to the total measurement of the sensor depending on the

Figure 3.1: A simplified version of the LoS calculation. Each square is simultaneously a pixel on the reconstructed image and an entry of matrix Ti. The grayscale represents the weight of that pixel to the measurement of a given sensor. The dashed line represents the LoS of that sensor.

solid angle subtended by the sensor as seen from that point. This reasoning is shown in figure 3.2. The fraction of light emitted from point P that reaches the detector corresponds to the fraction that is emitted along the solid angle Ω. This is the solid angle at point P subtended by surface S (yellow).

pinhole lid _sensor

point P Ω

S

Figure 3.2: Pinhole and sensor system. The solid angle Ω subtended by surface S dictates what fraction of light reaches the detector emitted from point P .

In general, the solid angle at point P , subtended by surface S is given by

Ω = Z Z

S ˆr · ˆn

r2 dS (3.2)

where ˆr = r/ris the unit vector that points from P to the surface element dS, and ˆnis the unit vector normal to the surface pointing away from P . This calculation can be carried out for every point in space and then used to compute matrix T.

Because of the tomography geometry, all the VoS are typically contained within a thin slab of the tokamak centered at the reconstruction plane. We begin by computing the solid angle at each point inside the slab. Dividing this number by 4π gives us the fraction of light that reaches a given sensor from that point. This number is a dimensionless quantity, commonly referred to as spat (sp). One sp is equivalent to 4π steradians. If it is assumed that the plasma is axisymmetric, at least inside the slab that contains the VoS, an integral can be taken in the toroidal coordinate leaving us with a number that has dimensions sp · m. The values obtained by this process can be put into the Timatrix after taking an average inside each pixel in the reconstruction.

of sight vertically symmetric. The units are sp.mm for the VoS and sp.mm2_{for the LoS matrices.}

If the LoS and VoS are correctly computed, then we should be able to verify the equality set by equation 2.3. This means that the integral of the VoS along a given direction should be constant and equal to the ´etendue which is the value stored in the LoS matrix. This a good sanity check since the two matrices are computed by different methods. Figure 3.3 shows the integration along the yy direction overlayed with the central axis of the VoS.

50 0 50 x (mm) 50 0 50 y (mm) 105 104 103 Ma tri x Ti / sp .m m (V oS ) & sp .m m 2 (L oS )

Figure 3.3: Graphical representation of matrix Ti. On the bottom is the line of sight. On top is the volume of sight overlayed with the integration along yy which reduces to the line of sight approximation.

The method here described for computing the VoS matrix has been developed independently, but it was later found that a similar approach had already been suggested [21] although not as thoroughly explained as in this work.

An alternative to explicitly computing the solid angle for each point in space is to use ray tracing. In ray tracing, one follows the path of the light rays that reach each detector. This has the added advantage of possibly including reflections on the vessel walls [22].

3.2

Algorithm Flowchart

In section 2.3.3 we discussed how the Minimum Fisher algorithm consists of iteratively solving equa-tion 2.19 that is repeated here for convenience. This secequa-tion will describe in detail the algorithm that was implemented.

g(n+1)= ( ˜TT· ˜T + αH(n))−1· ˜TT· ˜f (3.3) The Minimum Fisher algorithm is composed of two nested loops.

The innermost loop (IL) consists of solving equation 3.3 which can be thought of as a fixed point method. The inner loop usually converges within 10 iterations or does not converge at all, depending on the choice of the regularization constant α, i.e. for too small values of the regularization constant the tomography problem recovers its ill-conditioned nature and the IL does not converge.

The outermost loop is responsible for finding the α that yields the best reconstruction in the since of the χ2criteria established by equation 2.10. If the standard deviations of the sensors are well accounted for, then χ2 _{should equal the number of measurements. The value of χ}2 _{is usually a monotonically} increasing function of α and the role of the outer loop (OL) is to find the value of α for which χ2_{= s, with} sthe number of measurements.

Figure 3.4 shows the flowchart of the outer loop of the Minimum Fisher algorithm. The inner loop (IL) is hidden in the blocks with the statement ”IL converged?”. For every value of α, equation 3.3 is solved until it either converges to its fixed point, diverges, or reaches an established limit of iterations.

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3.3

Camera Calibration

As discussed previously in section 2.1, each sensor has a specific ´etendue that depends on its relative position to the pinhole, which is a measure of ”how much light” reaches that sensor. By the conservation of the ´etendue, if a homogeneous and isotropic light-emitting surface illuminates the pinhole camera, the distance between the two will not matter, as long as the surface occupies the entire field of view of the camera.

In this work, to accurately measure the ´etendue of each sensor, it is proposed the use of a uniform and isotropic light-emitting surface that can be easily achieved using a light source and a diffuser. This surface should be placed in front of the detector occupying the entire field of view of the sensor(s) to be calibrated.

The measurements obtained can be seen as the relative ´etendue between sensors and should be used to correct the geometry matrix. Furthermore, the measured values will also take into account sensor dependent calibration factors such as sensitivity and angular dependent efficiency.

As a sanity check, the measured values should be compared against the ones predicted by the geometry.

3.4

Validation with Physical Phantoms

Apart from a correct calibration, a way to experimentally validate the diagnostic is desirable because the plasma emissivity is usually unknown.

In this work, a movable light source is used to test the correct implementation of the diagnostic. The light source consists of a cylindrical cold cathode lamp with a length of 50 mm and a diameter of 4 mm. The lamp is placed inside a replica of the tokamak aligned along the toroidal direction using a plastic structure. This structure allows the placement of the source at different radial and angular positions.

The lamp has an expected point-like emissivity for pixel sizes larger than its diameter. Knowing the expected emissivity profile, it is possible to evaluate the quality of the reconstructions performed by the algorithm. The structural similarity index [23] is here proposed as a tool to compare the reconstructions of the physical phantom with the expected point-like emissivity.

In the next chapter (sections 4.1.1 and 4.3), we shall see exactly how the camera calibration and the physical phantoms were applied to the particular case of ISTTOK. Although these methods were developed for the ISTTOK diagnostic, they can easily be applied to other devices.

Chapter 4

Application to ISTTOK

This chapter describes how the methods explained in the previous chapters were applied to ISTTOK to produce a functioning tomography reconstruction algorithm.

We begin by presenting the hardware involved, namely the cameras used at ISTTOK. A brief overview is given regarding the camera electronics and more details are given in the appendices.

We then proceed to detail the algorithm implementation including the computation of the geometry matrix, calibration, and testing. The geometry matrix is computed using the two methods (LoS and VoS) described in the previous chapter for the geometry of ISTTOK.

We finally use the diagnostic to study the time evolution of the plasma position during a shot and to observe MHD instabilities.

4.1

ISTTOK Pinhole Cameras

The tomography set up at ISTTOK consists of three pinhole cameras fabricated in-house. Each camera contains a linear array of 16 photodiodes behind a pinhole plate. The photodiodes output a current that is converted to a voltage reading using transimpedance amplifiers. The spectral range of the photodiodes is from the soft X-rays to the infrared and no optical filtering is used, meaning that virtually all the power radiated by the plasma will be picked up by the tomography diagnostic. Appendix A.1 contains the datasheet of the photodiodes with the detailed dimensions and frequency response.1

The camera circuit is based on Burnay’s work [24]. Each photodiode is connected to a transimpedance amplifier with a 3.3MΩ resistor. The signals are then fed through a vacuum compatible DB-25 connector to the tokamak exterior. The 16 signals are then transmitted to the acquisition system by 16 twisted pairs and stored on the database.

The amplification stage acts also (unintentionally) as a lowpass filter with a cutoff frequency of around 100kHz[24]. The signals can be stored in the database at a frequency of 10kHz or 2MHz. The cameras’ schematic and the cables that connect to the acquisition system are documented in the appendices A.2 and A.3.

1_{The photodiode array AXUV20ELG has 20 diodes of which we only use the middle 16 due to hardware constraints and}

A picture of the tomography cameras is shown in figure 4.1. The pinhole is centered with the array of photo-diodes and placed at a certain distance above these (9mm for the top and side cameras and 13 for the bottom camera).

(a) (b)

Figure 4.1: (a) Uncovered camera showing the photodiode array mounted on top of the printed circuit board. (b) Camera with the pinhole plate in place.

The ISTTOK tokamak has a circular cross-section with a major radius of 46 cm and a minor radius of 10 cm with a limiter of 8.5 cm radius. The tomography diagnostic is installed in a cross-section of the tokamak with three ports positioned at poloidal angles 0◦, 90◦ and −82.5◦ from the low-field side (LFS). Figure 4.2a shows a drawing of the cross-section where the diagnostic is installed and figure 4.2b shows the lines of sight spanned by the three cameras.

During this work, the bottom camera was unavailable due to a broken photodiode array. For this reason, the bottom camera is absent from all the experimental results presented next.

4.1.1

Calibration Results for ISTTOK

The methods proposed in section 3.3 were used to measure the relative ´etendue of each sensor and perform a calibration of the tomography cameras.

A negatoscope (or lightbox) was used as a uniform and isotropic light-emitting surface2. The negato-scope consisted of a circular fluorescent lamp behind a diffuser. The quality of the light source was verified in terms of uniformity and isotropy.

The sensor measurements were acquired with the cameras facing the negatoscope. Signals were acquired at different distances from the negatoscope and different angles, with no changes to the mea-surements, confirming again that the source is well suited in terms of uniformity and isotropy, and that the surface occupies the entire field of view of the sensors.

(a) Tomography port (b) Lines of sight

Figure 4.2: Cross sectional view of the ISTTOK port used by the tomography diagnostic and the available lines of sight (LoS)

Figure 4.3 shows the measured signals with the cameras facing the calibration lightbox. It is visible that the peripheral sensors have lower ´etendue than what would be expected. This is mainly caused by (i) the thickness of the pinhole - this cuts out light entering the pinhole at directions different from the perpendicular direction (ii) the reflection on the sensor surface - because the sensors have a glassy coating, they act as partial mirrors for large incidence angles.

The relative ´etendue between sensors needs to be added as a correction to the geometry matrix T. For the LoS approach, each line tican be directly multiplied by the values measured by each sensor as in table 4.1. On the VoS case, each VoS already has its ´etendue accounted for. To apply the correction, each line ti has to be multiplied by the calibration factors of table 4.1 that are obtained by dividing the values measured by the sensor by the normalized theoretical ´etendue.

2 4 6 8 10 12 14 16 sensor 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 V top camera theoretical étendue out camera theoretical étendue

Figure 4.3: Voltage readings from the sensors versus the expected ´etendue for the outer and top cam-eras.

Sensor # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Top mV 32 44 56 69 96 113 130 148 150 141 123 105 88 62 53 55 ´etendue 56 69 83 99 115 130 141 148 148 141 130 115 99 83 69 56 cal. factor 0.56 0.64 0.68 0.69 0.84 0.87 0.92 1.00 1.02 1.00 0.95 0.92 0.89 0.75 0.77 0.98 Outer mV 2 6 12 19 27 37 45 50 51 48 41 32 23 16 11 9 ´etendue 22 27 32 39 45 51 55 58 58 55 51 45 39 32 27 22 cal. factor 0.10 0.23 0.37 0.49 0.61 0.73 0.81 0.87 0.89 0.87 0.81 0.72 0.59 0.49 0.40 0.40

Table 4.1: Voltage readings from the camera calibration measurements with background subtracted, and calibration factors.

4.2

Algorithm Implementation

The implementation of a tomography reconstruction algorithm begins with the computation of the geom-etry matrix T. As discussed in section 3.1 two approaches to this problem are possible, the line of sight (LoS) approximation and the volume of sight (VoS) complete description.

Figure 4.4 contains the representation of the matrices Tion a color scale for the geometry of ISTTOK using the LoS approach. The bottom camera shows smaller angle of the fan of LoS due to the increased distance between pinhole and photodiode array. Because of this increased distance, the LoS of the bottom camera also have smaller ´etendue which is visible by the change in color between the LoS of each camera. The outer camera also has smaller ´etendue, but this time due to the smaller pinhole diameter. 50 0 50 x (mm) 50 0 50 y (mm) 105 104 103 Ma tri x Ti (s p. m m 2) 50 0 50 x (mm) 50 0 50 y (mm) 105 104 103 Ma tri x Ti (s p. m m 2) 50 0 50 x (mm) 50 0 50 y (mm) 105 104 103 Ma tri x Ti (s p. m m 2)

Figure 4.4: Representation of matrices Ti on a color scale. From left to right: matrices T1−16, T17−32 and T33−48. Each matrix Ti contains one single line of sight but multiple matrices are represented per figure.

4.2.1

Geometry Matrix

As discussed earlier in section 2.1, the line of sight calculation is but an approximation to the real geometry of a tomography set up. To better resemble reality, one should take into account the finite dimension of the pinhole and sensor that compose the camera. This is achieved by going to each point in space and computing the fraction of the solid angle subtended by each of the sensors.

The computational steps for the VoS computation performed at ISTTOK are explained next:

90 60 30 0 30 60 90 r (mm) 90 60 30 0 30 60 90 z (mm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 En try va lue s f or m at rix Ti (s p m ) 1e 7 (a) pinhole_0.8mm 90 60 30 0 30 60 90 r (mm) 90 60 30 0 30 60 90 z (mm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 En try va lue s f or m at rix Ti (s p m ) 1e 7 (b) pinhole_0.5mm

Figure 4.5: VoS of the sixth sensor of the outer camera with (a) 0.8mm of pinhole diameter and (b) 0.5mm of pinhole diameter. The white lines represent the border of the VoS.

cylindrical slab with the same radius as the tokamak vessel, with a height of 9cm and centered at the poloidal plane of the ports used by the diagnostic. These 9cm correspond to the maximum width of the VoS in the toroidal direction.

2. Discretize volume V into a 3D grid. The size of the grid should in principle be at least as small as the minimum width of the VoS which is 1mm near the limiter region.

3. For every point in the grid, compute the solid angle subtended by the pinhole plus sensor system as is indicated in figure 3.2 and equation 3.2. In this work, the solid angle values are also divided by 4π rad to convert to sp units.

4. Perform an integration in the zz direction by summing all grid points with equal z and multiplying by the discretization step. This gives values with units sp.mm. Integration in the zz axis is performed with the assumption that the plasma will be locally axisymmetric such that we need not account for changes in this direction.

5. Choose the goal resolution for the reconstruction by setting the dimensions of each Timatrix. For example, if the desired reconstruction is 50 by 50 pixels and represents an area of −100mm < x, y < 100mm, then the top-left value on matrix Ti will be the average solid angle in −100mm < x < −96mmand 100mm > y > 96mm.

Figure 4.5 shows the VoS computation result for matrix T21. Sensor number 21 is in the outer camera which has a pinhole of 0.5mm. The result of an 0.8mm pinhole is also shown for comparison. The 0.8mm pinhole lets in roughly 2.5 times more light than the 0.5mm pinhole and the VoS for the 0.5mm pinhole is slightly more narrow as can be seen in the figure. Figure 4.6 shows all the matrices Ti for the VoS method.

4.2.2

Regularization Strength

A comparison that can be done between the LoS and VoS approaches is to determine how they influence the regularization.

50 0 50 x (mm) 50 0 50 y (mm) 105 104 103 Ma tri x Ti (s p. m m ) 50 0 50 x (mm) 50 0 50 y (mm) 105 104 103 Ma tri x Ti (s p. m m ) 50 0 50 x (mm) 50 0 50 y (mm) 105 104 103 Ma tri x Ti (s p. m m )

Figure 4.6: Representation of matrices Ti on a color scale. From left to right: matrices T1−16, T17−32 and T33−48. Each matrix Ti contains one single volume of sight but multiple matrices are represented per figure.

We compared both matrices by reconstructing a tomogram for a real shot.3 _{Figure 4.7 shows the} results of the reconstruction with both the VoS and LoS matrices. Both reconstructions were done with the same regularization constant α. The LoS tomogram appears to be less smooth than the VoS one. Referring to equation (2.11), the VoS tomogram has a regularization term ||gT_{· H · g||}2_{that is 25} times smaller than the same term in the LoS case, indicating that it is more smooth. Meanwhile, both reconstructions have a similar χ2_{, with only a 15% difference. This suggests that there is an intrinsic} regularization inherent to matrix T in the VoS case, as would be expected by the broader nature of the volumes of sight. a) VoS (a) b) LoS (b)

Figure 4.7: Plasma reconstructions made with (a) volume of sight matrix and (b) line of sight matrix. Bar plots show, for each camera, the signals measured by the detectors and the retrofit signals computed with equation (2.9).

4.2.3

Boundary Conditions

The reconstructed emissivity is expected to be non-negative everywhere and zero outside the vessel. These conditions have to be imposed either by adding them to the regularization functional or by chang-ing the reconstruction itself after or durchang-ing the iterative process. To force the non-negativity the values of the emissivity are clipped to a small positive number (10−10_{), this also prevents the division by a} too-small emissivity in equation 2.18. Forcing the emissivity to zero outside the vessel is done by having a term in the regularization functional that is proportional to the norm of the emissivity taken outside the vessel. The value of the proportionality constant has to be tuned to prevent the reconstruction from yielding non-zero values outside the vessel.

Figure 4.8 shows three different reconstructions done with the same signals and the same χ2 _but with different weights of the norm outside the vessel. It can be seen that the smallest weight leads to non-zero values outside the vessel. On the other hand, having a weight that is too large will focus the emissivity on the center of the vessel because the regularization also imposes a smooth transition across the vessel outline.

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Figure 4.8: Reconstructed emissivity with different weights of the norm outside the vessel represented by the white circle.

4.3

Physical Phantom Reconstructions

As discussed in section 3.4 a movable light source was used as a way to experimentally validate the diagnostic. Figure 4.9a shows the experimental set-up used to hold the cold cathode lamp inside the ISTTOK replica at different positions. Besides validating the diagnostic, the data from these experiments were also used to compare the LoS and the VoS approach systematically by performing reconstructions using both matrices.

The pixel size used here was chosen to be the diameter of the lamp (around 4mm). At this resolution, the lamp is expected to act as a point source. The chosen pixel size also needed to allow a clear distinction between the VoS and LoS matrices. This translates into having more than one pixel defining each volume of sight. In this case, the volumes of sight are three pixels wide on average.

(a) (b)

Figure 4.9: (a) Cold cathode lamp and supporting structure inside the vessel replica; (b) real vs. expected signals from the VoS and LoS geometries.

the LoS and the VoS approaches. These results were then compared to the actual signals obtained from the experiment. Figure 4.9b shows an example of the real data versus the data predicted by the two T matrices for two different lamp positions. The mean squared difference between expected and measured signals was 70% larger in the LoS case compared to the VoS case, evidence that the VoS matrix is indeed a more accurate representation of the set up geometry.

Reconstructions were performed with the acquired signals to determine the influence of each ge-ometry matrix on the quality of the reconstructed profiles. To evaluate the resemblance between the reconstructed profile and the expected point-like emissivity, we computed the structural similarity index (SSIM) [23] between them. An SSIM index of 1.0 means that two images are identical, whereas a value of 0.0 means that they are unrelated.

The VoS approach performed better in 50% of the cases, while the LoS performed better on 25% of the cases. The remaining 25% of the cases correspond to an equal performance of both methods.

The better performance of the LoS approach was observed in instances where the lamp was aligned with a single sensor in each camera. In these cases, the algorithm yielded a sharp emissivity profile, which scored better than the somewhat more diffuse solution provided by the VoS approximation. This is probably due to the larger area of the volumes of sight compared to the lines of sight.

Figures 4.10a and 4.10b show the reconstructions from an experiment with single sensor illumina-tion. Here, the LoS approach leads to a better reconstruction in the sense that it is closer to a point-like source. On the other hand, figures 4.10c and 4.10d show the results of multiple-sensor illumination. In this case, the LoS yields a broader profile because consecutive lines are further apart than consec-utive volumes. In our case, consecconsec-utive volumes have a small overlap. For these experiments, the regularization parameter α was adjusted to yield the best possible reconstructions.

VoS (a) LoS (b) VoS (c) LoS (d)

Figure 4.10: Single sensor illumination a) & b) vs. multiple sensor illumination c) & d) for the LoS and the VoS approximations. Bar plots show, for each camera, the signals measured by the detectors and the retrofit signals computed with equation (2.9).

4.4

High Field Side Activity

We now move onto discussing two know issues of the current diagnostic. These issues should be addressed in a future intervention.

The first problem is that the photodiode array that is installed in the bottom camera is damaged and has a leakage current. We believe this is due to sputtered material on the surface of the diodes from their previous life at the COMPASS tokamak from where they were borrowed. To recover the bottom camera, another photodiode array should be acquired and the respective camera should be calibrated.

The second problem is related to the geometry of the set up. As it is currently designed, the diag-nostic has poor coverage of the high field side (HFS) that is aggravated by the absence of the bottom camera.

cameras. The outer camera observes a lot more light than the top camera, especially through channels 5 to 16.

Looking at the lines of sight in figure 4.11b we can see that the top camera does not cover the HFS very well, apart from channel 16 which has a very large reading. The larger signals detected by the outer camera can thus be explained by plasma in the HFS region.

This discrepancy in the intensities measured by the two cameras proved difficult to work around using the MFI since it lacks information from the HFS. An ad hoc solution is to divide all the signals from the outer camera by some value to increase the MFI performance. This hides the phenomena taking place at the HFS. 1 5 10 15 1 5 10 15 LoS number 0 10 20 30 40 50

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(a) 100 50 0 50 100 x (mm) 100 50 0 50 100 y (mm) ₁ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 1112 131415 16

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Figure 4.11: (a) Line-averaged emissivity for typical ISTTOK operation; (b) Lines of sight for the top and bottom cameras (numbered clockwise).

4.5

Plasma Position

A straightforward application of the reconstructed plasma profile is using it to compute the position of the plasma centroid. Magnetic probes are commonly used to estimate the magnetic center of the plasma which is the center of the nested magnetic flux surfaces.

The plasma centroid computed through tomography is the center of mass of the emissivity profile which can be a useful variable for the control system. Because MFR is usually too slow for real-time applications, there is an interest in adaptations that can be faster [25]. We can bypass the reconstruction process and get a centroid estimate directly from the sensor measurements by finding an average line of sight on each camera and determining their intersection. The accuracy of this approximation needs to be verified using the output of the reconstruction algorithm.

Figure 4.12 compares the centroid estimates from both tomography and the magnetic diagnostic for shot #47220. The figure shows the horizontal displacement in the r coordinate and the vertical displacement in the z coordinate from the center of the vessel. The crosses represent the centroid estimate from the reconstructed profile and the dashed line represents the estimate from the average

line of sight. For this shot, the setpoint for the magnetic center was (r = 2.5, z = 1.0). The correlation between the line of sight estimate and the magnetic diagnostic is 93% for the data shown in the figure.

There is a clear shift between the magnetic center and the emission center of mass, this shift is especially pronounced in the r displacement. The same result has also been found in a similar analysis at the TCV [26] and at JET [4] where it has been suggested to arise from the Shafranov shift [27]. The Shafranov shift can be understood as an outward shift of the inner flux surfaces that makes them not concentric with the poloidal cross-section. Although the magnetic flux surfaces are displaced, this does not necessarily lead to a visible shift in the emissivity center of mass.

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4.6

MHD Activity

Figure 4.13: Fluctuations of the plasma toroidal current, responsible for generating the poloidal magnetic field. Taken from [28].

Another application of plasma tomography is in measuring magnetic instabilities or other types of magnetohydrodynamic (MHD) activity. The common approach to MHD analyses using tomography is

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Figure 4.14: Spectrogram of the signal from the first sensor in the outer tomography camera (top), and for one of the magnetic probes (middle). Plasma current (bottom). Shot #47309.

through singular value decomposition [29]. Singular value decomposition is rather hard to perform, espe-cially given the constraints discussed in section 4.4. Nevertheless, we show that the current tomography set up can pick up the signal from MHD activity in the ISTTOK tokamak.

A common instability that can be observed in ISTTOK is a cylindrical rotating mode with m = 2 and n = 1[30]. This rotating mode is illustrated in figure 4.13. The figure shows a cross-sectional view of the plasma where we can see that the plasma current has two maxima and two minima. This gives rise to two maxima and minima of the poloidal magnetic field which can be easily picked up by the magnetic probes. These current fluctuations occur simultaneously as density fluctuations which means they can in principle be picked up by the tomography diagnostic.

0 50 100 150 200 250 300 Poloidal Angle 40 30 20 10 0 10 20 30 40 Tim e ( s) [+ 10 9m s] 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8

(a) Angular position of the maxima and minima of the rotating mode, measured at 109ms for shot #47309.

30 20 10 0 10 20 30 Delay ( s) 1.0 0.5 0.0 0.5 1.0 Correlation corr(1, 1) corr(1, 7) corr(1, 16)

(b) Cross-correlation between the signals from LoS 1, 7, and 16 of the outer camera. Measured at 109ms for shot #47309.

Figure 4.15

these instabilities can be picked up by both the magnetic diagnostic and the tomography cameras. The rotating mode appears in the frequency spectrum as a bright spot between 50kHz − 80kHz.

Using the magnetic diagnostic, one can easily compute the angular position of the maxima and minima of the rotating mode as a function of time. Figure 4.15a shows the angular position of the maxima (in blue) and the minima (in red) as a function of time4_{. For any given instant, there are always} two maxima 180◦ apart and two minima in between the maxima. This confirms that the mode has a poloidal number m = 2.

To perform a similar analysis using the tomography cameras we take the cross-correlation between sensors corresponding to different lines of sight. If the signals are in phase with each other it means they are both looking at maxima and minima at the same time. If they are in phase opposition it means that one is looking at a maximum while the other is looking at a minimum and vice-versa. Figure 4.15b shows the cross-correlation between LoS 1 and LoS 7 and 16 from the outer camera (refer to figure 4.11b). LoS 1, 7, and 16 look respectively at the bottom, middle, and top of the tokamak. While LoS 1 and 16 have similar phases, LoS 7 is roughly in phase opposition with the former two. Thus allowing the conclusion that the poloidal mode number corresponds to m = 2.

4_{The figure actually shows the cross-correlation between the magnetic probes depending on their angular position, but for the}