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ISTTOK Plasma Tomography Using Minimum Fisher Regularization

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ISTTOK Plasma Tomography Using Minimum Fisher Regularization 10 3 50 ) 2 m m . p s ( 0 i 4 T y (mm) 10 x i r t a M 50 10 5 50 0 50 x (mm) 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 November 2019 ii 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 inversao˜ 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 sao˜ validados com fantomas artificiais ou com informac¸ao˜ proveniente de outros diagnosticos.´ 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 tambem´ a comparac¸ao˜ entre duas implementac¸oes˜ diferentes no que toca a` descric¸ao˜ matematica´ da amostragem espacial. Uma poss´ıvel aplicac¸ao˜ do diagnostico´ tomografico´ e´ demonstrada calculando a posic¸ao˜ do plasma e observando o desvio de Shafranov. Palavras-chave: Tomografia de plasma, M´ınimo de Fisher, Regularizac¸ao˜ Tikhonov, Toka- mak, Diagnosticos´ de Plasmas. iii iv 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: Plasma Tomography, Minimum Fisher, Tikhonov Regularization, Tokamak, Plasma Diagnostics v vi Contents Resumo................................................. iii Abstract.................................................v List of Figures............................................. ix Glossary................................................ xi 1 Introduction 1 1.1 Plasma Tomography in Tokamaks...............................2 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 4.1.1 Calibration Results for ISTTOK............................. 22 vii 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 A.3 Cabling.............................................. 47 viii List of Figures 1.1 Bremsstrahlung radiation profile computed using equation 1.2 with typical ISTTOK values 18 −3 (ne = 3 × 10 m , Te = 150eV, Zeff = 1:5, g¯ff = 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 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....................................... 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 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....................................... 26 ix 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 A.1 Pinout of the tomography cameras. View from the tokamak flange.............. 47 x 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 Tecnico´ 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
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