Bayesian Analysis of Electron Cyclotron
Emission Measurements at Wendelstein 7-X
vorgelegt von
M.Sc.
Udo Höfel
ORCID: 0000-0003-0971-5937
von der Fakultät II – Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
— Dr. rer. nat. —
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Holger Stark
Gutachter: Prof. Dr. Dieter Breitschwerdt
Gutachter: Prof. Dr. Robert Wolf
Gutachter: Prof. Dr. Hans-Jürgen Hartfuß
Tag der wissenschaftlichen Aussprache: 4. Februar 2020
Berlin, 2020
I sought the fount of fire in hollow reed
Hid privily, a measureless resource
For man, and mighty teacher of all arts.
Aeschylus, Prometheus Bound
Abstract
Electron cyclotron emission
spectroscopy (ECE) is a standard diagnos-
tic technique on the optimized stellarator Wendelstein
7-X
(
W7-X
) that can
record data with a high resolution in time. The spatial assignment via the black-
body emission of a plasma layer depends on the optical depth and thus plasma
parameters as well as the magnetic field along the line of sight. The measure-
ments with a multichannel radiometer contain a large amount of information
about the electron temperature profile, as well as being very sensitive to the
magnetohydrodynamic equilibrium at W7-X.
First, the diagnostic was comissioned and absolutely calibrated. At
W7-X
this is achieved by an optical system identical to the plasma measuring system,
which alternately measures room temperature and liquid nitrogen temperature
of a microwave blackbody radiator by means of a rotating mirror. The signal dif-
ference associated with the temperature change then permits the determination
of the calibration factors, the accuracy of which represents the most important
source of uncertainty for the diagnostic. In order to allow a systematic treat-
ment of the uncertainties, a completely new, general Bayesian forward model
of a calibration unit with rotating mirror was developed and tested within the
Bayesian modeling framework Minerva.
The calibrated data then allow to obtain a radiation temperature spectrum.
The actual desiderata, i. e. the sought-after quantities, however, are the elec-
tron temperatures on the effective plasma radius. Traditionally, the emission
region is approximated by the cold resonance location. However, this method
is inaccurate if, for example, relevant plasma pressure is reached that leads to a
modification of the magnetic field along the line of sight. For the more precise
determination of the emission region and the underlying electron temperature
profiles, forward modelling must therefore be carried out taking into account
the radiation transport along the line of sight. Furthermore, the ray should
be determined by raytracing, since ray deflection via the plasma parameter-
dependent refractive index can have a serious influence on the model predic-
v
Abstract
tions, especially at higher densities. Both is achieved by incorporating the tra-
cing visualized (TRAVIS) code into Minerva, in which the forward model of
the electron cyclotron emission (ECE) for
W7-X
is written. The model includes
a prediction of line-integrated electron density via interferometry. One of the
advantages of this completely new model is that it is relatively general and
should allow easy transferability to other machines, as well as compatibility
with the »plug’n’play« neural network generator currently in development.
As examples for applications, the model is used to obtain information about
the absolute values of the electron density profile during low and high density
plasma discharges, which is a good addition to the already existing possibilities
of density measurements by, for example, the Thomson scattering diagnostic.
Finally, the ECE data is used on a simple Bayesian heatwave analysis model in
an attempt to obtain the electron heat diffusivity.
vi
Zusammenfassung
Die Elektronen-Zyklotron-Emissionsspektroskopie
(ECE) ist eine
Standarddiagnostik am optimierten Stellarator Wendelstein
7-X
(
W7-X
),
die zeitlich hochaufgelöste Messdaten aufnehmen kann. Ihre räumliche Zu-
ordnung über die Schwarzkörperemission einer Plasmaschicht hängt von der
optischen Tiefe und damit Plasmaparametern sowie dem Magnetfeld entlang
der Beobachtungsrichtung ab. Die Messungen mit einem Vielkanalradiometer
enthalten einen großen Informationsanteil über das Elektronentemperaturpro-
fil, auch ist die Diagnostik eine sehr empfindliche Messmöglichlichkeit des
Magnetohydrodynamikequilibriums an W7-X.
Zu Beginn wurde die Diagnostik neu in Betrieb genommen und absolut kali-
briert. Am
W7-X
wird dies durch eine zur Plasmamessung identisch aufgebaute
Optik erreicht, die mittels einem rotierenden Spiegel abwechselnd Raumtempe-
ratur und Flüssigstickstofftemperatur eines Mikrowellenschwarzkörperstrah-
lers misst. Die mit der Temperaturveränderung einhergehende Signaldifferenz
erlaubt dann die Bestimmung der Kalibrierfaktoren, deren Genauigkeit die be-
deutendste Fehlerquelle der Diagnostik darstellt. Um eine systematische Be-
handlung der Unsicherheiten zu erlauben, wurde ein komplett neues, generel-
les Bayessches Vorwärtsmodell einer Kalibriereinheit mit rotierendem Spiegel
entwickelt und im Rahmen des Bayesschen Modellierungsframeworks Minerva
getestet.
Die kalibrierten Daten erlauben dann die Messung eines Strahlungstempera-
turspektrums. Der eigentlich gewünschte Wert ist jedoch die Elektronentem-
peratur in Abhängigkeit vom effektiven Plasmaradius. Traditionell erfolgt die
Approximation des Emissionsbereiches über den Ort der kalten Resonanz, al-
lerdings ist diese Methode ungenau, wenn beispielsweise relevante Werte des
Plasmadrucks erreicht werden, was zu einer Modifikation des Magnetfeldes ent-
lang der Sichtlinie führt. Für die genauere Bestimmung des Emissionsbereiches
und der zugrundeliegenden Elektronentemperaturprofile ist darum eine Vor-
wärtsmodellierung unter Berücksichtigung des Strahlungstransportes entlang
vii
Zusammenfassung
der Sichtlinie durchzuführen. Der Strahlverlauf im Plasma sollte des Weiteren
mittels Raytracing bestimmt werden, da die Strahlablenkung über den plasmapa-
rameterabhängigen Brechungsindex insbesondere bei höheren Dichten die Mo-
dellvorhersagen gravierend beeinflussen kann. Dies wird durch einbinden des
Strahlungstransport-Raytracing-Codes tracing visualized (TRAVIS) in Minerva
erreicht, in welchem auch das Vorwärtsmodell der ECE für W7-X geschrieben
ist. Das Modell beinhaltet des Weiteren eine Vorhersage der linienintegrierten
Elektronendichte via Interferometrie. Einer der Vorteile dieses komplett neuen
Modells ist, dass es relativ allgemein gehalten ist, und somit eine leichte Über-
tragbarkeit auf andere Maschinen erlauben sollte, sowie die Kompatibilität mit
dem zurzeit in Arbeit befindlichen »Plug’n’Play«-Neuronale-Netze-Generator.
Die Diagnostik wird beispielsweise angewandt um zu versuchen bei Niedrig-
und Hochdichteplasmaentladungen Informationen über die absoluten Werte
des Elektronendichteprofils zu erlangen, was eine gute Ergänzung zu den bereits
existierenden Möglichkeiten der Dichtemessungen durch beispielsweise die
Thomsonstreuungsdiagnostik darstellen würde. Schließlich werden die ECE-
Daten verwendet um mittels eines simplen Bayesschen Wärmewellenmodells
auf die Elektronenwärmediffusivität zu schließen.
viii
Contents
Abstract v
Zusammenfassung vii
1. Introduction 1
2. Electron cyclotron emission spectroscopy 7
2.1. General principle . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2. Tracing visualized code . . . . . . . . . . . . . . . . . . . . . . . 16
2.3. ECE diagnostic at W7-X . . . . . . . . . . . . . . . . . . . . . . 17
3. Bayesian modelling 25
3.1. Gaussian processes . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2. Maximum a posteriori . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3. Markov chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . 31
3.4. Minerva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4. ECE diagnostic calibration 39
4.1. Minerva implementation . . . . . . . . . . . . . . . . . . . . . . 40
4.2. Model formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5. Plasma profile inversion 63
5.1. Minerva implementation . . . . . . . . . . . . . . . . . . . . . . 64
5.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6. Higher level applications of ECE 93
6.1. Application on a High Density Plasma . . . . . . . . . . . . . . 93
ix
Contents
6.2. Application on a Low Density Plasma . . . . . . . . . . . . . . . 104
6.3. A Bayesian approach to heatwave analysis . . . . . . . . . . . . 106
6.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7. Résumé 113
A. Temperature dependence of the calibration model prediction 117
B. ECE calibration code 119
C. ECE profile inversion code 125
Acknowledgements 133
Statutory declaration 135
List of Figures 137
List of Codes 139
Publications as first author 141
Publications as coauthor 143
Acronyms 147
Bibliography 149
x
1
Introduction
The primary electricity production has grown on average
3.6%
per year between
1985 and 2011 [1]. Conservative estimates expect a further threefold increase
between 2000 and 2100 [2]. As mankinds hunger for energy keeps growing,
climate change poses serious threats to the stability of societies and the environ-
ment [3]. Using renewable energy sources helps alleviating CO
2
emissions, but
has drawbacks, for example with respect to the thus required energy storage
capabilities [4].
Nuclear fusion is a new primary, clean energy source under investigation. It
is expected to contribute to the energy mix in the second half of the century [5].
The advantages of nuclear fusion include large amounts of fuel, lack of both
CO2emissions and long-lived radioactive waste [6].
However, for fusion to happen, a deuterium-tritium plasma with temper-
atures exceeding 100 million kelvin needs to be confined contactlessly in a
vacuum chamber with a confinement time on the order of seconds. The two
main magnetic confinement concepts are the tokamak [7, 8] and the stellar-
ator [9]. Both have a toroidal magnetic field, but different ways to generate
an additional poloidal magnetic field. Tokamaks often use the ohmic current
induced by a transformer for this, while stellarators rely on external coils. The
superposition of the magnetic fields leads to twisted magnetic field lines. The
ratio of poloidal to toroidal turns is called rotational transform,
-ι
. To achieve
sufficient confinement, closed and nested magnetic flux surfaces with a finite
-ι
are necessary. Flux surfaces are defined as a surface for which
B⋅n=
0with
B
the magnetic field and
n
the normal to the surface holds. The force balance
equation,
∇p=ȷ×B,(1.1)
1
Chapter 1. Introduction
allows to get the static, ideal magnetohydrodynamic (MHD) equilibrium with
p
the pressure and
ȷ
the plasma current. As
∇p⟂B
, the pressure is constant
along a field line. Hence, having nested flux surfaces made up of field lines
is important for analyses of magnetic confinement plasmas, as they allow to
reduce the dimensionality that has to be handled within the flux surfaces from
3D to 1D. In a tokamak the poloidal magnetic field is generated by the plasma
current, which allows to keep the plasma toroidally symmetric. Tokamaks suffer
from disruptions [10] and plasma pressure limiting neoclassical tearing modes
that degrade the plasma energy and can initiate current driven instabilities [11,
12]. Further, it is difficult to get tokamaks with steady state capability due to
their reliance on the plasma current to get the desired shape of the magnetic
cage. Recently, notable progress in non-inductive operation has been achieved
at ASDEX Upgrade (AUG) [13]. The magnetic cage of stellarators is in contrast
provided entirely by external coils, thus easing steady state operation due to
easier control of the current in external coils. Roughly similar construction costs
are anticipated for tokamaks and stellarators on comparable reactor scales [14].
Table 1.1.:
Typical (plasma) para-
meter values at Wendelstein
7-X
.
parameter typical value
B02.5 T
Te,0≈4 keV
Ti,0≈1.5 keV
ne≈5×1019 m−3
τE≈150 ms
tpuls ≈10 s
For stellarators the energy confinement
time
τE
, representative of the quality of the
thermal insulation, can be described with
scaling laws, i. e. the international stellar-
ator scaling 1995 (ISS95) [15], or more re-
cently, the international stellarator scaling
2004 (ISS04) [16]. ISS04 predicts
τE∝n0.54
e,(1.2)
with
ne
the line averaged electron density.
Thus, high densities are beneficial, as they
increase the energy confinement time. Neo-
classical transport, i. e. transport consider-
ing geometrical effects neglected in classical
transport theory, is detrimental to good confinement. Consequently, neoclas-
sical optimization [17] aims to lessen the impact of neoclassical transport [18].
It necessitates high densities as well [19]. Furthermore, high edge densities are
mandatory for safe divertor operation as they help to alleviate the heat load the
divertor has to withstand [20, 21]. The densities achievable in stellarators are
limited by heating power and radiation losses [22–24], in contrast to tokamaks,
which have the so-called Greenwald limit [25] as the upper density limit.
2
The most modern, optimized stellarator is Wendelstein
7-X
(
W7-X
) [24, 26–
29], cf. figure 1.1 and table 1.1.
W7-X
makes use of modular coils, in contrast to
»classical« stellarators, reducing stress between the coils. This thesis focusses
on the analysis at
W7-X
, although the findings should apply to tokamaks as well.
At
W7-X
, the backbone of the plasma heating is done via electron cyclotron
resonance heating (ECRH) [30] that allows practical heating up to electron den-
sities of about
1020 m−3
in extraordinary mode (X mode), or about
2×1020 m−3
in ordinary mode (O mode) for up to
30 min
, which is a necessity to show the
intended steady state capability. In principle, neutral beam injection (NBI) is
also available and has no upper density limit, but in contrast to the ECRH this
system, at least as currently implemented in
W7-X
, is not steady state cap-
able. Additionally, ECRH offers the option to drive plasma currents (i. e. the
so-called electron cyclotron current drive (ECCD) [31–33]) and shape the pro-
files of plasma parameters, which can be an important aspect in controlling the
plasma [34]. For the first time plasma heating using only the second harmonic
O mode has been achieved with the ECRH system at
W7-X
[35], thus opening
the operational window to higher electron densities.
The complementary physical process to ECRH is electron cyclotron emis-
sion (ECE), which can be detected via microwave spectroscopy and can be used
to obtain electron temperature profiles with high temporal resolution [36, 37].
In a first, simple approach, the measured ECE intensity can be considered to
stem from a blackbody, which is a reasonable first estimation for typical plasma
parameters at
W7-X
(cf. table 1.1) given a carefully chosen line of sight allowing
to obtain the electron temperature. The location where the plasma emission
originates from can be approximated by the so-called cold resonance such that
radially resolved electron temperature profiles can be obtained. This approxim-
ation neglects relativistic effects, consequently the radiation originates solely
from electrons emitting at their cyclotron frequency. More details about ECE
and the system at
W7-X
can be found in chapter 2. Bayesian modelling, sum-
marized in chapter 3, is used to optimally combine information from different
measurements, and to properly formalize the physics models. Thus, a model for
the absolute intensity calibration of the ECE has been developed and deployed,
cf. chapter 4. However, the calibrated intensities cannot always be equated with
the electron temperature directly as there are a number of effects that may com-
plicate the interpretation. For instance, the locations from which the emission
originates depend on the radiation transport, potentially deviating notably from
the cold resonance. In addition, the electron energy distribution function may
3
Chapter 1. Introduction
Figure 1.1.:
Schematic structure of
W7-X
. The half transparent blue structure
indicates the plasma (corresponding to
≈30 m3
), red and orange the modular
and planar coils and grey the outer vessel including the ports. © IPP
contain a sufficient number of electrons in the high energy tail that contribute
to spectral intensities at significantly lower frequencies and thus hamper the
simple cold resonance approximation. These electrons emit at lower frequencies
than naïvely to be expected due to their relativistic mass increase. Especially in
plasmas with lower electron densities this leads to the so-called »shinethrough«
of the emission that stems from hot core electrons, effectively preventing ECE
measurements to be interpreted as electron temperatures. Furthermore, at high
electron densities refraction of the line of sight starts to play a role. Above a
certain electron density, the ECE cutoff conditions are fulfilled for the probed
frequencies, preventing the microwave emission and no longer contributing dir-
ectly to the measured intensity. For the interpretation of the ECE it is therefore
preferable to not rely on the blackbody cold resonance ansatz due to the afore-
mentioned effects. This is especially the case when approaching operational
parameters at which the classical approach starts to break down. At
W7-X
this
is frequently the case, as it aims at high electron densities, near or above the
cutoff densities of the second harmonic X mode. Consequently, a model that
4
combines the full radiation transport calculations for the ECE system with the
line integrated electron density from a single channel dispersion interferometry
system, particularly for
W7-X
, is introduced in chapter 5. This Bayesian model
in turn allows to push the model to its high density limits as detailed in sec-
tion 6.1. The full potential of ECE as a diagnostic makes use of its high temporal
resolution, allowing for measurements of dynamic plasma phenomena. As an
example, ECE can be used to infer information about the electron heat trans-
port in a plasma, for example via heat pulse modulation experiments [38] or via
transfer entropy analysis [39]. Examples of this are highlighted in section 6.3.
Finally, the conclusions in chapter 7 wrap the thesis up.
5
2
Electron cyclotron emission
spectroscopy
One of the advantages of ECE spectroscopy is that the functionality of the
optical equipment needed near the torus, i. e. mirrors, horns and waveguides, is
not sensitive to neutron radiation. The analysis of ECE radiation is a standard
tool [36, 40] to obtain with a passive measurement, i. e. a measurement that
does not perturb the plasma, information about the electron temperature at a
high sampling rate. Thus, this chapter will first explain the general principles
at work, where the first part tries to give a more intuitive understanding and
the second part briefly goes through the relevant equations describing ECE
quantitatively. Then, the tracing visualized (TRAVIS) code used to calculate
the quantitative equations, its advantages and limitations are shortly explained.
Finally, the ECE at
W7-X
, especially the experimental setup and the calibration
unit, are introduced. Parts of this chapter have been published in [41].
2.1. General principle
This section aims to give an overview of the governing principles relevant for
ECE. First, a more qualitative description of the physics is given in section 2.1.1.
Then, section 2.1.2 describes the physical processes quantitatively.
2.1.1. Qualitative description
As typical plasma experiments aiming to pave the way to a fusion reactor are op-
erated with strong magnetic fields, the charged particles making up the plasma
7
Chapter 2. Electron cyclotron emission spectroscopy
are subject to the Lorentz force, therefore leading to gyrations around the mag-
netic field lines. For the radius of the gyration one finds the so-called Larmor
radius,
rL=me,0v⟂
eB =v⟂
ωc,0
.(2.1)
Here,
me,0
is the electron rest mass,
v
is the velocity of the electron,
B
the
magnetic field,
e
the electron charge and
ωc,0
the cyclotron frequency. Due to
the finite Larmor-radius effects in magnetized plasmas the emission near the
harmonics of the cyclotron frequency can be calculated via
fℓ,0=ℓωc,0
2π =ℓeB
2πme,0
(2.2)
with ℓcorresponding to the harmonic.
According to equation (2.2) the emission frequency is tied to a magnetic field.
The magnetic field should be strictly monotonically increasing along the line
of sight of the ECE (looking towards the emission) to allow an unambiguous
interpretation
1
. Therefore, the intensity collected at a higher frequency will
be emitted at a greater distance from the antenna. The presence of a plasma
modifies the magnetic field strength locally, e. g. by the diamagnetic effect and
the Shafranov shift [42], which can be described e. g. via a variational moments
equilibrium code (VMEC) equilibrium. Hence, a VMEC equilibrium provides a
relationship between the magnetic field strength in real space and a normalized
effective radius coordinate system describing the plasma within the last closed
flux surface. The effective radius coordinate system is important to ease analysis
by lowering the dimensionality.
Thus, solving equation (2.2) for
B
ties an ECE channel frequency via VMEC
to an effective radius ρ. This is the so-called cold resonance,
ρCR(f)=gVMEC�2πfℓ,0me,0
eℓ�(2.3)
In principle this is not tied to VMEC as any method providing a similar mapping
can be used.
1
While the
W7-X
ECE system’s low field side antenna fulfills this condition, the antenna on
the high field side (HFS) does not. However, as the HFS antenna was not used anywhere
throughout the thesis, the description henceforth deals only with the case of a monotonically
increasing magnetic field strength along the line of sight.
8
2.1. General principle
R0R1
5.65.75.8 5.9 6 6.1
2.2
2.4
2.6
2.8
major radius Rin m
B(R)in T
⋮
reff
radiometer
f1,Δf
f0
f1
140 134.4
frequency in GHz
intensity If
Figure 2.1.:
Illustration of the principle of ECE. The magnetic field corresponds
to the equilibrium used in chapter 5 and the frequencies correspond to real
values of ECE channels at
W7-X
, while the shape of the intensity curve is
purely schematic. Based on [36].
Figure 2.1 shows the general principle that an ECE spectroscopy system in
a plasma fusion experiment makes use of. The minor effective radius is de-
noted
reff
(approximately
0.52 cm
in
W7-X
), the probing ECE channels center
frequency is called
f1
with a bandwidth of Δ
f
broad enough to collect meaning-
ful intensities. For this example the chosen frequency corresponds to channel 9.
The plasma center is roughly at
140 GHz
, denoted by
f0
. In the top part of the
picture one can see the magnetic field strength over the major radius of W7-X
9
Chapter 2. Electron cyclotron emission spectroscopy
along the ECE line of sight. The plasma center is roughly at
R0≈5.81 m
. Further-
more, channel 9 is chosen exemplarily on the gradient region of the profile. In
the top part of the figure channel 9 corresponds to
R1≈5.91 m
. The middle part
of the figure shows the plasma center and channel 9 indicated by dashed lines
in an idealized cut through the torus. Channel 9 corresponding to frequency
f1
has a finite bandwidth Δ
f
, thus the radiation collected by the radiometer
stems from the orange region, provided the relativistic effects in a hot plasma
are neglected. The lower part of the figure shows schematically the intensit-
ies as measured by the radiometer. For radiometers like the one at
W7-X
, the
intensity is, in a first approximation for sufficiently dense and hot plasmas, pro-
portional to the electron temperature. This assumes that the plasma acts like
a blackbody, in which case the Rayleigh-Jeans approximation is valid. Thus, a
calibration with other blackbody sources of known temperatures is possible.
This calibration (cf. chapter 4) allows to transform the measured intensity into
a so-called radiation temperature which is the temperature a blackbody would
have to have to produce the measured intensities. Much of the following deals
with deviations of the radiation temperature from the ideal blackbody case (cf.
chapter 5).
Figure 2.2 shows the electron cyclotron frequencies for the first three har-
monics. Due to the large aspect ratio of
W7-X
, which is approximately
10
, the
different harmonics are well separated. This allows an easier interpretation
of the ECE spectrum, as the radiation collected by a frequency channel stems
from one place in the plasma (assuming sufficient optical thickness and neglect-
ing Doppler and relativistic effects) and is not a mixture of contributions from
different harmonics.
As one can see from equation (2.2) the relativistic mass gain of the electrons
is not taken into account. However, in a typical plasma experiment aiming at
nuclear fusion electrons in the high energy tail of the electron energy distri-
bution function reach velocities at which the relativistic mass gain is notable.
Furthermore, depending on the angle, Doppler shifts can play a relevant role
for the frequencies observed in the laboratory coordinate system. These effects
can be taken into account by
fℓ=fℓ,0√1−(v/c)2
1−(v∥/c)cos(θ),(2.4)
with
fℓ,0
the nonrelativistic electron cyclotron frequency (cf. equation (2.2)),
v
the electron velocity,
v∥
the electron velocity along the magnetic field lines,
c
10
2.1. General principle
5.4 5.5 5.65.75.8 5.9 6 6.1 6.2
50
100
150
200
major radius Rin m
frequency in GHz
ℓ=1
ℓ=2
ℓ=3
ECE CR
Figure 2.2.:
First, second and third harmonics of electron cyclotron frequencies
(cf. equation (2.2)) along the line of sight of the ECE at
W7-X
. The cold res-
onance (CR) positions are depicted with purple crosses. One can see that the
harmonics are well separated. Hence there is no harmonic overlap, thus this
poses no problem for the localisation of the ECE channels. The equilibrium
used is the same as the one used for the analysis in chapter 5.
the speed of light and
θ
the angle of the observer relative to the magnetic field
lines. The relativistic mass increase, represented by the term in the numerator,
leads to a red shift. This is schematically shown in figure 2.3 (red line). In this
example, radiation emitted at the major radius of
5.77 m
is (partially) redshifted,
such that it appears to be a contribution from around
5.9 m
(corresponding to
f1
). This complicates the interpretation of the obtained spectrum and highlights
the need for a way to take relativistic downshifts into account. For the Doppler
shift, corresponding to the term in the denominator, one can see that it can be
minimized by choosing a sightline perpendicular to the magnetic field lines,
which is well satisfied for the low field side antenna of the ECE at W7-X.
The given explanations consider single electrons. Taking into account that
the electrons at each location exhibit a distribution of velocities leads to a rel-
atively broad range of major radii where the electron population includes elec-
trons that emit radiation with the same frequency, as shown by the marine
curve in figure 2.3. However, much of the radiation is reabsorbed by resonant
electrons along the line of sight, effectively narrowing the broad spatial emissiv-
ity profile. For an example, see the curve filled with orange in figure 2.3. The
propagation of radiation through the plasma is described by radiation transport.
11
Chapter 2. Electron cyclotron emission spectroscopy
R0
R1
relativistically
downshifted
emission
2.2
2.4
2.6
2.8
B(R)in T
f0f1
5.65.75.8 5.9 6 6.1
major radius Rin m
emissivity βf
Figure 2.3.:
Illustration of the effect of radiation transport on the ECE. The
magnetic field and major radii correspond to real values of
W7-X
, while the
shape of the emissivity curve is purely schematic.
It is important to note that the description of the wave emission of gyrating
electrons in a magnetized plasma within the »cold« plasma approximation al-
lows for two solutions for lines of sight perpendicular to the magnetic field; one
solution has the electric field of the wave parallel to the magnetic field, yielding
the so-called O mode, and the other solution has the electric field perpendicular
to the magnetic field, yielding the so-called X mode. Consequently, their cutoff
densities in a plasma appear at different frequencies. The O mode and X mode
have their cutoff densities at
ncut
e,O=(2πfℓ,0)2ε0me,0
e2
W7X
≈2.4×1020 m−3for fobs =140 GHz,(2.5a)
ncut
e,X=ne,O�1−fc,0
fobs �W7X
≈1.2×1020 m−3for fobs =140 GHz at B=2.52 T,
(2.5b)
12
2.1. General principle
with
ε0
the vacuum permittivity,
fobs
the observation frequency and the approx-
imate values specific for the standard ECE at
W7-X
. The optical thickness
τ
, a
measure for the opacity of the plasma for the wave (cf. section 2.1.2 and equa-
tion (2.23)), is different for the two modes. It depends largely on the electron
temperature, density and the gradient length of the magnetic field. This means
that the plasma can be optically thick for the X mode, so one pass through
the plasma is enough to obtain radiation temperatures reasonably close to the
electron temperatures, while the O mode may still see an optically thin plasma.
Therefore, an appropriate description of the O mode would require to take re-
flections at the vessel wall into account. A model assuming infinite reflections
between vessel walls, acting as mirrors perpendicular to the wave propagation
direction, is an often used approximation for many ECE systems. According to
the infinite reflection model the intensity is [40],
I=I0
1−exp(−τ)
1−Rwexp(−τ),(2.6)
with
I0
the intensity one would expect from an optically thick plasma,
τ
the
optical thickness,
Rw
the wall reflection coefficent and
I
the intensity corrected
for optical thickness effects.
To relax the assumptions mentioned above required for the infinite reflection
model, which are clearly violated in
W7-X
due to its complex 3D geometry of
the plasma vessel, a complete 3D ray tracing is beneficial. For practical usage,
a quantitative description of ECE spectroscopy is required, thus the necessary
formulæ are outlined in the next subsection.
2.1.2. Quantitative description
A quantitative description of the major physical processes leading to ECE spec-
troscopy measurements is crucial to obtain detailed and reliable electron tem-
perature profiles. This subsection follows [40, 43] and uses the notation of [43].
The radiation transport along a path sthrough the plasma is described by
N2
r
d
ds�Iω
N2
r�=βω−αωIω,(2.7)
with
Nr
the ray refractive index,
Iω
the wavelength dependent intensity,
βω
the
emissivity and
αω
the cyclotron absorption coefficient. Each of these variables
13
Chapter 2. Electron cyclotron emission spectroscopy
will be explained subsequently. New paragraphs indicate where the next of these
variables is defined due to the amount of space required for each explanation.
Ray refractive index First, the ray refractive index is given by
N2
r=N2�cos(ϑ)dΩ
dΩk�−1
(2.8)
with Nthe refractive index. Furthermore,
cos(ϑ)=(k⋅F)/(kF),(2.9)
the ray solid angle is d
Ω
and the wave vector solid angle is d
Ωk
[44]. The wave
vector is specified by
k
and the dimensionless normalized power flux density
by F(ω,k).
Emissivity Second, for the emissivity one finds
βω=N2
r
meω3
8π2c2|F(ω,k)|ω2
p,e
ω2∑
ℓ�du∥[𝒟ql fe]γ=γres,(2.10)
with the electron plasma frequency given by
ωp,e=√nee2
ε0me
,(2.11)
ℓ
the harmonics number,
u∥
the momentum per unit mass along the ray, the
normalized quasilinear diffusion coefficient
𝒟ql =(u⟂/c)2|Πℓ|2,(2.12)
which in turn contains the complex polarization vector
Πℓ=e−𝒥ℓ−1(k⟂rL)+e+𝒥ℓ+1(k⟂rL)+e∥u∥𝒥ℓ(k⟂rL)/u⟂(2.13)
with 𝒥Bessel functions of the first kind,
e±=ex±iey,(2.14)
14
2.1. General principle
as well as
k⟂rL=N⟂u⟂ω
ωc,0c(2.15)
with the Larmor radius (cf. equation (2.1)),
u⟂/∥ =γv⟂/∥
and
N⟂
the refractive
index perpendicular to the magnetic field. For the electron velocity distribution
function feoften a normalized relativistic Maxwellian is assumed,
fe(u)=�√π
2μ
exp(−μ)
𝒦2(μ)�μ3/2
√2π exp�−μ(γ−1)�,(2.16)
wherein
μ=mec2/Te,(2.17)
with
Te
in units of
keV
. The Bessel function of the second kind is denoted by
𝒦
.
As the integral of equation (2.10) is performed along the resonance lines, the
integrand is only evaluated where γ=γres holds. Therein,
γ=√1+(u/c)2,(2.18)
γres =(ωℓ,0/ω)+N∥u∥/c.(2.19)
Note that the integral in equations (2.10) and (2.20) is just over
u∥
, thus the
starting and end points of the integral can be determined by solving
γ=γres
for u⟂=0.
Cyclotron absorption coefficient
Third, the cyclotron absorption coeffi-
cient can be calculated via
αω= − πω
c2|F(ω,k)|ω2
p,e
ω2∑
ℓ�du∥[𝒟ql ℒ(fe)]γ=γres.(2.20)
In there, the quasilinear differential operator is defined by
ℒ= ∂γ+cN∥∂u∥.(2.21)
One can find that for a Maxwellian given by equation (2.16)
αω
and
βω
satisfy
Kirchhoff’s blackbody radiation law,
βω
αω
=N2
r
ω2
8π3c2Te.(2.22)
15
Chapter 2. Electron cyclotron emission spectroscopy
The cyclotron absorption coefficient allows the calculation of the optical thick-
ness along the ray,
τ=�αωds.(2.23)
Spectral intensity
Forth, solving equation (2.7) at the point where the ray
leaves the plasma on its way to the antenna,
b
, leads to the following definition
of the spectral intensity at the microwave receiver optics
Iω(b)=Iinc
ωexp�τω(a)−τω(b)�+�b
a
ds′βω
N2
r
exp�−τω(b)+τω(s′)�,(2.24)
where
Iinc
ω
is the spectral intensity at the point at which the ray ending in
b
entered the plasma. This point is described by
a
. It is clear that for optically
thick plasmas
Iinc
ω
is neglectable as the exponential term will damp its contri-
butions strongly. Note that
Nr(s=a,b)=
1needs to be fulfilled as a boundary
condition, effectively requiring the electron density and temperature to vanish
at the plasma edge.
Finally, one can predict the radiation temperature to be expected from a
plasma with given electron temperature and density with
Trad(ω)=8π3c2
ω2∑
r
wrIω,r(b)τ≫1
≈Te,(2.25)
wherein the sum collects contributions from multiple rays
r
(only required if
the beam has been split in multiple rays beforehand) reaching the antenna,
each weighted by
wr
. Given a large optical thickness
τ
,
Te
can be very well
approximated by
Trad
, which is the basis for the »classical« ECE analysis. For
practical application, Marushchenko and Turkin developed an optimized code
for these calculations which is shortly explained in the next subsection.
2.2. Tracing visualized code
TRAVIS is a Fortran code that is used for calculations for the ECE and ECRH sys-
tems, as well as for reflectometer systems, mostly at
W7-X
, but calculations for
the International Thermonuclear Experimental Reactor (ITER) have also been
16
2.3. ECE diagnostic at W7-X
done. This section is based heavily on [43] and summarizes the assumptions
that are relevant for the ECE mode of TRAVIS, as well as the (dis)advantages.
TRAVIS contains a ray tracer that takes the electron density and temperat-
ures, as well as the magnetic equilibrium provided for example via VMEC or
EFIT into account and implements the theoretical calculations from section 2.1.2.
To account for the finite width of the antenna pattern of the ECE diagnostic
TRAVIS can automatically create multiple appropriately weighted rays located
at concentric circles around the central ray. However, each ray is an independ-
ent calculation, such that the computational cost of such an approach can be
prohibitive for calculations that require a large number of iterations, like for
example a Markov chain Monte Carlo (MCMC) (cf. section 3.3). For such cases
using only the central ray offers an approximation that yields good results for
beams which do not diverge to a significant extent and the plasma density is
far enough from the cutoff density.
In a second step, the radiation transport equation (cf. equation (2.7)) is then
solved in reverse direction along the calculated rays such that a radiation tem-
perature for each frequency channel is predicted. Do note that TRAVIS does
its calculations either for the X mode or the O mode. This implies that two
separate calls to TRAVIS are required to predict both modes. While TRAVIS can
handle wall reflections in full 3D, which is required for stellarators due to their
complex geometry, transported radiation that would reenter the plasma in the
wrong mode is discarded. Computationally, it is expensive to search for the in-
tersections of rays with the wall polygons. For diagnostics like the ECE at
W7-X
this poses no problem if electron densities above approximately
1019 m−3
and
temperatures above about
1 keV
are achieved, as the X mode is then optically
thick enough that a second plasma pass and thus reflections are not required.
The TRAVIS code is provided in the general
W7-X
service-oriented architec-
ture (SOA) (cf. [45, 46]) as a webservice to decrease maintenance and improve
accessibilty for the W7-X team.
2.3. ECE diagnostic at W7-X
At
W7-X
, an ECE heterodyne radiometer
2
is the main diagnostic for obtaining
electron temperature core profiles with a continuous high temporal resolution
2
That is, the input signal is mixed with the signal of a local oscillator, leading to a signal at
the sum and the difference of the two signals.
17
Chapter 2. Electron cyclotron emission spectroscopy
of up to
2 MHz
. In contrast, the other main temperature diagnostics, Thomson
scattering [47] and X-ray imaging crystal spectroscopy [48] have sampling rates
of up to
30 Hz
respectively
500 Hz
, although the Thomson scattering system
can reach sampling rates of up to
10 kHz
in »burst« mode. However, the »burst«
mode can only be used for
1.2 ms
with a repetition rate of
5 Hz
[49] and thus
is not capable of providing data with high temporal resolution continuously.
Thus, the ECE is crucial in understanding e. g. fast plasma phenomena at
W7-X
.
2.3.1. Setup
The setup of the standard ECE system at
W7-X
[37] can be found in figure 2.4.
It measures at the outboard, low field side of the torus
3
. As
W7-X
has a five-
fold symmetry, there are 10 toroidal angles
4
at which the whole line of sight
of the ECE could have been truly perpendicular to the magnetic flux surfaces.
However, an ECE measurement near the plasma center would be completely
masked by strong non-absorbed ECRH radiation, as the ECRH heats the plasma
in these symmetry planes. The ECE line of sight was chosen to have a small
toroidal component, while still being near one of the 10 toroidal angles men-
tioned above
5
. As a consequence, the ECE does measure a Doppler shift, and
refraction does play a role for higher densities (see figure 6.3). However, the
sightline has been optimized to reduce the influence of both these effects as
much as possible [50]. Note that the line of sight in figure 2.4 does not appear
to be perpendicular to the flux surfaces, which is an artifact from the projec-
tion into the
(R,z)
plane in polar coordinates. The VMEC calculated magnetic
flux surfaces for a high mirror configuration
6
for operation phase (OP) 1.2b
are shown for no plasma beta. A wideband optical Gauss telescope system (i. e.
with a wavelength independent waist position [36]) with an 1
/
e
2
beam intens-
ity width of approximately
20 mm
is used. The variation of the width over the
sightline in the plasma (±2 mm) can be considered small.
3The exact toroidal angle is ϕ≈223°.
4
To be precise: There are five »triangular« planes and five »bean shaped« planes, i. e. the
magnetic flux surfaces take on the shape of a bean, similar to the magnetic flux surfaces
shown in figure 2.4, but symmetric around a horizontal cut at z=0. The triangular planes
are not well suited for ECE measurements from the low field side as the magnetic field is
not monotonously increasing along the potential line of sight.
5
The toroidal angle
ϕ=216°
corresponds to the bean shaped plane due to the fivefold sym-
metry of W7-X (ϕ=0°,72°,144°,216°,…).
6KJM, see w7x_ref_66.
18
2.3. ECE diagnostic at W7-X
noise source
δ
polarization
matching
140 GHz
waveguide
attenuator
mixer
PLL stabilized
122 GHz
coaxial att.
32-way
power
divider
1
2
32
Δf≤1.4 GHz
radial resolution
⋮⋮
550 kHz
temporal resolution
⋮⋮
ADC
2 MS s−1
DAQ
M
room temperature
LN2
calibration unit
W7-X plasma
vessel at ϕ≈223°
microwave absorber
legend
magnetic flux surfaces
invessel part
noise source
δ
polarization
matching
140 GHz
waveguide
attenuator
mixer
PLL stabilized
122 GHz
coaxial att.
32-way
power
divider
1
2
32
Δf≤1.4 GHz
radial resolution
⋮⋮
550 kHz
temporal resolution
⋮⋮
ADC
2 MS s−1
DAQ
M
room temperature
LN2
calibration unit
W7-X plasma
vessel at ϕ≈223°
microwave absorber
legend
magnetic flux surfaces
invessel part
Figure 2.4.:
Setup of the standard ECE system at
W7-X
. Black lines around
the shown magnetic flux surfaces indicate accurately the vessel cross section.
The magnetic flux surfaces correspond to the VMEC equilibrium used for the
analysis in chapter 5. Also, the angle of the line of sight through the plasma
is accurate.
19
Chapter 2. Electron cyclotron emission spectroscopy
To calibrate the system, exactly the same optical system including the va-
cuum windows is used below the
W7-X
experiment in the calibration unit,
which is explained in more detail in section 2.3.2. The only difference between
the two optical systems is the small tilt of the last plane in-vessel mirror to
achieve a sightline perpendicular to the magnetic flux surfaces to suppress
Doppler shift contributions. Instead of the inner plasma vessel, the calibration
unit has its sightline leading to a rotating mirror, which is surrounded by a
highly microwave absorbent foam (ECCOSORB®) at room and liquid nitrogen
temperature, see also section 2.3.2 and figure 2.6. A wire grid separates the
incoming radiation in X mode and O mode, for which two separate but similar
transmission lines exist. An oversized waveguide of approximately
23 m
length,
including two tapers, 11 mitre-bends and one polarization tuner transmits the
radiation from either the calibration unit or the plasma to the detection system
outside the torus hall, allowing easy access during operation. The overall loss
of this transmission line is
13.3 dB
. The polarization tuner allows to adopt the
mode to the radiometer input. Cross polarization coupling of the transmission
line is on the order of
1%
to
2%
[37] as has been measured with a polarized
signal source. A calibrated noise source can be selected by a waveguide switch
instead of the transmission line, which allows to calibrate with a higher sig-
nal to noise ratio at the expense of not taking the influence of components
in front of the noise source switch into account. A Bragg reflection notch fil-
ter with at least
55 dB
damping within
(140.0±0.5)GHz
and approximately
5.3 dB
insertion loss outside was used to block non-absorbed
140 GHz
ECRH ra-
diation [51]. Afterwards, the signal is down-converted to
4 GHz
to
40 GHz
via a
broadband low noise mixer feeded by a phase locked loop (PLL) stabilized local
oscillator (LO) at
122.06 GHz
[52]. Using several staged amplifiers and power
dividers the signal is split into 32 channels, which subsequently are band-pass
filtered with center frequencies between
4.4 GHz
and
39.6 GHz
and bandwidths
of
0.25 GHz ≤
Δ
f≤1.4 GHz
, with a resolution in real space between
0.5 cm
and
1.5 cm
[50]. Then, the signal passes through the detection diodes and fixed
postdetection amplifiers. Highly linear amplifiers with a variable gain and an
adjustable DC offset allow to choose a reasonable signal amplitude for each
plasma discharge, therefore making maximum use of the analog digital con-
verter (ADC) range. Subsequently, a low-pass filter with a
3 dB
point at
550 kHz
determines the maximum temporal resolution and ensures that no aliasing oc-
curs. Finally, a
16 bit
ADC with a sampling rate of up to
2 MS s−1
is used, before
the data are stored in the central
W7-X
database by the data acquisition (DAQ).
20
2.3. ECE diagnostic at W7-X
0 100 200 300 400 500 600 700 800 900
0
0.5
1
1.5
time in s
signal in V
channel 10
Figure 2.5.:
Example of the drifts of an ECE channel during a few minutes.
Channel 10 has a signal difference between the two reference temperatures
of around 4.7 mV. Only every 1000th point is shown.
2.3.2. Calibration unit
Many ECE radiometers that are absolutely calibrated use a rotating blade to
switch between two reference temperatures [53, 54]. Another method to switch
between the reference temperatures is given by a rotating mirror, as described
by Hartfuß et al.[36], and is sketched in figure 2.6. The advantage of these two
methods in contrast to just recording data at one reference temperature for
several minutes and then at another reference temperature for several minutes
(both without using a chopper) is the much lower measurement signal drift sens-
itivity, which, if the drifts are not strictly linear, would change the ratio of the
bit signal corresponding to the reference temperatures over time. The rotating
mirror and the subsequently applied conditional averaging act as a bandpass
which suppresses drifts on timescales larger than a rotation period, as the drifts
correspond to a low frequency contribution. As drifts on timescales above 10 s
are not negligible at
W7-X
(see figure 2.5) despite having the electronics in a
temperature controlled rack, the rotating mirror method has been chosen for
W7-X
as the temperature control allows only for temperature stability of the
21
Chapter 2. Electron cyclotron emission spectroscopy
electronics on the order of
1 K
to
2 K
. The advantage of the rotating mirror over
a rotating chopper lies in the better symmetry of the intermediate temperatures
that are measured when radiation from multiple radiation sources at different
temperatures is collected. Drifts on the magnitude observed here pose a con-
siderable problem for long term plasma operation. Either further measures to
suppress drifts have to be taken, or a regularly repeated offset determination
within a discharge has to be performed. For
W7-X
, it is planned to repeatedly
close the shutter in the planned
30 min
plasma discharges to correct the offset.
horn
φgeo
2
φgeo
1
φ(t)
φ=0∘
schematic representation
of effective temperature
gold-coated
rotating mirror
microwave black
body source
LN2
Figure 2.6.:
Schematic drawing of the ECE calibration
unit at W7-X.
The calibration unit of
the ECE at
W7-X
contains
a gold-coated brass mir-
ror rotating steadily with
approximately
3.6 Hz
, see
figure 2.6 for a schematic
illustration. Arranged cyl-
indrically around the mir-
ror, a microwave absorber
guarantees a black body
emitter at room temper-
ature
TRT
, which is kept
at
(294.45 ±3.50)K
in the
torus hall. However, a small
part at the lower side
of this cylinder is cut to
allow the observation of
a stainless steel container
thermally insulated by styrofoam. The stainless steel container’s inner wall
is lined with a pyramidical microwave absorber. The optical system between
the rotating mirror part of the calibration unit and the microwave antenna,
which is identical to the invessel optical system, is not shown. Light orange
indicates the line of sight of the microwave antenna. The microwave antenna
is characterized by a Gaussian beam. The beam is reflected at the gold-coated
mirror and finally »sees« either room or liquid nitrogen temperature, the lat-
ter being produced by a liquid nitrogen tank underneath the rotating mirror.
The cold reference temperature is not directly given by the temperature of
the liquid nitrogen,
TLN2=(77.2±0.5)K
, as water vapour (assumed to be at
TH2O=(280 ±10)K
with an uniform emissivity of
0.01 <εH2O<0.03
) accu-
22
2.3. ECE diagnostic at W7-X
mulates above the liquid nitrogen reservoir. Moreover, the temperature of the
mirror needs to be taken into account (corresponding to room temperature de-
scribed above, with an emissivity
0.01 <εmirror <0.03
). This leads to an effective
temperature difference between the hot and the cold source of about
208 K
and
an associated uncertainty on the order of
4 K
that is used in further calculations.
Due to the finite size of the beam, the effectively measured temperatures are
smeared out at the hot/cold edges, as different parts of the beam »see« different
temperatures. Details about the effective temperature estimation are given in
section 4.2.1.
As stated beforehand, the analysis of the calibration process allows to obtain
radiation temperature spectra (cf. section 2.1).
W7-X
is planned to have most
of its standard analyses done in a Bayesian way in the long term. Therefore,
the next chapter introduces the mathematical and philosophical concepts of
Bayesian analysis.
23
3
Bayesian modelling
The actual science of logic is conversant at present only with
things either certain, impossible, or entirely doubtful, none of
which (fortunately) we have to reason on. Therefore the true
logic for this world is the calculus of probabilities, which takes
account of the magnitude of the probability which is, or ought
to be, in a reasonable man’s mind.
(James C. Maxwell)
Most analyses in science want to test how well a model describes some obser-
vation. For physics that typically means that a forward model (or a set thereof)
predicting the observation needs to be constructed. Using Bayes’ formula [55,
56],
P(F|D)
�����
posterior
=
likelihood
�����
P(D|F)prior
�
P(F)
P(D)
���
evidence
,(3.1)
originating from the product rule of probability theory allows to infer the de-
siderata (i. e. the sought-after quantities). In equation (3.1)
P(⋅)
represents a
probability,
F
the free parameters to be inferred and
D
the data. A priori know-
ledge about the free parameters is encoded in the prior probability distributions.
The probability of the measured data given the free parameters is called the
likelihood. The normalization factor in the denominator, often called evidence,
is important for model comparison. While calculable via marginalisation,
P(D)=�P(D|F)P(F)dF,(3.2)
25
Chapter 3. Bayesian modelling
this integral is in general difficult to compute. However, there are methods avail-
able that can reconstruct the posterior without the explicit need to calculate
the evidence, for example with the MCMC algorithm described in section 3.3.
The posterior is the term we want to calculate, as it yields the probability dis-
tribution of the sought-after free parameters given the data. As the evidence
does not change the shape of the posterior for a given model, one can neglect
that term if one is not interested in comparing models explicitly. For comparing
models, however, the evidence is required, and it can be shown that this term
penalizes complexity (that is, applying Occam’s razor), as the probability mass
of more complex models is spread over a larger hypervolume. If no forward
model is used the difficulty to obtain consistent results if multiple observations
are available is severly increased. An example from plasma physics: Predicting
the measurements of X-ray imaging crystal spectroscopy and ECE from a com-
mon
Te
profile is straightforward, while a routine that calculates
Te
separately
for both diagnostics will hardly get consistent Teprofiles as a result.
The philosophy underlying the use of Bayesian probability theory in this
thesis (and in general in Minerva, see section 3.4) does refute the common
notion of measurement error in favour of putting the uncertainties on the model.
Thus, observations in the models in this thesis are assumed to be exact. This has
no directly obvious consequences, as the uncertainty distributions are usually
assumed to be symmetric. However, transfering the uncertainties from the
observations to the predictions allows to treat the uncertainties as any other
part of the model, practically acting as a »uncertainty submodel«. In particular,
one can multiply the predicted uncertainties with a free parameter which can
be optimized by maximizing the probability density1.
A specific and very powerful technique used throughout the thesis are so-
called virtual observations. They are useful for imposing certain constraints on
the prior and work in the following way: First, the equation describing the prior
constraint is reorganized such that the terms that can be calculated from the
forward model are on the left hand side. The result that one would expect to
get if the equation is exactly fulfilled corresponds then to the right hand side
of the equation. However, now an uncertainty on the prediction is introduced.
Thus, these prior constraints act like constraints imposed by observations from
1
Note that this works because larger uncertainties correspond to a higher complexity, as more
models can predict the observation well. This corresponds to a spreading of the probability
mass over a larger hypervolume mentioned above.
26
3.1. Gaussian processes
some diagnostic. However, as they do not relate to data obtained by a real
world measurement device, they are called »virtual«. Note that this implies
that whatever equation is implemented this way is thus only probabilistically
enforced.
An important part of a typical plasma model is the way the plasma profiles,
for example the electron density profile, is described. Thus the next section will
specify how the plasma profiles are described in this thesis.
3.1. Gaussian processes
One of the typical problems with descriptions of physics problems is that the
exact analytic formula to represent, for example, an electron temperature pro-
file is not known. One can choose a simple parameterization, for example the
classical approach for the electron temperature profile,
Te(ρ)=Te,0(1−ρβ),(3.3)
wherein
ρ
is the effective radius,
β
a parameter determining the shape and
Te,0
is the electron temperature at the core. However, the choice of a specific
parameterization can put severe constraints on the posterior that might hinder
proper physical interpretation, here for example of the profile shape. With these
parameterized methods, one can optimize their corresponding parameters. Ex-
emplarily, choosing equation (3.3) for the electron temperature profile implicitly
puts heavy assumptions on the transport processes in the plasma, as the para-
meterization influences the gradients that are themselves a result of transport
occuring in the plasma. Therefore, it would be good to have a nonparametric
way to constrain the posterior less.
A nice method to achieve this is given by Gaussian processes [57]. In contrast
to the parameterized methods one does not optimize in the parameter space
directly, but rather in a function space determined by hyperparameters. As a
consequence, a far greater freedom in function shapes is allowed. A Gaussian
process is defined as
f(x)∼𝒢𝒫�m(x),k(x,x′)�,(3.4)
wherein
x
corresponds to the finite number of support points
2
,
m(x)
to the
2
In the case of an electron temperature profile these are e. g. the effective radius positions at
which the Gaussian process is evaluated.
27
Chapter 3. Bayesian modelling
mean function and
k(x,x′)
to the covariance function
3
, specifying the covari-
ance of random variables. Thus, a stochastic process is a Gaussian process if
the values
f(x)
are elements of a multivariate normally distributed vector. The
hyperparameters only determine the mean and covariance of the multivariate
normal, not the realisation the function takes at a specific point, which in a
practical setting has to be determined as well. The covariance function corres-
ponds to the so-called kernel. An often used kernel is the squared exponential
kernel [58]. However, such a stationary (i. e. invariant under translations) kernel
has the disadvantage that the covariance, that is the assumed a priori smooth-
ness, is the same for all points. Thus, a stationary kernel will struggle to describe,
for example, a electron density profile that is flat throughout the core region but
contains notable gradients towards the plasma edge, as different smoothnesses
would be necessary. A solution to this is to use a kernel that depends on the
position. Consequently, the Gaussian processes used throughout this thesis use
the nonstationary kernel by Paciorek [59–61]
4
and a correlation length defined
by a hyperbolic tangent length scale function [62],
ℓ(x)=ℓ1+ℓ2
2−ℓ1−ℓ2
2tanh�x−x0
xw�,(3.5)
in which
ℓ1
is the core saturation value of the correlation length,
ℓ1
is the
saturation value of the correlation length at the edge,
x0
the point at which the
two length scales coalesce and
xw
the characteristic width of the coalescent
region. This allows, for example, to have a flat electron density profile in the
core while still getting a steep gradient at the edge required to adequately
describe observed plasma profiles. Even though a hyperbolic tangent length
scale function is used is the profile shape not locked into a hyperbolic tangent.
An example to foster intuition on how the points correlation in a Gaussian
process influences the shape is depicted in figure 3.1 with samples drawn from
a Gaussian process prior. For example, the points 1. and 2. on one of the prior
samples show similar
Te
values, indicating that the points are, probably, highly
correlated. The
Te
values between point 1. and 3. show a larger
Te
difference,
indicating that the correlation between the points is lower as compared to
3
Here,
x′
is used to indicate that the covariance function describes the correlations of each
component of xwith each other component of x.
4
Note that Gibbs derived almost the same kernel in his PhD thesis in 1997, only differing by a
factor of two in the exponential. The generalized kernel by Paciorek is based on Higdon et
al. and was used here.
28
3.2. Maximum a posteriori
0 0.2 0.40.60.81
0
1
2
33.
1. 2.
effective radius ρ
Tein keV
Figure 3.1.:
Shown are some examples for profile shapes that can be obtained
by sampling from a Gaussian process prior with fixed hyperparameters (cf.
the cyan
Te
node in figure 5.1). Corresponds to the
Te
Gaussian process used
for the profile inversion in chapter 5.
points 1. and 2. Be aware however that the samples are drawn probabilistically,
therefore drawing conclusions from a single curve has to be taken cum grano
salis.
With the knowledge about Gaussian processes the models used throughout
this thesis can be constructed. In what follows the techniques that will be used
to analyse the models will be introduced.
3.2. Maximum a posteriori
A typical problem in model analysis is the determination of the parameter space
point with the highest probability, that is the maximum a posteriori (MAP),
FMAP =arg max
F[P(F|D)].(3.6)
Note that usually not the probability density directly, but rather the so-called
joint defined by
P(D|F)P(F)
, which is proportional to the probability density
if no hyperparameters are changed, is calculated. There are various methods
available to find that spot, for example gradient based search algorithms [63].
Another class of algorithms use patterns, amongst which the Hooke and Jeeves
algorithm [64] is well established. This particular algorithm was used extens-
29
Chapter 3. Bayesian modelling
0. 1.
2.
3. 4.
5.
6.
7.
PDF isolines
max(PDF)
successful move
unsuccessful move
pattern
pattern move a
b
Figure 3.2.:
Example of the modus operandi of an maximum a posteriori (MAP)
in 2D based on the Hooke and Jeeves pattern search algorithm [64].
ively throughout this thesis. The way the algorithm works is shown schematic-
ally for two parameters
a
and
b
in figure 3.2. The starting point of the analysis is
at 0. Then, the algorithm takes a step along the axis of parameter
a
and checks
whether this increases the joint probability of the model. If this is the case the
reference point moves there. As it is indeed the case in the shown example, 1. is
made the new reference point. Subsequently, the algorithm moves along the
second parameter axis,
b
. Again the probability density increases, thus the pos-
ition is moved to 2. Moves along all dimensions have hence been undertaken.
The vector that combines the successful moves of this iteration corresponds to
the found pattern. Applying the pattern to 2. moves the point to 3. Then the
next iteration begins, probing a new position along the
a
axis which is, here,
again improving the joint probability, thus moving to 4. Probing a new position
along the
b
axis would move to 5., but the joint probability is lower there, thus
the point is discarded. Consecutively, the step along the
b
axis is attempted
in the other direction which would lead to 6. As the joint probability is again
lower at this point it is rejected as well. Thus, the pattern found for this iteration
leads to a move to point 7. Continuing these iterations
5
will yield a parameter
space position nigh the maximum of the probability densitiy function (PDF).
One criteria that can be used to determine when to end the chain is to define
a maximum number of iterations in which the reference point does not move,
and abort if this number is exceeded. It is worthy to note that the Hooke and
5
This includes, for the analyses performed in this thesis, a halving of the stepsize in each
iteration in which no move was found, until the stepsize falls below 10−5.
30
3.3. Markov chain Monte Carlo
posterior π
α(θ∗
b|θt)
1.
2.
θ∗
aθtθ∗
b
parameter θ
PDF
current pos.
proposed pos.
example α
accepted
rejected
Figure 3.3.:
Example of the modus operandi of an MCMC based on the Metro-
polis-Hastings algorithm.
Jeeves algorithm might get stuck in a local maximum. Even if not stuck in a local
maximum, it is desirable to find not only the maximum of the posterior, but
to reconstruct the shape of the posterior. If it is available, potential nontrivial
dependencies between the parameters can be found. Also, one can see from the
shape how stable the found MAP is. An MAP estimation yields a point estimate
and is thus not able to reconstruct the posterior. A technique with which the
shape of the posterior can be found is introduced in the next section.
3.3. Markov chain Monte Carlo
Markov chain Monte Carlo (MCMC) is a useful method to obtain the posterior
shape. A Markov chain allows to make a probabilistic statement about the next
step in the chain, wherein the knowledge of the full history does not yield an
advantage over knowing only the current state, i. e. the chain has no memory.
The goal of an MCMC analysis is often to sample from the posterior, which
corresponds to the stationary distribution of the Markov chain. As an example,
the Metropolis-Hastings algorithm [65, 66] is described, corresponding to the
algorithm used throughout the thesis. Figure 3.3 serves as a visualization of
some of the steps. First, a proposal distribution
6q
is used to obtain a candidate
6
The proposal distribution is the distribution from which a sample determines where the next
proposed parameter position will be.
31
Chapter 3. Bayesian modelling
with a new parameter space position,
θ∗
t+1
, from the current parameter space
position,
θt
. This allows the computation of the acceptance probability of the
new parameter space position, given by
α(θ∗
t+1|θt)=min�π(θ∗
t+1)q(θt|θ∗
t+1)
π(θt)q(θ∗
t+1|θt),1�,(3.7)
with
π(⋅)
the (unnormalized) posterior probability of the given position. Note
that this implies that the evidence
P(D)
(cf. equation (3.1)) does not need to
be calculated explicitly, as only the ratio
π(θ∗
t+1)/π(θt)
is needed – only the
relative change of »height« is required. In figure 3.3 two examples are shown;
one point jumps to a position with a higher posterior probability,
θ∗
a
, and an-
other jumps to a position with lower posterior probability,
θ∗
b
. In the case of
the former, the acceptance probability is
100 %
, in the latter it is a bit over
50 %
.
Subsequently, a random number from a uniform distribution between
0
and
1
is drawn, here denoted by
α
. This value is then compared to the acceptance
probability, if
α≤α
the new position is accepted, else it is rejected and the
current position is kept. To help to get an intuition for the modus operandi of
an MCMC, example random numbers are drawn to compare the outcome for
the examples in figure 3.3. The proposed jump to
θ∗
a
is accepted for any value
of
α
, as the posterior probability of
θ∗
a
is larger than the one at
θt
. For the other
proposed point at
θ∗
b
, 1. corresponds again to an
α
below the posterior curve
and thus
θ∗
b
would be accepted, while for 2. the posterior curve is below
α
and
the jump is thus rejected. A notable simplification of the algorithm is achieved
if the proposal distribution is symmetric as the
q
terms in equation (3.7) cancel
then. Many proposal distributions do satisfy this criterion, but for example a
proposal distribution with a fixed position generally does not. In this case, the
q
terms remove the influence of the proposal distribution, which otherwise
would lead to an over- or underweighing of certain parameter space points of
the posterior. The efficiency of the MCMC depends on the shape of the pro-
posal distribution, although in principle any proposal distribution will work.
Thus, adaptive changes of the proposal distribution are a way to automatically
improve the efficiency of the MCMC [67]. Noteworthily, adapting the adapta-
tion method itself might help in further speeding up convergence [68], which
however has not been used in this thesis. However, to ensure that the posterior
remains the stationary distribution that is converged to, the adaptive changes
to the proposal distribution need to diminish.
32
3.4. Minerva
A burn-in is the practice of running a specific number of MCMC iterations
at the beginning of an MCMC and use the resulting point as a new start point,
while discarding the results from the burn-in samples. Strictly speaking, a burn-
in is not necessary given the MCMC is run long enough. However, it can be a
way to get to a reasonably likely starting point, as one does not want to start in
the tails of the probability distribution due to the limited number of samples that
will be drawn during the MCMC. Do note that the MCMC in use in this thesis
are essentially black box MCMCs, thus the only way to ensure convergence
instead of just pseudo-convergence is by using sufficiently long runs [69]. In
general, the optimal acceptance rate depends on the exact problem, but for a
wide variety of models 0.234 is optimal7.
As MCMC samples are typically autocorrelated, a thinning is done, respect-
ively only every third point is kept within the MCMC runs of this thesis, which
reduces the autocorrelation. The posterior can be obtained from the samples
for example by using a kernel density estimate [72, 73] on the MCMC samples.
3.4. Minerva
Minerva [74] is a Java based, modular, general Bayesian modelling framework
that can handle arbitrarily complex models and is used among several other
large fusion experiments around the world, for example at the Joint European
Torus (JET) [75] and the Mega-Ampere Spherical Tokamak (MAST) [76] and is
the main inference framework at
W7-X
. It is designed to work for all kind of
inference problems and is not limited to physics or fusion problems. In contrast
to many other frameworks the methods used to infer the desiderata are kept
completely orthogonal to the modelling part based on graphical models. This
means that the program parts handling the inference are decoupled from the
program parts describing the models.
Graphical models are a powerful tool to describe the conditional dependency
structure of a probabilistic model [77]. A Minerva graphical model is a Bayesian
7
Roberts et al. [70] showed that running an MCMC with an acceptance rate of
0.234
is optimal,
if one uses a multivariate Gaussian proposal distribution and a high-dimensional, independ-
ent and identically distributed target (posterior) distribution. However, if the convergence
rates of some of the posterior dimensions deviate significantly when compared to other
dimensions,
0.234
is usually no longer optimal and can be much smaller [71]. This applies
to most of the MCMCs ran during the preparation of this thesis.
33
Chapter 3. Bayesian modelling
network, more precisely, a directed acyclic graph. Examples of Minerva graphs
can be found in figures 4.1 and 5.1. It consists of deterministic and probabilistic
nodes, which are connected via edges. The graphical model encodes the whole
joint probability of the free parameters and the data. The joint probability can
be calculated by adding up the logarithms of the probabilities of all probabilistic
nodes conditioned on their parent, i. e. the nodes describing the priors (colored
cyan in the aforementioned figures) and the nodes describing the observations
and predictions (colored grey in the aforementioned figures). Based on these
models, sample predictions that may be used for example for neural net training
sets [78] can be generated, as well as inversions based on different techniques,
such as MAP and MCMC methods. Scientific traceability of the models can be
ensured easily by serializing the model structure to an .xml file.
Some basic concepts used for the implementation at
W7-X
that help to ease
the handling of multiple models and their analysis are described henceforth.
As these concepts are relatively generic they probably can be used at other
experiments as well.
3.4.1. Datasources
A Minerva datasource is a node in a graphical model. Datasources should be the
only experiment specific nodes in the whole graphical model to allow an easy
adaptation of the model to another experiment by exchanging the datasources.
More specifically, a datasource loads the data from an immutable database
and does some preprocessing, if necessary. Furthermore, it provides metadata
like lines of sight or frequencies. There have been many attempts to provide
metadata in a generalized manner. The problem with that lies in the pleth-
ora of types of metadata that is required for different usecases. An example
from W7-X: The ECE datasource contains the metadata for its calibration unit
(cf. section 2.3.2), e. g. its diameter. However, not all diagnostics, let alone all
datasources require »calibration unit« type metadata. This makes it difficult to
generalize without burdening datasources that have no need for this type of
metadata in the first place. Thus, all the metadata provided by datasources is
handcrafted and specifically tailored for each datasource. This is time consum-
ing, but still the most pragmatic way the author knows to ensure a satisfying
user experience. An example of how a datasource is used in the construction
of a graphical model can be seen in code 3.1.
34
3.4. Minerva
Code 3.1:
Example code from the plasma profile inversion model, see chapter 5.
Code corresponds to [79] with small adjustments to improve readability.
239 // the graphical model
240 GraphicalModel g=new GraphicalModel("EceInterfModel");
242 // this node handles the time to be analyzed
243 W7xProgramTimeToNanos nanosecond =new W7xProgramTimeToNanos(g
,"time");
244 nanosecond.setTime(201808023016002, 4.45);
⋮
276 // provides ECE data and metadata
277 W7xEceQmeDataSource ds =new W7xEceQmeDataSource(g,"ds");
278 ds.setNanosecond(nanosecond); ⋮
640 // connection to the TRAVIS webservice
641 Travis travis =new Travis(g,"travis");
⋮
644 travis.setFrequencies(new Source(ds,"getFrequenciesGHz"));
In line 240 the model to be constructed is instantiated, lines 243 and 244 instan-
tiate the node describing the discharge time and set the time to be analyzed.
Exemplarily, the second scenario of the 16
th
experiment on the 23
rd
August 2018,
4.45 s
after the heating was switched on, is selected. Line 277 instantiates an ECE
datasource for
W7-X
. Subsequently, line 278 connects the
W7-X
time node to
the ECE datasource node. Similarly, line 641 instantiates a TRAVIS node, while
line 644 connects the ECE datasource node to the TRAVIS webservice wrapped
in the aforementioned node. This is done to ensure that TRAVIS utilizes the
frequencies, line of sight and further diagnostic specific settings used during ex-
periments for its calculations. Note that the TRAVIS node is not
W7-X
specific:
The model could be easily ported to e. g. JET, if the ECE datasource and the
node indicating the time to be analyzed get substituted with their corresponding
counterparts.
In the course of this thesis datasources for the ECRH, ECE, X-ray imaging
crystal spectroscopy, snifferprobe, single channel dispersion interferometry,
NBI and Langmuir probe systems have been written for
W7-X
and made avail-
able within the Minerva framework in preparation of a combined Bayesian
analysis on a large number of diagnostics.
35
Chapter 3. Bayesian modelling
3.4.2. Evaluation strategies
The evaluation strategy in Minerva is generally the following: First get reason-
able estimates for the starting parameters via the datasource. Consecutively, the
analysis method has to be selected. If the model is linear, the typical analysis
method chosen is the linear Gaussian inversion explained in more detail below.
For nonlinear problems, the standard procedure continues with an MAP inver-
sion
8
, which, if not trapped in a local maximum will generally yield a result
where relevant amounts of probability mass can be found. Consequently, the
MAP result is used as a starting point for the MCMC inversion with a sufficient
burn-in and an adaptive Metropolis adapter [67] that can be deactivated once
the MCMC chain is stable.
The linear Gaussian inversion [80–82] (for the notation and full derivation
see [82]) is a very powerful technique to determine the full posterior distribu-
tion, which can be applied if the model is linear and both the likelihood and the
priors can be expressed as multivariate normal distributions,
P(D|μ)=𝒩(D;Mμ+C,σD)(3.8)
P(μ)=𝒩(μ;μp,σp).(3.9)
with
D
the data vector,
μ
the free parameter vector,
σD
some known variance,
the mean of the likelihood a linear combination given by
Mμ+C
wherein
M
is the response matrix and
C
the constants vector. Similarly, the prior is
also given by a multivariate Gaussian distribution, with mean
μp
and variance
σp
. Note that »;« stands for »parameterized by«. This allows to determine the
posterior exactly, which is also given by a multivariate normal with a mean and
covariance of
σ=(MTσ−1
DM+σ−1
p)−1,(3.10)
μ0=σ[MTσ−1
D(D−C)+σ−1
pμp](3.11)
Note that throughout this thesis matrices are indicated by bold italic symbols,
whereas vectors are indicated by bold upright symbols.
A particularly powerful approach here lies in combining the linear Gaussian
inversion with the Hooke and Jeeves pattern search algorithm (cf. section 3.2)
8
For example using the Hooke and Jeeves pattern search algorithm mentioned above, optim-
izing with respect to the joint probability density.
36
4
ECE diagnostic calibration1
A common way to absolutely calibrate the radiometers is the hot/cold calibra-
tion method, which uses two reference temperatures to determine the sensit-
ivities respectively the calibration factors [36], although often only a relative
calibration with respect to a Michelson interferometer or a Thomson scattering
diagnostic is done [53]. As described in section 2.3.2 the reference temperatures
at
W7-X
are given by a microwave absorber at room temperature and a cold
source at liquid nitrogen temperature. In principle, a hot source could also be
used, as planned for ITER [83]. The signal chain from the horn to the DAQ
system contains many components. For a complete modelling of the ECE, one
would like to have individual uncertainties for each component, but the large
number of components involved effectively prohibit this approach. This makes
it challenging to get appropriate uncertainties. Realistic uncertainties are espe-
cially important for modelling of multiple diagnostics, as too small or too big
uncertainties will artificially bias the result. Good estimates for the overall un-
certainties of the calibration are obtained by inverting the forward model of the
calibration process described in this chapter (cf. figure 4.1 for the correspond-
ing graphical model). Easy automation of the whole calibration procedure is
possible as the implementation is capable of finding reasonable defaults where
necessary.
This chapter will begin by giving an overview of the available models, and
their limitations in section 4.1. Thereafter, the formulæ constituting the models
are layed out in section 4.2, first for the specific case at
W7-X
, then generalised
for arbitrary calibration units. Then, the results are discussed in section 4.3 and
conclusions are drawn in section 4.4.
1Most of this chapter has been published in [41]; the format has been adapted to the thesis.
39
Chapter 4. ECE diagnostic calibration
4.1. Minerva implementation
To obtain the physical quantities that are of interest for the evaluation of an
ECE calibration, namely the calibration factor of each individual channel (that is
the inverse sensitivity) and the effective beam width, the calibration procedure
itself is modeled. To evaluate multiple ECE channels in a consistent way, a
forward model predicting the different channel sensitivities has to be used. See
also section 3.4 for the required Minerva background.
For comparison of different modeling approaches, different descriptions of
the calibration process were implemented. The following properties of the
model have been combined, creating a total of eight structurally different mod-
els: i) Evaluation of individual channels respectively multiple channels simul-
taneously (indicated from here on with the keywords »single« respectively
»multi«), ii) with and without the use of a channel specific scaling factor
αi
to
scale the variance of the observed conditionally averaged signal (cf. the foot-
note in chapter 3) and iii) by having a channel specific beam width or a beam
width fixed for all channels by a scaling following
w(f)=w(140 GHz)√140 GHz/f,(4.1)
as expected from broadband Gauss telescope optics. Henceforth, models using
a channel specific beam width are denoted »shared«, while the models using
channel specific beam widths are denoted »individual«.
In the case of the datasource (cf. section 3.4.1) used for the evaluations presen-
ted here, the preprocessing consists of the following steps: First, the chopper
channel data are used to automatically identify the individual mirror rotation
periods. Subsequently, the mean of each period is subtracted to avoid an overes-
timation of the uncertainties. This also removes the explicit dependence on the
absolute values of the reference temperatures (cf. section 4.2.1 and appendix A).
Then, each period is binned to avoid influences by variations of the motor speed.
Finally, a coherent averaging is done to get a signal with a better signal to noise
ratio.
Due to the small signal to noise ratio, many rotations of the rotating mirror
need to be acquired to be able to extract enough information for meaningful
results. The preprocessing is necessary as working with the full data is unprac-
tical due to the sheer volume of raw data. In the case studied in this chapter,
this decreases the number of effective temperatures to predict from about
108
to
40
4.1. Minerva implementation
less than
104
for the full combined model. An example of such a preprocessed
signal for a channel with good sensitivity can be seen in figure 4.4.
The advantage of evaluating all channels simultaneously is the consistency
gained for channel independent parameters, namely of the angles
φgeo
1
and
φgeo
2
that determine where the radiometer starts respectively stops to collect
radiation mainly from the hot source (cf. section 2.3.2), and, depending on the
model, the beam width. The disadvantage that comes with evaluating multiple
channels simultaneously lies in the curse of dimensionality, i. e. the evaluation
time increases notably. This »curse« can be understood in the following way:
Consider that the posterior is normalized, i. e. that the integral over its PDF is
exactly
1
. Consequently, as its dimensionality increases the probability density
at any given parameter point shrinks. Therefore the information contributed
by one point to an accurate reconstruction of the posterior, e. g. via an MCMC,
diminishes accordingly, effectively requiring much more evaluations of the PDF
to achieve the same level of accuracy compared to a lower dimensional case.
Exemplarily, the graphical model for the multi-channel case with variance
scaling factors
αi
and channel specific beam widths
wi
is shown in a simplified
way in figure 4.1. The user only has to set which channel contains the chopper
signal and the calibration segments (i. e. the time windows that contain the
data), as the datasource then automatically fetches the radiometer settings and
the raw data from the
W7-X
archive database (ArchiveDB) and preprocesses the
data, including the merging of the calibration segments, if multiple segments are
given.
TLN2
represents the liquid nitrogen temperature,
TH2O
the temperature of
the water vapour that forms above the liquid nitrogen with
εH2O
the emissivity
of the water vapour,
TRT
the room temperature and
εMirror
the emissivity of the
mirror. Δ
T
is the effective temperature difference between the hot and the cold
source. As stated above,
φgeo
1
and
φgeo
2
denote the mirror angles between which
the radiation collected by the radiometer originates predominantly from the hot
source, cf. figure 2.6. Together with the Gaussian beam width of the microwave
antenna characteristic
wi
, the number of bins and the effective hot and cold
temperatures the predicted effective temperatures
Teff,i(φ)
can be calculated.
The calibration sensitivity
ηi
as calculated from the bit change Δ
si
and the tem-
perature difference Δ
Teff
can be multiplied with
Teff,i(φ)
to obtain the prediction
fpred
i(φ)
of the measurement signal for channel
i
. For the uncertainty of the
prediction the variance as estimated from the conditional averaging, scaled by
a factor
αi
, is used. This variance scaling factor allows to find the uncertainties
matching the predictive capabilities of the model, as too small uncertainties are
41
Chapter 4. ECE diagnostic calibration
calibration segments
chopper channel
user set values
W7-X ArchiveDB
radiometer settings
datasourcei
αicond. avg.
fmeas
i(φ)
αiσ2
i
observationi
fpred
i(φ)
Teff,i(φ)
1/ηi
λi
bi
Δsi
1/ηi
wi
φgeo
2
φgeo
1
Thot
eff
Tcold
eff
ΔTeff
TLN2
TH2OεH2Oεmirror
channel
{i|i∈ℕ\{0}∧i≤#channels}
node
uniform
legend normal
observed
Figure 4.1.:
The simplified Minerva graph showing the model dependencies.
Free parameters have a cyan background and their shape, as specified in the
legend, corresponds to the prior distribution used. Note that the nodes with
a hatched background sample completely randomly from their distribution,
thus propagating the corresponding uncertainties without increasing dimen-
sionality by parameters which do not influence the prediction, cf. appendix A.
penalized by increasing the relative distance between prediction and observa-
tion, while too large uncertainties correspond to a higher complexity and are
thus penalized by the implicit penalty for complexity in Bayes’ formula.
The conditionally averaged and binned signal
fmeas
i(φ)
in bit constitutes the
observation. In combination with the bit to Volt conversion factor
bi
, and the
factor taking the measured differences between the invessel and calibration
optics into account, λi, the calibration factor 1/ηican be calculated.
The single channel evaluation will be compared with the multi-channel eval-
uation in section 4.3.1. The evaluation strategy is described in section 3.4.2. This
42
4.1. Minerva implementation
allows for a realistic estimation of the uncertainties of the drop of the measured
signal associated with the decrease in radiation temperature as produced by
the switch from room temperature to the cold temperature. It has to be noted
that the measured signal stored in the database is the raw bit signal, which
is also what the model predicts. The trace of the logarithm of the joint (i. e.
prior times likelihood) probability density, which is a measure for how prob-
able a specific realisation is, of an MCMC of the full model, and example plots
of each the calibration and scaling factor traces are shown in figure 4.2. All
MCMCs have been run with a burn-in of at least
106
iterations, until the MCMC
traces of the free parameters did not show notable drifts of the running aver-
age of the logarithm of the joint probability density. For all MCMCs ran so far,
convergence was observed after at most
8×105
MCMC iterations, the single
channel evaluations typically needed less than
104
iterations. This evaluation is
fully automatized and writes the results to the central
W7-X
database, making
automatic calibration overnight feasible. However, in practice the lack of liquid
nitrogen availability in the torus hall prevents regular overnight calibrations.
4.1.1. Limitations
Practical limitations are given for example by the number of bins as well as the
number of channels. If the sampling rate notably exceeds
1 kHz
the evaluation
slows down considerably as well, as the data that have to be loaded for the pre-
processing increases accordingly. A practical number of bins can be determined
automatically, which gives a value close to the average number of data points
per mirror rotation. As there are computationally expensive steps involved for
each bin, increasing the number of bins also leads to an increase of required com-
putation time. The full 98D model (made up of
φgeo
1
,
φgeo
2
and 32 channels, each
with free parameters for the beam width, variance scaling and bit dip) includes
all 32 channels and the scaling of the variance, and takes roughly 230 hours
with a Intel® Xeon® central processing unit (CPU) E5-2660 v4 at
2 GHz
on a
virtualized linux server for a roughly one hour long calibration. By evaluating
each channel independently the required CPU time is reduced to about 21 hours,
such that with full parallelization the evaluation time can go down to around
40 minutes. This simplified model comes at the price of a generally smaller
consistency and larger uncertainties for the weaker channels. However, the
differences are negligible for reasonably sensitive channels (calibration factor
changes are typically below 1%). For less sensitive channels the difference can
43
Chapter 4. ECE diagnostic calibration
20
40
(a)
log(joint)−18 500
0.45
0.5
0.55
(b)
η−1
23 in keV V−1
012345 6 78 9 10
1
1.2
1.4
1.6
(c)
iteration in 104
α23
Figure 4.2.: (a)
shows the logarithm of the joint (i. e. prior times likelihood) of
an MCMC of the full model with variance scaling factors.
(b)
and
(c)
show
the MCMC traces of the calibration factor
η−1
23
and the variance scaling factor
α23
. The values do not show jumps to a notably different phase space part,
indicating that the chain converged. Only every 100
th
datapoint is shown to
keep the plot size reasonable.
44
4.2. Model formalism
reach about
10 %
. This is due to the more sensitive channels keeping the geomet-
rical factors (
φgeo
1
and
φgeo
2
) more or less fixed, such that the impact on channels
where the geometrical information is more concealed in noise profit the most.
The single channel evaluation routine provides a pragmatic approach to obtain
calibration factors if time requirements hinder the full model use.
4.2. Model formalism
This section introduces the required formulæ, first for
W7-X
and then for a
generalized model with arbitrary geometries and effective temperatures.
4.2.1. Model for W7-X
To calculate the temperature difference between the room temperature and the
effective cold temperature, one needs to take the influence of the water vapour
emissivity and the mirror emissivity into account. The effective temperature
after the radiation passes through the water vapour forming above the liquid
nitrogen is calculated via
Tvapour
eff =TLN2+εH2O(TH2O−TLN2),(4.2)
where
εH2O
is the emissivity of the water vapour. The effective cold temperature
after the mirror is given by
Tcold
eff =Tvapour
eff +εmirror(Thot
eff −Tvapour
eff ).(4.3)
The emissivity of the mirror only adds to the effective cold temperature and not
the effective hot temperature, as the mirror is already at the same temperature
as the effective hot temperature. In case the hot reference temperature would be
at a temperature different from the mirror temperature it would have to be taken
into account there as well. Note that this works because the signals are recorded
in the frequency range where the Jeans law approximation holds. This allows
the calculation of the channel specific calibration sensitivity (respectively, the
sensitivity
ηi
scaled with the attenuation and postdetection amplification as
chosen for this channel during the calibration)
ηi=Δsi
ΔTeff
=Δsi
Thot
eff −Tcold
eff
,(4.4)
45
Chapter 4. ECE diagnostic calibration
wherein Δ
si
represents the signal change in bits caused by the temperature
difference Δ
Teff
. The index
i
denotes the channel specific parameters. Moreover,
the channel specific expected effective temperature, depending on the mirror
position, is given by
Teff,i(φ)=∫
Teff(φ)gφ(wi,φ,φ)dφ
∫gφ(wi,φ,φ)dφ,(4.5)
where
Teff
is the effective temperature at a given mirror position without taking
the finite width of the Gaussian beam, defined by the horn characteristic, into
account,
Teff(φ)=�Tcold
eff for φ<φgeo
1
Thot
eff for φgeo
1≤φ≤φgeo
2
Tcold
eff for φ>φgeo
2
,(4.6)
and
gφ
the weight of each
Teff
assuming a perfect Gaussian beam horn charac-
teristic.
This in turn allows to predict the measured bit signal in dependence of the
mirror angle,
fpred
i(φ)=Teff,i(φ)ηi.(4.7)
In practice, it can be useful to subtract the mean of the predicted signal to
avoid dependencies on the effective temperatures where not necessary, see
appendix A.
Scaling the variance
σ2
i
with a channel specific parameter
αi
on the obser-
vation node allows for a realistic estimation of the uncertainties by taking the
predictive capability of the model into account, see also chapter 3.
The calibration factor is given by
1
ηi
=λibiGi
ηi
10−(RRF+RIF)/10.(4.8)
This takes the following quantities into account: The measured differences
between the invessel and calibration optics,
λi
, the measured bit to volt con-
version factor,
bi
, the post detection amplification chosen during calibration,
Gi
, the setting of the radiofrequency waveguide attenuator right in front of
46
4.2. Model formalism
the radiometer in decibel,
RRF
, and the setting of the attenuator at the inter-
mediate frequency device in decibel,
RIF
. For calibrating plasma measurements
one needs to rescale this factor with the appropriate gains and attenuator set-
tings used during the measurement. The reason for including this branch in
the model is twofold: i) it allows direct extraction of the sought after quantity,
without having to implement separate uncertainty propagation for
λi
and
bi
and ii) preparing for future evaluations of multi diagnostic calibration factors,
in which case one can simply extend this branch by supplying ECE raw plasma
data to get an electron temperature profile that can also be supplied for example
via Thomson scattering [47]. This would not be a simple cross calibration, but
would rather combine the diagnostic specific calibration models and their cor-
responding plasma forward models, thus taking all information optimally into
account. The calibration factors obtained that way are inherently consistent
within the frame of the model.
In the Bayesian formalism this leads to (for the model shown in figure 4.1)
P(φgeo,Δs,w,α|D)=P(D|φgeo,Δs,w,α)P(φgeo,Δs,w,α)
P(D),(4.9)
with
φgeo =(φgeo
1,φgeo
2),(4.10)
Δs =(Δs1,…,Δsn),(4.11)
w=(w1,…,wn),(4.12)
α=(α1,…,αn),(4.13)
D=(DECE,1,…,DECE,n),(4.14)
where
n
corresponds to the number of ECE channels,
φgeo
to the angles at
which the central line of sight switches from the cold source to the hot source
and vice versa,
Δs
to the channel specific change in the bit signal observed
when switching the temperature sources,
w
to the channel specific Gaussian
beam width,
α
to the channel specific variance scaling factors and
D
to the
channel specific conditionally averaged measured data. The conditional aver-
aging is done in the following way: First, the average number of data in one
mirror rotation is calculated. The time series is then split at each falling edge of
the chopper signal. Subsequently, resulting individual rotation measurements
have their mean removed, are rescaled and sorted into the number of bins de-
termined in the first step. Finally, dividing by the number of rotations yields
47
Chapter 4. ECE diagnostic calibration
the conditionally averaged data required for the observation (cf. the grey node
in figure 4.1). Note that the other models differ, for example by not using the
variance scaling factors.
4.2.2. Generalised model
The model described previously uses some simplifications that can easily be
dropped to generalise the model. For instance, one can drop the assumption
that the problem is one dimensional and that there are only two reference tem-
peratures. This allows easy extension to three (or more) reference temperatures
for example by adding a hot ceramics hot source. Switching to cylindrical co-
ordinates is a sensible approach for a geometry similar to the one presented
here, thus introducing
z
along the horizontal axis of the cylinder shown in
figure 2.6, and
r
as the radius. Assuming
i×j
reference temperatures leads to
a definition of the effective temperature
Teff(φ,z)=Teff,ij
, where
Teff,ij
is the
effective temperature valid for
φgeo
i≤φ≤φgeo
i+1
and
zgeo
j≤z≤zgeo
j+1
. The second
dimension is represented by
z
. As in the model described above,
Teff,ij
can be
the result of multiple layers contributing to the effective temperature at the
selected coordinates, such that a dependency on
r
might occur as well. The
calibration sensitivity
η(T)
does not necessarily have to be linear, however, one
will have to use free parameters for the temperatures and emissivities in this
case, as the prediction will no longer be independent of these parameters. In
general, any instrument function
g
can be used to calculate the appropriate
weighted effective temperature that the radiometer would see by looking at
(ϕ,z), therefore
Teff(φ,z)=∬
Teff(φ,z)g(φ,φ,z,z)dφdz
∬g(φ,φ,z,z)dφdz,(4.15)
which in combination with the calibration sensitivity allows the calculation of
the prediction.
4.3. Results
A typical excerpt of the calibration timetrace for a sensitive channel can be seen
in figure 4.3b, while the corresponding chopper signal is shown in figure 4.3a.
There are three points to consider: i) the drop in the chopper signal does not
48
4.3. Results
0
10
20
(a)
chopper in kbit
0 0.1 0.2 0.3 0.40.5 0.60.70.8 0.9
−4.5
−4
−3.5
−3
−2.5
(b)
time tin s
channel 23 in kbit
Figure 4.3.: (a)
shows the chopper signal measured by a photo diode,
(b)
the
raw signal of channel 23, which is a sensitive channel. The signal to noise
ratio is typically of the order of
1/50
. The background signal originates in
broadband noise of the intermediate frequency (IF) amplifiers right after the
mixer, subsequently measured for each frequency bin by the detector diode.
correspond to the full width of the cold phase, the real hot/cold duty cycle is
approximately
0.2
, given by the calibration unit geometry, increasing that value
further would require a significant modification of the calibration unit, ii) even
for the most sensitive channel the signal difference associated with the chopper
channel cannot be seen directly, confirming that more elaborated analysis tech-
niques are necessary and iii) no relevant drift within one rotation period can be
observed. As said before, notable drifts were seen in some cases on timescales
on the order of
10 s
(cf. figure 2.5). The conditionally averaged signal in fig-
ure 4.4 supports the conclusion that there is no relevant drift within one period.
The orange curve in figure 4.4a) corresponds to the measured and subsequently
conditional averaged signal of the sensitive channel 23,
fmeas
23 (φ)
. Furthermore,
the graph has been set to the mean values obtained from the previously run
MCMC. Then, Monte Carlo samples (orange curves) have been drawn from
49
Chapter 4. ECE diagnostic calibration
inferred φgeo
1inferred φgeo
2
(a)
−60
−40
−20
0
20
signal in bit
fpred
23 (φ)
fmeas
23 (φ)
(b)
0π/2π3π/22π
−20
−10
0
10
φin rad
signal in bit
fpred
11 (φ)
fmeas
11 (φ)
00.05 0.10.15 0.20.25
avg. time tin s
Figure 4.4.: (a)
shows the sensitive channel 23, while
(b)
depicts the weakly
sensitive channel 11.
φ=
0corresponds to a mirror position »looking« at the
center of the LN
2
cold source. One can see the measured (orange) conditionally
averaged and binned signal for both channels. The predictions (cyan) are
100
Monte Carlo samples, taken after the free parameters of the graph have
been set to their mean values (i. e. a point estimate of the posterior predictive).
the prediction with its multivariate normal uncertainty. Consequently, these
samples are predictions
fpred
23 (φ)
from a point estimate of the posterior predict-
ive. Correspondingly, figure 4.4b) shows the weakly sensitive channel 11 with a
barely noteable signal step between the not and cold phase. It is important to
note that these samples are calculated from the model that allowed the scaling
of the prediction variance, so that the uncertainties match the predictive cap-
ability of the model. Each cyan point in figure 4.4 corresponds to a predicted
effective temperature scaled by the calibration sensitivity, with the offset of a
whole period being removed.
50
4.3. Results
100 150 200 250 300
−20
−10
0
10
Teff in K
ECE signal in mV
channel 23
measured
η−1
23
Figure 4.5.:
Shown are the voltage values in dependence of the effective temper-
ature. Do note that the plotted voltage uncertainties correspond to
b23√α23σ2
23
with
b23
the bit to Volt conversion factor. The temperature uncertainties cor-
respond to the uncertainties given by the priors.
The measured bit values at each mirror angle
φ
are illustrated in figure 4.5,
where the bit values have been scaled to represent voltages and the mirror
angles were converted to their corresponding effective temperature. It should
be noted that the uncertainties on the voltage axis are scaled with the channel
specific variance scaling factor
αi
. The orange curve shows the sensitivity with
its uncertainties as calculated from the graphical model (cf. figure 4.1). Remain-
ing deviations might be caused by
50 Hz
noise (or its higher harmonics) that are
not completely notched out by the bandpass filter properties of the conditional
average. The plot highlights the advantage of this analysis method: While no
other radiometer calibration approach known to the author uses the data that
are taken when the horn pattern collects radiation from more than one refer-
ence temperature, this method allows to estimate the effective temperature (and
corresponding uncertainties) reducing the overall uncertainty. However, one
has to keep in mind that this is valid only as long as the Jeans law approximation
is valid. For the radiometer and the reference temperatures used here this is a
very good approximation. Furthermore, the frequency dependent beam width
is predicted. That implies that if a measurement of the beam width is available,
it can be easily supplied to the model as another observation, thus decreasing
uncertainties of the sensitivities.
51
Chapter 4. ECE diagnostic calibration
0.40.45 0.5 0.55 0.6
0
5
10
15
calibration factor η−1
23 in keV/V
probability density, channel 23
channel 23
channel 11
1 2 345
0
0.2
0.4
0.6
calibration factor η−1
11 in keV/V
probability density, channel 11
Figure 4.6.:
The kernel density estimates of the MCMC samples of the calib-
ration factors of the sensitive channel 23,
η−1
23
, and the insensitive channel 11,
η−1
11 . It is visible that the distribution of the insensitive channel deviates from
a normal distribution by having a pronounced tail towards larger calibration
factors. In contrast, the sensitive channel closely follows a normal distribution.
The kernel density estimates (i. e. the marginal posteriors) of the calibration
factors as obtained from the MCMC are shown in figure 4.6. One of the strengths
of an MCMC based evaluation is that one can get posterior distributions that
are non-Gaussian as well. However, in the case shown here a Gaussian fit is a
reasonable approximation to the posterior for the sensitive channel 23, while
for channel 11 the posterior deviates notably from a Gaussian distribution. Nev-
ertheless, for the sake of simplicity, the results of Gaussian fits to the marginal
calibration posteriors are currently used for all higher level analyses. A gain in
consistency could be achieved, especially for the weakly sensitive channels, if
the asymmetric uncertainties originating from the MCMC would be used. To
quantify the deviation for channel 11: The mean of a Gaussian fit to the ker-
nel density estimate is at
(2.95 ±0.61)keV V−1
, while the absolute maximum
of the kernel density estimate is at
2.77+0.58
−0.78 keV V−1
, corresponding to roughly
93.9%
of the Gaussian mean. However, the standard deviation shows a consid-
erable difference to the asymmetric uncertainty. The predictive capability of the
model is taken into account by scaling the variance via
αi
, thus an appropriate
estimation of the calibration factor uncertainties is achieved.
52
4.3. Results
0.40.45 0.5 0.55 0.6
20
22
24
26
calibration factor η−1
23 in keV/V
beam width in mm
channel 23
0
2
4
6
8
10
log(PDF)
Figure 4.7.:
The posterior distribution for the sensitive channel 23 with the
single channel evaluation model. One can see that the distribution is reason-
ably close to a 2D Gaussian distribution.
As the samples from the MCMC also allow to reveal correlations between
different free parameters, figure 4.7 shows exemplarily the posterior distribution
for the sensitive channel 23 for the the beam width and the calibration factor.
The distribution is reasonably close to a 2D Gaussian distribution.
An example of a posterior revealing more complex relationships between
two parameters can be seen in figure 4.8. There, the PDF of the angle that
determines the beginning of the hot reference temperature measurement phase
and the calibration factor is plotted. One can clearly see the deviation from a
Gaussian distribution. It should be noted that the channel chosen here shows
a relatively low signal to noise ratio. For channels with a better signal to noise
ratio, the distribution resembles again a Gaussian distribution.
4.3.1. Comparison of single and multichannel evaluation
As it is not ab initio clear how large the differences between the models of
varying complexity are, and thus which model is appropriate for practice, a
53
Chapter 4. ECE diagnostic calibration
2 3 4 5 6
0.2
0.4
0.6
0.8
1
calibration factor η−1
11 in keV/V
angle φgeo
1in rad
channel 11
0
1
2
3
4
log(PDF)
Figure 4.8.:
Example for a posterior distribution between the mirror angle
φgeo
1
and the calibration factor η−1
11 using the single channel evaluation.
careful comparison is shown in this subsection. Figure 4.9a shows the calib-
ration factor, which is the inverse sensitivity, for each channel. One can see
that the calibration factors vary over more than two orders of magnitude. A
single mixer is used for the whole spectrum to allow for a better correlation
analysis [52]. This is unusual as many ECE systems use multiple mixers to avoid
frequencies above
18 GHz
after mixing [84]. The single mixer approach leads
to intermediate frequencies up to
40 GHz
which need to be detected. The low
sensitivities for higher frequencies might at least partially originate from dif-
ferent cables used for frequencies below
18 GHz
(corresponding to frequencies
below
140 GHz
in the shown spectrum) and above
18 GHz
. The cable frequency
response damps higher frequencies more [36]. The conversion efficiency of the
extreme broadband mixer also drops for frequencies above
140 GHz
. Individual
diode sensitivities are expected to play an important role as well [37].
Figure 4.9b shows the uncertainties of the different models, normalised to
the multi shared model. Going from the single channel evaluation to a com-
bined model yields substantial decreases in the calibration factor uncertainties
of insensitive channels, although the uncertainties for these channels remain
54
4.3. Results
(a)
10−1
100
101
102
calib. factor in keV/V
single
multi individual
multi shared
(b)
125 130 135 140 145 150 155 160 165
1
2
3
frequency fin GHz
rel. unc. change
single
multi individual
Figure 4.9.: (a)
shows the frequency dependent calibration factors as obtained
for single channel evaluations and the multi-channel analysis with and
without a shared free parameter for the beam width. The uncertainties are only
shown for the multi-channel analysis with a shared (and appropriately scaled)
beam width, as this is the reference value for the relative uncertainty changes
shown in
(b)
. The standard deviation of the specified models is normalised
by the standard deviation of the multi shared model. The given uncertainties
correspond to one standard deviation as calculated from the MCMC samples.
very large. This phenomenon is most likely caused by the additional informa-
tion about the geometrical properties
φgeo
that is mainly provided by stronger
channels, helping the less sensitive channels to determine the begin and end
of the hot/cold phases. Using a single beam parameter leads for a few chan-
nels to a small shift of the calibration factor, also reducing the uncertainties
slightly, but less drastic than the switch from the single channel evaluation to
a multi-channel evaluation model.
Figure 4.10 shows the inferred intensity Gaussian beam width for each ECE
channel. The beam width has been measured in the lab with a 140 GHz source
attached at the receiver end of the antenna and an 2D array of 440 intens-
ity measurements in various fixed distances to the minimum beam waist. A
simple Bayesian Gaussian squared forward model that adapts the prediction
55
Chapter 4. ECE diagnostic calibration
125 130 135 140 145 150 155 160 165
10
20
30
40
frequency fin GHz
beam width win mm
single multi individual
multi shared direct meas.
Figure 4.10.:
The inferred Gaussian beam width of the microwave antenna char-
acteristic for each channel. Reliable values can be obtained if the model using a
commonly scaled beam width according to equation (4.1) is used (multi shared).
The shaded areas correspond to one standard deviation (from MCMC).
uncertainties was used to infer the beam width from the intensity measure-
ments with 10 million MCMC iterations. The inferred beam width was roughly
(20.4±0.2)mm
, with
37.5 cm
distance between the minimum beam waist and
the array measurements. This does not exactly match the distance at which the
microwave foam is located relative to the minimum beam waist (which would
be roughly
26 cm
to
32 cm
), but due to the small divergence of the beam width
the introduced error is small. One can see that switching from the single chan-
nel analysis to a combined channel analysis slightly decreases the uncertainties
for some less sensitive channels. In most cases the beam width shifts slightly
towards values closer to the directly measured width. If only a single beam
width, scaled according to equation (4.1), is used, the uncertainties get drastic-
ally reduced. The measured width is roughly
40 µm
away from the predicted
value by the model with a single beam parameter, with prediction uncertainties
on the order of 0.4 mm.
The variance scaling factors for the different channels are shown in figure 4.11.
One can see that the values are not too far away from
1
, which indicates that
the most relevant physic effects are considered. These values were reduced
from values typically around
2.6
at the begin of the first
W7-X
experimental
campaign by two changes of the setup: i) the horn was changed to include all
polarizations and ii) the container for the liquid nitrogen had a round aperture
56
4.3. Results
125 130 135 140 145 150 155 160 165
1
2
3
LFS HFS
frequency fin GHz
scaling factor α
single
multi shared
Figure 4.11.:
The inferred variance scaling factors
α
for the covariance is a meas-
ure of the model uncertainty for each channel. For the »multi individual« case
there is no visible difference to the »multi shared« case. Note that the chan-
nel at
149.36 GHz
shows
150 Hz
noise that dominates the Fourier spectrum
even after applying the conditional averaging. Subsequent calibration meas-
urements without this noise yielded α25 ≈0.99.
that was changed to a rectangular aperture. The round aperture system was
more sensitive to misalignments of the mirror-antenna system. Indeed an offset
of around
3.5 cm
of the beam on the mirror was measured for the calibration
used for the first experimental campaign. This offset complicates the geometry
and was not reflected in the model. As the physics for each channel should
be similar,
α
should have similar values for the different channels. A notable
difference between the
α
values is an indication that different physics effects
play a role, or at least that these different effects are of different importance. One
could expect that for channels with a small sensitivity electronics effects are of
larger importance, therefore changing the variance scaling. Notable differences
between the models that couple the beam width of different channels directly
and those that do not, and between the individual and multi-channel evaluations
are not observed. The values on the low field side (below
140 GHz
) scatter more
and tend to be larger. This indicates that the uncertainties are underestimated
for these channels, respectively, that the explanatory power of the model is
smaller than for channels on the high field side. A potential source for this
behaviour can be ascribed to implicit hardware assumptions. For example, these
can be violated more strongly for low field side channels, although currently
57
Chapter 4. ECE diagnostic calibration
125 130 135 140 145 150 155 160 165
0
2
4
6
frequency in GHz
radiation temperature in keV
#20171207.006.002
t=0.04 s
t=3.6 s
Figure 4.12.:
Exemplary radiation temperature spectrum from a centrally ECRH
heated
W7-X
plasma discharge, calculated with the calibration factors as ob-
tained from the Minerva model that incorporates variance scaling. The chan-
nel marked in orange corresponds to the ECE timetrace shown in figure 4.13.
no such problem is observed. In summary, one can see that the single channel
evaluation is satisfactory in most cases. However, if time is not a critical factor
it is still beneficial to use a model combining all ECE channels in one Bayesian
model.
4.3.2. ECE spectra
From the calibration procedure radiation temperature spectra can be derived
from measurements done during a plasma discharge. An example is shown in
figure 4.12. The data originate from a
3.8 s
long plasma discharge that was cent-
rally heated with ECRH [85, 86]. Time traces of the main plasma paremeters are
shown in figure 4.13. The ECRH power was
2.5 MW
in the first phase and was in-
creased to roughly
5 MW
shortly after pellet fuelling started. The line averaged
electron density as measured by a single channel dispersion interferometer [87]
rose during pellet fuelling up to about 7×1019 m−3.
Channel 15 (
138.26 GHz
) and 16 (
139.06 GHz
) show a very low sensitivity, lead-
ing to radiation temperatures above
20 keV
and uncertainties of several hundred
percent. Consequently, they were omitted in this plot. For channel 16 this is
expected, as the channels frequency band locates it in the slope of the notch
58
4.3. Results
(a)
pellets
0
2
4
6
8
temperature in keV
#20171207.006.002
Thomson scattering
ECE (136.3 GHz)
(b)
00.511.522.533.544.5
0
2
4
6
time tin s
nein 1019 m−3
interf.
ECRH
0
2
4
6
8
#20171207.006.002
00.511.522.533.544.5
0
2
4
6
time tin s
power in MW
Figure 4.13.: (a)
shows the calibrated signal of an ECE channel close to the
plasma core, compared with a central channel from the Thomson scattering
system [47] for the plasma discharge also shown in figure 4.12. The lines at
0.04 s
and
3.6 s
indicate the spectra shown in figure 4.12.
(b)
depicts the ECRH
and line averaged density from the single channel dispersion interferometry.
The ECRH blips are necessary for the collective Thomson scattering (CTS)
diagnostic [88].
filter. Above roughly
155 GHz
contributions from the third harmonic X mode
emission start to play a role. The spectrum from
0.04 s
shows »shinethrough«
from hot core electrons below
131 GHz
. The reconstruction of the electron tem-
perature profile from the spectrum will be discussed in chapter 5. However,
figure 4.13 shows a comparison of the ECE timetrace of a channel close to the
core to a Thomson scattering channel close to the core. The deviations in the
first second are probably caused by the selection of filters used in the poly-
chromators of the Thomson scattering system that lead to big uncertainties
for electron temperatures above around
7 keV
. The remaining seconds of the
59
Chapter 4. ECE diagnostic calibration
discharge show a good agreement for a wide range of electron densities and
multiple power levels. One of the optimization criteria of
W7-X
is the minim-
ization of Pfirsch-Schlüter currents, which increase with plasma pressure [89,
90]. The Pfirsch-Schlüter currents give rise to the so-called Shafranov shift mov-
ing the magnetic axis outwards. Hence, having the Shafranov shift minimized
makes the magnetic configuration less sensitive to plasma pressure changes.
For the ECE a small Shafranov shift implies that the positions of the channels
are less sensitive to pressure changes than they would be in an unoptimized
stellarator. Nevertheless, the Shafranov shift combined with the diamagnetic
shift (i. e. the finite plasma
β
effect) of the cold resonance positions of the ECE
channels for the experiment shown in figure 4.13 amounts to a maximum of
approximately
5.5 cm
, or
0.35
in effective radii. Thus, depending on the pressure
profile the ECE channel appropriate for a plasma center measurement might
change. This implies that the ECE at
W7-X
is a diagnostic rather sensitive to
the total plasma pressure.
4.4. Conclusions
The Bayesian Minerva model developed for the multi-channel consistent cal-
ibration of a generic microwave radiometer provides the sensitivities in an
automated fashion, insensitive to signal drifts on timescales larger than 1 s. By
using an explicit model for the calibration, the analysis can be done in a more
formalized way. This also allows to obtain feedback on how well the modelled
physical system is understood. In addition, the beam width can be inferred ac-
curately. Also, classical error propagation is linear while a Bayesian analysis
can handle nonlinearities.
Moreover, it allows to obtain non-Gaussian posterior distributions for the
calibration factors, although for the sensitive channels in the radiometer studied
a Gaussian distribution provides an excellent fit to the posterior distribution.
The use of a variance scaling factor allows to obtain uncertainties matching
the predictive capability of the model.
The ECE spectra obtained from typical
W7-X
plasmas do not show unexpec-
ted or unphysical features. For reasonably sensitive channels the uncertainties
are typically on the order of
6%
. Note that the contribution of statistical noise
is small compared to the uncertainties arising from the calibration. While the
high dimensional (98D) model provides in principle the highest consistency,
60
4.4. Conclusions
a comparison to the much quicker and parallelized single channel evaluation
(4D for each channel) yields only little differences, justifying the use of the
simpler approach in most cases. Nevertheless, the comparison with the high
dimensional model was a good consistency check.
Further improvements could be achieved by applying neural networks, as
described in Pavone et. al. [78] to speed up the evaluation. It also would be
interesting to use the calibration model in combination with a plasma model
containing predictions for Thomson scattering and ECE, guaranteeing con-
sistent calibration factors across different diagnostic systems with the models
presented in this chapter providing the tested and prepared ECE calibration
branch.
Appendix B shows the code, respectively keywords, that allow customization
of the developed standalone Java executable.
61
5
Plasma profile inversion
Typically, many ECE analyses rely on the underlying assumption of a black
body emitting plasma with clearly localized ECE positions, which is not neces-
sarily fulfilled over the full plasma radius. In particular, for
W7-X
, the lowest
frequencies probe nominally the plasma edge at the low field side. There, the op-
tical depth can be low (i. e.
τ<3
), with the relativistically downshifted emission
from hot core electrons superposed to the thermal emission from the edge. This
makes the direct use of
Trad
as a surrogate for
Te
impossible in these regions. In
W7-X
this »shinethrough« effect can be on the order of several
keV
for some
channels and thus become dominant, for example during plasma buildup in
figure 4.12 at
0.04 s
below
131 GHz
. However, the »shinethrough« effect occurs
in plasmas with a low electron density, or narrow plasmas with sufficiently
hot electrons, and is not notably visible in the examples shown in this chapter.
An example is depicted in section 6.2. Quantitative analysis of the underlying
electron cyclotron emission and absorption and the corresponding radiation
transport is handled by the TRAVIS code [43] (see also section 2.2). Note that
the model itself is by no means bound to TRAVIS; any ray tracing and radiation
transport code could be implemented straightforwardly.
This chapter is structured as follows: In section 5.1, a detailed description
of the Minerva model and a subsequent description of the methods used to
obtain the initial guess, for both the electron temperature and density profile,
is given. Therein, the approach presented here is also compared to other ways
used to obtain
Te
profiles. Thereafter, section 5.2 shows and discusses results for
an example timepoint in a typical
W7-X
discharge, while section 5.2.3 debates
the influence of NBI heating on the electron temperature and density profiles
as inferred from ECE. Do note that none of the chosen timepoints do exhibit
63
Chapter 5. Plasma profile inversion
notable »shinethrough« behaviour, as this is behaviour is no longer the norm
in discharges from OP 1.2a onwards. Nevertheless, the code can fully handle
»shinethrough« effects, as also discussed in section 5.1. Finally, section 5.3 con-
cludes the results and gives an outlook.
5.1. Minerva implementation
Sophisticated ECE analyses are also used at other plasma experiments, for ex-
ample at JET [91–93] and AUG [94–96], both tokamaks. At JET, Minerva is
also used as the embedding framework for the ECE model, while instead of the
raytracing code TRAVIS the SPECE code is used [97]. A comparison to these
approaches can be found at the end of this section.
The available Minerva models are shown in a simplified manner in figure 5.1.
Subsequently, the structure of the graphical model will be briefly explained.
The descriptions of the continuous electron temperature and density profiles
are done in one of two ways in the model: The first approach consists of the
parametrization that is often used in conjunction with TRAVIS,
f(ρ)=a0�g−h+(1+h−g)(1−ρp)q+h�1−exp �−ρ2/w2���,(5.1)
with
ρ
the variable normalized effective radius,
ρ∈[
0
,
1
]
. The parameters
are:
a0
the value at
ρ=
0,
g
the ratio of edge and core value (
f(
1
)/f(
0
)
),
p
and
q
that dominate the profile gradients,
h
the profile hollowness and
w
the
profile hollowness width, the latter two being useful for hollow profile shapes
as well as peaked profiles. Figure 5.2 illustrates some example profile shapes
obtained with the parameterization of equation (5.1). The second approach uses
a cubic interpolation between the evaluation points of a Gaussian process (cf.
section 3.1) to obtain a continuous profile. For an idea of how possible profile
shapes look like if a Gaussian process is used, see figure 3.1 as an example of
Te
profiles.
To be able to flexibly penalize deviations from zero gradients in the core
virtual observations (cf. chapter 3) have been added. Note that decreasing the
uncertainty of the predicted gradients allows to increasingly force the gradients
to be close to zero – at the cost of a narrowed posterior. Subsequently, the
Te
and
ne
profiles are fed to the TRAVIS nodes for the X mode and O mode predictions.
These TRAVIS nodes get the required ECE metadata, like the line of sight or the
64
5.1. Minerva implementation
ρ
σf
σy
GP
Te
ℓ(x)
xw
x0
ℓ1
ℓ2
Te(GP)
w
h
q
p
Te,0
g
Te, eq. (5.1)
σf
σyGP
ne
ℓ(x)
xw
x0ℓ1
ℓ2
ne(GP)
wh
qp
ne,0g
ne, eq. (5.1)
Te(ρ)ne(ρ)
(∂ρTe)|ρ=0=0
(∂ρne)|ρ=0=0
time
coils
ECE interf. TS, …
VMEC
TRAVIS X mode
TRAVIS O mode
interf. obs.
interf. pred.
interferometry
legend
node
uniform
normal
observed
X mode contrib.
X mode prediction O mode prediction
O mode contrib.
ECE prediction
ECE observation
α
ασ2
Figure 5.1.:
Simplified Minerva ECE profile inversion model, cf. section 3.4.
Either the Gaussian process implementation is used to describe the
Te
and
ne
profiles, or equation (5.1), which can also be used directly for TRAVIS. The
edges do not represent data flow.
65
Chapter 5. Plasma profile inversion
0 0.2 0.40.60.81
0
2
4
6
effective radius ρ
Tein keV
Figure 5.2.:
Shown are some examples for profile shapes that can be obtained
using the TRAVIS parameterisation given in equation (5.1). It is not possible
to get a profile with multiple bumps.
beam diameter from the ECE datasource. The magnetic configuration, respect-
ively the VMEC equilibrium, can be obtained in three ways: i) from a VMEC run
directly provided by the user through a VMEC run identifier of the webservice,
ii) from a dedicated VMEC run based on the settings of a user specified VMEC
identifier, but taking the coil currents of the selected experiment directly from
the corresponding coil current datasource and iii) from the VMEC identifier
supplied by a datasource, also written during this thesis, that provides the best
guess of the equilibrium from a precomputed set of reference equilibria, taking
into account the diamagnetic energy [98]. Note that there is no automated equi-
librium reconstruction running at
W7-X
at the moment. The resulting X mode
and O mode predictions are then rescaled each by a uniform free parameter
between
0
and
1
. This is done to take into account that we do always measure
a mixture of X mode and O mode, and, especially for the O mode, to at least
partially account for contributions from multiple plasma passes. Thus, these
scaling factors should not be interpreted to reflect exactly the X mode or O mode
content. Experimentally, it is expected to measure more than about
96 %
X mode
and correspondingly less than
4%
O mode while using the experimental setup
for the X mode. Adding the rescaled X mode and O mode predictions yields
then the overall ECE prediction which is compared to the measured ECE spectra
obtained via the datasource. A scaling parameter
α
on the variance
σ2
can be
used to have the uncertainties match the predictive capability of the model, see
66
5.1. Minerva implementation
Start
initial guess
𝒪(minutes)
write initial
guesses to database
MAP
𝒪(hours)
write most probable
results to database
MCMC
𝒪(days)
write results with
uncertainties to database
Stop
Figure 5.3.:
First, an initial guess is calculated by fitting the cold resonance
Trad
profile (which is assumed to be
Te
). This takes on the order of few minutes.
Subsequently an MAP inversion is done, finding the most probable values for
the free parameters. This usually takes up to
24 h
. Finally, an MCMC inversion
is done, such that samples drawn from the posterior allow an estimation of
non-Gaussian correlated uncertainties.
chapter 3. Furthermore, the electron density profile is integrated over the line
of sight of the interferometer and the result is compared to the line integrated
density as measured by interferometry, thus keeping ECE and interferometry
consistent by predicting both diagnostic measurements simultaneously.
In figure 5.3 the general ECE profile analysis scheme is shown. The analysis
chain is as follows: i) the initial guess is calculated after discarding channels that
are most likely affected by »shinethrough«
1
, followed by ii) an MAP inversion
with a maximum of
150
iterations and finally iii) an MCMC inversion with a
burn-in of
105
iterations and
3×104
samples drawn afterwards. After each stage
in the analysis chain, the results are written to the central ArchiveDB [99] to
make them available to the W7-X team.
1
As a reminder: This typically affects edge channels with an optical depth
τ<
3, for which
relativistically downshifted emission from the center is not fully reabsorbed. For an example
see figure 4.12.
67
Chapter 5. Plasma profile inversion
To determine the initial guesses in the case the
Te
and
ne
profiles are defined
by Gaussian processes, the following steps are performed. The prior distribu-
tions for the
Te
and
ne
profile distributions are multivariate normals with a
mean of zero and a covariance given by the hyperparameters
ℓi
1
,
ℓi
2
,
xi
0
,
xi
w
and
σi
f
[62] with
i
the values for either the electron temperature or density profiles,
cf. section 3.1. The prior values are typically obtained in the following way:
First, a simple Gaussian fit to the radiation temperatures at the cold resonance
positions of the ECE frequencies for the given VMEC equilibrium is done. This
Gaussian fit is used to determine channels which are likely to be affected by
»shinethrough« by excluding channels whose radiation temperature exceeds
the fit value by more than one standard deviation (dominated by the calibra-
tion uncertainty for the channel as calculated by the analysis in chapter 4)
2
.
The radiation temperatures of these channels are then put as observed values
in a model with the same Gaussian process setup as in the main ECE infer-
ence model (cf. figure 5.1), so an identical length scale function, depending on
the same hyperparameters is used. The only difference is the always activated
variance scaling of the prediction (cf. chapter 3), which has the advantage to
reduce the susceptibilty to deviations of the used VMEC equilibrium to the true
plasma equilibrium. As this simpler model is linear, one can use linear Gaussian
inversions (cf. section 3.4.2) to solve the problem analytically for fixed hyper-
parameters. This allows for a Hooke and Jeeves optimization (cf. section 3.2)
using a cost function that combines the probability densities of the evidence
with the hyperparameter prior. Effectively, this leads to an optimization of the
Gaussian process and its hyperparameters by applying Occam’s razor, therefore
taking model complexity into account by penalizing more complex models, see
chapter 3. The resulting hyperparameters are then put in the main ECE model
as values for the prior
Te
hyperparameter distributions. Correspondingly, the
resulting
Te
values of the truncated multivariate normal are put in the
Te
distri-
bution of the main model as well. To make the initial guess even more robust
against equilibria that for example do not take plasma pressure into account,
one can specify a shift of the initial cold resonance positions, thereby changing
the fitted electron temperature profile. Practically that means that the predic-
tions of the spectrum are calculated for shifts ranging from
−
Δ
ρ
to Δ
ρ
in steps of
2
Two further conditions have to be met: i) the channels effective radius exceeds
ρ>0.5
and ii)
all channels further out (i. e. with a larger effective radius) were identified as »shinethrough«
affected, although ii) is applied separately for the low field side and high field side.
68
5.1. Minerva implementation
δ
ρ
, comparing the logarithm of the joint (i. e. prior times likelihood) probability
density of the ECE observation for the shifts and selecting the shift that gives
the best prediction. While this does cost some time, the additional computing
time (on the order of seconds to few minutes) is small compared to the overall
analysis time (on the order of days), making the gain in robustness worthwhile
if the available VMEC equilibrium is expected to deviate notably from the real
equilibrium. Usually, the reference VMEC equilibrium found automatically
3
is
good enough that shifting the profile is not necessary. Exemplarily, the resid-
uals of the initial guess of the time point analysed in section 5.2 are shown in
figure 5.6. Note that there are other options available to determine the initial
Te
profile, although not regularly used. They consist of: i) the same procedure
described above, but using only channels on the low field side, ii) the same
procedure described above, but using only channels on the HFS, iii) using the
Te
values as determined by the Thomson scattering system to fit a Gaussian
process to, iv) fitting the Gaussian process to a manually provided
Te
profile and
v) fitting the Gaussian process to a Teprofile parameterized by equation (5.1).
The Thomson scattering
ne
profile [47] provides the observed values for
the same Gaussian process procedure described above, and the result is sub-
sequently rescaled such that the integral over the density profile along the inter-
ferometer line of sight
4
matches the measured line integrated electron density
value by interferometry. Do note that the deviation between the Thomson scat-
tering and interferometry system can be caused for example by calibration
issues of the Thomson scattering system. It is known that misalignments of the
Thomson scattering lasers occuring occasionally also contribute significantly
to a distortion of the Thomson scattering
ne
profile. This is further complicated
by the Thomson scattering system using three different lasers from OP 1.2b
onwards, such that potential misalignments of multiple lasers have to be taken
into account if averaging over a time window long enough to contain multiple
Thomson scattering laser pulses is desired. Sticking to one laser avoids prob-
3
The reference equilibrium is found by first comparing the experimental coil currents with
the coil currents from a precalculated set of equilibria and selecting all equilibria that have
a reasonably close magnetic configuration. Amongst these equilibria, the one with the
smallest deviation of their integrated plasma pressure profile to the maximum experimental
diamagnetic energy during the discharge is selected.
4
Technically, the interferometer has two slightly different lines of sight, as the laser passes
through the plasma, gets reflected and passes the plasma again. The exact lines of sight are
used for the calculations.
69
Chapter 5. Plasma profile inversion
lems with different laser alignments, but reduces the frequency with which
usable data is available. The rescaling is an easy method to make the
ne
profile
consistent with the line integrated electron density as observed by the interfer-
ometer, such that any deviation from this initial density profile in the inference
is driven by information from the ECE. Other options available as reference
for the initial density guess are: i) unscaled Thomson scattering profiles, ii)
default profiles scaled by interferometry, iii) manually chosen profiles and iv)
manually chosen profiles scaled to the line integrated density value measured
by interferometry.
It is important to put the zero gradient constraints via virtual observations
at the core already on the initial guesses. If one does not put them there, the
low probability density for the virtual observation nodes is going to dominate
the joint probability density. Hence, obtaining quickly a satisfying prediction is
much more difficult as the optimization strives to reduce the virtual observation
penalty, more or less regardless of the remaining profile shape.
While the option to let the Gaussian process hyperparameters (i. e. the para-
meters that determine the prior distribution of other parameters) free is fully
implemented in the main model, typically one will keep them at their initial
guesses, as the computational cost of using them as free parameters is con-
siderable. If TRAVIS and/or the analysis routines, like the MCMC, are sped
up sufficiently, leaving the hyperparameters free could help less constraining
the posterior. The evaluation strategies for the MAP and MCMC inversion are
detailed in section 3.4.2.
5.1.1. Limitations
Five systematic uncertainties have been identified which hamper the ECE spec-
trum interpretation and thus the profile inference: i) the effect of electron dens-
ity outside the last closed flux surface, ii) multipass microwave radiation trans-
port, iii) using only a central ray instead of multiple rays, iv) noise levels re-
spectively parasitic gyrotron mode effects for individual channels and v) errors
in the VMEC equilibrium.
As the density does not drop to zero at
ρ=1
, the best approach would be to
treat the density outside the last closed flux surface fully three dimensional to
avoid unphysical refraction patterns caused by abrupt changes in the refractive
index. However, as there is no density measurement along the line of sight
of the ECE, and the computational cost of a 3D treatment would be high, a
70
5.1. Minerva implementation
workaround will be used in the future: A simple extrapolation of the density
profile as measured by the Thomson scattering profile outside of the last closed
flux surface will allow a smooth density drop off, most likely reducing the error
that is made with respect to the ECE beam refraction.
Another problem is the handling of multiple passes of the beam through
the plasma. After e.g. the X mode passed through the plasma once, it does get
reflected. Upon hitting the plasma edge again, the polarisation does not entirely
match the one that would be required for the ray to continue as pure X mode.
Hence, one would have to continue with part of the power in X mode and
part in O mode. In practice, TRAVIS neglects the power that would go to the
O mode and continues just with the X mode. This applies to each further plasma
entrance as well. In addition, the requirement for the angle with which the beam
hits the plasma vessel after the first plasma pass being accurate and precise
grows with the number of reflections taken into account, as slightly different
angles may lead to completely different beam paths after several reflections due
to the complex 3D shape of the
W7-X
vessel. This makes it impossible to obtain
an accurate O mode prediction if more than two reflections are required. How
large this effect in W7-X is remains an issue for further studies.
Furthermore, due to the 3D geometry of
W7-X
, many rays instead of just
the central ray would be necessary to take into account reflections at the wall
realistically. However, the calculations were done with a single ray passing once
through the plasma due to the accompanying computational cost of multiple
rays. Note especially that using
i
rays with
j
reflections each would increase the
computational cost already by a factor of
ij
, even though the contributions from
the ray polygon intersection tests that have to be performed for each reflection
(cf. section 2.2) are still neglected.
It should be noted that in the last operational campaign, OP 1.2b, some chan-
nels in some discharges showed highly anomalous behaviour, most likely caused
by radiation from parasitic modes from the gyrotrons [100] several GHz away
from
140 GHz
and thus not filtered out by the ECE notch filter. However, this
is a problem that in the future will be solved on the hardware side by slightly
shifting the frequency bands of the ECE channels, hopefully avoiding these
parasitic mode effects.
Another huge factor are errors in the VMEC equilibrium. Due to the shal-
low gradient of the magnetic field strength the ECE at
W7-X
is highly sensit-
ive to small changes in the absolute magnetic field. If the VMEC equilibrium
e.g. fails to describe the plasma pressure profile accurately the positions from
71
Chapter 5. Plasma profile inversion
where the radiation is expected can shift notably, cf. the discussion at the end of
section 4.3.2. Currently no automatic (Bayesian) equilibrium reconstruction is
running at
W7-X
, although preparations are being made in the Minerva frame-
work.
5.1.2. Comparison to other ECE inference procedures
In order to be able to place the analysis presented here into context, differences
and similarities to the analysis approaches used at JET as well as AUG are
discussed.
The inference procedure used at JET uses Minerva as well, but uses SPECE in-
stead of TRAVIS [91–93]. Solver for plasma electron cyclotron emission (SPECE)
is specific for tokamaks: Wall reflections are approximated by assuming parallel
walls on the inner- and outerboard side, whereas TRAVIS can handle 3D geomet-
ries and is thus suitable for stellarators as well as for tokamaks (cf. section 2.3.2).
Further differences between the Minerva approach chosen at JET and at
W7-X
can be found in the different kernels of the Gaussian processes. The model at
JET uses a generalized squared exponential kernel [58, 93], while at
W7-X
the
kernel based on [61] (cf. section 3.1) is used. Also, the model at JET includes a
reflectometer, two Martin-Puplett interferometers and a radiometer, while the
W7-X
model includes the heterodyne radiometer together with the single chan-
nel dispersion interferometer. However, due to the implementation of TRAVIS
within Minerva, adding reflectometers to the
W7-X
model is straightforward,
as TRAVIS can be used to predict this diagnostic as well. In addition, virtual
observations to penalize non-zero
Te
and
ne
gradients in the core are used in
the
W7-X
model and the models to obtain the initial guess differ substantially.
Extensive details of the JET model can be found in [93].
At AUG, data from the ECE system, the lithium beam emission spectroscopy
(LiBES) system and deuterium cyanide laser interferometry (DCI) system are
combined [94–96]. Uncertainties are determined via
χ2
binning, while the ap-
proach presented in this thesis uses an MCMC to obtain samples from the pos-
terior. The ECE uncertainties for the radiation temperature at the two machines
are given by
ΔTAUG
rad =7%Trad,dat +(Trad,dat/fSNR)2+σstat +15 eV,(5.2)
ΔTW7X
rad =√σ2
calib +σ2
stat,(5.3)
72
5.2. Results
where
7%Trad,dat
includes the systematic calibration uncertainties,
fSNR
is a
factor considering the channel specific signal to noise ratio steming from the
calibration,
σstat
are statistical uncertainties from the chosen time window. The
digitization error is specified to be
15 eV
. The calibration uncertainties at
W7-X
are denoted by
σcalib
and are determined as described in chapter 4. For the ini-
tial guess of the electron density profile at AUG the LiBES and DCI diagnostics
are used, and by default a shift determined by the electron density profile at
the maximum curvature shifted to the point where the electron temperature
equals
100 eV
. The energy flux density used to calculate the absorption coeffi-
cient in the AUG model uses the electromagnetic energy only, while TRAVIS
also accounts for the »sloshing« energy (i. e. a flux of kinetic energy caused
by coherent particle motion in the ECE wave [40]). Currently, wall reflections
at AUG are handled in 1D with a reflection coefficient that can be chosen as
a model parameter, while TRAVIS works in 3D and reflection coefficients also
defined in 3D, but not used as model parameters. Further assumptions at AUG
include
Te(ρ>1.02)<50 eV
and
Te(ρ>ρWall)<2 eV
. However, the evaluation
at AUG takes considerably less time compared to the full analysis at
W7-X
(
𝒪(min)
for the initial guess,
𝒪(hour)
for the MAP and
𝒪(day)
for the MCMC).
5.2. Results
The previously described model was used to analyse exemplarily the plasma
at two times in a high mirror configuration in #20180823.016.002. Figure 5.4
depicts an overview of that discharge. In the first second, the ECRH delivers
approximately
2 MW
, the line averaged density rises to about
3×1019 m−3
, a
Thomson scattering system channel and an ECE channel similarly close to the
plasma core show electron temperatures in reasonable agreement. From
1 s
to
2 s
the plasma is relatively stationary. After the ECRH power has been increased
to
4 MW
at
1 s
, the electron temperature and density rise to an electron temper-
ature of circa
3 keV
and a line averaged density of approximately
4.6×1019 m−3
at
2 s
. From
2 s
to
4 s
the ECRH is modulated with a frequency of
17 Hz
to allow
for heat pulse propagation measurements. The first profile analysed within this
chapter was taken at
4.45 s
, with an averaging over
50 ms
to have the statistical
contribution to the ECE uncertainty vanish, while the contribution of the cal-
ibration to the ECE uncertainty remains unchanged. Until that time, the line
averaged density decreased slightly to about
4.1×1019 m−3
. At
4.5 s
NBI heating
73
Chapter 5. Plasma profile inversion
(a)
heat pulse
modulation
0
2
4
temperature in keV
#20180823.016.002
Thomson scattering
ECE (136.3 GHz)
(b)
012 3 4 5 6
0
2
4
time tin s
nein 1019 m−3
interf.
ECRH
NBI
0
2
4
#20180823.016.002
012 3 4 5 6 0
2
4
time tin s
power in MW
Figure 5.4.:
Shown is the plasma discharge also shown in the other plots in
chapter 5.
(a)
shows the calibrated signal of an ECE channel close to the core,
compared with a central channel from the Thomson scattering system [47].
A comparison to the data at
5.15 s
, indicated by another vertical black line, is
done in section 5.2.3.
(b)
depicts the ECRH and NBI heating power as well as
the line averaged density from the single channel dispersion interferometry.
The vertical black line at
4.45 s
indicates the time at which the analyses shown
in figures 5.5 to 5.8 and 5.11 to 5.14 was performed.
was switched on, providing roughly
3 MW
more input power. In addition, the
NBI acts as a considerable particle source. This leads to an increase in the line
averaged density to circa
5.4×1019 m−3
. Note that not all additional particles
stem from the NBI beam itself, but also from the beam duct connecting the
NBI source to the plasma vessel. Consequently, these ions carry less energy.
Also, the electrons contributed from the NBI beam are rather cold. In sum with
potential heat transport changes due to the increased power that are currently
under investigation, the electron temperature dropped to around
2.5 keV
. The
74
5.2. Results
0 100 200 300 400
−280
−260
−240
−220
−200
1.
2.
3.
time in h
log(joint)
#20180823.016.002 at 4.45 s
MAP
MCMC burn in
MCMC
Figure 5.5.:
The logarithm of the joint (i. e. prior times likelihood) of the profile
inversion model with fixed hyperparameters for the whole analysis. Finding
the MAP took around
4.4 h
. One can see that more than
100 h
are required
to get the MCMC reasonably stationary. 1. corresponds to the initial value in
figure 5.6b), 2. to the MAP result and 3. to the MCMC samples.
electron temperatures and densities were roughly stationary around
5.15 s
, at
which the second profile for this chapter has been analysed. The effects of NBI
heating on the plasma parameters are briefly discussed in section 5.2.3. Finally,
the discharge ends after the NBI has been switched off at
5.3 s
and the electron
density and temperature were relaxing back to their pre NBI heating values.
The quality of the analysis routine is described before discussing the analysis
results of the spectrum taken at
4.45 s
. Figure 5.5 shows the trace of the logar-
ithm of the joint (i. e. prior times likelihood) probability density of the model
during the evaluation, which serves as a quantification of the quality of the
overall fit. The logarithm of the joint probability density is calculated by taking
all probability nodes in the model, free and observed, and adding up the logar-
ithms of their individual probability densities. It is clear that the MAP inversion
is useful to move the initial guess of the MCMC to reasonably likely values. One
can see that the number of samples used for the burn-in of the MCMC, which
is used to optimize the proposal distribution adaptively as well as the location
in the probability landscape, is chosen reasonably large with
100 000
samples.
By eye, the chain looks converged. After the burn-in, the adaptive proposal
adapter was deactivated and
30 000
iterations of the MCMC have been used to
75
Chapter 5. Plasma profile inversion
estimate the uncertainties shown in figures 5.6 to 5.8 and 5.11 to 5.14. However, a
total analysis time of more than
400 h
is a considerable drawback, making this
analysis unsuitable for a quick analysis between discharges. This highlights
the need for optimization on the TRAVIS side, which will be parallelized on
a frequency as well as on a ray basis, potentially providing a 32 fold speedup.
Another twofold speedup could be achieved by parallelizing the calculations for
the X mode and O mode, requiring a webservice with a properly implemented
enterprise service bus (ESB). The inversion methods, namely the MCMC, can
also be optimized in various ways, for example by multilayer MCMC sampler
adaption showing substantial speedups in benchmarks [68].
5.2.1. Plasma profiles and observations
The observed spectrum is compared to the predicted X mode and O mode meas-
urements in figure 5.6. This is the most crucial plot of the analysis, as it allows
the clearest possible assessment of whether the result is satisfactory. In fig-
ure 5.6a) a direct comparison of the predicted radiation temperatures for the
X mode and O mode to the measured values is shown. For the X mode one can
see that for most of the frequencies the predictions are very close to the ob-
served values, thus the O mode contribution is small, which was expected as the
radiometer was setup to predominantly measure X mode. A notable deviation
can be seen above
157 GHz
, where TRAVIS does not find a resonance within the
used line of sight through the plasma. The fact that the observed values are lar-
ger than zero in that frequency region are probably due to ECE of other toroidal
locations and cannot be easily modelled. For the channel located at
145.76 GHz
,
a seemingly systematically to high value was observed throughout this day.
No reason for this behaviour is currently known. The prediction uncertainties
for the channels around the plasma core near
135.6 GHz
have uncertainties be-
low
10 %
of their radiation temperature value. Nevertheless, the initial guess
determined a length scale for the Gaussian process that leads to the relatively
smooth spectrum in that frequency range, even though this means some of the
observed values differ by more than a standard deviation from the predicted
values. If the hyperparameters were free in the analysis, the MCMC samples
would most probably be more spread out, so the width of the sample spread
shown here is a lower bound. The O mode reaches its maximum contribution
near
132 GHz
. While the MAP inversion result yielded no contribution from
O mode, the MCMC predicts a typical O mode contribution of few tens of eV.
76
5.2. Results
0
1
2
3
(a)
Trad in keV
#20180823.016.002 at 4.45 s
X mode
O mode ×10
observed
125 130 135 140 145 150 155 160 165
−0.5
0
0.5
(b)
frequency fin GHz
residual in keV
result
init.
Figure 5.6.: (a)
shows the comparison of the predicted and observed
Trad
spectra
for the first vertical black line in figure 5.4. The thick, darker lines represent
the MAP inversion result, while the thin lines are MCMC samples to give an
idea about the uncertainties. The remaining residuals are shown in
(b)
with
the predicted uncertainties in marine. TRAVIS could not find any resonance
for the channels with grey background, so they should be neglected. That the
ECE signal for these channels does not completely go back to zero is probably
due to multiple reflections at the wall.
In order to be able to better assess systematic effects, figure 5.6b) shows the
residuals of the combined X mode and O mode prediction together with the
prediction uncertainties mainly dominated by the calibration uncertainties. The
green line reflects the residuals of the initial guess, while the purple lines stem
from the MAP (thick) and MCMC inversion (thin). One can see that there are
no huge differences between the initial guess and the MAP or MCMC inversion
result. For analyses that take place between discharges in the control room, the
quickly available initial guess thus offers a good approximation for the com-
plete Minerva analysis. Below
131 GHz
and above
147 GHz
the residuals of the
inversions show a slight improvement over the initial guess. The only notable
deviation in these frequency ranges is the three channels at the HFS for which
77
Chapter 5. Plasma profile inversion
0 0.2 0.40.60.811.2
0
2
4
effective radius ρ
Tein keV
#20180823.016.002 at 4.45 s
ECE
Thomson scattering
XICS
Figure 5.7.:
The plot shows the electron temperature profile as resulting from
the inference done on the ECE model, as well as the values obtained from the
Thomson scattering and the X-ray imaging crystal spectroscopy (XICS) [48]
diagnostics. Thin lines represent MCMC samples and the darker, thick lines
the MAP results. The Thomson scattering positions outside the last closed
flux surface are calculated via a simple extrapolation scheme.
TRAVIS, as mentioned above, found no resonances. Besides the systematic de-
viation at
145.76 GHz
, some channels show residuals of up to about
500 eV
. A
possible explanation might be systematic differences between the invessel part
and the calibration unit part of the ECE, although much care has been taken to
make the parts as similar as possible, cf. section 2.3.1.
The derived
Te
profile is plotted in figure 5.7 along with the results from the
Thomson scattering and X-ray imaging crystal spectroscopy (XICS) systems.
The ECE samples are taken from the corresponding MCMC run (cf. figure 5.5).
An MCMC run on the XICS Minerva model [48] produced the samples for the
XICS system. The Thomson scattering data are also evaluated with an MCMC
within the Minerva framework, but as only summary statistics are stored on the
archive, no samples from the posterior can be shown. The uncertainties of the
Thomson scattering system do not take into account calibration and systematic
uncertainties, but only the statistical errors. ECE and Thomson scattering show
a good agreement between an effective radius of
0.3
and
1
. In the core the
difference is more notable, but the inferred uncertainties are bigger for both
diagnostics as well. Note that the ECE channels in the core show in general a
78
5.2. Results
0 0.2 0.40.60.811.2
0
2
4
effective radius ρ
nein 1019 m−3
#20180823.016.002 at 4.45 s
ECE
Thomson scattering
Figure 5.8.:
The plot shows the electron density profile as resulting from the
inference done on the combined ECE and interferometry model, as well as the
values obtained from the Thomson scattering system. For the ECE the thin
lines represent MCMC samples and the thick line the MAP result. Note that
the Thomson scattering system is not part of the model and consequently not
part of the inversions.
larger prediction uncertainty, thus, a more flat profile shape corresponding to a
lower complexity is found. XICS shows in general systematically higher values
than the ECE, but the shape of the profile is similar. However, XICS does match
Thomson scattering better in the core. The decrease of the XICS MAP electron
temperature outside an effective radius of
0.8
is due to a virtual observation that
encourages small values, while the large spread of XICS posterior samples in
this region indicates that the XICS raw data contains only a small information
content. This is due to the decreasing density of ionization states of the ions
being detected by XICS. It is currently not known why there is a systematic
deviation between XICS and the other two diagnostics.
The electron density profile as inferred is shown in figure 5.8. Keep in mind
that »ECE« represents the combined model of the ECE and the single channel
dispersion interferometry. As most of the ECE channels are reasonably optic-
ally thick and the electron density is far enough away from the cutoff of the
ECE channels, the inference of the electron density profile yields a broad pos-
terior. Interferometry ensures that the line integrated electron density remains
approximately the same (cf. figure 5.9). The inferred
ne
profiles exhibit a rel-
79
Chapter 5. Plasma profile inversion
1.08 1.11.12 1.14
0
10
20
∫nedlin 1020 m−2
probability density
#20180823.016.002 at 4.45 s
interf. pred.
interf. obs.
Figure 5.9.:
The kernel density estimate (i. e. the marginal posterior) of the
interferometry prediction as calculated from a thousand MCMC samples.
atively linear shape as this corresponds to a comparatively low complexity of
the profile shape, and due to the hyperparameters being fixed for this analysis.
This is the reason that the uncertainties appear to be lower in the middle of
the profile. In comparison to the electron density as inferred by Thomson scat-
tering measurements the profile by ECE and interferometry is systematically
higher. This discrepancy might be caused by the Thomson scattering laser be-
ing misaligned, thus leading to a loss in collected scattered photons creating
correspondingly inaccurate absolute electron density measurements including
a potentially wrong profile shape. Furthermore, spectral calibration uncertain-
ties of the Thomson scattering system are not taken into account. Chapter 6
discusses in detail the influence that densities above the cutoff of some ECE
channels have on the inference of the density profile.
Figure 5.9 shows the marginal posterior estimated by the kernel density es-
timate of the prediction for the single channel dispersion interferometer, i. e.
the electron density integrated along the line of sight of the interferometer.
Due to the retroreflector used, the line of sight crosses the plasma twice, thus
creating the seemingly high values of the observed value. The uncertainties
of the interferometry prediction were fixed at
2×1018 m−3
, which is estimated
from calibration uncertainties and drifts. Newer estimates suggest a prediction
uncertainty of
6×1018 m−3
to be more appropriate. One can see that there is a
good agreement between the predicted and observed values.
80
5.2. Results
−2−1012
0
0.2
0.4
0.6
0.8
inverse core gradient scale length in cm−1
probability density
#20180823.016.002 at 4.45 s
1/LTeprior
1/LTeposter.
1/Lneprior
1/Lneposter.
Figure 5.10.:
The kernel density estimates (i. e. the marginal posteriors) of the
virtual observation predictions as calculated from a thousand MCMC samples.
Note that for the representation the more intuitive inverse core gradient scale
length was chosen over the more technical quantities used in the model.
Figure 5.10 shows the priors and marginal posteriors of the virtual obser-
vation predictions that were used to constrain the gradients in the core to be
close to zero. The gradient was calculated by having one Gaussian process point
directly in the core, and another one very closeby such that the gradient scale
lengths at the core can be approximated by
1
Lq
=∇q
q=Δq
qaΔρwith q∈{Te,ne}.(5.4)
The minor plasma radius is denoted with
a
. Since the evaluations in this chapter
have been done the gradient as calculated directly from the cubic interpolation
is used. It should be noted that the gradient prediction has been provided with
a large uncertainty (one standard deviation) of
100 keV
or
100 ×1019 m−3
, re-
spectively. It can be seen that the posterior is for both the electron temperature
and density inverse core gradient scale lengths much more narrow than their
corresponding priors. This corresponds to smaller gradients in the plasma core.
Potentially, the priors could have been chosen much more narrow, reducing the
prior-posterior discrepancy. If desired, the uncertainty of the predicted gradi-
ents can be reduced, reducing the inferred gradients at the price of a more
sharply defined model posterior.
81
Chapter 5. Plasma profile inversion
optically »thick«
125 130 135 140 145 150 155 160 165
0
5
10
frequency fin GHz
optical depth τ
#20180823.016.002 at 4.45 s
X mode
O mode ×10
Figure 5.11.:
Shown is the optical depth spectra of the X mode and O mode, the
latter one upscaled by a factor of
10
to make it more visible. The thin lines
represent MCMC samples and the darker, thick lines the MAP results. TRAVIS
could not find any resonance for the channels with grey background, so they
should be neglected.
5.2.2. Details of ECE specific parameters
Figure 5.11 shows the optical depth (cf. equation (2.23)) spectra of the X mode
and O mode, the latter one upscaled by a factor of
10
to make it more visible.
Optical depth is a measure of how much radiation is absorbed. One can see
that the optical depth for the X mode reaches values above
3
. This is used as
a threshold for sufficient optical thickness to consider the plasma opaque at a
given frequency. The threshold stems from estimations of the error of
Te/⟨Te⟩
to be lower than
15 %
compared to the corresponding radiation temperature
quantities. Here,
Te
represents the variation of
Te
and
⟨Te⟩
the time average of
Te
.
For estimations of
Te
directly an optical thickness of
0.7
is sufficient [101]. This
can give a useful estimation on which channels can be used for the heatwave
analysis preceding the timepoints analysed within this chapter. If a proper
profile reconstruction for each point during the heatwave modulation would be
done, one could avoid using the optical depth dependent radiation temperature
spectrum as a surrogate for the electron temperature spectrum altogether. Thus
no requirements on the optical depth would remain (but a low optical depth
would produce a larger uncertainty in the electron temperature profiles). The
82
5.2. Results
plasma for the O mode is not opaque, indicating that multiple passes of the
beam through the plasma should be taken into account. The X mode plateau
seen between roughly
127 GHz
and
144 GHz
at circa
9
is an analysis artifact
caused by TRAVIS stopping the ray tracing once no relevant contributions will
come from further following the ray. While this can be easily deactivated, it
does give some small perfomance gains and is thus usually used. The optical
depth of the second harmonic X mode is [36]
τX2 ∝FX(q)Teωc,0LB(ωp,e/ωc,0)2,(5.5)
FX(q)=�12 −8q+q2
12 −4q�1/2�6−q
6−2q�2
,(5.6)
q=(ωp,e/ωc,0)2.(5.7)
For electron densities below approximately
1020 m−3
increases in the electron
density will enhance the optical depth
τX2
. However, larger densities lead to
FX(q)
getting smaller. Due to the direct proportionality a decrease in the elec-
tron temperature and the gradient length of the magnetic field
LB
will diminish
the optical depth. This is the cause of the dwindling optical depth on the ECE
HFS at
W7-X
manifested in figure 5.11. The posterior samples indicate that the
X mode optical depth for a channel is uncertain by up to
1
. The optical depths
of the ECE channels below
149 GHz
exceed
3
even taking the uncertainties
into account. Thus, they can be considered optically thick (for measurements
done predominantly in X mode), which is important to know e. g. for heatwave
analyses.
Figure 5.12 shows the electron momentum spectra normalized to the thermal
electron momentum at the points where the radiation emission has its center
of mass. The electron momentum per unit mass (i. e. a velocity) is defined as
u=vγ (5.8)
with
v
the electron velocity and
γ
the relativistic correction. The thermal elec-
tron momentum per unit mass is defined as
uth =√2Te/me(5.9)
with
Te
the electron temperature at the center of mass of the emission and
me
the electron mass. A value of approximately
u/uth ≈1
indicates that the
83
Chapter 5. Plasma profile inversion
125 130 135 140 145 150 155 160 165
1
2
3
4
frequency fin GHz
u/uth
#20180823.016.002 at 4.45 s
X mode
O mode
Figure 5.12.:
The plot shows the electron momentum spectra normalized to the
thermal electron momentum at the point where the radiation emission has its
center of mass. The thin lines represent MCMC samples and the darker, thick
lines the MAP results. The shaded range (only shown of the MAP result) show
the normalized momenta at the lower and upper integration boundary of the
radiation emission. TRAVIS could not find any resonance for the channels
with grey background, so they should be neglected.
main radiation contribution stems from the thermalized bulk plasma, while
higher values hint at nonthermal contributions. One can see that the MCMC
samples are hardly visible except for some O mode channels, thus showing
that the results are very stable
5
. For the X mode most of the radiation is from
thermalized plasma, notably between roughly
130 GHz
to
143 GHz
. This fre-
quency range is slightly smaller than the one derived from the optical thickness
considerations discussed above. Outside of that frequency range the values of
the normalized momenta increase up to about
1.6
, with an increasing range of
normalized momenta contributing to the predicted radiation. O mode channels
show a normalized momentum component ranging from
1.75
to
2.8
at their
radiation emission center of mass, indicating that there are considerable con-
tributions from hotter core electrons. Also, the lower and upper integration
boundaries show a broad range of normalized momenta contributing, from ap-
5
The shaded range correspond to the normalized momenta at the lower and upper integration
boundary (cut where
5%
and
95 %
of the emission are contained, respectively) in normalized
momentum space, here shown for the MAP result.
84
5.2. Results
1.00.50.00.51.0
0
1
2
3
outboard/LFS inboard/HFS
effective radius ρ
Trad in keV
#20180823.016.002 at 4.45 s
X mode
O mode ×10
Figure 5.13.:
The plot shows the MAP predicted X mode and O mode radiation
temperatures mapped to the effective radius position at which the center
of mass of the radiation emission occurs. The thin lines represent MCMC
samples and the darker, thick lines the MAP results. TRAVIS could not find
any resonance in the plasma for the X mode channel that »jumped« back to
ρ≈
0. LFS represents the outerboard low field side, HFS the inboard high field
side.
proximately
1
to
3
, making clear that the radiation temperature predicted for
a channel should not be directly understood as the electron temperature. Note
that these integration boundaries are, as all results shown, calculated for the
central frequency only. Taking into account the finite bandwidth of the ECE
channels would change accordingly the range of momenta contributing to the
signal.
Figure 5.13 shows the radiation temperature profiles mapped via TRAVIS and
the best known VMEC equilibrium for that discharge, hence with radiation
transport taken into account. The effective radius uncertainties correspond to
the effective radius of the lower and upper integration boundary of the radiation
emission
6
. One can see that the X mode channels exhibiting »shinethrough«
6
However, they are hardly visible, as these uncertainties are shown for the MAP result only,
corresponding to the thick lines. The integrations thresholds are chosen such that the integ-
ral over the emission includes
90 %
of the intensity, such that the lower integration boundary
cuts off at
5%
, and
95 %
for the upper integration boundary respectively, cf. footnote 5 on
page 84.
85
Chapter 5. Plasma profile inversion
on the edge of the low field side (outboard side) have the main intensity con-
tribution coming from the core, although the broad region from which they
collect radiation prevents a reasonable localisation. The O mode predictions
show for most low field side channels the major contribution coming from the
core with bad localisation. Taking multiple plasma passes into account would
further impair the localisation due to the wall being nonperpendicular to the
line of sight, as the reflected ray potentially will collect radiation originating
from the resonance layer at a different effective radius. This mapping was calcu-
lated only for the central frequencies, larger bandwidths correspond to worse
localization as radiation from a broader part of the plasma is collected.
The difference between the emission positions as determined from the cold
resonance (cf. equation (2.3)) and by TRAVIS is shown in figure 5.14a). One can
see that for the X mode both ways to determine the position yield similar results.
Deviations are still visible, which correspond to the
1 cm
to
2 cm
that the emis-
sion is coming from behind the cold resonance, cf. section 2.1 and figure 5.14b).
For many applications it can be expected that the cold resonance mapping is
sufficient for a rough guess of the ECE channels emission position. This is also
important for the analysis in this chapter, as too large deviations would prevent
the ECE from being used to determine the initial electron temperature profile
guess. In principle, one could use the electron temperature profile from the
Thomson scattering system as a replacement, but as it has a sampling rate of at
most
30 Hz
and collects the intensity only over few nanoseconds, changes of the
equilibrium profile on the order of few milliseconds might not be visible in the
Thomson scattering profile. The O mode emission of the low field side comes
dominantly from the core, virtually approaching the cold resonance position on
the inboard side with vanishing electron temperatures. However, both modes
were calculated using only one 3D plasma pass. As the O mode has a low optical
depth (considering one plasma pass), these effective radius values only possess
indicative character. Large deviations can occur e.g. for plasmas with a very
low or very high electron density. The former requires, due to low optical depth,
reflections at the wall to be taken into account and multiple plasma passes, ef-
fectively collecting radiation from potentially many different plasma locations.
The latter might lead to refraction such that radiation from a completely differ-
ent part of the plasma might be collected. Note that the channel marked with
a grey background in figure 5.14b) shows a deviation of approximately 1.25 cm
probably due to numerical inaccuracies of the magnetic field strength as cal-
culated by VMEC near the core, as only few flux surfaces are calculated in the
86
5.2. Results
(a)
effective radius ρCR
effective radius ρECE
#20180823.016.002 at 4.45 s
(b)
effective radius ρCR
Δrin cm
1.00.50.00.51.0
0
1
2
outboard inboard
X mode
1.00.50.00.51.0
1.0
0.5
0.0
0.5
1.0
outboard inboard
outboard inboard
X mode
O mode
CR
Figure 5.14.: (a)
shows the deviations of the ECE channel positions as calculated
from the model compared to the cold resonance (CR) positions. One can see
that the X mode cold resonance is a good estimate for the emission position
calculated by TRAVIS. The broad O mode low field side (outboard) emission
stems largely from the plasma core, the high field side (inboard) approaches
the cold resonance but measures very low intensities.
(b)
shows the difference
of the X mode between the model and cold resonance channel positions Δ
r
, cf.
section 2.1. The grey area in the core is numerically unstable. The thin lines
represent MCMC samples and the darker, thick lines the MAP results.
vicinity (in effective radius) of the magnetic axis. The factors, with which the
calculated X mode and O mode intensities are scaled to add up to the final
Trad
prediction, are free parameters due to the aforementioned reasons and small
inaccuracies of the polarisation setting of the ECE system with respect to the
inclination angle of the magnetic field angle in the used VMEC equilibrium.
The marginal posterior probability densities of the scaling factors for the cal-
culated X mode and O mode intensities (called »X mode contrib.« and »O mode
contrib.« in figure 5.1) are depicted in figure 5.15, where they are called »X mode«
and »O mode«. As only one plasma pass with one central ray was used in the
calculations, the contribution for the O mode is an upper boundary. If multiple
plasma passes would be used, the predicted radiation intensity would increase,
87
Chapter 5. Plasma profile inversion
0 0.2 0.40.60.81
0
5
10
15
20
contribution
probability density
#20180823.016.002 at 4.45 s
X mode
O mode
exp. estimates
est. ideal O mode
est. O mode multipass
Figure 5.15.:
The kernel density estimates (i. e. the marginal posterior) of the
X mode and O mode contributions as calculated from 103MCMC samples.
therefore allowing for a smaller O mode scaling factor. Thus, these free paramet-
ers allow to estimate how large of an effect the single plasma pass restriction
has. This is why the X mode and O mode contributions have been chosen to
not necessarily add up to
1
. However, the X mode is for the most part optically
thick. Hence, the X mode scaling factor is approximately the same as one would
expect if the TRAVIS prediction is correct (in the sense that no intensity is »lost«
due to missing reflections, etc.) and scaled down according to the polarisation
matching of the ECE diagnostic. This also allows to estimate the O mode factor
one would expect if multiple plasma passes were taken into account, see the
marine dashed curve (»est. ideal O mode«). One infers the most probable contri-
bution of the X mode to the overall intensity to be circa
98.8%
with the center
of mass at
97.5%
. From physics considerations (i. e. the position of the wire grid
selecting the mode) one would expect the O mode contribution to be lower than
4%
and the X mode contribution higher than
96 %
, cf. the vertical, dotted black
lines and section 5.1. As there is a considerable probability density mass of the
O mode marginal posterior above
4%
, it is clear that the reconstruction would
profit from calculations including multiple plasma passes. While this poses no
technical problem and is fully implemented, it is computationally expensive
to take multiple plasma passes into account, thus slowing down the inference,
cf. section 5.1.1. Another prohibitive factor is to get the line of sight accurately
after multiple reflections, which is practically not possible.
88
5.2. Results
0 0.2 0.40.60.811.2
0
1
2
3
effective radius ρ
Tein keV
#20180823.016.002
NBI off, 4.45 s
NBI on, 5.15 s
Thomson scattering
Figure 5.16.:
Comparison of inferred
Te
profiles with NBI heating on and off.
The thin lines represent MCMC samples and the darker, thick lines the MAP
results. The chosen timepoints are indicated by vertical black lines in figure 5.1.
5.2.3. Effects of NBI
In this subsection, the effect of NBI heating on the electron temperature and
density profile is discussed. An overview of the discharge with the two time
points being compared is shown in figure 5.4. The analyses were done while the
plasma was reasonably stationary, i. e. no big changes in plasma temperature or
density were occuring. Figure 5.16 shows the electron temperature profiles as
inferred before and after NBI heating was active. Outside of an effective radius
of
0.4
there is virtually no difference between the two profiles. In the very core,
the electron temperature decreases by about
1 keV
when the NBI heating is
active, although the NBI increases the total heating power by about
3 MW
. This
is caused by the NBI heating also providing notable amounts of particles, thus
increasing the density. Note that the difference between the MAP result and
the MCMC result at
5.15 s
hints at a non-Gaussian posterior, that is most of the
probability mass is found at electron temperatures above the MAP solution.
The density profiles as inferred are shown in figure 5.17. As expected the
electron density profiles show large uncertainties, which is to be expected as
neither »shinethrough« nor cutoff effects play a role that would increase the
information that the ECE measurements contain with respect to the density
profile. The total electron density increases when NBI heating is switched on,
89
Chapter 5. Plasma profile inversion
0 0.2 0.40.60.811.2
0
2
4
6
effective radius ρ
nein 1019 m−3
#20180823.016.002
NBI off, 4.45 s
NBI on, 5.15 s
Thomson scattering
Figure 5.17.:
Comparison of inferred
ne
profiles with NBI heating on and off.
The thin lines represent MCMC samples and the darker, thick lines the MAP
results. The chosen timepoints are indicated by vertical black lines in figure 5.1.
as it acts as a particle source. A slight steepening of the profile is observed with
the NBI being active. Note that the apparent smaller uncertainties around an
effective radius of
0.5
stems from the combination of a (more or less) fixed total
electron density by the interferometer and fixed hyperparameters, such that
near
ρ=0.5
a kind of »pivot point« forms. Chapter 6 discusses the effects that
ECE channels in cutoff have on the inference of the electron density profile.
5.3. Conclusions
The holistic evaluation scheme presented in this chapter showed an improve-
ment over the traditional evaluation schemes. Compared to the spline based
integrated data analysis (IDA) analysis done at AUG (cf. [102]), the approach in
this chapter is more generic as Gaussian processes put less constraints on the
profile shape. Furthermore, TRAVIS works both for stellarators (using a VMEC
equilibrium) and tokamaks (using an EFIT equilibrium or providing the neces-
sary quantities directly) and can handle wall reflections in full 3D, adding to the
generality. Compared to the Minerva approach undertaken at JET, the model
presented in this chapter allows the full 3D treatment required for a stellarator,
adding to the generality of the approach.
90
5.3. Conclusions
As the time an evaluation takes is too long for many practical purposes, ways
to speed up the evaluation are highly anticipated. The plan is to take advant-
age of the generic Minerva model structure to train a neural net in a similar
fashion as was done for X-ray imaging crystal spectroscopy at
W7-X
[78]. The
author regards optimizations specific for this ECE model as problematic, logic-
ally optimizations should be done either within TRAVIS to speed up forward
calculations or within the inversion methods of Minerva.
A possible improvement of the interpretation/modelling of the ECE could be
achieved by using a viewing dump, as noted by [101], as reflections and mode
scrambling effects at the wall could be removed which in turn eases the forward
calculation. Given the steady-state requirements of
W7-X
, the feasibility of a
microwave beam dump is problematic with the current line of sight, as the
target tiles of the ECE suffer already potentially from high heat loads.
Appendix C shows part of the code, or rather keywords, that allow custom-
ization of the developed standalone Java executable. The model detailed in this
chapter is used to analyse special cases with very low and high electron densi-
ties in the next chapter, which encompass »shinethrough« effects in the former
and partial cutoff effects in the latter case.
91
6
Higher level applications of ECE
The inference of the electron temperature and density profiles discussed in
chapter 5 extended the domain in which an ECE analysis is possible to regions
with more complex radiation transport behaviour. Examples are i) the recon-
structions at densities approaching the ECE cutoff (at
W7-X
: refraction starts to
play a role above approximately
8×1019 m−3
and ultimatively reflection at den-
sities above approximately
12 ×1019 m−3
) and ii) the reconstructions with low
electron density (at
W7-X
: below approximately
1019 m−3
) as redshifted emission
of strongly relativistic core electrons starts to get relevant for some channels. Ex-
tremely high electron temperatures and distortions of the Maxwellian electron
energy distribution function are expected to also hamper analysis considerably.
In the following, section 6.1 will explore ECE spectroscopy at high densities as
a direct application of this analysis to
W7-X
high density discharges, a major
goal of this project. Section 6.2 will shortly present and discuss findings at low
plasma densities, while section 6.3 will give an outlook on possible Bayesian
high level physics approaches, here specifically for a heatwave analysis to ob-
tain heatpulse diffusivities.
6.1. Application on a High Density Plasma
This section contrives an example analysis of a measurement, with some ECE
frequency channels exhibiting cutoff effects. The improvement of the uncer-
tainty of the inferred electron density profile due to the channels being close
to their cutoff is assessed. The limitations and complications coming with the
analysis of this edge case are, amongst others: i) the line of sight may no longer
be a straight line with the magnetic field strength increasing monotonically
93
Chapter 6. Higher level applications of ECE
along it, ii) the lines of sight might differ substantially between the X mode and
the O mode, iii) the lines of sight for different frequencies can differ signific-
antly and iv) the strong influence of the electron density profile on the optical
thickness.
The profile inversion model explained in chapter 5 was used to analyse ex-
emplarily a plasma discharge with electron densities high enough to achieve
(partial, i. e. limited to a few channels) second harmonic X mode cutoff for the
ECE channels. This particular experiment, #20171115.039.002, was the first time
that a high performance plasma was sustained by 2
nd
harmonic O mode ECRH
alone [35]. An overview of the discharge showing the traces of the electron
temperature, density and heating is shown in figure 6.1. After
2 s
all gyrotrons
were heating in O mode. This plasma scenario reached one of the highest dia-
magnetic energies in OP 1.2a, around
800 kJ
at around
4.5 s
. One can see that
the agreement between a Thomson scattering channel and an ECE channel,
both measuring around an effective radius of
0.1
, is reasonably good through-
out the lower density parts of the discharge. A possible reason for remaining
deviations could be found in the VMEC equilibrium, as one single equilibrium
was used throughout the whole discharge. It is optimized for the plasma at the
high density phase, around
3.75 s
. Thus, the ECE channel best suited to match
a corresponding Thomson scattering channel might differ in the course of the
discharge. Via a pellet injection system [49, 103] a density ramp is driven, such
that the selected ECE channel shown in the graph is pushed into cutoff around
2.85 s
, with its signal returning at around
4.15 s
. The remaining ECE signal can
stem from multiple origins, amongst them: i) a small chunk will originate from
O mode contributions due to the inherently imperfect selection of the X mode
due to the finite polarization mismatch at the last closed flux surface, ii) the
refracted ECE line of sight might have to consider Doppler broadening as the
ray is no longer perpendicular to the flux surfaces, iii) potentially a different
region in normalized momentum space is sampled and iv) the X mode might
get reflected from the cutoff layer in the plasma, again reflected at the vessel
wall, and, due to the line of sight now »looking« at another part of the plasma
where the resonance surface is at another electron density, radiation intensity
can be collected again. Note that in a stellarator, due to its helical axis and mag-
netic mirror fields, a toroidal shift of the ECE line of sight changes the relation
between observation frequency and observed flux surface. The Thomson scat-
tering system delivers electron temperatures reliably even if the ECE is in cutoff
as expected. The huge increase in particles through pellet injection lowers the
94
6.1. Application on a High Density Plasma
(a)
0
2
4
6
temperature in keV
#20171115.039.002
Thomson scattering
ECE (143.9 GHz)
pellets
(b)
00.511.522.533.544.5 5
0
5
10
15
time tin s
nein 1019 m−3
interf.
ECRH
0
2
4
6
#20171115.039.002
00.511.522.533.544.5 5 0
2
4
6
time tin s
power in MW
Figure 6.1.:
High performance plasma discharge with 2
nd
harmonic O mode
ECRH.
(a)
shows the radiation temperature of an ECE channel close to the
core, compared with a central channel from the Thomson scattering sys-
tem [47]. The vertical black line indicates the time at which the analysis shown
in section 6.1 was performed. (b) depicts the ECRH heating power as well as
the line averaged density from the single channel dispersion interferometry.
No other heating method was used.
electron temperature from about
4 keV
to about
2 keV
, as the ECRH power is
increased only from circa
4.5 MW
to
5.4 MW1
. A profile inversion of the com-
bined ECE interferometry model was performed at the time point indicated
by the vertical black line. The electron density decrease during the post pellet
injection phase offers a plasma with less abrupt changes when compared to the
pellet injection phase in which the dynamics of the pellet fuelling complicate
1
One can see the power drop at around
100 ms
. This were two gyrotrons in X mode for plasma
startup, which were shut down to change their polarisation to O mode. They came back
online at 3 s.
95
Chapter 6. Higher level applications of ECE
3.8 4 4.2 4.44.64.8
0
1
2
3
4
time in s
radiation temperature in keV
#20171115.039.002
136
140
144
148
frequency in GHz
Figure 6.2.:
ECE channels coming out of cutoff during the density decrease
in the high performance discharge #20171115.039.002, see figure 6.1. One can
see that first a channel with a low frequency comes out of cutoff, followed
sequentially by high frequency channels with decreasing frequencies. The
black vertical line indicates the time point of the profile inversion.
the plasma behaviour. Hence, the former is better suited for an assessment of
how much electron density profile information is concealed in an ECE spectrum
in partial cutoff.
Figure 6.2 shows the timetraces of the ECE channels between
135.5 GHz
and
150.6 GHz
during the time window in which the electron density drops gradually
from around
1.09 ×1020 m−3
to around
0.65 ×1020 m−3
, until no ECE channel is
in cutoff anymore. The black vertical line indicates at which timepoint the
profile inversion as described in this section was ran. One can see that the first
channel to come out of the cutoff is the channel with the lowest frequency.
However, subsequently channels at the upper end of the displayed frequency
range drop out of the cutoff first. Two factors play a role for this to happen:
i) at high densities the line of sight might be deflected strongly such that the
resonance layer is never hit (at least on the first plasma pass), even though the
density at the resonance layer is below the cutoff density and ii) the density
profile is not just a box, therefore it does play a role at which effective radius
a channel is collecting radiation as further out the density tends to be notably
lower.
96
6.1. Application on a High Density Plasma
5.4 5.5 5.65.75.8 5.9 6 6.1 6.2
0.1
0.2
0.3
0.4
0.5
0.6
radius Rin m
height zin m
#20171115.039.002 at 3.75 s and f=150.56 GHz
X mode LOS
O mode LOS
vacuum LOS
flux surfaces
vessel
Bres
Figure 6.3.:
Ray traced ECE channel with an electron density profile that reaches
values close to the cutoff density of the channel. The channel corresponds to
the highest frequency channel shown in figure 6.2. To account for the finite
beam width the beam is split in multiple rays.
To understand these phenomena it is helpful to visualize the X mode and
O mode lines of sight for a specific frequency in the experiment investigated in
this section. Figure 6.3 shows exemplarily how the beam of channel 26, split up
in 120 rays for each the X mode and the O mode, propagates through the plasma.
It should be noted that the X mode rays are deflected not only along
R
and
z
,
but also along the toroidal direction,
ϕ
. A widening of the beam can be observed
for the X mode, while the O mode does not show a strong deviation from the
vacuum line of sight. One can imagine now, that, if the frequency bandwidth is
large enough and the electron density is close to the point where the ray gets
reflected, small changes of the electron density can lead to drastically different
lines of sight of the rays representing the beam. Even for a single frequency, the
ECE signal does get highly sensitive to small electron density changes under
these circumstances. Therefore, it is very difficult to obtain a satisfactory initial
guess, as the ECE itself cannot be used in such high density scenarios for the
initial guess of the electron temperature; and the Thomson scattering system at
W7-X
provides an electron density and temperature profile only every
33 ms
to
97
Chapter 6. Higher level applications of ECE
(a)
cutoff
130 135 140 145 150 155 160
2
4
6
8
frequency fin GHz
electron density in 1020 m−3
ncut
e,X
ncut
e,O
ncrit
e,0,X
ncrit
e,0,O
(c)
5.4 5.6 5.8 6 6.2
−0.5
0
0.5
1
radius Rin m
height zin m
(b)
0 0.2 0.40.60.81
0
1
2
effective radius ρ
nein a.u.
Tein keV
Figure 6.4.: (a)
shows the critical central electron densities and cutoff densities
(cf. equation (2.5)) for the ECE at
W7-X
for profile shapes shown in
(b)
cor-
responding to #20171115.039.002 at
3.75 s
where many channels are deflected
due to cutoff effects. To indicate the central density reached in this discharge
a grey dashed line is used, which also signifies the ECE X mode channels to
be expected in cutoff. The line of sight of the three colored vertical lines in
(a) is shown in (c).
100 ms
. This in turn leads to problems in the analysis; the MAP often gets stuck
in a local maximum and the MCMC will take an inconveniently long time to
get its chain stationary.
Figure 6.4a) gives an estimation on which ECE channels can be expected to
be in cutoff. The calculations used the profiles shown in figure 6.4b), where
the electron temperature profile corresponds approximately to the one of the
discharge used within this section, #20171115.039.002, at
3.75 s
. For the electron
density profile the approximate shape at that time point was taken, with a to-be-
fitted factor scaling the density profile up respectively down. Then, the factor
(that is, the central electron density, henceforth called »critical central density«)
for which the scaled profiles deflect the ray so much that it does not reach the
cold resonance surface
2
is determined by a simple optimization scheme. Cor-
2
For the intersection tests a modified Möller–Trumbore ray-triangle intersection algo-
rithm [104] is used.
98
6.1. Application on a High Density Plasma
respondingly, figure 6.4c) shows the flux surfaces for the VMEC equilibrium
used in this section, as well as the rays (solid lines) and resonance layers at
the magnetic axis (dotted) for three frequencies (low field side: dark gray, cen-
ter: yellow, high field side: green) of the X mode. The straight, light grey line
indicates the vacuum line of sight. While the dark gray ray, corresponding to
the vertical line at
133.76 GHz
, is reflected back directly, the yellow ray, corres-
ponding to the vertical line at
142.86 GHz
is deflected upwards. The yellow ray
at the top of figure 6.4c) looks like it intersects the resonance layer, but that
is due to the projection onto the
(R,z)
plane as the ray changes the toroidal
angle sufficiently to see a slightly differently positioned resonance layer at this
toroidal angle. The same holds true for the green ray (cf. the vertical line at
153.56 GHz
in figure 6.4a), which is deflected strongly in the toroidal direction,
an effect that is not visible in this
(R,z)
projection. By using a central electron
density circa corresponding to #20171115.039.002 at
3.75 s
we can estimate which
frequencies of the ECE are in cutoff. This density is indicated in figure 6.4a)
by the horizontal grey dashed line. It intersects the calculated critical central
densities of the X mode shown in marine at
136.78 GHz
and
150.49 GHz
, so that
the frequencies in between can be expected to be in cutoff, indicated by the
grey background. The O mode critical central densities, indicated by the purple
curve and filled squares, do not show any intersections, hence no channel’s
O mode contribution is expected to vanish. Thus, the ECE spectrum for that
time is expected to exhibit a crater like structure, being in cutoff only in central
channels. It also explains why the channels subsequently coming out of cutoff
in figure 6.2 are not monotonically increasing in frequency. Furthermore, fig-
ure 6.4a) (or correspondingly newly calculated graphs if different temperature
or density profile shapes are required) allows to tailor discharges where certain
ECE channels are close to the cutoff, but still provide useful X mode information.
Similarly, the graph can be used for estimations which channels do not reach
the O mode cold resonance surface, although this is only relevant at extremely
high electron densities above 20 ×1019 m−3.
As expected after these simulations, the observed radiation temperature spec-
trum shows a crater-like structure, see figure 6.5. The ECE spectrum was av-
eraged over
5 ms
. Note that a VMEC equilibrium manually tailored for this
particular timepoint was used
3
, that uses the experimental coil currents instead
of the coil currents of the precalculated equilibrium. However, the pressure pro-
3EJM, see minerva-vmec-a87872131bcf5422ac2b858a41ceb7e5.
99
Chapter 6. Higher level applications of ECE
0
0.5
1
(a)
Trad in keV
#20171115.039.002 at 3.75 s
X mode
O mode
observed
125 130 135 140 145 150 155 160 165
−0.2
0
0.2
0.4
(b)
frequency fin GHz
residual in keV
result
init.
Figure 6.5.: (a)
shows the comparison of the predicted and observed
Trad
spec-
tra for the time defined by the vertical black line in figures 6.1 and 6.2. The
thick, darker lines represent the MAP inversion result, while the thin lines
are MCMC samples to give an idea about the uncertainties. The remaining
residuals are shown in
(b)
with the predicted uncertainties in marine. For the
channels with a grey background the resonance surface was not hit or their
optical depth is below
10−5
, so they should be neglected. That the ECE signal
for these channels does not completely go back to zero is probably due to
multiple reflections at the wall.
files from the reference equilibrium (cf. footnote 5 on page 84) are used, which
worked well enough for this case. Furthermore, in contrast to the analysis in
chapter 5, one reflection of the O mode at the inner torus wall was taken into ac-
count. This was done as the intensity from the X mode was expected to be much
lower in the cutoff region, increasing the relative importance of the O mode
contribution, which has a single pass absorption at these densities above
30 %
.
As the calibration factors of channels 15 and 16, corresponding to
138.26 GHz
and
139.06 GHz
, possess very high uncertainties they were neglected during the
analysis. The O mode prediction captures the central channels, in which the
X mode is in cutoff and thus not contributing directly, reasonably well, although
100
6.1. Application on a High Density Plasma
with degrading quality above
145 GHz
. However, it has to be noted, that the
O mode prediction serves as a proxy here: If the X mode gets reflected at the
plasma, hits the vessel wall and is reflected back to the plasma, it can poten-
tially reach the cold resonance surface, as the reflection might have redirected
the line of sight to some other part of the plasma. This implies that even if a
channel is in cutoff for the first plasma pass, it might still collect radiation after
taking its reflections into account. As no reflections for the X mode were taken
into account in the TRAVIS calculations, potential X mode contributions are
subsummed in the O mode proxy. Note that taking reflections for the X mode
into account is difficult as, depending on the density, large parts of the plasma
vessel can potentially act as a mirror due to the high sensitivity of the X mode
on the electron density. Hence, more polygons to describe the plasma vessel are
needed if an X mode reflection needs to be taken into account when compared
to an O mode reflection. In turn, more ray-polygon intersection checks are
necessary, which would make an analysis with X mode reflections taken into
account slower. The contribution of the O mode proxy was fixed at
100 %
, and
the values match at least roughly quantitatively. The low field side channels
below the ECE channels that went into cutoff are decently predicted by the
X mode, although there are deviations especially at the transition into cutoff.
On the high field side, where the ECE channels are not yet in cutoff the devi-
ations between predictions and observations are stronger. This might very well
hint at the MCMC not being completely converged.
Figure 6.6 shows the trace of the logarithm of the joint (i. e. prior times
likelihood) of the inversion at
3.75 s
. There are several things to note: i) The
MAP failed to find the global maximum, as the logarithm of the joint increased
further during the MCMC, ii) even though a burn-in of
125 000
iterations was
chosen, it is doubtful that the MCMC completely converged
4
. As a consequence,
the results obtained from such a MCMC have to be taken cum grano salis, even
though the electron density profile discussed later has proved to be remarkably
stable throughout the burn-in and thereafter. Unfortunately, due to multiple
server restarts and other events that required to pick up the analysis at the
last saved point, it is not possible to use accurate times on the abscissa. The
total analysis time was over three months, mainly due to the inclusion of one
4
While the MCMC is very robust and will eventually converge, it is usually slow to find the
MAP, especially if the MAP analysis providing the starting position got stuck in a local
maximum.
101
Chapter 6. Higher level applications of ECE
0 0.2 0.40.60.811.2 1.41.6
−1.6
−1.4
1.
2. 3.
iteration in 105
log(joint)in 103
#20171115.039.002 at 3.75 s
MAP
MCMC burn in
MCMC
Figure 6.6.:
The
log(joint)
, where the joint is the product of prior and likelihood,
of the profile inversion done for the spectrum in figure 6.5a). The MAP inver-
sion got stuck in a local maximum. Hence, the MCMC continued to climb up
the joint probability and did not get stationary. 1. corresponds to the initial
value in figure 6.5b), 2. to the MAP result and 3. to the MCMC samples.
(potential) reflection for the O mode. This requires costly collision checks to find
the spot at which the ray hits the vessel wall, even though the representation of
the carbon wall panel was optimized beforehand to contain only 54 polygons.
The semi-automatic optimization was performed by selecting only polygons
that with some probability are relevant to reflections, effectively reducing the
number of polygons by more than
50 %
. Note further that the small hole in the
wall panel on the high field side
5
was discarded, as calculations with multiple
rays were computationally prohibitively expensive and using the hole might
have caused jumps in the signal depending on the distortion of the line of sight
6
(either the ray »disappears« in the high field side horn, or it does get reflected
and contributes to the predicted signal if the ray is slightly more distorted and
hits the wall around the horn). To address the long evaluation time a Minerva
model based on figure 5.1 was developed to be able to generate training data
for a convolutional neural network by sampling from the priors [106]. Once
trained, the convolutional neural network will help to reduce the analysis time
by several orders of magnitude, making the analysis much more feasible.
5This hole corresponds to the high field side ECE horn [105].
6The resulting file can be found here: SAVE.
102
6.1. Application on a High Density Plasma
0 0.2 0.40.60.811.2
0
5
10
15
effective radius ρ
nein 1019 m−3
#20171115.039.002 at 3.75 s
ECE
Thomson scattering
Figure 6.7.:
The plot shows the electron density profile as resulting from the
inference done on the combined ECE and interferometry model, as well as the
values obtained from the Thomson scattering system. For the ECE the thin
lines represent MCMC samples and the thick line the MAP result.
Nevertheless, the inferred electron density profile as shown in figure 6.7 in
particular yielded an interesting result. The MAP and MCMC results are sim-
ilar, although there is a notable deviation between an effective radius
ρ
of
0.6
to
0.9
. The main difference to the Thomson scattering measurement points is
probably again caused by the different line integrated electron densities, which
may originate in systematic problems in the alignment of the Thomson scat-
tering system. For the future, it is planned to calibrate the Thomson scattering
system with respect to slightly different laser lines of sight, which would allow
to interpolate the required calibration more appropriately. Another source of
error, however, is the density outside the last closed flux surface, which is cur-
rently neglected in the combined ECE and interferometry model. This can lead
to an overestimation of the density. There is a distinct contrast between the
electron density profile as inferred during plasma scenarios in which neither a
partial cutoff nor noteworthy »shinethrough« occurs and the electron density
as inferred during partial ECE cutoff; In the former case, a large uncertainty of
the electron density profile is found (cf. figure 5.17), while in the latter the uncer-
tainty is reduced manifold. Therefore, the ECE, in combination with the single
channel interferometer, can provide useful information about the electron dens-
ity profile, given the right circumstances – which is also the justification for
103
Chapter 6. Higher level applications of ECE
the extensive and comparably slow analysis of the complex model in figure 5.1.
However, it is clearly necessary to increase the speed of a forward calculation
to be able to reach a safely converged MCMC within reasonable timespans. For
this, the convolutional neural network for which generation of the training
data has been started, is essential, although a parallelization of TRAVIS could
already provide major speedups. There are two options what one can do with
the training set, once available: i) one can use the convolutional neural network
trained on the training data to approximate the forward model (of the part in-
volving TRAVIS), which is the more flexible option, as the rest of the model and
its analysis methods can be changed, ii) one can train the convolutional neural
network on the inversion, which would be faster overall, but more locked into
the exact model structure. For a first approach the former option is advisable,
as one can compare the results directly with corresponding TRAVIS runs and
no comparisons with full inversions, which might require an MCMC run and
are consequently slow, are necessary.
6.2. Application on a Low Density Plasma
In contrast to the application on a high density plasma as shown in the previous
section, this short section focusses on a proof of principle example of a plasma
with low electron density. In low density plasmas, »shinethrough« effects (cf.
chapter 5) start to play a role and can dominate the intensity for several channels.
As a reminder, the effect called »shinethrough« corresponds to the relativistic-
ally downshifted core plasma emission that is not (completely) reabsorbed on
its way to the ECE horn due to the low optical depth at the emitted frequency.
An example spectrum exhibiting this behaviour was shown in figure 4.12. The
analysis of this spectrum is shown in figure 6.8. Note that the absolute values of
the spectrum shown in figure 4.12 and the spectrum shown in figure 6.8 differ
by a small amount, as both the averaging time and the calibration were slightly
different. The huge discrepancy at
129.16 GHz
is an indication that the MAP did
not converge and thus only represents a local maximum of the posterior. Note
that the channel at
129.16 GHz
is located exactly in the steep edge region of
the gradient of the optical depth (grey shaded), such that small changes in the
electron temperature and density profile can inflict large changes in the amount
of »shinethrough« expected to contribute to the signal. Note further, that the
other channel showing a large discrepancy at
136.26 GHz
is similarly located in
104
6.2. Application on a Low Density Plasma
0
2
4
6
(a)
Trad in keV
#20171207.006.002 at 0.04 s
X mode
O mode ×10
observed
−2
0
2
(b)
residual in keV
result
init.
125 130 135 140 145 150 155 160 165
0
5
10
(c)
frequency fin GHz
optical depth
X mode
O mode ×10
Figure 6.8.: (a)
shows the comparison of the predicted and observed
Trad
spectra
for the time defined by the left vertical black line in figure 4.13. The thick, dark
lines represent the MAP inversion result. The remaining residuals are shown
in
(b)
with the predicted uncertainties in marine.
(c)
shows the optical depth.
the gradient region of the optical depth, again shaded in grey. Inbetween these
two channels the plasma is optically thick as
τ>
3, cf. section 5.2.2. The optical
depth gradient at the side of the channel at
136.26 GHz
is less steep than on the
other side, hence explaining the smaller, but notable deviations starting to build
up from
134.36 GHz
onwards. The large sensitivity on the optical depth, which
depends notably on the electron density in the shown example, implies that
the electron density information content in the ECE signal is comparably large.
An MCMC was started, but did not finish in time to make it into the thesis.
Therefore, no uncertainties can be provided.
105
Chapter 6. Higher level applications of ECE
6.3. A Bayesian approach to heatwave analysis
An important quantity to determine the quality of the confinement is the (elec-
tron) heat diffusivity
χ
in
m2s−1
. One way to determine this value experiment-
ally is by so-called heat pulse propagation (HPP) experiments [38, 107]. In these
experiments, the power of at least one of the heating sources is modulated. The
temporally varying heating power at the ECRH deposition region acts as a small
local perturbation. This local perturbation causes »ripples« on top of the equi-
librated electron temperature profile that propagate through the plasma. These
»ripples« are called heatpulses/heatwaves. The speed and amplitude of these
pulses yields information about the heat diffusivity,
χHP
. A heating method es-
pecially well suited for HPP experiments is ECRH, as it allows for a well defined
power deposition region within an effective radius of
reff =1 cm
. At
W7-X
, the
NBI system deposits the power so broadly and acts so strongly as a particle
source that one cannot use it for HPP, at least not for the model presented in
this section. Note that the frequency of the modulation determines whether
the observations are better suited to infer the heat diffusivity or the power de-
position profile of the ECRH. The former corresponds at
W7-X
to modulation
frequencies below
100 Hz
(typically about
17 Hz
are used), while the latter ne-
cessitates frequencies around
500 Hz
to be able to explicitly neglect transport
effects. Heat pulses originating from modulation frequencies above 100 Hz are
damped more and hence the analysis of heatwaves across the effective radius
becomes more and more problematic, as less and less ECE channels are cap-
able of detecting the resulting heat pulse. ECE is apt for measuring HPP as
the sampling rate can reach up to
2 MHz
at
W7-X
. Note that the low temporal
resolution of the other main electron temperature diagnostics at
W7-X
, the
Thomson scattering and X-ray imaging crystal spectroscopy systems, prevents
them from observing heat waves in a way that can be easily analysed.
The theory of HPP is a comprehensively developed fieldb. Subsequently, a
schematic derivation of a simple equation governing heat transport is shown
that closely follows [108, 109]. First, energy conservation in cylindrical geo-
metry mandates
3
2
∂
∂t(neTe)+∇⋅qe=S,(6.1)
in which
t
is the time,
ne
the electron density,
Te
the electron temperature,
qe
the electron heat flux and
S
the effective source term. Now, the quantities are
106
6.3. A Bayesian approach to heatwave analysis
split in »background« quantities that do not change with time and a superposed
time dependent quantity indicated by a tilde over the variable,
ne=ne,0(r)+ne(r,t),(6.2)
Te=Te,0(r)+
Te(r,t),(6.3)
qe=qe,0(r)+qe(r,t),(6.4)
S=Se,0(r)+Se(r,t).(6.5)
In there,
r
refers to the radius in cylindrical geometry. Expanding the ECRH
modulation depth ϵ=Se/Sein first order allows to find
3
2ne,0
∂
Te
∂t+∇⋅ qe=Se.(6.6)
Finally, away from the sources (so that
S
can be neglected) and switching to
coordinates of the effective radius ρyields
ne,0
3
2
∂
Te
∂t+1
V′
∂
∂ρ�ne,0V′⟨|∇ρ|2⟩χHP ∂
Te
∂ρ�=0,(6.7)
in which
V′= ∂ρV
and
⟨⋅⟩
denotes a flux surface average. The electron heat
diffusivity in m2s−1as obtained from HPP corresponds to χHP.
Figure 6.9 shows the simplified version of the model used to gain radially
resolved profiles of the electron heat diffusivity. The model works as follows.
First the user provides the time, i. e. the start and end within a discharge. These
time points are provided to the ECRH and ECE datasources. Then, the power
trace as fetched from the ECRH is upsampled, as it has a lower sampling rate
than the ECE radiation temperature traces. Subsequently, a »jump finder« se-
lects each falling flank in the power trace, allowing for a conditional averaging
of the temporal dependence of the radiation temperatures of each ECE channel
during the ECRH modulation cycles. As the offset is removed in this procedure,
the result is taken to represent the time dependent perturbation of the elec-
tron temperature. Note that, in principle, a profile inversion as described in
chapter 5 would be ideal to be performed for each point in time. However, the
computational cost prohibits this approach. Thus, the radiation temperature
perturbation is used as a surrogate for the electron temperature perturbation.
As the ECE sampling rate is high, even the conditionally averaged signal can
107
Chapter 6. Higher level applications of ECE
time
ECRH
resampler cond. avg. CR
ECE VMEC
ℓ1
ℓ(x)
GP
χHP
χHP(ρ)
eq. (6.7)
α
σf
grid
ℓ2
xw
x0
σy
heat transport
σf
σy
grid
Te(ρ,t)
α
σx
GP
Te
obs.
perturbed Te
legend
node
uniform
normal
observed
Figure 6.9.:
The simplified Minerva heat transport model showing the depend-
encies used in the model. Free parameters have a cyan background, their shape
as specified in the legend corresponds to the prior distribution used. Edges
indicate the dependencies between nodes. Note that
α
and
σf
are different for
each subgraph.
contain several thousand data points. The sheer amount of data can slow the
analysis down considerably, which is why a downsampled signal is used in the
heat transport model.
A 2D squared exponential kernel is used for the Gaussian process to describe
the electron temperature perturbation,
Te
. It requires
σx
, i. e. the respective ker-
nel scale length in both the effective radius and the time direction, the electron
temperature perturbation variance
σf
and
σy
, which is set to
1‰
of
σf
and is
used to avoid kernel degeneration. Furthermore, a grid provides the points in
effective radius and time at which the kernel is evaluated. By default, a 20
×
20
108
6.3. A Bayesian approach to heatwave analysis
grid is used. The free multivariate normal of the electron temperature perturb-
ation at the grid points uses the 2D squared exponential kernel and a mean of
zero. The values (not the mean!) of the multivariate normal are combined with
the grid to do a bicubic interpolation, which provides the continuous
Te(ρ,t)
function. In the observation node the conditionally averaged signals at the cold
resonance positions, determined from the ECE channels’ center frequencies
and a VMEC equilibrium, are compared to the predicted bicubically interpol-
ated Gaussian process. The prediction uncertainties are based on the estimated
uncertainties of the conditional averaging, scaled with a variance scaling factor
α
. Note that the posterior is sensitive to the chosen interpolation method, as
the virtual observation depends on the first and second derivatives.
The Gaussian process describing the heatpulse diffusivity
χHP
is structurally
the same as used for the electron temperature and density profiles in chapter 5,
see section 3.1 for an explanation of the hyperparameters. By default, a grid
of
20
points is used. A free truncated multivariate normal using the kernel
defined above and a mean of zero allows to infer the heat diffusivity at the
grid positions. The continuous
χHP(ρ)
profile is obtained via a cubic interpola-
tion. This allows to calculate the prediction, which corresponds to the left hand
side of equation (6.7). Consequently, the virtual observation corresponds to the
right hand side and is consequently set to zero. The prediction uncertainties
are based on a fixed value scaled with a variance scaling factor α. To take into
account that equation (6.7) is only valid where no heating occurs, the following
improvement is suggested: The mirror positions of the ECRH allow, combined
with an electron temperature and density profile, to predict the power depos-
ition profile via TRAVIS. As, by default, we use the virtual observations with
the same fixed (albeit scaled) value for all radial positions alike, equation (6.7) is
enforced with the same strength within the power deposition region as outside
of it. The violation of it will be most pronounced within the power deposition re-
gion. Thus, one can use the power deposition profile as an additional scaling for
the prediction uncertainties of the virtual observation, such that the variance
is no longer
ασ2
, but rather
αPdepσ2
, automatically giving less importance to
the virtual observations where the expected violations are strongest. Note that
this would, in principle, also allow to infer at least the rough power deposition
region by having the variance scaling depend on the effective radius.
As the analysis of the model with all 430 free parameters is computation-
ally expensive, the following evaluation strategy that differs from the approach
described in section 3.4.2 is used. First, the free parameters in the heat trans-
109
Chapter 6. Higher level applications of ECE
0 10 20 30 40 50 60
−0.1
0
0.1
approx. phase shift
unphysical
behaviour
time in ms
Tein keV
#20160302.008.002, avg. heatpulse
obs.
pred.
ECRH
0 10 20 30 40 50 60
−0.3
−0.2
−0.1
0
0.1
0.2
ECRH power modulation in MW
#20160302.008.002, avg. heatpulse
Figure 6.10.:
Shown are the observed (from the conditional average) and pre-
dicted (from the Gaussian process) electron temperature perturbations for the
analysed ECE channels, as well as the conditionally averaged ECRH trace.
One can see that the plasma responses of the different channels show phase
(cf. the dashed black curve) and amplitude differences. The red ellipse shows
an example for unphysical behaviour in the fit.
port submodel are deactivated, such that they keep their momentarily value.
The remaining free parameters are all located in the electron temperature sub-
model. They are linear with respect to the corresponding observation, with
σf
,
σx
and
α
being treated as hyperparameters. This allows to use the power of the
linear Gaussian inversion combined with the Hooke and Jeeves algorithm (cf.
section 3.2) operating on the posterior probability, cf. section 3.4.2. The default
number of iterations amounts to
103
and is automatically cut short if 15 sequen-
tial iterations yield an improvement of the logarithm of the posterior probability
of less than 1.
The average heatpulse responses obtained during an exemplary HPP experi-
ment are shown in figure 6.10. Here, a total of 57 modulation cycles was used for
the conditional averaging. Thus, the
x
-axis represents the duration of the aver-
age modulation cycle. One can see that the modulated fraction of the ECRH was
switched off at
0 ms
. It was switched back on at
20 ms
, leading to an increase
110
6.3. A Bayesian approach to heatwave analysis
of the heating power by about
450 kW
. Correspondingly, the measurements of
the selected ECE channels
7
depicted in cyan show a declining radiation tem-
perature. Channels with larger amplitudes correspond to channels closer to the
power deposition region. Noteworthily, a slight shift by about
10 ms
in the min-
imum of the channels can be seen, indicating the delayed response of channels
further away from the perturbation. This delay contains information about the
heat transport in the plasma. Note that the traces of the selected ECE channels
had to be effectively downsampled. In principle, one could use also the raw data
directly. As the conditional average uses the mean number of data points per
period for the number of bins, the number of predictions grows to such large
values that Java cannot handle the arrays without specific implementations of
the linear Gaussian inversion, which has not been attempted during this thesis,
effectively necessitating the use of downsampled data. The fit on the down-
sampled data via the method described above provides a decent continuous
electron temperature perturbation fit and is shown in orange.
One could assume that this fit is good enough to continue directly with the
electron heat diffusivity fit and an deactivated electron temperature perturba-
tion subgraph to keep the dimensionality low. However, some of the features
visible in figure 6.10 are unphysical, e.g. the sudden drop in the electron tem-
perature shown in the red ellipse. This leads to deviations from the virtual
observation (cf. equation (6.7)) that cannot be avoided by only changing
χHP
.
Hence, to avoid unphysical behaviour, it is necessary to optimize the whole
model at once. In the example shown in figure 6.10 this would probably im-
ply a stronger smoothing of the traces, such that behaviour like the transient
drop in the electron temperature rise indicated by the red ellipse during the
equilibration at the higher heating power level is supressed. However, it is still
a reasonable approach to optimize
Te(ρ,t)
separately first to get a reasonable
starting point. In the future the analysis will be done with an MCMC to avoid
getting trapped in local maxima. This implies, due to the then high dimensional
model (430 free parameters) to be analyzed, that a lot of iterations are probably
necessary until convergence is reached. First tests indicate that after more than
2.2×105
MCMC iterations the chain is still moving up the posterior and no
convergence has been reached.
7
Shown are the channels
20
to
25
(corresponding to
143.9 GHz
to
149.4 GHz
, or cold reson-
ance positions in effective radii between
0.35
to
0.67
), hence all are on the high field side.
Consequently, both O mode and »shinethrough« related effects are neglectable.
111
Chapter 6. Higher level applications of ECE
6.4. Conclusions
The evaluation of the ECE as an electron density diagnostic (even if in combina-
tion with an interferometer) was, to the best knowledge of the author, attempted
for the first time. Here, the inferred density profiles agree well with the over-
all shape of the density profile determined by local measurements as available
from the Thomson scattering diagnostic. It is clear that work on the speed of the
forward calculations is still required, as analysis times on the order of months
is unpractical for most usecases. An artificial neural network is planned to help
to alleviate the long analysis times. The model for generating the training data
is prepared, although getting the artificial neural network to behave sensibly
in the vicinity of densities where »shinethrough« and cutoff effects play a role
will be challenging. Rough estimations for the required time with an artificial
neural network range on the order of minutes, at most hours, allowing to po-
tentially make practical use of the ECE as a density diagnostic. If possible this
would make a very valuable contribution to at least
W7-X
, as the Thomson
scattering system as the main provider of electron density profiles operates at
most at a frequency of
30 Hz
, while an ECE based system would allow far finer
temporal control, potentially reaching
MHz
. However, the ECE can only fulfill
this role while the electron density is in a range close to the cutoff densities for
several channels – or the electron density is so low that »shinethrough« starts
to become dominant in some channels.
As an outlook on a Bayesian analysis of the electron heat diffusivity a model
has been implemented, but not yet comprehensively analysed.
112
7
Résumé
In the context of furthering the understanding of magnetically confined plas-
mas it is important to model the available diagnostics comprehensively, and
combine them via Bayesian forward modelling. The knowledge gained by the
analysis of these models will help to design future experiments and improve
theoretical understanding. Thus, in this work, the electron cyclotron emission
(ECE) diagnostic, which is one of the main diagnostics (especially if high tem-
poral resolutions are desired) for the electron temperature profile is modelled.
For the first time, the modelling of the ECE absolute calibration is done in a
Bayesian way (cf. chapter 4) and includes the geometry of the calibration unit
(cf. section 2.3.2). The classical approach only takes into account the measured
signal difference during observation of two black body emitters at different
reference temperatures. The periodic change of the emission source allows
averaging over the performed cycles to improve the signal to noise ratio. In
addition to that allows the developed model to use the whole time series of
the average cycle, including the phases that collect radiation from multiple
reference temperature radiation sources due to the finite width of the ECE line of
sight. Furthermore, it allows to infer the sensitivities, i. e. the calibration factors,
for all ECE channels simultaneously, allowing to calibrate weakly sensitive
channels and to reduce the uncertainties. In addition, the width of the Gaussian
antenna characteristic can be inferred, which deviates by less than
1 mm
from
the independently inferred beam width for the case shown in section 4.3.1. Note
that the analysis of the model also provides the marginalised posteriors for the
sensitivities, allowing to reconstruct asymmetric uncertainties.
The application of the determined calibration factors allows to obtain only
radiation temperature spectra, however, in general, the desideratum is the elec-
113
Chapter 7. Résumé
tron temperature profile. Hence, a Bayesian forward model of the ECE radiation
transport in the plasma was constructed (cf. chapter 5 and figure 5.1). Internally,
the ray tracing code TRAVIS is used via a webservice. The forward model per-
mits the inference of a multitude of parameters, including the posteriors of the
electron temperature profile, the optical depth spectra and the normalized elec-
tron momentum spectra. Furthermore, the model is kept generic, such that it
would be applicable for tokamaks as well. Noteworthily, the radiation transport
depends on the electron density profile as well. Consequently, the ECE meas-
urements contain information about the density as well, although the amount
of information is small for optically thick plasmas in the »classical« operational
regime (i. e. at
W7-X
:
1019 m−3<ne,0<8×1019 m−3
). To allow a more robust
inference without the knowledge of the exact electron density profiles in these
cases, the interferometer was added to the ECE model, constraining primarily
the line integrated electron density. Hence, the electron density profile is one
of the parameters that can be inferred.
Finally, chapter 6 presents applications of the ECE system at
W7-X
. In the
first part, the ECE model (cf. figure 5.1) is aiming to push to the boundaries of
the operational space where ECE may be interpreted by applying it to a plasma
with high electron density, such that the ECE is partially in X mode cutoff. The
inferred electron density profile exhibits little variance as the channels close to
their cutoff provide a lot of information on the density due to the high sensitivity
of their lines of sight on the exact density. The main drawback of this analysis
is the required time, which is well over three months. However, the ground-
work for generating an artificial neural network of the time critical part of the
model, namely the TRAVIS ray tracing and radiation transport calculations, has
been layed. If further pursued this should bring enough speed improvements
to eventually open the possibility to use the ECE (in combination with the in-
terferometer) in partial cutoff as a density diagnostic for the first time. A proof
of concept for an analysis with low electron density plasmas was shown and
dicussed as well. Subsequently, a simple heat transport model was implemented,
although the long analysis time caused by the high dimensionality of the model
(430 free parameters) prohibited an analysis within the scope of this thesis.
Analyses of Bayesian forward models have been performed exemplarily for
the ECE in this thesis. Such models allow easy combination of measurements
from a variety of diagnostics and the inference of the posteriors of the corres-
ponding plasma parameters. In preparation for the Bayesian analysis of models
that combine several diagnostics multiple datasources have been written dur-
114
ing this thesis, notably the ECRH, ECE, X-ray imaging crystal spectroscopy,
snifferprobe, single channel dispersion interferometry, NBI, Langmuir probe
system and VMEC reference identifier datasources for
W7-X
. Generic nodes
for obtaining the electron temperature and density profiles were added to the
Minerva framework at
W7-X
as well. The general model for
W7-X
, combining
nine diagnostics, has been created, although the analysis of it remains an open
issue.
115
A
Temperature dependence of the
calibration model prediction
For a specific channel
i
the predicted signal in bit is described via equation (4.7).
Given a linear sensitivity, this equation can be rewritten to
fpred
i(φ)=Teff,i(φ)ηi=ΔsiTeff,i(φ)
Thot
eff −Tcold
eff
.(A.1)
As we do subtract the offset of the measured signal, we end up with
fpred
i(φ)−fpred
i=Δsi�Teff,i(φ)
Thot
eff −Tcold
eff
−�β−γ
n�
�����
offset �,(A.2)
where
β=Thot
eff
Thot
eff −Tcold
eff
,
γ
is the number of entries for the cold vector and
n
the number of entries for
the cold and the hot vector. As all effective temperatures are between
Tcold
eff
and
Thot
eff
, they can be expressed by
β−δ
with
δ∈[
0
,
1
]
. Thus, the predicted signal
can be written as
fpred
i(φ)−fpred
i=Δsi�β−δ−�β−γ
n�� (A.3)
=Δsi�γ
n−δ�.(A.4)
Thus, the predicted signal does not depend on the absolute values of the effective
temperature for a linear sensitivity.
117
B
ECE calibration code
Here, the main class for performing an ECE calibration at
W7-X
from within
Java is explained. The code is taken from
Calibrate.java
in the correspond-
ing Minerva package and corresponds to [110]. Obviously, not all code of the cal-
ibration analysis can be explained here. Nevertheless, the code is fully Javadoc
documented, so should be accessible to the interested reader.
The implementation of the
main
method as the entry point for the Java
Calibrate
class is presented in the subsequent code snippets. Code B.1 starts
with the beginning of the main method.
Code B.1:
Instantiation of calibration class, and first setter of the ECE Minerva
calibration code.
27 /**
28 * Intended to run the selected radiometer calibration model
with the specified
29 * settings.
30 *
31 * @param args empty
32 */
33 public static void main(String[] args) {
35 W7xEceQmeCalibrate calib =new W7xEceQmeCalibrate();
37 calib.setCalibrationSegments(QME_20181115_2());
Line 35 creates an instance of the calibration class. The setter in line 37 takes a
long[][]
array, such that multiple calibration segments (as
W7-X
time stamps)
119
Appendix B. ECE calibration code
can be passed to the inference routines. This allows »stitching together« mul-
tiple calibration segments which is useful for example if one had to refill li-
quid nitrogen during the calibration and wants to discard the associated data.
QME_20181115_2()
is just an example method that provides the
W7-X
time
stamps.
Code B.2 shows the setters related to the hardware setup during the calibra-
tion and the logging settings.
Code B.2: Hardware and logging setters of the ECE Minerva calibration code.
39 calib.setChannel(8);
40 calib.setChopperChannel(16);
41 calib.setChopperZoom(false);
42 calib.setLogToFile(true);
43 calib.setMode(Modes.HotColdLoad);
Line 39 sets the channel to evaluate. This setter is only of importance if a
model is selected that evaluates a single channel. Counting starts at
1
. The
channel containing the chopper signal is specified via the setter in line 40.
Additionally, line 41 allows to specify whether the chopper channel is located in
the zoom or the standard ECE system. Line 42 allows to specify whether logging
information should be written to a file. This is particularly recommended for
evaluations that take a long time to get a quick overview of the current status.
The mode can be selected via the setter in line 43. The
Modes
enum has two
possible values,
HotColdLoad
and
NoiseSource
. Note that the
NoiseSource
mode is experimental, and not as deeply tested as the
HotColdLoad
mode. In
principle, the
NoiseSource
mode allows an absolute (sic!) calibration of the
ECE, although at the moment the effect of the transmission line is not taken
into account therein.
Code B.3 shows how to select the desired model. As explained in section 4.1,
there is a plethora of calibration models to choose from.
Code B.3:
The calibration model selection of the ECE Minerva calibration code.
45 calib.setModel(RadiometerHotColdLoadModel.STANDARD_HYPER);
In fact, due to technical reasons, there are more models available than de-
scribed above. As the zoom system can in principle be calibrated the same
way as the standard ECE system, options are also available to calibrate the
120
former, or both systems combined. Do note that the setter takes an interface
object of type
EceCalibration
as an argument, which is implemented by both
RadiometerHotColdLoadModel
and
RadiometerNoiseSourceModel
. Again, the code
of the noise source models is kept similar to the code of the hot cold load models
(not the structure of the model, though), but it is less thoroughly tested which
is why chapter 4 does not mention the noise source models. However, it could
be worthwhile to finish the implementation of the noise source models in the
future, as this would allow a much quicker calibration procedure (on the order
of few minutes). The disadvantage of the noise source models are currently
twofold: i) the noise source is located after the transmission line (cf. figure 2.4),
thus not taking the transmission line into account, and ii) the accuracy of the
excess noise ratio of the current noise source is only certain within
1 dB
, thus
leading inevitably to large uncertainties in the resulting calibration factors.
Code B.4 highlights the available analysis settings. The settings shown in
code B.4 correspond to a calibration performed after OP 1.2b of W7-X.
Code B.4: Setters for the analysis of the ECE Minerva calibration code, part I.
47 calib.setNumBins(0);
48 calib.setMultithreading(false);
49 calib.setNumThreads(4);
50 calib.setNumIntegralSteps(0);
51 calib.setUseMap(true);
52 calib.setNumMapIterations(500);
53 calib.setUseLaplaceApproximation(false);
54 calib.setUseMcmc(true);
55 calib.setNumMcmcBurnIn(500_000);
56 calib.setNumMcmcKeepEvery(3);
57 calib.setNumMcmcIterations(50_000);
58 calib.setScan(true);
59 calib.setNumScanPoints(100);
60 calib.setUseSinglePeriodAnalysis(false);
61 calib.setCompareToReferenceDischarges(true);
62 calib.setValidSince("2018-11-15");
63 calib.setReason("calibration after op1.2b");
64 calib.setNumSamplesToPlot(100);
Line 47 allows you to select the number of bins to be used for the conditionally
averaged signal. A reasonable default is chosen if
0
is the argument. Be aware
121
Appendix B. ECE calibration code
that for high sampling rates the default will determine a large number of bins,
which will slow down the analysis correspondingly. If a model has been chosen
that does multiple single channel evaluations, line 48 allows to activate multi-
threading. The corresponding number of threads can be chosen in line 49. The
integrals in equation (4.5) are performed numerically, thus the number of steps
to be done can be chosen via the setter in line 50. Again,
0
chooses a default
that performed well for the calibrations done for this thesis. Choosing to few
steps can lead to a posterior with »steps«. This is not relevant for noise source
models. If an MAP inversion should be performed, the setter in line 51 should
be set to
true
. This is generally recommended. The corresponding number of
iterations is specified in line 52. In general, a value around
500
seems to behave
well. Do note that internally the MAP methods are stopped once either these
number of iterations have been performed, or the change in the logarithm of the
prior times likelihood probability after
10
iterations is less than
1
. If the setter
in line 53 is set to
true
a Laplace approximation (LA) is attempted to estimate
the initial proposal distriubution for the MCMC. While fully implemented, this
has not been tested in practice and thus should be considered an experimental
option. Similar to line 51, the MCMC inversion can be activated by the setter
in line 54. The number of burn-in iterations is specified in line 55. Typically,
500_000
should be more than enough. As there can be a slight correlation
between subsequent points in an MCMC (cf. section 3.3), one can set a number
of points to skip before the next value gets stored. This can be done as shown
in line 56. The number of MCMC iterations to be performed after the burn-in
can be specified via the setter in line 57. If the setter in line 58 is set to
true
a
scan is done in which the free parameters are set to the mean values obtained
from an MCMC performed beforehand. Therein, each free parameter is scanned
independently from
−
3
σ
to
+
3
σ
around the MCMC mean, with the number of
points for the scan determined via the setter in line 59. Thus, the distributions
obtained that way will be generally too narrow, as uncertainty contributions
from the other free parameters are neglected. Usually, the scan is not necessary.
Also only relevant for very specific usecases is the single period analysis that
can be activated via the setter shown in line 60. In this analysis the inversions
are executed as usual. Subsequently, the free parameters in the model are set to
their MCMC means and are deactivated. The only parameters that remain free
are the bit dip and the variance scaling (if active). Then, another short MAP and
MCMC inversion is done for each period in the calibration segment. The inferred
sensitivites are stored. This analysis method can be used to detect some sort of
122
sensitivity drifts, although it is generally better to still use multiple periods as
without averaging there will be a huge spread of values, and the analysis will
take much longer. To see quickly after a calibration which channels changed,
an option to compare previously calculated ECE spectra with spectra calculated
with the results from this analysis, see line 61. The validity of the calibration
values can be set as shown in line 62. This is not automized, as the validity might
predate the actual calibration, if the hardware was left unchanged beforehand.
Line 63 contains the code to give a reason for the calibration (analysis) that is
saved to the
W7-X
database if writing to it is enabled. If plots are to be saved
the number of samples used can be specificated as shown in line 64.
Code B.5: Setters for the analysis of the ECE Minerva calibration code, part II.
65 calib.setOutputFolder(null);
66 calib.setPath(null);
67 calib.setPlot(true);
68 calib.setReducedNumSamples(50);
69 calib.setRemarks(null);
70 calib.setRemarksZoom(null);
71 calib.setWriteToDatabase(true);
72 calib.setDeactivateNonnecessaryFreeParameters(true);
The output folder can be specified with the setter shown in line 65. If you set
the value to
null
a sensible default folder is created. Line 66 shows the setter
to specify the name of the log file. The path of the output folder is prepended.
Again,
null
provides a default. Saving plots automatically can be done via the
setter in line 67. As the calibration values stored on the
W7-X
database do not
require all MCMC samples for most subsequent analyses, a reduced number
of samples as specified in line 68 is stored directly with the calibration values.
The full MCMC samples are not discarded, but kept separately to keep access
time for the mean calibration values small. For each calibration uploaded to the
W7-X
database remarks can be added for the standard ECE system (cf. line 69)
as well as for the zoom system (cf. line 70). If, like for the example shown here,
the setters set the remarks to
null
, no remarks are added. Via the setter in
line 71 one can determine whether to write the results automatically (including
the full model structure) to the immutable
W7-X
database. A huge speedup can
be achieved be deactivating free parameters on which the prediction does not
explicitly depend (cf. appendix A), as this decreases the effect of the »curse of
123
Appendix B. ECE calibration code
dimensionality« notably, especially for the combined models. The uncertainty
of these deactivated free parameters is still taken into account for the MCMC,
as their values are sampled anew for each MCMC iteration.
Finally, code B.6 shows in line 73 the method to run the calibration with the
options selected above. Line 74 closes the main method.
Code B.6:
The method to run the
W7-X
ECE calibration and the end of the
main
method begun in code B.1.
73 calib.run();
74 }
All the setters shown here have analogous keywords that can be set for the
standalone executable version directly via the command line or can be provided
via a .properties file (example: SAVE), allowing a high degree of flexibility.
124
C
ECE profile inversion code
Here, the main class for performing an ECE profile inversion at
W7-X
from
within Java is explained. The code corresponds to [111] from
History.java
and
Inference.java
in the corresponding Minerva package.
History.java
contains the settings for previously run inferences with manually set settings
to allow for easy reproducibility. Obviously, not all code of the profile inversion
analysis package can be explained here. Nevertheless, the code is fully Javadoc
documented to be accessible to the interested reader. Example settings for a
profile inference are shown in code C.1 which starts with the enum that lists
analyses done for a specific person.
Code C.1: Enum with evaluated shots.
920 /**
921 * Evaluated profiles for Udo Hoefel.
922 *
923 * @author Udo Hoefel
924 */
925 enum UdoHoefel implements EvaluatedProfiles {
⋮
937 shot_20180823_016_time_4_45,
938 shot_20180823_016_time_5_15,
939 shot_20181009_034_time_3_00,
940 ;
There is an enum for each of the people that requested analyses. The begin-
ning of the implementation of the
EvaluatedProfiles
interface is shown in
code C.2.
125
Appendix C. ECE profile inversion code
Code C.2: Implementation of the EvaluatedProfiles interface, part I.
942 @Override
943 public W7xEceQmeProfileInversion getSettings() {
944 W7xEceQmeProfileInversion inference =new
W7xEceQmeProfileInversion();
946 switch (this) { ⋮
1521 case shot_20180823_016_time_4_45: {
1522 inference.setYear(2018);
1523 inference.setMonth(8);
1524 inference.setDay(23);
1525 inference.setExperiment(16);
1526 inference.setScenario(2);
1527 inference.setTime(4.45);
1529 inference.setTimeWindowForAveraging(50);
1530 inference.setDataVersion(0);
1532 inference.setEceChannelsToDeactivate(16);
1533 inference.
setEceChannelsToDeactivateForInitialGuess(16);
1535 inference.setTeInitialGuess(TeInitialGuess.ECE);
1536 inference.setNeInitialGuess(NeInitialGuess.
THOMSON_SCATTERING_SCALED);
The year of the discharge to be analyzed is given in line 1522, the month in
line 1523 and the day in line 1524. On the specified date, the lines 1525 to 1527
allow to determine the nanosecond at which to analyze the combined ECE
and interferometry data. Line 1529 contains the command to set the time win-
dow over which the ECE and interferometry signal is averaged. An example
of how to use the most recent data version is shown in line 1530. As this is
not guaranteed to be reproducible the version that the most recent version
corresponds to is written explicitly to the output files. Sometimes some ECE
channels are problematic due to various problems. Specific ECE channels can
be deactivated for the inference as shown in line 1532 or only for the initial
guess as shown in line 1533. The initial guess for the electron temperature pro-
126
file is set exemplarily in line 1535. The enum
TeInitialGuess
allows to select
the
ECE
to use all ECE channels for the initial guess,
ECE_LFS
restricts the
channels for the initial guess to the low field side,
ECE_HFS
correspondingly
to the channels on the HFS. If
THOMSON_SCATTERING
is selected the initial
guess uses the measurements from the Thomson scattering system. Further-
more,
TRAVIS_PARAMETERIZATION
uses equation (5.1) to represent the profile.
The parameters can be set via
setTeTravisParamsInitialGuess
. If a manual
profile should be applied use
MANUAL
. The
setManualTeInitialGuess
meth-
od allows to set the corresponding profile and requires a ScalarFunction1D.
Correspondingly, line 1536 shows how to set the initial guess for the electron
density profile. Similar as for the electron temperature profile initial guess, an
enum, NeInitialGuess, allows the selection of the type of initial electron dens-
ity profile guess. Possible options are
THOMSON_SCATTERING
using the density
profile as measured by Thomson scattering,
THOMSON_SCATTERING_SCALED
,
which uses the same profile as for
THOMSON_SCATTERING
but scaled to match
the line integrated density as measured by the single channel dispersion inter-
ferometry. A default profile scaled to the interferometry measurements can be
used via
INTERFEROMETRY_SCALED_DEFAULT_PROFILE
if Thomson scatter-
ing data are not available. Similar to the initial guess for the electron temper-
ature profile,
MANUAL
and
TRAVIS_PARAMETERIZATION
are available for the
electron density profile initial guess and can be used with their corresponding
setters. To scale a manual profile such that consistency with the line integrated
electron density measured by the single channel dispersion interferometer is
guaranteed one can use
INTERFEROMETRY_SCALED_MANUAL_PROFILE
. The
next part of the implementation is shown in code C.3.
Code C.3: Implementation of the EvaluatedProfiles interface, part II.
1538 inference.setNumRho(20);
1539 inference.setAdditionalPointNearCore(true);
1540 inference.setDensityProfileActive(true);
1541 inference.setUseHyperparams(false);
1542 inference.setDeltaRho(0.0);
1543 inference.setStepsizeRho(0.005);
1544 inference.setInitialGuessIterations(0);
1546 inference.setModel(EceProfileInversion.
ECE_PROFILE_INVERSION_OX);
127
Appendix C. ECE profile inversion code
The number of points at which the Gaussian processes are evaluated is set in
line 1538. Increasing this number increases the dimensionality of the model.
By setting the setter in line 1539 to true, one will get an additional Gaussian
process evaluation close to the core (at
ρ=10−3
), such that the virtual obser-
vations that limit the gradients in the core do not have to depend as much
on the interpolation mode used between the Gaussian process points. How-
ever, this increases the dimensionality of the model by 2. In line 1540 it is
shown how one can deactivate the density profile and just keep it at the initial
guess. Due to the computational cost that free hyperparameters entail, they
are normally deactivated. In case they are still wanted as free parameters,
one can set the setter in line 1541 to
true
. If a shift of the initial guess of
the electron temperature profile is desired, line 1542 allows to specify by how
much the profile is shifted. The stepsize for this shift is set in line 1543, such
that effectively the prior times likelihood of the whole model is checked for
an initial guess electron temperature profile shifted by
−
Δ
ρ, −
Δ
ρ+
δ
ρ,…,
Δ
ρ
and the one with the highest probability is used for the inference. Iterating
on the initial electron temperature profile guess will be done the number of
times specified in line 1544. Usually, this can be left at zero, more than one
iteration typically make the initial guess worse. Selecting the type of model
can be done like is shown in line 1546. The enum
EceProfileInversion
con-
tains the available models. The standard model
ECE_PROFILE_INVERSION_OX
uses Gaussian processes,
ECE_PROFILE_INVERSION_OX_HYPER
uses in ad-
dition a variance scaling factor. For the parameterization described by equa-
tion (5.1) two corresponding models are available: The first available model is
ECE_PROFILE_INVERSION_OX_TRAVIS_PARAM
. The one including the vari-
ance scaling is
ECE_PROFILE_INVERSION_OX_HYPER_TRAVIS_PARAM
. The
settings for the MAP and MCMC inversion are specified in code C.4.
Code C.4: Implementation of the EvaluatedProfiles interface, part III.
1548 inference.setUseMap(true);
1549 inference.setNumMapIterations(150);
1550 inference.setMapAutoConverge(true);
1551 inference.setUseMcmc(true);
1552 inference.setNumMcmcBurnIn(100_000);
1553 inference.setNumMcmcKeepEvery(3);
1554 inference.setNumMcmcIterations(10_000);
128
If an MAP inversion should be done the setter in line 1548 has to be set to
true
.
The maximum number of MAP iterations done can be set as shown in line 1549.
If one wants to stop the MAP automatically once the logarithm of the prior
times likelihood does not change by more than 1over 7 iterations the setter in
line 1550 should be set to
true
. This might lead to fewer MAP iterations being
done than specified in line 1549. To perform an MCMC inversion set the setter
in line 1551 to
true
. The number of MCMC iterations to be used for the burn-in
is specified as shown in line 1552. To make sure that the MCMC samples are
uncorrelated, you can choose to keep only every
nth
sample, cf. line 1553. The
number of MCMC iterations to be done aside the burn-in iterations is set as
shown in line 1554. Further settings are shown in code C.5.
Code C.5: Implementation of the EvaluatedProfiles interface, part IV.
1557 W7xVmecIdDataSource vmecId =new
W7xVmecIdDataSource();
1558 ArchiveDBFetcher adb = ArchiveDBFetcher.
defaultInstance();
1559 long t1 =adb.quiet().getT1("20180823.016.002");
1560 vmecId.setNanosecond(t1 + 4_450_000_000L);
1561 inference.setFixedVmecId(vmecId.getBestVmecId());
1562 VmecInfoType type =vmecId.getBestVmecIdType();
1563 double scaling =vmecId.getMagneticFieldScaling(
type);
1564 inference.setMagneticFieldRescaling(scaling);
1566 inference.setMirrorId("W7X-Vessel/
W7X_ECE_targetTileOnly_noHFShorn.xml");
1568 inference.setFixedXModeContribution(null);
1569 inference.setFixedOModeContribution(null);
Line 1557 instantiates a datasource that can provide automatically select best
guess VMEC equilibria for a given timepoint. Then, line 1558 instantiates the
class used for fetching data. Subsequently, line 1559 fetches the T1 trigger of,
exemplarily, #20180823.016.002 in
W7-X
nanoseconds. Thereafter, the VMEC
equilibrium datasource is set to
T1 +4.5 s
in line 1560. This allows to set the pro-
file reconstruction to the best automatically determinable VMEC equilibrium,
see line 1561. As there may be different VMEC equilibria supplied by different
129
Appendix C. ECE profile inversion code
analysis types, the analysis type that is estimated by the VMEC datasource to
match best to the experimental data is fetched in line 1562. But the precalculated
VMEC equilibrium may not have the appropriate absolute
B
field to which the
ECE at
W7-X
is highly sensitive due to the shallow magnetic field gradient
along the line of sight of the ECE. Therefore, a factor that rescales the magnetic
field to match to the field as expected from the experiment is fetched in line 1563.
The shape of the flux surfaces stays the same. If reflections, for example of the
O mode, are to be taken into account TRAVIS needs to have a way to determine
where the ray gets reflected, what the reflection coefficient is and so on. The
file that contains this information is supplied as shown in line 1566. X mode and
O mode contributions are typically free parameters, in which case the setters
in lines 1568 and 1569 have to be set to
null
. If they are set to any number
between
0
and
1
they are no longer treated as free parameters and kept fixed.
The remaining settings for the inference are shown in code C.6.
Code C.6: Implementation of the EvaluatedProfiles interface, part V.
1572 inference.setInterpolationMode(InterpolationMode.
CUBIC);
1573 inference.setLogToFile(true);
1574 inference.setShowPlots(false);
1575 inference.setPlotLevel(Level.INFO);
1576 inference.setNumSamplesToPlot(100);
1577 inference.setNumSamplesToSave(100);
1578 inference.setOutputFolder(null);
1579 inference.setPath(null);
1580 inference.setNoOutput(false);
1581 inference.setPlot(true);
1582 inference.setBeamInfo(true);
1583 inference.setWriteToDatabase(false);
1584 inference.setDatabase(Database.W7X_ARCHIVE);
1585 inference.setWriteAliases(false);
1586 break;
1587 }⋮
1696 return inference;
1697 }
1698 }
130
The interpolation mode can be set to either
LINEAR
or
CUBIC
, see line 1572 and
determines how the inferred Gaussian process points get turned to a continuous
profile. If one wants to have the logging messages written to a log file, the
setter in line 1573 should be set to
true
(recommended). Showing each plot
interactively (which blocks the evaluation until closed) can be achieved by
setting the setter in line 1574 to
true
. The plot level can be set as shown in
line 1575. This allows to avoid plots with lower importance. The number of
samples to be plotted can be set as shown in line 1576. Correspondingly, the
number of samples to be saved is determined by the setter in line 1577. Usually,
the output folder is determined automatically. However, the setter in line 1578
can be used to use a different output folder. If a file is used to store the logged
messages its path can be specified as shown in line 1579,
null
uses a default.
For automatic routines that upload for example the initial guess to the archive,
automatically created local plots and files are usually just a waste of time. Thus,
all output can be deactivated if the setter in line 1580 is set to
true
. In the case
that only plots should be avoided set the setter in line 1581 to
false
. For a very
detailed analysis of a time point, activate additional information about the ECE
beam by setting the setter in line 1582 to
true
. The results can be automatically
written to the database if the setter in line 1583 is set to
true
. Line 1584 shows
the setter allowing to select to which database to write results to. Possible
options are
W7X_ARCHIVE
for the productive archive and
W7X_TEST
for the
test archive. Finally, one can force rewrite the aliases pointing to the data if
the setter in line 1585 is set to
true
. By default, the aliases will be written
automatically if required. In the
Inference
class one can then call the code
shown in code C.7 in the main method.
Code C.7: Code to run the profile inversion from within Java.
37 UdoHoefel.shot_20180823_016_time_4_45.run();
An example input file for the compiled Java executable as used for the analysis
in section 5.2 can be found here: SAVE.
131
Acknowledgements
First I have to thank Prof. Wolf for giving me the opportunity to do this PhD
thesis at IPP.
I am deeply indebted to Matthias Hirsch as well, as I could always ask for
advice, always got a lot of useful comments and was very gently directed back
to not stray away too much from the topics at hand. I also have to mention Tor-
sten Stange. Not only did he introduce me to IPP, but he also always helped me
whenever I needed help and the light in his office was often the only »guiding
star« when I left IPP late. Nikolai Marushchenko and Yuriy Turkin were also
very helpful as they provided theoretical insights, the webservice for TRAVIS
and never seemed to get tired of my questions. A large thank you is also de-
served by Michael Grahl who always quickly helped if I didn’t understand
something with the archive or the webservices. Keeping my sanity was made
a lot easier by the shared suffering of the quirks of the archive by Kai-Jakob
Brunner and especially Oliver Ford, who were also fostering my physics under-
standing tremendously.
A thank you also goes out to Fabian Wilde (I am curious how long it will take
you to get the Tesla) and Peter Traverso. The office was never again so much
fun after you left the office!
I already miss the Squash sessions with Adnan Ali, Daniel Böckenhoff and
occasionally Tullio Barbui and Lukas Rudischhauser. The gatherings with Va-
leria Perseo and Priyanjana Sinha were amongst the most enjoyable time I had
in Greifswald. Merci beaucoup and mea culpa for not coming to Lindy Hop any-
more. The discussions with Humberto Trimiño Mora were amazing, although
sometimes slightly scary. You are too good at them!
A special thanks goes out to Jakob Svensson. Starting the PhD I did not expect
major revisions of my attitude towards how an analysis should be done to be
on the table, but lo and behold, your view on Bayes was able to do so. The other
Minervarians, especially Sehyun Kwak (or was it Kwak Sehyun?) and Andrea
Pavone are both amazingly kind and clever. I learned a lot from you. I am happy
133
that I had the opportunity to work with Jonathan Schilling, whose speed and
efficiency is simply stunning.
Many of the people above I consider good friends, I am glad that I met you.
Of course I also have to mention my family which, to a large part, made me
who I am today. Also, everyone very much appreciated the »care packages«,
even though they were not really necessary
SMILE-WINK
. And last but not least: Thank
you for everything, Sandy.
Statutory declaration
I hereby declare in accordance with the examination regulations that I myself
have written this document, that no other sources as those indicated were used
and all direct and indirect citations are properly designated, that the document
handed in was neither fully nor partly subject to another examination procedure
or published and that the content of the electronic exemplar is identical to the
printing copy.
Greifswald, 24th February 2020
Udo Höfel
135
List of Figures
1.1. Schematic structure of W7-X . . . . . . . . . . . . . . . . . . . . 4
2.1. Illustration of the principle of ECE . . . . . . . . . . . . . . . . 9
2.2. Harmonics of the electron cyclotron frequencies at W7-X . . . 11
2.3. Radiation transport effects influencing ECE . . . . . . . . . . . 12
2.4. Setup of the standard ECE system at W7-X . . . . . . . . . . . 19
2.5. Example drifts of the standard ECE system at W7-X . . . . . . 21
2.6. Schematic drawing of the W7-X ECE calibration unit. . . . . . 22
3.1.
Examples of electron temperature profile shapes obtained via a
GP .................................. 29
3.2. Modus operandi of an MAP based on Hooke and Jeeves . . . . 30
3.3. Modus operandi of an MCMC . . . . . . . . . . . . . . . . . . . 31
4.1. Simplified ECE calibration Minerva graph . . . . . . . . . . . . 42
4.2. Example MCMC traces of the calibration model . . . . . . . . . 44
4.3. Raw chopper and channel signal during an ECE calibration . . 49
4.4. Preprocessed ECE calibration data . . . . . . . . . . . . . . . . 50
4.5. Voltage vs. effective temperature of the ECE calibration . . . . 51
4.6. The KDEs of two example ECE channel calibration factors . . . 52
4.7. The posterior distribution for the sensitive channel 23 . . . . . 53
4.8. The posterior distribution for the insensitive channel 11 . . . . 54
4.9.
Comparison of different Minerva models with respect to the
calibration factor . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.10.
Comparison of different Minerva models with respect to the
beam width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.11.
Comparison of different Minerva models with respect to the
variance scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.12. Calibrated ECE spectra from W7-X . . . . . . . . . . . . . . . . 58
137
List of Figures
4.13.
Comparison of the calibrated ECE signal to a corresponding TS
channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1. Minerva ECE profile inversion model . . . . . . . . . . . . . . . 65
5.2. Examples of the parameterization used also with TRAVIS . . . 66
5.3. Evaluation scheme for the ECE analysis at W7-X . . . . . . . . 67
5.4. Overview of discharge #20180823.016.002 . . . . . . . . . . . . 74
5.5. The log(joint) of the whole profile inversion model . . . . . . . 75
5.6. Comparison of predicted and observed Trad spectra . . . . . . . 77
5.7. Comparison of Teprofiles . . . . . . . . . . . . . . . . . . . . . 78
5.8. Comparison of neprofiles . . . . . . . . . . . . . . . . . . . . . 79
5.9. The marginal posterior of the interferometry prediction . . . . 80
5.10. The marginal posteriors of the virtual observation predictions . 81
5.11. Spectra of the optical depth from the inverted model . . . . . . 82
5.12.
Spectra of the electron momentum normalized to the thermal
electron momentum . . . . . . . . . . . . . . . . . . . . . . . . 84
5.13. Mapped Trad profiles . . . . . . . . . . . . . . . . . . . . . . . . 85
5.14. Comparison of effective radii from CR and TRAVIS calculations 87
5.15. The KDEs of the X mode and O mode contributions . . . . . . 88
5.16. Comparison of Teprofiles with NBI heating on and off . . . . . 89
5.17. Comparison of neprofiles with NBI heating on and off . . . . . 90
6.1. Overview of the high performance discharge #20171115.039.002 95
6.2.
ECE channels coming out of cutoff during the high performance
discharge #20171115.039.002 . . . . . . . . . . . . . . . . . . . . 96
6.3. Ray traced ECE channel at densities near its cutoff . . . . . . . 97
6.4. critical and cutoff densities for the ECE at W7-X . . . . . . . . 98
6.5.
Comparison of predicted and observed
Trad
spectra during par-
tial cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.6. The log(joint) of the whole profile inversion model . . . . . . . 102
6.7. High density neprofile . . . . . . . . . . . . . . . . . . . . . . . 103
6.8.
Comparison of predicted and observed
Trad
spectra with »shine-
through« effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.9. Simplified Minerva heat transport model . . . . . . . . . . . . . 108
6.10. Comparison of predicted and observed
Te. . . . . . . . . . . . 110
138
List of Codes
3.1. Example code from the main plasma profile inversion model . . 34
B.1. Instantiation of calibration class, and first setter . . . . . . . . . 119
B.2. ECE calibration hardware and logging setters . . . . . . . . . . 120
B.3. ECE calibration model setter . . . . . . . . . . . . . . . . . . . . 120
B.4. ECE calibration analysis setters, part I . . . . . . . . . . . . . . . 121
B.5. ECE calibration analysis setters, part II . . . . . . . . . . . . . . 123
B.6. ECE calibration run method . . . . . . . . . . . . . . . . . . . . 124
C.1. Enum with evaluated shots. . . . . . . . . . . . . . . . . . . . . 125
C.2. Implementation of the EvaluatedProfiles interface, part I. . . 126
C.3. Implementation of the EvaluatedProfiles interface, part II. . 127
C.4. Implementation of the EvaluatedProfiles interface, part III. . 128
C.5. Implementation of the EvaluatedProfiles interface, part IV. . 129
C.6. Implementation of the EvaluatedProfiles interface, part V. . 130
C.7. Running the profile inversion . . . . . . . . . . . . . . . . . . . 131
139
Publications as first author
Peer-reviewed articles
[1] U. Höfel
et al. »Bayesian modelling of the electron cyclotron emission
diagnostic at Wendelstein
7-X
«. In: Nuclear Fusion, (2020). In prepara-
tion.
[2] U. Höfel
et al. »Bayesian Modelling of Microwave Radiometer Calib-
ration on the example of the Wendelstein
7-X
Electron Cyclotron Emis-
sion diagnostic«. In: Review of Scientific Instruments, Vol. 90.4, Art. 04
3502 (11th Apr. 2019). doi:10.1063/1.5082542.
141
Publications as coauthor
Peer-reviewed articles
[1]
T. Stange et al., amongst them
U. Höfel
. »First demonstration of
fully equilibrated ions and electrons in a Magnetically Confined High
Temperature Plasma sustained by 2nd harmonic O-mode polarized ECRH
only«. In: Physical Review Letters, (2020). Submitted.
[2]
N. A. Pablant et al., amongst them
U. Höfel
. »Investigation of the
neoclassical ambipolar electric field in ion-root plasmas on W7-X«. In:
Nuclear Fusion, (22nd Jan. 2020). doi:10.1088/1741-4326/ab6ea8.
[3]
J. W. Oosterbeek et al., amongst them
U. Höfel
. »Michelson In-
terferometer design in ECW heated plasmas and initial results«. In:
Fusion Engineering and Design, Vol. 146 (Sept. 2019), pages 959–962. doi:
10.1016/j.fusengdes.2019.01.124.
[4]
J. Baldzuhn et al., amongst them
U. Höfel
. »Pellet fueling experi-
ments in Wendelstein
7-X
«. In: Plasma Physics and Controlled Fusion,
Vol. 61.9, Art. 095012 (13
th
Aug. 2019). doi:
10.1088/1361-6587/
ab3567.
[5]
R. C. Wolf et al., amongst them
U. Höfel
. »Performance of Wendel-
stein
7-X
stellarator plasmas during the first divertor operation phase«.
In: Physics of Plasmas, Vol. 26.8, Art. 082504 (13
th
Aug. 2019). doi:
10.1063/1.5098761.
[6]
D. Zhang et al., amongst them
U. Höfel
. »First Observation of a
Stable Highly Dissipative Divertor Plasma Regime on the Wendelstein
7-X
Stellarator«. In: Physical Review Letters, Vol. 123.2, Art. 025002 (9
th
July
2019). doi:10.1103/PhysRevLett.123.025002.
143
Publications as coauthor
[7]
A. Pavone et al., amongst them
U. Höfel
. »Neural network approx-
imation of Bayesian models for the inference of ion and electron tem-
perature profiles at W7-X«. In: Plasma Physics and Controlled Fusion,
Vol. 61.7, Art. 075012 (30
th
May 2019). doi:
10.1088/1361-6587/
ab1d26.
[8]
A. Dinklage et al., amongst them
U. Höfel
. »Plasma termination by
excess pellet fueling and impurity injection in TJ-II, the Large Helical
Device and Wendelstein 7-X«. In: Nuclear Fusion, Vol. 59.7, Art. 07601
0 (24th May 2019). doi:10.1088/1741-4326/ab17fd.
[9]
S. C. Liu et al., amongst them
U. Höfel
. »The effects of magnetic
topology on the scrape-off layer turbulence transport in the first divertor
plasma operation of Wendelstein 7-X using a new combined probe«. In:
Nuclear Fusion, Vol. 59.6, Art. 066001 (24
th
Apr. 2019). doi:
10.1088/
1741-4326/ab0d29.
[10]
M. Hirsch et al., amongst them
U. Höfel
. »ECE Diagnostic for the
initial Operation of Wendelstein
7-X
«. In: European Physical Journal
Web of Conferences, Vol. 203, Art. 03007 (25
th
Mar. 2019). doi:
10 .
1051/epjconf/201920303007.
[11]
M. Zanini et al., amongst them
U. Höfel
. »ECCD operations in the
second experimental campaign at
W7-X
«. In: European Physical Journal
Web of Conferences, Vol. 203, Art. 02013 (25
th
Mar. 2019). doi:
10.105
1/epjconf/201920302013.
[12]
N. Chaudhary et al., amongst them
U. Höfel
. »Investigation of
Optically Grey Electron Cyclotron Harmonics in Wendelstein
7-X
«. In:
European Physical Journal Web of Conferences, Vol. 203, Art. 03005
(25th Mar. 2019). doi:10.1051/epjconf/201920303005.
[13]
J. W. Oosterbeek et al., amongst them
U. Höfel
. »Assessment of
ECH stray radiation levels at the
W7-X
Michelson Interferometer and
Profile Reflectometer«. In: European Physical Journal Web of Conferences,
Vol. 203, Art. 03010 (25
th
Mar. 2019). doi:
10.1051/epjconf/2019
20303010.
[14]
G. G. Plunk et al., amongst them
U. Höfel
. »Stellarators Resist Turbu-
lent Transport on the Electron Larmor Scale«. In: Physical Review Letters,
Vol. 122.3, Art. 035002 (25
th
Jan. 2019). doi:
10.1103/PhysRevLett.
122.035002.
144
Peer-reviewed articles
[15]
D. Moseev et al., amongst them
U. Höfel
. »Collective Thomson
scattering diagnostic at Wendelstein
7-X
«. In: Review of Scientific In-
struments, Vol. 90.1, Art. 013503 (8
th
Jan. 2019). doi:
10 . 1063 / 1 .
5050193.
[16]
R. C. Wolf et al., amongst them
U. Höfel
. »Electron-cyclotron-res-
onance heating in Wendelstein
7-X
: A versatile heating and current-
drive method and a tool for in-depth physics studies«. In: Plasma Phys-
ics and Controlled Fusion, Vol. 61.1, Art. 014037 (27
th
Nov. 2018). doi:
10.1088/1361-6587/aaeab2.
[17]
A. Langenberg et al., amongst them
U. Höfel
. »Impurity trans-
port studies at Wendelstein
7-X
by means of x-ray imaging spectro-
meter measurements«. In: Plasma Physics and Controlled Fusion, Vol. 61.1,
Art. 014030 (23rd Nov. 2018). doi:10.1088/1361-6587/aaeb74.
[18]
G. Fuchert et al., amongst them
U. Höfel
. »Global energy confine-
ment in the initial limiter configuration of Wendelstein
7-X
«. In: Nuclear
Fusion, Vol. 58.10, Art. 106029 (29
th
Aug. 2018). doi:
10.1088/1741-
4326/aad78b.
[19]
A. Pavone et al., amongst them
U. Höfel
. »Bayesian uncertainty
calculation in neural network inference of ion and electron temperature
profiles at
W7-X
«. In: Review of Scientific Instruments, Vol. 89.10, Art. 1
0K102 (6th Aug. 2018). doi:10.1063/1.5039286.
[20]
A. Dinklageet al., amongst them
U. Höfel
. »Magnetic configuration
effects on the Wendelstein
7-X
stellarator«. In: Nature Physics, Vol. 14.6
(21st May 2018). doi:10.1038/s41567-018-0141-9.
[21]
B. P. van Milligen et al., amongst them
U. Höfel
. »Study of radial
heat transport in
W7-X
using the Transfer Entropy«. In: Nuclear Fusion,
(19th Apr. 2018). doi:10.1088/1741-4326/aabf5d.
[22]
N. A. Pablant et al., amongst them
U. Höfel
. »Core radial electric
field and transport in Wendelstein
7-X
plasmas«. In: Physics of Plasmas,
Vol. 25.2, Art. 022508 (12th Feb. 2018). doi:10.1063/1.4999842.
[23]
S. C. Liu et al., amongst them
U. Höfel
. »Observations of the effects
of magnetic topology on the SOL characteristics of an electromagnetic
coherent mode in the first experimental campaign of
W7-X
«. In: Nuclear
145
Publications as coauthor
Fusion, Vol. 58.4, Art. 046002 (7
th
Feb. 2018). doi:
10.1088/1741-
4326/aaa98e.
[24]
T. Stange et al., amongst them
U. Höfel
. »Advanced electron cyclo-
tron heating and current drive experiments on the stellarator Wendel-
stein
7-X
«. In: European Physical Journal Web of Conferences, Vol. 157,
Art. 02008 (23
rd
Oct. 2017). doi:
10.1051/epjconf/20171570200
8.
[25]
S. A. Bozhenkov et al., amongst them
U. Höfel
. »The Thomson
scattering diagnostic at Wendelstein 7-X and its performance in the
first operation phase«. In: Journal of Instrumentation, Vol. 12.10, Art. P
10004 (10th Oct. 2017). doi:10.1088/1748-0221/12/10/P10004.
[26]
H.
-
S. Bosch et al., amongst them
U. Höfel
. »Final integration, com-
missioning and start of the Wendelstein
7-X
stellarator operation«. In:
Nuclear Fusion, Vol. 57.11, Art. 116015 (3
rd
Aug. 2017). doi:
10.1088/
1741-4326/aa7cbb.
[27]
R. C. Wolf et al., amongst them
U. Höfel
. »Major results from the
first plasma campaign of the Wendelstein
7-X
stellarator«. In: Nuclear
Fusion, Vol. 57.10, Art. 102020 (27
th
July 2017). doi:
10.1088/1741-
4326/aa770d.
[28]
M. Hirsch et al., amongst them
U. Höfel
. »Confinement in Wendel-
stein
7-X
limiter plasmas«. In: Nuclear Fusion, Vol. 57.8, Art. 086010
(14th June 2017). doi:10.1088/1741-4326/aa7372.
[29]
D. Moseev et al., amongst them
U. Höfel
. »Inference of the mi-
crowave absorption coefficient from stray radiation measurements in
Wendelstein
7-X
«. In: Nuclear Fusion, Vol. 57.3, Art. 036013 (22
nd
Dec.
2016). doi:10.1088/1741-4326/aa4f13.
[30]
M. Krychowiak et al., amongst them
U. Höfel
. »Overview of diag-
nostic performance and results for the first operation phase in Wendel-
stein
7-X
(invited)«. In: Review of Scientific Instruments, Vol. 87.11, Art. 1
1D304 (27th Oct. 2016). doi:10.1063/1.4964376.
[31]
A. Köhn et al., amongst them
U. Höfel
. »Schemes of microwave
heating of overdense plasmas in the stellarator TJ-K«. In: Plasma Physics
and Controlled Fusion, Vol. 55.1, Art. 014010 (17
th
Dec. 2012). doi:
10.
1088/0741-3335/55/1/014010.
146
Acronyms
ADC analog digital converter
AUG ASDEX Upgrade
CPU central processing unit
CTS collective Thomson scattering
DAQ data acquisition
DCI
deuterium cyanide laser interfer-
ometry
ECCD
electron cyclotron current
drive
ECE electron cyclotron emission
ECRH
electron cyclotron resonance
heating
ESB enterprise service bus
HFS high field side
HPP heat pulse propagation
IDA integrated data analysis
IF intermediate frequency
ISS04
international stellarator scaling
2004
ISS95
international stellarator scaling
1995
ITER
International Thermonuclear Ex-
perimental Reactor
JET Joint European Torus
LA Laplace approximation
LiBES
lithium beam emission spectro-
scopy
LO local oscillator
MAP maximum a posteriori
MAST
Mega-Ampere Spherical Toka-
mak
MCMC Markov chain Monte Carlo
MHD magnetohydrodynamic
NBI neutral beam injection
OP operation phase
O mode ordinary mode
PDF probability densitiy function
PLL phase locked loop
SOA service-oriented architecture
SPECE
solver for plasma electron
cyclotron emission
TRAVIS tracing visualized
VMEC
variational moments equilib-
rium code
W7-X Wendelstein 7-X
X mode extraordinary mode
147
Bibliography
[1]
T. Hamacher et al. »Nuclear fusion and renewable energy forms: Are
they compatible?« In: Fusion Engineering and Design, Vol. 88.6-8 (Oct.
2013), pages 657–660. doi:10.1016/j.fusengdes.2013.01.074.
[2]
J. Ongena and G. Van Oost. »Energy for Future Centuries: Will
Fusion Be an Inexhaustible, Safe, and Clean Energy Source?« In: Fu-
sion Science and Technology, Vol. 45.2T (Mar. 2004), pages 3–14. doi:
10.13182/fst04-a464.
[3]
J. Scheffran. »Climate Change and Stability: The Case of the Mediter-
ranean Region«. In: International Federation of Automatic Control Pro-
ceedings Volumes, Vol. 43.25 (Oct. 2010), pages 35–40. doi:
10.3182/
20101027-3-xk-4018.00009.
[4]
A. A. Solomon et al. »How much energy storage is needed to incorpor-
ate very large intermittent renewables?« In: Energy Procedia, Vol. 135
(Oct. 2017), pages 283–293. doi:10.1016/j.egypro.2017.09.520.
[5]
D. J. Ward. »Fusion as a future energy source«. In: Europhysics News,
Vol. 47.5-6 (21
st
Nov. 2016), pages 32–34. doi:
10.1051/epn/2016505
.
[6]
»Advantages of Fusion«. url:
https : / / www . iter . org / sci /
Fusion (visited on 21st Apr. 2019).
[7]
P. H. Rebut,R. J. Bickerton and B. E. Keen. »The Joint European
Torus: installation, first results and prospects«. In: Nuclear Fusion, Vol. 25.9
(Sept. 1985), pages 1011–1022. doi:10.1088/0029-5515/25/9/003.
[8]
H. Shirai et al. »Recent progress of the JT-60SA project«. In: Nuclear
Fusion, Vol. 57.10, Art. 102002 (June 2017). doi:
10 . 1088 / 1741 -
4326/aa5d01.
149
Bibliography
[9]
M. Hirsch et al. »Major results from the stellarator Wendelstein 7-
AS«. In: Plasma Physics and Controlled Fusion, Vol. 50.5, Art. 053001
(10th Mar. 2008). doi:10.1088/0741-3335/50/5/053001.
[10]
A. H. Boozer. »Theory of tokamak disruptions«. In: Physics of Plasmas,
Vol. 19.5, Art. 058101 (May 2012). doi:10.1063/1.3703327.
[11]
R. J. Buttery et al. »Neoclassical tearing modes«. In: Plasma Physics
and Controlled Fusion, Vol. 42.12B (Dec. 2000), B61–B73. doi:
10.1088/
0741-3335/42/12b/306.
[12]
R. J. La Haye. »Neoclassical tearing modes and their control«. In: Phys-
ics of Plasmas, Vol. 13.5, Art. 055501 (May 2006). doi:
10.1063/1.
2180747.
[13]
A. Bock et al. »Advanced tokamak investigations in full-tungsten AS-
DEX Upgrade«. In: Physics of Plasmas, Vol. 25.5, Art. 056115 (May 2018).
doi:10.1063/1.5024320.
[14]
F. Warmer et al. »System Code Analysis of HELIAS-Type Fusion Re-
actor and Economic Comparison With Tokamaks«. In: IEEE Transac-
tions on Plasma Science, Vol. 44.9 (Sept. 2016), pages 1576–1585. doi:
10.1109/tps.2016.2545868.
[15]
U. Stroth et al. »Energy confinement scaling from the international
stellarator database«. In: Nuclear Fusion, Vol. 36.8 (Aug. 1996), pages 1063–
1077. doi:10.1088/0029-5515/36/8/i11.
[16]
H. Yamada et al. »Characterization of energy confinement in net-
current free plasmas using the extended International Stellarator Data-
base«. In: Nuclear Fusion, Vol. 45.12 (Dec. 2005), pages 1684–1693. doi:
10.1088/0029-5515/45/12/024.
[17]
C. Beidleret al. »Physics and Engineering Design for Wendelstein
VII-X
«.
In: Fusion Technology, Vol. 17.1 (1990), pages 148–168. doi:
10.13182/
FST90-A29178.
[18]
F. L. Hinton and R. D. Hazeltine. »Theory of plasma transport in
toroidal confinement systems«. In: Reviews of Modern Physics, Vol. 48.2
(Apr. 1976), pages 239–308. doi:10.1103/RevModPhys.48.239.
150
Bibliography
[19]
R. C. Wolf et al. »Wendelstein
7-X
Program—Demonstration of a Stel-
larator Option for Fusion Energy«. In: IEEE Transactions on Plasma Sci-
ence, Vol. 44.9 (Sept. 2016), pages 1466–1471. doi:
10.1109/tps.2016.
2564919.
[20]
K. Lackner and R. Schneider. »The role of edge physics and confine-
ment issues in the fusion reactor«. In: Fusion Engineering and Design,
Vol. 22.1-2 (Mar. 1993), pages 107–116. doi:
10.1016/s0920-3796(05)
80012-2.
[21]
D. Zhang et al. »First Observation of a Stable Highly Dissipative Di-
vertor Plasma Regime on the Wendelstein
7-X
Stellarator«. In: Physical
Review Letters, Vol. 123.2, Art. 025002 (9
th
July 2019). doi:
10.1103/
PhysRevLett.123.025002.
[22]
S. Sudo et al. »Scalings of energy confinement and density limit in
stellarator/heliotron devices«. In: Nuclear Fusion, Vol. 30.1 (Jan. 1990),
pages 11–21. doi:10.1088/0029-5515/30/1/002.
[23]
P. Zanca et al. »A unified model of density limit in fusion plasmas«. In:
Nuclear Fusion, Vol. 57.5, Art. 056010 (Mar. 2017). doi:
10.1088/1741-
4326/aa6230.
[24]
T. Sunn Pedersen et al. »First results from divertor operation in
Wendelstein
7-X
«. In: Plasma Physics and Controlled Fusion, Vol. 61.1,
Art. 014035 (27th Nov. 2018). doi:10.1088/1361-6587/aaec25.
[25]
M. Greenwald. »Density limits in toroidal plasmas«. In: Plasma Phys-
ics and Controlled Fusion, Vol. 44.8 (Aug. 2002), R27–R53. doi:
10.1088/
0741-3335/44/8/201.
[26]
J. Nührenberg and R. Zille. »Stable stellarators with medium
β
and
aspect ratio«. In: Physics Letters A, Vol. 114.3 (10
th
Feb. 1986), pages 129–
132. doi:10.1016/0375-9601(86)90539-6.
[27]
T. Klinger et al. »Performance and properties of the first plasmas of
Wendelstein
7-X
«. In: Plasma Physics and Controlled Fusion, Vol. 59.1,
Art. 014018 (18
th
Oct. 2016). doi:
10 . 1088 / 0741 - 3335 / 59 / 1 /
014018.
151
Bibliography
[28]
M. Krychowiak et al. »Overview of diagnostic performance and res-
ults for the first operation phase in Wendelstein
7-X
(invited)«. In: Re-
view of Scientific Instruments, Vol. 87.11, Art. 11D304 (27
th
Oct. 2016).
doi:10.1063/1.4964376.
[29]
R. C. Wolf et al. »Major results from the first plasma campaign of the
Wendelstein
7-X
stellarator«. In: Nuclear Fusion, Vol. 57.10, Art. 10202
0 (27th July 2017). doi:10.1088/1741-4326/aa770d.
[30]
V. Erckmann et al. »Electron Cyclotron Heating for W7-X: Physics
and Technology«. In: Fusion Science and Technology, Vol. 52.2 (1
st
Aug.
2007), pages 291–312. doi:10.13182/fst07-a1508.
[31]
N. J. Fisch and A. H. Boozer. »Creating an Asymmetric Plasma Res-
istivity with Waves«. In: Physical Review Letters, Vol. 45.9 (1
st
Sept. 1980),
pages 720–722. doi:10.1103/physrevlett.45.720.
[32]
Y. Turkin et al. »Current Control by ECCD for W7-X«. In: Fusion
Science and Technology, Vol. 50.3 (Oct. 2006), pages 387–394. doi:
10.
13182/fst06-5.
[33]
M. Zanini et al. »ECCD operations in the second experimental cam-
paign at
W7-X
«. In: European Physical Journal Web of Conferences, Vol. 203,
Art. 02013 (25
th
Mar. 2019). doi:
10.1051/epjconf/20192030201
3.
[34]
J. I. Paley et al. »From profile to sawtooth control: developing feedback
control using ECRH/ECCD systems on the TCV tokamak«. In: Plasma
Physics and Controlled Fusion, Vol. 51.12, Art. 124041 (12
th
Nov. 2009).
doi:10.1088/0741-3335/51/12/124041.
[35]
T. Stange et al. »First demonstration of fully equilibrated ions and elec-
trons in a Magnetically Confined High Temperature Plasma sustained
by 2nd harmonic O-mode polarized ECRH only«. In: Physical Review
Letters, (2020). Submitted.
[36]
H. J. Hartfuss and T. Geist.Fusion Plasma Diagnostics with mm-
Waves: An Introduction. Weinheim: Wiley-VCH, Sept. 2013. doi:
10 .
1002/9783527676279.
152
Bibliography
[37]
M. Hirsch et al. »ECE Diagnostic for the initial Operation of Wendel-
stein
7-X
«. In: European Physical Journal Web of Conferences, Vol. 203,
Art. 03007 (25
th
Mar. 2019). doi:
10.1051/epjconf/20192030300
7.
[38]
A. Jacchia et al. »Determination of diffusive and nondiffusive trans-
port in modulation experiments in plasmas«. In: Physics of Fluids B:
Plasma Physics, Vol. 3.11 (1
st
Nov. 1991), pages 3033–3040. doi:
10.1063/
1.859781.
[39]
B. P. van Milligen et al. »Study of radial heat transport in
W7-X
using the Transfer Entropy«. In: Nuclear Fusion, (19
th
Apr. 2018). doi:
10.1088/1741-4326/aabf5d.
[40]
M. Bornatici et al. »Electron cyclotron emission and absorption in
fusion plasmas«. In: Nuclear Fusion, Vol. 23.9 (Sept. 1983), pages 1153–
1257. doi:10.1088/0029-5515/23/9/005.
[41]
U. Höfel et al. »Bayesian Modelling of Microwave Radiometer Calibra-
tion on the example of the Wendelstein
7-X
Electron Cyclotron Emission
diagnostic«. In: Review of Scientific Instruments, Vol. 90.4, Art. 043502
(11th Apr. 2019). doi:10.1063/1.5082542.
[42]
V. D. Shafranov. »Equilibrium of a toroidal pinch in a magnetic field«.
In: Soviet Atomic Energy, Vol. 13.6 (Nov. 1963), pages 1149–1158. doi:
10.1007/bf01312317.
[43]
N. B. Marushchenko,Y. Turkin and H. Maassberg. »Ray-tracing
code TRAVIS for ECR heating, EC current drive and ECE diagnostic«.
In: Computer Physics Communications, Vol. 185.1 (2014), pages 165–176.
doi:10.1016/j.cpc.2013.09.002.
[44]
D. Walsh. »On Zheleznyakov’s Equation of Radiative Transfer in a
Magnetoactive Plasma«. In: The Astrophysical Journal, Vol. 159 (Feb.
1970), page 733. doi:10.1086/150349.
[45]
S. Bozhenkov et al. »Service oriented architecture for scientific ana-
lysis at
W7-X
. An example of a field line tracer«. In: Fusion Engineering
and Design, Vol. 88.11 (1
st
Aug. 2013), pages 2997–3006. doi:
10.1016/
j.fusengdes.2013.07.003.
153
Bibliography
[46]
M. Grahl et al. »Archive WEB API: A web service for the experiment
data archive of Wendelstein
7-X
«. In: Fusion Engineering and Design,
Vol. 123 (1
st
Mar. 2017). Proceedings of the 29th Symposium on Fusion
Technology, pages 1015–1019. doi:
10.1016/j.fusengdes.2017.
02.047.
[47]
S. A. Bozhenkovet al. »The Thomson scattering diagnostic at Wendel-
stein 7-X and its performance in the first operation phase«. In: Journal
of Instrumentation, Vol. 12.10, Art. P10004 (10
th
Oct. 2017). doi:
10.
1088/1748-0221/12/10/P10004.
[48]
A. Langenberg et al. »Forward Modeling of X-Ray Imaging Crystal
Spectrometers Within the Minerva Bayesian Analysis Framework«. In:
Fusion Science and Technology, Vol. 69.2 (Apr. 2016), pages 560–567. doi:
10.13182/FST15-181.
[49]
H. Damm.Upgrade of the Wendelstein
7-X
Thomson Scattering Diagnostic
to Study Short Transient Plasma Effects. A Demonstration on Pellet Injec-
tion for Stellarator Fuelling. Master’s thesis. University of Greifswald,
5th June 2018, pages 46–49. 199 pages. doi:10.17617/2.2596741.
[50] S. Schmuck et al. »Design of the ECE diagnostic at Wendelstein 7-X«.
In: Fusion Engineering and Design, Vol. 84.7–11 (June 2009). Proceeding
of the 25th Symposium on Fusion Technology, pages 1739–1743. doi:
10.1016/j.fusengdes.2008.12.094.
[51]
D. Wagner et al. »Sub-THz notch filters based on photonic bandgaps in
overmoded waveguides«. In: 40th International Conference on Infrared,
Millimeter, and Terahertz waves (IRMMW-THz). IEEE, Aug. 2015. doi:
10.1109/irmmw-thz.2015.7327739.
[52]
C. Fuchs and H.
-
J. Hartfuss. »Extreme broadband multichannel ECE
radiometer with “zoom” device«. In: Review of Scientific Instruments,
Vol. 72.1 (3rd Jan. 2001), pages 383–386. doi:10.1063/1.1309005.
[53]
X. Liu et al. »Absolute intensity calibration of the 32-channel hetero-
dyne radiometer on experimental advanced superconducting tokamak«.
In: Review of Scientific Instruments, Vol. 85.9, Art. 093508 (Sept. 2014).
doi:10.1063/1.4896047.
154
Bibliography
[54]
J. L. Ségui et al. »An upgraded 32-channel heterodyne electron cyclo-
tron emission radiometer on Tore Supra«. In: Review of Scientific In-
struments, Vol. 76.12, Art. 123501 (14
th
Dec. 2005). doi:
10.1063/1.
2140225.
[55]
E. T. Jaynes.Probability theory. The Logic of Science. Edited by G. L.
Bretthorst. Cambridge University Press, 10
th
Apr. 2003. 727 pages.
doi:10.1017/CBO9780511790423.
[56]
D. Sivia and J. Skilling.Data Analysis. A Bayesian Tutorial. 2
nd
ed.
Oxford University Press, 11th June 2006. 260 pages.
[57]
J. Svensson and JET EFDA contributors.Non-parametric Tomo-
graphy Using Gaussian Processes. Research report. Abingdon, United
Kingdom: Culham Centre for Fusion Energy, 2011. 15 pages. url:
http:
/ / www . euro - fusionscipub . org / wp - content / uploads /
eurofusion/EFDP11024.pdf (visited on 20th Feb. 2019).
[58]
C. E. Rasmussen and C. K. I. Williams.Gaussian Processes for Ma-
chine Learning. MIT Press, 23
rd
Nov. 2005. 272 pages. url:
https://
mitpress.mit.edu/books/gaussian-processes-machine-
learning (visited on 23rd Feb. 2019).
[59]
M. N. Gibbs.Bayesian Gaussian Processes for Regression and Classific-
ation. PhD thesis. University of Cambridge, Sept. 1997. 128 pages. url:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=
10 . 1 . 1 . 147 . 1130&rep = rep1&type = pdf
(visited on 26
th
Mar.
2019).
[60]
D. Higdon,J. Swall and J. Kern. »Non-Stationary Spatial Modeling«.
In: Bayesian Statistics. Proceedings of the Sixth Valencia International
Meeting. Edited by J. M. Bernardo et al. Vol. 6. Oxford University
Press, 12
th
Aug. 1999. url:
https://global.oup.com/academic/
product/bayesian-statistics-6-9780198504856
(visited on
22nd Feb. 2019).
[61]
C. J. Paciorek.Nonstationary Gaussian processes for regression and
spatial modelling. PhD thesis. Carnegie Mellon University, May 2003.
236 pages. url:
https://www.stat.berkeley.edu/~paciorek/
diss/paciorek-thesis.pdf (visited on 25th Feb. 2019).
155
Bibliography
[62]
M. A. Chilenski et al. »Improved profile fitting and quantification of
uncertainty in experimental measurements of impurity transport coeffi-
cients using Gaussian process regression«. In: Nuclear Fusion, Vol. 55.2,
Art. 023012 (23
rd
Jan. 2015). doi:
10 . 1088 / 0029 - 5515 / 55 / 2 /
023012.
[63]
M. Avriel.Nonlinear Programming: Analysis and Methods. Dover Pub-
lications, 11th Sept. 2003. 512 pages.
[64]
R. Hooke and T. A. Jeeves. »“Direct Search” Solution of Numerical
and Statistical Problems«. In: Journal of the Association for Computing
Machinery, Vol. 8.2 (Apr. 1961), pages 212–229. doi:
10.1145/321062.
321069.
[65]
N. Metropolis et al. »Equation of State Calculations by Fast Comput-
ing Machines«. In: The Journal of Chemical Physics, Vol. 21.6 (1
st
June
1953), pages 1087–1092. doi:10.1063/1.1699114.
[66]
W. K. Hastings. »Monte Carlo sampling methods using Markov chains
and their applications«. In: Biometrika, Vol. 57.1 (1
st
Apr. 1970), pages 97–
109. doi:10.1093/biomet/57.1.97.
[67]
H. Haario,E. Saksman and J. Tamminen. »An adaptive Metropolis
algorithm«. In: Bernoulli, Vol. 7.2 (Feb. 2001), pages 223–242. url:
http
s://projecteuclid.org:443/euclid.bj/1080222083
(visited
on 3rd Jan. 2019).
[68]
D. Nguyen et al. »Automatic adaptation of MCMC algorithms«. In:
ArXiv e-prints, (24th Feb. 2018). arXiv: 1802.08798 [stat.ME].
[69]
C. J. Geyer. »Introduction to Markov Chain Monte Carlo«. In: Hand-
book of Markov Chain Monte Carlo. Edited by S. Brooks et al. Hand-
books of Modern Statistical Methods. Chapman and Hall/CRC, 10
th
May
2011. Chapter 1. doi:10.1201/b10905-2.
[70]
G. O. Roberts,A. Gelman and W. R. Gilks. »Weak convergence
and optimal scaling of random walk Metropolis algorithms«. In: The
Annals of Applied Probability, Vol. 7.1 (Feb. 1997), pages 110–120. doi:
10.1214/aoap/1034625254.
156
Bibliography
[71]
M. Bédard. »Optimal acceptance rates for Metropolis algorithms: Mov-
ing beyond
0.234
«. In: Stochastic Processes and their Applications, Vol. 118.12
(31
st
Dec. 2008), pages 2198–2222. doi:
10.1016/j.spa.2007.12.
005.
[72]
E. Parzen. »On Estimation of a Probability Density Function and Mode«.
In: The Annals of Mathematical Statistics, Vol. 33.3 (Sept. 1962), pages 1065–
1076. doi:10.1214/aoms/1177704472.
[73]
B. W. Silverman.Density Estimation for Statistics and Data Analysis.
1
st
ed. Chapman & Hall/CRC Monographs on Statistics & Applied Prob-
ability. Taylor & Francis, 1
st
Apr. 1986. 176 pages. url:
https://www.
crcpress.com/Density-Estimation-for-Statistics-and-
Data-Analysis/Silverman/p/book/9780412246203
(visited
on 17th Mar. 2019).
[74]
J. Svensson,A. Werner and JET EFDA contributors. »Large Scale
Bayesian Data Analysis for Nuclear Fusion Experiments«. In: 2007 IEEE
International Symposium on Intelligent Signal Processing (3
rd
–5
th
Oct.
2007). Alcalá de Henares, Madrid, Spain, Oct. 2007. doi:
10.1109/
WISP.2007.4447579.
[75]
J. Svensson et al. »Modelling of JET Diagnostics Using Bayesian Graph-
ical Models«. In: Contributions to Plasma Physics, Vol. 51.2-3 (16
th
Mar.
2011), pages 152–157. doi:10.1002/ctpp.201000058.
[76]
M. J. Hole et al. »The use of Bayesian inversion to resolve plasma
equilibrium«. In: Review of Scientific Instruments, Vol. 81.10, Art. 10E12
7 (27th Oct. 2010). doi:10.1063/1.3491044.
[77]
C. M. Bishop.Graphical Models. In: Pattern Recognition and Machine
Learning. 1
st
ed. Information Science and Statistics. Springer-Verlag New
York, 17
th
Aug. 2006. Chapter 8, pages 359–362. url:
http://www.
ebook.de/de/product/5324937/christopher_m_bishop_
pattern_recognition_and_machine_learning.html.
[78]
A. Pavone et al. »Bayesian uncertainty calculation in neural network
inference of ion and electron temperature profiles at
W7-X
«. In: Review
of Scientific Instruments, Vol. 89.10, Art. 10K102 (6
th
Aug. 2018). doi:
10.1063/1.5039286.
157
Bibliography
[79]
Version svn checkout -r13781. url:
https://svn.minerva-centra
l.net/minerva/software/minerva/minervaIII/applicati
ons/minerva-ece-w7x-profile-app.
[80]
A. Tarantola.Inverse Problem Theory and Methods for Model Para-
meter Estimation. Other Titles in Applied Mathematics. Society for In-
dustrial and Applied Mathematics, 2005. doi:
10.1137/1.97808987
17921.
[81]
J. Svensson,A. Werner and JET-EFDA Contributors. »Current
tomography for axisymmetric plasmas«. In: Plasma Physics and Con-
trolled Fusion, Vol. 50.8, Art. 085002 (29
th
May 2008). doi:
10.1088/
0741-3335/50/8/085002.
[82]
O. P. Ford.Tokamak Plasma Analysis through Bayesian Diagnostic Mod-
elling. PhD thesis. University of London, 8
th
Nov. 2010. 179 pages. url:
http://www.oliford.co.uk/phys/thesis-final.pdf
(visited
on 24th Feb. 2019).
[83]
G. Taylor et al. »Update on the status of the ITER ECE diagnostic
design«. In: European Physical Journal Web of Conferences, Vol. 147,
Art. 02003 (24
th
July 2017). Edited by B. K. Shukla,H. B. Pandya
and S. L. Rao.doi:10.1051/epjconf/201714702003.
[84]
X. Han et al. »Design and characterization of a 32-channel heterodyne
radiometer for electron cyclotron emission measurements on experi-
mental advanced superconducting tokamak«. In: Review of Scientific
Instruments, Vol. 85.7, Art. 073506 (July 2014). doi:
10 . 1063 / 1 .
4891040.
[85]
T. Stange et al. »Advanced electron cyclotron heating and current
drive experiments on the stellarator Wendelstein
7-X
«. In: European
Physical Journal Web of Conferences, Vol. 157, Art. 02008 (23
rd
Oct.
2017). doi:10.1051/epjconf/201715702008.
[86]
H. P. Laqua et al. »Overview of
W7-X
ECRH Results«. In: EPJ Web of
Conferences, Vol. 203 (25
th
Mar. 2019). Edited by E. Poli,H. Laquaand J.
Oosterbeek, page 02002. doi:
10.1051/epjconf/201920302002
.
158
Bibliography
[87]
K.
-
J. Brunner et al. »Real-time dispersion interferometry for density
feedback in fusion devices«. In: Journal of Instrumentation, Vol. 13.09,
Art. P09002 (3
rd
Sept. 2018). doi:
10.1088/1748-0221/13/09/
P09002.
[88]
J. van den Berg et al. »Fast analysis of collective Thomson scatter-
ing spectra on Wendelstein
7-X
«. In: Review of Scientific Instruments,
Vol. 89.8, Art. 083507 (17th Aug. 2018). doi:10.1063/1.5035416.
[89]
R. C. Wolf and the Wendelstein
7-X
team. »A stellarator reactor
based on the optimization criteria of Wendelstein
7-X
«. In: Fusion En-
gineering and Design, Vol. 83.7-9 (Dec. 2008), pages 990–996. doi:
10.
1016/j.fusengdes.2008.05.008.
[90]
R. C. Wolf et al. »Electron-cyclotron-resonance heating in Wendel-
stein
7-X
: A versatile heating and current-drive method and a tool
for in-depth physics studies«. In: Plasma Physics and Controlled Fu-
sion, Vol. 61.1, Art. 014037 (27
th
Nov. 2018). doi:
10 . 1088 / 1361 -
6587/aaeab2.
[91]
S. Schmuck et al. »Bayesian derivation of electron temperature profile
using JET ECE diagnostics«. European Physical Society, 38
th
Conference
on Plasma Physics. Strasbourg, France, 2011. url:http://ocs.ciem
at.es/EPS2011PAP/pdf/P5.046.pdf (visited on 23rd Apr. 2018).
[92]
S. Schmuck et al. »Electron Temperature and Density Inferred from
JET ECE Diagnostics«. European Physical Society, 41
st
Conference on
Plasma Physics. Berlin, Germany, 2014. url:
http://ocs.ciemat.
es/EPS2014PAP/pdf/P1.025.pdf (visited on 23rd Apr. 2018).
[93]
S. Schmuck et al. »Bayesian Inference Using JETs Microwave Diagnos-
tic System«. In: Nuclear Fusion, (2019). Submitted.
[94]
S. K. Rathgeber et al. »Estimation of edge electron temperature pro-
files via forward modelling of the electron cyclotron radiation transport
at ASDEX Upgrade«. In: Plasma Physics and Controlled Fusion, Vol. 55.2,
Art. 025004 (Feb. 2013). doi:10.1088/0741-3335/55/2/025004.
[95]
S. S. Denk et al. »Radiation transport modelling for the interpreta-
tion of oblique ECE measurements«. In: European Physical Journal Web
of Conferences, Vol. 147, Art. 02002 (24
th
July 2017). Edited by B. K.
159
Bibliography
Shukla,H. B. Pandya and S. L. Rao.doi:
10. 1051 /epjconf/
201714702002.
[96]
S. S. Denk et al. »Analysis of electron cyclotron emission with extended
electron cyclotron forward modeling«. In: Plasma Physics and Controlled
Fusion, Vol. 60.10, Art. 105010 (7
th
Sept. 2018). doi:
10.1088/1361-
6587/aadb2f.
[97]
D. Farina et al. »SPECE: a code for Electron Cyclotron Emission in
tokamaks«. In: AIP Conference Proceedings, Vol. 988.1 (2008), pages 128–
131. doi:10.1063/1.2905053.
[98]
K. Rahbarnia et al. »Diamagnetic energy measurement during the
first operational phase at the Wendelstein
7-X
stellarator«. In: Nuclear
Fusion, Vol. 58.9, Art. 096010 (4
th
July 2018). doi:
10.1088/1741-
4326/aacab0.
[99]
M. Grahl et al. »Archive WEB API: A web service for the experiment
data archive of Wendelstein
7-X
«. In: Fusion Engineering and Design,
Vol. 123 (Nov. 2017), pages 1015–1019. doi:
10.1016/j.fusengdes.
2017.02.047.
[100]
A. Schlaich.Time-dependent spectrum analysis of high power gyrotrons.
PhD thesis. Karlsruhe: Karlsruhe Institute of Technology, 6
th
Aug. 2015.
286 pages. doi:10.5445/ksp/1000046919.
[101]
M. Peters,G. Gorini and P. Mantica. »Optical thickness corrections
to transient ECE temperature measurements in tokamak and stellarator
plasmas«. In: Nuclear Fusion, Vol. 35.7 (1995), page 873. doi:
10.1088/
0029-5515/35/7/I10.
[102]
S. K. Rathgeber.Electron Temperature and Pressure at the Edge of
ASDEX Upgrade Plasmas. Estimation via electron cyclotron radiation and
investigations on the effect of magnetic perturbations. PhD thesis. Ludwig-
Maximilian-Universität München, 28
th
Feb. 2013. 104 pages. URN:
urn:
nbn:de:bvb:19-157614.
[103]
M. Dibon et al. »Blower Gun pellet injection system for W7-X«. In:
Fusion Engineering and Design, Vol. 98-99 (2015): Proceedings of the 28th
Symposium On Fusion Technology (SOFT-28), pages 1759–1762. doi:
10.
1016/j.fusengdes.2015.01.050.
160
Bibliography
[104]
T. Möller and B. Trumbore. »Fast, Minimum Storage Ray-Triangle
Intersection«. In: Journal of Graphics Tools, Vol. 2.1 (1997), pages 21–28.
doi:10.1080/10867651.1997.10487468.
[105]
M. Hirsch et al. »Microwave and Interferometer Diagnostics prepared
for first Plasma Operation of Wendelstein
7-X
«. In: 1st EPS conference
on Plasma Diagnostics (Frascati, Italy, 14
th
–17
th
Apr. 2015). Vol. 240. Pro-
ceedings of Science. 26th Oct. 2016. doi:10.22323/1.240.0111.
[106]
A. Pavone et al. »Neural network approximation of Bayesian models
for the inference of ion and electron temperature profiles at W7-X«. In:
Plasma Physics and Controlled Fusion, Vol. 61.7, Art. 075012 (30
th
May
2019). doi:10.1088/1361-6587/ab1d26.
[107]
C. M. Bishop and J. W. Connor. »Heat-pulse propagation in tokamaks
and the role of density perturbations«. In: Plasma Physics and Controlled
Fusion, Vol. 32.3 (Mar. 1990), pages 203–211. doi:
10 . 1088 / 0741 -
3335/32/3/005.
[108]
K. K. Kirov et al. »Analysis of ECRH switch on/off events in ASDEX
Upgrade«. In: Plasma Physics and Controlled Fusion, Vol. 48.2 (Jan. 2006),
pages 245–262. doi:10.1088/0741-3335/48/2/006.
[109]
G. M. Weir.Thermal Transport by Heat Pulse Propagation on HSX. Mo-
tivation, Simulation, and Experimental Preparation. Master’s thesis. Uni-
versity of Wisconsin-Madison, 2011. 60 pages.
[110]
Version svn checkout -r13803. url:
https://svn.minerva-centr
al.net/minerva/software/minerva/minervaIII/applicat
ions/minerva-ece-w7x-apps.
[111]
Version svn checkout -r13862. url:
https://svn.minerva-centr
al.net/minerva/software/minerva/minervaIII/applicat
ions/minerva-ece-w7x-profile-app.
161
