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Tomography of fast-ion velocity-space distributions
from synthetic CTS and FIDA measurements
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IOP PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY NUCLEAR FUSION
Nucl. Fusion 52 (2012) 103008 (11pp) doi:10.1088/0029-5515/52/10/103008
Tomography of fast-ion velocity-space
distributions from synthetic CTS and
FIDA measurements
M. Salewski
1
, B. Geiger
2
, S.K. Nielsen
1
, H. Bindslev
3
,
M. Garc
´
ıa-Mu
˜
noz
2
, W.W. Heidbrink
4
, S.B. Korsholm
1
,
F. Leipold
1
,F.Meo
1
, P.K. Michelsen
1
, D. Moseev
2,5
, M. Stejner
1
,
G. Tardini
2
and the ASDEX Upgrade team
2
1
Association Euratom-DTU, Technical University of Denmark, Department of Physics,
DTU Risø Campus, DK-4000 Roskilde, Denmark
2
Association Euratom-Max-Planck-Institut f
¨
ur Plasmaphysik, D-85748 Garching, Germany
3
Faculty of Sciences and Technology, Aarhus University, DK-8000 Aarhus C, Denmark
4
Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA
5
Association Euratom-FOM Institute DIFFER, 3430 BE Nieuwegein, The Netherlands
E-mail: msal@fysik.dtu.dk
Received 10 May 2012, accepted for publication 1 August 2012
Published 21 August 2012
Online at
stacks.iop.org/NF/52/103008
Abstract
We compute tomographies of 2D fast-ion velocity distribution functions from synthetic collective Thomson scattering
(CTS) and fast-ion D
α
(FIDA) 1D measurements using a new reconstruction prescription. Contradicting conventional
wisdom we demonstrate that one single 1D CTS or FIDA view suffices to compute accurate tomographies of arbitrary
2D functions under idealized conditions. Under simulated experimental conditions, single-view tomographies do
not resemble the original fast-ion velocity distribution functions but nevertheless show their coarsest features. For
CTS or FIDA systems with many simultaneous views on the same measurement volume, the resemblance improves
with the number of available views, even if the resolution in each view is varied inversely proportional to the number
of views, so that the total number of measurements in all views is the same. With a realistic four-view system,
tomographies of a beam ion velocity distribution function at ASDEX Upgrade reproduce the general shape of the
function and the location of the maxima at full and half injection energy of the beam ions. By applying our method to
real many-view CTS or FIDA measurements, one could determine tomographies of 2D fast-ion velocity distribution
functions experimentally.
(Some figures may appear in colour only in the online journal)
1. Introduction
Fast ions play a key role in high performance plasmas:
they mediate energy from external heating sources or fusion
reactions to the bulk plasma and so maintain the high
temperatures typical for fusion-relevant plasmas. The fast-
ion orbits can be perturbed by fluctuations in the plasma, and
the ions can then be prematurely ejected from the plasma,
leading to undesired local heating of the first wall instead of
plasma heating. Several types of modes selectively deplete
or reorganize fast ions in particular velocity-space regions,
for example sawteeth [1–3], Alfv
´
en eigenmodes [4–6] and
neoclassical tearing modes [7]. Turbulence also ejects ions
selectively depending on their energy [8, 9]. In particular,
it is this selectivity of fast-ion depletion or reorganization
in velocity space that can be quantified with velocity-space
tomography. Additionally, velocity-space tomography could
be used to monitor phase-space engineering of fast-ion velocity
distribution functions which has enabled control of sawteeth
and of neoclassical tearing modes [10]. We show velocity-
space tomographies using parameters typical for the ASDEX
Upgrade collective Thomson scattering (CTS) [11–15] and
fast-ion D
α
(FIDA) diagnostics [16].
CTS and FIDA diagnostics are sensitive to 1D functions
g of local fast-ion velocity distribution functions f in
magnetically confined plasmas. The spatial resolution of the
CTS diagnostic at ASDEX Upgrade is about 10 cm, and the
measurement location can be moved freely in the plasma core
by means of steerable antennas. The time resolution has often
been set to 4 ms. CTS diagnostics are sensitive to the 1D
projection of f onto the wave vector
k
δ
= k
s
− k
i
which is
the difference between the wave vectors of scattered radiation
k
s
and incident radiation k
i
. The most important angle to
describe the pre-selected projection direction given by
k
δ
is
0029-5515/12/103008+11$33.00 1 © 2012 IAEA, Vienna Printed in the UK & the USA
Nucl. Fusion 52 (2012) 103008 M. Salewski et al
the projection angle φ
CTS
=
(k
δ
, B) where B is the magnetic
field. In CTS experiments the ions leave spectral signatures
in the scattered radiation. A frequency shift ν
δ
of scattered
radiation can be related to an ion velocity
v projected onto k
δ
:
ν
δ
= ν
s
− ν
i
≈ v · k
δ
/2π = uk
δ
/2π (1)
where u is the projected velocity and k
δ
=|k
δ
|. We define here
a CTS measurement as detection of the fast-ion phase-space
density in a particular interval in u that is related to an interval in
ν
δ
via equation (1). We define a view as a set of measurements
taken in a projection direction described by φ
CTS
. A second
CTS receiver has been installed at ASDEX Upgrade in 2012,
so that two simultaneous views with independently variable
projection angles φ
CTS
are available.
The location of a FIDA measurement is determined by the
intersection of the injected neutral beam (NBI) and the line-
of-sight (LOS) of the optical head. The spatial resolution of
the FIDA diagnostic at ASDEX Upgrade is about 7 cm, and
the time resolution is 2 ms. Beam source S3 is observed in the
plasma core at two different fixed angles φ
FIDA
=
(k
LOS
, B)
where
k
LOS
represents the wave vector along the LOS of the
optical heads. The toroidal LOS has an angle of φ
FIDA
= 11
◦
,
and the new poloidal LOS has φ
FIDA
= 64
◦
. The angles φ
CTS
and φ
FIDA
are analogue and will hereafter simply be called φ.
FIDA diagnostics are also sensitive to 1D functions of f as the
fast ions likewise leave a spectral signature in the detected light
by Doppler shift and Stark splitting. For FIDA diagnostics
no simple relation between the projected velocity u and the
wavelength λ exists, so we define here as FIDA measurement
the detection of Doppler- and Stark-shifted light in a particular
wavelength interval.
Computed tomography in real space is used in many
applications, for example in medical imaging in x-ray
computed axial tomography (CAT or CT) scanners, positron
emission tomography (PET) scanners or magnetic resonance
imaging (MRI) scanners [17, 18]. It is also widely used in
nuclear fusion research [19, 20]. We give a new prescription
for tomographic reconstruction in velocity space that is
analogue to those in real space. The prescription is based
on CTS or FIDA weight functions [21–23] which were not
available in previous work [24]. In [24] reconstructions
from two and three synthetic CTS views have been shown to
contain salient features of the underlying 2D fast-ion velocity
distribution functions in idealized situations. It has since
become conventional wisdom that a 2D velocity distribution
function could not be found from one single 1D CTS or FIDA
view and that at least two CTS or FIDA views with different
projection directions would be necessary for that [12, 22–32].
We demonstrate that in fact just one single 1D CTS or FIDA
view theoretically suffices to compute tomographies of almost
the entire discrete 2D velocity distribution function under
idealized conditions. Nevertheless, in simulated tokamak
experiments with many CTS or FIDA views, the resemblance
of tomographies and the original functions improves with
the number of available views. Several tokamaks have
been equipped with multiple FIDA views, for example DIII-
D[33], NSTX [34], MAST or ASDEX Upgrade which
is now also equipped with two CTS receivers. With our
prescription we can compute tomographies for any set of fast-
ion measurements, in particular those obtained with CTS or
FIDA or other fast-ion charge exchange spectroscopy (FICXS)
that detects other light than D
α
. A mix of diagnostics would
also be possible as will be relevant to the CTS/FIDA system
at ASDEX Upgrade, the CTS/FICXS system at LHD [35, 36]
and the proposed two-view CTS system for ITER [37–40]in
particular if it can be combined with FICXS [32]. However,
only one of the two CTS views is an enabled ITER diagnostic.
One could also include neutral particle analysers (NPAs) or
other fast-ion diagnostics in such mixes. We will study
tomographies from such diagnostic mixes elsewhere.
In section 2 we will argue that one single 1D set of CTS
measurements at different frequencies in fact theoretically
suffices to reconstruct the original 2D velocity distribution
function under ideal conditions. As weight functions form
the core of our tomographic reconstruction prescription to be
presented in section 4, we briefly review their meaning and
use in section 3. Tomographic reconstructions of a variety of
functions from synthetic CTS measurements under idealized
conditions are demonstrated in section 5 and under simulated
experimental conditions in section 6. In section 7 we show
that tomographies can likewise be computed from synthetic
FIDA measurements. We discuss the analogy of velocity-
space tomography to real-space tomography in section 8 and
draw conclusions in section 9.
2. Velocity-space tomography gedankenexperiment
First we perform a gedankenexperiment to motivate how one
single 1D projection can in fact contain enough information
to reconstruct the underlying 2D velocity-space distribution
function in discrete problems. Suppose that Alice has a way
to construct a 2D velocity-space distribution function f ion
by ion and that Bob has a way to measure the 1D velocity
distribution function g by CTS every time a new ion has been
added. Bob will only know his own measurements obtained
in a single CTS view. Alice adds an ion at some coordinate
pair (v
,v
⊥
) of her choice, for example at the location chosen
in figure 1(a). Bob then measures g which would have the
characteristic hammock shape shown in figure 1(b)[23, 41].
Bob can now work out the (v
,v
⊥
)-coordinates using
u = v
cos φ + v
⊥
sin φ cos γ, (2)
where γ is the gyrophase of the ion [23]. Since cos γ takes
values from −1 to 1, the width of the interval in which
Bob detects the ion is 2v
⊥
sin φ. The centre of the interval
is v
cos φ. Knowing his projection angle φ and the width
and centre of his measured function g, he can tell at which
coordinates (v
,v
⊥
) Alice has added the ion. Alice then adds
a second ion at a velocity-space location of her choice, and
Bob again measures g by CTS. Now the function g looks more
complicated but Bob can subtract his previous function g and
has again a simple hammock-shaped function from which he
can deduce the location of the second ion. This procedure can
be repeated until the entire 2D velocity distribution function is
constructed ion by ion, and Bob will know the entire function
exactly, looking just at his 1D measurements. Alice could
also construct f by adding collections of ions with identical
velocities instead of single ions. Bob could then tell how many
2
Nucl. Fusion 52 (2012) 103008 M. Salewski et al
-3 -2 -1 0 1 2 3
0
1
2
3
v
||
[10
6
m/s]
v
⊥
[10
6
m/s]
0
5
10
(
a
)
f
-4 -2 0 2 4
0
1
2
3
u [10
6
m/s]
g [10
12
s/m
4
]
(
b
)
g
Figure 1. (a) Example function f consisting of a single pixel in arbitrary units. (b) Projection g of the pixel function for a projection angle
of φ = 70
◦
.
ions have been added since the integral over u is proportional
to the number of ions:
n =
g du =
f dv
dv
⊥
. (3)
This gedankenexperiment shows that one single 1D CTS
view can in fact contain enough information for accurate
reconstruction firstly in simple situations and secondly also
in arbitrarily complicated situations if the complexity is added
step by step. In real experiments only the complicated situation
can be generated, and it is not immediately obvious that
the 1D function g can contain enough information about
the 2D function f . But we will demonstrate that we can
compute accurate tomographies from one single CTS or FIDA
view using our tomography reconstruction prescription if just
enough information is available.
3. Discrete weight functions for CTS and FIDA
Discrete weight functions will lead to the tomographic
reconstruction prescription presented in section 4. The
reconstruction prescription in [24] did not use weight functions
and was made tractable by expansion of the 1D (synthetic)
measurements as well as the 2D fast-ion velocity distribution
functions into orthonormal sets of base functions. Bessel
functions have been used but other choices would be possible
[24]. Exploiting CTS or FIDA weight functions [21–23]
we will give a simpler reconstruction prescription that is
inherently tractable and obviates the use of such expansions.
Weight functions have previously been used in an alternative
reconstruction prescription where the tomography was found
by iteration. This has the disadvantage that the solution
depends on the arbitrary start conditions of the iteration [23].
The new prescription we present gives unique solutions. In
this section we define weight functions in discrete form.
Assuming f to be rotationally symmetric about the v
-
axis, weight functions describe the mapping from 2D velocity-
space distribution functions f to 1D functions g that are
measured with CTS [23]orFIDA[22]. We here treat a
discrete tomography problem and so also deal with discrete
functions. The coordinates (u, φ, v
,v
⊥
) are discretized in
(u
i
,φ
j
,v
k
,v
⊥l
) where the subscripts i, j, k, l run from 1 to
the corresponding upper case letter I, J, K,L. I is the number
of measurements at different u
i
in a CTS or FIDA view, J is
the number of available views, and (K, L) are the number
of grid points in (v
,v
⊥
), respectively. g
ij
= g(u
i
,φ
j
)
is a matrix of discrete 1D functions in u
i
for each viewing
angle φ
j
. f
kl
= f(v
k
,v
⊥l
) is the discrete 2D velocity-space
distribution function. g
ij
and f
kl
are related by discrete CTS
or FIDA weight functions w
ij kl
analogue to the continuous
weight functions [23] so that
g
ij
=
K
k=1
L
l=1
w
ij kl
f
kl
v
⊥
v
. (4)
Weight functions pick out and assign weights to the velocity-
space interrogation region that is observed for a particular pro-
jection angle φ
j
and a projected velocity range at u
i
(observed
in a frequency range at f
i
) for CTS or a wavelength range at λ
i
for FIDA. In (v
,v
⊥
)-coordinates CTS weight functions have
a nearly triangular shape as shown in figure 2 for u
i
= 2 ×
10
6
ms
−1
and four typical projection angles φ
j
. Weight func-
tions describing CTS measurements quantify the probability
that a gyrating ion with velocity (v
,v
⊥
) is observed in a partic-
ular projected velocity range at u
i
for a given projection angle
φ
j
. The scattering must always originate from the coloured
triangular region. A comprehensive discussion of weight func-
tions for fast-ion CTS measurements is given elsewhere [23].
The weight functions describing FIDA measurements are more
complicated and account for the charge exchange probability,
the probability of photon emission from atomic level n = 3
to n = 2, Doppler shift of radiation originating from a gyr-
ating particle, Stark splitting of the deuterium Balmer alpha
line, and the instrument function of the FIDA spectrome-
ter [16, 21, 22, 26, 29]. The Doppler shift part of FIDA weight
functions is analogous to the CTS weight functions [23].
4. Tomographic reconstruction prescription
To find tomographies from CTS or FIDA measurements, we
rewrite equation (4) to formulate a linear algebra problem of
the form
W
mn
F
n
= G
m
. (5)
The matrix elements G
m
, F
n
and W
mn
are, respectively,
obtained from the matrix elements g
ij
, f
kl
and w
ij kl
by
G
m
= g
ij
(6)
F
n
= f
kl
(7)
W
mn
= w
ij kl
(8)
3
Nucl. Fusion 52 (2012) 103008 M. Salewski et al
–4 –2 0 2 4
2
4
v
||
[10
6
m/s]
v
⊥
[10
6
m/s]
–1.5
–1
–0.5
–4 –2 0 2 4
2
4
v
||
[10
6
m/s]
v
⊥
[10
6
m/s]
–1.5
–1
–0.5
–4 –2 0 2 4
2
4
v
||
[10
6
m/s]
v
⊥
[10
6
m/s]
–2
–1
–4 –2 0 2 4
2
4
v
||
[10
6
m/s]
v
⊥
[10
6
m/s]
–2
–1.5
Figure 2. Gyromotion weight functions w for u = 2 × 10
6
ms
−1
and various projection angles φ. The colourbar shows the base 10
logarithm.
using the assignment rules
m = (i − 1) × J + j (9)
n = (k − 1) × L + l. (10)
F is a column matrix of size N × 1 obtained from the
discrete 2D fast-ion velocity distribution function described
by N = K × L points. G is a column matrix of size M × 1
obtained from the discrete 1D functions measured with CTS
or FIDA. If J views are available and I measurements in u
i
(CTS) or λ
i
(FIDA) are taken in each view, then the total
number of measurements is M = I × J . W is then a transfer
matrix of size M × N taking F into G. The prescription
given here corresponds to stacking lines or rows on top of
each other but the order of this reorganization of the matrices
is arbitrary as long as we obey equation (4). The forward
problem to determine g from f or equivalently G from F is
straightforward given that w and consequently W are known.
An example of the action of the transfer matrix W on a pixel
function F is illustrated in figure 1. The projection angle φ
j
of
this single-view example (J = 1) is set to 70
◦
, and we compute
a weight function for each u
i
to obtain the value of G from the
inner product WF. The 1D function G for a pixel function
has the characteristic hammock shape shown in figure 1. The
inverse problem to determine f from g or equivalently F from
G is more complicated: we have to find an optimum solution
F
+
to the under- or overdetermined system of linear equations
(equation (5)) where W and G are known. We then also know
f
+
because we know F
+
and the reorganization procedure.
We find an optimum solution to WF = G for any size
of W from the Moore–Penrose pseudoinverse or generalized
inverse W
+
under positivity constraint. W
+
is a unique N × M
matrix [42–44]. It can be computed from the singular value
decomposition (SVD) of W :anM × N matrix W can always
be decomposed uniquely as
W = UV
T
(11)
where U is the normalized eigenvector matrix of WW
T
(an
orthogonal M × M matrix), V is the normalized eigenvector
matrix of W
T
W (an orthogonal N ×N matrix), V
T
denotes the
transpose of V , and is a diagonal (but rectangular) M × N
matrix [44]. The diagonal entries σ
1
,σ
2
, ..., σ
R
are the singular
values of W , and R is the rank of W . The other entries of
are zero. The Moore–Penrose pseudoinverse is then
W
+
= V
+
U
T
(12)
+
is a diagonal (but also rectangular) N × M matrix, and the
diagonal entries are 1/σ
1
, 1/σ
2
, ..., 1/σ
R
, i.e. the reciprocals
of corresponding entries of . The other entries of
+
are
zero. The computed tomography is then
F
+
= W
+
G. (13)
This is the equation from which we could determine F
+
from
actual measurements. If W is invertible, then W
+
is identical
to the inverse W
−1
. But W is generally a rectangular M × N
matrix that cannot be inverted. If the system WF = G is
overdetermined, F
+
gives the minimum 2-norm of the residual
|WF − G|
2
. If the system WF = G is underdetermined,
F
+
is the particular solution with minimum 2-norm |F |
2
out of infinitely many solutions (the one with no nullspace
component).
5. Tomographies under ideal conditions
In this section we firstly demonstrate that our prescription
for computed tomography in velocity space can reproduce a
variety of functions—any function we tested—in an idealized
situation. Secondly, we also demonstrate that just one single
synthetic CTS or FIDA view on that function suffices to
construct an accurate tomography. We assume that the function
can be described accurately on a numerical 2D grid, i.e. the
grid size is so fine that even features on the smallest scale
are accurately described. We also assume that there is no
4
