PAPER
Tomog aphy o as -ion eloci y-space dis ibu ions
om syn he ic CTS and FIDA measu emen s
To ci e his a icle: M. Salewski e al 2012 Nucl. Fusion 52 103008
View he a icle online o upda es and enhancemen s.
Rela ed con en
Combina ion o as -ion diagnos ics in
eloci y-space omog aphies
-
Measu emen o a 2D as -ion eloci y
dis ibu ion unc ion by omog aphic
in e sion o as -ion D-alpha spec a
-
On eloci y space in e oga ion egions o
as -ion collec i e Thomson sca e ing a
ITER
-
Recen ci a ions
Collec i e Thomson Sca e ing Diagnos ic
o Wendels ein 7-X a 175 GHz
D. Mosee e al
-
Collec i e Thomson sca e ing wi h 77,
154, and 300 GHz sou ces in LHD
M. Nishiu a e al
-
Collec i e Thomson sca e ing diagnos ic
o he GDT open magne ic ap
A G Shalasho e al
-
This con en was downloaded om IP add ess 87.218.223.151 on 18/08/2020 a 11:22
IOP PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY NUCLEAR FUSION
Nucl. Fusion 52 (2012) 103008 (11pp) doi:10.1088/0029-5515/52/10/103008
Tomog aphy o as -ion eloci y-space
dis ibu ions om syn he ic CTS and
FIDA measu emen s
M. Salewski1, B. Geige 2, S.K. Nielsen1, H. Bindsle 3,
M. Ga c´
ıa-Mu˜
noz2, W.W. Heidb ink4, S.B. Ko sholm1,
F. Leipold1,F.Meo
1, P.K. Michelsen1, D. Mosee 2,5, M. S ejne 1,
G. Ta dini2and he ASDEX Upg ade eam2
1Associa ion Eu a om-DTU, Technical Uni e si y o Denma k, Depa men o Physics,
DTU Risø Campus, DK-4000 Roskilde, Denma k
2Associa ion Eu a om-Max-Planck-Ins i u ¨
u Plasmaphysik, D-85748 Ga ching, Ge many
3Facul y o Sciences and Technology, Aa hus Uni e si y, DK-8000 Aa hus C, Denma k
4Depa men o Physics and As onomy, Uni e si y o Cali o nia, I ine, CA 92697, USA
5Associa ion Eu a om-FOM Ins i u e DIFFER, 3430 BE Nieuwegein, The Ne he lands
E-mail: [email p o ec ed]
Recei ed 10 May 2012, accep ed o publica ion 1 Augus 2012
Published 21 Augus 2012
Online a s acks.iop.o g/NF/52/103008
Abs ac
We compu e omog aphies o 2D as -ion eloci y dis ibu ion unc ions om syn he ic collec i e Thomson sca e ing
(CTS) and as -ion Dα(FIDA) 1D measu emen s using a new econs uc ion p esc ip ion. Con adic ing con en ional
wisdom we demons a e ha one single 1D CTS o FIDA iew su ices o compu e accu a e omog aphies o a bi a y
2D unc ions unde idealized condi ions. Unde simula ed expe imen al condi ions, single- iew omog aphies do
no esemble he o iginal as -ion eloci y dis ibu ion unc ions bu ne e heless show hei coa ses ea u es. Fo
CTS o FIDA sys ems wi h many simul aneous iews on he same measu emen olume, he esemblance imp o es
wi h he numbe o a ailable iews, e en i he esolu ion in each iew is a ied in e sely p opo ional o he numbe
o iews, so ha he o al numbe o measu emen s in all iews is he same. Wi h a ealis ic ou - iew sys em,
omog aphies o a beam ion eloci y dis ibu ion unc ion a ASDEX Upg ade ep oduce he gene al shape o he
unc ion and he loca ion o he maxima a ull and hal injec ion ene gy o he beam ions. By applying ou me hod o
eal many- iew CTS o FIDA measu emen s, one could de e mine omog aphies o 2D as -ion eloci y dis ibu ion
unc ions expe imen ally.
(Some igu es may appea in colou only in he online jou nal)
1. In oduc ion
Fas ions play a key ole in high pe o mance plasmas:
hey media e ene gy om ex e nal hea ing sou ces o usion
eac ions o he bulk plasma and so main ain he high
empe a u es ypical o usion- ele an plasmas. The as -
ion o bi s can be pe u bed by luc ua ions in he plasma, and
he ions can hen be p ema u ely ejec ed om he plasma,
leading o undesi ed local hea ing o he i s wall ins ead o
plasma hea ing. Se e al ypes o modes selec i ely deple e
o eo ganize as ions in pa icula eloci y-space egions,
o example saw ee h [1–3], Al ´
en eigenmodes [4–6] and
neoclassical ea ing modes [7]. Tu bulence also ejec s ions
selec i ely depending on hei ene gy [8,9]. In pa icula ,
i is his selec i i y o as -ion deple ion o eo ganiza ion
in eloci y space ha can be quan i ied wi h eloci y-space
omog aphy. Addi ionally, eloci y-space omog aphy could
be used o moni o phase-space enginee ing o as -ion eloci y
dis ibu ion unc ions which has enabled con ol o saw ee h
and o neoclassical ea ing modes [10]. We show eloci y-
space omog aphies using pa ame e s ypical o he ASDEX
Upg ade collec i e Thomson sca e ing (CTS) [11–15] and
as -ion Dα(FIDA) diagnos ics [16].
CTS and FIDA diagnos ics a e sensi i e o 1D unc ions
go local as -ion eloci y dis ibu ion unc ions in
magne ically con ined plasmas. The spa ial esolu ion o he
CTS diagnos ic a ASDEX Upg ade is abou 10 cm, and he
measu emen loca ion can be mo ed eely in he plasma co e
by means o s ee able an ennas. The ime esolu ion has o en
been se o 4 ms. CTS diagnos ics a e sensi i e o he 1D
p ojec ion o on o he wa e ec o kδ=ks−kiwhich is
he di e ence be ween he wa e ec o s o sca e ed adia ion
ksand inciden adia ion ki. The mos impo an angle o
desc ibe he p e-selec ed p ojec ion di ec ion gi en by kδis
0029-5515/12/103008+11$33.00 1© 2012 IAEA, Vienna P in ed in he UK & he USA
Nucl. Fusion 52 (2012) 103008 M. Salewski e al
he p ojec ion angle φCTS =(kδ,B)whe e Bis he magne ic
ield. In CTS expe imen s he ions lea e spec al signa u es
in he sca e ed adia ion. A equency shi νδo sca e ed
adia ion can be ela ed o an ion eloci y p ojec ed on o kδ:
νδ=νs−νi≈ ·kδ/2π=ukδ/2π(1)
whe e uis he p ojec ed eloci y and kδ=|kδ|. We de ine he e
a CTS measu emen as de ec ion o he as -ion phase-space
densi y in a pa icula in e al in u ha is ela ed o an in e al in
νδ ia equa ion (1). We de ine a iew as a se o measu emen s
aken in a p ojec ion di ec ion desc ibed by φCTS. A second
CTS ecei e has been ins alled a ASDEX Upg ade in 2012,
so ha wo simul aneous iews wi h independen ly a iable
p ojec ion angles φCTS a e a ailable.
The loca ion o a FIDA measu emen is de e mined by he
in e sec ion o he injec ed neu al beam (NBI) and he line-
o -sigh (LOS) o he op ical head. The spa ial esolu ion o
he FIDA diagnos ic a ASDEX Upg ade is abou 7 cm, and
he ime esolu ion is 2 ms. Beam sou ce S3 is obse ed in he
plasma co e a wo di e en ixed angles φFIDA =(kLOS,B)
whe e kLOS ep esen s he wa e ec o along he LOS o he
op ical heads. The o oidal LOS has an angle o φFIDA =11◦,
and he new poloidal LOS has φFIDA =64◦. The angles φCTS
and φFIDA a e analogue and will he ea e simply be called φ.
FIDA diagnos ics a e also sensi i e o 1D unc ions o as he
as ions likewise lea e a spec al signa u e in he de ec ed ligh
by Dopple shi and S a k spli ing. Fo FIDA diagnos ics
no simple ela ion be ween he p ojec ed eloci y uand he
wa eleng h λexis s, so we de ine he e as FIDA measu emen
he de ec ion o Dopple - and S a k-shi ed ligh in a pa icula
wa eleng h in e al.
Compu ed omog aphy in eal space is used in many
applica ions, o example in medical imaging in x- ay
compu ed axial omog aphy (CAT o CT) scanne s, posi on
emission omog aphy (PET) scanne s o magne ic esonance
imaging (MRI) scanne s [17,18]. I is also widely used in
nuclea usion esea ch [19,20]. We gi e a new p esc ip ion
o omog aphic econs uc ion in eloci y space ha is
analogue o hose in eal space. The p esc ip ion is based
on CTS o FIDA weigh unc ions [21–23] which we e no
a ailable in p e ious wo k [24]. In [24] econs uc ions
om wo and h ee syn he ic CTS iews ha e been shown o
con ain salien ea u es o he unde lying 2D as -ion eloci y
dis ibu ion unc ions in idealized si ua ions. I has since
become con en ional wisdom ha a 2D eloci y dis ibu ion
unc ion could no be ound om one single 1D CTS o FIDA
iew and ha a leas wo CTS o FIDA iews wi h di e en
p ojec ion di ec ions would be necessa y o ha [12,22–32].
We demons a e ha in ac jus one single 1D CTS o FIDA
iew heo e ically su ices o compu e omog aphies o almos
he en i e disc e e 2D eloci y dis ibu ion unc ion unde
idealized condi ions. Ne e heless, in simula ed okamak
expe imen s wi h many CTS o FIDA iews, he esemblance
o omog aphies and he o iginal unc ions imp o es wi h
he numbe o a ailable iews. Se e al okamaks ha e
been equipped wi h mul iple FIDA iews, o example DIII-
D[33], NSTX [34], MAST o ASDEX Upg ade which
is now also equipped wi h wo CTS ecei e s. Wi h ou
p esc ip ion we can compu e omog aphies o any se o as -
ion measu emen s, in pa icula hose ob ained wi h CTS o
FIDA o o he as -ion cha ge exchange spec oscopy (FICXS)
ha de ec s o he ligh han Dα. A mix o diagnos ics would
also be possible as will be ele an o he CTS/FIDA sys em
a ASDEX Upg ade, he CTS/FICXS sys em a LHD [35,36]
and he p oposed wo- iew CTS sys em o ITER [37–40]in
pa icula i i can be combined wi h FICXS [32]. Howe e ,
only one o he wo CTS iews is an enabled ITER diagnos ic.
One could also include neu al pa icle analyse s (NPAs) o
o he as -ion diagnos ics in such mixes. We will s udy
omog aphies om such diagnos ic mixes elsewhe e.
In sec ion 2we will a gue ha one single 1D se o CTS
measu emen s a di e en equencies in ac heo e ically
su ices o econs uc he o iginal 2D eloci y dis ibu ion
unc ion unde ideal condi ions. As weigh unc ions o m
he co e o ou omog aphic econs uc ion p esc ip ion o be
p esen ed in sec ion 4, we b ie ly e iew hei meaning and
use in sec ion 3. Tomog aphic econs uc ions o a a ie y o
unc ions om syn he ic CTS measu emen s unde idealized
condi ions a e demons a ed in sec ion 5and unde simula ed
expe imen al condi ions in sec ion 6. In sec ion 7we show
ha omog aphies can likewise be compu ed om syn he ic
FIDA measu emen s. We discuss he analogy o eloci y-
space omog aphy o eal-space omog aphy in sec ion 8and
d aw conclusions in sec ion 9.
2. Veloci y-space omog aphy gedankenexpe imen
Fi s we pe o m a gedankenexpe imen o mo i a e how one
single 1D p ojec ion can in ac con ain enough in o ma ion
o econs uc he unde lying 2D eloci y-space dis ibu ion
unc ion in disc e e p oblems. Suppose ha Alice has a way
o cons uc a 2D eloci y-space dis ibu ion unc ion ion
by ion and ha Bob has a way o measu e he 1D eloci y
dis ibu ion unc ion gby CTS e e y ime a new ion has been
added. Bob will only know his own measu emen s ob ained
in a single CTS iew. Alice adds an ion a some coo dina e
pai ( ,
⊥)o he choice, o example a he loca ion chosen
in igu e 1(a). Bob hen measu es gwhich would ha e he
cha ac e is ic hammock shape shown in igu e 1(b)[23,41].
Bob can now wo k ou he ( ,
⊥)-coo dina es using
u= cos φ+ ⊥sin φcos γ, (2)
whe e γis he gy ophase o he ion [23]. Since cos γ akes
alues om −1 o 1, he wid h o he in e al in which
Bob de ec s he ion is 2 ⊥sin φ. The cen e o he in e al
is cos φ. Knowing his p ojec ion angle φand he wid h
and cen e o his measu ed unc ion g, he can ell a which
coo dina es ( ,
⊥)Alice has added he ion. Alice hen adds
a second ion a a eloci y-space loca ion o he choice, and
Bob again measu es gby CTS. Now he unc ion glooks mo e
complica ed bu Bob can sub ac his p e ious unc ion gand
has again a simple hammock-shaped unc ion om which he
can deduce he loca ion o he second ion. This p ocedu e can
be epea ed un il he en i e 2D eloci y dis ibu ion unc ion is
cons uc ed ion by ion, and Bob will know he en i e unc ion
exac ly, looking jus a his 1D measu emen s. Alice could
also cons uc by adding collec ions o ions wi h iden ical
eloci ies ins ead o single ions. Bob could hen ell how many
2
Nucl. Fusion 52 (2012) 103008 M. Salewski e al
-3 -2 -1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
(
a
)
-4 -2 0 2 4
0
1
2
3
u [106 m/s]
g [1012 s/m4]
(
b
)
g
Figu e 1. (a) Example unc ion consis ing o a single pixel in a bi a y uni s. (b) P ojec ion go he pixel unc ion o a p ojec ion angle
o φ=70◦.
ions ha e been added since he in eg al o e uis p opo ional
o he numbe o ions:
n=gdu= d d ⊥.(3)
This gedankenexpe imen shows ha one single 1D CTS
iew can in ac con ain enough in o ma ion o accu a e
econs uc ion i s ly in simple si ua ions and secondly also
in a bi a ily complica ed si ua ions i he complexi y is added
s ep by s ep. In eal expe imen s only he complica ed si ua ion
can be gene a ed, and i is no immedia ely ob ious ha
he 1D unc ion gcan con ain enough in o ma ion abou
he 2D unc ion . Bu we will demons a e ha we can
compu e accu a e omog aphies om one single CTS o FIDA
iew using ou omog aphy econs uc ion p esc ip ion i jus
enough in o ma ion is a ailable.
3. Disc e e weigh unc ions o CTS and FIDA
Disc e e weigh unc ions will lead o he omog aphic
econs uc ion p esc ip ion p esen ed in sec ion 4. The
econs uc ion p esc ip ion in [24] did no use weigh unc ions
and was made ac able by expansion o he 1D (syn he ic)
measu emen s as well as he 2D as -ion eloci y dis ibu ion
unc ions in o o hono mal se s o base unc ions. Bessel
unc ions ha e been used bu o he choices would be possible
[24]. Exploi ing CTS o FIDA weigh unc ions [21–23]
we will gi e a simple econs uc ion p esc ip ion ha is
inhe en ly ac able and ob ia es he use o such expansions.
Weigh unc ions ha e p e iously been used in an al e na i e
econs uc ion p esc ip ion whe e he omog aphy was ound
by i e a ion. This has he disad an age ha he solu ion
depends on he a bi a y s a condi ions o he i e a ion [23].
The new p esc ip ion we p esen gi es unique solu ions. In
his sec ion we de ine weigh unc ions in disc e e o m.
Assuming o be o a ionally symme ic abou he -
axis, weigh unc ions desc ibe he mapping om 2D eloci y-
space dis ibu ion unc ions o 1D unc ions g ha a e
measu ed wi h CTS [23]o FIDA[22]. We he e ea a
disc e e omog aphy p oblem and so also deal wi h disc e e
unc ions. The coo dina es (u, φ, ,
⊥)a e disc e ized in
(ui,φ
j,
k,
⊥l)whe e he subsc ip s i, j, k, l un om 1 o
he co esponding uppe case le e I,J,K,L.Iis he numbe
o measu emen s a di e en uiin a CTS o FIDA iew, Jis
he numbe o a ailable iews, and (K, L) a e he numbe
o g id poin s in ( ,
⊥), espec i ely. gij =g(ui,φ
j)
is a ma ix o disc e e 1D unc ions in ui o each iewing
angle φj. kl = (
k,
⊥l)is he disc e e 2D eloci y-space
dis ibu ion unc ion. gij and kl a e ela ed by disc e e CTS
o FIDA weigh unc ions wij kl analogue o he con inuous
weigh unc ions [23] so ha
gij =
K

k=1
L

l=1
wij kl kl ⊥ .(4)
Weigh unc ions pick ou and assign weigh s o he eloci y-
space in e oga ion egion ha is obse ed o a pa icula p o-
jec ion angle φjand a p ojec ed eloci y ange a ui(obse ed
in a equency ange a i) o CTS o a wa eleng h ange a λi
o FIDA. In ( ,
⊥)-coo dina es CTS weigh unc ions ha e
a nea ly iangula shape as shown in igu e 2 o ui=2×
106ms
−1and ou ypical p ojec ion angles φj. Weigh unc-
ions desc ibing CTS measu emen s quan i y he p obabili y
ha a gy a ing ion wi h eloci y ( ,
⊥) is obse ed in a pa ic-
ula p ojec ed eloci y ange a ui o a gi en p ojec ion angle
φj. The sca e ing mus always o igina e om he colou ed
iangula egion. A comp ehensi e discussion o weigh unc-
ions o as -ion CTS measu emen s is gi en elsewhe e [23].
The weigh unc ions desc ibing FIDA measu emen s a e mo e
complica ed and accoun o he cha ge exchange p obabili y,
he p obabili y o pho on emission om a omic le el n=3
o n=2, Dopple shi o adia ion o igina ing om a gy -
a ing pa icle, S a k spli ing o he deu e ium Balme alpha
line, and he ins umen unc ion o he FIDA spec ome-
e [16,21,22,26,29]. The Dopple shi pa o FIDA weigh
unc ions is analogous o he CTS weigh unc ions [23].
4. Tomog aphic econs uc ion p esc ip ion
To ind omog aphies om CTS o FIDA measu emen s, we
ew i e equa ion (4) o o mula e a linea algeb a p oblem o
he o m
WmnFn=Gm.(5)
The ma ix elemen s Gm,Fnand Wmn a e, espec i ely,
ob ained om he ma ix elemen s gij , kl and wij kl by
Gm=gij (6)
Fn= kl (7)
Wmn =wij kl (8)
3
Nucl. Fusion 52 (2012) 103008 M. Salewski e al
–4 –2 0 2 4
2
4
|| [106 m/s]
⊥ [106 m/s]
–1.5
–1
–0.5
–4 –2 0 2 4
2
4
|| [106 m/s]
⊥ [106 m/s]
–1.5
–1
–0.5
–4 –2 0 2 4
2
4
|| [106 m/s]
⊥ [106 m/s]
–2
–1
–4 –2 0 2 4
2
4
|| [106 m/s]
⊥ [106 m/s]
–2
–1.5
Figu e 2. Gy omo ion weigh unc ions w o u=2×106ms
−1and a ious p ojec ion angles φ. The colou ba shows he base 10
loga i hm.
using he assignmen ules
m=(i −1)×J+j(9)
n=(k −1)×L+l. (10)
Fis a column ma ix o size N×1 ob ained om he
disc e e 2D as -ion eloci y dis ibu ion unc ion desc ibed
by N=K×Lpoin s. Gis a column ma ix o size M×1
ob ained om he disc e e 1D unc ions measu ed wi h CTS
o FIDA. I J iews a e a ailable and Imeasu emen s in ui
(CTS) o λi(FIDA) a e aken in each iew, hen he o al
numbe o measu emen s is M=I×J.Wis hen a ans e
ma ix o size M×N aking Fin o G. The p esc ip ion
gi en he e co esponds o s acking lines o ows on op o
each o he bu he o de o his eo ganiza ion o he ma ices
is a bi a y as long as we obey equa ion (4). The o wa d
p oblem o de e mine g om o equi alen ly G om Fis
s aigh o wa d gi en ha wand consequen ly Wa e known.
An example o he ac ion o he ans e ma ix Won a pixel
unc ion Fis illus a ed in igu e 1. The p ojec ion angle φjo
his single- iew example (J=1) is se o 70◦, and we compu e
a weigh unc ion o each ui o ob ain he alue o G om he
inne p oduc WF. The 1D unc ion G o a pixel unc ion
has he cha ac e is ic hammock shape shown in igu e 1. The
in e se p oblem o de e mine om go equi alen ly F om
Gis mo e complica ed: we ha e o ind an op imum solu ion
F+ o he unde - o o e de e mined sys em o linea equa ions
(equa ion (5)) whe e Wand Ga e known. We hen also know
+because we know F+and he eo ganiza ion p ocedu e.
We ind an op imum solu ion o WF =G o any size
o W om he Moo e–Pen ose pseudoin e se o gene alized
in e se W+unde posi i i y cons ain . W+is a unique N×M
ma ix [42–44]. I can be compu ed om he singula alue
decomposi ion (SVD) o W:anM×Nma ix Wcan always
be decomposed uniquely as
W=UVT(11)
whe e Uis he no malized eigen ec o ma ix o WWT(an
o hogonal M×Mma ix), Vis he no malized eigen ec o
ma ix o WTW(an o hogonal N×Nma ix), VTdeno es he
anspose o V, and is a diagonal (bu ec angula ) M×N
ma ix [44]. The diagonal en ies σ1,σ
2, ..., σRa e he singula
alues o W, and Ris he ank o W. The o he en ies o 
a e ze o. The Moo e–Pen ose pseudoin e se is hen
W+=V+UT(12)
+is a diagonal (bu also ec angula ) N×Mma ix, and he
diagonal en ies a e 1/σ1,1/σ2, ..., 1/σR, i.e. he ecip ocals
o co esponding en ies o . The o he en ies o +a e
ze o. The compu ed omog aphy is hen
F+=W+G. (13)
This is he equa ion om which we could de e mine F+ om
ac ual measu emen s. I Wis in e ible, hen W+is iden ical
o he in e se W−1. Bu Wis gene ally a ec angula M×N
ma ix ha canno be in e ed. I he sys em WF =Gis
o e de e mined, F+gi es he minimum 2-no m o he esidual
|WF −G|2. I he sys em WF =Gis unde de e mined,
F+is he pa icula solu ion wi h minimum 2-no m |F|2
ou o in ini ely many solu ions ( he one wi h no nullspace
componen ).
5. Tomog aphies unde ideal condi ions
In his sec ion we i s ly demons a e ha ou p esc ip ion
o compu ed omog aphy in eloci y space can ep oduce a
a ie y o unc ions—any unc ion we es ed—in an idealized
si ua ion. Secondly, we also demons a e ha jus one single
syn he ic CTS o FIDA iew on ha unc ion su ices o
cons uc an accu a e omog aphy. We assume ha he unc ion
can be desc ibed accu a ely on a nume ical 2D g id, i.e. he
g id size is so ine ha e en ea u es on he smalles scale
a e accu a ely desc ibed. We also assume ha he e is no
4

Nucl. Fusion 52 (2012) 103008 M. Salewski e al
-3 -2 -1 0 1 2 3
0
1
2
3
||
[106 m/s]
⊥ [106 m/s]
0
0.5
1
Figu e 3. The o iginal checke boa d unc ion shown he e is
digi ized in N=30 ×61 pixels. Typical 1D p ojec ions a e shown
in igu e 4. Tomog aphies a e shown in igu e 5.
noise. The e ec s o insu icien esolu ion and noise will
be discussed in sec ion 6. Unde hese idealized condi ions,
we se he nume ical g id o he omog aphy equal o ha o
he o iginal unc ion. As will be shown in sec ion 6, hese
assump ions will no gi e a ealis ic pic u e o he eco e able
in o ma ion in eal expe imen s. Ne e heless, p e ious wo k
used iden ical g ids o omog aphy and he o iginal unc ion
[23,24], and he esul s ound in his sec ion gi e an uppe limi
o he quali y ha can be achie ed and demons a e ha one
single iew is enough unde ideal condi ions. Ou p esc ip ion
immedia ely sugges s ha he omog aphy should be e y
accu a e in his case i jus M>N(mo e measu emen s han
pixels). I he nume ical g ids o he o iginal unc ion and he
omog aphy a e equal, we can gi e a simple ela ion be ween
Fand F+. One can subs i u e o Gand use he o hogonali y
o U.
F+=W+G=W+WF =V+UTUVTF=V+VTF
(14)
+has as Rones on he diagonal and o he wise ze os.
The e o e, only he i s Rcolumns and ows o Vand
VTwill be used in he econs uc ion. In ha sense he
econs uc ion o iden ical nume ical g ids is analogue o lossy
da a comp ession using SVD [44].
Unde hese assump ions, we econs uc a checke boa d
unc ion ( igu e 3) and a pacman unc ion ( igu e 6) using
jus one single iew. We choose hese es unc ions because
i is easy o spo di e ences be ween he o iginal unc ion
and he omog aphy. The checke boa d pa e n in igu e 3
co e s he eloci y-space egion o −3.5×106ms
−1<
<3.5×106ms
−1and 0 <
⊥<3.5×106ms
−1
and is digi ized in N=30 ×61 =1830 pixels. This
esolu ion is ypical o simula ed as -ion eloci y dis ibu ion
unc ions oday. We dis ibu e Mmeasu emen s e enly in
he in e al −5×106ms
−1<u<5×106ms
−1 o ensu e
comple e co e age o he eloci y-space egion we show he e
o any φ. Syn he ic measu emen s in one single iew o
φ=30◦and M=101 o M=2501 a e illus a ed in
igu e 4. By inc easing he esolu ion one can cap u e an
inc easingly mo e ine-g ained s uc u e o g ha con ains
eco e able in o ma ion abou he 2D unc ion . We s ess
ha he noisy looking cu e (M=2501) is he accu a e one
whe eas he smoo h looking cu e (M=101) con ains he
leas in o ma ion. Ac ually he smoo h cu e has a la ge noise
le el o igina ing om he disc e iza ion. The esolu ion in
he ucoo dina e o M=101 co esponds oughly o he
esolu ion o mos o he channels o he ASDEX Upg ade
CTS ecei e s. O e 2500 measu emen s in one iew seem
possible in high- equency esolu ion measu emen s ha we e
demons a ed a TEXTOR [45–47].
Single- iew omog aphies compu ed om Msyn he ic
measu emen s such as hose in igu e 4a e p esen ed in igu e 5.
Fo any esolu ion hey con ain a ine-g ained s uc u e ha is
simila o ha in he o iginal in igu e 3. E en o M∼N/20
(M=101), he omog aphy con ains e enly dis ibu ed small-
scale s uc u es bu hey a e la ge han hose in he o iginal
by a ac o o wo. The checke boa d pa e n a a co ec
scale begins o eme ge when M∼N/2(M=1001). Fo
M∼N he omog aphy closely esembles he o iginal wi h
mino de ec s, and o M∼4N/3 hey a e indis inguishable.
The econs uc ion p esc ip ion in p e ious wo k [24] ailed
o econs uc he o iginal unc ion o low ⊥co esponding
o abou ⊥<106ms
−1in ou g aphs. The checke boa d
pa e ns in igu e 5demons a e ha ou p esc ip ion wo ks
o all ⊥abou e enly.
Figu e 7shows single- iew omog aphies o he pacman
unc ion ( igu e 6), which we conside o be qui e complex, o
a a ious numbe o measu emen s M. F om he e on we do
no use measu emen s in he in e al −0.7×106<u<0.7×
106ms
−1. CTS due o bulk ions makes unambigous de ec ion
o as ions e ydi icul i no impossible in hisin e al, and so
we block i in he syn he ic diagnos ic. This loss o in o ma ion
esul s in he appea ance o iangula egions ha a e no
expe imen ally accessible ( igu es 7(a)–(d)). The shape o
such iangles depends on he p ojec ion angle φ. The sides o
hese iangles a e gi en by ⊥=(cons × h ± cos φ)/sin φ
and ⊥=0[23]. The o iginal pacman unc ion con ains
complica ed s uc u es wi h a scale sepa a ion o one o de o
magni ude be ween he la ge-scale s uc u es (pacman head,
spook) and he small-scale ine de ails (eyes and mou h, zick
zack pa e n o he spook inge). The omog aphy o he
pacman unc ion is also an accu a e ep oduc ion o he o iginal
unc ion i Mis la ge enough. The equi ed numbe o
measu emen s M o accu a e omog aphies is simila o he
equi ed M o he checke boa d—and in ac o any unc ion
we es ed—and does no signi ican ly depend on whe he an
in e al in uhas been blocked.
Las ly, we no e ha he p ojec ion angle φis no e y
impo an in he idealized si ua ion excep o a φ=90◦when
all in o ma ion abou is los and a φ=0◦whe e he weigh
unc ions a e singula . No ad an age is gained om ha ing
many iews o an equal o al numbe o measu emen s Min
he idealized si ua ion. Fo example a omog aphy om wo
iews each wi h 1000 measu emen s (M=2×1000 =2000)
oughly esembles he o iginal as much as he omog aphy
om one iew wi h M=2000 measu emen s. Likewise, he
angles a e no impo an o many- iew sys ems ei he i jus
he esolu ion o he measu emen s is high enough. The quali y
o he omog aphy o any numbe o iews depends mos ly on
he o al numbe o measu emen s Min he idealized si ua ion.
6. Tomog aphies o hea ily unde -diagnosed
as -ion dis ibu ion unc ions
The p e ious sec ion demons a ed ha ou omog aphy
p esc ip ion will wo k in an idealized si ua ion. The
o iginal unc ion had he same numbe o g id poin s as
5
Nucl. Fusion 52 (2012) 103008 M. Salewski e al
–4 –2 0 2 4
0
1
2
3
u [106 m/s]
g [1012 s/m4]
101
501
2501
1.3 1.4 1.5 1.6
1.8
2
2.2
2.4
2.6
u [106 m/s]
g [1012 s/m4]
101
501
2501
Figu e 4. P ojec ions o he checke boa d unc ion ( igu e 3) o φ=30◦wi h M=101, 501 and 2501 measu emen s in one iew g.(a)
Zoomed ou showing he en i e unc ions g.(b) Zoomed in showing ha ine-scale s uc u e can be esol ed wi h M=2501 measu emen s,
which is su icien o compu e an accu a e omog aphy.
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
Figu e 5. Single- iew omog aphies o a checke boa d unc ion in N=30 ×61 =1830 pixels gi ing only Mmeasu emen s a φ=30◦.
The Mmeasu emen s a e e enly spaced in −5×106ms
−1<u<5×106ms
−1. The numbe o measu emen s is a ied om M=101 o
M=2501. The axes and colou ba s a e iden ical o hose o he o iginal in igu e 3.
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
1
2
3
Figu e 6. The o iginal pacman unc ion shown he e is digi ized in
N=30 ×61 =1830 pixels. Tomog aphies a e shown in igu e 7.
he econs uc ions as in p e ious wo k [23,24]. The 2D
eloci y dis ibu ion unc ion in an ac ual okamak expe imen
will ha e a ine-g ained s uc u e. I is hen p ac ically
impossible o make enough CTS o FIDA measu emen s o
ca y all in o ma ion abou he ine-g ained . To simula e
expe imen al condi ions, we i s cons uc a 1D p ojec ion
gwi h Msyn he ic measu emen s om a inely esol ed 2D
dis ibu ion:
G=W1F. (15)
He e we disc e ize in N1=350 ×701 ∼250 000 g id
poin s and ake M∼3400 o M∼340 measu emen s lea ing
unde -diagnosed by a ac o on he o de o 100 o 1000,
espec i ely. Accu a e omog aphies a e impossible o such
unde -diagnosed . Then we compu e a omog aphy wi h a
much lowe numbe o g id poin s han N1:N2=30 ×61 =
1830 N1.
F+=W+
2G. (16)
Subs i u ion o Gnow gi es
F+=V2+
2UT
2W1F. (17)
Subs i u ion o he SVD o W1=U11VT
1does no lead
o simpli ica ions as in equa ion (14) since UT
2U1does no
disappea . I he g ids o he omog aphy and he o iginal
unc ion a e no iden ical, i is necessa y o unca e he SVD
and use only singula alues abo e a selec ed le el. This is
e ec i ely also a lossy da a comp ession echnique since we
ind a lowe ank app oxima ion o he ans e ma ix W2 ha
has abou ank R∼1700 in ou example. Re e ence [24] used
such a lossy da a comp ession echnique o simula e he e ec s
o noise, no ing ha noise dec eases he in o ma ion con en o
he smalles singula alues. The e ec o noise and o unde -
diagnosing, i.e. compu ing he omog aphy on a much coa se
g id han he o iginal, a e simila . G=W1Fis di e en om
G+=W2F+, and his di e ence can be in e p e ed as noise
o igina ing om he disc e iza ion.
Figu e 8shows a ypical beam ion eloci y dis ibu ion
unc ion a ASDEX Upg ade esol ed on N1=350 ×701
g id poin s o which we p esen omog aphies om CTS
6
Nucl. Fusion 52 (2012) 103008 M. Salewski e al
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
–3 –2 –1 0 1 2 3
0
1
2
3
Figu e 7. Single- iew omog aphies o a pacman unc ion in 1830 pixels gi ing only Mmeasu emen s a φ=30◦. The Mmeasu emen s
a e e enly spaced in −5×106ms
−1<u<−0.7×106ms
−1and 0.7×106ms
−1<u<5×106ms
−1. The numbe o measu emen s is
a ied om M=88 o M=2868. The axes and colou ba s a e iden ical o hose in igu e 6.
-3 -2 -1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 105
Figu e 8. Typical beam ion dis ibu ion unc ion o beam S3 a
ASDEX Upg ade compu ed wi h TRANSP/NUBEAM. The
dis ibu ion unc ion is shown on a g id wi h 350 ×701 pixels.
measu emen s he e. The o iginal unc ion has peaks a ull
and hal injec ion ene gy o 60 keV in deu e ium. We plo
omog aphies (N2=30 ×61 =1830 g id poin s) o he
o iginal unc ion in igu e 9 o a a ious numbe o a ailable
iews Jand measu emen s M. The h ee- iew and ou -
iew CTS omog aphies a e p oxies o mixed CTS/FIDA
omog aphies ha can be econs uc ed om he wo a ailable
CTS iews and he wo a ailable FIDA iews a ASDEX
Upg ade. The combina ion o di e en diagnos ics in ou
me hod will be discussed elsewhe e. We se he numbe o
measu emen s pe iew in e sely p opo ional o he numbe
o iews so ha he o al numbe o measu emen s Mis almos
he same in each column o igu e 9. The le column shows
omog aphies o one o ou iews wi h abou M∼340
measu emen s in o al (M<N), and he igh column wi h
abou M∼3400 measu emen s in o al (M>N). In he
idealized si ua ion he numbe o iews Jis unimpo an ;
only he numbe o measu emen s Mma e s. The e o e, jus
one iew su ices o accu a e omog aphies in he idealized
si ua ion. Howe e , unde simula ed expe imen al condi ions,
he numbe o iews Jis highly impo an o he ele ance
o he omog aphy o he o iginal unc ion. The single-
iew omog aphies do no esemble he o iginal unc ion bu
hey esemble a he he weigh unc ions, and aking mo e
measu emen s Min ha one iew does no help signi ican ly.
Ne e heless, he lopsidedness owa ds nega i e eloci ies is
co ec ly econs uc ed in he omog aphies. Fo wo iews he
egion o he beam ions is oughly iden i iable, and wo maxima
eme ge. The ou - iew omog aphies esemble he o iginal
unc ion bes . To quan i y he di e ence be ween he o iginal
unc ion and he omog aphy, we de ine an e o measu e as
Q om =1
n as | − +|d d ⊥(18)
n as = d d ⊥(19)
which is a single numbe quan i ying he esemblance o he
omog aphy wi h he o iginal unc ion. Q om =0 means ha
he ma ch is pe ec , and Q om ∼1 means ha +does no
esemble he o iginal unc ion . In his example Q om =1
o one iew, Q om =0.8 o wo iews, Q om =0.7 o
h ee iews and Q om =0.5 o ou iews, bu he pa icula
alues depend on he pa icula dis ibu ion unc ion and he
diagnos ic se up. This measu e should be use ul o u u e
op imiza ion s udies. Compa ing he low esolu ion column
(M<N) wi h he high esolu ion column (M>N), we ind
ha aking en imes mo e measu emen s pe iew does no
help imp o ing he omog aphies much whe eas adding ex a
iews does. Fo he high esolu ion cases wi h M∼3400 only
340 singula alues a e use ul whe eas abou 300 a e use ul in
he low esolu ion cases wi h M∼340.
We now illus a e omog aphies o a simple unc ion om
syn he ic CTS measu emen s. Figu e 11 shows omog aphies
o a d i ing Maxwellian unc ion wi h N1=350 ×701 pixels
( igu e 10) ha we hen diagnose in one o ou iews, and we
seek omog aphies wi h N2=30 ×61 =1830 pixels. E en
hough he numbe o measu emen s M∼2000 is again almos
he same o he ou cases, he omog aphies imp o e wi h he
numbe o iews. One iew is no enough o gi e omog aphies
ha esemble he o iginal. Ne e heless, he omog aphy o
jus one single iew co ec ly iden i ies he loca ion o he
Maxwellian peak, so we can conclude ha measu emen s in
one single iew con ain ele an in o ma ion abou e en
unde simula ed expe imen al condi ions.
7. Tomog aphies om FIDA measu emen s
So a we ha e buil he ans e ma ix W om gy omo ion
weigh unc ions, and hese a e su icien o desc ibe CTS
measu emen s. Analy ic exp essions o hese CTS weigh
unc ions a e a ailable [23]. Weigh unc ions ele an o
FIDA measu emen s a e mo e complica ed and a e calcula ed
7
Nucl. Fusion 52 (2012) 103008 M. Salewski e al
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 105
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 10
5
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 105
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 10
5
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 105
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 10
5
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 10
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
0
5
10
15
x 10
5
Figu e 9. Tomog aphies (N2=30 ×61 pixels) o a ypical beam ion dis ibu ion unc ion (N1=350 ×701 pixels) o a ious numbe s o
iews and measu emen s M. The o al numbe o measu emen s Mis simila in each column. In he le column (a), (c), (e), (g)M∼340
whe eas in he igh column (b), (d), ( ), (h)M∼3400. The iewing angles a e φ=20◦ o one iew, φ=(10◦,80◦) o wo iews,
φ=(10◦,40◦,80◦) o h ee iews and φ=(10◦,30◦,60◦,80◦) o ou iews.
–3 –2 –1 0 1 2 3
0
1
2
3
|| [106 m/s]
⊥ [106 m/s]
2
4
6
8
Figu e 10. A d i ing Maxwellian esol ed in 350 ×701 pixels.
by coun ing pho ons in he di e en wa eleng h in e als.
FIDA weigh unc ions he e o e con ain nume ical noise ha
dec eases wi h he squa e oo o he compu e ime allowed o
hei compu a ion. We use φ=(11◦,64◦) o he wo FIDA
iews a ailable a ASDEX Upg ade. The measu emen s M
a e e enly dis ibu ed in he wa eleng h in e als 649 nm <
λ<654 nm and 659 nm <λ<663 nm. FIDA ligh
canno be obse ed in he wa eleng h in e al 654 nm <λ<
659 nm due o beam emission and halo neu als [16], and
so we exclude his wa eleng h ange also in he syn he ic
measu emen s. Figu e 12 shows a omog aphy o he o iginal
unc ion ( igu e 8) om syn he ic measu emen s o he wo-
iew FIDA sys em a ASDEX Upg ade and demons a es ha
ou p esc ip ion also wo ks o FIDA measu emen s. The
o iginal has N1=350 ×701 g id poin s which was he e
diagnosed by M=2×90 =180 measu emen s, and he
omog aphy in igu e 12 has N2=30×61 =1830 g id poin s.
We he e use he la ges 80 singula alues o he compu a ion
o he Moo e–Pen ose pseudoin e se.
8