Re . Sci. Ins um. 81, 10D929 (2010); h ps://doi.o g/10.1063/1.3483214 81, 10D929
© 2010 Ame ican Ins i u e o Physics.
2D elec on cyclo on emission imaging a
ASDEX Upg ade (in i ed)
Ci e as: Re . Sci. Ins um. 81, 10D929 (2010); h ps://doi.o g/10.1063/1.3483214
Submi ed: 14 May 2010 . Accep ed: 19 July 2010 . Published Online: 28 Oc obe 2010
I. G. J. Classen, J. E. Boom, W. Su op, E. Schmid, B. Tobias, C. W. Domie , N. C. Luhmann, A. J. H. Donné,
R. J. E. Jaspe s, P. C. de V ies, H. K. Pa k, T. Munsa , M. Ga cía-Muñoz, and P. A. Schneide
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2D elec on cyclo on emission imaging a ASDEX Upg ade „in i ed…a…
I. G. J. Classen,1,2,b兲J. E. Boom,2W. Su op,1E. Schmid,1B. Tobias,3C. W. Domie ,3
N. C. Luhmann, J .,3A. J. H. Donné,2,4 R. J. E. Jaspe s,4P. C. de V ies,2
H. K. Pa k,5T. Munsa ,6M. Ga cía-Muñoz,1and P. A. Schneide 1
1Max Planck Ins i u ü Plasmaphysik, 85748 Ga ching, Ge many
2FOM-Ins i u e o Plasma Physics, Rijnhuizen, 3430 BE Nieuwegein, The Ne he lands
3Uni e si y o Cali o nia a Da is, Da is, Cali o nia 95616, USA
4Eindho en Uni e si y o Technology, 5600 MB Eindho en, The Ne he lands
5POSTECH, Pohang, Gyeongbuk, 790-784, Sou h Ko ea
6Uni e si y o Colo ado, Boulde , Colo ado 80309, USA
共P esen ed 18 May 2010; ecei ed 14 May 2010; accep ed 19 July 2010;
published online 28 Oc obe 2010兲
The newly ins alled elec on cyclo on emission imaging diagnos ic on ASDEX Upg ade p o ides
measu emen s o he 2D elec on empe a u e dynamics wi h high spa ial and empo al esolu ion.
An o e iew o he echnical and expe imen al p ope ies o he sys em is p esen ed. These
p ope ies a e illus a ed by he measu emen s o he edge localized mode and he e e sed shea
Al én eigenmode, showing bo h he ad an age o ha ing a wo-dimensional 共2D兲measu emen , as
well as some o he limi a ions o elec on cyclo on emission measu emen s. Fu he mo e, he
applica ion o singula alue decomposi ion as a powe ul ool o analyzing and il e ing 2D da a
is p esen ed. © 2010 Ame ican Ins i u e o Physics. 关doi:10.1063/1.3483214兴
I. INTRODUCTION
Many p ocesses and ins abili ies in okamak plasmas ex-
hibi wo-dimensional 共2D兲dynamics, he eby making hem
di icul o diagnose by one-dimensional 共1D兲diagnos ics.
The s eng h o 2D elec on cyclo on emission imaging
共ECEI兲was i s demons a ed a he TEXTOR okamak.1I
p o ided aluable physics insigh in he esea ch o , among
o he s, he saw oo h ins abili y2,3and ea ing modes.4,5An
o e iew o he de elopmen o ECEI sys ems since he i s
TEXTOR diagnos ic can be ound in he li e a u e.6
Recen ly an ECEI sys em has been ins alled a ASDEX
Upg ade in o de o in es iga e he 2D elec on empe a u e
dynamics o o he eac o ele an ins abili ies, including
edge localized modes 共ELMs兲and neoclassical ea ing
modes, no easily accessible a TEXTOR. This sys em im-
ages a plasma a ea o app oxima ely 12 by 40 cm 共8by16
channels兲. An o e iew o he echnical and expe imen al
p ope ies o his ASDEX Upg ade ECEI sys em is p esen ed
he e. Apa om gi ing a desc ip ion o he sys em 共Secs. II
and III兲, his pape also co e s an o e iew o he expe i-
men al anges and limi a ions om he pe spec i e o a use
共Sec. IV兲. These p ope ies will be illus a ed by he mea-
su emen s o he ELM ins abili y, showing bo h he ad an-
age o ha ing a 2D measu emen , as well as some o he
limi a ions o elec on cyclo on emission 共ECE兲measu e-
men s 共Sec. V兲. Finally, a singula alue decomposi ion
共SVD兲is used o o e come he limi a ions se by he mal
noise, which allowed he isualiza ion o Al én wa es
共Sec. VI兲.
II. THE PRINCIPLE OF ECEI
A s anda d 1D ECE adiome e measu es he elec on
empe a u e a di e en adial loca ions along a single line o
sigh , making use o he magne ic ield dependence o he
elec on cyclo on equency
␻
ec=eB/medue o he a ia ion
o he magne ic ield wi h majo adius B⬇B0R0/Ras is seen
in Fig. 1共a兲. In he he mal equilib ium, he measu ed in en-
si y a a gi en esonan equency 共and co esponding adius兲
is gi en by he Rayleigh–Jeans limi o Planck’s law
IBB共
␻
兲=
␻
2kBTe/共8
␲
3c3兲, so he plasma beha es as a black-
body and he in ensi y is p opo ional o he elec on em-
pe a u e. The e o e, a equency esol ed measu emen o
he elec on cyclo on emission in ensi y cons i u es a adi-
ally esol ed measu emen o he elec on empe a u e.
The same p inciple is employed in a 2D ECEI diagnos-
ic, excep ha mul iple lines o sigh a e simul aneously
quasiop ically imaged on o a linea a ay o diode de ec o s
关see Fig. 1共b兲兴. Each o he lines o sigh is ea ed as a 1D
ECE adiome e . This gi es a di ec 2D measu emen o he
elec on empe a u e; adially esol ed due o he equency
esol ed measu emen o he ECE in ensi y along each line
o sigh , e ically esol ed due o he mul iple lines o sigh
co esponding o he de ec o s on he a ay.
III. ASDEX UPGRADE ECEI
In Fig. 2, a schema ic ep esen a ion o he a ious com-
ponen s o he ASDEX Upg ade ECEI sys em is gi en. ECEI
uses he same 36 cm diame e qua z acuum window as he
1D ECE adiome e 共“manhole” in sec o 9兲, and also sha es
pa o he 共newly designed兲op ics wi h his diagnos ic. The
wo sys ems a e sepa a ed by a beamspli e 共a hin dielec ic
oil兲 ha has an app oxima ely 50/50% ansmission/
e lec ion o e he ull ange o equencies, wi h he ECEI
op ical pa h in e lec ion. The ECEI op ics ac ually consis o
a兲In i ed pape , published as pa o he P oceedings o he 18 h Topical
Con e ence on High-Tempe a u e Plasma Diagnos ics, Wildwood, New
Je sey, May 2010.
b兲Au ho o whom co espondence should be add essed. Elec onic mail:
[email p o ec ed].
REVIEW OF SCIENTIFIC INSTRUMENTS 81, 10D929 共2010兲
0034-6748/2010/81共10兲/10D929/6/$30.00 © 2010 Ame ican Ins i u e o Physics81, 10D929-1
wo b anches, he on side op ics ha images he plasma
adia ion on he de ec o a ay, and he LO side op ics 共sepa-
a ed om he i s one by a u he beamspli e 兲 ha images
he local oscilla o BWO beam on he a ay. All ECEI com-
ponen s 共excep he compu e s兲a e posi ioned in he ASDEX
Upg ade o us hall. The a ious componen s o he sys em
a e desc ibed in he ollowing pa ag aphs.
The ASDEX Upg ade ECEI sys em u ilizes he de ec o
a ay and in e media e equency 共IF兲elec onics o he p e-
ious TEXTOR ECEI sys em. The de ec o a ay employs 16
dual dipole an ennas wi h an a ay o 16 minia u e ellip ical
subs a e lenses9共minilenses兲in on o hem, yielding
highly Gaussian, pa allel ield pa e ns. In Fig. 3, a schema ic
layou o he ECEI elec onics o a single an enna is shown.
The signal is downcon e ed wo imes. The i s downcon-
e sion occu s a he a ay by mixing he plasma signal wi h
he BWO signal 共 unable be ween 90 and 140 GHz兲. This is
a single sideband downcon e sion; he lowe sideband is e-
jec ed by using quasiop ical high pass il e s 共dich oic pla es兲
in on o he a ay. The downcon e ed b oadband signal o
each o he 16 an ennas is ansmi ed o he 16 IF elec onic
modules by mic owa e cables.
In he IF modules, he signal is di ided in o eigh po -
ions, which a e subsequen ly mixed wi h eigh local oscilla-
o signals 共 ol age con olled oscilla o s a LO=2.4, 3.2,
4.0, 4.8, 5.6, 6.4, 7.2, and 8.0 GHz兲in he second downcon-
e sion s ep. The equency di e ence be ween hese local
oscilla o s de e mines he adial in e channel spacing o
ECEI and is hence 800 MHz. This ypically co esponds o
1.5 cm in he plasma, depending upon he g adien o B. The
ull adial co e age o ECEI is hence app oxima ely 12 cm.
The esul ing signals a e band pass il e ed 共double side-
band兲, wi h an in e media e equency bandwid h BIF o
700 MHz. This de e mines he equency bandwid h o he
ECE adia ion selec ed by each channel and, along wi h
b oadening e ec s, he adial esolu ion. A e de ec ion, he
signals a e low pass il e ed 共wi h a en h o de il e 兲wi h
adjus able ideo bandwid h BV=25–400 kHz, de e mining
he ime esolu ion o ECEI. The sampling equency is no -
mally se o 共a leas 兲 wice BV. Finally, he signals a e con-
e ed o a digi al signal by 12 bi digi ize s, capable o s o -
ing 2 Msamples pe channel a a maximum sampling a e o
2 Msamples/s. Mo e echnical de ails conce ning he a ay
and elec onics can be ound in he li e a u e.7–9
The on side op ics, shown in Fig. 4, consis o a se o
high densi y polye hylene 共HDPE兲lenses, pa o which is
sha ed wi h a 60 channel 1D ECE adiome e . The line o
sigh o he 1D sys em is designed o coincide wi h he op-
ical axis o ECEI, simul aneously p o iding in he same
poloidal plasma c oss sec ion a measu emen o he ull 1D
elec on empe a u e p o ile as well as he de ailed 2D co -
e age o a selec ed a ea o abou 12 by 40 cm 共8by16
channels兲. This also makes c oss calib a ion wi h he abso-
lu ely calib a ed 1D sys em easie .
ECEI lines o sigh ha e adjus able 共al hough no inde-
penden 兲 oci. This is made possible by ansla ing one o he
lenses o shi he posi ion o he ocal plane o he desi ed
adius 共Fig. 4兲. This makes i possible o always ope a e
ECEI in ocus 共unlike 1D ECE, which has a ixed ocus兲.
Also, he con ocal pa ame e 共abou 40 cm兲is la ge han he
adial co e age, ensu ing all ECEI channels a e in ocus.
The op ics ha e been designed such ha he loca ion
FIG. 3. Schema ic ep esen a ion o he elec onics o a single de ec o o a
ypical ECEI sys em.
FIG. 4. 共Colo online兲Op ics o he ASDEX Upg ade ECEI sys em. Pa o
he op ics a e sha ed wi h he 60 channel 1D ECE adiome e . The posi ion
o he ocal plane can be shi ed by ansla ing one o he HDPE lenses.
FIG. 1. Compa ison be ween 共a兲a s anda d 1D ECE adiome e and 共b兲a
2D ECEI sys em. In an ECEI sys em, mul iple lines o sigh a e imaged on o
a 1D a ay o de ec o s.
FIG. 2. 共Colo online兲Diag am showing all majo componen s o ECEI.
10D929-2 Classen e al. Re . Sci. Ins um. 81, 10D929 共2010兲
along he beam pa h whe e all 16 beams om he de ec o s
exac ly o e lap each o he in space 共 he Fou ie plane o he
op ics兲lies close o he window ape u e. In his way, all
beams use as much o he ape u e as possible, he eby op i-
mizing he pe pendicula spa ial esolu ion. The window ap-
e u e is hence he limi ing ac o de e mining he 共di ac-
ion limi ed兲Gaussian beam wais w0=2d␭/
␲
D, whe e dis
he dis ance om he window ape u e 共1.8 m a he cen e o
he plasma兲and D=0.36 m is he window ape u e diame e
共see Fig. 4兲. This gi es a smalles possible spo size o abou
w0=15 mm in he plasma, which is also achie ed by he
op ics. The an enna pa e n o each o he de ec o s 共includ-
ing hei associa ed minilens兲can be app oxima ed by a
Gaussian beam wi h a wais o 7 mm in he objec plane 共a
he de ec o 兲, and he a ay o 16 de ec o s has a leng h o
20 cm. The op ics images his objec plane 共a ay兲on o he
ocal plane in he plasma. No e ha he spo size and e ical
co e age in he plasma canno be independen ly imaged.
Consequen ly, he ideal spo size o 15 mm in he plasma
共abou a ac o o 2 la ge han he spo size a he a ay兲, also
ixes he e ical co e age in he plasma, which is abou
40 cm 共 wice he size o he a ay兲. The e o e, wi h he cu -
en a ay, a smalle e ical co e age is no possible 共 he
beams would no i h ough he window ape u e兲, and a
la ge co e age would lead o a p opo ionally la ge spo
size.
The local oscilla o o he i s downcon e sion s ep is a
backwa d wa e oscilla o 共BWO兲. This is a unable mic o-
wa e sou ce ha ypically deli e s a 50 mW mic owa e
beam a an adjus able equency be ween 90 and 140 GHz
共F-band兲wi h a equency s abili y o 0.02%. The BWO op-
ics image he adia ion quasiop ically on o he de ec o a ay
as an elonga ed Gaussian beam, in o de o illumina e all 16
de ec o s. The BWO is enclosed ina1cm hick i on casing
o s ay magne ic ield shielding.
The a ailable cu o equencies o he dich oic pla es
o he sideband ejec ion a e cu en ly 99.1, 105.0, 111.7,
115.6, and 120.2 GHz. The il e has o be chosen such ha
he lowe sideband plasma ligh is blocked, while pe mi ing
bo h he uppe sideband and BWO equencies o pass.
The e o e, in p ac ice, he il e mus ha e a cu o equency
jus below he BWO equency, he eby limi ing he numbe
o use ul BWO equency se ings o he a ailable dich oic
pla es.
The ope a ion du ing 140 GHz elec on cyclo on eso-
nance hea ing 共ECRH兲is possible. In o de o p o ec he
a ay agains s ay ECRH powe , h ee quasiop ical no ch
il e s wi h a 共combined兲60 dB ejec ion a 140 GHz 共3dB
ejec ion a abou 127 GHz兲a e used. No no ch il e s a e
cu en ly a ailable o o he ECRH equencies a he
ASDEX Upg ade. Ope a ion wi h a dich oic pla e cu o e-
quency abo e he ECRH equency is always possible. In
o he cases, he sys em will be shielded wi h a shu e .
The ECEI diagnos ic has been designed o be ully e-
mo ely con olled. The posi ion o he ocusing lens, he e-
quency and powe o he BWO and he selec ion o he di-
ch oic pla es 共using a s eppe mo o con olled slide wi h i e
il e s and a shu e 兲a e all emo ely con olled. Also, he
sys em can be swi ched on and o emo ely. No en y o he
To us Hall is equi ed a any s age du ing ope a ion, g ea ly
enhancing he lexibili y o he sys em.
IV. OPERATIONAL RANGE AND LIMITS
The ange o adii o which ECEI can be uned o du ing
expe imen depends on he o oidal magne ic ield and on he
equency ange o he diagnos ic. Figu e 5p o ides a
g aphical ep esen a ion o his ope a ional ange. The ho i-
zon al shaded a ea ep esen s he accessible plasma egion
共 he second ha monic ECE adia ion om he HFS is
blocked by he hi d ha monic esonance a he low ield
side兲. The il ed shaded a ea ep esen s he ange ECEI can
be uned o as a unc ion o B0, limi ed by he equency
ange o he BWO. In eali y, no all BWO equency se ings
can be used, due o he limi ed numbe o a ailable dich oic
pla es o ejec he lowe sideband. The da k shaded a ea
gi es he limi a ion se by he cu en ly a ailable dich oic
pla es. The e o e, cu en ly he plasma cen e is accessible
be ween B0=1.8 and 2.25 T and he low ield side 共LFS兲
edge be ween 2.35 and 2.9 T.
Ano he limi a ion is se by cu o . Fo a second ha -
monic X-mode ECE diagnos ic, he cu o occu s a he igh
hand cu o equency10
␻
R=1
2关
␻
ec+共
␻
ec
2+4
␻
p
2兲1/2兴, whe e
␻
p=冑nee2/共␧0me兲is he densi y dependen plasma e-
quency. Figu e 6gi es he densi y abo e which he cu o
occu s as a unc ion o adius and o oidal magne ic ield.
No e ha measu emen s a high densi y gene ally equi e a
high o oidal magne ic ield. When in cu o , he second ha -
monic X-mode adia ion canno each he obse e , so signal
le els d op owa d ze o, al hough some second ha monic
O-mode adia ion is no mally s ill isible due o he ini e
sensi i i y o O-mode o any eal diagnos ic. Ano he ela ed
e ec o be awa e o is he bending o he 共mainly o -axis兲
lines o sigh due o e ac ion. Close o cu o , he index o
e ac ion o he plasma becomes no iceably smalle han
FIG. 5. Range o possible measu emen posi ions ECEI can be uned o as a
unc ion o he o oidal magne ic ield B0. The da kes shaded a ea ep e-
sen s he 共cu en 兲ope a ional ange.
10D929-3 Classen e al. Re . Sci. Ins um. 81, 10D929 共2010兲
uni y ha e ac s he lines o sigh away om he plasma
cen e . This e ec apidly becomes smalle when he plasma
is u he om cu o .
Two o he p ope ies o ECE measu emen s o be sho ly
add essed a e he op ical hickness and b oadening e ec s.
The op ical hickness o he second ha monic X-mode a-
dia ion is 共 o ypical ASDEX Upg ade pa ame e s B0
=2.5 T and R esonance=2 m兲app oxima ely gi en by
␶
X2
⬇3.3⫻Te关keV兴⫻ne关1019兴共see Re . 10 o he exac o -
mula兲. I he op ical hickness is oo low 共compa able o o
below uni y兲, he plasma is pa ially anspa en and he ECE
measu emen s 共 he adia ion empe a u e TR兲can no longe
be ega ded as a di ec elec on empe a u e measu emen
共also densi y plays a ole兲. Neglec ing wall e lec ions, he
adia ion empe a u e is gi en by10 TR=Te关1−exp共−
␶
兲兴,
whe e
␶
is a unc ion o bo h empe a u e and densi y. Fo
he no mal ASDEX Upg ade H-mode plasmas, hese only
play a ole a he oo o he pedes al 共see Sec. V兲.
The equency b oadening o he ECE adia ion due o
ela i is ic e ec s10,11 ⌬
␻
el⬇n
␻
ec共 h/c兲2共whe e h
=冑2kBTe/me兲, and Dopple b oadening10,12 ⌬
␻
D⬇2 h/w0
共which o pe pendicula obse a ion can be ega ded as an
e ec o he ini e ansi ime o he emi ing elec ons
h ough he Gaussian spo o size w0兲, esul in an emi ing
laye wi h signi ican adial ex en 共 o ECEI a 1 keV abou
3 cm ull wid h a hal maximum兲. Howe e , in he op ically
hick case, mos o he adia ion o he b oadened emission
laye is eabso bed be o e i eaches he obse e , and only
he las laye 共 ypically na owe han 1 cm兲, closes o he
obse e , con ibu es o he ECE signal, and de e mines he
adial spa ial esolu ion. In he op ically hin plasma edge
共see Sec. V兲 he ull emission laye is obse ed.
V. 2D DYNAMICS: EDGE LOCALIZED MODES
One o he main mo i a ions o ins alling ECEI a he
ASDEX Upg ade was he s udy o ELMs, which is one o
he mos impo an and s ill no ully unde s ood ins abili ies
in usion de ices. The necessi y o a 2D diagnos ic is illus-
a ed in Fig. 7, whe e he ECEI measu emen s o he LFS
plasma edge du ing di e en phases o an ELM c ash a e
p esen ed. The da a ha e been il e ed 共using singula alue
decomposi ion, see Sec. VI兲 o educe he he mal noise. The
obse ed p onounced 2D dynamics would no ha e been ac-
cessible wi h a con en ional 1D ECE adiome e . Figu es
7共b兲–7共d兲a e aken du ing he ea ly s ages o he c ash 关 ime
indica ed in Fig. 7共a兲兴whe e a clea de o ma ion o he em-
pe a u e p o ile due o 共p ecu so 兲mode ac i i y is isible,
showing a a he complica ed and pa ly incohe en poloidal
mode s uc u e. In Fig. 7,
0共2.747 165 s兲is de ined as he
ime o he i s isible de ia ion om he 共p ec ash兲smoo h
empe a u e p o ile. Figu es 7共e兲and 7共 兲show images du -
ing a la e phase o he c ash 关 ime indica ed in Fig. 7共a兲兴,
whe e poloidally localized s uc u es 共 hough o be ela ed
o he ELM ilamen s obse ed by many o he diagnos ics兲
a e seen o o a e poloidally 共upwa d in he igu e兲. To en-
hance hese s uc u es in he images, no he absolu e 共 adia-
ion兲 empe a u e, bu TR-TR,a is shown, whe e TR,a is he
a e age empe a u e 共calcula ed sepa a ely o each ECEI
channel兲a e he ELM c ash 关shaded a ea in Fig. 7共a兲兴.
These ilamen s ha e a adia ion empe a u e o abou
200 eV.
Whe he one can in e p e he obse ed ilamen s as
elec on empe a u e luc ua ions depends upon he op ical
hickness o hese s uc u es. Figu e 8共c兲shows he op ical
hickness o ypical elec on densi y and empe a u e p o-
iles 关shown in Figs. 8共a兲and 8共b兲兴a di e en phases du ing
he ELM cycle. Fo bo h he p e-ELM and pos -ELM cases,
he op ical hickness d ops below uni y a oughly he sepa-
a ix posi ion, wi h apidly ising alues inside he sepa a-
ix. In he case whe e he elec on empe a u e p o ile has
been modi ied o include a ilamen 关inspi ed by he adia ion
empe a u es seen in Figs. 7共e兲and 7共 兲兴bu he elec on
densi y p o ile is le unchanged, i u ns ou ha he op ical
hickness o his ilamen is oughly uni y. The assump ion o
simul aneously inc eased densi y could d as ically inc ease
his numbe . Consequen ly, he op ical hickness is ma ginal
共no mally alues abo e 2 o 3 a e conside ed ully op ically
FIG. 6. 共Colo online兲Cu o densi y 共solid cu e兲as a unc ion o majo
adius R o a ious o oidal magne ic ields B0共labels兲. The dashed cu e
gi es a ypical example o a densi y p o ile. In his example, pa s o he
plasma would be in cu o o B0⬍2.5 T.
FIG. 7. 共Colo online兲Du ing an ELM c ash, a p onounced 2D s uc u e is
obse ed a he plasma edge. The ime aces in 共a兲show a ull ELM cycle.
Images 共b兲–共d兲show he “p ecu so ” phase, and images 共e兲and 共 兲show he
mo emen o a “ ilamen .” The a ows indica e he poloidal mo emen o
he s uc u es.
10D929-4 Classen e al. Re . Sci. Ins um. 81, 10D929 共2010兲

hick兲, bu su icien ly la ge o assume he adia ion empe a-
u e TRis s ill compa able o he elec on empe a u e Te
共no e ha TRⱕTeso he ac ual empe a u e o he ilamen s
migh be highe 兲.
Figu e 8共d兲shows he b oadened emission p o iles o
he ilamen case. The solid line shows he cold esonance,
and he dashed and do ed lines ep esen he 共95%兲bound-
a ies o he b oadened egions o ECEI and 1D ECE, e-
spec i ely. A low alues o op ical hickness, he emission
om he ull b oadened p o ile eaches he obse e 共 eab-
so p ion does no play a ole兲, possibly educing he adial
esolu ion. Howe e , in he egion whe e he op ical hick-
ness is low, he wid h o he emi ing laye is seen o be abou
1 cm. The adial esolu ion in hese edge measu emen s is
hence compa able o ha o co e measu emen s.
I should be no ed ha he e ec o shine h ough13 共an
inc eased adia ion empe a u e jus ou side he sepa a ix,
hough o be a nonlocal e ec caused by downshi ed adia-
ion om u he inside he plasma兲, o en obse ed by he
1D ECE adiome e , has so a no been obse ed wi h ECEI.
Al hough no ully unde s ood, he di e ence migh be ex-
plained by he di e en spo sizes o he wo diagnos ics
共 ypically w0=15 mm o ECEI and 35 mm o 1D ECE兲,
esul ing in a 2.5 imes la ge Dopple b oadening o ECEI,
and hence di e en emission p o iles 关see Fig. 8共d兲兴.
VI. SINGULAR VALUE DECOMPOSITION
The accu acy o he ECEI measu emen s is limi ed by
he mal noise, inhe en o all ECE diagnos ics.14 The 共 he -
mal兲noise le el is de e mined by he bandwid hs o he sys-
em, and o ECEI a ull sampling a e his amoun s o a
ela i e noise le el o a ound 3%
冑具dTR
2典/具TR典=冑2BV/BIF.共1兲
A dis inc i e ad an age o a 2D sys em such as ECEI is
he ac ha i p o ides many closely spaced da a channels
ha , in many applica ions, will show simila da a. This high
deg ee o edundancy can be aken ad an age o by using
SVD as a powe ul ool o analyzing and il e ing he 2D
da a. SVD is a well known ma hema ical echnique o de-
compose wo-dimensional ma ices in o pai s o eigen ec-
o s. The ma hema ical de ails can be ound in he
li e a u e,15 bu he gene al idea becomes clea by looking a
he applica ion o SVD o il e ing ECEI da a. In p inciple,
ECEI da a a e h ee-dimensional: he e a e wo space dimen-
sions, and a ime dimension. Howe e , o he SVD decom-
posi ion, we will ega d ECEI da a as ha ing one ime di-
mension 共o leng h m, ypically m=1000兲, and only one
spa ial dimension 共o leng h n=128兲, ea ing he 16 by 8
wo-dimensional space as a one-dimensional a ay o leng h
128. The e o e, he ECEI da a a e ep esen ed by a wo-
dimensional m-by-nma ix X. The SVD decomposi ion o
he da a can now be w i en as
X=USVT,共2兲
whe e Uis an m-by-nma ix whose 共leng h m兲columns uk
ep esen ime eigen ec o s 共some imes called “ch onos”兲.
VTis an n-by-nma ix whose 共leng h n兲 ows k
T ep esen
spa ial eigen ec o s 共“ opos”兲.Sis an n-by-nma ix whose
only nonze o elemen s a e on he diagonal, so S
=diag共s1,....,sk兲, whe e sk, he so called singula alues, a e
he eigen alues belonging o he opos-ch onos pai s o
eigen ec o s. By con en ion, he ec o s a e o de ed om
la ge o small sk. The e o e, in he case o ECEI, he SVD
esul s in 128 pai s o eigen ec o s 共ukand k
T兲wi h co e-
sponding dec easing eigen alues sk. The il e ed ECEI da a
Xlwould hen be ep esen ed by
Xl=兺
k=1
l
uksk k
T,共3兲
whe e lis he numbe o 共mos signi ican 兲eigen ec o pai s
e ained in he summa ion. The in o ma ion con ained in he
eigen ec o pai s wi h k⬎lis hence los . Ma hema ically, Xl
共 he unca ed SVD兲is he closes ank lma ix o he o igi-
nal ma ix Xand is hence he bes app oxima ion o X, pos-
sible by only using leigen ec o pai s. No e ha SVD il e -
FIG. 9. 共Colo online兲SVD il e ing o emo e 共 he mal兲noise. In 共a兲, he
spec og am o he da a o a single ECEI channel 共channel 8-4, sho 25525兲
con aining RSAEs is shown. 共b兲The same da a a e SVD il e ing 共 un-
ca ed a e en eigen ec o pai s兲.共c兲The esidue ha has been emo ed in
he il e ing.
FIG. 8. 共Colo online兲Typical p o iles 共based on measu emen s兲o 共a兲
elec on densi y, 共b兲elec on empe a u e, and 共c兲op ical hickness jus be-
o e and jus a e an ELM c ash. Also he case wi h a simula ed ilamen is
shown 共 ilamen only in empe a u e, no in densi y兲. The b oadening o he
second ha monic ECE esonance o he ilamen case is shown in 共d兲 o
bo h ECEI and 1D ECE.
10D929-5 Classen e al. Re . Sci. Ins um. 81, 10D929 共2010兲
ing does no educe he spa ial o ime esolu ion o he
il e ed da a, as many o he il e ing me hods 共such as Fou-
ie il e ing兲do.
Cohe en oscilla ions 共p esen on mul iple channels wi h
simila dynamics兲will esul in much la ge singula alues
han incohe en 共noise兲oscilla ions. The e o e, he s ong
and cohe en oscilla ions a e s ill p esen in Xl, whe eas he
esidue X-Xlcon ains almos exclusi ely incohe en oscilla-
ions. This is o cou se only ue i lis chosen su icien ly
la ge, so ha he cohe en pa o he da a can be desc ibed
by a ank lma ix. In p ac ice, lis no mally chosen o be
be ween 10 and 20 in he analysis o ECEI da a. Fo his
pu pose, he ECEI da a a e di ided in o da a segmen s o
ypically 1000 ime poin s. Fo each o hese segmen s, he
unca ed SVD is calcula ed. The leng h o he segmen s is
bo h de e mined by compu a ional limi s, and he equi e-
men ha he oscilla ions in he da a do no show oo la ge a
ime e olu ion 共in which case mo e eigen ec o pai s would
be needed兲.
An example o SVD il e ing is shown in Fig. 9. Shown
a e h ee spec og ams o he da a om a single ECEI chan-
nel 共channel 8-4, sho 25525兲 ha con ains Al én wa e ac-
i i y, so called e e sed shea Al én eigenmodes 共RSAEs兲.
These cohe en oscilla ions ha e ela i e empe a u e luc-
ua ion ampli udes o a ound 1%, and a e hence below he
he mal noise le el. Figu e 9共a兲shows he spec og am o he
aw da a 共X兲. Figu e 9共b兲shows he SVD il e ed da a 共Xl,
whe e l=10兲, and Fig. 9共c兲shows he esidue X-Xl. The colo
scaling o all h ee spec og ams is equal, so his clea ly
demons a ed ha he SVD il e ing signi ican ly enhances
he cohe en in o ma ion in he da a, and he esidue p ac i-
cally only con ains noise. As in p ac ice almos all oscilla-
ions o in e es a e a o below he he mal noise le el, SVD
il e ing has p o en o be almos essen ial o any di ec plo -
ing 共s ills o mo ies兲o 2D ECEI da a 共like also he images
in Fig. 7兲.
Figu e 10 shows examples o he spa ial s uc u e o
RSAEs as measu ed by ECEI. The Fou ie ampli ude in Fig.
10共b兲and mode s uc u e 共o mo e accu a ely he eal pa o
he complex Fou ie ampli ude兲in Fig. 10共c兲show a adially
localized oscilla ion wi h a poloidal mode numbe o a ound
m=8. An RSAE wi h a second adial ha monic is shown in
Figs. 10共d兲and 10共e兲. This analysis made use o bo h SVD
il e ing o ge id o he majo i y o he noise, and subse-
quen Fou ie equency selec ion o pick ou he sepa a e
modes. These i s , local, 2D ECE measu emen s o he mode
s uc u e o Al én wa es e eal mo e in o ma ion han p e-
ious 1D measu emen s,16 and could con ibu e o he unde -
s anding o hese modes and he associa ed as pa icle
anspo . Simila esul s ha e in he mean ime been
achie ed wi h ano he ECEI sys em on DIII-D.17
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FIG. 10. 共Colo online兲共a兲Spec og am o sho 25528 showing many
RSAEs. In 共b兲and 共c兲, espec i ely, he ampli ude and mode s uc u e o one
o he RSAEs is shown. An example o an RSAE wi h a highe adial
ha monic is shown in 共d兲and 共e兲.
10D929-6 Classen e al. Re . Sci. Ins um. 81, 10D929 共2010兲