Two- and Three-Dimensional Tomography of Radiated Power using Imaging Bolometers in Toroidal Devices
Byron J. Peterson
National Institute for Fusion ScienceSOKENDAI
R. Sano (QST), J. H. Jang, S. Oh (NFRI), K. Mukai (NIFS, SOKENDAI), S. N. Pandya (IPR),
W. Choe (KAIST), N. Iwama (NIFS)
The 3rd IAEA Technical Meeting on ‘Fusion Data Processing, Validation and Analysis’, 28th to 31st May, 2019, Vienna, Austria. O-16
Outline
• IRVB concept
• Tomography
• 3D tomography in LHD (R. Sano)
• Bolometer channels and plasma grid definition
• Initial numerical test with standard linear solver
• Extension of linear system with prior information
• 3D reference profile function and iterative optimizer
• Numerical test and application to experimental data
• 2D tomography on KSTAR (J. H. Jang)
• KSTAR IRVB setup
• Tomography technique
• Phantom reconstruction tests
• Experimental results
2-D
Laplacian
Imaging Bolometer (IRVB) Concept
2lkt
P
f
radrad =
1)( 4
0
4
−
= −
f
BSbb
kt
TT
foil thermal
diffusivity
Calculate Prad from foil T using 2D heat diffusion equation
black body cooling term
plasma radiated power
is determined by solving
heat diffusion equation
foil thermal
conductivity foil
thickness
bolometer
pixel area
IRVB pinhole camera
B.J. Peterson et al., Rev. Sci. Instrum. 74 (2003) 2040.
2
2
2
21
y
T
x
T
t
Tbbrad
+
=
++−
ther
ma
l
dif
fusi
vio
n
to f
ram
e
IR measured
by camera
copper frame
Th
in f
oil
copper frame
lig
ht
shie
ld
Line of Sight Integration(forward problem)
Tomography(Reconstruction)(inverse problem)
Tomography problem: line averaged local information
Pi: Line-integrated detector Field of View (FoV) dataHij: Projection matrix (i detectors x j plasma voxels)Sj: 1D, 2D or 3D Plasma parameter distribution grid
Projection matrix, Hij = Vijij/4pV - intersecting volume of FoV i and S grid jij = Ai/lij
2 - detector area, Ai, solid angle, ij
⇒Relation between P-S ⇒ Must be inverted to get S from P
Plasma parameter distribution (unknown)
Line-integrated data(measurement output)
𝐒 𝐏
j = 1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
i = 1 2 3
4
5
6
7
8
9
1012 11
l5,10
Outline
• IRVB concept
• Tomography
• 3D tomography in LHD (R. Sano)
• Bolometer channels and plasma grid definition
• Initial numerical test with standard linear solver
• Extension of linear system with prior information
• 3D reference profile function and iterative optimizer
• Numerical test and application to experimental data
• 2D tomography on KSTAR (J. H. Jang)
• KSTAR IRVB setup
• Tomography technique
• Phantom reconstruction tests
• Experimental results
Plasma voxel(grid element)(cylindrical coordinate)Horizontal(R): 5 cm
54 divisions (2.4 m < R < 5.1 m)Vertical (Z): 5cm
52 divisions (-1.3 m < Z < 1.3 m)Toroidal(φ): 1 degree
360 divisionsTotal number of voxels: 1,010,880
ROI (Region of Interest) voxel (tomography target): masking process, helical periodicity (n=10)
⇒ 16,188 voxels (0°<φ<18°)
Diagnostics(IRVB × 4)
IRVB channels: 1008+1008+560+620=3,196ch
𝐏 (3196 ch)𝐒 16188 voxelsH (3196 × 16188)Helical periodicity
IRVBs and plasma voxels are designed as 3D tomography system
Sight line without periodicitySight line with periodicity
10-O6-T
6.5-U
6.5-L
𝐏 = H𝐒
RoI
Model (EMC3-EIRENE), S
Reconstructed , መ𝑆 (with 10% noise)
Synthetic image, P
Reconstructed P
3D reconstruction is achieved by standard linear solver for tomography→ But, large number of artifacts should be reduced
Radiation power density (W/cm3)
Radiation power density (W/cm3)
Φ=0.5 Φ=9.5 Φ=17.5
Φ=0.5 Φ=9.5 Φ=17.5
Numerical test
Standard linear solver is extended by prior information
Lagrange function
Λ 𝐒 = γ C𝐒 2 +H𝐒 − 𝐏 2
M
H:Projection matrix
P: Integrated data
S :Plasma parameter distribution
M:Total number of diagnostics
channels
γ : Regularization parameter
C : Identity matrix
Extended solver (with prior information )
Λ 𝐒 = γ C (𝐒 − α𝐦) 2 +H𝐒 − 𝐏 2
M
𝐦:Reference profile
(prior information)
α:Weighting factor
Euclidean distance between reference profile and reconstructed profile
Lagrange function
Standard linear solver (Tikhonov regularization)
Extended by 𝐦
R. Sano et al., Rev. Sci. Instrum. 87, 11D440 (2016).
Flow of 3D tomography with prior information
Reference profile is processed as a rough estimationby diagnostics data and knowledge of plasma
IRVB imagesReference
processing
(fitting)
3D reference profile, 𝐦
Reconstruction (extended)Prior information
Reconstruction (standard)Weighting factor 𝛼 is fixed to 0.5(by numerical test results)
𝐏
Model
PP r, z, ϕ =r − Rcenter cos ϕ − z sin ϕ 2
b2+
r − Rcenter sin ϕ + z cos ϕ 2
a2
1 − w 2 ≤ PP r, z, ϕ ≤ 12
Smodel r, z, ϕ = c 1 + γin−outcosR − Rcenter
2 Rcenter − Redge
π 1 + γpeakXX r, z, ϕ exp −(PP − 1 − Τw 2 2)2
( Τw 2)2
Definition of ellipse
Asymmetry part X-point peaking parta
bw
(Rcenter, 0)
(PP = 1)
(PP = 1 − w 2)
XX r, z, ϕ = exp −R − Rcenter cos ϕ − z ± a 1 − dXw sin ϕ
2
2 dXwa2
3D model function is employed for 𝐦 (Reference profile)
b: semi minor radiusa: semi major radius
w:width of radiation region (ratio)
γin−out: asymmetric factor (inboard − outboard)γpeak:Magnitude of local peak
c: radiation intensityRcenter: center of radiation region
dX: Location of local peak
8 free parameters
Simulation
Initial parameters
Normalized experimentalImage (IRVB data)
Normalized model projection (Model)
Reference profile is processed to fit IRVB image to model projection
Mean square error (normalized)
Number of calculation for 1 iteration = 3𝑛2 + 2𝑛 (𝑛 = 5)
Parameter scanning is individually carried out for each IRVB (U, L)
𝑏 1,𝑎
1,𝑤
1,𝛾
𝑖−𝑜1,𝛾
𝑝1,𝑅
1,𝑐1,𝑑
𝑋1
(𝑏0, 𝑎0, 𝑤0, 𝛾𝑖−𝑜0, 𝛾𝑝0, 𝑅0 ,𝑐0, 𝑑𝑋0)
R scan(n)
c scan (n)
𝛾𝑝 − 𝑑𝑋scan (n x n)
w-𝛾𝑎 scan (n x n)(𝑏0, 𝑎0, 𝑤1, 𝛾𝑖−𝑜1, 𝛾𝑝0, 𝑅0,𝑐0, 𝑑𝑋0)
b-a scan (n x n)(𝑏1, 𝑎1, 𝑤0, 𝛾𝑖−𝑜0, 𝛾𝑝0, 𝑅0 ,𝑐0, 𝑑𝑋0)
(𝑏0, 𝑎0, 𝑤0, 𝛾𝑖−𝑜0, 𝛾𝑝0, 𝑅1,𝑐0, 𝑑𝑋0)
(𝑏0, 𝑎0, 𝑤0, 𝛾𝑖−𝑜0, 𝛾𝑝1, 𝑅0, 𝑐0, 𝑑𝑋1)
(𝑏0, 𝑎0, 𝑤0, 𝛾𝑖−𝑜0, 𝛾𝑝0, 𝑅0, 𝑐1, 𝑑𝑋0)
𝑎𝑏
2
, 𝜀𝑛2,
H𝒎𝐏
Iteration until convergence
1st 2nd 3rd
5th 7th 9th
𝜙 = 0.5° 𝜙 = 9.5°
Incident radiation power (W)5.0x10-6 1.0x10-5 1.5x10-5 2.0x10-50.0
Radiation power density (W/cm3)
𝜙 = 0.5°
𝜙 = 9.5°
3D source profile (EMC3-EIRENE)
Tomography(Standard)
δ2 =𝐒 − 𝐒𝟎
2
𝐒𝟎2
Characteristics of source is recovered (numerical test)
𝜙 = 0.5° 𝜙 = 9.5°
Radiation power density (W/cm3)
Reconstructed profile δ2 = 0.674
inboard outboard
inboard outboard
𝜙 = 0.5° 𝜙 = 9.5°
Radiation power density (W/cm3)
Reconstructed profile δ2 = 0.717
Tomography(with prior)
inboard outboard
δ2 (reconstruction error)
is reduced by prior information
Magnetic axis
Near X-pointFar X-point
Experimental reconstruction become easily understandable by prior information (artifacts are suppressed)
Edge radiation
Core radiation (Radiation collapse)
Standard solver (Tikhonov) Extended solver (with prior )
Edge radiation
Core radiation (Radiation collapse)
Negative values:Number=3299 voxelsTotal =-111.00
Negative values:Number=2731 voxelsTotal =-60.46
Negative values:Number=2684 voxelsTotal =-52.74
Negative values:Number=2493 voxelsTotal =-39.14
Outline
• IRVB concept
• Tomography
• 3D tomography in LHD (R. Sano)
• Bolometer channels and plasma grid definition
• Initial numerical test with standard linear solver
• Extension of linear system with prior information
• 3D reference profile function and iterative optimizer
• Numerical test and application to experimental data
• 2D tomography on KSTAR (J. H. Jang)
• KSTAR IRVB setup
• Tomography technique
• Phantom reconstruction tests
• Experimental results
KSTAR IRVB setup
KSTAR IRVB hardware[2]
[1] B. J. Peterson, “Development of Imaging Bolometers at NIFS and NFRI”
[2] J. Jang et al, 2018 Curr. Appl. Phys, 18 461
IRVB field of view[1]
IR camera : FLIR SC7600
✓ NETD : < 25mK
✓ Frame rate : 105 Hz
✓ Resolution : 512 x 640
pixels
Foil : Pt
✓ Size : 0.002 x 70 x 90 mm
✓ Double side carbon
coating
✓ Photon energy range :
Eph < 7.5 keV
Bolometer
✓ Time resolution : 10 ms
✓ Aperture : 3.5 mm x 3.5 mm
✓ 24(tor) x 32 (pol) = 384 ch
Tomographic reconstruction technique
Phillips-Thikhonov method*
• Minimizing J = mean squared error +
signal variation𝑱 = 𝒇 −𝑾 ∙ 𝒈 𝟐/𝑴 + 𝜸 𝑳 ∙ 𝒈 𝟐
• P-T solution (𝜕𝑱/𝜕𝒈𝒊 = 𝟎)
𝐠 𝛄 = 𝐖𝐓 ∙ 𝐖 +𝐌𝛄𝐋𝐓 ∙ 𝐋−𝟏
∙ 𝐖𝐓 ∙ 𝐟
g : Reconstructed image
γ : Optimal regularization parameter
L : Laplacian
• GCV statistics : optimized γ
→ accuracy vs smoothness
→ high γ : smooth, inaccurate
→ low γ : accurate, unstable to noise
Ill-posed problem
𝐟 = 𝐖 ∙ 𝐠f : Line-integrated image
W : Weight matrix
g : Local emission profile (2-D)
Tangential line integration Local emission profile
(poloidal cross-section)
3-D geometry
(with toroidal symmetry)
IRVB image
Line-integration
Reconstruction
Tomography: A non-invasive imaging tool for observing the inner structure of
the plasmas
▪ Line-integrated raw data → tomography is essential
KSTAR IRVB Tomography setup
✓ Reconstruction grid : 1.2 < R < 2.4 m, -1.5 < Z < 1.5 m
→ Divided by 63 x 150 plasma pixels (2 cm x 2 cm for each)
✓ First wall geometry of KSTAR is applied in tomography code
KSTAR vacuum
vessel
Bolometer pixelPlasma pixel
FoilAperture
(b)(a)
Boundary condition of
KSTAR IRVB tomography
Open
Closed
Region between
Passive Stabilizers
(-0.5m < Z < 0.5m)
εb ≠ 0
No emission
on boundary
εb = 0
KSTAR IRVB line of sight
(description for plasma pixel and bolometer pixel)
Phantom reconstruction tests (1)
✓ Accuracy of KSTAR IRVB tomography is validated by reconstruction of
various synthetic images (phantoms)
✓ Total radiated power
𝑃𝑟𝑎𝑑 = 𝐴𝑝𝑖𝑥𝑒𝑙 × 2𝜋σ𝑅 ∙ 𝜖 𝑅, 𝑍
( 𝜖 𝑅, 𝑍 : 𝑒𝑚𝑖𝑠𝑠𝑖𝑣𝑖𝑡𝑦 𝑎𝑡 (𝑅, 𝑍) )
✓ Reconstruction error
erecon (%) =𝜖𝑝ℎ𝑎𝑛𝑡𝑜𝑚−𝜖𝑟𝑒𝑐𝑜𝑛
𝜖𝑝ℎ𝑎𝑛𝑡𝑜𝑚× 100
Synthetic image for
continuum radiation in KSTAR H-mode
→ Reconstructed well
Reconstruction erecon (%)Phantom
Reconstruction erecon (%)Phantom
More complicated phantoms[1] can also be
reconstructed well (Above : inter-ELM radiation
pattern, below : during ELM)
[1] J. Jang et al, 2018 Curr. Appl. Phys, 18 461
Reconstruction erecon (%)
d)
b)a)
c)
Phantom erecon (%)ReconstructionPhantom
a) D-shape + Hot spots in b) X-point, c) inboard and d) outboard divertor *
(10% noise added to line-integrated signal)*J. Jang et al, Rev. Sci. Instrum. (2018)
✓ High reconstruction accuracy near first wall in KSTAR
→ useful for impurity seeding exp. or plasma-divertor detachment
Phantom reconstruction tests (2)
Spatial resolution of IRVB tomography
✓ Spatial resolution of IRVB tomography
~ 9 cm
: Two gaussian peak with 9 cm gap
can be distinguished
(10% noise added to line-integrated
signal)
ReconstructionPhantom
ReconstructionPhantom
▪ Phantom
▪ Reconstruction
50% of
peak
Exp1) ELM mitigation by Kr seeding
✓ 1.7x1019 Kr particles injected
✓ ELM suppression (~5𝜏𝐸)
✓ ELM mitigation (~10𝜏𝐸 to the end of shot)
• 50% reduction in ∆𝑊𝐸𝐿𝑀 × 𝑓𝐸𝐿𝑀
ത𝑛𝑒(1019 m-2)
Te,ECE
(keV)
Dα
(1019)
Wdia
(MJ)
Prad
(MW)
Kr
H89L
time (s)
Prad
Prad,core
core
pedestal
Supp. Miti.
Kr signal
(a.u.)
Plasma
radiation
(kW/m3)
6.8s (before Kr seeding)
7.8s (natural ELM)
8.7s (ELM suppression)
9.4s (ELM mitigation)
Kr ion
density
(nKr) (m-3)
8.7s (Supp) 9.4s (Miti)6.8s 7.8s
IRVB plays crucial role in impurity study in KSTAR
7 8 9 10
Kr
time (s)
Prad
Prad,core
core
pedestal
Supp.
ത𝑛𝑒(1019 m-2)
Te,ECE
(keV)
Dα
(1019)
Wdia
(MJ)
Prad
(MW)
H89L
Kr signal
(a.u.)
Exp2) ELM suppression by Kr seeding
✓ 3.5x1019 Kr particles
injected
✓ ELM suppression (~4 𝜏𝐸)
✓ H-L transition
Plasma
radiation
(kW/m3)
Kr ion
density
(nKr) (m-3)
6.8s (before Kr seeding)
7.8s (natural ELM)
8.4s (ELM suppression)
8.4s (Supp)6.8s 7.8s
IRVB plays crucial role in impurity study in KSTAR
Kr density can be estimated from
Kr radiation (D.E. Post et al, 1977)
𝑃𝑟𝑎𝑑 𝜌 = 𝑛𝑒 𝜌 𝒏𝒁 𝝆 𝐿𝑧(𝑇𝑒(𝜌))
𝐿𝑧 : radiative cooling rate
(coronal equilibrium)
7 8
Exp 3) ITB formation by Kr seeding
✓ total 5.2x1019 Kr particles injected
✓ ELM mitigation → H-L back transition → ITB formation
✓ Ti and Te profiles shows strong core peaking
✓ Relation between Kr and ITB formation is still under investigation
Te (keV)
Ti (keV)
𝜌
5.9s (before Kr seeding)
9.0s (before ITB formation)
11.0s (before ITB formation)
ITB
ITB
Kr ITB
ത𝑛𝑒(1019/m2)
Ip (kA)
Wtot
(kJ)
Dα
(1019)
Prad
(kW)
Te,ECE
(keV)
PNBI
(MW)
time (s)
𝜌
9.0s
(Before ITB)
11.0s
(After ITB)Plasma
radiation
(kW/m3)
Kr ion
density
(m-3)
Conclusions
• IRVBs can be applied to toroidal devices for the purpose of 2D and 3D tomography of radiated power
• In helical devices 3D tomography is desirable, but difficult• Improvements can be made using prior information
• In a tokamak with a tangential view• 2D profiles can be obtained with good spatial resolution in core and
divertor by assuming toroidal symmetry
• Also total power estimates can be provided.
• In KSTAR IRVB is the only bolometer and is providing both 2D profiles and total power estimates.
Future work
• 2D tomography of radiation in helical devices using IRVBs assuming radiation is constant on a field line• Should be applicable to both LHD and W7-X
Noise and signal can be estimated.
Ssignal =Psignal
Abol=AbolAap cos4 JPradlplasma
Abol4plap- f2 Vplasma
32
4
3
3
2
4
24
cos
24
cos
bolbol
IRIR
plasmafapIRf
applasmarad
bolf
bolIRIR
plasmafapIRf
plasmarad
IRVB
signal
fN
Nf
Vlkt
AlP
fA
ANf
Vlkt
lP
S
SSNR
−−
===p
p
Foil properties (Pt):
k = 0.716 W/cmK – foil thermal cond.
= 0.2506 cm2/s – foil thermal diffusivity
tf – foil thickness
Af = 48 cm2 – utilized area of the foil
IR camera properties:
IR = 15 mK – IR camera NET
fIR – frame rate of IR camera
NIR – number of IR pixels
IRVB properties:
Abol – pixel area
fbol – frame rate of IRVB
Nbol – # of bolometer pixels
SIRVB – IRVB noise equivalent power density
hIRVB – IRVB noise equivalent power
Plasma parameters:
Lplasma – length sight line in plasma
Prad – total radiated power
Vplasma – plasma volume
Pinhole camera properties:
Aap = (1.4)2* Abol – area of aperture
lap-f – distance from foil to aperture
Q = 10 - 20 – angle between sightline and
aperture
Ssignal – estimated radiated power density on foil
S/N = Ssignal /SIRVB – signal to noise ratio
2
3
2
3
5
10
h bolbol
f
bolbol
IRIR
IRf
f
bolIRVBIRVB
fN
A
fN
Nf
kt
A
NS +==
small
[4] B.J. Peterson et al., Rev. Sci. Instrum. 74 (2003) 2040.
[4]
Phantom reconstruction tests (2)
Reconstruction erecon (%)
d)
b)a)
c)
Phantom erecon (%)ReconstructionPhantom
a) D-shape + Hot spots in b) X-point, c) inboard and d) outboard divertor *
*J. Jang et al, Rev. Sci. Instrum. (2018)
✓ High reconstruction accuracy near first wall in KSTAR
→ useful for impurity seeding exp. or plasma-divertor detachment
Reconstruction erecon (%)
d)
b)a)
c)
Phantom erecon (%)ReconstructionPhantom
a) D-shape + Hot spots in b) X-point, c) inboard and d) outboard divertor *
*J. Jang et al, Rev. Sci. Instrum. (2018)
✓ High reconstruction accuracy near first wall in KSTAR
→ useful for impurity seeding exp. or plasma-divertor detachment
Phantom reconstruction tests (+10% noise)
Spatial resolution of IRVB tomography
✓ Spatial resolution of IRVB tomography
~ 9 cm
: Two gaussian peak with 8 cm gap
can be distinguished
ReconstructionPhantom
ReconstructionPhantom
▪ Phantom
▪ Reconstruction
50% of
peak