European Joint PhD Programme, Lisboa, 10.2.2009 Diagnostics of Fusion Plasmas Tomography Ralph Dux
European Joint PhD Programme, Lisboa, 10.2.2009
Diagnostics of Fusion Plasmas
Tomography
Ralph Dux
Tomography
The goal of tomography is to reconstructfrom a number of line-integrated measurements of radiation or density the local distribution
typical diagnostics:
1D,2D:Bolometer total radiation
Soft X-ray cameras radiation with energies > 1kev (typical value)
1D:interferometric density measurements
spectroscopic measurements
110 lines-of-sight
Basic law of photometry
The power d12 emitted by a source with radiance L and area dA1 onto the detectorwith area dA2
dA1 dA2
sourcedetector
1n
2n
122
211
212
221112
2
cos
cos
coscos
ddAL
ddAL
r
dAdALd
symmetric in source and detector• contains Lambert law and 1/r2 decay
projected area of source x solid angle of detector as seen from the sourceprojected area of detector x solid angle of source as seen from detector
sr
LddA
dL
2
2
m
W
cos
radiance = emitted power per projected unit area and unit solid angle
Plasmas are (nearly always) opically thin
Plasmas are radiating in the volume (not just at the surface) and we define a volumequantity the emission coefficient as the change of the radiance per length elementdue to spontaneous emission:
emission sspontaneoudl
dL
The other changes of L due to absorption or induced emission can usually be neglected, i.e. we assume an optically thin plasma.
Furthermore, for the application in mind we can assume the plasma radiationto be isotropic (does not hold for very specific cases like when looking at a -transition of a B-field splitted line).
gdV
d
4
1
4
1
The quantity g is the emitted power density due to spontaneous emission.
The detector has a certain area Adet and can detect radiation emitted within a solid angle det that is defined by the aperture area and the distance between detector and aperture. The product of projected detector area and solid angle is called the ettendue Edet. Line-of-sight approximation:• The plasma fills the whole solid angle of the detector (contributing plasma area r2). • At a certain distance l along the line-of-sight the radiance does not strongly vary in the direction perpendicular to the line-of-sight (at most linear).
detected power = ettendue x line-integrated emitted power / 4:
The line-of-sight approximation
LOSLOS
dllg
EdLAP
4
)(cos detdetdetDet
DetAapA
d
2d
AapDet
ldl
The ettendue for two types of pinhole cameras
2
apdet
2
apdet
detdetDet
coscos
cos
r
AA
r
AA
AE
ap
single detectors on a circle around the apperture(det=0, ap= )
flat detector array behind apperture(det=, ap= , r=d /cos )
2
4apdet
2
2apdet
detdetDet
cos
cos
cos
d
AA
r
AA
AE
r
ap
det
dLOS
r
ap
det
LOS
Soft X-ray: The detection efficiency
SXR-cameras use filters usually made of Be (d=10-250m) to stop the low energy photons ( typ. < 1keV)
The detection efficiency () depends on the absorption in the Be filter and the absorptionlength of the photons in the detecting Si-Diode...
Thus, it is not the total emitted power per volumeg but a weighted average that we will get:
For the circular camera type with circular Be-filter does not depend on , however, for a flat cameradesign with flat Be filter the dependence will becomemore and more critical with rising . Since we do notknow the plasma spectrum it might prevent the useof the edge channels.
r
ap
det
dLOS
r
ap
det
LOS
Be filter
Be filter
dg
g
sxr
The Radon transform
),(
detDet ),(
4
pLOS
dlrgE
P
det
Det 4),(E
Ppf
),(
),(),(
pLOS
dlrgpf
A LOS is uniquely described by theimpact radius p (the distance betweenthe LOS and the plasma axis) and thepoloidal angle of this point.
Power on the detector for a LOS
transform it into a ‘chord brightness’ (independent of detector quantities)
integral relation between ‘chord brightness’and power density:
This is a Radon transform (Radon is anAustrian mathematician 1887-1956).For an emission distribution which is zero outside a given domain g(r,) can be calculated if the function f(p,) is known.
The back transformation
Two classical papers:
A.M. CormackJ.Appl.Phys. 34 (2722) 1963.J.Appl.Phys. 35 (2908) 1964.‘Representation of a function by its line integrals..’
that give the back transformation.
The back transformation
Two classical papers:
A.M. CormackJ.Appl.Phys. 34 (2722) 1963.J.Appl.Phys. 35 (2908) 1964.‘Representation of a function by its line integrals..’
give the back transformation.
No -dependence of g leads tothe Abel inversion:
22
1)(
rp
dppf
dr
drg
a
r
The back transformation
But:• The back transformation needs completeknowledge about the function f(p,) to construct the g-function.
• It is not sufficient to measure at rightangles with high precision.
1.5 0.5
0.5 1.5
2 2
2
2
0.5 1.5
1.5 0.5
2 2
2
2
The back transformation
But:• The back transformation needs completeknowledge about the function f(p,) to construct the g-function.
• It is not sufficient to measure at rightangles with high precision.
• LOS under all angles are needed.
1.5 0.5
0.5 1.5
2 2
2
2
0.5 1.5
1.5 0.5
2 2
2
21.52
3212 0.521.52
0.52
The back transformation
But:• The back transformation needs completeknowledge about the function f(p,) to construct the g-function.
• In medical applications, we have about 300000 LOS for an area of 30cmx30cm (often on a regular grid)
The back transformation
But:• The back transformation needs completeknowledge about the function f(p,) to construct the g-function.
• In fusion plasmas, we have at most 200 samples on a non-regular grid in p,-space. the achievable spatial resolution is quite low
Old SXR-setup at ASDEX Upgrade
Create virtual LOS to get higher resolution
In order to reach a higher resolution, virtual LOS can be created by movingthe object in front of the given LOS
examples:• move plasma up, down, left, right and make sure the plasma does not change to much (used for bolometric measurements in ASDEX Upgrade divertor)
Create virtual LOS to get higher resolution
In order to reach a higher resolution, virtual LOS can be created by movingthe object in front of the given LOS:
examples:• move plasma up, down, left, right and make sure the plasma does not change to much (used for bolometric measurements in ASDEX Upgrade divertor)
Create virtual LOS to get higher resolution
In order to reach a higher resolution, virtual LOS can be created by movingthe object in front of the given LOS
examples:• move plasma up, down, left, right and make sure the plasma does not change to much (used for bolometric measurements in ASDEX Upgrade divertor)
• rotation tomography: an island rotates with constant angular frequency on the ‘straight field line angle’ data taken at different times can be combined to create virtual LOS
Create virtual LOS to get higher resolution
In order to reach a higher resolution, virtual LOS can be created by movingthe object in front of the given LOS
examples:• move plasma up, down, left, right and make sure the plasma does not change to much (used for bolometric measurements in ASDEX Upgrade divertor)
• rotation tomography: an island rotates with constant angular frequency on the ‘straight field line angle’ data taken at different times can be combined to create virtual LOS
contours of straight field line angle *and flux surface label
Create virtual LOS to get higher resolution
In order to reach a higher resolution, virtual LOS can be created by movingthe object in front of the given LOS
examples:• move plasma up, down, left, right and make sure the plasma does not change to much (used for bolometric measurements in ASDEX Upgrade divertor)
• rotation tomography: an island rotates with constant angular frequency on the ‘straight field line angle’ data taken at different times can be combined to create virtual LOS
contours of straight field line angle *and flux surface label
and a few LOS
Create virtual LOS to get higher resolution
In order to reach a higher resolution, virtual LOS can be created by movingthe object in front of the given LOS
examples:• move plasma up, down, left, right and make sure the plasma does not change to much (used for bolometric measurements in ASDEX Upgrade divertor)
• rotation tomography: an island rotates with constant angular frequency on the ‘straight field line angle’ data taken at different times can be combined to create virtual LOS
the LOS in the *-space
The finite element approach
We subdivide the plasma cross section intoa rectangular grid with n=nxxny grid points.
For each grid point we calculate its contribution to the m LOS (most simplythe dl going through the small square around the point). We obtain an mxn contribution matrix T.
The m LOS integrals in this finite elementapproach are then obtained by matrix multiplication of the contribution matrix withthe vector g containing the emissivities at each pixel.
The inverse of T delivers the emissivity distribution. Thus, the tomographic reconstruction is often called inversion.
But direct inversion almost always impossibleFor n>m: less equations than unknowns.For n=m: badly conditioned problem (small changes in f produce large changes in g)For n<m: a least-squares fit can be used to obtain g
gfT
mf
f
f
f...
2
1
ng
g
g
g...
2
1
gf
1T
Least-Squares Fit with Regularisation
A pure least-squares fit worksonly for fewer free parameters nthan data points m (n<m)
For n>>m, we can always achieveoverfitting, i.e. 2=0.
In this case another functional isminimized which contains an extra regularizing functional R of g, that tests how rough/irregular g is.
R can be based on:the gradients, the curvature, the entropy,weaker gradients parallel to B than perp. to B
The value of defines the influence ofthe regularization. Often, is set asto get a 2 of about 1. The maximum entropy algorithms also yield the rightchoice for based on Bayesian probabilitytheory.
m
ii
n
jjij
i
fgm 1
2
12
2 11T
R 2
2
1
Least-Squares Fit with Regularisation
2.5% noise
no noise
Anton, PPCF 38 (1849) 1998.
110 LOS
Least-Squares Fit with Regularisation
Flaws, PhD Thesis, LMU München (2009).
inversion of the island structurewith reduced number of LOSwith ME and ME + virtual LOS
1D inversion
The inversion is considerably simplified,when g can be assumed to be constanton magnetic flux surfaces:
• transport coefficients along B much larger than perpendicular to B• no gradients of density, temperature, impurity density on the flux surface• inside the separatrix• measurements may not be too fast, i.e. they have to average over several cycles of MHD modes which might be present (typ. 1ms) in order to smear out poloidal asymmetries
We include the known flux surface geometryassuming that g=g() where is a flux surfacelabel.
Even just 1 camera will deliver good images.
1D inversion
One possible approach is:• parametrize g by a function depending on a few parameters p1,p2 ..pN
where the number of free parameters N<<m• the function should have zero gradient at =0• the function should not allow negative values• the exponential of splines are very handy: a regularization can be build in by using only a higher density of spline knots in the region where you expect strong gradients
• subdivide each LOS in equal length elements and calculate for each LOS and length element the of the flux surface • find the minimum of 2 with the Levenberg-Marquardt algorithm (for non-linear dependence of the line integrals on the parameters pn)
• the uncertainty of g at a certain radius can be estimated from the curvature matrix of 2 and the uncertainties of the parameters
Npppgg ,..,, 21
m
iiN
i
fdlppgm 1
2
LOSith
122 ,..,,
11
N
jpjpi
ji
N
i jig p
g
p
g
pp1 1
222
1D inversion
Result for different g-profiles:1. triangular profile (typical for soft X-ray)2. hollow profile (typical for total radiation)3. very peaked profilewith 10% relative uncertainty of the measuredline integrals
the emission in the centre has alwayshighest uncertainty, since only a few LOSgo through the centre and since only a smalllength is contributing to the signal
the relative uncertainty of the central emissionbecomes even larger when there is a ring withstrong radiation at the plasma edge
a bolometer is not very good to measurein the centre