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.


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


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



• 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



• In medical applications, we have about 300000 LOS for an area of 30cmx30cm (often on a regular grid)



• 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)


In order to reach a higher resolution, virtual LOS can be created by movingthe object in front of the given LOS:





• 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





contours of straight field line angle *and flux surface label

and a few 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


2.5% noise

no noise

Anton, PPCF 38 (1849) 1998.

110 LOS


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

european joint phd programme, lisboa, 10.2.2009 diagnostics of fusion plasmas tomography ralph dux

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