Kein Folientitel

1
Diagnostics of Fusion Plasmas Tomography
Ralph Dux
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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 ?gt 1kev (typical
value) 1D interferometric density
measurements spectroscopic measurements
110 lines-of-sight
3
Basic law of photometry
detector
source
dA2
dA1
The power d?12 emitted by a source with radiance
L and area dA1 onto the detector with area dA2

• 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 source
projected area of detector x solid angle of
source as
seen from detector
radiance emitted power per projected
unit area and unit solid angle
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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
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).
The quantity g is the emitted power density due
to spontaneous emission.
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The line-of-sight approximation

• 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?

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The ettendue for two types of pinhole cameras
single detectors on a circle around the
apperture(??det0, ?ap ?)
flat detector array behind apperture(??det?,
?ap ?, rd /cos ?)
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Soft X-ray The detection efficiency
SXR-cameras use filters usually made of Be
(d10-250??m) to stop the low energy photons (
typ. lt 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 volume g 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 camera design 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.
Be filter
det
ap
LOS
Be filter
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The Radon transform
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
A LOS is uniquely described by the impact radius
p (the distance between the LOS and the plasma
axis) and the poloidal angle ? of this point.
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.
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The back transformation
Two classical papers A.M. Cormack J.Appl.Phys.
34 (2722) 1963. J.Appl.Phys. 35 (2908)
1964. Representation of a function by its line
integrals.. that give the back transformation.
10
The back transformation
Two classical papers A.M. Cormack J.Appl.Phys.
34 (2722) 1963. J.Appl.Phys. 35 (2908)
1964. Representation of a function by its line
integrals.. give the back transformation. N
o ?-dependence of g leads tothe Abel inversion
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The back transformation

• But
• The back transformation needs complete
• knowledge about the function f(p,?) to
• construct the g-function.
• It is not sufficient to measure at rightangles
  with high precision.

12
The back transformation

• But
• The back transformation needs complete
• knowledge 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.

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The back transformation

• But
• The back transformation needs complete
• knowledge 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)

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The back transformation
Old SXR-setup at ASDEX Upgrade

• But
• The back transformation needs complete
• knowledge 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

15
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)

16
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)

17
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

18
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

19
Create virtual LOS to get higher resolution
contours of straight field line angle ?and flux
surface label ?? and a few LOS

• 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

20
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
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The finite element approach
We subdivide the plasma cross section into a
rectangular grid with nnxxny 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 element approach 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 impossible For
ngtm less equations than unknowns. For nm badly
conditioned problem (small changes in f produce
large changes in g) For nltm a least-squares fit
can be used to obtain g
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Least-Squares Fit with Regularisation
A pure least-squares fit works only for fewer
free parameters n than data points m (nltm) For
ngtgtm, ?we can always achieve overfitting, i.e.
?20. In this case another functional ?
is minimized 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 of the
regularization. Often, ? is set asto get a ?2 of
about 1. The maximum entropy algorithms also
yield the right choice for ? based on Bayesian
probabilitytheory.
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Least-Squares Fit with Regularisation
Anton, PPCF 38 (1849) 1998.
no noise
2.5 noise
110 LOS
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Least-Squares Fit with Regularisation
Flaws, PhD Thesis, LMU München (2009).
inversion of the island structurewith reduced
number of LOS with ME and ME virtual LOS
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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 geometry
• assuming that gg(?) where ? is a flux surface
• label.
• Even just 1 camera will deliver good images.

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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 Nltltm
• 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

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1D inversion

• Result for different g-profiles
• triangular profile (typical for soft X-ray)
• hollow profile (typical for total radiation)
• very peaked profile
• with 10 relative uncertainty of the measured
• line integrals
• the emission in the centre has always
• highest uncertainty, since only a few LOS
• go through the centre and since only a small
• length is contributing to the signal
• the relative uncertainty of the central emission
• becomes even larger when there is a ring with
• strong radiation at the plasma edge? a bolometer
  is not very good to measurein the centre