European Joint PhD Programme, Lisboa, 10.2.2009

1. European Joint PhD Programme, Lisboa, 10.2.2009 Diagnostics of Fusion Plasmas Tomography Ralph Dux

2. 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

3. Basic law of photometry detector source dA2 dA1 The power d12emitted 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

4. 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.

5. 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:

6. det ap LOS det ap LOS The ettendue for two types of pinhole cameras single detectors on a circle around the apperture(det=0, ap= ) flat detector array behind apperture(det=, ap= , r=d /cos )

7. det ap LOS Soft X-ray: The detection efficiency SXR-cameras use filters usually made of Be (d=10-250m) 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 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

8. 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.

9. 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. No -dependence of g leads tothe Abel inversion:

11. 2 2 2 2 1.5 0.5 0.5 1.5 2 2 0.5 1.5 1.5 0.5 2 2 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. 2 2 2 2 1.5 0.5 0.5 1.5 2 2 1.52 0.5 1.5 1.5 0.5 0.52 2 2 12 32 1.52 0.52 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.

13. 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)

14. 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. 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

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

21. The finite element approach We subdivide the plasma cross section into a 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 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 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

22. Least-Squares Fit with Regularisation A pure least-squares fit works only for fewer free parameters n than data points m (n<m) For n>>m, we can always achieve overfitting, i.e. 2=0. 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.

23. Least-Squares Fit with Regularisation Anton, PPCF 38 (1849) 1998. no noise 2.5% noise 110 LOS

24. 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

25. 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 g=g() where  is a flux surface • label. • Even just 1 camera will deliver good images.

26. 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

27. 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