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

Tomography The goal of tomography is to reconstruct from 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 d  12 emitted by a source with radiance L and area dA 1 onto the detector with area dA 2 dA 1 dA 2 source detector symmetric in source and detector contains Lambert law and 1/r 2 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

Plasmas are (nearly always) opically thin Plasmas are radiating in the volume (not just at the surface) and we define a volume quantity the emission coefficient  as the change of the radiance per length element due 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 radiation to 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.

The detector has a certain area A det 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 E det. Line-of-sight approximation: The plasma fills the whole solid angle of the detector (contributing plasma area  r 2 ). 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

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 ) ap det LOS ap det LOS

Soft X-ray: The detection efficiency SXR-cameras use filters usually made of Be (d=  m) to stop the low energy photons ( typ. < 1keV) The detection efficiency  ( ) depends on the absorption in the Be filter and the absorption length 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 become more and more critical with rising. Since we do not know the plasma spectrum it might prevent the use of the edge channels. ap det LOS ap det LOS Be filter

The Radon transform 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. 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 an Austrian mathematician ). 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. Cormack J.Appl.Phys. 34 (2722) J.Appl.Phys. 35 (2908) ‘Representation of a function by its line integrals..’ that give the back transformation.

The back transformation Two classical papers: A.M. Cormack J.Appl.Phys. 34 (2722) J.Appl.Phys. 35 (2908) ‘Representation of a function by its line integrals..’ give the back transformation. No -dependence of g leads to the Abel inversion:

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 right angles with high precision

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 right angles with high precision. LOS under all angles are needed  2 3  2 1     2

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 LOS for an area of 30cmx30cm (often on a regular grid)

The back transformation 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 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 moving the 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 moving the 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 moving the 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 moving the 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 moving the 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 moving the 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 into a rectangular grid with n=n x xn y grid points. For each grid point we calculate its contribution to the m LOS (most simply the 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 with the 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

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 as to get a  2 of about 1. The maximum entropy algorithms also yield the right choice for  based on Bayesian probability theory.

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

Least-Squares Fit with Regularisation Flaws, PhD Thesis, LMU München (2009). inversion of the island structure with reduced number of LOS with ME and ME + virtual LOS

1D inversion The inversion is considerably simplified, when g can be assumed to be constant on 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.

1D inversion One possible approach is: parametrize g by a function depending on a few parameters p 1,p 2..p N 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 p n ) the uncertainty of g at a certain radius  can be estimated from the curvature matrix of  2 and the uncertainties of the parameters

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 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 measure in the centre