N.Shi et al., address: Division seminar, April 9, 2014, Institute of Plasma Physics, Academy of Sciences. A novel electron density reconstruction method for asymmetrical toroidal plasmas 基于复杂磁面的电子密度重建方法 N.Shi 1, S.Ohshima 1, K.Tanaka 2, T.Minami 1, K.Nagasaki 1, S.Yamamoto 1, Y.Ohtani 3, L.Zang 3, T.Mizuuchi 1, H.Okada 1, S.Kado 1, S.Kobayashi 1, S.Konoshima 1, N.Kenmochi 3, and F.Sano 1 1 Institute of Advanced Energy, Kyoto University, Japan 2 National Institute for Fusion Science, Japan 3 Graduate School of Energy Science, Kyoto University, Japan

N.Shi et al., address Outline  Background  Reconstruction method (SVD-GCV)  Evaluation of SVD-GCV on different devices  Summary

N.Shi et al., address Outline  Background  Reconstruction method (SVD-GCV)  Evaluation of SVD-GCV on different devices  Summary

N.Shi et al., address Background  Particle transport of bulk ions and electrons is one of the most important issues for thermonuclear fusion research, since the density profile affects fusion performance and consequently plasma profile and stability.  FIR laser interferometry is a routine diagnostic technique for measuring the electron density with a high time resolution. However, it can only provide the line-integrated density.  The Heliotron J device at the Institute of Advanced Energy in Kyoto University is a flexible helical-axis heliotron with a highly asymmetrical poloidal cross section. H.Okada, et al., the 18 th International Toki Conference,2008.

N.Shi et al., address Magnetic flux surface T. Mizuuchi, et al., AIP conference Proceedings,2008. Vol. 996, #14.5 port  Different position has different magnetic flux surface in Heliotron-J. #14.5 port is the corner section, as shown in the right figure.  We have two choices to arrange the light path: i.one is using the vertical port  single channel FIR system. ii.the other is using the horizontal port  multi- channel FIR system. Plasma R=1.18 R=1.28 Plasma Vertical case Horizontal case

N.Shi et al., address Comparison  Because of the strongly shaped plasma configurations and limitation of the available line integrated data (ill-posed problem), the normal inversion method cannot offer the correct density profile.  One possible approach is to reduce the number of degrees of freedom by expanding the electron density profile into a series of orthogonal functions.  the lack of density profile measurements.  For example, Abel inversion is based on the circular plasma; Slice and stuck technique need interpolation and directly matrix inversion calculation, which might be acceptable on slight asymmetrical plasma but it is not feasible on Heliotron J case. The propagation of the errors are enlarged by strong shaped magnetic flux surface. Comparison of conventional and SVD-GCV method for reconstructing peaked and hollow profiles.

N.Shi et al., address Outline  Background  Reconstruction method (SVD-GCV)  Evaluation of SVD-GCV on different devices  Summary

N.Shi et al., address Reconstruction method  The electron density is discretized on each magnetic surface and the line-density integral equations can be shown as: and can be arranged in matrix form: (1) (2)  Zeroth order regularization Best-approximate solution, which is defined as the least-squares solution of minimal norm.... ρ=1 ρ=0.9 ρ=0.95 L 11 L 22 L 21 L 33 L 32 L 31 n e1 n e3 n e2 …

N.Shi et al., address SVD  Let’s firstly consider the the weighted least-squares fitting, which means minimize Χ 2 with getting a least-squares solution. The expression of and in Eq. (3) is N i /σ i and L ij /σ i, respectively, where σ i is the standard deviation of N i.  Then, in order to obtain the best-approximate solution, SVD is supposed to be introduced on : U k [k=1,2,…,n]: the orthonormal eigenvectors of (the exponent T denotes transposition) with the n largest eigenvalues; V k [k=1,2,…,n]: the orthonormal eigenvectors of ; w k [k=1,2,…,n]:the positive square roots of the eigenvalues of, called also singular values. (3) (4)

N.Shi et al., address Solution with SVD  With the help of equation (4), the solution which minimizes Χ 2 could be written as.  The error components in, which correspond to large singular values w i, are harmless. However, the error components which correspond to small w i are amplified by the factor of 1/ w i, so that those are dangerous.  To obtain a unique and sensible solution, special features (i.e., smoothness, stabilities) should be introduced.  Regularization method as a remedial process could provide this kind of additional information through using regularization functional. (5)

N.Shi et al., address First-order linear regularization  We suppose the prior feature is a credible solution which is not too different from a constant. It means minimize the density gradients. A reasonable functional to minimize is where C is the (n-1)×n first difference matrix.  Introducing Lagrange multiplier and using the Eq. (6) as the regularization functional, the minimization of R under the constraint of a constant mean- squared error is equivalent to the minimization of where γ (γ>0) is the reciprocal of the Lagrange multiplier. (6) (7) γ  : the smoothing becomes more important in determining the solution. γ  : fitting is the dominant role in the solution. regularization parameter, the weighting between the goodness of fit and the smoothness

N.Shi et al., address Stability  The solution could be obtained with the help of the SVD on,  Consider the stability of the Eq. (8) by comparing with Eq. (5):  It clearly shows the stabilization comparing with the solution in Eq. (5). Errors in are not propagated with the factors 1/ w i, but only with the factors w i /( w i 2 +mγ) into the solution. (8) (5)

N.Shi et al., address GCV  To obtain the optimum value of γ, the GCV method is used. The GCV is a rotation-invariant version of Allen’s predicted residual sum of squares (PRESS), based on the mean squared error between the predicted and observed data.  Allen’s PRESS estimate of γ is the minimizer of  The GCV estimate of γ corresponds to the minimized value of V(γ), given by where, and I denotes the m×m unit matrix.  When the SVD of is substituted into Eq. (10), it becomes (9) (10) (11)

N.Shi et al., address An example of GCV estimate  An example of GCV estimate on simulated data:  The γ corresponding to the smallest V(γ) is substituted into Eq. (8) as follow: (8)

N.Shi et al., address Outline  Background  Reconstruction method (SVD-GCV)  Evaluation of SVD-GCV on different devices  Summary

N.Shi et al., address Evaluation of SVD-GDV  Factors affecting the accuracy on Heliotron-J  Error analysis  Preliminary test on LHD

N.Shi et al., address Factors affecting the accuracy  Factors affecting the accuracy:  The shape of the electron density profile  The number of channels  4-7 channels  The channel spacing The peak position for a hollow profile is at r/a>0.5; the ratio between the center and peak densities is greater than 0.5. Equal spacing: the channels are equally spaced from the core to one edge of the flux surface; Unequal spacing: more channels are used in regions where the flux surfaces are least circular.  For the peaked profile, good reconstruction are obtained regardless of the number of channels.  For the hollow profiles, the number of channels has a clear effect on the accuracy of the reconstruction, with better results being obtained as the channel number is increased.

N.Shi et al., address  For unequal spacing case, it means higher priority was placed on obtaining a more accurate reconstructed density profile than on determining the central line-integrated plasma density.  For the peaked profiles, good reconstruction results are obtained regardless of the number of channels used. In addition, for the hollow profiles, the results are significantly improved for the case of using five or six channels, although they are still poor for four channels.  The main reason for the failure in the four channel case is that this number is insufficient to follow the changes in the density profile. If sufficient channels are available, the effects of the profile shape and channel position can be ignored.

N.Shi et al., address Error analysis  Two kinds of error sources were considered. The first is due to mechanical vibrations and electrical pickup noise, whose effects can be tested by adding random errors to the line-integrated density.  100 sets of simulations of measurement with random errors (standard deviation is 1%) were constructed.  The error bars in left figure represent the standard deviation of the obtained values for each radial point.  The errors produced are seen to be larger near the core and the edge region. At the core, this is the result of the limited number of channels. At the edge, it is due to the highly asymmetrical shape of the flux surfaces.

N.Shi et al., address  The second source is uncertainty in the channel position, whose effects can be tested by adding random errors to the channel position. The uncertainty is determined by the installed accuracy of mirrors along the light path of the FIR system and the consideration about the effects of beam width.  100 sets of channel position with random errors (corresponding to a distance deviation 10mm) were constructed.  The resulting errors are seen to be largest for r/a values from 0.6 to 1.0. This is because in this region, small variations in the channel position give rise to the largest changes in the path length, and bring the largest errors to the line-integrated density accordingly.  It is known that, the density gradient is more important for the study of particle transport. A rough estimation has been done. The standard deviation of the density gradient is around 13% for the line-integrated error analysis and 14% for the position error analysis.

N.Shi et al., address Preliminary test on LHD Min(GCV)=1.984E-3 Gamma=1.285E-4  Test with LHD data, 4.3s. Comparison with Thomson data. Flux surfaces Gamma

N.Shi et al., address Error analysis ＃ gamma=5 error=0.02 gamma=0.5 error 0.02 gamma=50 error 0.02 gamma=500 error 0.02  CO 2 imaging interferometer reconstructed results (fix the gamma in advance). Magnetic surfaces Error =0.02 Error bar with random error. (100 times) The gamma value is around 0.1~1 during 100 times calculations.

N.Shi et al., address Particle transport analysis Amplitude reconstruction Phase reconstruction Integrated amplitude Integrated phase Diffusion coefficient Convection velocity Based on radial density fitting. Based on integrated density fitting.

N.Shi et al., address Outline  Background  Reconstruction method (SVD-GCV)  Evaluation of SVD-GCV on different devices  Summary

N.Shi et al., address Summary  A novel electron density reconstruction method is developed for strongly asymmetrical toroidal plasmas. It is based on a regularization technique, with aid of SVD, GCV function is used to optimize the regularization parameter.  Several factors that may affect the reconstruction results are discussed, including the number of measurement channels, the channel positions, and the density profile shapes.  For the beam path of the FIR system in the Heliotron J, a seven-channel layout gives the best reconstruction and the effects of the mentioned above can be ignored. However, if the channel positions are suitably optimized, sufficiently accurate reconstructions can be achieved even with five channels.  In the near future, the light path system of the multi-channel FIR laser interferometer will be designed based on the results of this study.

N.Shi et al., address Thank you!

N.Shi et al., address Matrix C  The Eq. (7) expresses the regularization in general form, and it is in standard form if C=I.  If using SVD on standard form, the matrix should be decomposed. Otherwise, the decomposition should be applied on.  On the other hand, in the sense of density reconstruction of FIR system, the last row expresses the operator applying on the density of the most outside flux surface. In this area, the density is closed to zero, corresponding to the smallest ||n e || 2. The first difference matrix. Prior requirement. Smallest solution.