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TJ-II:Tomography: Difference between revisions
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:<math>E(\rho,\theta) = \sum_{n,m}{C_{nm}f_{n}(\rho) \exp(im\theta)}\,\!</math> | :<math>E(\rho,\theta) = \sum_{n,m}{C_{nm}f_{n}(\rho) \exp(im\theta)}\,\!</math> | ||
Separating out the '' | Separating out the ''m'' = 0 terms: | ||
:<math>E(\rho,\theta) = \sum_{n=0}^N{C^0_{n0} f^0_{n}(\rho)} + \sum_{n,m=1}^{N,M}{f^1_{n}(\rho)\left [C^1_{nm}\cos(m\theta) + C^2_{nm}\sin(m\theta) \right ]}</math> | :<math>E(\rho,\theta) = \sum_{n=0}^N{C^0_{n0} f^0_{n}(\rho)} + \sum_{n,m=1}^{N,M}{f^1_{n}(\rho)\left [C^1_{nm}\cos(m\theta) + C^2_{nm}\sin(m\theta) \right ]}</math> | ||
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:<math>\int_0^1{f_{n}(\rho)f_{n'}(\rho) \rho d\rho} = c_n \delta_{nn'}</math> | :<math>\int_0^1{f_{n}(\rho)f_{n'}(\rho) \rho d\rho} = c_n \delta_{nn'}</math> | ||
where δ<sub>ij</sub> is the Kronecker delta. The [[:Wikipedia: | where δ<sub>ij</sub> is the Kronecker delta. The [[:Wikipedia:Fourier–Bessel_series|Fourier-Bessel]] functions satisfy this requirement. | ||
== The reconstruction algorithm == | == The reconstruction algorithm == | ||
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where the ''C<sub>μ</sub>'' are the mode coefficients. This is a linear set of equations that can be solved using standard techniques, allowing fast recovery of the mode coefficients from the detector signals. The detector signals ''I<sub>ν</sub>'' are assigned weights ''w<sub>ν</sub>'' inversely proportional to the square of the measurement error of measurement ''I<sub>ν</sub>''. Thus, any channel ''ν'' can be excluded from contributing to the result by setting ''w<sub>ν</sub>'' = 0. | where the ''C<sub>μ</sub>'' are the mode coefficients. This is a linear set of equations that can be solved using standard techniques, allowing fast recovery of the mode coefficients from the detector signals. The detector signals ''I<sub>ν</sub>'' are assigned weights ''w<sub>ν</sub>'' inversely proportional to the square of the measurement error of measurement ''I<sub>ν</sub>''. Thus, any channel ''ν'' can be excluded from contributing to the result by setting ''w<sub>ν</sub>'' = 0. | ||
The regression technique used is a combined Gauss-Newton and modified Newton gradient descent algorithm that minimizes both the reconstruction error and the mode amplitudes. | The regression technique used is a combined Gauss-Newton and modified Newton gradient descent algorithm that minimizes both the reconstruction error and the mode amplitudes.<ref>J.A. Alonso, J.L. Velasco, I. Calvo, T. Estrada, J.M. Fontdecaba, J.M. García-Regaña, J. Geiger, M. Landreman, K.J. McCarthy, F. Medina, B.Ph. Van Milligen, M.A. Ochando, F.I. Parra, the TJ-II Team and the W-X Team, ''Parallel impurity dynamics in the TJ-II stellarator'', [[doi:10.1088/0741-3335/58/7/074009|Plasma Phys. Control. Fusion '''58''' (2016) 074009]]</ref> | ||
== An example == | == An example == | ||
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[[File:18370.gif|center|A rotating m=3 mode observed at TJ-II]] | [[File:18370.gif|center|A rotating m=3 mode observed at TJ-II]] | ||
== See also == | |||
* [[TJ-II:Pellet injector]] | |||
== References == | |||
<references /> | |||