TJ-II:Tomography - Revision history

Admin at 13:42, 20 February 2017

2017-02-20T13:42:38Z

← Older revision		Revision as of 15:42, 20 February 2017
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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 />

Admin at 09:54, 26 September 2014

2014-09-26T09:54:47Z

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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:~~Fourier\%E2\%80\%93Bessel\_series~~\|Fourier-Bessel]] functions satisfy this requirement.		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 ==

Admin at 18:00, 21 February 2010

2010-02-21T18:00:28Z

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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 ''n'' = 0 terms:		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>

Admin: /* An example */

2010-02-21T17:56:49Z

An example

← Older revision		Revision as of 19:56, 21 February 2010
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	== An example ==		== An example ==

	The rotation of an ''m'' = 3 mode observed by applying the above reconstruction algorithm to the high-pass filtered [[TJ-II:Bolometry\| multichannel bolometry]] ~~signals~~ at [[TJ-II]]:		The rotation of an ''m'' = 3 mode (or [[Magnetic island\|magnetic island]]) observed by applying the above reconstruction algorithm to the high-pass filtered [[TJ-II:Bolometry\| multichannel bolometry]] signals at [[TJ-II]]:

	[[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]]

Admin: /* An example */

2009-10-21T08:38:23Z

An example

← Older revision		Revision as of 10:38, 21 October 2009
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	The rotation of an ''m'' = 3 mode observed by applying the above reconstruction algorithm to the high-pass filtered [[TJ-II:Bolometry\| multichannel bolometry]] signals at [[TJ-II]]:		The rotation of an ''m'' = 3 mode observed by applying the above reconstruction algorithm to the high-pass filtered [[TJ-II:Bolometry\| multichannel bolometry]] signals at [[TJ-II]]:

	[[File:18370.gif\|center\|A rotating m=2 mode observed at TJ-II]]		[[File:18370.gif\|center\|A rotating m=3 mode observed at TJ-II]]

Admin: Created page with 'A tomography program has been developed at TJ-II for the reconstruction of the local emission of radiation from a set of line-integrated measurements in a single poloidal pla…'

2009-10-20T09:26:41Z

Created page with 'A tomography program has been developed at TJ-II for the reconstruction of the local emission of radiation from a set of line-integrated measurements in a single poloidal pla…'

New page

A tomography program has been developed at [[TJ-II]] for the reconstruction of the local emission of radiation from a set of line-integrated measurements in a single poloidal plane (φ = constant).
The local emission ''E'' is expressed as a function of the vacuum [[Toroidal coordinates|flux coordinates]] ''(ρ,θ)'' (from [[VMEC]]) in the form of a series expansion:

:<math>E(\rho,\theta) = \sum_{n,m}{C_{nm}f_{n}(\rho) \exp(im\theta)}\,\!</math>

Separating out the ''n'' = 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>

For an optimal reconstruction process, one would like the expansion functions to form an orthogonal set (thus avoiding statistical collinearities). The orthogonality should apply with respect to the ''area integral'', since this is what contributes to the measured emission. This leads to:

:<math>\int_0^1{f_{n}(\rho)f_{n'}(\rho) \rho d\rho} = c_n \delta_{nn'}</math>

where δij is the Kronecker delta. The [[:Wikipedia:Fourier\%E2\%80\%93Bessel\_series|Fourier-Bessel]] functions satisfy this requirement.

== The reconstruction algorithm ==

The local emission ''E(ρ,θ)'' is calculated on a sufficiently fine grid ''(ρi,θj)'', ''i = 1,...,Nρ, j = 1,...,Nθ''. The emission of a grid element (with area ''daij'') to detector number ''ν'' is evaluated taking into account the distance ''dνij'' of the grid element ''(i,j)'' to the detector:

:<math>I^\nu_{ij} = \frac{E(\rho_i,\theta_j)}{(d^\nu_{ij})^2} \delta^\nu_{ij} da_{ij}</math>

Of course, only grid elements lying inside the detector beam viewing angle are contemplated;
this is expressed by the factor ''δνij'' which equals 1 when the grid element lies inside the beam, and 0 otherwise (partial overlap between the viewing beam and the grid element is not contemplated since the grid is assumed to be sufficiently fine).
By integrating over all grid elements ''(i,j)'', the total emission received by detector ''ν'', ''Iν'' can be calculated.

Since the local emission ''E(ρ,θ)'' is expanded as a series, this integration can be performed for each mode separately. Labeling the modes ''(n,m)'' by a single index ''μ'', we can write the relation between mode amplitudes and detected intensities as:

:<math>I_\nu = A_{\nu\mu} C_\mu \,\!</math>

where the ''Cμ'' 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 ''Iν'' are assigned weights ''wν'' inversely proportional to the square of the measurement error of measurement ''Iν''. Thus, any channel ''ν'' can be excluded from contributing to the result by setting ''wν'' = 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.

== An example ==

The rotation of an ''m'' = 3 mode observed by applying the above reconstruction algorithm to the high-pass filtered [[TJ-II:Bolometry| multichannel bolometry]] signals at [[TJ-II]]:

[[File:18370.gif|center|A rotating m=2 mode observed at TJ-II]]