# TJ-II:Tomography

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 flux coordinates *(ρ,θ)* (from VMEC) in the form of a series expansion:

- $ E(\rho,\theta) = \sum_{n,m}{C_{nm}f_{n}(\rho) \exp(im\theta)}\,\! $

Separating out the *m* = 0 terms:

- $ 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 ]} $

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:

- $ \int_0^1{f_{n}(\rho)f_{n'}(\rho) \rho d\rho} = c_n \delta_{nn'} $

where δ_{ij} is the Kronecker delta. The Fourier-Bessel functions satisfy this requirement.

## The reconstruction algorithm

The local emission *E(ρ,θ)* is calculated on a sufficiently fine grid *(ρ _{i},θ_{j})*,

*i = 1,...,N*. The emission of a grid element (with area

_{ρ}, j = 1,...,N_{θ}*da*) to detector number

_{ij}*ν*is evaluated taking into account the distance

*d*of the grid element

^{ν}_{ij}*(i,j)*to the detector:

- $ I^\nu_{ij} = \frac{E(\rho_i,\theta_j)}{(d^\nu_{ij})^2} \delta^\nu_{ij} da_{ij} $

Of course, only grid elements lying inside the detector beam viewing angle are contemplated;
this is expressed by the factor *δ ^{ν}_{ij}* 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*,

*I*can be calculated.

_{ν}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:

- $ I_\nu = A_{\nu\mu} C_\mu \,\! $

where the *C _{μ}* 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*are assigned weights

_{ν}*w*inversely proportional to the square of the measurement error of measurement

_{ν}*I*. Thus, any channel

_{ν}*ν*can be excluded from contributing to the result by setting

*w*= 0.

_{ν}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.^{[1]}

## An example

The rotation of an *m* = 3 mode (or magnetic island) observed by applying the above reconstruction algorithm to the high-pass filtered multichannel bolometry signals at TJ-II:

## See also

## References

- ↑ 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*, Plasma Phys. Control. Fusion**58**(2016) 074009