Response Matrix Generation

The response matrix describes the relationship between the n inversion grid cells and m line integrated measurements given by the demodulated brightness and ¯v images. We wish to construct response matrices for both emissivity and flow recon-structions, which will be denoted E and V respectively. The measurements were typically binned 4x4 (to 256x256 pixels) before inversion, to reduce the size of the required response matrix and therefore the computational requirements of the in-version process. Due to the spatial resolution of the instrument discussed in section 4.4, no spatial resolution is lost in the direction perpendicular to the fringes due to this binning, although spatial resolution is lost in the direction parallel to the fringes.

4.6. Tomographic Inversion 88

R (m)

Z (m)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6

Figure 4.11: Poloidal cross-section of the MAST divertor showing a typical recon-struction grid for emissivity and flow tomography. The grey grid is the reconstruc-tion grid, and the blue curved lines are camera sight-lines from a particular image column projectsed on to the grid. The blue shaded area represents the approximate shape of the divertor plasma from which emission is expected to be observed.

4.6. Tomographic Inversion 89 In calculating the system response the finite depth-of-field of the imaging system is ignored, and each line of sight is assumed to collect light from a narrow ‘pencil’

beam through the plasma, with the collection power constant along the line-of-sight. This is a reasonable approximation for inversions of the MAST CIS images, because 1) After the 4x4 binning of the images, the binned pixel size is larger than the expected blur radius for out-of-focus areas of the plasma, and 2) Due to the tangential view of the plasma, the pixels contributing most information to the inversion are dominated by light emitted close to the in-focus object plane.

Given this pencil beam model, and without concern for absolute calibration, the i^th brightness measurement is given by:

where e_0,j is the plasma emissivity in the j^th inversion grid cell and L_ij is the length of the i^th sight-line which falls within that grid cell. This equation constitutes one row of the response matrix, which is given by E_ij = L_ij if all detector pixels have the same effective collecting power (or equivalently if the data images are flat field corrected before inversion). The response matrix is calculated by finely discretising each sight-line j into line segments along its length and calculating the R, Z coordinates of each segment’s centre point, to determine which reconstruction cell the segment falls within. The length L_ij is then given by L_ij =P

kl_k, where the sum is over all sight-line segments k which fall within the j^th grid cell and l_k is the length of the k^th line segment. Since a simplified vacuum vessel geometry is used in the sight-line calculation, some sight-lines which are unobstructed in the sight-line model are in fact blocked by in-vessel components in the real measurements. To account for this, image masks are created manually which indicate which sight-lines are affected by this. These sight-lines are then excluded from the fitting process, i.e. they are not included as rows in the response matrix. In addition some grid cells do not encounter any diagnostic sight-lines and are completely unconstrained (e.g. some grid cells fall within machine components or within vessel walls). While these are not explicitly excluded from the matrix generation or fitting, they are represented by a row of zeros in the response matrix and are not involved in the inversion. The brightness response matrix is only a function of the viewing geometry, and therefore needs to be recalculated only when the viewing geometry is changed.

For flow reconstruction the instrument response is more complex, as we see from equation (4.2.4) that the line-integrated flow measurement depends on both the magnitude and direction of the local flow, in addition to the local emissivity. With

4.6. Tomographic Inversion 90 a single camera, insufficient information is available to recover both the magnitude and 3D direction of the flow. Instead, we assume that the impurity flow is primarily parallel to the magnetic field due to the confining effect of the field, i.e. v ≈ v||B,ˆ where v||is the parallel flow speed and ˆB is a unit vector in the direction of the local magnetic field. Since the magnetic field on MAST is routinely reconstructed from magnetic probe measurements using the EFIT code, this can provide a constraint for the flow direction and we only have to solve for the scalar quantity v||. This is the same approach as used in Howard et al. [2010a]. The response matrix elements can then be calculated as:

V_ij = eo,j

Pn

j⁰=1E_ij⁰e_0,j⁰ X

k

Bˆ_k· l_k, (4.6.6)

where again the sum over k is over each sight-line segment falling within the j^th reconstruction grid cell, l_k is the vector line segment along the line of sight, and Bˆ_k is a unit vector in the direction of the magnetic field at the centre point of l_k. Note that the emissivity e₀ and magnetic field direction ˆB_k are time-varying properties of the plasma; this means that individual flow response matrices must be generated for each image we wish to invert. The values of e0 are obtained from inversion of the emissivity, and therefore any reconstruction errors in the emissivity cause inaccurate calculation of the flow response matrix leading to further errors in the flow inversion. Furthermore, not all measurement sight-lines from the line-integrated image contain useful flow information. Specifically, sight-lines where no light is detected cannot carry any spectral information, and must be excluded from the fitting (along with sight-lines with sufficiently low SNR that the flow information is dominated by noise). Therefore only sight-lines which show brightness above a threshold value are included when generating the response matrix.