Results with all measurements

Figure 3.10: Linear and log plot of eigenvalue (variance) versus PC index for all available magnetic measurements
Figure 3.11: The leading pair of eigenvectors as an impulse plot for all 8 experimental signals (diamagnetic coil near $\phi =0^\circ$ and $\phi =36^\circ$, the 3 saddle coils, the $\cos(2\theta)$ coil and $\cos(m=1)$ and $\cos(m=2)$/$\cos(m=1)$ from the poloidal field coil array, respectively)

To begin with, we present results by showing the recovery accuracy attained in database regressions using all the simulated magnetic diagnostic signals. In addition to the coil current ratios and limiter position, the predictor set used thus consists of signals from the diamagnetic coils, the 3 independent saddle coils, the $\cos(2\theta)$ coil and two signals from the poloidal field coil array, $\cos(m=1)$ and $\cos(m=2)$/$\cos(m=1)$. This set is bound to contain redundancies (e.g., many of the signals depend strongly on $W_{\mathrm{p}}$) which unnecessarily inflates the model size and may have a negative impact on the stability of regressions as explained previously. We thus perform a principal component analysis (PCA) on the magnetic signals, transforming the original measurements to principal components (PCs) which are statistically independent within the database sample. It is unnecessary to include the current ratios and limiter position in the PCA, as these are independent of the plasma measurements due to the random database design, and any accidental correlations within the database are merely due to the finite number of cases. We remark that it is convenient to base the PCA on the correlation matrix, since this effectively means that each measurement is normalized to its own spread and is thus dimensionless. This neatly side-steps the requirement in a covariance-based PCA of assigning weights to the measurements to account for their differing dimensions (here, most are in Weber but the $\cos(2\theta)$ signal is a line integrated quantity and $\cos(m=2)$/$\cos(m=1)$ is already dimensionless). A covariance-based PCA could be more suited to the situation where all measurements were of identical type and dimension, since it is not invariant under transformations of the measurements to different systems of units.

The pattern of eigenvalues (or variances) versus PC index is shown in figure 3.10. We find that there are essentially two large leading PCs, in qualitative agreement with the findings of Jiminez et al [55]. The eigenvalues decay roughly exponentially with increasing PC index and are listed in table 3.6. For completeness, table 3.7 gives the eigenvectors (weightings of each signal) of the PCs.

Table 3.6: PCA eigenvalues for all magnetic signals

PC#	Eigenvalue	% of total variance
1	6.99426	87.428
2	0.80563	97.499
3	0.10464	98.807
4	0.06247	99.588
5	0.02685	99.923
6	0.00539	99.991
7	0.00065	99.999
8	0.00011	100.00

Table 3.7: PCA eigenvectors for all magnetic signals

Signal	PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8
$s_{\mathrm{1}}$	0.365	-0.154	0.350	0.770	0.162	0.219	0.203	0.111
$s_{\mathrm{2}}$	0.365	0.003	0.741	-0.306	-0.360	-0.225	-0.180	-0.105
Loop3	0.376	0.032	-0.248	0.056	0.175	-0.813	0.148	0.282
Loop2	0.375	-0.074	-0.306	0.076	-0.086	-0.032	0.083	-0.859
Loop1	0.373	-0.090	-0.274	-0.224	-0.534	0.325	0.473	0.336
$\cos(2\theta)$	0.370	-0.120	0.074	-0.492	0.714	0.289	0.083	0.0277
$\cos(m=1)$	0.375	-0.084	-0.305	0.098	-0.093	0.174	-0.816	0.2122
$\cos(m=2)$/$\cos(m=1)$	0.186	0.970	-0.004	0.054	0.045	0.141	0.018	-0.004

Physically interpreting the PCs is not normally straightforward, however in this case the first is almost perfectly correlated (99.5%) with the plasma energy content. Examining the eigenvector structure, we find that the first PC weights every signal roughly equally except for the $\cos(m=2)/\cos(m=1)$ signal. This is reasonable, since the magnitude of the plasma magnetic signals increases in proportion to $W_{\mathrm{p}}$, whereas the latter signal ratio has this dependence normalized out through the 1/$\cos(m=1)$ factor. The second PC gives a low weighting to all other signals, being almost perfectly correlated with $\cos(m=2)/\cos(m=1)$. This confirms the presence of independent information in this signal ratio as shown in the earlier correlation analysis. We speculate that the second PC may be linked with the radial position of the plasma magnetic axis, since $\cos(m=2)/\cos(m=1)$ was shown in the analytic model of section 3.4 to be related to the centroid of the Pfirsch-Schlüter currents.

Knowledge of expected noise levels in the diagnostic signals enables the selection of a cut-off point, beyond which PCs can be neglected, since any information they hold is too small to be reliably resolved from the noise. However, we postpone the choice of cut-off until after preliminary examination of the effects of varying the numbers of retained PCs and noise levels for several parameters. We regard the baseline error for any parameter as the recovery error for a model containing only the experimentally measurable vacuum configuration parameters ( $i_{\mathrm{s}}$, $i_{\mathrm{tor}}$, $i_{\mathrm{vert}}$, ${Z_{\mathrm{lim}}}$ and no PCs of the magnetic measurements) -- this provides a consistent reference for subsequent improvement in parameter identification as plasma information is added to the model in the form of PCs of the magnetic signals. The plots below show the percentage recovery error for the baseline model containing the vacuum information together with differing numbers of PCs (up to the maximum of 8) and with simulated random Gaussian noise levels of 0-20%. Where the recovery error drops sharply once the first PC is added to the baseline model, we have truncated the plot to increase detail on the recovery accuracy with 1 or more retained PCs.

Figure: Recovery RMSE for ${r_{\mathrm{eff}}}$ (left) and ${R_{\mathrm{geo}}}$ (right)

Figure: Recovery RMSE for $W_{\mathrm{p}}$ (left) and $\beta_{\mathrm{axis}}$ (right)

The recovery error for the plasma effective radius ${r_{\mathrm{eff}}}$ and the average major radius of the geometric centre of the boundary flux surface ${R_{\mathrm{geo}}}$ is shown in figure 3.12. Geometric parameters such as these which relate to the bulk plasma are quite robustly recoverable even with few PCs and with high levels of signal noise. Other quantities such as $W_{\mathrm{p}}$ (figure 3.13, left), which depend only on the volume-integrated magnetic field, are also well recovered in the database. Indeed, the error for $W_{\mathrm{p}}$ drops to 2% for a single added PC, which is consistent with the findings of the correlation analysis above. Moreover, a model with a single PC where the signals contain 20% random noise still predicts $W_{\mathrm{p}}$ to better than 10% accuracy.

However, on the right of figure 3.13 is shown the error for a parameter local to the plasma core, the $\beta$ value at the magnetic axis. In contrast with $W_{\mathrm{p}}$, the error stagnates at around 30% for models with 2 or more PCs. This is due to the fact that the local $\beta$ value is derived from the pressure profile shape, which is extremely difficult to infer from remote magnetic measurements. This behaviour is also true of other parameters linked to the pressure profile shape, a particular example being the rotational transform $\iota$, which depends on the local pressure gradient. The sharp rise in error for 0-2% noise that is evident for models with 5 or more PCs is due to the rapid noise degradation of the information contained in the weaker PCs.

Figure 3.14: Recovery RMSE for $R_{0,0}$ (left) and $R_{0,1}$ (right)
Figure 3.15: Recovery RMSE for $R_{0,2}$ (left) and $Z_{0,1}$ (right)

Figure 3.16: Recovery RMSE for $R_{1,0}$ (left) and $Z_{1,0}$ (right)
Figure 3.17: Recovery RMSE for $R_{2,0}$ (left) and $Z_{2,0}$ (right)

Turning our attention to the 3-D flux surface recovery, we present a selection of similar plots for some leading-order Fourier coefficients $R_{0,0}, R_{0,1}, R_{0,2}$ and $Z_{0,1}$ which relate to the position of the flux surface centroids ($Z_{0,0}$ vanishes due to symmetry), $R_{1,0}$ and $Z_{1,0}$ which describe the average flux surface ellipticity, and $R_{2,0}$ and $Z_{2,0}$ which describe the average triangularity. Since these are the coefficients with the largest magnitudes, they form the skeleton of the geometry and are correspondingly important for the accuracy of the overall flux surface reconstruction. We remark that due to the model used, the fit is made simultaneously over all radii, thus the error here is an average figure. Investigation of the radial behaviour of the error reveals that the error is maximal towards the magnetic axis ($s=0$) and minimal towards the boundary ($s=0$). This is because the external magnetic measurements react only to the integrated field and thus do not give precise information regarding the centre of the plasma.

Figure 3.14 and 3.15 show the errors for the leading order $m_{\mathrm{pol}}=0$ Fourier coefficients. These are particularly important in the reconstruction as they represent the correct centring of the flux surfaces. They are strongly influenced by the $\beta$-induced Shafranov shift, which is dependent on the local pressure. Their associated errors are quite reasonable at between 5% and 8% for models with 2 or more PCs and noise levels around 5%. It is noteworthy that the most dramatic improvement results from the addition of just the first two PCs, again in accordance with [55].

Figure 3.16 shows the $m_{\mathrm{pol}}=1$ coefficient recovery. Because these describe the poloidal dimensions of the flux surface, they are constrained by the limiter position and are thus already well determined with no PCs added to the baseline model. This is equivalent to the statement that the normalized differential volume element does not deviate strongly from unity, or if $V(s)$ and $V_\mathrm{max}$ are the flux surface volume and its maximum, that $s \approx V(s)/V_\mathrm{max}$.

In figure 3.17, good recovery of the $m_{\mathrm{pol}}=2$ coefficients is quite evident, again with the strongest gain in accuracy achieved for just two PCs.

Having examined the behaviour of recovery error for some of the important parameters, we are in a position to choose both an appropriate cut-off point for retaining PCs and a suitable level of noise to use in regressions. In general, both scalar parameters and also coefficients in the Fourier decomposition of the flux surface geometry continue to show improved recovery errors up to 5 retained PCs and this improvement remains quite stable in the presence of noise up to the maximum considered level of 20%. Noting that the biggest impact of noise on the recovery is invariably in the 0-2% range, we select 5% as a safe level of noise to stabilize the recovery without placing over-optimistic demands on the accuracy of the experimental signals. Note that the number of PCs required varies for different parameters, so in some cases, we could reduce the model size further. The presence of noise in the regressions means that any extra parameters will not destabilize the recovery. The leading 5 PCs account for over 99.9% of the total variance in the magnetic signals, with a signal-to-noise variance ratio of 11 for the weakest retained PC.

Table 3.8 summarizes the parameter recovery errors using our chosen model with 5 PCs and 5% noise. For comparison, we also show results for just 2 PCs with the same noise level. We stress that due to the model used, the recovery of the Fourier coefficients is performed simultaneously for all $s$ values, therefore the figures quoted for each coefficient are profile averaged.

Table 3.8: Summary statistics of FP for 2 and 5 PC models with 5% noise

General				2 PCs		5 PCs
Parameter	Units	Mean	Spread	RMSE	$\varepsilon$(%)	RMSE	$\varepsilon$(%)
$V$	$m^3$	0.9684	0.1992	0.0052	2.63	0.0046	2.11
$\beta_{\mathrm{axis}}$	%	1.808	0.911	0.296	32.47	0.288	30.35
$W_{\mathrm{p}}$	kJ	6.400	3.600	0.070	1.95	0.064	1.67
${r_{\mathrm{eff}}}$	cm	15.511	1.598	0.042	2.63	0.037	2.17
${\Phi_{\mathrm{edge}}}$	mWb	96.57	22.38	0.59	2.65	0.52	2.10
${B_{\mathrm{0}}}$	T	1.2650	0.1004	0.0051	5.04	0.0050	4.78
$R_{0,0}$	cm	205.73	2.26	0.22	9.91	0.19	8.62
$R_{0,1}$	cm	-6.185	0.592	0.063	10.69	0.055	9.32
$R_{1,-2}$	cm	-0.243	0.266	0.010	3.65	0.008	3.05
$R_{1,-1}$	cm	-1.026	1.104	0.029	2.66	0.026	2.33
$R_{1,0}$	cm	7.872	8.453	0.047	0.56	0.040	0.48
$R_{1,1}$	cm	3.453	3.706	0.022	0.61	0.019	0.51
$R_{1,2}$	cm	0.568	0.617	0.016	2.53	0.013	2.14
$R_{2,-2}$	cm	-0.0085	0.0139	0.0019	13.35	0.0016	11.30
$R_{2,-1}$	cm	-0.2812	0.3310	0.0065	1.97	0.0055	1.65
$R_{2,0}$	cm	0.581	0.681	0.043	6.25	0.037	5.40
$R_{2,1}$	cm	0.3298	0.3836	0.0085	2.22	0.0070	1.84
$R_{2,2}$	cm	0.2142	0.2504	0.0054	2.17	0.0044	1.74
$Z_{0,1}$	cm	-1.068	0.997	0.073	7.35	0.064	6.45
$Z_{1,-2}$	cm	-0.119	0.161	0.012	7.26	0.01	5.93
$Z_{1,-1}$	cm	-0.732	0.872	0.033	3.73	0.029	3.28
$Z_{1,0}$	cm	14.83	15.86	0.12	0.74	0.10	0.65
$Z_{1,1}$	cm	-3.208	3.425	0.046	1.35	0.041	1.20
$Z_{1,2}$	cm	-0.511	0.561	0.010	1.85	0.009	1.58
$Z_{2,-2}$	cm	0.0243	0.0345	0.0030	8.71	0.0026	7.56
$Z_{2,-1}$	cm	-0.095	0.108	0.010	9.25	0.009	8.02
$Z_{2,0}$	cm	-0.481	0.559	0.021	3.70	0.018	3.21
$Z_{2,1}$	cm	-0.273	0.328	0.008	2.57	0.006	1.91
$Z_{2,2}$	cm	-0.1163	0.1401	0.0043	3.05	0.0033	2.33

Figure 3.18: Flux surface reconstruction in $\phi =0^\circ ,18^\circ ,36^\circ$ planes (FP surfaces solid, NEMEC surfaces dashed) with overall RMSE (mm) versus $s$

The FP coefficients generated above were subsequently used in parameter recoveries on a smaller, independent set of 80 test equilibria which were not used in the FP regressions. It was verified that the recovery RMSE was similar over both databases, thereby demonstrating adequate robustness of the FP regressions.

It is difficult to visualize how the Fourier coefficient errors relate to real errors in the flux surfaces (i.e. deviations of the FP recovery from the NEMEC flux surfaces), thus in figure 3.18 we show a reconstruction of an equilibrium from outside the database using its simulated magnetic data against the NEMEC calculation in 3 different poloidal planes ( $\phi =0^\circ ,18^\circ ,36^\circ$) along with the overall deviation from the NEMEC flux surfaces versus $s$. The parameters for this equilibrium were: $I_{\mathrm{mod}}$ = 15.5 kA, $I_{\mathrm{s}}$ = 18.47 kA, $I_{\mathrm{tor}}$ = 2.17 kA, $I_{\mathrm{vert}}$ = 158 A, ${Z_{\mathrm{lim}}}$ = 28.47 cm, $V = 1.08~\mathrm{m}^3$, ${r_{\mathrm{eff}}}$ = 16.2 cm, $W_{\mathrm{p}}$ = 15.4 kJ, $\beta_{\mathrm{axis}}$ = 2.54%. The RMS deviation of the flux surfaces are representative of the average in the database, which decreases from roughly 7 mm at the centre to 1.5 mm at the edge. We remark that the time taken for reconstruction of the Fourier coefficients of the 11 flux surfaces illustrated here (for $s = 0.0, 0.1, \ldots, 1.0$) is less than 10 ms on a 300MHz UltraSPARC workstation, compared to the 60 minutes of Cray CPU time required for the NEMEC calculation itself. The reconstructions were performed with a hand-optimized computer code written in C++ that was written specifically for the task.