Results with selected measurements

It is clear from the PCA of all the available magnetic measurements in figure 3.10 that much of the variance is explained by just the first two PCs. The eigenvector weightings in figure 3.11 indicate that the first PC is a measure of the plasma energy content (or equivalently $\cos(m=1)$, the dipole moment of the Pfirsch-Schlüter currents) and the second is determined by the $\cos(m=2)/\cos(m=1)$ signal ratio. We redo the earlier regressions using just these predictors in order to demonstrate that comparitively little information is lost.

In order to confirm that our choice of signals is favourable, we also tabulate recovery results for a model with the largest practical number of PCs of the individual poloidal field coil signals; it turns out that 3 PCs can be retained. As before, we include 5% pseudo-random measurement noise throughout. Table 3.9 shows that despite the extra fitted parameter of the 3 PC model over the $\cos(m=1)$ and $\cos(m=2)/\cos(m=1)$ model, the results are on average slightly inferior to the latter case. This is particularly noticeable for the flux surface Fourier coefficients. We remark that it is more straightforward in practice to measure the cosine moments in hardware, since constructing them by software post-processing places higher demands on the accuracy of the individual signal measurements. It also requires the digitizing and storage of fewer signals and thus a more compact diagnostic setup.

Table: Summary statistics for reduced models with (a) 3 PCs of the poloidal field measurements and (b) $\cos(m=1)$ signal and $\frac{\cos(m=2)}{\cos(m=1)}$ ratio, with 5% noise in both cases

General				3 PCs of $B_\theta$ signals		$\cos(m=1)$ & $\frac{\cos(m=2)}{\cos(m=1)}$
Parameter	Units	Mean	Spread	RMSE	$\varepsilon$(%)	RMSE	$\varepsilon$(%)
$V$	$m^3$	0.9684	0.1992	0.0052	2.63	0.0051	2.56
$\beta_{\mathrm{axis}}$	%	1.808	0.911	0.308	33.82	0.00299	32.79
$W_{\mathrm{p}}$	kJ	6.400	3.600	0.104	2.88	0.112	3.10
${r_{\mathrm{eff}}}$	cm	15.511	1.598	0.043	2.69	0.041	2.56
${\Phi_{\mathrm{edge}}}$	mWb	96.57	22.38	0.57	2.57	0.58	2.59
${B_{\mathrm{0}}}$	T	1.2650	0.1004	0.0052	5.23	0.0051	5.08
$R_{0,0}$	cm	205.73	2.26	0.24	10.62	0.23	9.99
$R_{0,1}$	cm	-6.185	0.592	0.067	11.32	0.063	10.75
$R_{1,-2}$	cm	-0.243	0.266	0.010	3.81	0.010	3.60
$R_{1,-1}$	cm	-1.026	1.104	0.031	2.79	0.029	2.62
$R_{1,0}$	cm	7.872	8.453	0.049	0.58	0.047	0.56
$R_{1,1}$	cm	3.453	3.706	0.023	0.62	0.022	0.60
$R_{1,2}$	cm	0.568	0.617	0.016	2.60	0.015	2.44
$R_{2,-2}$	cm	-0.0085	0.0139	0.0019	13.93	0.0019	13.56
$R_{2,-1}$	cm	-0.2812	0.3310	0.0072	2.14	0.0066	2.00
$R_{2,0}$	cm	0.581	0.681	0.046	6.70	0.042	6.24
$R_{2,1}$	cm	0.3298	0.3836	0.0086	2.25	0.0085	2.22
$R_{2,2}$	cm	0.2142	0.2504	0.0058	2.30	0.0055	2.20
$Z_{0,1}$	cm	-1.068	0.997	0.076	7.61	0.074	7.40
$Z_{1,-2}$	cm	-0.119	0.161	0.012	7.31	0.011	7.02
$Z_{1,-1}$	cm	-0.732	0.872	0.033	3.80	0.032	3.64
$Z_{1,0}$	cm	14.83	15.86	0.13	0.80	0.12	0.74
$Z_{1,1}$	cm	-3.208	3.425	0.049	1.42	0.046	1.34
$Z_{1,2}$	cm	-0.511	0.561	0.011	1.88	0.010	1.82
$Z_{2,-2}$	cm	0.0243	0.0345	0.0032	9.38	0.0030	8.81
$Z_{2,-1}$	cm	-0.095	0.108	0.011	9.88	0.0010	9.23
$Z_{2,0}$	cm	-0.481	0.559	0.022	3.91	0.021	3.70
$Z_{2,1}$	cm	-0.273	0.328	0.008	2.56	0.008	2.62
$Z_{2,2}$	cm	-0.1163	0.1401	0.0044	3.15	0.0044	3.11

This is a good indicator that for equilibrium reconstructions based on magnetic measurements, the diagnostic setup of choice would be an array of local probe measurements, possibly with an additional diamagnetic coil to provide an independent check of the plasma energy.