Recovery of the pressure PCs

It is instructive to examine the recovery of individual principal components of the pressure profile in order to better understand the results above. The leading 4 eigenvectors of the pressure profile PCA were previously shown in figure 4.2, where they have been scaled to their standard deviation (i.e. the square root of their eigenvalues) to give a correct impression of their relative magnitude. As described in chapter 4, they are polynomial-like, with the first representing the average profile, the second the peakedness or hollowness (one zero), the third a higher order modulation with two zeroes and so on. We remark again that the first principal component is almost perfectly correlated with the plasma kinetic energy content.

In the same way as with the pressure itself, we can model the pressure profile PCs with the PCs of the dense magnetic measurements. The results for the same model of the vacuum parameters with 9 PCs of the magnetic measurements (1% noise) are illustrated in table 5.2, where the PC mean is omitted since it is zero by construction.

Table 5.2: Recovery statistics for the leading pressure PCs

PC#	Spread	RMSE	$\mathrm{R}^2$
1	11.70	0.24	0.9996
2	3.53	0.38	0.9884
3	2.37	0.86	0.8681
4	1.14	0.97	0.2844

As expected, the first PC is recovered almost perfectly, however the second PC already has an $\mathrm{R}^2$ of 0.9884 which corresponds to an 11% error. Beyond this, the PCs are essentially undetermined. Thus, we can conclude that although there is reasonable success in diagnosing peakedness or hollowness, there is little chance of detecting higher-order structure on the pressure profile in the database, even using extremely dense and practically noise-free external magnetic measurements. The situation can only deteriorate in an experimental scenario, where the measurements will inevitably be both fewer and more noisy.

It is interesting to draw an analogy with the situation on tokamaks for elongated plasmas, where it is possible to separately diagnose two profile parameters, the poloidal beta $\beta_\mathrm{p}$ (the ratio of plasma pressure to the magnetic field pressure from the plasma current) and internal inductance $l_\mathrm{i}$ (a parameter describing the peakedness of the current profile) from external measurements. Our equilibrium database investigations for W7-AS show parallels in that the two leading PCs of $p(s)$ are recoverable with the magnetic measurements. From figure 4.2, it is clear that the first PC is proportional to the energy or $\beta_\mathrm{p}$ and the second represents a peaking or broadening of the average database profile form.