FP using kinetic data

Overall, the procedure is very similar to that already used in chapter 3. Instead of magnetic measurements forming part of the set of independent predictor variables, we use parameters describing ${p_{\mathrm{eq}}}(s)$, the equilibrium pressure profile as a function of normalized toroidal flux. The parameterization in terms of $({P_{\mathrm{axis}}}, a,b,c)$ chosen for the generation of the FP database (equation 3.1), although compact, is an unsuitable choice of parameter set for describing the plasma information in FP models, since their relationship with the pressure profile $p(s)$ is highly non-linear. Using them would add an avoidable burden to the task of the interpretive algorithm, which we prefer to keep as simple as possible. Instead, predictors are constructed by carrying out a PCA on $p(s)$ evaluated on a radial $s$-grid and retaining enough principal components (PCs) to reconstruct $p(s)$ to a level of accuracy comfortably within that ultimately achieved by the interpretive procedure.

Once again the current ratios and limiter position are excluded from the PCA for the same reasons as given in chapter 3, that they are independent of $p(s)$ and any chance correlations thus purely due to the finite size of the database. We remark that, since all of the predictors have the same dimension and are measurements of the same quantity over the profile, it is appropriate to base the PCA on the covariance matrix rather than the correlation matrix used for the (mixed) magnetic measurements. This gives a more natural weighting to the eigenvectors and favours absolute rather than relative change in the pressure.

Figure 4.1: $\rm{Log}_{10}$ plot of eigenvalue magnitude versus PC number for the first few pressure profile PCs

Figure 4.2: The leading 4 eigenvectors of the pressure PCA, scaled by the square root of the eigenvalue, as a function of normalized toroidal flux

Table 4.1: PC eigenvalue magnitude and cumulative percentage of total variance explained for the first few pressure profile PCs

PC #	Eigenvalue [kPa$^2$]	% variance
1	136.834	87.552
2	12.429	95.505
3	5.605	99.091
4	1.305	99.926
5	0.095	99.987
6	0.019	99.999
7	0.002	100.000

Examining the $\log_{10}$ plot of PC eigenvalue (variance) for the first few PCs (figure 4.1), we see that the variance decays roughly exponentially with PC index. Ideally, we should retain all PCs with non-zero variance to explain 100% of the variance in the original parameters. However, this is impractical as PCs with very small eigenvalues are highly sensitive to noise and are too unstable to use in parameter recovery with real data, unless the modelling incorporates noise filtering [30].

It is also instructive to look at the eigenvectors themselves: because the variables are of the same dimension and taken on a regular radial grid, they are far easier to interpret than in the case of the PCA of the magnetic measurements. Figure 4.2 shows the leading radial eigenfunctions of the decomposition, scaled by the square root of the eigenvalue to provide a correct magnitude reference. They are polynomial-like in that the $i^\mathrm{th}$ eigenvector intersects the $y$-axis exactly $i$ times. The first eigenvector corresponds to a zeroth order variation (i.e. an average profile), the second superimposes a peaking or broadening and so on.

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)

Figure 4.5: Recovery RMSE for $R_{0,0}$ (left) and $R_{0,1}$ (right)
Figure 4.6: Recovery RMSE for $R_{0,2}$ (left) and $Z_{0,1}$ (right)

Figure 4.7: Recovery RMSE for $R_{1,0}$ (left) and $Z_{1,0}$ (right)
Figure 4.8: Recovery RMSE for $R_{2,0}$ (left) and $Z_{2,0}$ (right)

We are again faced with the decisions of both a cut-off point beyond which we can neglect PCs in the FP models and a suitable level of simulated noise to add to the pressure data for the purposes of stabilizing the regressions. As an aid to determining the cut-off point, we now examine identical parameter recoveries as before in chapter 3. Note that we will never require the experimental determination of the pressure data; they are simply treated as parameters through which we can conveniently vary the equilibrium. We thus need only include a nominal level of noise to ensure that the regressions are well-conditioned and stable.

Figures 4.3 and 4.4 show results for ${r_{\mathrm{eff}}}$, ${R_{\mathrm{geo}}}$, $W_{\mathrm{p}}$ and $\beta_{\mathrm{axis}}$ for models with up to 8 retained PCs and 0-20% noise. Recovery of purely geometric parameters is good, with errors typically of the order of 1-2% of the spread. These are the most stable output parameters from NEMEC, generally suffering less from factors such as poor convergence than other quantities. Integrated magnetic field related quantities are also well recovered with similar accuracy. The error is naturally highest for $\beta_{\mathrm{axis}}$ which is a local quantity and thus relatively difficult to diagnose.

Improvement in the recovery of all quantities is observable up to the addition of the fourth PC. In contrast with the sharp rise in recovery error for the 0-5% added noise range with the magnetic data, here the results degrade only slowly with increasing noise in the pressure data. Nonetheless, we will still include a nominal 1% noise level in regressions. One curiosity in the results is that the error for $W_{\mathrm{p}}$ stagnates at around 2% even for low noise and many retained PCs, whereas this error was substantially lower in the magnetic case at around 0.5% (figure 3.13). Moreover, the corresponding error for $\beta_{\mathrm{axis}}$ is actually marginally lower in figure 4.4, whereas this was virtually undetermined in the magnetic case. Far from being an anomaly, this is merely a consequence of our choice of rather artificial predictor variables. The central $\beta$ value is determined purely by the ratio of the pressure and magnetic field at the axis, both of which are well recovered from the PCs of $p(s)$, so it is not surprising that $\beta_{\mathrm{axis}}$ is almost perfectly recovered. $W_{\mathrm{p}}$, on the other hand, is the integral of the pressure profile with respect to the volume: our pressure predictors are chosen with respect to the normalized toroidal flux, however, and the only volume information is given by ${Z_{\mathrm{lim}}}$. The magnetic data also included diamagnetic signals which are practically linear in $W_{\mathrm{p}}$ by themselves, thus it is plausible that recovery might be better for the magnetic case. We also observe that the error in $W_{\mathrm{p}}$ using the kinetic data is better than that achieved with $\cos(m=1)$ and $\cos(m=2)/\cos(m=1)$, which further reinforces our claims above. Using $p(R)$ to recover $W_{\mathrm{p}}$ produces similar results to the magnetic case, however. In any case, the error here is sufficiently low that our requirement of an accurate equilibrium parameterization is fulfilled.

Results for the flux surface geometry Fourier coefficient recovery are shown in figures 4.5-4.8. The same overall trend is evident, with generally very accurate recovery of quantities for 4 PCs. The maximum recovery error for the Fourier coefficients is typically less than a millimetre, with adjusted $\mathrm{R}^2$ values of about 0.999 or greater, showing that the FP recovery of individual Fourier harmonics is indeed successful. However, it is not immediately obvious how these errors will affect the final flux surface reconstruction since the coefficients must first be summed (see section 2.1.4) and the errors in the summed harmonics will be determined by their cross correlations. The worst case would be for all errors to be perfectly correlated and thus to add: this gives an upper estimate of the RMSE as 1-2 millimetres.

As shown in table 4.1, the first 4 PCs account for over 99.75% of the total variance and we retain only these 4 for use as predictors in the regression analysis. By coincidence, this is also the number of free parameters that were varied in generating the pressure profiles. Table 4.2 summarizes results for the $p(s)$-based FP modelling.

Table 4.2: Summary statistics for $p(s)$-based FP models with 4 PCs with 1% noise

Parameter	Units	Mean	Spread	RMSE	$\varepsilon$(%)
$V$	$m^3$	0.9684	0.1992	0.0033	1.64
$\beta_{\mathrm{axis}}$	%	1.81	0.91	0.02	2.36
$W_{\mathrm{p}}$	kJ	6.400	3.600	0.085	2.37
${r_{\mathrm{eff}}}$	cm	15.511	1.598	0.021	1.30
${\Phi_{\mathrm{edge}}}$	mWb	96.57	22.38	0.40	1.85
${B_{\mathrm{0}}}$	T	1.2650	0.1004	0.0003	0.33
$R_{0,0}$	cm	205.73	2.26	0.03	1.45
$R_{0,1}$	cm	-6.185	0.592	0.007	1.27
$R_{1,-2}$	cm	-0.243	0.266	0.005	1.70
$R_{1,-1}$	cm	-1.026	1.104	0.017	1.51
$R_{1,0}$	cm	7.872	8.453	0.023	0.27
$R_{1,1}$	cm	3.453	3.706	0.015	0.40
$R_{1,2}$	cm	0.568	0.617	0.010	1.66
$R_{2,-2}$	cm	-0.0085	0.0139	0.0012	8.46
$R_{2,-1}$	cm	-0.2812	0.3310	0.0034	1.02
$R_{2,0}$	cm	0.581	0.681	0.013	1.92
$R_{2,1}$	cm	0.3298	0.3836	0.0030	0.80
$R_{2,2}$	cm	0.2142	0.2504	0.0022	0.86
$Z_{0,1}$	cm	-1.068	0.997	0.013	1.29
$Z_{1,-2}$	cm	-0.119	0.161	0.008	4.81
$Z_{1,-1}$	cm	-0.732	0.872	0.025	2.92
$Z_{1,0}$	cm	14.83	15.86	0.037	0.23
$Z_{1,1}$	cm	-3.208	3.425	0.029	0.85
$Z_{1,2}$	cm	-0.511	0.561	0.007	1.23
$Z_{2,-2}$	cm	0.0243	0.0345	0.0013	3.87
$Z_{2,-1}$	cm	-0.095	0.108	0.003	2.85
$Z_{2,0}$	cm	-0.481	0.559	0.004	0.69
$Z_{2,1}$	cm	-0.273	0.328	0.004	1.21
$Z_{2,2}$	cm	-0.1163	0.1401	0.0020	1.45

We remark that the time taken for reconstruction of the Fourier coefficients for 11 flux surfaces (for $s = 0.0, 0.1, \ldots, 1.0$) is less than 10 ms on a 300 MHz UltraSPARC workstation. In terms of floating point operations, reconstruction requires 225 multiplications for each of the 256 Fourier coefficients, or 57,600 multiplications per flux surface. Coupled with the overall accuracy of the regressions, this means that our kinetic-based FP analysis constitutes a very accurate and compact parameterization of the NEMEC equilibria which is ideal for use as the core of an interpretive procedure.