Pressure profile recovery with idealized measurements

Here, we wish to probe the limits of the profile information that can be inferred from external magnetic measurements. This was previously investigated in [54] and [56] using analytic models, however this is the first time to our knowledge that such work has been performed in the context of a large equilibrium database. Our strategy is as follows: the magnetic field is firstly evaluated at many sample points outside the plasma. These data contain much redundancy, however this is necessary to ensure that all of the plasma-induced variations in the external field are captured. A PCA is performed on the large set of (highly correlated) measurements in order to condense the information into a few linear combinations of the raw data and the resulting PCs are used together with the vacuum information to recover $p(s)$ at 21 discrete $s$-values ( $s=0.00, 0.05, \ldots, 1.00$). This is a direct means of testing whether external magnetic measurements are capable of predicting profile information under idealized circumstances, something that was hitherto impossible to show due to the lack of a suitable equilibrium database. The limiting accuracy with which the usual set of equilibrium parameters can be identified with the same predictors is also shown.

A small modification to the DIAGNO code enables the evaluation of all 3 components of the magnetic field for each equilibrium in the existing FP database at points which densely cover the exterior of the vacuum vessel. This information can be thought of as being gathered from a massed array of the ``3-D local probes'' mentioned in [56]. We choose measurements from local probes rather than from larger coils for the reasons outlined in [56], namely that coils with large extent tend to measure (albeit very robustly) only bulk quantities and are ill-suited to resolving fine details of the magnetic field. Note that even with a very dense array of probes such as this, proximity of the probes to the plasma is critical for resolving high order harmonics of the magnetic field. This is because their strength decays as the power of the mode number. We have attempted to be conservative by placing our probes just outside the vacuum vessel, but in fact it is also possible to place coils inside, one example being the array of $B_\theta$ probes on W7-AS which was described in section 2.4.1.

Our initial set of 720 measurements consists of the 3 components of the external magnetic field over one half period of W7-AS (i.e. at 10 values of the toroidal angle, $\phi = 0^\circ,4^\circ,8^\circ \ldots 36^\circ$) and at 24 poloidal locations ( $\theta = 0^\circ,15^\circ,30^\circ \ldots 330^\circ$). We remark that it is only necessary to consider one half period due to the stellarator symmetry. The 3 components are chosen in the radial (normal to the vacuum vessel), poloidal (tangential to the vacuum vessel) and toroidal directions. Note that here, the PCA is based on the covariance rather than the correlation matrix. This is an appropriate choice since all the measurements are of identical type and dimension, so that the PCs preferentially weight signals with higher variance.

The PCA eigenvalue profile of the 720 raw measurements is shown in fig. 5.1, for the noise-free case and also where the measurements were perturbed with 1% and 2% simulated random noise. The observed stagnation of the eigenvalues of the noisy data marks the point where the variance becomes noise-dominated. Strictly speaking, only those PCs with a strong signal to noise variance ratio should be included in parameter models if the regressions are to be robust enough to use with imperfect data. This can be estimated for any PC by comparing its variance to the steady state value of the noisy eigenvalues. However, as this only constitutes an assessment of the extent of $p(s)$-relevant information in the magnetic measurements and is not intended for use outside the database, we include only a nominal 1% level of noise for stability of the regression.

The regression model used to recover $p(s)$ at fixed $s$ points is the standard second order polynomial in the vacuum parameters and the retained magnetic PCs, as described for scalar parameters in section 3.5. The decision of how many PCs to retain was made by considering the trend of the average pressure profile recovery error $\langle \delta_\mathrm{p} \rangle$ versus number of PCs in the model for various levels of noise, shown in fig. 5.2. Here, if RMSE$_{\,i}$ is the usual recovery error of the pressure at the $i^\mathrm{th}$ of $n=21$ $s$-values, then $\langle \delta_\mathrm{p} \rangle$ is defined as:

$$\langle \delta_\mathrm{p} \rangle = \sqrt{ \frac{1}{n} \sum_{i=1}^n (\mathrm{RMSE}_{\,i})^2 }$$ (5.1)

The improvement in $\langle \delta_\mathrm{p} \rangle$ that is evident up to the $9^{\mathrm{th}}$ PC for noise levels up to 6% leads us to retain 9 PCs.

Figure 5.1: PCA eigenvalues for a measurement set consisting of all 3 components of $B$ at 10 values of the toroidal angle $\phi$ and 24 equally spaced points in the poloidal angle $\theta$, with 0%, 1% and 2% noise
Figure 5.2: RMS $p(s)$ profile recovery RMSE $\langle \delta_\mathrm{p} \rangle$ versus retained PCs of the dense magnetic measurements for differing levels of added noise (baseline error with 0 PCs = 2.63 kPa)

Figure 5.3: Left axis: p(s) recovery using baseline model with 9 PCs of the dense magnetic measurements and 1% simulated noise. Error bars indicating the recovery RMSE are superimposed on the average database profile versus $s$, with solid lines indicating the profile spread (1 standard deviation) over the database. Right axis: crosses show % error $\varepsilon$

We show the recovery results for $p(s)$ in fig. 5.3 using the leading 9 PCs of the dense magnetic measurements in our regression models. This is presented as the regression RMSE at each $s$-value about the mean $p(s)$ profile in the database. The solid lines indicate the profile spread (one standard deviation) above and below the mean. On average, $p(s)$ is predicted with reasonably tight error bars, indicating that the profile form is quite well recoverable from the magnetic information alone for our 4-parameter choice of profiles (for further clarification of this see section 5.1.1). The percentage error displays non-monotonic behaviour across the profile, with a local maximum at the plasma core, a minimum near $s=0.1$ and rises steadily towards the boundary. A discussion of two possible explanations for this, one artificial and one physical, is appropriate.

The first is simply that it may be an artifact of the particular parameterization used in generating the database, i.e.  $p(s) = p_0 (1-s^2) \exp(as + bs^2 + cs^3)$. Towards $s=0$, the shape parameters $b$ and $c$ have little or no influence on the profile form and there are thus only 2 degrees of freedom here, whereas towards the $s=1$ all 4 parameters have equal effect. The degree of regularization across the profile thus decreases with $s$ and from this we would expect the percentage recovery error $\varepsilon$ to be relatively flat in the centre and to increase monotonically towards the boundary. This is indeed the case except for at $s=0$ itself, where there is a pronounced local maximum in $\varepsilon$. This is difficult to explain in these terms since there is rigorously only a single varying quantity $p_0$ here, however it may be connected with the fact that the NEMEC radial grid has relatively fewer spatial points here than elsewhere on the profile. Proving the variable regularization of $p(s)$ to be responsible for the rise in error towards the boundary would necessitate cross-checking with another database with a flat ``regularization density'', perhaps a $p(s)$ representation using spline function with knots at fixed $s$ values.

A further complication in $\varepsilon$ behaviour may be due to the trade-off between the proximity of the measurements and the decreasing of the plasma pressure towards the boundary. The influence of the pressure gradient on the external field increases with its absolute value, but also decreases with distance of the flux surface from the measurement contour, so the minimum in $\varepsilon$ may be due to competition between these two effects.

Note that $p(s)$ is completely decorrelated from the vacuum field parameters, thus a baseline regression using only the coil current ratios and limiter position yields error bars as large as the spread of the profile. In contrast, the vacuum parameters determine at least the plasma boundary and so a similar baseline regression for the flux surface Fourier coefficients yields errors which are already much smaller than the spread.

Table 5.1: Comparison of parameter recovery accuracy using a baseline model of only the vacuum parameters (``Vacuum''), a $p(s)$-based model as in chapter 4 (``Kinetic'') and a model with 9 PCs of the magnetic measurements from the idealized dense probe array with nominal 1% simulated noise (``Magnetic'').

General				Vacuum		Kinetic		Magnetic
Variable	Units	Mean	Spread	RMSE	$\varepsilon$(%)	RMSE	$\varepsilon$(%)	RMSE	$\varepsilon$(%)
$V$	m$^3$	0.9684	0.1992	0.0085	4.26	0.0033	1.64	0.0031	1.57
$\beta_{\mathrm{axis}}$	%	1.81	0.91	0.83	91.17	0.02	2.36	0.18	20.19
$W_{\mathrm{p}}$	kJ	6.400	3.600	3.039	84.39	0.085	2.37	0.010	0.26
${r_{\mathrm{eff}}}$	cm	15.511	1.598	0.086	5.39	0.021	1.30	0.026	1.66
${\Phi_{\mathrm{edge}}}$	mWb	96.57	22.38	1.4	6.17	0.40	1.85	0.31	1.40
${B_{\mathrm{0}}}$	T	1.2650	0.1004	0.0155	15.40	0.0003	0.33	0.0029	2.97
$R_{0,0}$	cm	205.73	2.26	1.28	56.69	0.03	1.45	0.10	4.60
$R_{0,1}$	cm	-6.185	0.592	0.357	60.56	0.007	1.27	0.030	5.03
$R_{1,-2}$	cm	-0.243	0.266	0.049	18.37	0.005	1.70	0.005	1.90
$R_{1,-1}$	cm	-1.026	1.104	0.116	10.51	0.017	1.51	0.018	1.65
$R_{1,0}$	cm	7.872	8.453	0.146	1.73	0.023	0.27	0.026	0.31
$R_{1,1}$	cm	3.453	3.706	0.100	2.69	0.015	0.40	0.015	0.40
$R_{1,2}$	cm	0.568	0.617	0.040	6.52	0.010	1.66	0.010	1.67
$R_{2,-2}$	cm	-0.0085	0.0139	0.0046	33.20	0.0012	8.46	0.0012	8.50
$R_{2,-1}$	cm	-0.2812	0.3310	0.0173	5.24	0.0034	1.02	0.0037	1.11
$R_{2,0}$	cm	0.581	0.681	0.114	16.72	0.013	1.92	0.019	2.76
$R_{2,1}$	cm	0.3298	0.3836	0.0344	8.96	0.0030	0.80	0.0043	1.13
$R_{2,2}$	cm	0.2142	0.2504	0.0272	10.87	0.0022	0.86	0.0028	1.11
$Z_{0,1}$	cm	-1.068	0.997	0.396	39.74	0.013	1.29	0.036	3.59
$Z_{1,-2}$	cm	-0.119	0.161	0.024	14.72	0.008	4.81	0.008	4.85
$Z_{1,-1}$	cm	-0.732	0.872	0.049	5.56	0.025	2.92	0.025	2.93
$Z_{1,0}$	cm	14.83	15.86	0.35	2.23	0.037	0.23	0.060	0.38
$Z_{1,1}$	cm	-3.208	3.425	0.128	3.74	0.029	0.85	0.030	0.88
$Z_{1,2}$	cm	-0.511	0.561	0.033	5.79	0.007	1.23	0.007	1.28
$Z_{2,-2}$	cm	0.0243	0.0345	0.0070	20.26	0.0013	3.87	0.0016	4.71
$Z_{2,-1}$	cm	-0.095	0.108	0.029	27.01	0.003	2.85	0.005	4.69
$Z_{2,0}$	cm	-0.481	0.559	0.058	10.42	0.004	0.69	0.010	1.75
$Z_{2,1}$	cm	-0.273	0.328	0.029	8.75	0.004	1.21	0.004	1.26
$Z_{2,2}$	cm	-0.1163	0.1401	0.0145	10.39	0.0020	1.45	0.0021	1.51

For completeness, we detail a side-by-side comparison of scalar parameters and flux surface Fourier coefficient recovery results (with the models already described in chapter 3, which implies that the Fourier coefficient errors are presented as averages over $s$) in table 5.1 for three distinct models consisting of (i) only the baseline vacuum configuration parameters, (ii) the baseline plus 4 PCs of the equilibrium pressure profile $p(s)$ (the same model as used in the section 4.2) and (iii) the baseline plus 9 PCs of the dense magnetic measurements including 1% measurement noise. The latter represents the highest accuracy likely to be attainable using magnetic diagnostic information alone (due to the large number of measurements and low error levels assumed) and provides a benchmark against which we can judge other diagnostic setups.

The overall results of (iii) are quite accurate in that both scalar and Fourier coefficient errors are significantly reduced when judged against the previous results of chapter 3 also using magnetic measurements. The comparison shows the errors to be generally only slightly inferior to (ii), with the greatest differences in precision noticeable for $\beta_{\mathrm{axis}}$ and the $m=0$ Fourier coefficients (which are also closely linked to the local $\beta(s)$ value, since it strongly influences the Shafranov shift of the flux surface). Such degradation is to be expected since the magnetic measurements are external to the plasma and lack the local information contained in $p(s)$. Indeed many errors in the Fourier coefficients are very similar in both cases.