Sensitivity to errors in the Thomson data

There are many sources of error in the Thomson scattering data that can lead to incorrect $T_{\mathrm{e}}$ and $n_{\mathrm{e}}$ profiles. The largest uncertainty in $T_{\mathrm{e}}$ is due to the difficulty in keeping the systematic errors in the complex detection and acquisition system of the many spectral channels sufficiently low. Other factors include possible mismatching of wavelength channels and temperatures to be measured in view of the large temperature ranges which can appear at a single spatial point (e.g. central temperatures range from 300 eV up to 6 keV). Also, the appearance of super-thermal electrons created by ECR heating can violate the assumption of a Maxwellian distribution for the evaluation of the scattering function.

The density is prone to further possible errors. For example, line-of-sight misalignments with the observation geometry reduces the intensity of the scattered light. This leads to an underestimate of the density, which is proportional to the absolute intensity, but has no effect on the spectrum of scattered light and therefore does not influence the temperature. Also, background radiation may drive the detection system out of the linear response region, falsifying $n_{\mathrm{e}}$. Another important source of errors is the rather subtle problem of properly calibrating the different channels to one other, which also affects the density but not the temperature. We do not attempt to correct for systematic errors which affect individual channels. However, the procedure is insensitive to systematic errors which affect the inboard and outboard sides equally. Systematic errors leading to rigid shifts in the data are considered later. To deal with random errors, we rely on the quoted experimental errors in each channel to reflect such deviations. The spline smoothing algorithm takes these errors into account in the fitting procedure, giving a higher weighting to those channels with smaller errors in the least squares minimization.

We thus proceed assuming that the dominant source of error affecting the procedure will be scatter in the Thomson electron pressure data, whereby each ${p_{\mathrm{e}}}(R)$ data-point has an associated uncertainty $\Delta{p_{\mathrm{e}}}(R)$. We investigate the robustness of the interpretation by simulating varying levels of experimental noise and examining the corresponding error in the interpreted $p(s)$.

To this end, over 300 FP-reconstructed equilibria with varying external coil current configurations, axis pressure values and $p(s)$ profile shapes (broad, linear and peaked) were used to generate idealized noise-free Thomson profiles, which were perturbed with pseudo-random Gaussian noise of equal relative magnitude at each channel. To quantify the input and output errors, we define $\varepsilon$, a representative fractional deviation of the noisy simulated Thomson input profile from the unperturbed one ( $\tilde{{p_{\mathrm{e}}}} (R)$ and ${p_{\mathrm{e}}}(R)$, respectively) as:

$$\varepsilon = \sqrt{\frac{\int{\Big(\tilde{{p_{\mathrm{e}}}}... ...R)\Big)^2 \ dR}} {\int{\tilde{{p_{\mathrm{e}}}}(R)^2 \ dR}}},$$ (4.3)

and $\eta$, the fractional deviation of the interpreted pressure $p(s)$ from the known equilibrium pressure ${p_{\mathrm{eq}}}(s)$ as:

$$\eta = \sqrt{\frac{\int{\Big(p(s) - {p_{\mathrm{eq}}}(s)\Big)^2 \ ds}} {\int{p(s)^2 \ ds}}}.$$ (4.4)

Figure 4.10: $\eta$ versus $\varepsilon$ trend averaged over several configurations, axis pressures and profile shapes. Error bars indicate one standard deviation of $\eta$.

The simulated Thomson errors are normally distributed, have zero mean and are uncorrelated; thus we can use the $\chi^2$ goodness-of-fit test to determine an optimal choice of smoothing spline to fit the data. The behaviour of $\eta$ versus $\varepsilon$ with error bars detailing the spread of errors encountered over the test equilibria is shown in figure 4.10. The trend that emerged shows no particular dependence on the configuration or the axis pressure and only weakly depends on the profile shape. On average, the fractional error in the interpreted profile has approximately the same magnitude as the input error, i.e. $\eta \simeq \varepsilon$.