Quantifying Resolution Limits in Pedestal Profile Measurements with Gaussian Process Regression

TL;DR

This work tackles the challenge of quantifying spatiotemporal resolution limits when inferring differentiable pedestal profiles from localized measurements in magnetically confined plasmas. It introduces Gaussian Process Regression (GPR) as a flexible, uncertainty-aware framework and derives two key contributions: an explicit low-pass filtering interpretation that links kernel hyperparameters to a cutoff frequency $\\xi_*$ and a signal-to-noise dependent trade-off, and a novel information-theoretic metric $N_{eff}$ that measures how many measurements effectively inform the inferred value and its derivatives. Together, these tools provide quantitative criteria to diagnose over-regularization and over-fitting, guiding hyperparameter choices and enabling credible interpretation of pedestal gradients and their time evolution. The methods are demonstrated on DIII-D pedestal data, including both ELM-averaged fits and inter-ELM evolution, revealing credible gradient features and their alignment with rational surfaces, and enabling analysis of how the pedestal evolves over an ELM cycle. The practical impact is a principled, quantitative framework for designing and assessing GPR-based profile inferences in fusion plasmas and beyond, with potential extensions to multi-diagnostic data and time-dependent transport studies.

Abstract

Edge transport barriers (ETBs) in magnetically confined fusion plasmas, commonly known as pedestals, play a crucial role in achieving high confinement plasmas. However, their defining characteristic, a steep rise in plasma pressure over short length scales, makes them challenging to diagnose experimentally. In this work, we use Gaussian Process Regression (GPR) to develop first-principles metrics for quantifying the spatiotemporal resolution limits of inferring differentiable profiles of temperature, pressure, or other quantities from experimental measurements. Although we focus on pedestals, the methods are fully general and can be applied to any setting involving the inference of profiles from discrete measurements. First, we establish a correspondence between GPR and low-pass filtering, giving an explicit expression for the effective `cutoff frequency' associated with smoothing incurred by GPR. Second, we introduce a novel information-theoretic metric, \(N_{eff}\), which measures the effective number of data points contributing to the inferred value of a profile or its derivative. These metrics enable a quantitative assessment of the trade-off between `over-fitting' and `over-regularization', providing both practitioners and consumers of GPR with a systematic way to evaluate the credibility of inferred profiles. We apply these tools to develop practical advice for using GPR in both time-independent and time-dependent settings, and demonstrate their usage on inferring pedestal profiles using measurements from the DIII-D tokamak.

Figures (11)

• Figure 1: (Top) Illustration of how the length scale $\ell$ parameter affects profiles (orange curve with $\pm34\%$ confidence intervals) inferred using GPR. The decomposition of this profile into scaled regression coefficient functions $Y_i \beta_i(x)$ (smooth curves, various colors) associated with the data points $Y_i$ (scattered points, various colors) is also shown. (Bottom) Plot of the $N_{eff}(x)$ metric defined in section \ref{['sec:neff']}. The colored vertical lines show the $x$ location of the data points. $N_{eff} \approx 1$ indicates only one measurement is contributing to the inference of the profile at a given point, indicating overfitting in the $\ell=0.12$ case.
• Figure 2: (Top) Matérn transfer functions $H(\xi)$ against frequency $\xi$ and (bottom) filter cutoffs $\xi_*$ against the signal to noise rate $S=\sigma^2 \ell / \sigma_\varepsilon \Delta x$ for different values of $\nu$. Frequencies are normalized to $2 \pi \ell$. GPR will heavily damp frequencies $\xi$ above the cutoff frequency $\xi_*$.
• Figure 3: Illustration of 'ringing' artifacts caused by the Gibbs phenomenon when using GPR to infer non-smooth functions. True profile (blue line) and data (blue scatter) are shown along with inferences (orange with $\pm 34\%$ confidence interval shaded) overplotted. The inverse filter cutoff $1/\xi_*$ is also shown.
• Figure 4: Illustration of effective low-pass filter behavior of GPR when applied to non-uniformly sampled data with non-identical error bars. Figure labels, as well as hyperparameters chosen for the GPR kernel, are the same as in Figure \ref{['fig:lowpass_ringing']}.
• Figure 5: GPR inferences for the electron temperature $T_e$ as a function of normalized poloidal flux $\psi$ for DIII-D shot #174864. (Top and middle rows) Data (blue scatter) are shown with corresponding inferred profiles (orange with $\pm 34\%$ confidence interval shaded). The inverse filter cutoff $1/\xi_*$ is also shown. Comparing to the width of the pedestal, the right-most profile shows signs of over-regularization. (Bottom row) $N_{eff}$ for the derivative profiles is shown as a function of $\psi$, with $N_{eff} \lesssim 2$ indicating overfitting.