Bayesian electron density inference from JET lithium beam emission spectra using Gaussian processes

Abstract

Abstract. A Bayesian model to infer edge electron density profiles is developed for the JET lithium beam emission spectroscopy (Li-BES) system, measuring Li I (2p-2s) line radiation using 26 channels with ∼ 1 cm spatial resolution and 10 ∼ 20 ms temporal resolution. The density profile is modelled using a Gaussian process prior, and the uncertainty of the density profile is calculated by a Markov Chain Monte Carlo (MCMC) scheme. From the spectra measured by the transmission grating spectrometer, the Li I line intensities are extracted, and modelled as a function of the plasma density by a multi-state model which describes the relevant processes between neutral lithium beam atoms and plasma particles. The spectral model fully takes into account interference filter and instrument effects, that are separately estimated, again using Gaussian processes. The line intensities are inferred based on a spectral model consistent with the measured spectra within their uncertainties, which includes photon statistics and electronic noise. Our newly developed method to infer JET edge electron density profiles has the following advantages in comparison to the conventional method: i) providing full posterior distributions of edge density profiles, including their associated uncertainties, ii) the available radial range for density profiles is increased to the full observation range (∼ 26 cm), iii) an assumption of monotonic electron density profile is not necessary, iv) the absolute calibration factor of the diagnostic system is automatically estimated overcoming the limitation of the conventional technique and allowing us to infer the electron density profiles for all pulses without preprocessing the data or an additional boundary condition, and v) since the full spectrum is modelled, the procedure of modulating the beam to measure the background signal is only necessary for the case of overlapping of the Li I line with impurity lines.

Overview of the JET preparation for deuterium–tritium operation with the ITER like-wall

“…Unlike a parametric model, which might severely constrain the posterior distribution, the Gaussian process, not depending on any specific parameterisation, puts less constraints on the posterior distribution. In nuclear fusion research, Gaussian processes were introduced by non-parametric tomography of the electron density and current distribution [13], followed by a number of applications [9,10,16,17,23,24,38]. Gaussian processes are also the standard way to model profiles in Minerva.…”

Imputation of faulty magnetic sensors with coupled Bayesian and Gaussian processes to reconstruct the magnetic equilibrium in real time

“…σ 2 n ∼ O(10 −4 ) is determined by treating it as a hyperparameter for the numerical stability during matrix inversion. 5,16,17 Recall that N i (N m ) is the total number of intact (missing) magnetic signals. Here, X ( * ) is the 2 × N i (N m ) matrix containing the physical positions of all the intact (missing) magnetic probes in two dimensional space, i.e., physical R and Z positions at a fixed toroidal location.…”

“…As searching for the hyperparameters may become time consuming, thus not applicable for real-time control, one can obtain these values beforehand using many existing plasma discharges as for the case of density reconstruction. 16 Once we have values for the hyperparameters, i.e., σ f ∼ O(10 −2 ) and R = Z ∼ O(10 −1 ) in this study, we use Eq. (6) to obtain the values of the missing magnetic signals B * , i.e., B * =K * K−1 B.…”

Development of a non-parametric Gaussian process model in the three-dimensional equilibrium reconstruction code V3FIT

“…Many equilibrium reconstruction codes, such as EFIT (Lao & Ferron 1990) and V3FIT (Hanson et al 2009), define the radial profiles using a parametric representation. The radial profiles are assumed to have a specified functional form characterized by multiple free parameters p. The best fit equilibrium is defined by the set of parameters that minimize the error between the observed diagnostic signals, S O , and modelled diagnostic signals S M (p).…”

“…This framework has been used to perform soft x-ray (SXR) tomographic analysis of W7-AS and TJ-II stellarator plasmas (Li et al 2013). GPR has also been used to infer edge density profiles on the joint European torus (JET) (Kwak et al 2017), and for uncertainty analysis in transport calculations (Chilenski et al 2015).…”