Implementing an X-ray tomography method for fusion devices

Abstract

In fusion devices, the X-ray plasma emissivity contains essential information on the magnetohydrodynamic activity, the magnetic equilibrium and on the transport of impurities, in particular for tokamaks in the soft X-ray (SXR) energy range of 0.1–20 keV. In this context, tomography diagnostics are a key method to estimate the local plasma emissivity from a given set of line-integrated measurements. Unfortunately, the reconstruction problem is mathematically ill-posed, due to very sparse and noisy measurements, requiring an adequate regularization procedure. The goal of this paper is to introduce, with a didactic approach, some methodology and tools to develop an X-ray tomography algorithm. Based on a simple 1D tomography problem, the Tikhonov regularization is described in detail with a study of the optimal reconstruction parameters, such as the choice of the emissivity spatial resolution and the regularization parameter. A methodology is proposed to perform an in situ sensitivity and position cross-calibration of the detectors with an iterative procedure, by using the information redundancy and data variability in a given set of reconstructed profiles. Finally, the basic steps to build a synthetic tomography diagnostics in a more realistic tokamak environment are introduced, together with some tools to assess the capabilities of the 2D tomography algorithm.

A Bayesian formulation for perturbed current tomography in tokamaks

“…This is a well studied process and as such there exist methods for fitting the smoothing parameter, which is analogous to our CAR variance, used in Tikhonov regularisation from only the measured data. Here we adapt one such method, called the L-curve method [12].…”

“…However, it does have a few problems. Most notably, the curves can be less L-shaped at times and therefore it can be difficult to determine the point of maximum curvature [12]. The other problem is that it lacks any formal theoretical justification, in our Bayesian case, and while it is not surprising that it does work, given the similarities between our Bayesian procedure and Tikhonov regularisation, we would still prefer a more complete Bayesian method whose use can be motivated from the principles of probability theory.…”

“…In [10] the electron density and temperature profiles are inferred from experimental data taken from the mega ampere spherical tokamak experiment. The implementation of an x-ray tomography method for the inference of the static emissivity profiles of a tokamak is presented in both [11,12]. Wang et al [11] adopts a Bayesian approach where Jardin et al [12] uses a traditional regularised fitting approach.…”

“…The implementation of an x-ray tomography method for the inference of the static emissivity profiles of a tokamak is presented in both [11,12]. Wang et al [11] adopts a Bayesian approach where Jardin et al [12] uses a traditional regularised fitting approach. The two methods are found to perform similarly in practice and therefore on existing evidence we are justified in adopting the Bayesian method.…”

Tikhonov Regularization and Perturbation-Level Tuning for the CNM in Pharmacokinetics

“…For that reason, we propose to use Tikhonov regularization to handle the overdetermined system for hyperplane fitting in Step 2.2 in the early CNM. Tikhonov regularization has been also applied in many medical optimization problems, such as in B-mode imaging [ [4]]-[ [6]], X-ray imaging [ [7]]-[ [9]], ultrasound tomography [ [10]]-[ [12]], and magnetic resonance imaging [ [13]]-[ [15]]. To the best of our knowledge, Tikhonov has been used to solve inverse problems in many areas, including pharmacokinetics [ [16]].…”