^{1}, J. Svensson

^{1}, H. Thomsen

^{1}, F. Medina

^{2}, A. Werner

^{1}and R. Wolf

^{1}

### Abstract

In this study, a Bayesian based non-stationary Gaussian Process (GP) method for the inference of soft X-ray emissivity distribution along with its associated uncertainties has been developed. For the investigation of equilibrium condition and fast magnetohydrodynamic behaviors in nuclear fusion plasmas, it is of importance to infer, especially in the plasma center, spatially resolved soft X-ray profiles from a limited number of noisy line integral measurements. For this ill-posed inversion problem, Bayesian probability theory can provide a posterior probability distribution over all possible solutions under given model assumptions. Specifically, the use of a non-stationary GP to model the emission allows the model to adapt to the varying length scales of the underlying diffusion process. In contrast to other conventional methods, the prior regularization is realized in a probability form which enhances the capability of uncertainty analysis, in consequence, scientists who concern the reliability of their results will benefit from it. Under the assumption of normally distributed noise, the posterior distribution evaluated at a discrete number of points becomes a multivariate normal distribution whose mean and covariance are analytically available, making inversions and calculation of uncertainty fast. Additionally, the hyper-parameters embedded in the model assumption can be optimized through a Bayesian Occam's Razor formalism and thereby automatically adjust the model complexity. This method is shown to produce convincing reconstructions and good agreements with independently calculated results from the Maximum Entropy and Equilibrium-Based Iterative Tomography Algorithm methods.

The authors would like to thank Dr. Oliver P. Ford for his helpful suggestions.

I. INTRODUCTION

II. METHOD

A. Soft X-ray diagnostics in W7-AS

B. Gaussian Process

C. A non-stationary covariance function

D. Bayesian formulae

E. Bayesian Occam's razor optimization

III. PERFORMANCE AND RESULTS

A. Error model

B. Benchmark using simulated data

C. Application on W7-AS

D. Application on TJ-II

IV. DISCUSSION

V. SUMMARY

### Key Topics

- Emissivity
- 25.0
- Gaussian processes
- 19.0
- Surface reconstruction
- 13.0
- Stellarators
- 12.0
- Soft X-rays
- 11.0

## Figures

A schematic view of the miniature soft X-ray system (MiniSoX diagnostic system) in Wendelstein 7-AS shows the eight compact detector arrays with a total of 256 sight lines in one poloidal cross-section, achieving a substantial coverage. 13,14 The dashed black lines indicate the magnetic flux surfaces of a typical plasma in Wendelstein 7-AS.

A schematic view of the miniature soft X-ray system (MiniSoX diagnostic system) in Wendelstein 7-AS shows the eight compact detector arrays with a total of 256 sight lines in one poloidal cross-section, achieving a substantial coverage. 13,14 The dashed black lines indicate the magnetic flux surfaces of a typical plasma in Wendelstein 7-AS.

Experimental data from W7-AS: time trace of signals from two different detectors during discharge #56316 shows the variance of data increase abruptly once the discharge starts.

Experimental data from W7-AS: time trace of signals from two different detectors during discharge #56316 shows the variance of data increase abruptly once the discharge starts.

Scatter plot showing the linear relation between the most likely standard deviation of one data set and the average of this data set. The possible values of the standard deviation are chosen to be within a certain maximum percentage level (from 2.5% to 25%) of the average of one data set. The most likely standard deviations of many data sets appear around the 12.5% of the average of data.

Scatter plot showing the linear relation between the most likely standard deviation of one data set and the average of this data set. The possible values of the standard deviation are chosen to be within a certain maximum percentage level (from 2.5% to 25%) of the average of one data set. The most likely standard deviations of many data sets appear around the 12.5% of the average of data.

A 2D emissivity distribution in the poloidal plane, where the MiniSoX-Tomography system was located, was simulated based on a typical magnetic configuration in W7-AS.

A 2D emissivity distribution in the poloidal plane, where the MiniSoX-Tomography system was located, was simulated based on a typical magnetic configuration in W7-AS.

To compare the different inversion methods, the artificial line integral data calculated from a simulated emissivity distribution is taken as input data of the different methods. Red dots: Artificial data without errors; green diamonds: artificial data with independently normally distributed random noise, used as input data.

To compare the different inversion methods, the artificial line integral data calculated from a simulated emissivity distribution is taken as input data of the different methods. Red dots: Artificial data without errors; green diamonds: artificial data with independently normally distributed random noise, used as input data.

Distribution of the reciprocal of local length scales inferred by a stationary Gaussian Process regression. The black line indicates the boundary of the vacuum vessel.

Distribution of the reciprocal of local length scales inferred by a stationary Gaussian Process regression. The black line indicates the boundary of the vacuum vessel.

Reconstructions by (a) non-stationary GP and (b) MaxEnt methods using the artificially noisy data, and the errors in both methods are exactly described as how they are added. The black contours show the simulated emissivity distribution for a clear comparison.

Reconstructions by (a) non-stationary GP and (b) MaxEnt methods using the artificially noisy data, and the errors in both methods are exactly described as how they are added. The black contours show the simulated emissivity distribution for a clear comparison.

The 1D plots about the profiles intercepted at (a) R = 2.0m and (b) R = 1.9m from the reconstructions by two methods. Red asterisks: simulated emissivity profiles; green dots: the non-stationary GP reconstruction with 95% confidence intervals; blue circles: the MaxEnt reconstruction.

The 1D plots about the profiles intercepted at (a) R = 2.0m and (b) R = 1.9m from the reconstructions by two methods. Red asterisks: simulated emissivity profiles; green dots: the non-stationary GP reconstruction with 95% confidence intervals; blue circles: the MaxEnt reconstruction.

Fit between the data predicted from reconstruction by non-stationary GP and the artificially noisy data. Green diamonds: artificially noisy data used for the inversion of reconstruction; red dots: predicted data with their 95% confidence intervals.

Fit between the data predicted from reconstruction by non-stationary GP and the artificially noisy data. Green diamonds: artificially noisy data used for the inversion of reconstruction; red dots: predicted data with their 95% confidence intervals.

Comparison of the reconstructions from shot number 56316 by (a) non-stationary GP and (b) MaxEnt using the experimental data from W7-AS. The black contours show the flux surface derived from the equilibrium calculation of the vacuum configuration.

Comparison of the reconstructions from shot number 56316 by (a) non-stationary GP and (b) MaxEnt using the experimental data from W7-AS. The black contours show the flux surface derived from the equilibrium calculation of the vacuum configuration.

The 1D plots about profiles intercepted at (a) R = 2.0m and (b) R = 1.9m from the reconstructions by two methods using the experimental data. The green dots show the reconstructed profile with 95% confidence intervals given.

The 1D plots about profiles intercepted at (a) R = 2.0m and (b) R = 1.9m from the reconstructions by two methods using the experimental data. The green dots show the reconstructed profile with 95% confidence intervals given.

Hundred samples of possible reconstructions, drawn from the multivariate normal posterior distribution to visualize the uncertainties of reconstruction.

Hundred samples of possible reconstructions, drawn from the multivariate normal posterior distribution to visualize the uncertainties of reconstruction.

Fit between the predicted data from the reconstruction by non-stationary GP and the used experimental data. Red dots show the predicted data with error bars

Fit between the predicted data from the reconstruction by non-stationary GP and the used experimental data. Red dots show the predicted data with error bars

Comparison of the reconstructions by two methods using the experimental data from shot number 53962 at time point 0.3741 s. For comparison, black contours show the flux surface derived from the equilibrium calculation of the vacuum configuration.

Comparison of the reconstructions by two methods using the experimental data from shot number 53962 at time point 0.3741 s. For comparison, black contours show the flux surface derived from the equilibrium calculation of the vacuum configuration.

Reconstructions at two different time points with high β0 in center from shot number 51755 in W7-AS, calculated by the non-stationary GP, shows a large horizontal shift, which also coincide with the equilibrium flux surfaces (black contours) calculated by free boundary VMEC calculations and also a large indentation frequently occurring in the inboard side. The green and red lines indicate the locations of the magnetic axis of the vacuum and finite β configurations, respectively. Specifically, a strongly inward axis of vacuum configuration is achieved by low magnetic fields and comparably higher vertical fields for high β experiments.

Reconstructions at two different time points with high β0 in center from shot number 51755 in W7-AS, calculated by the non-stationary GP, shows a large horizontal shift, which also coincide with the equilibrium flux surfaces (black contours) calculated by free boundary VMEC calculations and also a large indentation frequently occurring in the inboard side. The green and red lines indicate the locations of the magnetic axis of the vacuum and finite β configurations, respectively. Specifically, a strongly inward axis of vacuum configuration is achieved by low magnetic fields and comparably higher vertical fields for high β experiments.

(a) Schematic diagram of the setup of a soft X-ray diagnostic system in TJII, which consists of five detector arrays with each array having 16 detectors. (b) Experimental data from a typical discharge.

(a) Schematic diagram of the setup of a soft X-ray diagnostic system in TJII, which consists of five detector arrays with each array having 16 detectors. (b) Experimental data from a typical discharge.

Comparison of the reconstructions by the non-stationary GP and EBITA methods using the experimental data from shot number 18272 in TJ-II. They both have similar shape and location. Black contours show the flux surface derived from the equilibrium calculation of the vacuum configuration.

Comparison of the reconstructions by the non-stationary GP and EBITA methods using the experimental data from shot number 18272 in TJ-II. They both have similar shape and location. Black contours show the flux surface derived from the equilibrium calculation of the vacuum configuration.

## Tables

Notations used in overall processes about the model assumption.

Notations used in overall processes about the model assumption.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content