Nonlinear bayesian tomography of ion temperature and velocity for Doppler coherence imaging spectroscopy in RT-1

Abstract

We present a novel Bayesian tomography approach for Coherence Imaging Spectroscopy (CIS) that simultaneously reconstructs ion temperature and velocity distributions in plasmas. Utilizing nonlinear Gaussian Process Tomography (GPT) with the Laplace approximation, we model prior distributions of log-emissivity, temperature, and velocity as Gaussian processes. This framework rigorously incorporates nonlinear effects and temperature dependencies often neglected in conventional CIS tomography, enabling robust reconstruction even in the region of high temperature and velocity. By applying a log-Gaussian process, we also address issues like velocity divergence in low-emissivity regions. Validated with phantom simulations and experimental data from the RT-1 device, our method reveals detailed spatial structures of ion temperature and toroidal ion flow characteristic of magnetospheric plasma. This work significantly broadens the scope of CIS tomography, offering a robust tool for plasma diagnostics and facilitating integration with complementary measurement techniques.

Figures (9)

• Figure 1: Conceptual illustration of the relationship between local variables and observations in CIS. $e(\vec{r})$, $T_\mathrm{i}(\vec{r})$, and $\vec{v}_\mathrm{i}(\vec{r})$ are local emissivity, local ion temperature, and local ion flow velocity, respectively. $S_\mathrm{CIS}(\vec{x})$ is the observed image measured with CIS. $\vec{r}$ is the coordinate in the plasma region, and $\vec{x}$ is the coordinate on the image sensor.
• Figure 2: Conceptual diagrams of the tomographic model for CIS. The first diagram is a directed Bayesian graphical model of the CIS projection equations, consisting of Eqs. \ref{['eq: def_of_a']}, \ref{['eq: discretized_g0']}, \ref{['eq: discretized_gc']}, \ref{['eq: discretized_gs']}, \ref{['eq: def_of_g0obs']}, and \ref{['eq: def_of_IRI_obs']}. Each node represents a random variable, and the blue nodes are observed data. Step 1 computes the posterior probability of the log-emissivity $\hat{\bm{e}}$ given the observed data $\bm{I}_0^\mathrm{obs}$. Step 2 marginalizes the variable $\hat{\bm{e}}$. Step 3 computes the posterior probabilities of $\hat{\bm{a}}$ and $\hat{\bm{v}}$. Step 4 marginalizes the variable $\hat{\bm{a}}$.
• Figure 3: (a) A conceptual diagram of the camera system in RT-1, where the CIS camera is installed tangentially to the plasma cross-section to detect toroidal flow. (b) Examples of projected lines of rays (the red lines) from the camera on the poloidal cross-section of RT-1. The number of rays is reduced for simplicity. (c) The distribution of scattered inducing points $\vec{r}^\mathrm{idc}$ (the blue points) and boundary points $\vec{r}^\mathrm{bd}$ (the orange points). The number of inducing points is 2041, and the number of boundary points is 229 in the setup. (d) The length scale function $\ell'(\vec{r})$ in RT-1. These conditions are consistent with previous work K_Ueda_2024.
• Figure 4: Phantom data for the test tomography and the corresponding projected images. The left column, (a), (b), and (c), are phantom distributions of local emissivity ($e_\mathrm{true}$), local temperature ($\hat{T}_\mathrm{true}$), and local velocity ($\hat{v}_\mathrm{true}$), respectively. Panel (d) is the projected emissivity, $\bm{I}_0^\mathrm{inp}$, using Eqs. \ref{['eq: discretized_g0']} and \ref{['eq: def_of_g0obs']}. Panels (e) and (f) are input images for CIS tomography generated by Eqs. \ref{['eq: discretized_gc']}, \ref{['eq: discretized_gs']}, and \ref{['eq: def_of_IRI_obs']}, corresponding to $\bm{I}_\mathrm{Re}^\mathrm{inp}$ and $\bm{I}_\mathrm{Im}^\mathrm{inp}$, respectively. Panels (g) and (h) are the projected temperature and projected velocity, respectively.
• Figure 5: Evidence maps for the CIS tomography model when the input images are shown in Fig. \ref{['fig: phantom_cis_GPT']}. (a) $\mathcal{L}^\mathrm{emit}$ and (b) $\mathcal{L}^\mathrm{CIS}$ are defined in Eqs. \ref{['eq: evidence_for_emission']} and \ref{['eq: evidence_for_CIS']}, respectively. The horizontal and vertical axes represent the length scale factor $\hat{\ell}_F$ and the sigma scale $\sigma_{g}$, which are defined in Eqs. \ref{['eq: length_scale']} and \ref{['eq: sigma_g']}, respectively. The red crosses indicate the maximum points.