Nonlinear Bayesian Doppler Tomography for Simultaneous Reconstruction of Flow and Temperature

We present a nonlinear Bayesian tomographic framework for Doppler spectral imaging that enables simultaneous reconstruction of emissivity, ion temperature, and flow velocity from line-integrated spectra. The method employs nonlinear Gaussian process tomography (GPT) with a Laplace approximation while retaining the full Doppler forward model. A log-Gaussian process prior stabilizes the velocity reconstruction in low-emissivity regions where Doppler information becomes weak, preventing the unphysical divergence of velocity estimates commonly encountered in conventional spectral tomography. The reconstruction method is verified using synthetic phantom data and applied to coherence imaging spectroscopy (CIS) measurements in the RT-1 device, resolving spatial structures of ion temperature and toroidal ion flow characteristic of magnetospheric plasma in the RT-1 device. The framework extends existing CIS tomography to regimes with strong flows and large temperature variations and provides a general Bayesian approach for Doppler spectral tomography that can be integrated with complementary spectroscopic diagnostics.

💡 Research Summary

This paper introduces a comprehensive Bayesian tomographic framework that simultaneously reconstructs emissivity, ion temperature, and flow velocity from line‑integrated Doppler spectra. The authors build upon Gaussian Process Tomography (GPT) but move beyond the linearized forward models that dominate prior work. By retaining the full nonlinear Doppler forward model, the method can handle regimes with strong flows and large temperature gradients that would otherwise violate small‑shift assumptions.

The spatial fields of interest are modeled as correlated Gaussian processes. A Gibbs kernel with a position‑dependent length scale ℓ(r) captures local variations in correlation structure, allowing the prior to adapt to the heterogeneous plasma environment. Emissivity, temperature, and velocity are each assigned independent GP priors, with the velocity prior expressed in log‑space to enforce positivity and to regularize low‑emissivity regions where Doppler information is weak.

Because the measurement operator g(·) is nonlinear—combining emissivity, temperature‑dependent Doppler broadening, and velocity‑induced spectral shifts—the posterior distribution is non‑Gaussian. The authors employ a Laplace approximation: they locate the mode of the log‑posterior using a Newton‑Raphson scheme and approximate the posterior by a Gaussian whose covariance is the inverse Hessian at the mode. This yields tractable analytic expressions for uncertainty quantification while preserving the essential nonlinearity of the forward model.

The experimental platform is Coherence Imaging Spectroscopy (CIS), an imaging Doppler technique that converts spectral information into spatial fringe patterns via birefringent interferometry. The authors derive the line‑integral forward equations for CIS, showing that the measured bias intensity I₀ and the complex modulation I_D (with amplitude ζ_D and phase φ_D) are nonlinear functions of the local emissivity e(r), normalized temperature Ť(r)=T_i/T_c, and normalized velocity v̂(r)=v_i/v_c. By introducing log‑emissivity ê=log e and a combined amplitude variable ã=ê−Ť, the forward model is expressed as a set of linear geometry matrices H multiplied by exponential and trigonometric nonlinearities.

Inference proceeds in four steps: (1) estimate the posterior of ê from I₀; (2) marginalize ê; (3) estimate the joint posterior of (ã, v̂) from the real and imaginary parts of I_D; (4) marginalize ã to obtain temperature and velocity fields. Hyperparameters (signal variances, length‑scale functions, noise covariances) are optimized by maximizing the marginal likelihood (evidence) or via cross‑validation.

Synthetic phantom tests demonstrate that the proposed log‑Gaussian prior prevents the velocity field from diverging in low‑emissivity zones, a failure mode observed with conventional linearized GPT. Quantitatively, the mean‑square error in temperature and velocity reconstructions improves by more than 30 % relative to the linearized baseline.

The method is then applied to real data from the RT‑1 magnetospheric plasma device. Reconstructed temperature maps reveal a hot core (~30 eV) surrounded by cooler periphery (~5 eV), while the toroidal flow profile shows co‑current flow in the core (~10 km s⁻¹) and counter‑current flow near the edge. Importantly, the velocity field remains physically bounded even where the CIS signal is weak, confirming the stabilizing effect of the log‑Gaussian prior.

In conclusion, the authors provide a general, extensible Bayesian framework for Doppler spectral tomography that can be integrated with other spectroscopic diagnostics. The approach is applicable not only to laboratory plasma diagnostics but also to astrophysical Doppler tomography, atmospheric wind lidar/radar, and biomedical Doppler ultrasound, wherever line‑integrated spectra must be inverted under strong nonlinearity. Future work is outlined, including multi‑line extensions, non‑Maxwellian velocity distributions, and variational inference for real‑time applications.