Bayesian plasma model selection for Thomson scattering.

Laser Thomson scattering (LTS) is a measurement technique that can determine electron velocity distribution functions in plasma systems. However, accurately inferring quantities of interest from an LTS signal requires the selection of a plasma physics submodel, and comprehensive uncertainty quantification (UQ) is needed to interpret the results. Automated model selection, parameter estimation, and UQ are particularly challenging for low-density, low-temperature, potentially non-Maxwellian plasmas like those created in space electric propulsion devices. This paper applies Bayesian inference and model selection to a Raman-calibrated LTS diagnostic in the context of such plasmas. Synthetic data are used to explore the performance of the method across signal-to-noise ratios and model fidelity regimes. Plasmas with Maxwellian and non-Maxwellian velocity distributions are well characterized using priors that span a range of accuracy and specificity. The model selection framework is shown to accurately detect the type of plasmas generating the electron velocity distribution submodel for signal-to-noise ratios greater than around 5. In addition, the Bayesian framework validates the widespread use of 95% confidence intervals from least-squares inversion as a conservative estimate of the uncertainty bounds. However, epistemic posterior correlations between the variables diverge between least-squares and Bayesian estimates as the number of variable parameters increases. This divergence demonstrates the need for Bayesian inference in cases where accurate correlations between electron parameters are necessary. Bayesian model selection is then applied to experimental Thomson scattering data collected in a nanosecond pulsed plasma, generated with a discharge voltage of 5 and 10 kV at a neutral argon background pressure of 7 Torr-Ar. The Bayesian maximum a posteriori estimates of the electron temperature and number density are 1.98 and 2.38 eV and 2.6 × 1018 and 2.72 × 1018 m-3, using the Maxwellian and Druyvesteyn submodels, respectively. Furthermore, for this dataset, the model selection criterion indicates strong support for the Maxwellian distribution at 10 kV discharge voltage and no strong preference between Maxwellian and Druyvesteyn distributions at 5 kV. The logarithmic Bayes' factors for these cases are -35.76 and 1.07, respectively.

Megahertz-rate background-oriented schlieren tomography in post-detonation blasts.

The understanding and predictive modeling of explosive blasts require advanced experimental diagnostics that can provide information on local state variables with high spatiotemporal resolution. Current datasets are predominantly based on idealized spherically symmetric explosive charges and point-probe measurements, although practical charges typically involve multidimensional spatial structures and complex shock-flow interactions. This work introduces megahertz-rate background-oriented schlieren tomography to resolve transient, three-dimensional density fields, as found in an explosive blast, without symmetry assumptions. A numerical evaluation is used to quantify the sources of error and optimize the reconstruction parameters for shock fields. Average errors are ∼3% in the synthetic environment, where the accuracy is limited by the deflection sensing algorithm. The approach was experimentally demonstrated on two different commercial blast charges (Mach ∼1.2 and ∼1.7) with both spherical and multi-shock structures. Overpressure measurements were conducted using shock-front tracking to provide a baseline for assessing the reconstructed densities. The experimental reconstructions of the primary blast fronts were within 9% of the expected peak values. The megahertz time resolution and quantitative reconstruction without symmetry assumptions were accomplished using a single high-speed camera and light source, enabling the visualization of multi-shock structures with a relatively simple arrangement. Future developments in illumination, imaging, and analysis to improve the accuracy in extreme environments are discussed.

Bayesian framework for THz-TDS plasma diagnostics.

Terahertz time-domain spectroscopy (THz-TDS) is an optical diagnostic used to noninvasively measure plasma electron density and collision frequency. Conventional methods for analyzing THz-TDS plasma diagnostic data often do not account for measurement artifacts and do not quantify parameter uncertainties. We introduce a novel Bayesian framework that overcomes these deficiencies. The framework enables computation of both the density and collision frequency, compensates for artifacts produced by refraction and delay line errors, and quantifies parameter uncertainties caused by noise and imprecise knowledge of unmeasured plasma properties. We demonstrate the framework with sample measurements of a radio frequency inductively-coupled plasma discharge.

Linear absorption tomography with velocimetry (LATV) for multiparameter measurements in high-speed flows.

We present a linear model for absorption tomography with velocimetry (LATV) to reconstruct 2D distributions of partial pressure, temperature, and streamwise velocity in a high-speed flow. Synthetic measurements are generated by multi-beam tunable diode laser absorption spectroscopy (TDLAS). The measurement plane is tilted relative to the streamwise direction and absorbance spectra are Doppler-shifted by the gas flow. Reconstruction comprises two stages. First, the thermodynamic state is obtained by reconstructing two or more integrated absorption coefficients and evaluating local Boltzmann plots. Second, the velocity field is directly reconstructed from absorbance-weighted linecenters. Absorbance data are inferred by Voigt fitting and reconstructions are quickly computed by matrix-vector multiplication. Nonlinear parameter combinations, such as the mass flow, are more accurate when computed by LATV than estimates obtained by assuming uniform gas properties along each beam.

Efficient Bayesian inference of absorbance spectra from transmitted intensity spectra.

High-resolution absorption spectroscopy is a promising method for non-invasive process monitoring, but the computational effort required to evaluate the data can be prohibitive in high-speed, real-time applications. This study presents a fast method to estimate absorbance spectra from transmitted intensity signals. We employ Bayesian statistics to combine a measurement model with prior information about the shape of the baseline intensity and absorbance spectrum. The resulting linear least-squares problem shifts most of the computational effort to a preparation step, thereby facilitating quick processing and low latency for any number of measurements. The method is demonstrated on simulated tunable diode laser absorption spectroscopy data with additive noise and a fluctuating fringe. Results were highly accurate and the method was computationally efficient, having a processing time of only 2 ms per spectrum.

General error model for analysis of laser-induced incandescence signals.

This paper presents a novel error model for TiRe-LII signals and illustrates how the model can be used to diagnose a detection system, quantify uncertainties in TiRe-LII, and characterize fluctuations in the measured process. Noise in a single TiRe-LII measurement shot obeys a Poisson-Gaussian noise model. Variation in the aerosol results in shot-to-shot fluctuations in the measured signals. These fluctuations induce a quadratic relationship between the mean and variance of a set of signals. We show how this model can elucidate aspects of the measurement system and fundamental properties of the aerosol, by comparing the noise model to four sets of experimental data.

Measurement-based meshing, basis selection, and prior assignment in chemical species tomography.

Gas distributions imaged by chemical species tomography (CST) vary in quality due to the discretization scheme, arrangement of optical paths, errors in the measurement model, and prior information included in reconstruction. There is currently no mathematically-rigorous framework for comparing the finite bases available to discretize a CST domain. Following from the Bayesian formulation of tomographic inversion, we show that Bayesian model selection can identify the mesh density, mode of interpolation, and prior information best-suited to reconstruct a set of measurement data. We validate this procedure with a simulated CST experiment, and generate accurate reconstructions despite limited measurement information. The flow field is represented using the finite element method, and Bayesian model selection is used to choose between three forms of polynomial support for a range of mesh resolutions, as well as four priors. We show that the model likelihood of Bayesian model selection is a good predictor of reconstruction accuracy.

Improving chemical species tomography of turbulent flows using covariance estimation.

Chemical species tomography (CST) experiments can be divided into limited-data and full-rank cases. Both require solving ill-posed inverse problems, and thus the measurement data must be supplemented with prior information to carry out reconstructions. The Bayesian framework formalizes the role of additive information, expressed as the mean and covariance of a joint-normal prior probability density function. We present techniques for estimating the spatial covariance of a flow under limited-data and full-rank conditions. Our results show that incorporating a covariance estimate into CST reconstruction via a Bayesian prior increases the accuracy of instantaneous estimates. Improvements are especially dramatic in real-time limited-data CST, which is directly applicable to many industrially relevant experiments.

Bayesian approach to the design of chemical species tomography experiments.

Reconstruction accuracy in chemical species tomography depends strongly on the arrangement of optical paths transecting the imaging domain. Optimizing the path arrangement requires a scheme that can predict the quality of a proposed arrangement prior to measurement. This paper presents a new Bayesian method for scoring path arrangements based on the estimated a posteriori covariance matrix. This technique focuses on defining an objective function that incorporates the same a priori information about the flow needed to carry out limited data tomography. Constrained and unconstrained path optimization studies verify the predictive capabilities of the objective function, and that superior reconstruction quality is obtained with optimized path arrangements.