Also at ]Oden Institute for Computational Engineering and Sciences, UT-Austin

# Automated Estimation of Plasma Temperature and Density from Emission
Spectroscopy

Todd A. Oliver<EMAIL_ADDRESS>Oden Institute for Computational
Engineering and Sciences
University of Texas at Austin  Craig Michoski<EMAIL_ADDRESS>Sapientai LLC  [  Samuel Langendorf<EMAIL_ADDRESS>Los Alamos
National Laboratory  Andrew LaJoie<EMAIL_ADDRESS>Los Alamos National
Laboratory

###### Abstract

This paper introduces a novel approach for automated estimation of plasma
temperature and density using emission spectroscopy, integrating Bayesian
inference with sophisticated physical models. We provide an in-depth
examination of Bayesian methods applied to the complexities of plasma
diagnostics, supported by a robust framework of physical and measurement
models. Our methodology is validated through experimental observations,
focusing on individual and sequential shot analyses. The results demonstrate
the effectiveness of our approach in enhancing the accuracy and reliability of
plasma parameter estimation, marking a significant advancement in the field of
emission spectroscopy for plasma diagnostics. This study not only offers a new
perspective in plasma analysis but also paves the way for further research and
applications in nuclear instrumentation and related domains.

##  I Introduction

Spectroscopy is a powerful and versatile analytical tool that plays an
increasingly critical role in a broad array of scientific and industrial
fields. For example, atomic emission spectroscopy, centered on the study of
emitted light spectra from excited atoms or molecules, offers a non-invasive
and highly sensitive method for the characterization and identification of
various substances. From environmental monitoring, medical diagnostics, and
forensic analysis, to astrophysics, material science, and industrial process
control, emission spectroscopy’s extensive applications underscore its
importance.

However, the quantities observed in emission spectroscopy—i.e., the
intensities of light at different wavelengths—are generally not directly of
interest in applications. Instead, it is common to extract the quantities of
interest (QoI’s) from these observables using a model. As an example, the
simplest way to extract information from spectra is by assuming local
thermodynamic equilibrium (LTE) conditions, and then leveraging the Boltzmann
distribution law, where populations of distinct atomic energy levels are
charted against their corresponding energies, yielding a graph whose slope
bears a reciprocal relationship to the temperature of the system. Such plots
are referred to as Boltzmann plots. However, in many application areas, such
as in transient high energy laser plasma Cristoforetti  _et al._ (2013), the
underlying LTE assumptions do not hold, and thus are generally considered
inadequate for effectively interpreting the measured spectra. Thus, in order
to accommodate for the added complexity present under non-LTE conditions (such
as the presence of large local gradients), various complex kinetic models of
the plasma have been developed and deployed, which effectively model the
additional, e.g., radiative, transient, and diffusive processes present in
these systems.

Thus, to extract information from the observed spectroscopic data, the data
must be compared against model outputs to determine the conditions that
generated the observations—i.e., the data processing takes the form of an
inverse problem. Because both the observation and the models describing the
physical process that generate it are complicated and include uncertainties
that are difficult to quantify—due to noise and observational errors but also
modeling approximations and assumptions—it is challenging to represent their
effects upon the resulting inferred quantities of interest.

While it is well within the scope and capabilities of statistical and
computational science to provide solutions to this problem, the complexity and
nuance of the problem itself alongside the significant laboratory challenge of
simply getting experiments to function properly, can lead to a type of extreme
instrumentalism Leplin (2017) in laboratory environments where the spectra and
models are, e.g., treated in potentially perilously reductive ways, such as by
hasty and subjective visual inspection performed ‘over-the-bench’ by machine
operators, to performing rudimentary (and potentially overly-simplistic)
deterministic regression techniques, such as least squares residual analysis
between the spectra and the kinetic model outputs. These interpretive
conditions are then often reported with weakly justified generic absolute
error bars in the literature, leading to a concern that the reported results
are not rigorously and reliably understood by the operators themselves, or,
for that matter, sufficiently aligned with reproducible scientific modes of
inquiry and compliance.

In this work, we focus on solution methods to this problem that utilize
Bayesian analysis Von Toussaint (2011) to provide principled and rigorous
statistical methodologies, with the motivating factors being to develop and
analyze a framework that allows for the automation of uncertainty
quantification in this specific context. A broad overview of how Bayesian
spectral fitting methods have been developed and implemented for spectroscopic
data analysis can be found in Cao and Sciortino (2019); Razul  _et al._
(2003), and more specifically the utilization of these types of models for
evaluating emission spectra in plasma discharges in Dodt  _et al._ (2006) and
fusion devices Fischer  _et al._ (2010); Bozhenkov  _et al._ (2017);
Verdoolaege  _et al._ (2006); Sciortino  _et al._ (2020). Similar approaches
can be found for gamma-ray spectroscopy Burr  _et al._ (2020), Nuclear
Magnetic Resonance spectroscopy Rubtsov and Griffin (2007); Matviychuk  _et
al._ (2017), transmission spectroscopy Pluriel  _et al._ (2022); Zingales  _et
al._ (2022), diffuse reflectance spectroscopy Hohmann  _et al._ (2021), and
quantum noise spectroscopy Ferrie  _et al._ (2018). Finally, mixture models
are considered in Park  _et al._ (2009), while automation techniques for these
types of techniques are addressed in Li  _et al._ (2008).

The specific motivating experiment for this work is the Plasma Liner
Experiment (PLX) at Los Alamos National Laboratory, and particularly, the
collection of emission spectra from the formed spherical plasma liners.
Emission spectroscopy is a powerful diagnostic technique for the PLX liner
plasmas, which are comprised largely of mid-density $10^{15}-10^{17}$ cm-3 low
temperature $1-5$ eV plasmas with typically argon species. The plasma emits a
rich series of lines in the visible spectrum, which are readily collected via
fiber-coupled spectrometers and digitized on CCD detectors, accumulating a
large volume of data in the form of images. Comparison of the emission lines
to atomic physics models contains useful information on the liner electron
temperature and ionization state evolution, but to date this comparison has
been performed manually, by generation of a large number of comparison plots
between observation and many candidate model spectra, and manual comparison to
determine estimates of the plasma parameters. This procedure has limited the
amount of information that can be gleaned from the spectra and left them
somewhat under-utilized on many experimental campaigns, motivating the
development of a more powerful and automated method.

The structure of this paper is organized as follows. Section II describes the
inference approach developed by integrating Bayesian inference techniques with
base physical models. Sections III discusses the details of the experimental
system under consideration, and Sections IV shows the results of the
application of the Bayesian inference approach to the experimental data
interpretation on this experimental system. Finally, we provide a conclusion
and future directions in Section V.

##  II Inference Approach

To determine the state of the plasma—including, potentially, the temperature,
density, and composition—a model relating these variables to the observed
spectral data is required. The plasma properties of interest are then inferred
based on what parameter values lead to agreement between the modeled and
observed spectra. This process—i.e., determining the value that minimizes the
misfit between the experiment and the model—naturally takes the form of an
optimization problem, but a straightforward optimization does not
automatically account for uncertainty, which is introduced both by
uncertainties inherent in any measurement process as well as by approximations
inherent in any modeling process. To account for these uncertainties, a
Bayesian inference approach is adopted here. In this approach, the solution of
the inverse problem is a probability distribution for the plasma properties of
interest. This probability distribution characterizes not only the most likely
values of the parameters but also the relative probability of other values,
accounting for the uncertainties in the measurements and models underlying the
analysis.

###  II.1 Bayesian Inference Overview

To be concrete, let $\theta$ denote the collection of plasma properties we
hope to infer. Further, let $y_{obs}$ denote the observations, and let
$\mathcal{M}$ denote a physical model that maps the parameters $\theta$ to the
observations $y_{obs}$. Given such a model, the approach here is to infer the
values of the parameters by finding those values that make the output from the
model best match the observations. If both the model and the observations were
perfect—i.e., free from errors and uncertainties—then there would exist values
of the parameters, denoted $\theta^{\ast}$, such that

$y_{obs}=\mathcal{M}(\theta^{\ast}),$

and we could discover such parameter values by solving the following
minimization problem:

$\theta^{\ast}=\arg\min_{\theta}\|y_{obs}-\mathcal{M}(\theta)\|,$

where $\|y_{obs}-\mathcal{M}(\theta)\|$ denotes a metric function (usually a
norm) measuring the discrepancy between the observations and the model.

Of course, in practice, it is almost never the case that both the observations
$y_{obs}$ and model $\mathcal{M}$ are entirely free from errors and
uncertainties. In this situation, if only the “best” parameter values are of
interest, then the optimization approach is still feasible. However, it is
often of interest to find not only the best fit but all values that lead to
adequate, in some sense, agreement between the measurements and the model and
to characterize the relative plausibility of these values. When such
uncertainties are of interest, the Bayesian statistical framework provides a
coherent and self-consistent methodology for such analysis.

Bayesian methods have become common in a wide range of scientific applications
involving inference problems Christian (2001); Jaynes (2003); Kaipio and
Somersalo (2005); Calvetti and Somersalo (2007); Von Toussaint (2011); Cheung
_et al._ (2011); Oliver  _et al._ (2015). In the Bayesian framework, the
solution of the inference problem is given by the probability density for the
parameters of interest $\theta$ conditioned on the observations $y_{obs}$—an
object known as the “posterior” probability density. Denoting this posterior
by $\pi(\theta|y)$, Bayes’ theorem states that

$\pi(\theta|y)=\frac{\pi(\theta)\,\pi(y|\theta)}{\pi(y)},$

where $\pi(\theta)$ is known as the “prior” and $\pi(y|\theta)$ is the
“likelihood”. The denominator $\pi(y)$ is a normalizing constant that ensures
that the posterior integrates to one and will not be discussed further here.
The prior characterizes any information about the parameters that is available
independent from the observations $y_{obs}$. For this work, the prior is taken
to be a uniform distribution over a set of possible plasma conditions selected
by the analyst. To complete the formulation, the likelihood must be specified.
The likelihood is based on two models: 1) the physical model $\mathcal{M}$,
which is described in section §II.2 and 2) a probabilistic model representing
all relevant uncertainties, which is described in section §II.3.

###  II.2 Physical Models

For the sake of exposition, the physical model under consideration here is
split into two components: 1) the _base physics model_ , which determines the
expected spectrum under given conditions, not accounting for measurement
imperfections, and 2) the _measurement model_ , which determines the expected
observed spectrum under given conditions and outputs of the base physics
model, and does account for potential measurement distortions.

####  II.2.1 Base physics model

Detailed modeling of emission spectra from plasmas is a broad and deep
subject, and several modeling paradigms can be (and have been) applied
depending on the plasma conditions of interest. One paradigmatic
categorization is whether the plasma can be described by certain equilibria,
such as local-thermal-equilibrium (LTE) conditions Fujimoto and McWhirter
(1990), which hold in dense plasmas when the plasma collisions are frequent
and fully govern electron energy state populations, and coronal-equilibrium
conditions Fujimoto (1973), which hold in rarefied plasmas where the dominant
de-excitation of electrons is radiative relaxation to the ground state. In
contrast, at intermediate densities, the modeling paradigm of interest
generally used to model the relative contributions of these mechanisms is a
collisional-radiative (CR) model, which solves for the populations of electron
energy states at given conditions and thus the relative strength of emission
lines Ralchenko (2016).

The simplest version of these models describe local (zero dimensional) light
emission from a point source at a given plasma condition, but additional
physical effects may occur en route to the observational instrument, such as
those caused by optical opacity of the plasma, etc. These effects can be
treated by techniques common in radiation transport, including, for example,
radiation diffusion and characteristics methods, which can be solved
simultaneously to determine the electron energy level populations. For the
plasma conditions of interest in this paper, the plasma density is low enough
that it can be assumed to be optically thin for the visible emission lines of
interest, and effects of radiation transport can subsequently be neglected.

A complication for the analysis of spectra of optically thin plasmas to
experimental measurements is that most practical experimental light collection
schemes lead to the collection of light from a finite volume of the plasma,
typically a narrow line-of-sight captured by a viewing lens or collimated-
fiber-optic that delivers light from the plasma to a spectrometer. Thus, the
recorded spectrum contains contributions from different spatial regions of the
plasma, which may in general correspond to different physical (local)
densities, temperatures, etc. As a consequence, a simplifying modeling
assumption is made along the spatial profile of the discrete plasma conditions
that assumes QoI homogeneity along the spectrometer’s line of sight, thus
allowing for the inverse problem to be solved over the entire line-integrated
volume of plasma observed. While it is possible to enrich this simplifying
assumption by either modifying the underlying model (and hence usually its
modeling assumptions) or by adding additional experimental diagnostics, the
resulting problem is still more than likely to remain largely under-determined
Craig and Brown (1976).

In this paper the atomic spectra are obtained using the CR model PrismSPECT
MacFarlane  _et al._ (2003). PrismSPECT is a non-local (or local)
thermodynamic equilibrium collisional-radiative model that calculates emission
and absorption spectrum assuming a uniform plasma given a fixed electron
temperature $T_{e}$ and density $\rho$ Sawada (2008); Han  _et al._ (2015).
The Bayesian inference model utilized by SpectAI is independent of the
underlying model generating the spectra that are being compared. Thus, in
principle, these inputs can be obtained using any atomic emission and
absorption spectra model. For example, other collisional radiative based
models that could, in principle, be used include FLYCHK Chung  _et al._ (2003,
2005), ATOMIC Hakel  _et al._ (2006), ABAKO Florido  _et al._ (2009), OPAL
Iglesias and Rogers (1991), OPAS Mondet  _et al._ (2015), SCO-RCG Nagayama
_et al._ (2016), etc., or see Piron  _et al._ (2017) for a broad overview and
comparison of available codes/models.

####  II.2.2 Measurement model

The _measurement model_ utilizes the _base physics model_ while accounting for
two additional separate issues. First, the instrument model introduces
spectral broadening present beyond the physical broadening mechanisms
represented in the base physics model, the instrument being limited in
spectral resolution for example depending on the groove density of its
diffraction grating. Second, the measurement model accounts for the data
conversion between what the spectrometer records intensity in (i.e. as a
function of pixel index), and the corresponding wavelength used for
interpreting the corresponding base physical model output. In principle, other
effects could be treated, but we restrict to these two in the present work.

These measurement features are essentially characteristics of the observation
equipment—i.e., the spectrometer and any associated infrastructure. Thus, in
principle, they can be determined by an instrument calibration process to a
known calibration source, which is commonly done. With a good calibration, one
can then assume that these features are simply fixed for the Bayesian analysis
described in §II.3, which we do here. However, since even small mismatches
between the observed and modeled spectra can significantly affect the
inference results, mismatches that could occur for example from a slight
rotation of the diffraction grating in the positioning system, we have found
it beneficial to include a method to estimate these features using the
observations of interest, rather than using separate calibration data. These
techniques are described below.

##### Instrument line broadening

The effect of instrument broadening is represented by convolving a Gaussian
function with the output of the model from section §II.2.1. Specifically,
given the output of the base physics model, $\mathcal{M}_{b}(\lambda;\theta)$,
the model predicts that the spectrum that should be observed when accounting
for instrument broadening is given by

$\mathcal{M}(\lambda;\theta,\sigma)=\int_{-\infty}^{\infty}\mathcal{M}_{b}(\tilde{\lambda};\theta)\,g(\lambda-\tilde{\lambda};\sigma)\,\mathrm{d}\tilde{\lambda},$
(1)

where

$g(s;\sigma)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(-\,\frac{1}{2}\frac{s^{2}}{\sigma^{2}}\right),$
(2)

and the parameter $\sigma$ is the standard deviation, which determines the
broadening width.

##### Pixel to wavelength mapping

The intensity measurement is recorded as a function of pixel index. In order
to compare this to the model $\mathcal{M}(\lambda;\theta)$, the pixel index
$p$ must be mapped to the corresponding wavelength. This is provided by a
generic function $\lambda(p)$, where the function $\lambda(p)$ is estimated
from measurements recorded for a known source. Thus, this map is typically
provided independent of the measurements of interest. In this work, we use a
linear mapping:

$\lambda(p)=mp+b,$ (3)

where the parameters $m$ and $b$ may be set by the analyst. However, it has
been observed that even small deviations in the parameters $m$ and $b$, or
small modeling errors that lead to shifts in the modeled spectrum, can
contaminate the inference process and lead to temperature inferences that are
implausible. To avoid this problem, the parameters $m$ and $b$ can also be
calibrated using the observed spectrum and the modeled spectrum at a selected
condition. As with $\sigma$, this calibration could be incorporated into the
Bayesian inference, but this complication is beyond the scope of the present
work. Instead, the parameters $m$ and $b$ are determined using a deterministic
least-squares solve using peak locations from a chosen experiment and single
simulation. That is, given the highest $N$ intensity peaks at locations
$\lambda_{k}$, $k=1,\ldots,N$ in the selected simulation and the corresponding
$N$ highest peaks at pixels $p_{k}$, $k=1,\ldots,N$ in the observed spectrum,
$m$ and $b$ are given by

$(m,b)=\arg\min_{\hat{m},\hat{b}}\sum_{k=1}^{N}\left(\lambda_{k}-\hat{m}p_{k}-\hat{b}\right)^{2},$
(4)

where the peaks are ordered according to increasing wavelength in the
simulation and increasing pixel in the observations. This aligns the peaks in
the selected simulation with those in the experiment as well as possible given
the linear fit. Example results for this inference are shown in Fig. 4

###  II.3 Probabilistic Model

The resulting likelihood from the Bayesian procedure encodes the plausibility
of the observation for given values of the parameters, accounting for
uncertainties in both the observations and the model. Thus, to formulate a
likelihood, we require a probabilistic model of the discrepancy between the
model and the observation. Such a model could be the source of nearly endless
development and debate. Here, we adopt a practical approach that does not
require detailed information about sources of error in either the observation
or the physical model. As such, it is applicable in situations where the
details of the precise sources of error are not well-understood, as is common
in practice with complex experiments and models. However, the approach does
account for known characteristics of data produced by emission spectroscopy.
Specifically, we assume that both the observations and the model predictions
consist of a set of strong “lines” embedded in a background signal. An example
of such an observation is shown in Figure 7. The likelihood model used here is
based upon a simple algebraic model of functions that have such features.

The algebraic spectrum model takes the following form:

$\mathcal{I}(\lambda)=\sum_{i=1}^{N}s_{i}g(\lambda;\lambda_{i},w_{i})+b(\lambda),$
(5)

where $\mathcal{I}$ is the spectral intensity, $\lambda$ is the wavelength,
$N$ is the number of lines, $s_{i}$ is the strength of the $i$th line, $g$ is
a line shape function, $\lambda_{i}$ is the location of the $i$th line,
$w_{i}$ is the width of the $i$th line, and $b$ is the background. For the
remainder of this work, we assume that the background is negligible or can be
exactly removed prior to the analysis, and thus it will be neglected going
forward.

In designing a likelihood model, the task is to assign probabilities to
observed differences. Toward this end, consider the difference between two
spectra. Using superscripts to distinguish between these two spectra and
assuming (5) is appropriate for both, their difference can be written as

$\delta\mathcal{I}(\lambda)=\mathcal{I}^{(1)}(\lambda)-\mathcal{I}^{(0)}(\lambda)=\sum_{i=1}^{N_{1}}s^{(1)}_{i}g^{(1)}(\lambda;\lambda^{(1)}_{i},w^{(1)}_{i})-\sum_{i=1}^{N_{0}}s^{(0)}_{i}g^{(0)}(\lambda;\lambda^{(0)}_{i},w^{(0)}_{i}).$

Spectra that are “close” will have the same number of lines, $N_{1}=N_{0}=N$,
and be described using the same line shape functions $g^{(1)}=g^{(0)}=g$. In
this case,

$\delta\mathcal{I}(\lambda)=\sum_{i=1}^{N}s^{(1)}_{i}g(\lambda;\lambda^{(1)}_{i},w^{(1)}_{i})-s^{(0)}_{i}g(\lambda;\lambda^{(0)}_{i},w^{(0)}_{i})$

Finally, we can further simplify by linearizing this result—i.e., by writing
spectrum (1) as a Taylor series expansion around spectrum (0) parameters and
dropping all terms higher than first order. Specifically,

$\displaystyle\delta\mathcal{I}(\lambda)$
$\displaystyle=\sum_{i=1}^{N}(s^{(0)}_{i}+\delta
s_{i})(g(\lambda;\lambda^{(0)}_{i}+\delta\lambda_{i},w^{(0)}_{i}+\delta
w)-s^{(0)}_{i}g(\lambda;\lambda^{(0)}_{i},w^{(0)}_{i})$
$\displaystyle\approx\sum_{i=1}^{N}\delta
s_{i}g(\lambda;\lambda_{i},w_{i})+s_{i}\frac{\partial
g}{\partial\lambda_{i}}\delta\lambda_{i}+s_{i}\frac{\partial g}{\partial
w_{i}}\delta w_{i},$

where $\delta s_{i}=s_{i}^{(1)}-s_{i}^{(0)}$,
$\delta\lambda_{i}=\lambda_{i}^{(1)}-\lambda_{i}^{(0)}$, $\delta
w_{i}=w_{i}^{(1)}-w_{i}^{(0)}$.

This development serves as the basis for the probabilistic model of the
discrepancy between the observed and modeled spectra. Specifically, the model
that follows is given by:

$\mathcal{I}_{obs}(\lambda)-\mathcal{M}(\lambda;\theta)=\sum_{i=1}^{N}\delta
s_{i}g(\lambda;\lambda_{i},w_{i})+s_{i}\frac{\partial
g}{\partial\lambda_{i}}\delta\lambda_{i}+s_{i}\frac{\partial g}{\partial
w_{i}}\delta w_{i}+\epsilon$ (6)

where

$\displaystyle\delta s_{i}\sim N(0,\eta^{2}),\quad\delta\lambda_{i}\sim
N(0,\ell^{2}),\quad\delta w_{i}\sim N(0,\sigma^{2}),$

and $\epsilon$ is a zero-mean white noise Gaussian process. Clearly this model
has a number of parameters: $N$; $s_{i}$, $\lambda_{i}$, and $w_{i}$ for
$i=1,\ldots,N$; $\eta$, $\ell$, $\sigma$, and $\beta$. Determination of these
hyperparameters is described below. Once these values are fixed, the
probabilistic model is entirely defined by the distributions for $\delta
s_{i}$, $\delta\lambda_{i}$, $\delta w_{i}$ and $\delta b$, which are
Gaussian, the form (6), which is linear in these parameters, and the additive
Gaussian noise $\epsilon$. Thus, the resulting probabilistic model of the
discrepancy and the resulting likelihood is a Gaussian process (GP).

###  II.4 Hyperparameters

The parameters of the likelihood model, i.e., the hyperparameters, can be
split into two qualitatively different groups. The first group includes $N$,
$s_{i}$, $\lambda_{i}$, and $w_{i}$. These parameters characterize the
original spectrum that the likelihood model is describing deviations from. As
such, these parameters are identified based on the original observation. We
use standard, “off-the-shelf” techniques to determine these parameters.
Specifically, we use the `find_peaks` function from the `scipy.signal` package
Virtanen  _et al._ (2020), that utilizes standard peak prominence heuristics
to determine peak locations. To avoid spurious peaks due to noise in the
observations, peaks are required to have a height that is at least 0.03 times
the maximum intensity in the observation window, and peaks are required to
have a prominence of 0.01 times the maximum intensity in the observation
window.

The second group of parameters includes $\eta$, $\ell$, $\sigma$, and $\beta$.
These parameters control how the stochastic model assigns probability around
the result of the deterministic model. These parameters are determined in a
preprocessing step that occurs prior to the Bayesian inference. First,
reasonable values of the physical model parameters are estimated using
standard least-squares minimization. Then, denoting these initial parameter
values by $\theta^{*}$, the misfit $\varepsilon^{*}$ is computed via
$\varepsilon^{*}=\mathcal{I}_{obs}-\mathcal{M}(\theta^{*})$. Given this
misfit, the parameters $\delta s_{i}$, $\delta\lambda_{i}$, $\delta w_{i}$,
and $\delta b$ can be determined by solving a linear least-squares problem to
minimize the discrepancy between the misfit model (6) and the computed
$\varepsilon^{*}$. Specifically, we solve the following optimization problem

$\delta s^{*}_{i},\delta\lambda^{*}_{i},\delta w^{*}_{i}=\arg\min_{\delta
s_{i},\delta\lambda_{i},\delta w_{i}}\|\varepsilon^{*}-e(\delta
s_{i},\delta\lambda_{i},\delta w_{i})\|,$ (7)

where

$e(\delta s_{i},\delta\lambda_{i},\delta w_{i})=\sum_{i=1}^{N}\delta
s_{i}g(\lambda;\lambda_{i},w_{i})+s_{i}\frac{\partial
g}{\partial\lambda_{i}}\delta\lambda_{i}+s_{i}\frac{\partial g}{\partial
w_{i}}\delta w_{i}.$ (8)

This leads to $N$ (the number of lines) values for each parameter. Then, the
standard deviation paramters $\eta$, $\ell$, and $\sigma$ are determined by
taking the observed standard deviation of this best fit over all the lines.

###  II.5 Summary

To summarize the overall approach, the solution of the inverse problem is
given by the posterior probability distribution for the plasma density and
temperature, $\rho$ and $T$, given the observed intensity, $I$:

$\pi(\rho,T|I)\propto\pi(\rho,T)\,\pi(I|\rho,T),$

where the prior $\pi(\rho,T)$ is uniform over a range of conditions and the
likelihood $\pi(I|\rho,T)$ which incorporates the physical, measurement, and
probabilistic models described in § II.2-§II.4. Since the prior is taken to be
uniform here, the posterior is proportional to the likelihood function. The
likelihood model is Gaussian in the intensity, but, because the forward model
relating the density and temperature to the intensity is nonlinear, the
resulting posterior probability over density and temperature is not
necessarily Gaussian and cannot be sampled easily. While specialized
algorithms are available for exploring such distributions, such as various
Markov chain Monte Carlo (MCMC) methods Martino (2018), these algorithms
require the evaluation of the forward model inside the sampling loop. Since
the forward model evaluation is the most computationally expensive component
of this approach, and since the space of interest is only two-dimensional, we
choose a grid-based approach instead. In this approach, the model is run on a
grid in density, temperature space a priori and then the posterior is
evaluated on that same grid. In this way, a single set of simulations can be
used to support the inference for a large number of experimental observations,
which effectively amortizes the computational effort over the entire
experimental campaign and dramatically reduces the time required for the
inference on any single experiment. Thus, given an observed spectrum, the
steps to evaluate this posterior probability are as follows:

  1. 1.

Run a set of physical model simulations—here using PrismSpect, as described in
Section II.2—to compute the predicted spectrum for a set of plausible
$\rho_{i},T_{i}$ conditions, assuming the gas composition is known.

  2. 2.

Determine the measurement model hyperparameters (see Section II.2.2),
including:

     * •

The line broadening parameter, $\sigma$, and

     * •

The pixel to wavelength mapping parameters, $m$ and $b$.

  3. 3.

Determine the probabilistic model hyperparameters, as discussed in Section
II.4.

  4. 4.

Evaluate the posterior at points where the simulations where run,
$\pi(\rho_{i},T_{i}|I)$, by evaluating the likelihood given the simulation
results and observations, as described in Section II.3.

As discussed previously, step 1 need only be performed once for a given set of
experiments at similar conditions.

##  III Experimental Observations

The inference framework described above is applied to the analysis of results
from a series of experiments at PLX. In these experiments, two unmagnetized
argon plasma jets are collided head-on at varied (relative fixed) velocity and
density, to study the collisionality of the supersonic plasmas under these
conditions. The plasma behavior under these conditions is relevant to the
prospects of PLX / PJMIF as a spherical compression driver, specifically as
related to occurrence of collisional plasma shocks vs. kinetic
interpenetration of the jets. These phenomena have been observed in previous
studies on PLX. Emission spectroscopy is collected from these experiments, and
is an important diagnostic in the overall experiment for constraining the
plasma electron temperature and average ionization state $\bar{Z}$.

Figure 1 shows a diagram of the experimental setup and spectroscopic
diagnostic layout. Light is collected from the central volume of the colliding
plasma jets and imaged onto the entrance slit of a Chromex 500is imaging
spectrometer, which is coupled to a PCO Dicam Pro intensified CCD camera. The
spatial direction resolved by the spectrometer is aligned with the jet
collision axis, and it was envisioned that this would provide spatially
resolved data across plasma shocks when they occurred. In implementation,
spatial resolution was significantly blurred by effects of perspective
distortion in the entrance optics viewing the transparent plasma, so effective
spatial resolution was not obtained. The data therefore constitute a
spatially-integrated measure over the plasma volume. Data are temporally
resolved by control of the camera exposure time and delay, acquiring one frame
of data per shot. Shots are repeated with varied camera delays to build up a
time-resolved evolution of the emission spectrum at each jet operating
condition.

Figure 1: A diagram of the experimental setup of the jet merging chamber that uses plasma guns from PLX. The purple highlighted region indicates the spectrometer field of view. Shot numbers | P inlet (psig) | V (kV) | avg. jet velocity (km/s)
---|---|---|---
7328-7365 | 50 | 4 | $16\pm 1.7$
7366-7398, 7497-7504 | 30 | 4.5 | $28\pm 4.1$
7399-7433 | 40 | 4.25 | $26\pm 5.6$
7434-7470 | 45 | 4.25 | $23\pm 3.7$
7471-7496 | 35 | 4.25 | $27\pm 5.9$
7505-7540 | 20 | 4.5 | $52\pm 1.5$
Table 1: Experimental shot settings, and average measured plasma jet velocity

A series of shots are performed spanning different jet operating conditions,
summarized in Table 1. The main experimental settings that are varied are the
charge voltage of the plasma gun capacitor bank, and the inlet pressure to the
gas puff valve. With higher bank voltage, more accelerating current is
delivered, resulting in a higher velocity. With higher inlet pressure, more
gas is injected into the gun, resulting in a heavier jet and ultimately a
slower velocity.

Figure 2 shows raw data obtained from an experimental shot, and the same data
processed compared to a Prismspect simulated spectrum. Strong lines of singly-
ionized argon are consistently observed across many experimental operating
conditions, with relatively subtle changes in the spectrum from shot to shot.
This is expected, as the radiative cooling from the ionized argon increases
significantly with temperature in this range $\sim 2$ eV, and acts to clamp
the temperature from rising greatly above this level. A colder liner
temperature is typically desireable in PLX / PJMIF to keep liner sound speed
and density expansion low, and to increase the liner Mach number. This dataset
is analyzed using the developed inference approach.

Figure 2: (a) Raw data image from shot 7337. (b) Processed and binned data
from shot 7337, overlaid with PrismSPECT simulated spectrum. Visible strong
lines of singly-ionized argon labeled.

##  IV Results

The analysis can be decomposed into two phases: (1) a pre-processing phase,
where the line broadening and pixel to wavelength map parameters (§II.2) and
probabilistic model hyperparameters (§II.4) are determined, and (2) the
Bayesian inference phase. Sample results from each phase are presented for
cases identified in Table 1.

###  IV.1 Pre-processing Phase Results

To determine the pixel to wavelength mapping, the least-squares problem
defined in (4) is solved. The data for this problem is determined using
standard peak finding techniques111Specifically, we use the find_peaks
function from the scipy.signal package. See
https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.find_peaks.html.
Figure 3 shows example results from the peak finding phase for

(a) Observed intensity versus pixel for PLX shot 7330.

(b) Modeled intensity versus wavelength example.

Figure 3: Intensity versus pixel and wavelength with peaks indicated by the
vertical dashed lines.

solving the resulting least-squares problem gives the pixel with wavelength
mapping shown in Figure 4. The fit succeeds for this case and there is no
suggestion that a richer parameterization of the mapping is necessary.

Figure 4: Pixel to wavelength mapping determined by fitting peak wavelengths
as a linear function of peak pixels.

###  IV.2 Bayesian Inference Results

The inference results are split into two sections. First, to give a sample of
the full picture of information that is obtained, joint posteriors for density
and temperature for individual shots are shown. The features observed in the
posterior as well as the fits between the PrismSpect predictions and the
measured spectra are examined. Second, we use the results to look for trends
in the measured temperature as a function of the experimental settings. These
results reinforce the importance of accounting for uncertainty in the analysis
of such measurements.

####  IV.2.1 Individual shot results

Figure 5 shows the joint posterior probability for six shots taken on October
1, 2021. These shots—specifically, 7330, 7331, 7337, 7338, 7343, and 7344—are
all at the same nominal conditions of 50 psig inlet pressure for the gas puff
valve and 4 kV voltage of the plasma gun capacitor bank. The spectra are
measured at different camera delay times, with 7330 and 7331 using a delay of
58 $\mu$s while 7337 and 7338 have a 52 $\mu$s delay and 7343 and 7344 have a
46 $\mu$s delay. Dependencies on camera delay time and other parameters are
examined more closely in Section IV.2.2. Here, these shots are used as
representative individual spectra, and the results show some notable features.
First, the uncertainty is not negligible. The range of non-negligible
posterior probability generally spreads over an interval of approximately 0.5
eV. Second, the posterior probability does not always concentrate around a
single peak. The results from shots 7337 and 7338 show this clearly. Since the
Bayesian approach provides a full probability distribution as the solution of
the inverse problem, such behavior is allowed when multiple conditions fit the
data similarly well, even if they are separated in parameter space.

Figure 5: Joint posterior probability distributions for temperature and
density inferred from spectra recorded during six PLX shots on 10/1/2021.

To get a better sense of the inferred temperature, since this is often the
primary quantity of interest of such measurements, the joint posterior is
marginalized over the density to give the marginal posterior distribution for
the temperature alone; specifically,

$\pi(T|I)=\int\pi(\rho,T|I)\,d\rho.$

Figure 6 shows these marginal distributions for the same shots as in Figure 5.

Figure 6: Marginal posterior probability distributions for temperature
inferred from spectra recorded during six PLX shots on 10/1/2021.

Again, the marginal probability results demonstrate that the uncertainties are
signficant. Qualitatively speaking, these result from the fact that model
results at multiple conditions provide similar fits to the measured spectrum.
Figure 7 demonstrates this fact using shot 7330 as an example. The figure
shows that over the range of highly probable conditions, as determined by the
inference process, the spectrum is qualitatively very similar. Thus, given the
inherent measurement precision and modeling errors, it is difficult to
determine the conditions more precisely given the available information.

Figure 7: Comparison of the measured and modeled spectrum for shot 7330.

####  IV.2.2 Shot sequence results

In addition to examining individual shots, the spectra obtained at different
conditions can be analyzed in an attempt to identify the dependency of the
state of the plasma upon various experimentally controllable parameters. As an
example, Figure 8 shows the dependence of the inferred temperature on the
camera delay time, which corresponds to the time in the evolution of the
plasma formed in the core of the merging jets, for the different inlet
pressure and capacitor bank voltages shown in Table 1. The results show that,
with the present measurements and models, it is very difficult to identify any
meaningful trends, because the uncertainty is large enough that it is unclear
whether changes in the most probable temperature between conditions are real.

Figure 8: Inferred temperature as a function of camera delay time for various
inlet pressures and capacitor bank volrages. The symbols indicate the maximum
a posteriori temperature and the uncertainty bar indicates the $95\%$
credibility interval for $T_{e}$. Some conditions are repeated, leading to
different results at the same nominal conditions.

To evaluate the uncertainty estimates, we note that the data set includes some
conditions that are repeated. These repeats lead to a scatter in the results
at the same delay times shown in the figure, but all the repeats fall within
the uncertainty of each other. This gives some confidence that the uncertain
estimates from the inference are not too small. Further, the scatter between
repeats is roughly of the same magnitude as the uncertainty, indicating that
our uncertainty estimates are not dramatically too large. Thus, we are forced
to conclude that any trends in the temperature are swamped by the uncertainty
and cannot be determined from these measurements alone.

##  V Conclusions

In this study, we have presented an approach for the automated estimation of
plasma temperature and density from emission spectroscopy, integrating
physical and probabilistic models by way of Bayesian inference. The robustness
of our approach lies in its ability to assimilate complex physical models with
probabilistic reasoning, offering a nuanced understanding of plasma
characteristics. The detailed overview of Bayesian inference, coupled with the
foundational physical models, set the stage for a methodological framework
that is both rigorous and adaptable.

Our experimental results on a series of experiments at PLX have yielded
promising results and consist of a pre-processing phase and a Bayesian
inference phase. The results from this inference, particularly the individual
shot and shot sequence analyses, have demonstrated the efficacy of our
approach in providing accurate and reliable estimations of plasma parameters.
These results underscore the potential of Bayesian methods in the field of
plasma diagnostics, often offering a significant leap over traditional
estimation techniques.

The current state of the technique offers many paths for possible extensions
and improvements. First, for practical reasons, the inferences of some
parameters—e.g., the hyperparameters of the probabilistic model—are conducted
separately from the primary inference for the plasma properties. These
inferences could be unified, allowing the effects of calibration uncertainties
to be included and quantified. Second, the process could be extended to enable
inference for trace species, enabling the identification of contaminants. More
broadly, there is significant potential for adoption of these techniques in
plasma research and other related fields. For instance, the ability to
accurately estimate plasma temperature and density is not only critical for
understanding plasma behavior but also pivotal for advancing nuclear
instrumentation and its applications.

In conclusion, this study contributes a novel and effective tool for the field
of plasma diagnostics, harnessing the power of Bayesian inference and
sophisticated physical modeling. It opens up new avenues for research and
application, promising to enhance the precision and understanding of plasma
processes, as well as finding application in other areas of experimental
analysis, such as analytical chemistry, etc. Future work will focus on
refining the models further and exploring the integration of additional data
sources, to extend the capabilities and applicability of the methodology.

##  VI Acknowledgements

This work was partially supported by U.S. ARPA-E Award No. DE-AR0001265. This
manuscript has been authored in collaboration with Los Alamos National
Laboratory/Triad National Security, LLC, Contract No. 89233218CNA000001, with
the U.S. Department of Energy/National Nuclear Security Administration. The
views and opinions of authors expressed herein do not necessarily state or
reflect those of the United States Government or any agency thereof.

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