Robust analysis of space-, time-, and energy-resolved soft x-ray measurements of magnetically confined fusion plasmas (invited) Free

The multi-energy soft x-ray (ME-SXR) project aims to develop a single versatile x-ray diagnostic suitable for multiple applications in the study of magnetically confined fusion plasmas. Developed in cooperation with the Princeton Plasma Physics Lab, the diagnostic is based on a compact pinhole camera using a custom calibration, which allows the lower threshold for photon detection to be adjusted on a per-pixel level. This allows for a novel synthesis of spatial resolution, spectral resolution, and versatility by carefully adjusting the pixel thresholds in situ to configure the detector to best suit a given physics task (Sec. II). Multiple applications of the detector are presented in this paper, including studying profile evolution (Sec. III), diagnosing impurities (Sec. IV), characterizing T_e (Sec. V), and analyzing non-thermal populations (Sec. VI). A method for integrating other ME-SXR diagnostics using a Bayesian framework is also presented (Sec. VII).

Following a proof-of-concept implementation at Alcator C-Mod,¹ the ME-SXR diagnostic was developed and tested on the Madison Symmetric Torus (MST).² The MST is a medium-sized reversed-field pinch (RFP) located at the campus of the University of Wisconsin-Madison. It features a unique conductive aluminum shell that permits self-organizing behavior. This also results in the presence of significant population of aluminum ions in the plasma bulk, affecting the soft x-ray spectrum. The defining characteristic of the reversed-field pinch (RFP) is that the magnetic field is mostly toroidal in the core, like the tokamak, but twists around as you move out in radius before reversing near the edge. This results in the presence of unstable m = 1 tearing modes in the core, which increases plasma stochasticity and transport. However, by making use of an improved confinement scenario called Pulsed Poloidal Current Drive^3,4 (PPCD), the MST can produce fusion-relevant plasmas, achieving temperatures as high as 2 keV and densities of ∼10¹⁹ m⁻³. A significant population of soft x rays (SXRs) are emitted in this regime, resulting primarily from bremsstrahlung and radiant recombination between the electron and ion populations. The presence of partially ionized aluminum ions from the shell also contributes strong emission lines to the spectrum at around 2 keV. These properties make the MST an interesting testbed for a new soft x-ray diagnostic. Additionally, a robust set of existing diagnostics allows for complementary measurements and useful comparisons.

Three additional papers relevant to the ME-SXR project are contained in these proceedings: Delgado-Aparicio et al. further consider the analysis of data on the MST; Chellai et al.⁵ describe the calibration of a ME-SXR detector for the WEST tokamak; and Barbui et al.⁶ describe the calibration of a multi-energy hard x-ray (ME-HXR) detector for WEST.

The ME-SXR diagnostic is based on PILATUS3 100K-M, a pixelated hybrid photon-counting x-ray detector produced commercially by DECTRIS Ltd. and specifically modified for the Princeton Plasma Physics Laboratory (PPPL). The device is composed of a single 450 μm Si sensor that absorbs incident photons, producing a cloud of photoelectrons with a total charge proportional to the photon energy.⁷ This charge is then transferred via a bump-bonded indium connection to one of the many charge-sensitive preamplifiers (CSAs) located on one of the 16 application-specific integrated circuits (ASICs) that compose the detector. The charge is converted to a pulse that is discriminated against a threshold by a comparator, rejecting photons with energies below the threshold. The threshold is controlled by a global comparator V_cmp setting but can be further adjusted, or trimmed, on an individual level by an additional setting stored in a six-bit register called the “trimbit” setting. Pulses that pass this threshold are recorded into a 20-bit counter and read out at pre-set intervals. These ASICs are arranged in an 8 × 2 grid, each containing an array of 60 × 97 individual pixels (each with their own CSA, comparator, trimbit setting, and counter), leading to a total of 480 × 194 = 93 120 pixels (often referred to as 100k). Individual pixels have an effective area of 172 × 172 μm². The detector is installed with a radial view of the plasma through a rectangular pinhole-like opening.

The silicon thickness sets an upper limit of about 30 keV, and the lower limit is set by placing a 25 μm Be filter over the pinhole. An additional 100 μm Mylar filter was also installed to reduce the overall photon flux and prevent saturation.⁸ The integration time can be set to as low as 1 ms, and the electronics readout takes an additional 1 ms, resulting in an effective time resolution of up to 500 Hz. Spatial resolution is limited by the pinhole geometry to about 1 cm in the core. An in situ spatial calibration was performed by exposing the detector to an Fe-55 source attached to a retractable probe.

Energy thresholds were calibrated via a custom procedure by exposing the detector to a series of fluorescence targets with strong lines at pre-selected energies and scanning the trimbit settings.⁹ These data were used to create a mapping between energy threshold (E_c) and trimbit setting for each pixel. Multiple custom multi-energy configurations were developed, including high signal-to-noise ratio (SNR) 1D imaging, high radial resolution 1D imaging, and 2D “metapixel” imaging.¹⁰ 

In normal operation, ME-SXR records a series of images at the selected cycle rate, usually 500 Hz. Once an image is recorded, the data can be separated into one-dimensional SXR emission profiles in the poloidal plane according to the threshold. The evolution of these profiles over time can provide significant insight into the heating and profile evolution of a plasma.

An example of data taken with eight simultaneous thresholds is shown in Fig. 1, which depicts the evolution of a single PPCD (improved confinement) plasma. Three full profiles are shown, with lower thresholds of 4, 5, and 6 keV, and the evolution of the core-most chord is shown for all eight thresholds. Initially, the SXR emissivity is very low, corresponding to a low T_e. However, once the PPCD period begins, driving J_∥ and thereby suppressing core tearing mode activity and reducing thermal transport, the plasma emissivity begins to increase rapidly. This emissive structure gradually broadens as the plasma bulk continues to heat. This comes to a sudden end at ∼21 ms, once the PPCD banks have discharged all of their available energy. The “improved confinement” period rapidly collapses as the return of tearing modes allows the stored thermal energy to rapidly transport out of the core, cooling the plasma and thereby terminating the emissivity structure.

The ME-SXR diagnostic can also be used to identify the presence of mid-Z impurities in the plasma. Such impurities will be partially ionized and typically feature strong emission lines comfortably within the SXR range, which can be discerned by a proper selection of thresholds. This was tested by injecting small amounts of argon gas into non-reversed plasmas as a pre-fill, allowing the gas to diffuse throughout the vacuum vessel before the plasma is formed. For typical MST parameters, Ar features strong emission lines around 3 keV.¹¹ The ME-SXR diagnostic allows us to set thresholds both above and below this energy allowing for a direct detection of these photons.

Figure 2 shows single-frame eight-color 1D measurements for two similar plasmas, except that (b) features the argon dopant and (a) is argon-free. The counts for E_c ≤ 3 keV increase substantially when argon is present, while, for higher energies, the measurements are comparable. This comparison is shown more clearly in Fig. 2(c), which shows the photon counts (averaged over five central chords) vs the threshold energy for each case. This clearly demonstrates that the increased signal is entirely due to Ar emission lines, meaning that the concentration of argon is low enough to not substantially change Z_eff (which would affect all thresholds). This demonstrates that the ME-SXR diagnostic can be used as an ad hoc spectrometer to diagnose the presence of mid-Z like Ar impurities in the plasma.

The difference in the measured spectra can be used to positively identify argon as the source of increased emissivity. Since the two spectra are very similar for E_c > 3 keV, we will assume that the entire difference in signal is due to the presence of an additional source of photons of a single characteristic energy, E₀. We will denote the two spectra shown in Fig. 2(c) as y_Ar and y₀ for the Ar-doped and clean measurements, respectively. Then, the difference, d = y_Ar − y₀, approximately forms an S-curve, just like those encountered during the energy calibration procedure.⁹ This is shown as the data points in Fig. 3. Points for E_c > 4 keV were set to zero since Ar lines will not contribute significant signal for these thresholds.

The S-curve traced out by d can be fit directly to an S-curve model like that used in energy calibration. The chosen model includes a simple linear model of charge-sharing⁸ and an S-curve width of σ_E = 0.3 keV, both determined by calibration. That leaves a model with two free parameters, the amplitude N₅₀ and the source energy E₀, given by

The charge-sharing slope k is related to the line energy k = −2f/E₀, where f = 0.266 is the ratio of the pixel area where charge sharing occurs to the total pixel area (see Delgado-Aparicio et al.’s³³ contribution to these proceedings for a more thorough description of this treatment).

In order to properly account for counting statistics, the model parameters were determined using Bayesian inference¹² (discussed in more detail in Sec. VII) with a Poisson likelihood function,

Here, d_i is the data point for the ith chord and the parameters N_i = N (E_c,i|N₅₀, E₀) are equated to Eq. (1) for the appropriate threshold. We will assume independent uniform priors, $N50∼U(300,700)$ and $E0(keV)∼U(2,5.5)$⁠, denoted π(N₅₀) and π(E₀). Then, the posterior distribution is given by Bayes’s rule as $p(N50,E0|d)∝L(N50,E0)π(N50)π(E0)$⁠. Since we are only interested in the source energy, we will marginalize N₅₀ out as a nuisance parameter to obtain

This distribution is shown by the red curve in Fig. 3.

The resulting posterior distribution is very narrowly peaked, with a 1σ credible interval of E₀ = 3.05 ± 0.02 keV, though this estimate does not account for variability between plasma discharges. This E₀ is very close to the energy of the brightest line of Ar⁺¹⁶. Since the He-like state is often the most common ionization state of mid-Z impurities in the plasma core, this measurement would be sufficient to identify the impurity (or at least narrow the range of candidates). Figure 3 also shows the fit between the model [Eq. (1)] and the measured data d, with error bars representing the 95% confidence level. Agreement is not perfect, but some discrepancy is expected given that the reference signal y₀ was taken from a different plasma discharge than y_Ar. In situations where the variability between plasma discharges can be further reduced, accuracy can presumably be improved. PILATUS3 can be calibrated for thresholds up to E_c > 20 keV, making this technique applicable to a wide range of impurities.

There are multiple techniques that could be employed to infer the electron temperature profile from ME-SXR data. In situations where circular symmetry can be assumed, a straightforward Abel inversion can be used to extract the emissivity profile¹³ for each threshold, which is then related to T_e directly via ratios. The method presented here is similar to this approach but employs a Bayesian methodology to systematically propagate uncertainty and account for non-cylindrical flux surface geometries. It relies upon prior knowledge of the profile shape, which can be a drawback in some situations.

In general, the count rate detected by a cluster of pixels indexed by i on ME-SXR with a shared threshold E_c and plasma volume is related to the local photon emissivity rate via line integral,

where p and ϕ are the tangency radius and angle that parameterize the chord, η_i is the pixel étendue,¹⁴ and z parameterizes the distance along the chord.

This equation can be recast as a matrix equation that maps the emissivity as a function of flux radius ɛ_γ(ρ) to the measured photon count rates for each cluster of pixels N_γ,i using a flux surface reconstruction code (such as MSTfit¹⁵).

The next step is to choose a model for the emissivity profile. We will select a three-parameter α − β profile that is commonly used for the RFP.¹⁶ This profile shape assumes a relatively flat core that then decreases monotonically around the mid-radius before going to zero at the edge,

This shape has the advantage that it forces the core of the profile to be flat, stabilizing the ratio. However, parameterizations of other forms could be used instead, or possibly even nonparametric models based on Gaussian processes.¹⁷ 

Next, the task is to constrain the profile parameters given the measured data by sampling the posterior distribution. Sampling also allows for a direct computation of profile uncertainty. This procedure was performed for a single frame recorded during 300 kA PPCD operations, and the resulting emissivity profiles for a single 300 kA PPCD plasma are shown in Fig. 4(a). The profile is flat in the core out to r/a ≈ 0.4 before decreasing to near-zero at around r/a ≈ 0.8.

These emissivity profiles, and their associated uncertainties, can be used to produce estimates of the electron temperature by considering the ratio of the observed emissivity at a given point in the radial profile relative to a chosen reference threshold. If we assume that the threshold energy is sufficiently high to justify ignoring emission lines and recombination steps, then the emissivity (up to a multiplicative constant) is given by the energy integral $I$

where R(E; E_c) is the total pixel response function that accounts for transmission through filters, absorption into Si, and energy discrimination due to detector electronics as a function of the chosen threshold (or “cutoff energy”) E_c. When taking the ratio of two emissivity profiles, the multiplicative constants cancel out, leaving a ratio of the energy integrals,

where E_ref is the threshold corresponding to the reference profile selected to serve as the denominator.

Using these ratio curves, the electron temperature may be directly inferred. The measured ratio for each threshold >3 keV was used to infer T_e, and these individual inferences were averaged (weighted by uncertainty) to produce an overall estimation of T_e and its uncertainty. This process was repeated at each radial point to produce a profile. The resulting profile is shown in Fig. 4(b) along with corresponding Thomson scattering points for reference. Comparison in the core to the mid-radius is generally favorable, but at r/a = 0.6, the profile begins to increase unphysically. However, soft x-ray emission from this far out in the radial profile tends to be very low due to the declining electron density, so this portion of the profile is mostly constrained by low count rates and is, therefore, subject to significant noise. As a result, values for T_e for ρ > 0.6 are omitted. Figure 4(c) extends this analysis to multiple time points during the improved confinement period, showing that, as expected, the core temperature is consistently increasing.

This methodology was used to individually analyze an ensemble of 35 individual 300 kA PPCD shots, each including three time points. The core T_e for each point was then compared against the corresponding Thomson scattering measurement. As suggested by the individual profile inversion shown in Fig. 4, the temperature inferred from ME-SXR data is systematically higher than the Thomson measurements by ΔT_e ≈ 180 eV on average. The source of the 180 eV offset has not been determined, though the effect of high-energy photons originating from a non-Maxwellian electron population (known to be present in PPCD¹⁸) is a possible candidate. Regardless of the source, because the offset is constant (and, therefore, independent of the incident photon rate), it is not a result of detector saturation. Therefore, we can treat ΔT_e as a calibration factor. Using this methodology, ME-SXR measurements can be used to effectively determine the evolution of T_e over time.

Although the most of this paper focuses on the diagnosis of plasma thermal properties, the ME-SXR diagnostic can also be used to characterize non-thermal properties. The 450 μm Si sensor is sensitive to photons up to around 30 keV, allowing higher-energy x rays emitted by runaway electrons to be detected. This section presents observations from one such study on the MST.

Under the conditions of high current and low density, RFP plasmas may spontaneously transition into a helical equilibrium. This state is characterized by a magnetic spectrum largely (though not entirely) dominated by a single toroidal tearing mode and is, therefore, called “Quasi-Single Helicity” (QSH).¹⁹ Previous studies on the RFX-mod RFP have shown that QSH is associated with improved thermal confinement and the formation of transport barriers.²⁰ The presence of runaway electrons in QSH plasmas in the MST was briefly studied in a 2010 paper by Clayton et al.²¹ 

These observations were extended by the ME-SXR diagnostic. The detector was configured into an eight-color high SNR configuration using the medium gain calibration,⁹ which allows for thresholds between 4 and 14 keV. The thresholds were set to E_c = 5, 6, 7, 8, 9, 10, 11, and 12 keV. This configuration allows for some sensitivity to thermal electrons on the low end but also extends into the hard x-ray range. For non-PPCD plasmas with T_e < 1 keV, the signal from thermal electrons above 5–6 keV is expected to be negligible. Simultaneous measurements were also made using a fast (sub-μs) energy-resolved single-chord detector based on a Si avalanche photodiode, referred to as the “fast x-ray” (FXR) detector,²² for comparison.

Figure 5 presents data from a 500 kA QSH plasma: (a) shows the development of the dominant mode, while (b) and (c) show the corresponding FXR and ME-SXR measurements. A strongly non-thermal population develops around 20 ms as the core-most resonant mode grows to dominate the spectrum, and this population is sustained for around 5 ms before terminating with a burst and then becoming mostly thermal again as the magnetic equilibrium evolves. Combined with simultaneous thermal measurements, these observations helped build an understanding of how confinement properties vary over the lifetime of a QSH plasma.⁸ Note the logarithmic scale on the axis of the ME-SXR data. The close correspondence between the FXR and the ME-SXR photon flux provides confidence in the accuracy of the measurements and in the usefulness of ME-SXR as a standalone runaway electron diagnostic.

The ME-SXR diagnostic was also used to study the runaway electrons present when the MST is operated like a tokamak. A clear correlation was observed between decreasing density and increasing photon flux. These observations are reported in Delgado-Aparicio’s³³ contribution to these proceedings.

Section V provided a straightforward technique to extract T_e measurements with minimal a priori assumptions. However, oftentimes additional information is available, such as concurrent measurements from other diagnostics, previous results, and knowledge of the underlying physics. This section presents a consistent way of incorporating all of this additional information into a single framework, called integrated data analysis (IDA).¹⁶ With this framework, we will be able to simultaneously produce measurements of the electron temperature and impurity ion densities, demonstrating the versatility of the ME-SXR diagnostic.

The IDA methodology demonstrated here is based on Bayesian inference.¹² This is an approach to probability that views probability as the quantification of the degree of certainty based on the available information, rather than the long-term frequency over many repetitions. The heart of Bayesian inference is Bayes’s rule,²³ $p(θ|d,I)∝p(d|θ,I)p(θ|I)$⁠, which states that the posterior p(θ|d, I) of a system being best-described by the parameter vector θ given some measured data d is proportional to the product of the likelihood p(d|θ, I) of having measured that data given the parameter vector and the prior information p(θ|I). The I in Bayes’s rule is meant to represent the additional information that has been incorporated into the analysis, such as the choice of a particular model.

Bayes’s rule is intended to be applied iteratively whenever new information is made available. The approach to IDA employed here takes advantage of this feature to iteratively apply Bayes’s rule with simultaneous measurements from independent diagnostics, taking advantage of all available information to make the best possible estimate of the model parameters.¹⁶ This approach allows the data analyst to exploit the ways in which different diagnostic measurements correlate the plasma properties, sometimes enabling estimates of properties (such as Z_eff in the RFP) that cannot be readily measured by a single dedicated diagnostic.²⁴ This approach will also provide a natural and consistent framework for propagating measurement uncertainty, including when the underlying distributions are not Gaussian.

The aim of this section is to provide a full, detailed example of how to implement the ME-SXR diagnostic into an IDA framework. Two versions of the analysis will be performed. The first will incorporate just the ME-SXR and Thomson scattering diagnostics (termed “ME-SXR + TS”), while the second will also include measurements from a complementary SXR tomography array²⁵ (termed “full IDA”). The first analysis will allow us to demonstrate the ME-SXR diagnostic’s sensitivity to mid-Z impurities (such as Al), while the second will refine these results to provide the best estimates of the plasma profiles given all available data. A PPCD plasma was chosen to be the test case since PPCD plasmas have been thoroughly studied, meaning we can make use of well-informed priors. Section VII A begins by describing the PPCD plasma model. Section VII B then goes through the process of correctly selecting likelihood and prior distributions based on the best available information. Section VII C presents the results of the ME-SXR + TS analysis, demonstrating good sensitivity to Z_eff. Finally, Sec. VII D presents the results of full IDA and draws comparisons.

Previous experience on the MST has found that the temperature and density profiles in high-current PPCD plasmas are well-described by the aforementioned “α–β” model. Additional features such as islands or hollow rings can be included as additional terms if needed. The temperature profile is modeled as

where ρ is the normalized MSTFit¹⁵ radial flux surface label. The value of β_T is typically fixed, as it tends to be semi-redundant with the α_T parameter. An MSTFit reconstruction will be used to provide the mapping between ρ and standard Cartesian spatial coordinates (x, y), as well as a reconstructed n_e(ρ) profile.

MST plasmas typically feature several impurity species in concentration high enough to measurably impact the SXR spectrum: N and O from air, C from graphite in the limiter, B from probes, and Al from the vacuum vessel wall. Ar can also be doped into the MST but is not otherwise present in significant concentrations. It has been established that, during PPCD discharges, these ions are subject to a classical transport effect known as temperature screening.²⁶ Essentially, the presence of an ion temperature gradient leads to an expulsion of impurity ions from the core of the plasma, resulting in a hollow profile shape that peaks at the outer mid-radius. These predictions have been experimentally confirmed with charge-exchange recombination spectroscopy measurements.²⁷ The PPCD plasma model accounts for this phenomenon by adding an additional “hollow bump” term to the standard “α–β” density profile, given by

where n_Z refers to the total impurity species density (including all ionization states), δn_Z refers to the increased accumulation at the hollow bump for the impurity species Z, δr/a is the normalized location of the bump, and w_r/a is the normalized width of the bump. α_Z, β_Z, δr, and w_r are assumed to be the same for all impurity species.

Rather than treating all impurity ion densities as free parameters, only n_C is treated as a free parameter, while the remaining low-Z impurities are constrained based on empirically established ratios:²⁴ n_O/n_C = 0.9, n_N/n_C = 0.3, and n_B/n_C = 0.3. As these species all tend to be fully ionized and do not feature recombination steps or lines in the SXR spectrum, they essentially just serve to contribute to the overall Z_eff. This means that any analysis routine using this model would not be able to meaningfully distinguish between, for instance, an increase in n_C and an increase in n_O. Al ions in the MST, however, are not fully ionized, so n_Al is left as an additional free parameter.

Finally, a fixed neutral density model was included based on previous measurements.²⁸ This profile is flat throughout the core but grows exponentially by two orders of magnitude near the edge. The presence of a finite neutral density has been found to significantly affect the ionization balance in MST plasmas.⁸ 

As implemented, the model has six free parameters represented in the parameter vector,$θ=(Te,0,αT,nAl,0,nC,0,δnAl,δnC)$⁠. The remaining parameters are held fixed at the default values. This set of parameters was settled upon as a compromise between model flexibility (including more parameters) and computational requirements (models with fewer parameters converge more quickly).

A summary of all the parameters which describe the PPCD plamsa model, their default values, and associated priors are available in the supplementary material. Most of the priors were taken to be uniform with bounds given by the (Min, Max) values listed in the table. This equates to an assertion that the probability of that parameter taking on any value within this range is equal, while any value outside of this range has a probability of zero. These bounds were mostly chosen by physical considerations (T_e,0 must be positive) or constrained by previous expectations (T_e,0 > 2 keV is implausible for these bank settings). The one exception is the core carbon density, which was assigned a Gaussian prior based upon an ensemble of previous charge exchange recombination spectroscopy measurements.²⁴ It is assumed to have a mean of μ_C = 2.8 × 10⁻³ and a standard deviation of σ_C = 1.4 × 10⁻³. The range of profiles permitted by this selection of priors is illustrated in Fig. 6(I).

A likelihood function was chosen for each diagnostic (ME-SXR, Thomson, and SXR tomography). Each likelihood model, which quantifies the probability that one might have measured the data d given some known parameters θ, was chosen by combining the relevant diagnostic forward model^10,29 with a statistical model of the measurement noise. For each case, the measurement noise was assumed to be Gaussian, giving a general form of

where f(p_i, θ) is the diagnostic forward model for the ith pixel given the chord radius p_i, and $σi2=σd,i2+σm,i2$⁠, where σ_d,i is the measurement noise and σ_d,i is the systematic uncertainty in the model. The uncertainty σ_m for the ME-SXR forward model was found to be around 15%, due mostly to uncertainty in the Mylar filter thickness.⁸ For the SXR tomography model, the uncertainty σ_m was taken to be 2%.³⁰ 

Although the noise model for the ME-SXR diagnostic is more properly described by a Poisson distribution, given the relatively high count rates during PPCD, the Gaussian approximation $σd,i≈N$ was considered to be adequate.

The likelihoods were combined via an iterative application of Bayes’s rule, yielding the posterior distribution. For the ME-SXR + TS analysis, this is given by

where D₁ = (d_MESXR, d_TS). For the full IDA framework (ME-SXR + TS + SXT), the likelihood is given by

where D₂ = (d_MESXR, d_SXT, d_N2, d_TS). The posterior distribution describes how well a given set of parameters θ describes the data D, so the process of model-fitting is replaced with drawing many samples from this distribution and analyzing their statistical properties.

This analysis was performed for a single time point (averaged over 1 ms) near the end of the enhanced confinement period of a 300 kA PPCD plasma. The posterior distribution p₁(θ|D₁, I) was sampled using the emcee Markov chain Monte Carlo sampling software³¹ until good convergence was achieved.

N samples were drawn from the posterior distribution, ${θ̂1,θ̂2,…,θ̂N}$⁠, which were used to produce an ensemble of N plasma profiles, i.e., ${Te(ρ|θ̂1)},{Te(ρ|θ̂2)},…,{Te(ρ|θ̂N)}$ for the temperature profile, each of which was calculated over a grid of 100 ρ points. This ensemble was used to estimate the median profile, ⟨T_e(ρ)⟩, and the 65%, 95%, and 99.7% credibility regions (the Bayesian equivalent to 1-σ, 2-σ, and 3-σ confidence intervals³²) at each point. These profile samples are shown in Figs. 6(II-a), 6(II-c), and 7(II-d). The ensemble of profiles can also be used to compute the profiles for derived quantities, such as n_D and Z_eff, shown in Figs. 7(II-b) and 7(II-d), respectively. These profiles are the primary result of this analysis.

The integration of ME-SXR and Thomson scattering data allows for a highly accurate estimate of the core electron temperature, T_e = 1125 ± 12 eV. The ion densities have also been well-constrained compared to the priors, with core values of n_Al = 1.99 ± 0.44 × 10¹⁶ m⁻³ and n_C = 7.01 ± 1.3 × 10¹⁶ m⁻³. However, because the n_C profile is actually a stand-in for all of the low-Z impurities (C, B, O, and N), its individual value is not necessarily physically meaningful. Instead, the important result is the Z_eff profile, which is well-constrained to Z_eff = 1.97 ± 0.08 in the core. This is consistent with previous estimates using other diagnostics.²⁴ 

This analysis does not, however, do a good job constraining the ion density profiles outside of the core. This is because the ME-SXR model, with the 100 μm filter, has difficulty discerning between n_Al and n_C in the lower-signal regions. This results in a strong correlation between n_Al, n_C, and δn_Al in the posterior distribution. The overall estimate for Z_eff, however, is relatively well-constrained. The high uncertainty in the individual measurements presents a significant opportunity for additional diagnostics to improve these results. Even so, the obtained profiles represent a significant improvement over the priors p(θ|I).

Overall, these results serve as a demonstration that the ME-SXR diagnostic, when combined with Thomson scattering, can be used to reconstruct T_e, n_Al, and Z_eff profiles that are well-constrained in the core. These results are both more accurate and more informative than the direct T_e inversion methods previously discussed, although they require more information and assumptions about the underlying plasma equilibrium. Uncertainty in the ion densities is still large in the edge, though, providing an opportunity for additional diagnostics to further constrain the results. This is accomplished in Sec. VII D by the inclusion of the SXR tomography diagnostic, which, equipped with 45/172 μm Be filter pairs, is expected to provide increased sensitivity to the Al line emissions.

The analysis was repeated with the full likelihood function p₂(θ|D₂, I), which incorporates the SXR tomography data and forward model. The posterior was again sampled, with the resulting profiles and credible intervals shown in Fig. 7(III). In comparison with the ME-SXR + TS analysis, the addition of new measurements has done little to affect the T_e profile. However, the ion density profiles have been significantly refined with n_C now clearly demonstrating a hollow profile while n_Al for nearly flat. The core values have also been further refined, especially n_Al. However, the change in the resulting Z_eff profile is modest. The new profiles are all within the uncertainty bands of the previous analysis [Fig. 7(II)].

The best way to assess the quality of a fit is to compare the output of the models with the original measured data. This too is accomplished through sampling. For each diagnostic, the forward model is calculated for each profile in the ensemble. This produces an ensemble of synthetic measurements for each channel, which are then analyzed statistically. Because these ensembles tend to be very nearly Gaussian, they are well-characterized by the mean and standard deviation. Figure 7 shows this data/model comparison for the ME-SXR, Thomson scattering, and SXR tomography diagnostics. The agreement, though not perfect, is generally good. Some level of disagreement is to be expected when simultaneously fitting multiple models, each with their own sources of uncorrelated systematic uncertainty. This is in-fact a feature of IDA and is the reason we included the σ_m terms in the likelihood functions. The fact that all three diagnostics are in good agreement with the synthetic measurements and that the resulting profiles are in good agreement with prior results lends a significant amount to confidence in the ME-SXR forward model.

The versatility of the ME-SXR diagnostic for characterizing multiple properties of a magnetically confined fusion plasma has been demonstrated. The ME-SXR diagnostic was used to observe plasma temporal evolution. Photon counts were seen to evolve appropriately as the plasma heats up and decrease as the plasma cools. A method for identifying the energy of an emission line was presented and tested using Ar-doped plasmas. A technique for determining the electron temperature profile by directly inverting the emissivity was shown to agree with Thomson scattering measurements up to an ∼180 eV offset, which may be due to the presence of background hard x rays from the runaway population known to develop during PPCD. Runaway electrons were observed to form during helical plasmas in the MST, and their population was correlated with the evolution of the magnetic equilibrium. The ME-SXR diagnostic was also incorporated into an integrated data analysis (IDA) framework based on Bayesian inference. Using this methodology, it was shown to be possible to simultaneously extract T_e and n_Z profiles from ME-SXR and Thomson scattering data. The resulting profiles can be further refined by including the SXR tomography diagnostic in the analysis. The ability to simultaneously fit multiple diagnostics to their respective data using physically reasonable profiles provides a high level of confidence for the ME-SXR forward model.

See the supplementary material for a table of the PPCD model parameters, units, and priors.

This work was supported by the U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences program, under Award Nos. DE-SC0018266 and DE-SC0015474.

The data that support the findings of this study are available from the corresponding author upon reasonable request.