Estimation of Stagnation Performance Metrics in Magnetized Liner Inertial Fusion Experiments using Bayesian Data Assimilation P. F. Knapp, M. E. Glinsky, M. A. Schaeuble, C. A. Jennings, M. Evans, J. Gunning, T. J. Awe, G. A. Chandler, M. Geissel, M. R. Gomez, K. D. Hahn, S. B. Hansen, E. C. Harding, A. J. Harvey-Thompson, S. Humane, B. T. Klein, M. Mangan, T. Nagayama, A. J. Porwitzky, D. E. Ruiz, P. F. Schmit, S. A. Slutz, I. C. Smith, M. R. Weis, D. A. Yager-Elorriaga, D. J. Ampleford, K. Beckwith, T. R. Mattsson, K. J. Peterson, and D. B. Sinars 1) Sandia National Laboratories, Albuquerque, NM 87185 2) University of Rochester Department of Physics and Astronomy, Rochester, NY 14627 3) CSIRO Energy, Clayton Victoria, 3168, Australia 4) Lawrence Livermore National Laboratories, Livermore, CA 94550 5) Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109 We present a new analysis methodology that allows for the self-consistent integration of multiple diagnostics including nuclear measurements, x-ray imaging, and x-ray power detectors to determine the primary stagnation parameters, such as temperature, pressure, stagnation volume, and mix fraction in Magnetized Liner Inertial Fusion (MagLIF) experiments. The analysis uses a simplified model of the stagnation plasma in conjunction with a Bayesian inference framework to determine the most probable configuration that describes the experimental observations while simultaneously revealing the principal uncertainties in the analysis. We validate the approach by using a range of tests including analytic and 3D MHD models. An ensemble of MagLIF experiments is analyzed and the generalized Lawson criterion χ is estimated for all experiments. Inertial Confinement Fusion (ICF) is the process by which fusion fuel is compressed to high density and temperature with the goal of producing an abundance of thermonuclear fusion reactions before the fuel disassembles. A variant of this approach, known as Magneto-Inertial Fusion (MIF), uses an embedded magnetic field to ease some of the requirements imposed by traditional ICF. In either ICF or MIF, accurately characterizing the plasma conditions during the period of neutron production (e.g. stagnation) is of paramount importance for understanding target performance, assessing the impact of target modifications, evaluating the relative importance of sources of degradation, and ultimately for charting progress towards high yield. Fusion experiments are highly integrated, with many interdependent processes occurring simultaneously, making cause and effect difficult to quantify. For this reason, multiphysics simulations are used to design and, in many cases, interpret the experimental results. Unfortunately, these simulations are extremely complex and expensive to perform. Furthermore, without a detailed understanding of the state of the plasma in the experiment, it is difficult to adequately constrain the simulations, making progress difficult. The Magnetized Liner Inertial Fusion (MagLIF) concept is an approach to MIF that is being pursued on the Z machine at Sandia National Laboratories. In this approach, a pre-imposed magnetic field is used to insulate a hot laser-heated fuel volume from an imploding cylindrical metal liner. If the fuel is kept sufficiently pure and is able to retain enough of its energy during the implosion, it will reach conditions where significant thermonuclear burn can occur. In order to understand the performance of a given target and how it compares to our expectations, we must understand the state of the fuel during neutron production. In these experiments many diagnostics are fielded to measure various physical quantities relating to the plasma conditions of the experiment. Unfortunately, most quantities of interest must be constrained using multiple diagnostic outputs. As an example, the fuel pressure cannot be directly measured, only inferred through measurements of other quantities like temperature and density. Furthermore, the inference of pressure will depend on the level of contaminants in the fuel which contribute extra electrons and is, itself, a difficult parameter to estimate. The measurement of pressure, therefore requires information from many diagnostics and is inherently a multi-objective inference. For these reasons we are pursuing an inversion technique based on Bayesian inference that uses multiple different diagnostic signatures to simultaneously constrain multiple quantities of interest. In this approach, sometimes referred to as data fusion or data assimilation, an appropriately parameterized model is used to generate synthetic diagnostic data which is compared with all of the experimental data simultaneously, incorporating known constraints on parameters. This method allows us to estimate the most likely configuration in the N-dimensional parameter space, the associated uncertainties, and correlation between both data and inputs. The remainder of this paper is organized as follows. In Sec. II a brief introduction to Bayesian inference is given along with the details of the algorithm developed for this application. In Sec. III the physics model is introduced along with the formalism used to form each of the synthetic diagnostics. In Sec. IV we present a series of validation tests, where data is produced using analytic models and multiphysics simulations to demonstrate the utility of the technique. In Sec. V we present the analysis of real experimental data and finally, in Sec. VI we discuss the outlook for using this methodology for experimental data analysis as well as planned extensions of the physics model to better capture experimental nuances. A. Bayesian Inference In recent decades, Bayesian inference has become a popular and versatile method used for data analysis in a wide range of fields. The application of this approach to ICF is fairly new, but is quickly gaining momentum. The basic problem we wish to solve is the following: With a model F(m) describing the physics and diagnostic observables of interest, we wish to find the set of model parameters m that best matches a set of observables x with associated uncertainties σ. Bayes' theorem states: P(m|x, A) = [P(x|m, A) P(m|A)] / P(x|A) where P(m|x, A) is the posterior, P(x|m, A) is the likelihood, P(m|A) is the prior, and P(x|A) is the evidence. In our case, we use the normal distribution as our likelihood: P(x|m, A) ∝ ∏_{i=1}^N exp( - (F_i(m) - x_i)^2 / (2 σ_i^2) ) B. Methodology The task is to estimate the posterior distribution P(m|x). We adopt a two-step procedure: the first step estimates a Gaussianized approximation of the posterior distribution using Levenberg-Marquardt (LM) least squares optimization with finite differences, and the second step optionally samples the nonlinear aspects of the posterior distribution using a classic Metropolis MCMC method starting from the maximum a posteriori (MAP) model. Many Gaussians are truncated using a Tobit model construction to prevent parameters from taking unphysical values. An outer loop is implemented where the mean of the prior for the next iteration is set to the MAP of the previous posterior distribution, converging within 2 to 3 iterations. A. The Hotspot Model The hotspot is defined by a series of one-dimensional, isobaric cylinders ("slices"), each with height δh and radius R_HS. For bulk stagnation metrics, N = 3 slices is used. Each slice is defined by 6 parameters: 1. Pressure: P_HS 2. Temperature: T_c 3. Mix Fraction: f_mix 4. Mix Charge: Z_mix 5. Hotspot Radius: R 6. Liner areal density: ρR_l The burn duration τ_b is taken as a constant for all slices and measured using x-ray power detectors. In the isobaric approximation: T(r) = T_c [ 1 - (r/R)^α (1 - T_b/T_c) ] with T_b/T_c = 0.1, and P_HS = n_e k_B T_e + n_i k_B T_i = (1 + <Z>) n_i k_B T. B. X-ray Emission The time-integrated x-ray emissivity is expressed as: ϵ_ν = τ_b exp(-ρR_l κ_ν) g_FF ∑_s f_s j_s where j_s = j_s^ff + j_s^bf. Rewriting with pressure gives: ϵ_ν = A_ff exp(-ρR_l κ_ν) τ_b [ P_HS^2 g_FF <Z> / ((1 + <Z>)^2) ] ∑_s [ f_s j̃_s exp(-hν/T) / T^(5/2) ] C. Neutron Emission The neutron emissivity is calculated as: ϵ_n = [ 1 / (1 + δ_{j,k}) ] τ_b f_j f_k n_i^2 <σv>_{j,k} = [ P_HS^2 τ_b / (1 + δ_{j,k}) ] [ f_j f_k <σv>_{j,k} / ((1 + <Z>)^2 T_i^2) ] The neutron spectrum dN_n/dE_n is evaluated using the exact relativistic expression from Ballabio et al. D. Synthetic Diagnostics Forward models compute: 1. Filtered x-ray yields Y_ν for PCDs across Kapton filters (25.4 µm, 254 µm, 508 µm). 2. Multi-channel filtered Time Integrated Pinhole Camera (TIPC) 1D axial profiles: Y_{ν,j}^n = C_IP Ω ∫ dhν A(hν) F_j(hν) ∫_{V_n} ϵ_{ν,n} dV. 3. Spherical crystal monochromatic x-ray imaging signal I^n(x). 4. Neutron time-of-flight (nTOF) signal convolved with instrument response function (IRF). A. Test Cases Validation was conducted using: 1. An analytic 1D profile test case verifying exact recovery of pressure, temperature, liner areal density, and fuel radius. 2. An ensemble of 50 synthetic hotspots generated by a 3D double-helix stagnation model incorporating helical morphology, axial brightness variations, and bifurcations. The 2D crystal images were straightened, cleaned, and partitioned into 3 slices. Comparisons showed that inferred emissivity-weighted temperature, pressure, and volume agreed well with true values, while liner areal density and mix fraction showed lower sensitivity and systematic bias. B. Comparison with Simulation Results 1D and 3D Gorgon magnetohydrodynamic (MHD) simulations were used to assess the stationary model assumption against dynamic implosions. Results demonstrated that inferred parameters correspond closely to x-ray and neutron emission-weighted conditions during peak compression. The Bayesian framework was applied to an ensemble of 36 MagLIF experiments performed on the Z facility dating back to 2015. The inferred parameters allowed estimation of the generalized Lawson criterion χ, defined as: χ = (2/3) [ f_D f_T / (1 + <Z>)^2 ] [ ϵ_α τ_E / V_HS ] ∫_V [ P_HS <σv>_DT / T^2 ] dV where ϵ_α = 3.5 MeV, assuming an equivalent 50:50 DT fuel mixture and τ_E ≡ τ_b. Three top-performing experiments reached χ values between 0.06 and 0.08 with DD yields between 0.5 - 1.1 × 10^13, stagnation pressures of 1.2 - 1.8 Gbar, and temperatures of 2.5 - 3.3 keV. TABLE I. Input and inferred parameters of the three experiments that achieved the highest values of χ (reported with ±1σ confidence interval): • Shot z3179: - I_peak: 15 MA - B_{z,in}: 15.8 T - ρ_gas: 0.7 mg/cm³ - E_PH: 0.95 kJ - Y_DD: 0.55 × 10¹³ - τ_b: 1.8 ns - T: 3.3 ± 0.1 keV - P_HS: 1.2 ± 0.07 Gbar - V_HS: 0.03 ± 0.002 mm³ - χ: (8.4 ± 0.9) × 10⁻² • Shot z3236: - I_peak: 15 MA - B_{z,in}: 10 T - ρ_gas: 1.05 mg/cm³ - E_PH: 1.35 kJ - Y_DD: 1.1 × 10¹³ - τ_b: 2.1 ns - T: 2.9 ± 0.1 keV - P_HS: 1.3 ± 0.07 Gbar - V_HS: 0.08 ± 0.004 mm³ - χ: (6.2 ± 0.6) × 10⁻² • Shot z3289: - I_peak: 20 MA - B_{z,in}: 15.9 T - ρ_gas: 1.05 mg/cm³ - E_PH: 1.15 kJ - Y_DD: 1.1 × 10¹³ - τ_b: 2.0 ns - T: 2.6 ± 0.3 keV - P_HS: 1.8 ± 0.16 Gbar - V_HS: 0.05 ± 0.007 mm³ - χ: (7.1 ± 1.8) × 10⁻² The Bayesian parameter estimation framework provides a consistent determination of plasma parameters with rigorous uncertainties. Future work will focus on 3D model improvements, evaluating the value of information from new spectroscopic diagnostics, and data-driven experimental design. Appendix A details posterior distribution corner plots comparing Gaussianized Levenberg-Marquardt estimates with Markov Chain Monte Carlo (MCMC) sampling for shots z3019 and z3179, showing that the Gaussian approximation provides conservative uncertainty estimates.