Non-parametric inference of impurity transport coefficients in the ASDEX Upgrade tokamak

PAPER • OPEN ACCESS Non-parametric inference of impurity transport coefficients in the ASDEX Upgrade tokamak To cite this article: T. Nishizawa et al 2022 Nucl. Fusion 62 076021 View the article online for updates and enhancements. You may also like Single hole doped strongly correlated ladder with a static impurity S Gayen - Pilot study on peptide purity—synthetic oxytocin R D Josephs, M Li, A Daireaux et al. - Electronic and Shallow Impurity States in Semiconductor Heterostructures Under an Applied Electric Field Hai-Yang Zhou, Shi-Wei Gu and Yao-Ming Shi - This content was downloaded from IP address 161.111.10.228 on 31/08/2022 at 11:48 International Atomic Energy Agency Nuclear Fusion Nucl. Fusion 62 (2022) 076021 (15pp) https://doi.org/10.1088/1741-4326/ac60e8 Non-parametric inference of impurity transport coefficients in the ASDEX Upgrade tokamak T. Nishizawa1,2,∗,R.Dux 1, R.M. McDermott1, F. Sciortino1, M. Cavedon1, C. Schuster1,3,E.Wolfrum 1, U. von Toussaint1, A.Jansen Van Vuuren4,5, D.J. Cruz-Zabala4,5, P. Cano-Megias5,6, C. Moon2 and the ASDEX Upgrade Teama 1Max Planck Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching, Germany 2Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan 3Physik-Department E28, Technische Universität München, James-Franck-Str 1, 85748 Garching, Germany 4Dept. of Atomic, Molecular and Nuclear Physics, University of Seville, Avda. Reina Mercedes, 41012 Seville, Spain 5Centro Nacional de Aceleradores (U. Sevilla, CSIC, J. de Andalucia), Seville, Spain 6Dept. of Energy Engineering, University of Seville, Spain E-mail: [email protected] Received 14 January 2022, revised 6 March 2022 Accepted for publication 24 March 2022 Published 4 May 2022 Abstract We present a non-parametric inference of impurity transport coefficients by using charge exchange recombination spectroscopy measurements of Ne X, Ne VIII, O VIII, and C VI lines. Due to their close atomic numbers, neon, oxygen and carbon impurity ions are assumed to have the same diffusion coefficient Dand convection velocity v. Unlike conventional techniques that modulate or perturb the impurity contents, we employ a quasi-stationary plasma with static impurity profiles. Since the ratio of vto Donly describes the equilibrated profile of the sum of all impurity charge states, steady-state measurements can still decouple D and vif different charge states are simultaneously observed. We have formulated a non-parametric analysis framework based on the Bayesian probability theory and conducted transport coefficient measurements for a Type III ELMy H-mode plasma at ASDEX Upgrade. The charge exchange reactions with the background neutrals, which are known to affect the impurity charge state balance, are taken into account by introducing additional free parameters. While Dat the pedestal is close to the neoclassical level (<1ms −2), a large diffusion coefficient and a strong outward convection are inferred right inside the pedestal top. Keywords: tokamak, impurity, Bayesian inference, charge exchange recombination spectroscopy (Some figures may appear in colour only in the online journal) ∗Author to whom any correspondence should be addressed. aSee Meyer et al 2019 (https://doi.org/10.1088/1741-4326/ab18b8)forthe ASDEX Upgrade Team. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1741-4326/22/076021+15$33.00 Printed in the UK 1 ©EURATOM 2022 Nucl. Fusion 62 (2022) 076021 T. Nishizawa et al 1. Introduction Controlling impurity ions is a critical requirement for developing a fusion reactor. Impurity accumulation in the core region hampers the fusion reactions through the fuel dilution and radiates energy away through line radiation and enhanced Bremsstrahlung. However, a proper amount of impurity ions is likely to be required in the edge region in order to mitigate the heat load on the plasma facing components [1,2]. Furthermore, impurities modify the heat transport and even improve the confinement under some circumstances [3–5]. As such, simply reducing the impurity contents is not sufficient, and achieving controllability of impurity ions by understanding their transport processes is necessary. Impurity transport is often modeled by using the diffusion coefficient Dand the convection velocity vin a simplified cylindrical geometry. Observing time transient impurity emission is a common technique to experimentally determine those transport coefficients. When there is a drastic change in the confinement property, e.g. ELM bursts or the formation of an internal transport barrier, impurity ions redistribute, leading to temporally evolving impurity emission [6–8]. In the case of quasi-stationary plasmas, time-evolving impurity profiles can be created by using techniques such as laser-blow-off [9–11] (LBO), impurity puffing [12], and ion cyclotron resonance frequency modulation [13]. The transient impurity emissions are typically modeled by using an impurity transport solver STRAHL [14], and the Dand vprofiles that reproduce the observations are inferred. When the measurements provide the time evolution of the total impurity density of one species and its local flux, Dand vcan be more directly determined from the transport equation [6,12]. Recently, a systematic study on the model selection in the Dand vmeasurements using LBO and x-ray spectroscopy has been conducted [9]. The study reports that the inference problem of Dand vis ill-posed in the assumed diagnostic setup, and the inference results strongly depend on the choice of the model functions used for representing the profiles. When the model functions are too simple, the inferred Dand vprofilescan be quite different from the true ones, and the uncertainties become unrealistically small even though the forward model is in a seemingly reasonable agreement with the data. These results indicate that addressing the model selection problem and properly estimating uncertainties are crucial in the impurity transport coefficient measurements. Reference [9] concludes that this can be achieved by a Bayesian approach. Several references [11,15–17] have already reported impurity transport studies by using the Bayesian formalism. In these approaches, the probability distribution for the Dand vprofiles, which is called the prior, is first defined without considering the data. Then, by multiplying this prior distribution by the likelihood, which depends on the data, we obtain the results of the Dand vprofile measurements, the posterior. Since the Bayesian method calculates the probability distribution for obtaining given pairs of Dand vprofiles, the robust uncertainty estimation is possible not only for Dand v,but also for their ratio by taking into account their correlation. In addition, the optimum model function can be selected among many candidates by evaluating the Bayes factors [9,11]. An additional advantage of the Bayesian approach is the coherent combination of multiple diagnostics, which is referred to as integrated data analysis (IDA) [18]. The sum of information from various diagnostics provides a more localized likelihood and improves the ill-posed conditions. References [15,16] combined charge exchange spectroscopy (CXRS), vacuum ultraviolet spectroscopy, and soft x-ray measurements andinferredtheDand vprofiles that are consistent with all the diagnostics. The model selection is also an important problem for more fundamental plasma parameters such as the electron density neand the electron temperature Te. For these parameters, nonparametric inference techniques based on the Gaussian processes are often used [19–21]. In a non-parametric approach, each spatial grid point has its own degree of freedom, and the prior specifies the correlation between spatial points by using a probability distribution such as the Gaussian process. By tuning this probability distribution the properties that plasma parameters are expected to have, e.g., the degree of smoothness and variation, can be imposed. For this reason, the posterior given by the non-parametric approach is not restricted to piecewise connected cubic functions, which is the case for cubic splines and contains all possible solutions that are consistent with the data and the prescribed properties of the profile. Due to its complete search of the solution space, even second derivatives of the original profile can be evaluated by using a non-parametric inference technique [22]. The difference between parametric and non-parametric inferences is further discussed in appendix A. High degrees of freedom, however, lead to a large computation cost, and the application of the non-parametric profile inference was originally limited to the cases where the posterior distribution can be calculated analytically. With the advent of efficient algorithms [23–25], non-parametric approaches have been applied to complex models in recent years [26,27]. This paper reports the non-parametric inference of the impurity transport coefficients for the first time. In order to make the complex high dimensional problem tractable, we employ an unconventional diagnostic setup. Instead of modulating the impurity contents, we utilize a plasma in a quasistationary state and measure multiple impurity emission lines from different charge states by using CXRS. When only a fully-ionized state is present in a steady state plasma, the continuity equation is given by: dnZ/dr=vnZ/D,wherenZis the impurity density, and ris the radius. Thus, we can only probe the ratio of vto D. However, when lower charge states have significant populations, source and sink terms cannot be neglected, and multiple continuity equations of different charge states, which couple to each other, need to be evaluated. In this case, Dand vcan be separated by measuring different charge states. Indeed, simultaneous modeling of D and vin quasi-steady state plasmas by using multiple charge state measurements has already been reported [28,29]. One of 2 Nucl. Fusion 62 (2022) 076021 T. Nishizawa et al the advantages in this technique is that the steady-state impurity profiles can be determined analytically for given background plasma parameters and impurity transport coefficients. A steady state also allows for time averaging over a long interval, and precise measurements become possible. In addition, we take into account the charge exchange reactions due to the background thermal neutrals, which has been shown to have a large influence in the charge state balance [17,30]. This paper is organized as follows. Section 2describes the experimental scenario and the diagnostic setup. The analysis procedure formulated within the Bayesian framework is presented in section 3. The results of the impurity transport coefficient inference are given in section 4. Finally, section 5 presents the conclusions. In this paper, we use a lower subscript and a vector to represent a radial position and a profile, respectively, e.g., ne,nis the value of neat the radial position of n,andneis the profile of ne. As for the impurity ions, which have multiple charge states, a vector without a charge state index represents the profiles of all charge states except the neutral state, e.g., nNe is a symbol for the neon ion profiles from nNe1+to nNe10+. 2. Impurity density measurements using charge exchange spectroscopy Charge exchange recombination spectroscopy (CXRS) [31] provides localized impurity density measurements. Impurity ions in the plasma undergo charge exchange reactions with energetic neutral hydrogen or deuterium atoms introduced by neutral beam injection (NBI). This electron transfer brings impurity ions into a lower charge state and leaves them in excited states. CXRS measures line emission associated with the relaxation of those impurity ions. Since this process occurs along the trajectory of the neutral beam, the emission intensity, impurity velocity, and temperature near the intersection between a line of sight (LOS) and the neutral beam can be obtained through line-shape analysis. The spectral radiance due to the beam-induced charge exchange reaction at wavelength λis given by: Lλ=los nzνλ zds,(1) where nzis the impurity density, and the integral los is along the LOS. The rate coefficient νλ zis a rather complicated function of local plasma parameters, the energy components of the neutral beam and their spatial profiles. We use the charge exchange impurity concentration analysis code COLRAD to calculate νλ z. COLRAD takes into account important effects: the beam divergence and attenuation, the halo contribution, and the populations of the beam neutrals in the excited states. The detailed description can be found in reference [32]. Using CXRS, we measured three emission lines Ne X 524.49 nm, Ne VIII 606.9 nm, and C VI 529.07 nm. In the vicinity of Ne VIII 606.9 nm, there is another emission line Table 1. Combinations of spectrometer, optical head and emission line in the charge exchange recombination spectroscopy measurements. Spectrometer Optical head Emission line(s) CER A 524 nm (Ne X), 529 nm (C VI) CMR B, C 524 nm (Ne X), 529 nm (C VI) CAR A 607 nm (Ne VIII +O VIII) CPR B, C 607 nm (Ne VIII +O VIII) O VIII 606.9 nm. These lines cannot be discriminated, and we can only calculate the sum of nNe8+and nO8+. Not that Ne VIII 606.9 nm and O VIII 606.9 nm have almost identical νλ z.The LOSs for CXRS and the magnetic geometry of the plasma are showninfigure1. A steady neon puff of 8.8×1020 atoms s−1 was applied to a Type-III ELMy H-mode discharge in the ASDEX Upgrade tokamak (AUG). The profiles of the electron density ne, the electron temperature Te, and the ion temperature Tiare shown in figure 3in section 4where the results of the transport coefficient measurements are discussed. Other main plasma parameters are Ip=0.6MA,κ=1.65, and δ=0.38 where Ip,κ,andδare the plasma current, elongation, and triangularity, respectively. The external heating of 7.5 MW was provided by NBI, and 2.5 MW by electron cyclotron heating. NBIs were modulated to allow for the subtraction of passive emission. When the NBI, on which the LOSs are focused, was off, another NBI was turned on such that the NBI heating power stayed constant. The plasma was quasi-stationary from 6.2 to 8.7 s. We calculated the time-averaged net intensities of active charge exchange emission (including Type-III ELMs) during this period by subtracting the passive intensities. Figure 1shows the viewing geometry of the CXRS measurements. The LOSs originate from three different optical heads A, B, and C, and each head houses a lens and a bundle of optical fibers. Different colors are used to trace the LOSs depending on their optical heads. At the marker locations, the LOSs intersect with the 60 keV neutral beam. The core regions are measured by head A with the spatial resolution of a few cm, while the pedestal region is probed by head B and head C with higher spatial resolution (sub-cm). In this measurement, we used four spectrometers: CER, CMR, CAR, and CPR. The setup of the spectrometers, optical heads and the measured emission lines are summarized in table 1.Wetake into account these combinations when estimating systematic uncertainties in section 3.5. The data points of CXR measurements are shown in figure 6in section 4together with the forward-modeled data. 3. Analysis 3.1. Modeling impurity transport In the confined region, the continuity equation of impurity ions with an atomic number Zis given by [28]: 3 Nucl. Fusion 62 (2022) 076021 T. Nishizawa et al Figure 1. Flux surfaces of the ASDEX Upgrade plasma in the Type-III ELM regime and lines of sight (LOS) for the charge exchange recombination spectroscopy measurements. The LOSs are projected onto the poloidal surface. The red solid line is the last closed flux surface. The blue lines are the lines of sight from optical head A while the green and the magenta lines are from optical head B and C, respectively. Markers are placed at the intersections between the neutral beam and the LOS. The shapes of the markers represent the spectrometers, to which the lines of sight are connected. More details about the CXR system at AUG are discussed in reference [33]. ∂nz ∂t=−1 r ∂ ∂r(rΓz) +Rz+1ne+Rcx z+1n0+ i Rcx, b z+1,inb 0,inz+1 −(Iz+Rz)ne+Rcx zn0+ i Rcx, b z,inb 0,inz +Iz−1nenz−1,(2) where tis time, ris the radius, and n0is the neutral density except the beam neutrals. z=2, 3, 4, ...,Z−1represents a charge state. The continuity equations for z=1andz= Zcan be obtained by removing interactions with lower and higher charge states from equation (2), respectively. In this one dimensional model, r≡V/(2π2Raxis), where Vis the volume enclosed by a flux surface, and Raxis is the major radius at the magnetic axis. Rzand Izare the effective recombination and ionization coefficients, respectively, which depend on neand Te. The charge-exchange recombination rate with thermal neutrals is given by Rcx z. An explicit expression of this rate coefficient is [34] Rcx z= q fq(ne,Te)Qq(Tred), (3) where Tred ≡(TimD+T0mZ)/(mD+mZ) is the reduced temperature. Tiand T0are the impurity ion temperature and the neural particle temperature, respectively. mDand mZare the masses of the deuterium atom and the impurity ion. fqis a fraction of n0in a quantum state q. For each q,Qqdefines the charge exchange recombination rate coefficient. nb 0,iand Rcx, b z,i are the beam neutrals and their recombination rate coefficient for the beam energy component iand the impurity charge state z, respectively. The energy components of NBIs and their profiles have already been characterized at AUG [32]. Since we can determine Rcx, b z,iby using the charge exchange cross section provided by reference [35], the recombination due to the beam neutrals is treated as a known quantity in the present work. We use the diffusion coefficient Dand the convection velocity vto model the particle flux Γzas follows: Γz=−D∂nz ∂r+vnz.(4) In this transport model, the same transport coefficients Dand vare used for all charge states of the three different impurity species: carbon, oxygen, and neon. In the SOL, we add a parallel loss term −nz/τto the rhs of equation (2), where τ−1 ≡2Mv L3Ti+Te mD .(5) Here, Mvis the Mach number, and Lis the connection length. We set Mv=0.1andL=25m,whichareoftenusedfor AUG plasmas [7,8]. While this modeling for the parallel loss effect is somewhat ad hoc, the inference results in the confined regionare relatively insensitive to the details in the SOL, which will be shown in section 4. 4 Nucl. Fusion 62 (2022) 076021 T. Nishizawa et al 3.2. Modeling the neutral profile We model the transport process of neutral particles using a Monte Carlo code in a cylindrical geometry [30]. Neutral deuterium atoms are released from the wall with the energy E0,wall. When ionization happens, tracking of the deuterium atom ends at that location. When it experiences a charge exchangerecombination event, the velocity of the tracking particle is replaced by that of the ion involved in the reaction. The frequencies of ionization and charge-exchange events at each radial location are determined by the background plasma profile. In this model, the profiles of n0and the neutral temperature T0are determined by E0,wall and the neutral density at the wall n0,wall. We use these two quantities as free parameters to describe n0 and  T0. 3.3. Prior We introduce a prior that defines the probability distribution of the parameters of interest before measurements are made.  T0and the relative profile shape of n0for a given energy at the wall E0,wall are uniquely determined by using the Monte Carlo code discussed in section 3.2. We calculate  T0and the profile shapes of n0for E0,wall =10, 20, 40, 80, and 120 eV. By interpolating those profiles, n0and  T0for arbitrary E0,wall between 10 and 120 eV can be obtained without running the Monte Carlo code again. We use the neutral density at the wall n0,wall to determine the absolute density, and define the prior distribution of n0and  T0as p(n0, T0|E0,wall,n0,wall)p(E0,wall)p(n0,wall), p(E0,wall)∝1/E0,wall 10 eV ⩽E0,wall ⩽120 eV, p(n0,wall)∝1/n0,wall 104m−3⩽n0,wall ⩽1018 m−3.(6) Here, we choose logarithmic priors for p(E0,wall)andp(n0,wall) to avoid a bias toward high amplitudes [36,37]. Since equation (2) is linear with respect to nz, the profile of each charge state for given  Dand v can be solved analytically. The analytical solution and its derivation are given in appendix B. We use the Gaussian process and introduce the probability distribution of  Das follows: p( D|ˆ Σ)∝exp −1 2  θˆ Σ−1 θ  θ=[log10 D1,log 10 D2,...,log 10 DN], (7) where ˆ Σn,n=σ2 Dexp −(rn−rn)2 2l2 co .(8) σD=1.(9) Note that the Gaussian process provides the profile of log10 D instead of D. This parameterization avoids a bias toward a larger scale and removes negative D, which is unphysical. Other impurity transport studies by using the Bayesian framework also infer the power of D[11,15,16]. The spatial correlation length lco showninfigure2is set to be proportional to the spacing between spatial grids. lco is reduced near the Figure 2. Correlation length lco used for equations (8)and(11)vs the radial grid points. The location of the last closed flux surface is shownbythereddashedline. pedestal region, where large variations in Dand vover small spatial scales are expected. Instead of defining the prior distribution of v directly, we introduce the probability distribution of r 0vdr/D.This parametrization leads to a significantly better performance of MCMC compared with the direct parametrization of v.In equation (B.4), vis contained only in the form of redge rvdr/D. Thus, the original parameter space becomes simpler when we use r 0vdr/D. The probability distribution of v can still be calculated when all constituents of the prior distributions are multiplied. First, we use the Gaussian process again to define: p( ψ|ˆ Ω)∝exp −1 2  ψˆ Ω−1 ψ, (10) where ˆ Ωn,n=F2exp −(rn−rn)2 2l2 co .(11) Using ψ, we define the discrete form of r 0vdr/Das follows: n  i=1 vi Di Δri=(ψn−ψ1)−(F+ψn−ψ1)·rn rN .(12) By this definition, r 0vdr/Dvaries from 0 to −F. The profile shape of Z z=1nzis proportional to exp (r 0vdr/D)inthe confined region.Thus, the sum of all charge states of one impurity species varies by a factor of e−Ffrom the core to the edge if there is no parallel loss term in the SOL. We use F=ln 2 in this inference problem. We found that equation (12) leads to a smaller autocorrelation of MCMC samples, and reduces the computation expense compared with the other parameterizations that had been tested. Using equations (10)–(12), we define the conditional probability density p(v|ˆ Ω, D). In addition to the rate coefficients,  D,andv, the boundary conditions for each charge state need to be specified in order to calculate impurity density profiles using equation (B.15). We use the ratios between charge states when there is no transport at the edge, and introduce the line-averaged highest charge state density ¯nZ≡redge 0nZdr/redge,forwhichwe have a good estimate from the CXRS measurements, as a parameter for specifying the absolute impurity densities. As discussed in appendix B, the edge boundary conditions have a small impact on the profile shapes in the confined region when 5 Nucl. Fusion 62 (2022) 076021 T. Nishizawa et al Figure 3. Profiles of the electron density: core (a1) and edge (a2), the electron temperature and the ion temperature: core (b1) and edge (b2) for a Type-III ELMy H-mode discharge (#38498, time: 7.060–7.075 s). Inferred neutral profile: core (c1) and edge (c2). Inferred diffusion coefficient profile: core (d1) and edge (d2). Inferred convection velocity profile: core (e1) and edge (e2). Inferred profile of the convection-velocity-to-diffusion-coefficient ratio: core ( f1) and edge ( f2). The blue shaded areas represent the ranges between the 16th and 84th percentiles at each radial point. The blue solid lines show the 50th percentiles at each radial point. The green dashed lines are the neoclassical prediction calculated by neoclassical code (NEO) while the purple diamonds are given by the quasi-linear gyro-fluid code (TGLF-SAT2). The number of samples and autocorrelation are 39 000 and <200 samples, respectively. equation (B.15) is solved. Thus, somewhat arbitrary choices for the boundary conditions can be tolerated. Finally, our prior distribution is given by: p( D,v,nC,nO,nNe,n0, T0| Iall) ∝p(nC| D,v,n0, T0,¯nC6+)p(nO| D,v,n0, T0,¯nO8+) ×p(nNe| D,v,n0, T0,¯nNe10+) ×p(v|ˆ Ω, D)p( D|ˆ Σ)p(n0, T0|E0,wall,n0,wall) ×p(¯nC6+)p(¯nO8+)p(¯nNe10+)p(E0,wall)p(n0,wall), (13) where Iall represents all the prior beliefs contained on the rhs. We use logarithmic priors for p(¯nC6+), p(¯nO8+), and p(¯nNe10+). 3.4. Likelihood We calculate the CX emission intensities for given impurity density distributions by using equation (1). The impurity density nzis given by the prior equation (13), and νλ zcan be calculated by using COLRAD. The main consideration in defining the likelihood is to take uncertainties into account. In this paper, we neglect uncertainties in the νλ z,ne,Te,andTi profiles and the atomic data. If these parameters are not fixed, the prior given by equation (13) needs to be modified, and calculating the posterior distribution becomes challenging. Note that since the decoupling of Dand vdepends solely on the charge state balance in this analysis, the inference results can be more susceptible to the atomic data compared with the conventional approaches. Characterizing this dependency remains future work. The NBI modeling using COLRAD has already 6 Nucl. Fusion 62 (2022) 076021 T. Nishizawa et al been validated against experimental data [32]. We determine the most probable ne,Te,andTiprofiles through IDA [18], which utilizes multiple diagnostics. Due to the long timeaveraging of the CX emission data (over 15 ms), the photon noise has insignificant contributions. Hence, we model correlated and uncorrelated systematic uncertainties within the Bayesian framework. One of the sources of the correlated uncertainties originates from the darkening of lenses in the optical heads. Access to the inside of the vacuum vessel is limited during an experimental campaign of AUG, which usually lasts around 7 months. We use the calibration data that were taken prior to the experimental campaign. Therefore, some degree of degradation in the transmission was expected when the measurement was carried out. It is expected that the LOSs that share the same optical head are affected by this effect in a similar fashion. In addition, the transmission of the optical heads and the sensitivities of the spectrometers were calibrated individually by using an integrating sphere. There could be different systematic errors when optical systems were irradiated, which also lead to correlated systematic uncertainties. We take into account these uncertainties by forward-modeling CX emission intensities as follows: Lmodel 524, j=ζk(524, j)ηl(524, j)los(524, j) nNe10+ν524 Ne10+ds, (14) Lmodel 529, j=ζk(529, j)ηl(529, j)los(529, j) nC6+ν529 C6+ds, (15) Lmodel 607, j=ζk(607, j)ηl(607, j)los(607, j) (nNe8++nO8+)ν607 Ne8+ds, (16) where ζkand ηlmodel the uncertainties in the absolute intensity calibration due to the optical heads and due to the spectrometers, respectively. The subscript k=A, B, C represents the optical heads, and the subscript l= CER, CMR, CAR, CPR is for the spectrometers. kand ldepend on the wavelength of an emission line λ=524, 529, and 607 nm and a LOS j. We estimate that the absolute intensity calibration of the optical heads and that of the spectrometers have an uncertainty of 10%. This uncertainty can be implemented by using N(1, 0.12) for the probability distribution of ζk(λ,j)andηk(λ,j)whereN(μ,σ2) is a normal distribution with a mean of μand a variance of σ2. The main source of uncorrelated uncertainties is an optical fiber coupler. The fibers housed in an optical head are connected to couplers on a fiber adapter panel. Depending on the needs of experiments, the connections between the LOSs and the spectrometers are configured differently. It is known that the transmission of a fiber coupler differs slightly each time, when a fiber is removed and reconnected. We estimate that this process leads to a 10 % error, which has no correlation between the spatial channels. Another source of uncorrelated uncertainties is the subtraction of the base line and passive emission intensities. We assume that this leads to the error of =1014 photons/(m2s sr), which is non-negligible only for 529 nm and 607 nm line measurements in the core region. Based on the considerations above, we introduce the following likelihood: p( L524, L529, L606|nC,nO,nNe) ∝ λ,j exp −(Lλ,j−Lmodel λ,j)2 2(0.1Lmodel λ,j+)2, (17) where Lmodel λ,jis given by equations (14)–(16). 3.5. Markov chain Monte Carlo method We numerically calculate the posterior probability distribution by using a Markov chain Monte Carlo (MCMC) method [38]. In our model, the probability distributions of equations (7)and (10) have the same number of degrees of freedom as that of the spatial grid points. Since the computation rapidly becomes more expensive as the dimensionality increases, an efficient MCMC algorithm is necessary in this inference problem. We employ No-U-Turn Sampler algorithm [25], which has a favorable scaling property for dimensionality compared with other MCMC algorithms, such as random-walk Metropolis [25,39,40]. Due to the complexity of the model, Markov chains often get trapped at local maxima before they reach a typical set when the likelihood equation (17) is evaluated in its original form. In order to facilitate the MCMC sampling, we rewrite equation (17) in the asymptotic form [27]: ˘p( L524, L529, L606|nC,nO,nNe) ∝ λ,j exp −|z(λ,j)|2 2·1+uv2|z(λ,j)| 1+v2|z(λ,j)|2/2, (18) where z(λ,j)=Lλ,j−Lmodel λ,j 0.1Lmodel λ,j+, (19) u=1, and v=0.2. This asymptotic form simplifies the probability distribution in the volume where the probability is extremely low while ˘p( L524, L529, L606|nC,nO,nNe)≈ p( L524, L529, L606|nC,nO,nNe) holds in the typical set. Markov chains robustly reach and explore the typical set when equation (18) is used for the likelihood. The MCMC computation for this model took <400 h to obtain around 200 independent samples by using an Apple M1 chip. 4. Results of impurity transport coefficient inference We have defined the prior equation (13) and the likelihood equation (17) in section 3. Using these probability distributions, the posterior is given by: p( D,v,nC,nO,nNe,n0, T0| L524, L529, L606, Iall) ∝p( L524, L529, L606|nC,nO,nNe) ×p( D,v,nC,nO,nNe,n0, T0| Iall).(20) 7 Nucl. Fusion 62 (2022) 076021 T. Nishizawa et al Figure 4. Profiles of the diffusion coefficient (a)andthe convection-velocity-to-diffusion-coefficient ratio calculated by marginalizing the prior equation (13). The blue shaded areas represent the ranges between the 16th and 84th percentiles at each radial point. The blue solid lines show the 50th percentiles at each radial point. The number of samples and autocorrelation are 40 000 and <200 samples, respectively. We calculate the posterior distribution using MCMC, the details of which are already discussed in section 3.5. The probability distribution of one plasma parameter profile can be obtained by marginalizing other parameters. For example, the probability distribution of the Dprofile is p( D| L524, L529, L606, Iall) =p( D,v,nC,nO,nNe,n0, T0| L524, L529, L606, Iall) ×dv dnCdnOdnNe dn0d T0.(21) The marginalized posterior distributions of the transport coefficients and the neutral profile for the type-III ELMy H-mode discharge are shown in figure 3together with the ne,Te,andTi profiles, which are treated as known parameters. The inferred neutral profiles are similar to the ones for the type-I ELM cases reported in reference [30]. Without taking into account the CX reactions with the thermal neutrals, the forward model is not able to reproduce the Ne X and Ne VIII intensities, which is already pointed out in reference [30]. The Dand vprofiles for neon calculated by a NEO and a quasi-linear gyro-fluid code (TGLF SAT-2) [11,17] are also plotted for comparison. Fast ions are neglected in both calculations while electromagnetic terms are included in the TGLF modeling. In the Figure 5. Examples of the impurity profiles taken from the posterior distribution. pedestal, the diffusion coefficient Dis below one, being close to the neoclassical level provided by NEO. The inferred v/D profile indicates that an inward pinch exists in this region. These observations are in a qualitative agreement with the Dand vmeasurements of type-I ELMy H-mode discharges [8,15,30]. Just inside the pedestal top ρpol =0.9–0.95, the diffusion coefficient peaks around D=20 m2s−1, and a strong outward convection ∼100 m s−1is seen. Neither NEO or TGLF modeling predicts such a magnitude of convection. The Type-III ELM crashes, which are included in the CXRS measurements but not in the NEO and TGLF calculations, may contribute to this discrepancy. Since the Type-III ELM consisted of major crashes with the frequency of around 200 Hz and minor crashes with higher frequencies, filtering out ELMs was difficult by using the current CXRS system. The posterior is proportional to the product of the likelihood and the prior. Therefore, it is important to inspect how much the likelihood updates the prior distribution. The profiles of Dand v/Dcalculated by running MCMC without the likelihood are shown in figure 4. The prior distribution of  Dgiven by equation (7) should produce the distribution with the mean of 1 and the standard deviation of a factor of 10 over all radial positions, which is shown in figure 4(a)as expected. Small deviations are due to the statical uncertainties in the MCMC sampling. Since we allocated shorter correlation lengths lco in the edge, the prior leads to a wider distribution of v/Das ρpol increases. The inferred Ddistribution in 8 Nucl. Fusion 62 (2022) 076021 T. 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