A new FILDSIM model for improved velocity-space sensitivity modelling and reconstructions

Plasma Physics and Controlled Fusion PAPER • OPEN ACCESS A new FILDSIM model for improved velocity-space sensitivity modelling and reconstructions To cite this article: Bo S Schmidt et al 2024 Plasma Phys. Control. Fusion 66 045004 View the article online for updates and enhancements. You may also like Experimental investigation of beam-ion losses induced by magnetic perturbations using the light ion beam probe technique in the ASDEX Upgrade tokamak J. Galdon-Quiroga, L. Sanchis-Sanchez, X. Chen et al. - Prompt non-resonant neutral beam-ion loss induced by Alfvén eigenmodes in the DIII-D tokamak X. Chen, W.W. Heidbrink, G.J. Kramer et al. - Velocity space resolved absolute measurement of fast ion losses induced by a tearing mode in the ASDEX Upgrade tokamak J. Galdon-Quiroga, M. Garcia-Munoz, L. Sanchis-Sanchez et al. - This content was downloaded from IP address 150.214.182.235 on 09/07/2024 at 11:44 Plasma Physics and Controlled Fusion Plasma Phys. Control. Fusion 66 (2024) 045004 (13pp) https://doi.org/10.1088/1361-6587/ad268f A new FILDSIM model for improved velocity-space sensitivity modelling and reconstructions Bo S Schmidt1,∗, Jes´ us Poley-Sanjuán2, José Rueda-Rueda3, Joaquín Galdon-Quíroga3, Marcelo Baquero-Ruiz2, Henrik Järleblad1, Bernard C G Reman1, Mads Rud1, Andrea Valentini1, Manuel García-Mu˜ noz3 and Mirko Salewski1 1Department of Physics, Technical University of Denmark, Kgs. Lyngby 2800, Denmark 2Swiss Plasma Center, ´ Ecole Polytechnique Fédérale de Lausanne, Lausanne CH-1015, Switzerland 3Department of Atomic, Molecular and Nuclear Physics, University of Seville, Seville 41012, Spain E-mail: [email protected] Received 4 November 2023, revised 18 January 2024 Accepted for publication 6 February 2024 Published 15 February 2024 Abstract We present a new version of the FILDSIM code (Galdon-Quíroga et al 2018 Plasma Phys. Control. Fusion 60 105005), which significantly refines the modelling of the fast-ion loss detector (FILD) signal. We demonstrate that the FILD weight functions computed using this new version of FILDSIM are more accurate relative to synthetic benchmarks than those computed using the previous version. Thus, the new version enables higher-quality velocity-space sensitivity modelling and reconstructions. We validate the improvements on experimental data from discharge #75620 at TCV. Additionally, we present a novel approach for characterizing FILDs through a gross FILD measurement and a gross weight function based on the calculations from the new version of FILDSIM. We use them to characterize the TCV FILD. Keywords: fast ion, fast-ion loss detector, FILDSIM, inverse problem, velocity space, reconstruction 1. Introduction Fast ions generated through high-energy neutral beam injection (NBI), electromagnetic wave heating in the ion cyclotron range of frequencies (ICRF), and fusion reactions with energies ranging from tens of keV to several MeV play a crucial role in the heating and stability of fusion plasmas. In particular, fusion-generated alpha particles are responsible for keeping the energy of the thermal plasma of future fusion reactions sufficiently high by transferring their energy to the thermal plasma via collisions. ∗Author to whom any correspondence should be addressed. 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. Thus, fast-ion transport and confinement are of particular concern. In magnetically-confined nuclear fusion experiments, fast ions can escape confinement and strike the plasmafacing components of the device, leading to material degradation and a decline in reactor efficiency [1,2]. This undesirable transport can be attributed to different mechanisms, including interactions with magnetohydrodynamic (MHD) activity [3,4] such as toroidicity-induced Alfvén eigenmodes (TAEs) [5–7] or edge-localized modes (ELMs) [8,9]. Given these challenges, a precise understanding of the fast-ion velocityspace distribution function is crucial. This understanding assists in identifying loss mechanisms, optimizing control algorithms, and customizing auxiliary heating methods to populate velocity-space regions where fast ions are less prone to loss. Such estimates also enable the validation of predictive models that can be applied to future devices and help guide material engineering solutions designed to withstand fast-ioninduced wear and tear. 1 © 2024 The Author(s). Published by IOP Publishing Ltd Plasma Phys. Control. Fusion 66 (2024) 045004 B S Schmidt et al Fast-ion loss detectors (FILDs) provide valuable information on the velocity space of lost fast ions through measurements close to the plasma edge (∼3 cm) [10,11]. The velocity space can be expressed in terms of the energy, E, and pitch, p, given by E=1 2mv2;p=sgn(IpBt)v∥ v.(1) Here, v= (v2 ∥+v2 ⊥)1/2,v∥is the ion velocity parallel to the magnetic field, and v⊥is the ion velocity perpendicular to the magnetic field. Since the Larmor radius ρ=mv⊥/qB is inversely proportional to the local magnetic field strength, FILDs act as magnetic spectrometers by measuring the energy and pitch of the lost fast ions; the ions follow magnetic field lines and strike a scintillator plate in a manner governed by the local magnetic field. Since FILDs provide essential data on lost fast-ion velocity distributions, they have been installed in numerous magnetically-confined fusion devices globally, including ASDEX Upgrade [11–13], DIII-D [14], EAST [15], JET [16], KSTAR [17], LHD [18], MAST-U [19,20], NSTX [21], and TCV [22]. The FILD designs for ITER [23] and JT60SA [24] underscore the importance of the diagnostic. The paper is structured as follows: section 2introduces the mechanics and design principles of FILDs and introduces the simulation code ‘FILDSIM’. Section 3describes our improvements to the current FILDSIM model. Section 4evaluates the new FILDSIM model against its predecessor and identifies key physical factors leading to the discrepancies between the two models. Section 5presents two novel ways of characterizing FILDs using the new FILDSIM model. Section 6conducts an evaluation of reconstructions of lost fast-ion velocity distributions, specifically for discharge #75 620 at TCV, to compare the performance of the new FILDSIM model against its predecessor. Section 7concludes the paper. 2. FILD mechanics and FILDSIM modelling In a scintillator-based FILD, ions pass through a pinhole connected to a 3D collimator in the FILD probe head. The probe head is placed close to the plasma, in the far scrape-off layer. After passing through the pinhole, the collimated ions strike a scintillator plate, triggering the emission of photons from the scintillating material. A specialized camera setup captures these photons. See figure 1(a) for an illustration of the working principles of a FILD, and figure 1(b) for the placement of the FILD at TCV. Computational modelling is invaluable for a more in-depth understanding of FILD measurements. The FILDSIM code [25] simulates trajectories of fast ions originating from the pinhole of the FILD and detects their collisions with either the scintillator plate or other components of the FILD probe head. A population of Nions with fixed energy and pitch is initialized at random locations within the spatial limits of the FILD pinhole and with random gyrophases within a predetermined gyrophase acceptance cone. The locations where the ions strike the scintillator plate are collectively used to define a strike-point distribution, reflecting where ions with specific energy and pitch may strike the scintillator plate. The value of Nis chosen to balance computational feasibility with the accuracy of the resulting strike-point distributions. A typical simulation uses N=3·104markers resulting in a computation time of approximately 10 min. Thus, ions with identical energy and pitch can strike the scintillator plate at different locations. This variability is due to two factors: the finite dimensions of the pinhole and the initial gyrophases of the ions. Consequently, a given set of ions with specific energy and pitch creates a distribution of strike points, effectively blurring the fast-ion velocity distribution. Let P(y,z|E′,p′)denote the strike-point distribution on the scintillator plate, conditioned on the ion energy E′and pitch p′. We establish a mapping ψ:S→V, where S⊂R2represents the scintillator plate and V⊂R2velocity space parametrized by (E′,p′), defined as ψ:EP(y,z|E′,p′)(y,z)7→(E′,p′).(2) In this mapping, the centroid of the strike-point distribution P(y,z|E′,p′)is mapped to the point (E′,p′)∈Vthrough ψ. This one-to-one mapping, traditionally used to analyze FILD measurements, is constructed by simulating the strikepoint distributions for various (E′,p′)pairs, associating each centroid with its corresponding point in velocity space, thus forming a bijection termed a ‘strike map’. The raw camera measurement Mraw =Mraw(y,z)is mapped to the fast-ion velocity distribution using the strike map. However, this mapped version termed the ‘mapped FILD measurement’ and denoted Mf, is not a one-to-one mapping onto the actual fast-ion velocity distribution f. The discrepancy arises from the spatial distribution of strike points on the scintillator for specific values of energy and pitch of the lost fast ions. The relation between the mapped FILD measurement and the fast-ion velocity distribution can be modelled as a Fredholm integral equation of the first kind Mf(E,p) = ˆ1 −1 dp′ˆ∞ 0 dE′W(E′,p′|E,p)f(E′,p′),(3) where the kernel Wis a weight function that weighs the contributions from all points in velocity space to a single point in the mapped FILD measurement [25,26]. Unprimed coordinates refer to the mapped FILD measurement, and primed coordinates to the lost fast-ion velocity space. The continuous integral equation (3) is recast into the matrix-vector equation b=Ax (4) using numerical quadrature. Specifically, we partition velocity space using a uniformly-distributed grid, approximating the value of each grid cell by its midpoint value and the integral by the midpoint sum [27]. The ‘system matrix’ A∈Rm×n contains rows of discretized weight functions. Here, mrepresents the number of grid points in the discretized mapped FILD measurement b, and nis the number of grid points in x. This framework is consistent with similar models for confined fastion diagnostics [26,28–39]. 2 Plasma Phys. Control. Fusion 66 (2024) 045004 B S Schmidt et al Figure 1. (a) FILD probe head schematic: Blue trajectories illustrate collimated lost fast ions passing through the FILD pinhole and striking a plate coated with a scintillating material (green). Photons (yellow) are emitted at the impact locations. The red trajectory illustrates a lost fast ion which, upon colliding with the collimator, is blocked from passing through the pinhole. (b) Trapped orbit of a lost fast ion measured by the TCV FILD (black box) located in the midplane of the vessel. 3. Improved FILDSIM modelling A fundamental challenge in interpreting FILD measurements exists due to the variability in strike points for ions with the same energy and pitch. Knowledge of the discharge conditions can partially assist in mitigating this issue; for example, during an NBI discharge with an injection energy of 30 keV, measurements from prompt losses at 30 keV and 15 keV are expected from the beam’s fulland half-energy components. While contextual knowledge aids in a nuanced interpretation of the FILD data, the most reliable inference is obtained from solving the inverse problem in (4). Prior FILDSIM simulations have used Gaussian distributions to model the strike-point distributions, fpand fE, in both pitch and energy. For the pitch, fp(p;p0,σp) = 1 √2πσp exp"−(p−p0)2 2σ2 p#,(5) where p0is the actual pitch of the ion and σpthe standard deviation of the pitch strike-point distribution. We denote such Gaussian distributions by Np;p0,σp. The energy was described by a skew-Gaussian distribution, given by fE(E;β,E0,σE) = N(E;E0,σE)Φ (E;β,E0,σE),(6) where Φ(x;β,E0,σE)is the skewness parameter function defined as Φ(x;β,E0,σE) = 1+erfβE−E0 √2σE.(7) Here, βis the parameter governing the degree of skewness, E0 is the energy of the ion, and σEis the standard deviation of the energy strike-point distribution. The skewness of the strikepoint distributions is linked to how fast ions traverse the collimator. This results in ions of identical energies striking the scintillator at varying locations due to limitations in collimation. Notably, the skewness increases with an increase in ion energy. This skewness is observed for all types of fast ions, irrespective of their generation method, whether it be NBI, ICRF, or fusion reactions. The collimator factor fcolE′,p′= ∆θNE′,p′/2πN′, accounting for the fraction of the number of ions striking the scintillator relative to the number of ions initialized in the pinhole within the gyrophase acceptance cone ∆θ/2πof the collimator, is also included in the FILDSIM model, and a yield function ϵ(E)describing the number of photons emitted by the scintillator plate as a function of the energy of the ion. In constructing the unified model, the equation for the strike-point distribution fpE directly multiplies the individual distributions, assuming statistical independence of the components: fpE (E,p;β,E0,p0,σE,σp) =ϵ(E)fcol (E′,p′)fE(E;β,E0,σE)fp(p;p0,σp) =∆θϵ(E)N(E′,p′) 4π2N′σpσE exp"−(p−p0)2 2σ2 p−(E−E0)2 2σ2 E# ×1+erfβE−E0 √2σE. (8) For the computation of the kernels, the three parameters σE,σp, and βare determined by minimizing the residual sum of squares between the model and the strike-point distributions obtained from FILDSIM. The fit is achieved through an iterative process over predefined parameter ranges. The computations are performed using N=3·104markers, a number that ensures sufficient statistical robustness for optimal parameter fitting. 3 Plasma Phys. Control. Fusion 66 (2024) 045004 B S Schmidt et al 3.1. Skewness and kurtosis The Gaussian and skew-Gaussian models exhibit several inaccuracies that warrant revisiting. Let XEand Xpbe random variables representing the ion energy and pitch. The skewGaussian model psG for XEdiverges significantly from the empirical probability density function (pdf) ˆ pE(x), particularly in the tails corresponding to higher energies. Similarly, the Gaussian model for Xpis too narrow, significantly underestimating the ion counts at pitches close to the ion’s true pitch value, and the tails are too broad. These shortcomings not only compromise the accuracy of the model but also introduce errors in the reconstructions. To address these issues, we investigate alternative distributions that more accurately represent the simulated strike-point distributions. In classical data analysis, statistical distributions are generally characterized by their first and second moments– the mean µand variance σ2. However, in systems exhibiting non-Gaussian behaviour, this characterization is insufficient. Consequently, there is a need to consider higher-order moments–specifically the third and fourth moments, termed ‘skewness’ and ‘kurtosis.’ Let Xbe a random variable whose moment-generating function MX(t) = EetXexists for some interval around t=0. The nth moment about the origin is then E[Xn] = dnMX dtnt=0 ,(9) where Eis the expectation value. From this, we calculate the skewness and kurtosis about the mean µas s= Eh(X−µ)3i σ3;k= Eh(X−µ)4i σ4.(10) Skewness quantifies the degree of asymmetry about the mean, while kurtosis quantifies the concentration of extreme values in the tails of the distribution. Importantly, skewness and kurtosis offer a more nuanced approach to characterizing nonGaussian aspects, enabling us to rectify the shortcomings in the Gaussian models by capturing the inherently non-Gaussian aspects. For pitch strike-point distributions, an ideal model would have zero skewness and a kurtosis significantly less– potentially half–than that of the previous FILDSIM model. A suitable pdf is Wigner’s semicircle, described by the equation W(p;p0,pR) = 2 πp2 Rqp2 R−(p−p0)2.(11) Here, p0is the true pitch of the ion, and pRis the radius of the semicircle, determined through an optimization process similar to that used for other model parameters. The distribution is supported on p∈p0−pR,p0+pR. This function offers desirable attributes for modelling the pitch distributions. Specifically, it has short tails, which help to avoid the overestimation problems inherent in the Gaussian model, and a larger width near the peak, providing a more accurate representation of the high-density regions around p0typical in the pitch distributions. From the analytical expressions for the pdfs of the Gaussian and Wigner’s semicircle distributions, we can calculate their skewness and kurtosis for a quantitative comparison. Their respective moment-generating functions MX,G(t) and MX,W(t)are [40] MX,G(t) = expµt+1 2σ2t2; MX,W(t) = 2I1(Rt) Rt , (12) where I1is the modified Bessel function of the first kind. Straightforward computations give sG=0;kG=3,(13a) sW=0;kW=2.(13b) The kurtosis values for the Gaussian and Wigner’s semicircle distributions are based on their well-established theoretical properties. These values are intrinsic to the distributions and do not depend on specific model parameters. The lower kurtosis of Wigner’s semicircle is advantageous as it mitigates the extended tails inherent in the Gaussian model, yielding a more localized representation around the peak pitch value p0. For energy strike-point distributions, the goal is to maintain a skewness of the candidate distributions close to that of the previous model while reducing the kurtosis by approximately 1–5%. This adjustment aims to provide a more accurate representation of the distribution’s behaviour at higher energies. With this goal in mind, we propose three candidate pdfs to model the energy distributions: the raised cosine distribution f, the Cauchy distribution g, and the logistic distribution h. These distributions were initially selected based on their statistical moments being similar but not identical to those of a Gaussian distribution. Their mathematical representations are f(x;µ,σ) = 1 2σ1+cosx−µ σπ,(14a) g(x;µ,γ) = 1 πγ 1 1+x−µ γ2,(14b) h(x;µ,s) = exp−x−µ s s1+exp−x−µ s2.(14c) Here, f(x;µ,σ)is supported on [µ−σ,µ +σ], 2γis the full width at half-maximum (FWHM) of the Cauchy distribution, and sis a scale parameter. These distributions are illustrated and compared with a Gaussian distribution in figure 2. The third and fourth moments of the three candidate distributions multiplied by the skewness parameter function do not have analytical expressions. However, analytic expressions can be determined for the expressions in (14), revealing their differences relative to a Gaussian distribution. By symmetry, the raised cosine distribution and the logistic distribution have a skewness of zero, whereas the Cauchy distribution 4 Plasma Phys. Control. Fusion 66 (2024) 045004 B S Schmidt et al Figure 2. The Gaussian distribution N(x;0,1)plotted with the candidate distributions: (a) the raised cosine distribution f(x;0,2.5), (b) the Cauchy distribution g(x;0,1), and (c) the logistic distribution h(x;0,1). The parameter values can be determined by comparing the expressions given here with those in (14). The difference between the Gaussian and candidate distributions is also plotted and indicated by the green curves. Note that the raised cosine distribution is the only distribution with shorter tails than the Gaussian distribution. does not have a well-defined third moment since the integrals ´∞ −∞ xkg(x;µ,γ)dx diverge for k⩾1. With the momentgenerating functions MX,rc (t) = π2sinhσt σt(π2+σ2t2)eµt,(15a) MX,log (t) = exp(µt)B(1−st,1+st)(15b) for the raised cosine distribution [41] and the logistic distribution [42], where B(x,y) = Γ(x)Γ(y) Γ(x+y)(16) is the beta function expressed in terms of the standard Gamma function, their kurtoses can be shown to be: kf=c1σ−4,kg=∞,kh=c2s−4(17) for some scalars c1and c2. It follows that distributions with a kurtosis different from kG=3 can be more accurately described in terms of the raised cosine and logistic distributions by optimally fitting σand s. To introduce controlled skewness, we multiply each distribution by the skewness parameter function Φ. Note that multiplying a distribution p(x) by Φchanges its moments, since the nth moment Mnis given by Mn=ˆ∞ −∞ dxxnp(x),(18) whereas M′ n=ˆ∞ −∞ dxxnp(x)Φ(x;µ,σ) =Mn+ˆ∞ −∞ dxxnp(x)erfβx−µ √2σ. (19) Still, the initial guesses of candidate distributions based on their similarity to the Gaussian distribution are also a good guess for their skewed variants. The candidate distributions are illustrated in figure 3for intervals relevant to FILD measurements at TCV. The illustrations concentrate solely on the candidate distributions, excluding the FILDSIM simulation histograms, to present a clear comparison of their skewness and tails relative to the skewGaussian distribution. The skewness and kurtosis of these distributions, calculated to assess their alignment with targeted Gaussian and skew-Gaussian properties, are tabulated in table 1. For energy strike-point distributions, the raised cosine distribution has the best fit, with its skewness identical to the skew-Gaussian’s at 1.8 and a kurtosis of 4.7, slightly lower than the skew-Gaussian’s 4.8. For pitch strike-point distributions, the Gaussian and Wigner’s semicircle distributions meet the targeted skewness of 0. However, Wigner’s semicircle markedly surpasses the Gaussian in kurtosis, having a value of 2.6 compared to the Gaussian’s 4.1. The elevated kurtosis of the Gaussian distribution, as seen in figure 3(d), is characterized by its more pronounced tails compared to those of the Wigner’s semicircle. This distinction underscores our selection of the Wigner’s semicircle for its relatively shorter tails. The discrepancy between the theoretical kurtosis values in (13a) and (13b) and the emipirical values in table 1can be attributed to several factors. Despite the large sample size, the kurtosis calculated by MATLAB is a sample-based estimate, which uses a specific estimator designed to correct for finite sample sizes. This estimator deviates from the theoretical population kurtosis and is subject to sample-to-sample variability. Moreover, MATLAB’s pseudo-random number generation algorithms, which underlie the simulation, may introduce additional approximation errors. Therefore, the values in table 1should be considered approximate, albeit closely aligned, representations of their theoretical counterparts, as they are influenced by the method of kurtosis calculation, random variability, and computational approximations. By incorporating the raised cosine and Wigner’s semicircle distributions, we have refined the FILDSIM model for fitting the strike-map distributions in energy and pitch. The mathematical representation of this new model is: f(E,p;β,E0,p0,σE,pR) = ∆θϵ(E)N(E′,p′) 2π2N′σEp2 Rqp2 R−(p−p0)2 ×1+cosE−E0 σE π ×1+erfβE−E0 √2σE.(20) 5 Plasma Phys. Control. Fusion 66 (2024) 045004 B S Schmidt et al Table 1. Skewness and kurtosis for the candidate distributions in figure 3. Energy Pitch Skew-Gaussian Raised cosine Cauchy Logistic Gaussian Wigner’s semicircle s1.8 1.8 2.3 2.0 0 0 k4.8 4.7 7.1 5.7 4.1 2.6 Figure 3. (a) Raised cosine with σ=9.1, (b) Cauchy with γ=0.62, and (c) logistic distributions with s=4.6 plotted with a skew-Gaussian distribution with σE=9.1 and β=2.9 to model the strike-point distribution for E=25 keV and p=0.57; see also figure 4. (d) Wigner’s semicircle with pR=0.012 and a Gaussian distribution with σp=0.007 for E=25 keV and p=0.57. This formulation meets our statistical criteria: it allows for manipulating skewness in energy through βand has a kurtosis less than the Gaussian model for both energy and pitch, in line with our initial objectives. 4. Evaluation of FILDSIM models at TCV We have implemented the proposed model for the TCV FILD. The energy and pitch strike-point distributions along with the skew-Gaussian/Gaussian and the skew-raised cosine/Wigner’s semicircle model fits are illustrated in figure 4, simulated with N=3·104markers, a value at which the residual errors and the χ2-values have saturated (see below). We will refer to these models as the ‘standard Gaussian model’ (SGM) and the ‘raised cosine model’ (RCM). Compared to the SGM, the RCM provides a more accurate fit for both energy and pitch strike-point distributions, specifically eliminating the gap in the upper tails of the energy distributions and removing the tails in the pitch distributions entirely. Recall the χ2statistic defined by χ2= k X i=1 (Oi−ξi)2 ξi ,(21) where Oiand ξidenote the observed and expected frequencies for measurement i, and kis the total number of measurements. For the purpose of comparing the SGM and RCM in fitting the energy strike-point distributions, both models yield comparable χ2-test statistics. Specifically, the average χ2values for 19 different energy levels ranging from 7 to 47 keV at a constant pitch p=0.57, the average χ2values are χ2SGM =22.6;χ2RCM =22.9.(22) In all cases, the χ2values fall below the critical value χcrit = 27.6 for a significance level of α=0.05, calculated for the degrees of freedom df =k−1−p, where pis the number of estimated parameters. The tests indicate that neither model is statistically preferable based on χ2alone. Hence, more sophisticated statistical tests are required to differentiate between the SGM and RCM energy strike-point distributions. The statistical characteristics are different for the pitch strike-point distributions. The χ2-test statistics averaged over 15 different pitch values in the range from 0.4 to 0.8, holding the energy constant at E=25 keV, are χ2SGM =53.9;χ2RCM =23.4,(23) and χcrit =27.6. The χ2values for the SGM are not significant for any of the 15 pitch strike-point distributions, whereas the χ2values for the RCM are significant in all 15 cases. 6 Plasma Phys. Control. Fusion 66 (2024) 045004 B S Schmidt et al Figure 4. (a) SGM and (c) RCM fits to synthetic strike-point distributions in energy at TCV for p=0.57. With E0indicated as a subscript, the parameter values for the fits in (a) are: (σE,β)7= (1.1,2.4),(σE,β)13 = (3.4,2.4),(σE,β)19 = (5.9,2.7),(σE,β)25 = (9.1,2.9). The parameter values for the fits in (c) are: (σE,β)7= (1.0,10),(σE,β)13 = (3.8,8.5),(σE,β)19 = (6.5,5.3),(σE,β)25 = (9.7,3.6). (b) SGM and (d) RCM fits to synthetic strike-point distributions in pitch at TCV for E=25 keV. The parameter values for (b) and (d) are indicated on the subfigures. To further quantify the improvement of the RCM over the SGM, we define the directional residual difference (DRD) as (DRDi)k1k2=|Rik1|−|Rik2|,(24) where Rik is the ith residual of model kjfor j=1,2. It follows that (DRDi)k1k2<0 when the ith residual of k1is closest to 0, and (DRDi)k1k2>0 when the ith residual of k2is closest to 0. The DRD for k1=RCM and k2=SGM are illustrated in figure 5. Since the improvements in the RCM compared to the SGM for the strike-point distributions in energy occur at the upper tails, only the DRDs for the upper 90th percentile are illustrated. For both energy and pitch strike-point distributions, almost all residuals of the RCM are closer to zero than those of the SGM, so DRDi<0, indicating an improved model. The SGM’s overestimation of the tails is clearly evident, manifesting as more negative DRD values in these regions. 4.1. SGM and RCM weight functions In this section, we contrast the SGM and RCM weight functions. Consider an ion initialized at (E′,p′). The velocityspace distribution of this ion is a delta function f=δE′,p′. All possible strike points from ions initialized at this point in velocity space form a strike-point distribution. Reshaping each strike-point distribution into a column vector and horizontally concatenating them results in a 2D array in which each column is a strike-point distribution corresponding to one point in velocity space, and the rows are weight functions. They indicate which points in velocity space can strike a specific point on the scintillator. In the following, we focus on the weight function corresponding to E=25 keV and p=0.57. This point was chosen as it is central to a measurement to be analyzed in section 6. The weight functions at other points in velocity space display the same characteristics. The numerically calculated weight function is used as a benchmark for comparison; see figure 6(a). The SGM and RCM weight functions are illustrated in figures 6(b) and (c). The RCM weight function closely aligns with the numerically calculated weight function in both pitch and energy. Specifically, the sharp cut-off in pitch and the behaviour of the upper tails are well-represented. On the other hand, the SGM weight function exhibits noticeably longer tails in both energy and pitch, and its high-intensity regions do not qualitatively mirror those of the numerically calculated weight function. To evaluate the accuracy of the SGM and RCM models, we introduce difference weight functions ∆Wℓdefined as ∆Wℓ=Wℓ−Wnum,(25) 7 Plasma Phys. Control. Fusion 66 (2024) 045004 B S Schmidt et al Figure 5. (a) DRD for energy and (b) DRD for pitch: Directional residual differences between SGM and RCM residuals for the strike-point distributions in figure 4. Most points fall below 0 for all energies and pitches, indicating that the RCM residuals are generally closer to zero, thus providing a better fit than the SGM. Figure 6. (a) Numerically calculated weight function using FILDSIM for E′=25 keV and p′=0.57. (b)–(c) Weight functions according to the SGM and RCM, respectively, under the same conditions as (a). The weight functions are zero in the white regions. where Wℓis the SGM or RCM weight function, and Wnum is the numerically calculated weight function considered to be the benchmark. The difference weight functions ∆WSGM and ∆WRCM are illustrated in figure 7. Visual inspection of ∆WSGM indicates a noticeably larger width in pitch and elongated tails compared to Wnum, as well as quantitatively larger deviations compared to ∆WRCM. Building on this definition, we introduce the absolute error EA,ℓ of model ℓto quantify the total deviation across all elements of the difference weight function: EA,ℓ = m X i=1 n X j=1∆Wℓ,ij,(26) where mand nare the number of rows and columns in ∆Wℓ. The absolute error for the difference weight functions under consideration according to the SGM and the RCM is EA,SGM =186;EA,RCM =95.(27) Thus, the RCM is a 49% improvement in the absolute error over the SGM. Importantly, this improvement is not a localized phenomenon restricted to the specific weight function under consideration; rather, similar improvements are observed in all weight functions. 5. Characterization of FILDs This section introduces two novel ways of characterizing FILDs: the gross FILD measurement Mgross and the gross weight function Wgross. To determine the regions in the FILD camera where the most counts would be generated given a uniform distribution of ions in velocity space, we integrate the strike-point distributions for all initial energies and pitches. The resulting integrated measurement, denoted as Mgross, is calculated by setting f=1 in (3) and evaluating the integrals: Mgross (E,p) = ˆ1 −1 dp′ˆ∞ 0 dE′W(E′,p′|E,p).(28) The gross FILD measurement is the mapped FILD measurement for uniformly distributed ions in velocity space. Since the individual strike-point distributions can overlap, the magnitude of the gross FILD measurement at a given point should 8