A genetic algorithm approach to reconstructing spectral content from filtered x-ray diode array spectrometers
G. E. Kemp, M. S. Rubery, C. D. Harris, M. J. May, K. Widmann, R. F. Heeter, S. B. Libby, M. B. Schneider, B. E. Blue
AA genetic algorithm approach to reconstructing spectral content from filtered x-raydiode array spectrometers a) G. E. Kemp, b) M. S. Rubery, C. D. Harris, M. J. May, K. Widmann, R. F. Heeter, S. B. Libby, M. B. Schneider, and B. E. Blue Lawrence Livermore National Laboratory, Livermore, CA 94550,USA Atomic Weapons Establishment, Reading, RG7 4RS, UK (Dated: 3 August 2020)
Filtered diode array spectrometers are routinely employed to infer the temporal evo-lution of spectral power from x-ray sources, but uniquely extracting spectral contentfrom a finite set of broad, spectrally overlapping channel spectral sensitivities is de-cidedly nontrivial in these under-determined systems. We present the use of geneticalgorithms to reconstruct a probabilistic spectral intensity distribution and compareto the traditional approach most commonly found in literature. Unlike many of thepreviously published models, spectral reconstructions from this approach are neitherlimited by basis functional forms, nor do they require a priori spectral knowledge.While the original intent of such measurements was to diagnose the temporal evo-lution of spectral power from quasi-blackbody radiation sources – where the exactdetails of spectral content was not thought to be crucial – we demonstrate that thisnew technique can greatly enhance the utility of the diagnostic by providing morephysical spectra and improved robustness to hardware configuration for even stronglynon-Planckian distributions. a) The following article has been submitted to by Review of Scientific Instruments. After it is published, itwill be found here. b) [email protected] a r X i v : . [ phy s i c s . c o m p - ph ] J u l . INTRODUCTION Filtered x-ray diode array spectrometers are routinely employed to characterize the x-rayemission vs. time from high-energy density plasmas, such as those generated from laser-matter interactions or in Z-pinch devices . The spectral response, R i ( E ) (typically inunits of V /GW ) for photon energy E , of each diode channel i is uniquely characterized bythe choice of filter, diode, and (for low energy channels) x-ray mirror design. By carefulpairing of materials, channels can be strongly sensitive in (relatively) narrow spectral bandsand an array of such channels can cover a broad spectral window from eV to keV photonenergies. The corresponding voltage trace recorded by channel i is therefore described by V i ( t ) = Ω i (cid:90) ∞ R i ( E ) S ( E, t ) dE, (1)where Ω i is the detector solid angle and S ( E, t ) is the incident spectral flux (typically in unitsof GW/eV /sr ) at time t . While conceptually straightforward to implement experimentally,uniquely recovering the incident spectral content from a finite set of channels is decidedlyless trivial.Several spectral reconstruction algorithms have been suggested but most involve generat-ing a finite set linear basis functions B such that S ( E ) = (cid:80) j B j ( E ) X j where the weightingvalues X j are either solved for iteratively, through matrix inversion, or with non-linear least-square fitting . Some of the more common forms of these basis functions include gaussians ,cubic-splines , or those generated through machine evolution . While numerically fast androbust, the reconstructed spectral features are limited by the complexity and number of thebasis functions and – when compared to higher resolution spectral data – can result inspurious spectral content.Other approaches suggest that more physical spectral reconstructions can be obtainedwhen a priori spectral information via candidate spectra – provided either through modeling,experimental measurements, or with analytic physics models – is incorporated into thereconstruction routine . The Levenberg-Marquardt method , for example, is routinelyemployed to analyze inertial confinement fusion hohlraums which models the x-ray emissionas a linear combination of analytic bremsstrahlung, M-band (approximated as a gaussianbump), and Planckian (thermal) forms. While the inclusion of a priori information likelyimproves the fidelity of the extracted spectrum, the generalization of the algorithm to time-2ependent or more complex x-ray sources becomes cumbersome.In this work, we introduce an alternative approach to extracting incident spectral con-tent from filtered x-ray diode arrays using a Genetic Algorithm (GA). Unlike many of thepreviously discussed routines, neither complex basis functional forms nor assumptions aboutthe spectral content are explicitly necessary. In Section II, we discuss the specifics of ourimplementation. Demonstration of the robustness and flexibility of the algorithm is providedthrough a variety of examples of physically representative x-ray sources using synthetic ( i.e. known) spectra and channel configurations in Section III. A comparison to the traditionalapproach most commonly employed in literature is also presented. We conclude with adiscussion of the results in Section IV. II. GENETIC ALGORITHM APPROACH:
PÚKA
Genetic algorithms describe a suite of commonly employed heuristic techniques thathave previously been demonstrated to be efficient at finding reasonable – albeit not al-ways optimal – solutions to optimization problems where multiple local optima can existand the parameter space is relatively large and unstructured : e.g. the traveling salesmanproblem . In essence, they mimic the evolutionary biological process of natural selection toiteratively produce a converged solution. The basic approach is as follows. First, an initialpopulation of potential solutions is generated – each individual solution in the populationhas its own genome, an a priori discretized representation of the solution domain. Next, agoodness of fit parameter is used to sort and rank the genomes. A subsequent generationof solutions is then populated using a combination of routines; some fraction of the highestranking genomes are selected to continue unaltered to next generation (known as elites) andthe rest are altered from the previous generation, with the better solutions being given moreopportunities to contribute genetic information to the next generation. Some genomes aremodified by combining two or more genomes (known as crossover) and others are muta-tions of a single parent genome – multiple techniques exist for crossover and mutation .Each successive generation proceeds identically until the desired fit parameter or maximumnumber of allowed generations is reached.Applying this technique to our filtered x-ray diode array spectrometer problem, the con-tinuous incident spectral flux S ( E, t ) becomes the genome. We first discretize the spectrum3nd channel responsivities into a finite number k of energy bins ( i.e. chromosomes), S k ( E k , t ) and R i,k ( E k ) , respectively, to keep the problem numerically tractable. The correspondingvoltages are now calculated as V i ( t ) = Ω i (cid:88) k R i,k ( E k ) S k ( E k , t ) ∆ E k , (2)where ∆ E k are the bin widths whose spacing is chosen in an a priori manner. The populationis initialized with mutations of the best-fitting Planckian distribution to the voltage traceswhere a variety of mutation routines ( e.g. single chromosome manipulation, chromosomepair flipping, add/subtract randomized gaussian and continuum spectral features, etc.) areemployed. The spectra are then ranked using a chi-squared fitness function given by χ ( t ) = 1 N N (cid:88) i =1 ( V i ( t ) − V mi ( t )) V mi ( t ) , (3)where V mi is the measured voltage of channel i for a system with N channels. For subsequentgenerations, elites typically constitute − of the top ranked genomes, the top ofthe best fitting solutions are crossed through a variety of two-parent crossover routines ( e.g. ,single and multiple point segment crossover, uniform mixing, weighted averaging, etc. ), andthe rest are generated through mutation of both the elites and better fitting solutions;the remaining lower subsequently become extinct and do not contribute to the nextgeneration. Although the number of generations required for convergence will vary greatlydepending upon the specifics of the GA implementation, we find convergence within roughly500 generations for genomes with logarithmically-spaced energy bins and a populationsize of 100 for any of the spectra considered in this work. We note that this binning ansatzreasonably resolves all the relevant spectral features ( e.g. spectral edges and regions withlarge gradients) across all the channel response functions for the cases considered in thiswork.Due to the randomized nature of machine evolution and the infinite number of degeneratesolutions that satisfy this under-determined – yet consistent – system of equations ( i.e. thenumber of energy bins exceeds the number of channels), the resultant spectra from separatereconstruction attempts never converge to a unique solution. Indeed, GAs have previouslybeen demonstrated to be capable of finding not just one, but many solutions to such systems4f equations (when they exist) . In this scenario, probabilistic methods become necessarywhen reporting solutions . Therefore, multiple populations are generated and evolved inparallel. The resulting optimized solutions from each population are then consolidated toproduce an averaged solution and standard deviation (both weighted by /χ ). In thismanner, only robust spectral features remain in the averaged result and uncertainties canbe reported on a bin-by-bin basis. In this work, converged statistics from 100 differentconverged populations are presented as the probabilistic solution . Due to the polymorphicnature of the routine, we have dubbed this algorithm púka after the mischievous shape-shifting creature of Celtic folklore . III. RECONSTRUCTIONS OF SYNTHETIC SPECTRA
Synthetic spectra from a variety of laser-matter interactions were generated to test therobustness of the genetic algorithm to representative, routinely diagnosed x-ray sources onboth the OMEGA and National Ignition Facility (NIF) laser facilities. The simulationswere performed in hydra – a multi-physics, multi-dimensional, arbitrary Lagrangian-Eulerian, radiation-hydrodynamics code – which incorporated non-local thermodynamicequilibrium (non-LTE) effects with detailed super-configuration accounting atomic mod-els from cretin . Modeling details typical of such interactions can be found in priorpublications . While physically meaningful spectra are not necessary to demonstrate thecapabilities of the approach, we note that the filtered diode array spectrometers employedon such facilities are meticulously designed to diagnose similar spectral content. For thepurposes of this study, we consider 3 test spectra of increasing spectral complexity: (i)non-Planckian continuum emission from a direct-drive exploding-pusher , (ii) non-LTE XeL-shell ( ∼ − keV ) source , and (iii) non-LTE Kr K-shell ( ∼ keV ) source . Dante is the primary filtered x-ray diode array spectrometer used on both NIF andOMEGA laser facility. Dante has 18 “XRD-31” x-ray diodes with Al, Ni, and Cr photo-cathodes. Several channels employ grazing incidence x-ray mirrors to eliminate sensitivityto high-energy photons. For the purposes of this study, we consider 3 NIF channel configu-rations: the first from shot N170924-003, the second from N180129-001, and the third fromN180604-002, as detailed in Table I, II, and III, respectively. The first is typically employedto quantify Xe L-shell flux ( ∼ − keV ), the second is traditionally fielded for diagnosing5 ABLE I.
Dante-1 channel configuration from N170924-003. The Ta grids are 25 µm thick with8.8% open area. Channel Diode Filters: Mirror: Solid angle Peak sensitivity:number material material/thickness [ µm ] material/angle [deg] [ × − sr ] energy/width [eV]1 Al Sc/0.8 + Ti/0.5 + Ta grid SiO /3.5 0.5786 352.86/57.392 Al B/1.0 + CH/0.8 + Ta grid aC/5 0.5929 167.65/26.633 Al Lexan/4.0 + Ta grid SiO /3.5 0.5786 248.86/45.954 Ni V/2.0 SiO /2.5 0.5603 476.28/48.015 Cr Co/1.4 + CH/0.2 + Ta grid - 0.6514 716.48/84.276 Cr Cu/1.1 + Ta grid - 0.6514 828.97/136.247 Ni Mg/5.0 + Zn/1.0 + CH/0.1 + Ta grid - 0.6514 948.75/90.388 Ni Mg/22 + Ta grid - 0.6514 1184.11/157.279 Ni Al/20 - 0.6514 1426.76/178.5210 Al Si/20 - 0.6514 1713.98/151.5211 Al Ti/26 - 0.6514 4187.27/837.4512 Al Saran/66 - 0.6514 2593.02/308.6513 Al Ag/4.4 - 0.6514 2987.79/481.1814 Al Lexan/2.0 + CaF /21.0 + CH/3.0 - 0.6514 3103.96/620.7915 Ni Fe/30 - 0.6514 6186.51/1200.2916 Ni Ti/26 - 0.6514 4166.37/833.2717 Cr Mn/26.4 + Ni/3.6 - 0.6514 6256.33/334.7118 Ni Mn/26.4 + Ni/3.6 - 1.773 5852.86/910.28 TABLE II.
Dante-1 channel configuration from N180129-001. The Ta grids are 25 µm thick with8.8% open area. Channel Diode Filters: Mirror: Solid angle Peak sensitivity:number material material/thickness [ µm ] material/angle [deg] [ × − sr ] energy/width [eV]1 Al Sc/0.8 + Ti/0.5 SiO /3.5 0.5740 349.85/60.322 Al B/1.5 + CH/1.2 aC/5 0.5881 172.49/20.323 Al Lexan/4.0 + Ta grid SiO /3.5 0.5740 249.55/45.334 Ni V/1.75 SiO /2.5 0.5558 472.28/53.05 Cr Co/1.4 + CH/0.2 + Ta grid - 0.6514 713.29/88.626 Cr Cu/2.0+ Ta grid - 0.6514 863.62/93.27 Ni Mg/5.0 + Zn/0.65 + CH/0.1 + Ta grid - 0.6514 947.78/90.898 Ni Mg/22 + Ta grid - 0.6514 1181.55/160.049 Ni Al/20 + Ta grid - 0.6514 1427.6/177.4410 Al Si/20 - 0.6514 1721.08/148.3111 Al Fe/1.0 + Cr/0.65 + Parylene/5.0 + Ta grid - 0.6514 3547.61/709.5212 Al Saran/66 - 0.6514 2577.59/328.5713 Al Ag/4.4 - 0.6514 3002.96/462.7414 Al Ti/13 - 0.6514 3785.2/757.0415 Ni Zn/25 - 0.6514 8998.97/750.8716 Ni Al/250 - 0.6514 14400/2874.4917 Cr Ni/10 - 0.6514 7173.62/1355.6518 Ni Fe/1.0 + Cr/0.65 + Parylene/5.0 - 1.773 3337.92/667.58 ABLE III.
Dante-1 channel configuration from N180604-002. The Ta grids are 25 µm thick with8.8% open area. Channel Diode Filters: Mirror: Solid angle Peak sensitivity:number material material/thickness [ µm ] material/angle [deg] [ × − sr ] energy/width [eV]1 Al Sc/0.8 + Ti/0.5 + Ta grid SiO /3.5 0.5740 353.32/57.222 Al B/1.0 + CH/0.8 + Ta grid aC/5 0.5929 166.7/27.723 Al Lexan/4.0 + Ta grid + Ta grid SiO /3.5 0.5786 249.55/45.334 Ni V/2.0 + Ta grid SiO /2.5 0.5603 475.46/49.085 Cr Co/1.4 + CH/0.2 + Ta grid - 0.6514 711.48/90.886 Cr Al/250 - 0.6514 14300/2854.727 Ni Mo/32 - 0.6514 15400 3087.018 Ni Y/50 + Al/10 - 0.6514 13000/2598.069 Ni Ge/42 C/375 - 0.6514 9746.42/1576.510 Al Si/20 - 0.6514 1707.2/158.3411 Al Mg/16.4 + Ni/3.6 - 0.6514 5599.46/1119.8912 Al Saran/66 - 0.6514 2585.91/317.7513 Al Ag/4.4 - 0.6514 2958.99/516.2414 Al Mg/22 + Ta grid - 0.6514 1156.77/191.715 Ni Al/20 - 0.6514 1427.94/176.7916 Ni Mg/5.0 + Zn/1.0 + CH/0.1 + Ta grid - 0.6514 944.71/97.0717 Cr Cu/1.1 + Ta grid - 0.6514 834.28/130.2218 Ni Al/250 - 1.773 14400/2887.03 Sp e c t r a l f l u e n c e [ k J / k e V / s r ] (a-I) (a-II) (a-III)0 5 10 15 20Energy [keV]0510152025 C u m u l a t i v e f l u e n c e [ k J / s r ] (b-I) 0 5 10 15 20Energy [keV](b-II) 0 5 10 15 20Energy [keV](b-III) incident UNSPECPUKA
FIG. 1. Reconstructed (a) spectral and (b) cumulative spectral fluence for the non-Planckiancontinuum spectrum (i) using channel configurations I, II, and III. Incident spectrum (red) isunknown to the reconstruction routines. Sp e c t r a l f l u e n c e [ k J / k e V / s r ] (a-I) (a-II) (a-III)0 5 10 15 20Energy [keV]01020304050 C u m u l a t i v e f l u e n c e [ k J / s r ] (b-I) 0 5 10 15 20Energy [keV](b-II) 0 5 10 15 20Energy [keV](b-III) incident UNSPECPUKA
FIG. 2. Reconstructed (a) spectral and (b) cumulative spectral fluence for the non-LTE Xe L-shellspectrum (ii) using channel configurations I, II, and III. Incident spectrum (red) is unknown to thereconstruction routines. Sp e c t r a l f l u e n c e [ k J / k e V / s r ] (a-I) (a-II) (a-III)0 5 10 15 20Energy [keV]01020304050 C u m u l a t i v e f l u e n c e [ k J / s r ] (b-I) 0 5 10 15 20Energy [keV](b-II) 0 5 10 15 20Energy [keV](b-III) incident UNSPECPUKA
FIG. 3. Reconstructed (a) spectral and (b) cumulative spectral fluence for the non-LTE Kr K-shellspectrum (iii) using channel configurations I, II, and III. Incident spectrum (red) is unknown to thereconstruction routines. ∼ − keV ), and the third is usuallyemployed to quantify Kr K-shell flux ( ∼ keV ).Synthetic voltages were generated using (2) for each of the synthetic x-ray spectra andresponse functions described above. Spectral reconstructions from púka and the traditionalgaussian-basis method (known as unspec ) are then compared and contrasted with theincident synthetic spectra (which is unknown to the algorithms); the results are shown inFigs. 1 and 2, and 3, respectively. The uncertainty bars plotted with the púka resultsindicate the /χ weighted average ± σ (standard deviation) for each energy bin for the (a)spectral fluence and (b) cumulative spectral fluence distributions. IV. DISCUSSION
Outside of an initial Planckian guess, the traditional unspec routine only modifies spec-tral energy content via the fixed-position, fixed-width gaussian bumps in order to fit theobserved voltages. While this algorithmic ansatz has been demonstrated to work quite wellfor quasi-blackbody sources (at least in terms of the original intent of recovering total spec-tral power), it is often incapable of recovering sufficient spectral detail to recover the fluencemetrics of interest for non-LTE spectra with bright bound-bound emission without meticu-lous channel design. Consider, for example, the Kr K-shell spectrum shown in Fig. 3; while unspec can recover the > keV K-shell fluence within 20% (a typical value prescribedto the reconstructed fluences given traceable
Dante calibration data ) with the speciallydesigned channel configuration III, there are no gaussian bumps above keV in configura-tion I and, as such, little-to-no energy is attributed to the K-shell region – an algorithmicfallacy to which púka isn’t subject. In general, we find that the reconstructed spectra from púka are (i) more robust to channel selection and (ii) both qualitatively and quantitativelycompare best with the incident spectral distribution and relevant broadband fluence met-rics for each of the sources considered (summarized in Table IV). Furthermore, the inherentprobabilistic nature of púka enables a dynamic and simultaneous assessment of the degreeto which which spectral regions are constrained for a given configuration.The púka uncertainty values are indicative of the variability in the 100 degenerate solu-tions rather than any kind of experimental uncertainty. Individual bin uncertainties largerthan the average indicate an inability of the algorithm to uniquely place energy – a result9 ABLE IV. Tabulated fluence metrics of typical interest for each type of x-ray source. Uncertaintyvalues for reconstructed unspec spectra are assumed to be ± . Reconstructed values in italic font indicate an inability to recover the desired quantity within quoted uncertainty. Spectrum Fluence Metric Incident [kJ/sr] Reconstructed [kJ/sr]Configuration púka unspec (i) Non-Planckian continuum Total ( − keV ) 18.8 IIIIII 18.8 ± ± ± ± ± ± − keV ) 38.5 IIIIII 38.2 ± ± ± ± ± ± − keV ) 5.0 IIIIII 5.06 ± ± ± ± ± ± − keV ) 40.2 IIIIII 40.4 ± ± ± ± ± ± . − keV ) 15.4 IIIIII 15.1 ± ± ± ± ± ± K-shell ( > keV ) 1.2 IIIIII 1.17 ± ± ± ± ± ± of the reconstructed spectra having a higher local bin resolution than the system ( i.e. thecombination of the diagnostic configuration and routine) is capable of robustly reconstruct-ing. Consider, for example, a region of the spectrum where the the response functions arelargely parallel. It is not possible to uniquely determine the local distribution of energy inthis scenario; the only way to better constrain the spectrum in this region is to modify thelocal channel responsivities (or provide additional constraints). While the resulting spectraluncertainties are inherently sensitive to choice of binning, the broadband fluence metricsof interest converge regardless of bin resolution (within reasonable limits). For example,running the same calculation with an order of magnitude less bins (30 vs. 300) does indeedsignificantly reduce the spectral uncertainty in some regions, but the cumulative spectralvalues and respective uncertainties are largely insensitive to this change. As such, we findthat cumulative spectral fluence distributions can be a more robust and physically intuitiveway of illustrating spectral uncertainty and overall energetic discrepancy with respect to theincident spectra. We note that these cumulative distributions are not generated from theaveraged spectra and resulting standard deviations ( i.e. through a quadratic addition of er-rors) but rather the weighted average of the individual cumulative distributions themselvesas the former cannot capture decreases in cumulative uncertainty.10espite being more robust to channel configuration than the traditional unspec routine,the fidelity of the púka reconstructions is still subject to the specifics of the spectral content.On occasion, púka has been observed to robustly invent features in regions where spectralsensitivity is insufficient ( e.g. the feature around ∼ keV in Fig. 3(a)), albeit decidedly lessspurious than some unspec reconstructions. We note that robustness of spectral featuresdoesn’t necessarily imply physicality. Without redesigning a tailored channel configuration,additional solution constraints (other than the only current requirement that spectral inten-sities be positive-definite) may become necessary to further suppress unphysical features. Inaddition to applying a priori / a posteriori spectral knowledge, an adaptive binning routine oradditional algorithmic constraints in low-sensitivity spectral regions may become necessaryfor some applications.While due diligence is still necessary when designing an optimized experimental configura-tion for the spectral metrics of interest, we anticipate that this approach may alleviate somehardware driven design constraints originally imposed by the unspec routine. As púka doesn’t necessitate narrow, spectrally independent channels, configurations with broadbandenergy responsivities that overlap significantly may improve the robustness of the configu-ration to the inevitability of experimental channel data loss – i.e. when calibrated responsefunctions are inconsistent with the resulting data ( e.g. broken filters) or when hardwareissues arise ( e.g. inadequate scope settings or high-voltage bias on the diodes not beingfielded as specified) – as compared to unspec . Despite the current narrowband Dante configuration design approach, we find that the current level of spectral responsivity over-lap can often be sufficient for determining some inconsistent channel data when a singlespectral distribution cannot simultaneously fit all the voltage data to within experimentaluncertainty.Although beyond the scope of this work, it is anticipated that high-resolution spectramay further improve the spectral reconstruction with this approach. As we envision, thehigh-resolution spectral content can simply overwrite a portion of the genome and púka willonly be allowed to modify other spectral regions. High resolution spectra can be providedeither as candidate spectra from atomic-kinetics models or from independent experimen-tal measurements, such as the time-integrated
Virgil crystal spectrometer ( . − keV )already implemented at NIF on Dante-1 . In addition to comparing and contrasting toalgorithms other than unspec , future work will also explore the robustness of the routine11o experimental uncertainty: e.g. those due to response function calibration uncertaintyor channel mistiming in time-resolved reconstructions. A Monte-Carlo approach would ap-pear well suited to this task – as has been previously implemented for unspec – but it isuncertain at this time how the degeneracy of the spectral solutions will also feed into thisconsideration. V. ACKNOWLEDGMENTS
The first author would like to acknowledge many enlightening conversations with D. Barnak,Y. Frank, D. A. Liedahl, C. W. Mauche, E. V. Marley, A. S. Moore, P. L. Poole, andC. A. Thomas.This work performed under the auspices of U.S. Department of Energy by Lawrence Liv-ermore National Laboratory under Contract DE-AC52-07NA27344. This document wasprepared as an account of work sponsored by an agency of the United States government.Neither the United States government nor Lawrence Livermore National Security, LLC,nor any of their employees makes any warranty, expressed or implied, or assumes any legalliability or responsibility for the accuracy, completeness, or usefulness of any information, ap-paratus, product, or process disclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product, process, or service bytrade name, trademark, manufacturer, or otherwise does not necessarily constitute or implyits endorsement, recommendation, or favoring by the United States government or LawrenceLivermore National Security, LLC. The views and opinions of authors expressed herein donot necessarily state or reflect those of the United States government or Lawrence Liver-more National Security, LLC, and shall not be used for advertising or product endorsementpurposes.
VI. DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authorupon reasonable request. 12
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