Abstract
In this study, we compared analytical and machine learning models for two-dimensional nondestructive localization of radioactive sources within a simulated waste drum. A 60Co gamma source was located at 72 different positions within the simulated drum. A reflector-coated liquid light guide (LLG) system detected the Cherenkov radiation generated by 60Co source. Time-of-flight measurements revealed that time differences of the source peaks correlated linearly with angular position, while their amplitudes exhibited complex dependencies on both radial and angular coordinates. For 24 arbitrary source positions, an analytical localization model achieved mean absolute errors of 2.024 cm for radial distance and 5.940° for angular position, with an overall Euclidean error of 2.518 cm. However, model performance degraded significantly at the LLG intersection region and at small radial distances where signal intensity decreased. To address these limitations, a multilayer perceptron (MLP) neural network was developed and reduced errors to 0.601 cm radially, 4.703° angularly, and 1.181 cm in Euclidean distance. The use of the MLP neural network represents improvements of 70.31%, 20.82%, and 53.10%, respectively. The neural network model successfully resolved localization challenges in geometrically complex regions by capturing nonlinear spectral-spatial relationships. These findings demonstrate that integrating machine learning with localization enhances radioactive waste characterization capabilities.
Full Text
Preamble
Comparative Study Analytical Machine Learning Models Gamma Source Localization Waste Drums Sangjun Seunghyeon Hyung Jinhong Seung Chulhaeng Bongsoo School Energy Systems Engineering, Chung-Ang University, Seoul, 06974, South Korea Department Materials Science Engineering, Soong-sil University, 06978, South Korea Department Radiation Oncology, Augusta University, Georgia, 30912, orresponding Email:
Abstract
study, compared analytical machine learning models two-dimensional nondestructive localization radioactive sources within simulated waste drum. gamma source located different position within simulated drum. reflector-coated liquid light guide (LLG) system detected Cherenkov radiation generated source.
Time-of- flight measurements revealed differences source correlated linearly angular position, while their amplitudes exhibited complex dependencies radial angular coordinates. arbitrary source position analytical localization model achieved absolute errors radial distance angular position, overall Euclidean error However, model performance degraded significantly intersection region small radial distances where signal intensity decreased. address these limitations, multilayer perceptron (MLP) neural network developed reduced errors radially, angularly, Euclidean distance
neural network represent improvements 70.31%, 20.82%, 53.10%, respectively. neural network model successfully resolved localization challenges geometrically complex regions capturing nonlinear spectral-spatial relationships.
These findings demonstrate integrating machine learning localization enhances radioactive waste characterization capabilities.
Keywords
Liquid light guide, Time-of-flight, osition-sensitive, Cherenkov radiation Reflector
1. Introduction
management continuous monitoring radioactive waste remain critical challenges context long-term storage environmental protection.
Radioactive waste generated nuclear power plants poses complex radiological risks during handling, transportation, storage. mitigate these risks, radioactive waste packaging maintain structural integrity, containment, confinement capabilities throughout interim storage permanent disposal provide adequate protection against radiological hazards [1-3].
Among radionuclides present radioactive waste, represents particularly significant isotope distinctive gamma energies emits gamma energies these high-energy gamma significant contributor external rates shielding requirements during storage.
These characteristic gamma energies enable reliable detection, thereby classifying easy-to-measure (ETM) nuclide. scaling factor
method
employs nuclide measurements primary indicators estimating difficult-to- measure (DTM) nuclide activities.
method
become standard
methodology
characterizing radionuclide inventories radioactive waste, particularly nuclides cannot readily quantified through direct measurement. approach exploits correlation between nuclides derive quantitative estimates nuclide activities
Precise determination information nuclide remains essential implementation method.
Disposal facilities require monitoring infrastructure ensure containment radioactive materials enable detection potential releases through continuous surveillance storage conditions.
Nondestructive assay (NDA) emerged internationally recognized standard achieving necessary measurement precision techniques significant classification verification radioactive waste without requiring invasive sampling could compromise containment integrity.
American Society Testing Materials (ASTM) International recognizes segmented gamma scanning (SGS) tomographic gamma scanning (TGS) standard techniques [7-9].
Alamos National Laboratory (LANL) constructed first commercial system However, these standard techniques certain limitations. assumes homogeneous distribution matrix material radioactive content throughout waste drum; deviations assumption produce assay errors exceeding [10]. addressed limitation developing
method
substantially improves measurement accuracy However, requires complex detection process longer measurement times [11-12].
Researchers explored promising solutions using time-of-flight (ToF)-based detection systems address these challenges [13-15].
methods
localize source positions measuring differences radiation-induced signals arriving detectors.
These techniques enable cost-effective monitoring large areas combined optical fibers. effective detectors plastic scintillating fiber (PSF). serve function: detecting radiation transmitting resulting light. maintain core-cladding structure conventional optical fibers incorporate fluorescent agents within material radiation interacts material scintillation light generated transmitted
through axis. combination radiation detection scintillation efficient optical transmission makes well-suited radiation monitoring applications [16-18].
Researchers investigated Cherenkov radiation parallel scintillation- based systems.
Cherenkov radiation typically considered unwanted
background
signal scintillation detectors lower intensity compared scintillation [19-21] However, Cherenkov radiation offers distinct advantages differentiated scintillation response times, directional emission, characteristic emission spectrum spanning visible wavelength range [22].
Plastic, quartz, glass optical fibers demonstrated effectiveness Cherenkov radiation sensors their optical transparency transmittance properties [23-25].
However, inherently lower light yield Cherenkov radiation necessitates fiber bundles achieve adequate signal levels. zones between fibers traditional fiber bundles
result
incomplete photon capture compromis light collection efficiency. address these limitations liquid light guides developed which increa effective detection without introducing inter-fiber zones. umerous studies investigated characteristics demonstrated their feasibility radionuclide localization through
methods
utilizing Cherenkov radiation [26-29]. consist liquid-filled enclosed within polymer featur substantially larger diameters conventional optical fibers. increased diameter extends interaction length charged particles thereby enhanc Cherenkov radiation generation incorporates fused-silica window open-pipe configuration, which eliminates zones inherent fiber bundles maximizes light collection efficiency.
Additionally, demonstrate distinct advantages conventional optical fibers: attenuation coefficients superior transmittance across UV-Vis wavelength range. these properties,
suitable radiation detection applications recent studies demonstrated effectiveness reflector-coated source localization. coating reflector coupling photomultiplier (PMT) opposite accurate one-dimensional localization achieved [30].
However, challenges remain radiation source located scenarios, Cherenkov intensity diminishes because fewer gamma reach inverse square resulting fewer Compton scattering interactions consequently fewer Cherenkov photons being generated.
Conventional analytical models curve fitting
methods
based geometrical assumptions often struggle interpret low-intensity signals accurately under these conditions. adoption machine learning -based approaches, which well-suited handling nonlinear data.
Neural network models localize source positions accurately conventional methods, particularly challenging measurement conditions [14,31]. study, compared analytical -based models two-dimensional nondestructive localization gamma-emitting source inside simulated radioactive waste drum. disk-type source located various positions inside emulate realistic waste geometries, wound around circumference simulated drum. source position, Cherenkov light collected reflector-coated system using measurements collected Cherenkov light generates position spectra, these spectra analyzed function source position using cylindrical coordinates.
These spectral variations formed basis developing analytical localization model. assess potential enhancement techniques, multilayer perceptron (MLP) neural network trained position spectral data. models validated using training subsequently tested arbitrary source positions included training data.
performance analytical localization model compared -based approach using neural networks performance models evaluated using absolute errors (MAEs) radial distance, angular position, Euclidean distance.
Methods
Cherenkov radiation charged particle travels through medium speed exceeding speed light medium, medium becomes polarized charged particle, polarized medium emits coherent, directional radiation phenomenon known Cherenkov radiation first characterized Pavel Cherenkov Cherenkov radiation become fundamental detection principle unique threshold behavior directional emission characteristics. charged particle possess sufficient kinetic energy exceeding specific threshold value Cherenkov radiation, which called Cherenkov threshold energy (CTE). medium constant refractive index electrons, denoted calculated using following relation:
� � ℎ , � = � � � 2 1 +
where electron rest-mass energy (0.511 [23]. materials typical optical fiber whose refractive indices calculated electrons approximately respectively.
These threshold energies determine whether secondary electrons produced gamma interactions generate Cherenkov radiation.
While electrically neutral gamma cannot produce Cherenkov radiation directly their
electric charge, charged particles alpha particles directly generate Cherenkov radiation.
However, gamma interact edium various mechanisms produce secondary charged particles. interaction resulting energy distribution secondary particles governed energy incident gamma atomic number target material. predominantly composed atomic-number elements, Compton scattering dominant interaction mechanism incident gamma rays.
During Compton scattering, incident gamma transfers portion energy orbital electron, ejecting ejected electron called Compton electron. gamma incident optical fiber, Cherenkov radiation emitted kinetic energy resulting Compton electron exceeds electrons medium.
Compton electron achieves maximum energy gamma scattered maximum energy Compton electron produced gamma energy calculated using following equation [32]:
� ��� , � = � � −
energy ncident gamma crucial factor determining whether Compton electrons exceed Cherenkov threshold. gamma minimum gamma energy required produce Cherenkov radiation, derived
combining Eq. 1 and Eq. 2:
� � ℎ , � =
gamma energies generate Compton electrons
maximum energ respectively. These energies exceed electrons typical optical fibers shown Cherenkov threshold energies electrons gamma function refractive index.
Experimental setup Experimental setup two-dimensional localization source using reflector- coated system.
Cherenkov light detection. (Series diameter, length, Lumatec) coated reflector (PF03-03-F01, Thorlabs), while opposite optically coupled module (H10721P-01, Hamamatsu Photonics) silicone-rubber-based optical interface (EJ-560, Eljen Technology) refractive index matching.
Blackout fabric (BK5, Thorlabs) cover eliminate ambient light interference. module converted incoming light electrical signals, which amplified amplifier (ABL0300-00- 4030, WENTEQ) subsequently recorded oscilloscope (SDS5104X, Siglent). oscilloscope operated constant sampling acquire signals temporal precision.
Detailed specifications oscilloscope module provided Tables respectively.
Properties Value Sample [GS/s] Bandwidth [GHz] Vertical resolution [bit] Maximum waveform capture [wfm/s] 500,000
Minimum time limit for trigger [ns]
Trigger resolution
Properties Value Spectral response sensitivity wavelength Effective Typical current Specific erties H10721P-01 localize radioactive sources within simulated waste drum, wound intersection region where crosses itself region, gamma generate Cherenkov radiation segments, producing signals positions simultaneously.
Since intersect itself point where starts completes circular path, experimental setup, represent physical position appear distinct peaks position spectrum. phenomenon presents challenge conventional analytical models, discussed Section intensity Cherenkov radiation plays critical detection. number Cherenkov photons length wavelength denoted expressed following relation:
���� =
where structure constant (1/137), electric charge particle units elementary charge electrons), ratio particle velocity speed light vacuum [33]. shown intensity Cherenkov radiation exhibits wavelength dependence.
Specifically, number photons proportional proportionality means higher photon yields occur shorter wavelengths resulting significantly enhanced emission UV-visible region compared longer wavelengths.
Consequently, performance optical sensing system highly sensitive spectral response characteristics constituent components, including reflector.
These components exhibit transmittance reflectance across UV-Vis wavelength range ensure efficient photon collection detection.
Spectral characteristics reflector, module UV-Vis wavelength range.
transmittance reflectance reflector, quantum efficiency module. which serves Cherenkov light generation medium light guide, demonstrates excellent optical transmittance wavelength range [34]. reflector exhibits reflectance greater wavelength range [35]. module detect light across broad spectral range exhibits quantum efficiency approximately making well-suited maximizing Cherenkov light collection [36].
Time-of-Flight
method
Cherenkov radiation generated within core, Cherenkov photons propagate bidirectionally along total internal reflection (TIR). fundamental optical principle which light traveling medium higher refractive index lower refractive index completely reflected interface occurs incident angle light exceeds critical angle interface.
Cherenkov radiation, characteristic angle Cherenkov photon emission, known Cherenkov angle, critical angle crucial parameters Cherenkov angle, denoted determined kinetic energy Compton electron shown
� � ℎ = arccos
�� = arccos
photons propagate along emission angle Cherenkov photons, determined Compton electron trajectory Cherenkov angle, satisfy condition core-cladding interface. critical angle depends refractive indices
cladding materials, specific refractive indices components provided manufacturer. source position within determined difference between signals detected module. photons reach module either directly after reflection far-end reflector. difference between Cherenkov photons arriving module depends path-length difference, shown ToF-based localization through total internal reflection using reflector-coated Cherenkov radiation generated radioactive source position along length photons traveling directly traverse distance while those reflected far-end reflector traverse distance Consequently, difference between signals, denoted expressed function
� � =
acquisition signal enters oscilloscope amplitude greater amplitude threshold oscilloscope recorded leading-edge timing. signals leading-edge threshold within defined window, oscilloscope captures signals simultaneously using interval triggering.
However, inherent limitation leading-edge timing susceptibility amplitude variations signal amplitudes affect which signal crosses amplitude threshold.
Furthermore, discrepancies between signals introduce additional timing errors.
These effects reduce accuracy time-difference measurements, thereby compromising localization accuracy. address these limitations, Amplitude Rise-time Compensation (ARC) triggering employed. triggering commonly crossover-timing
method
converts signals bipolar waveforms improve timing precision. triggering offers immunity amplitude fluctuations improves precision time-difference measurements regardless variations signal [37]. signal attenuated through multiplication constant (where attenuated signal delayed fixed interval summed signal. delay chosen signal ensure resulting signal crosses before reaching maximum amplitude. signals captured simultaneously oscilloscope converted bipolar waveforms through triggering, difference between crossings waveforms measured. implementation triggering illustrated
Amplitude Rise-time Compensation triggering precise time-difference measurements. two-dimensional source localization using cylindrical coordinates, measurement points arranged regular intervals radial distance angular position. uncollimated
60 Co
disk-type source activity positioned measurement points: radial distances increments, covering angular range intervals. source located center inner radial distance respectively other words, radial distance refers distance between source center intersection region, angular position represents angular position increases counterclockwise direction position, 10,000 paired signals differences acquired.
These signals constituted training dataset, which derive analytical localization model through curve fitting train model through neural network additional 5,000 paired signals collected arbitrary positions evaluate localization accuracy Model performance evaluated using absolute error
radial distance, angular position, Euclidean distance between estimated actual source positions.
Discussion
difference spectral characteristics position spectra dependent source position provide foundation spatial localization methods.
Position spectra acquired using 10,000 paired signals obtained source positions their width position spectra different radial distances angular positions compared investigate influence source position spectral characteristics. consistency, position spectrum normalized total count Position spectra various angular positions along radial distance radial
distance spectrum consistently exhibits three distinct peaks. varies angular position defined source peak, indicating actual source position. shown difference source clearly shifts changes angular position. fixed radial distance, amplitude source greatest angular position which directly opposite intersection region decreases angular position deviates particular, decreases abruptly source approaches intersection region. intersection region, normalization process distributes total counts between angular positions resulting characteristic low-intensity peaks.
Nevertheless, source peaks, including these low-intensity peaks, demonstrate temporal stability maximum standard deviation across repeated experiments.
Table summarizes differences source their standard deviations.
Source position [degrees] difference StDev Table difference standard deviation according source position. peaks appearing consistently spectra referred reflection
peaks. These reflection peaks arise reflections ends, which analyzed prior studies [25-27]. contrast angular-dependent characteristics source peak, amplitudes reflection peaks across spectra significant angular dependence.
Position spectra function radial distance fixed angular position influence radial distance spectral characteristics examined. shows position spectra various radial distances fixed angular position shown difference source remained unchanged although radial distance varied. invariance difference radial distance indicates difference source depends solely angular position. contrast, amplitude source decreased width broadened decreasing radial distance. radial distance decreases, source moves farther causing gamma spread wider area. wider distribution increases range differences, thereby leading broadening. broadening hinders accurate source localization, particularly intersection region
center drum. However, reflection peaks showed radial dependence, similar their angular independence, maintaining consistent differences, widths, amplitudes across spectra Comparison position spectra sources positioned radial distances angular positions spectra radial distances angular positions position spectra (diamond markers) closely match those (solid lines), showing minimal variation despite difference radial distance. similarity between spectra different radial distances poses challenges distinguish actual radial distance source position. limitatio reduc localization accuracy small radial distances where broadening obscures radial distance information.
Variation source amplitude respect source position Cartesian coordinates Cartesian coordinates. shown amplitude source source located decreas progressively radial distance approaches Additionally, intersection region correspond angular positions exhibits markedly reduced amplitude despite relatively source amplitudes observed other positions.
Despite these limitations observed small radial distances intersection region, spectral dependence source radial distance angular position provide sufficient information constructing localization model based spectral characteristics.
Analytical localization model Quantifying relationship between source position spectral dependence source essential model construction. construct analytical localization model, source
position expressed function difference amplitude source peak, which exhibit different dependencies source coordinates. result, difference amplitude source expressed functions angular position radial distance. demonstrated Figs. difference source depends solely angular position, remaining invariant radial distance. illustrates relationship, showing difference varies linearly angular position.
Relationship between differences source angular position.
While difference characteristics enable angular localization, amplitude characteristics essential radial distance determination. amplitude source varies changes radial distance angular position. shown amplitude depends strongly radial distance exhibits weaker dependence angular position. dependency indicates amplitude source represented function radial distance angular position Therefore, bivariate functional model essential
accurate radial distance estimation. amplitude source exhibits strong correlation radial distance demonstrating approximately proportional relationship. amplitude source reaches maximum decreases symmetrically approaches assuming dependence proportional account interactions between radial angular dependencies, model incorporating interaction terms developed. 11(a) resulting optimal model residuals, respectively.
Optimal analytical model amplitude source function radial distance angular position residuals Based relationships shown Figs. localization model estimating coordinates constructed function difference source amplitude source (33.331
� ( � � , � ) =
2 (0 ≤ � ≤ 2 휋
0.104+
process analytical model construction, functions derived using position spectra radial distances ranging insufficient spectral variation range, shown Extrapolation applied estimate source positions radial distances evaluation process involves validation using training predictions arbitrary positions assess model performance, illustrated Performance analytical localization model: validation using training prediction arbitrary source positions. validation
results
reveal capabilities limitations analytical localization approach. 12(a) shows analytical localization model cannot accurately determine actual source position intersection region where sharp decrease amplitude source occurs simultaneously Although amplitude variations provide useful localization information, abrupt decrease intersection region challenges conventional analytical approaches based functional relationships. errors Euclidean distance caused
phenomenon evident 12(b) Additionally, trend observed center where extrapolation applied.
These errors highlight inherent limitations analytical approach regions where signal characteristics become complex, signal counts insufficient. standard deviations arbitrary source positions radial distance, angular position respectively overall standard deviation Euclidean distance maximum minimum Euclidean distance errors These
results
demonstrate while analytical model achieves reasonable accuracy regions, performance degradation intersection central regions indicates sophisticated approaches handle complex signal characteristics. localization model neural network employed develop -based localization model addresses limitations identified analytical approach. neural network implemented MATLAB nonlinear regression. architecture designed capture complex nonlinear relationships between spectral features source positions. network architecture consisted hidden layers layer contain 1,436 nodes Rectified Linear (ReLU) activation function mitigate vanishing-gradient issues improve training efficiency [38].
Optimization performed using optimizer, which combines adaptive learning rates momentum-based updates accelerate convergence enhance training bility [39]. model trained maximum 1,000 epochs, training randomly shuffled epoch. prevent overfitting, early stopping implemented terminate training validation failed improve specified number consecutive iterations.
training insufficient, augmentation
methods
effective obtaining
results
through neural networks. augmentation
methods
increase amount training while preserving features original position spectra. augmentation method, sliding window technique applied increase number spectra derived source positions thereby constructing training sliding window technique divide difference sequences overlapping segments Before dividing them, time-difference sequences source position first truncated minimum valid count identified across positions. truncated sequences extended appending portion sequence length equal sliding window.
These extended sequences segmented using sliding window advanced position spectrum obtained extended sequences excluded because overlapped position spectrum obtained original sequences. result, sliding window technique constructed training data, resulting total number position spectra equal minimum valid count divided size. minimum valid signal count identified 7,000 among 10,000 signals acquired position.
Using sliding window length 5,000
method
generated position spectra source position, yielding 1,008 total spectra training model. augmented select optimal model through 7-fold cross-validation, which ensures model generalizability prevents overfitting [40]. fold, spectra randomly assigned validation while remaining spectra training. squared error (MSE) function quantify prediction accuracy during training validation. model achieving lowest validation across cross-validation folds selected final localization model.
Performance localization model: validation using training prediction arbitrary source positions. proposed localization model improved localization accuracy compared analytical model. particular, ability handle complex nonlinear relationships evident regions where analytical model showed significant errors. 13(a) shows localization model successfully addresses shortcomings analytical model, specifically degraded performance intersection region center. predictions arbitrary positions, shown 13(b), improvements pronounced regions where analytical model encountered challenges geometric complexity signal amplitudes. radial distance, localization model reduced representing 70.31% improvement. angular position, decreased achieving 20.82% improvement.
Euclidean distance dropped showing 53.10% enhancement spatial accuracy.
Table summarizes localization
results
analytical models.
Actual position Analytical model model theta theta Error theta Error 7.71, 4.02, 8.78, 3.93, 8.09, 4.57, 9.55, 5.82, 10.36, 7.94, 9.42, 8.68, 10.45, 8.95, 10.7, 12.98, 11.79, 12.8, 13.51, 12.87, 12.9, 11.85, 13.75, 11.71, 11.65, 13.5, 10.67, 12.86, 13.6, 13.21, 13.78, 13.6, 18.19, 14.34, 13.9, 12.46, 14.04, 14.2, 18.33, 15.27, 14.3, 15.64, 14.11, 15.2, 13.55, 13.87, 15.2, 15.08, 14.69, 15.3, 14.74, 14.45, 17.1, 16.17, 16.56, 17.1, 17.08, 16.95, 17.8, 18.22, 17.55, 18.6, 11.62, 19.19, Error standard deviation according model.
4. Conclusion
study, demonstrated feasibility two-dimensional nondestructive localization using reflector-coated wrapped around simulated waste drum, source located various points inside acquire time-difference signals obtained signals generated position spectra
results
tendencies across position spectra revealed distinct spatial dependencies characteristic source position. difference source peaks exhibited linear correlation angular position, providing reliable angular localization independent radial distance.
However, their amplitude showed complex dependencies radial distance angular position, strong radial dependence weaker angular dependence. analytical localization model constructed based mathematical relationships between spectral features source source positions.
However, radial distance less, spectral changes radial distance observed.
Consequently, analytical model employed extrapolation distances Based predictions arbitrary source positions, analytical model achieved radial distance angular position.
While analytical model demonstrated feasibility nondestructive localization model exhibited reduced accuracy specific regions: intersection region where overlaps itself, center where distance between source increases number gamma interactions within decreases, thereby reducing Cherenkov signal intensity.
These regions exhibited amplitude reduction broadening, which limited accuracy analytical model. -based localization model developed overcome these limitations.
implementation neural network substantially improved localization accuracy training model actual source positions, radial distance decreased 70.31% angular position decreased 20.82% compared analytical model. model demonstrated superior performance handling nonlinear relationships complex signal characteristics, particularly challenging regions. overall Euclidean distance improved representing 53.10% enhancement spatial accuracy. future studies, experiments conducted under conditions closely simulate actual radioactive waste environments which includ multiple radioactive sources.
Additionally, current two-dimensional localization model extended three-dimensional imaging through development artificial intelligence -based reconstruction algorithms volumetric source mapping.
Acknowledgements supported Korea Institute Energy Technology Evaluation Planning (KETEP) grant funded Ministry Trade, Industry Energy (MOTIE) Republic Korea RS-2024-00398867 supported National Research Foundation Korea (NRF) grant funded Korean government (MSIT) 2020M2D2A2062457).
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