Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks
EEPJ manuscript No. (will be inserted by the editor)
Muon Radiography to Visualise Individual Fuel Rods in SealedCasks
Thomas Braunroth , Nadine Berner , Florian Rowold , Marc P´eridis , and Maik Stuke [email protected], GRS gGmbH, Schwertnergasse 1, 50667 Cologne, Germany [email protected], [email protected], GRS gGmbH, Boltzmannstr. 14, 85748 Garching, Germany fl[email protected], GRS gGmbH, Kurf¨urstendamm 200, 10719 Berlin, Germany [email protected], BGZ mbH, Dammstraße, 84051 Essenbach, Germanythe date of receipt and acceptance should be inserted later Abstract.
Cosmic-ray muons can be used for the non-destructive imaging of spent nuclear fuel in sealeddry storage casks. The scattering data of the muons after traversing provides information on the therebypenetrated materials. Based on these properties, we investigate and discuss the theoretical feasibilityof detecting single missing fuel rods in a sealed cask for the first time. We perform simulations of avertically standing generic cask model loaded with fuel assemblies from a pressurized water reactor andmuon detectors placed above and below the cask. By analysing the scattering angles and applying asignificance ratio based on the Kolmogorov-Smirnov test statistic we conclude that missing rods can bereliably identified in a reasonable measuring time period depending on their position in the assembly andcask, and on the angular acceptance criterion of the primary, incoming muons.
PACS.
XX.XX.XX No PACS code given
The operation of nuclear power plants generates high-levelradioactive wastes which need to be stored and disposedof. A well-established part of nuclear waste management isthe dry storage of used fuel assemblies in designated casks.Depending on the availability of a final repository, the fuelassemblies might remain inside the casks placed in interimstorage facilities for decades. The casks are designed to en-close the high-level radioactive waste and separate it fromthe biosphere.Inspecting the interior of a storage cask directly would re-quire an opening of the cask, a difficult task due to theradiation. So far, conventional radiography using neutronsor photons could not be applied successfully, partly due tothe rich scattering history of traversing particles as a di-rect result of the dimensions of the storage cask [1]. Othermethods such as three-dimensional temperature field mea-surements or antineutrino monitoring [2] require a detailedknowledge of the fuel history or are not suitable for theassessment of individual storage casks. Cosmic muons, cre-ated directly or indirectly in the atmosphere by the inter-action of cosmic radiation with particles, have been alsoused for imaging purposes ( muography ) [3]. These muonsand anti-muons (muons in the following) are characterizedby a broad energy spectrum spanning several orders ofmagnitude and have a mean momentum of approximately4 GeV/ c [4]. For muons with an absolute momentum above1 GeV/ c the integral vertical intensity I V is approximately 70 m − s − sr − [5,6] and the decrease of the flux intensityscales approximately with cos( θ ), with θ being the anglewith respect to the vertical [4]. In experimental physics, avalue of I V ≈ − min − has been established for work-ing with horizontal detectors.Alvarez and colleagues were one of the first to use muonsfor non-invasive imaging and published their landmark pa-per on the search for hidden chambers in the pyramids ofGiza in 1970 [7]. These first studies were based on the mea-surement of the attenuation of the cosmic muon flux andprovided two-dimensional projection images ( muon radio-graphy ). Since then, muon radiography has been used invarious fields such as the study of volcanoes [8], geologicalapplications [9], the identification of cavities in archaeol-ogy [10] as well as in industrial applications [11].In 2003, a Los Alamos research group proposed using thescattering angle of the outgoing muons as the basic infor-mation for imaging [12]. This approach requires the mea-surement of the incoming as well as the outgoing trajecto-ries of the muons and allows to obtain three-dimensionalimages ( muon tomography ) of volumes not exceeding tensof meters. This technique has already been used in variousfields, such as nuclear control [13], transport control [14]and the monitoring of historical buildings [15]. In additionto experimental studies, Monte-Carlo simulations play animportant role in muon imaging, e.g. in terms of feasibilitystudies or with respect to the detector design.The application of muography for the purpose of non-invasive control and monitoring of the interior of dry stor- a r X i v : . [ phy s i c s . i n s - d e t ] F e b Thomas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks age casks has gained an increasing interest and fosteredexperimental and simulation studies. Besides the funda-mental suitability of muography for this purpose, theseefforts also addressed methodological and time require-ments. Durham et al. [16] applied muon scattering radio-graphy to a Westinghouse MC-10 cask and showed ex-perimentally that cosmic muons can indeed be used todetermine if spent fuel assemblies are missing withoutthe need to open the cask. A number of simulation stud-ies were performed using two planar tracking detectorsplaced on opposite sides of an object, each focusing ondifferent aspects, for example: Jonkmans et al. [17] inves-tigated the capabilities of muons to image the contentsof shielded containers to detect enclosed nuclear materi-als with high- Z . Clarkson et al. [18] performed Geant4 simulations of a scintillating-fibre tracker for tomographicscans of legacy nuclear waste containers. Chatzidakis [19]applied a Bayesian approach to monitor sealed dry casksto infer on the amount of spent nuclear fuel and to inves-tigate the limitiations of this approach. Using the attenu-ation and scattering characteristics of the muons derivedfrom
Geant4 simulations, Ambrosino et al. [20] foundthat a 10 cm Uranium block inside a concrete structurecould be identified after a one-month period of measure-ment. Poulsen et al. [21] were the first to apply filteredback projection algorithms to muon tomography imagingof dry storage casks using simulated data and could showthat this technique can be applied to the detection of miss-ing fuel assemblies. In a more recent work, Poulsen et al. [22] used the experimental data of Ref. [16] for a numeri-cal study using
Geant4 to distinguish different loads ofa cask. The study indicates that a one-week muon mea-surement for the given experimental setup is sufficient todetect a missing fuel assembly or to identify a dummy as-sembly made out of iron or lead.With respect to the resolving power, both experimentaland simulation studies have been focused on the level offuel assemblies so far. In addition, the majority of studiesare based on a transversal configuration, where the detec-tors are placed on the sides of the cask.In this study, we use Monte-Carlo simulations to investi-gate a longitudinal configuration, with the detectors placedabove and below the cask. We will investigate if muogra-phy allows detecting individual missing fuel rods. To un-ravel insights independent of reconstruction algorithms,this work will focus on radiographic images based on trans-mission ratios as well as scatter-angle information.The guiding questions of this work are as follows: Is itpossible to even detect individual missing fuel rods withmuon radiography? If so, are there any constraints or re-quirements with respect to the experimental setup andhow much time does a measurement require? What canbe used as a significance measure to detect a missing fuelrod? Does the significance depend on the relative positionof the considered fuel rod within the fuel assembly and isthe significance dependent on the number of consideredevents?This contribution is structured as follows: In Sec. 2 we pro-vide a detailed description of the simulation tool as well as the investigated geometry. Moreover, we describe thedata processing and aspects related to the validation ofthe simulation. Sec. 3 features the analysis and discussion.We address two levels of detail concentrating on the recog-nition of (missing) fuel assemblies and individual fuel rods.This is followed by a summary and conclusion in Sec. 4.We end with a short outlook in Sec. 5.
Simulations were performed with a dedicated tool basedon the Monte-Carlo toolkit
Geant4 [23,24,25].
Geant4 allows simulating the passage of particles through matterand has been used for numerous applications, includinghigh energy physics, nuclear physics, accelerator physicsand others.In this section we describe the key aspects of the tool, i.e.the geometry (Sec. 2.1), the treatment of primary particleproperties (Sec. 2.2), aspects related to physics and track-ing (Sec. 2.3) as well as optimization strategies (Sec. 2.4)to reduce the computational time.The tool was compiled against v10.06.p2 of
Geant4 andallows using multithreading. The results are written event-by-event into
ROOT [26,27] container files, which allowsperforming post-processing in a flexible manner. Finally,Sec. 2.5 discusses the validation of the code by compar-ing simulated and tabulated (or empirically established)energy losses and angular straggling for different targetmaterials and projectile energies.
The key component of the geometry is a generic caskmodel (referred to as generic model in the following) whichmimics the features of the CASTOR ® V/19 cask [28], e.g.in terms of major components, dimensions, materials andmasses. The CASTOR ® V/19 cask is used for transportas well as storage purposes and is designed to carry up to19 fuel assemblies from pressurized water reactors (PWR).All information on geometries and materials specifying thegeneric model was derived from public available sourcessuch as Refs. [28,29].A visualisation of the generic model was generated withthe
Geant4
OpenGL interface and is depicted in Fig. 1.The individual components of the generic model can beidentified in the exploded view shown in Fig. 2.A comparison of some key properties of the generic modelon the one hand and the CASTOR ® V/19 cask on theother hand can be found in Table 1 and highlights themutual similarities.Each of the 19 fuel compartments can be occupied withone (modelled) fuel assembly. Each modelled fuel assem-bly comprises top- and bottom-nozzle, fuel rods as well ascontrol rods. The fuel rods consist of nuclear fuel (UO )and cladding tubes (Zirconium alloy). A complete 18x18-24 fuel assembly consists of 300 fuel rods and 24 control homas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks 3 Table 1.
Comparison of key dimensions between the genericmodel and the CASTOR ® V/19 cask (in storage configura-tion).Property Generic Model CASTOR ® Model V/19 [28]Overall Height 594 cm 594 cmOuter Diameter 244 cm 244 cmCavity Height 502.5 cm 503 cmCavity Diameter 148 cm 148 cm rods. The arrangement of fuel and control rods within acomplete fuel assembly is shown in Fig. 3. The modelledtop and base components of the assembly are simplified.They are assumed to be box-like, with heights chosen tocomply with real masses. Basic properties related to themodelled fuel assemblies are summarized in Table 2.All parts of the
Geant4 model were derived from basic
Geant4 solid objects and refined with Boolean opera-tions. Each component can be switched on and off by acommand-based user interface, which easily allows per-forming simulations for different target geometries. In ad-dition, it is possible to remove arbitrary components froma fuel assembly, e.g. specific fuel rods at specific slot po-sitions. This allows, among others, for investigating thecontribution of specific fuel rods to radiographic (or to-mographic) images in more detail.
Instead of the generic model, it is possible to generate abox-like object, whose basic properties - i.e. dimensions,placement and material - can be specified individually foreach simulation run. All interactions within this box arerecorded within
ROOT container files, particularly usefulfor validation purposes.
The z -axis coincides with the symmetry axis of the genericmodel and is oriented upwards, i.e. its orientation is se-lected so that the z -coordinate of the model’s top is larger Table 2.
Properties of the modelled fuel assemblies.Value/Parameter PropertyMaterial - Head and Base Stainless SteelMaterial - Cladding Zirconium AlloyMaterial - Fuel Uranium DioxideMaterial - Control Rods Zirconium AlloyLength - Fuel Rod 4407 mmLength - Active Length 3900 mmNumber of Control Rods 24Number of Fuel Rods 300Total Weight 845 kg
Fig. 1.
Visualisation of the generic model and its upward ori-entation within the present work. The coordinate system asit is used in this study is provided in the lower right corner.The origin of the coordinate system coincides with the centerposition of the bottom part of the model. The red and greenareas above and below the model indicate the incoming andoutgoing detectors. than the z -coordinate of its bottom, z top > z bottom . The x and y axis are orientated in such a way that the ( x, y, z )-coordinate system generates a right-handed euclidean space.The orientation of the individual axes is indicated in Fig. 1.The angle θ reflects the angle of a given vector (cid:126)r and theinverted z -axis. Using this convention, a muon from thezenith is characterized by θ = 0 ◦ . The angle ϕ is the anglebetween the projection of a vector (cid:126)r onto the ( x, y )-planeand ˆ e x . Detector systems are mimicked by two rectangular de-tector planes (vanishing thickness, area of (3 ×
3) m ),whose normal vectors are parallel to ˆ e z . The detector planeplaced above the generic model is called incoming detectorwhereas the detector plane placed below the generic modelis called outgoing detector. The gap between the detectorsurface and the top (or the bottom) of the generic cask is Thomas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks
111 234567 8 9 10 1233 3
Fig. 2.
Exploded view of the generic model showing its individual components: Monolithic body (1), basket with 19 fuelcompartments (2), trunnions (3), primary lid (4), polyethylene plate (5), secondary lid (6), protection plate (7), inner moderatorrods and plugs (8), outer moderator rods and plugs (9), polyethylene plate (10), base plate (11) and a representative fuel assembly(12). See text for details. ≈
10 cm. Muons traversing these planes are tracked andkey information is determined, see Sec. 2.3.This approach gives access to at least as many proper-ties as a real detection system for muon tomography mayprovide. Since the simulation allows determining all prop-erties precisely on an event-by-event base, it benefits froman infinite resolving power . In this regard, it provides abest-case scenario and can be considered as a first im-portant theoretical step towards substantiated feasibilitystudies. Primaries are generated by a user-defined class based onthe G4VUserPrimaryGeneratorAction class provided by
Geant4 .The Monte-Carlo approach is realized by using uniformrandom number generators with limits a , b ( U [ a, b ]) to de-termine for each event k the particle properties, i.e. parti-cle type (muon ( µ − ) or anti-muon ( µ + )), initial momen- In principle, the present approach allows to include resolu-tion effects in the post-processing without the need to repeattime-consuming simulations. tum information ( p µ k , θ k , ϕ k ) as well as initial positioninformation ( x k , y k ). Details are described in the follow-ing. Particle Type
All primaries are muons or anti-muons, assuming a chargeratio µ + /µ − equal to 1.28 [30]. The particle type for theevent k is determined assuming U [0 , .
28] - if the gener-ated random number is smaller than (or equal to) 1.28,the generated primary for this event will be an anti-muon( µ + ), otherwise it will be a muon ( µ − ). Initial Particle Momentum
The description of the initial muon momentum is basedon an empirical and parametric approach [31], accordingto which the muon intensity at any given momentum andangle to the vertical intensity I V is given by I ( p µ , θ ) = cos ( θ ) · I V ( ζ ) , ζ = p µ · cos ( θ ) (1)Here, I V is given by the Bugaev parametrisation [32]: I V ( p µ ) = c · p − ( c + c · log ( p µ )+ c · log ( p µ )+ c · log ( p µ )) µ (2) homas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks 5 The parameters c to c were determined in Ref. [31] by afitting approach and are quantified as: c = 0 . c = 0 . c = 1 . c = − . c = 0 . Geant4 , the analytical description is discretized in atwo-dimensional pattern as follows.Firstly, the polar component of the angular spectrum rep-resented by θ in , ranging from θ min = 0 ◦ to an upper limitof θ max = 25 ◦ , is split into n bins b θ,i ( i = 1 , ..., n ). Thequoted upper limit of 25 ◦ limits the contributions of tra-jectories that geometrically can only be detected by one ofthe detectors. All b θ,i span over identical angular ranges.For each angular bin b θ,i , the lower and upper limits arethen given by θ ll i [ ◦ ] = 25 ( i − n and θ ul i [ ◦ ] = 25 in .The function I i ( p µ , θ i ), for i = 1 , .., n , can then be as-sociated to each of these bins of the angular spectrum,where θ i is the arithmetic mean angle of the specific bin.These assigned functions I i are then integrated numeri-cally within the kinetic energy range of T ∈ [1 GeV , w i for each of the angular bins b θ,i : w i = (cid:90) p µ ( T =1 TeV) p µ ( T =1 GeV) dp (cid:48) µ I i ( p (cid:48) µ , θ i ) (3)Information on θ ll i , θ ul i and w i are stored in a dedicatedtext file that is used as an input parameter to the sim-ulation. U [0 , (cid:80) i w i ] uses these weights w i to determinefor each event k the proper angular bin b θ,k . In a nextstep, the proper polar angle θ k for the present event k isdetermined using U [ θ ll k , θ ul k ]. The azimuthal angle ϕ k is de-termined for each event k using U [ − π, π ]. Fig. 3.
Left : Top-view on the arrangement of fuel rods (yel-low), cladding tubes (green) as well as control rods (blue) incase of a complete fuel assembly. Each element can be spec-ified unambiguously based on its slot position given by x id , y id . Right : Same as left figure, but several elements along thediagonal are missing ( x id /y id = 1 , , Fig. 4.
Sample distributions of the initial angle θ in ( left ) and ofthe absolute muon momentum p ( right ) as determined with theincoming detector. The red curve shown in the left spectrum isproportional to cos ( θ ), while the red curve shown in the rightspectrum is proportional to Eq. 1 with θ = 12 . ◦ . Secondly, the absolute momentum spectra are treated com-parably. The momentum axis from p µ ( T = 1 GeV) to p µ ( T = 1 TeV) is discretized into m bins b p µ ,j ( j = 1 , ..., m )with increasing bin sizes. Each bin b p µ ,j is characterizedby the following integral: v i,j = (cid:90) p ul µ,j p ll µ,j dp (cid:48) µ I i ( p (cid:48) µ , θ i ) (4)Here, p ll µ,j and p ul µ,j denote the lower and upper limits ofthe bin b p µ ,j .For each angular bin b θ,i , values for p ll µ,j , p ul µ,j and v i,j arestored in dedicated text-files which are used as mandatoryinput information for the simulation code. U [0 , (cid:80) j v i,j ] uses these weights v i,j to determine for eachevent k the proper momentum bin b p µ ,k . Finally, the proper p µ,k is determined using U [ p ll µ,k , p ul µ,k ]. Within the presentwork, all numerical integrations were performed with GNUOctave [33].Fig. 4 shows histograms of the simulated angular and mo-mentum distributions for θ min = 0 ◦ , θ max = 25 ◦ , n = 10and m = 172.In addition to this distribution-based approach describedabove, the tool also allows performing simulations withmono-energetic and mono-directional muons. Initial Particle Position
The initial positions of the muons are selected as follows.The z -coordinate is a fixed value and selected so that itis ensured that the muon is created just above the incom-ing detector. The x - and y -coordinates are selected fromuniform distributions covering ranges from x min to x max ( U [ x min , x max ]) and y min to y max ( U [ y min , y max ]), respec-tively. The associated limits can be specified in the inputfile of the simulation. This code uses the modular physics list FTFP BERTimplemented in
Geant4 , which is recommended by the
Thomas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks
Geant4 developers for high-energy physics.The incoming and outgoing detector volumes are linked toa dedicated tracking class. This allows tracking and stor-ing information on the particle properties needed for thepost processing to generate radiographic (or tomographic)images.
Muons may interact with the material of the generic modelaccording to the chosen physics list in various ways. Someof these interactions are irrelevant to the present work. Forexample, some interactions may generate secondary parti-cles - e.g. photons or neutrons - whose tracking consumescomputational time without any benefits or consequencesto the generated images. To avoid an unnecessary increaseof the computational time, the trajectories of such sec-ondary particles are annihilated at the end of their firststeps. It must be stressed that this does not concern thepossible interactions of muons with matter - all interac-tions which may lead to such secondary particles still takeplace.In addition and as already described in Sec. 2.2, the muonproperties - with respect to the angle θ and kinetic energy T - were cut to avoid the simulation of muon trajectoriesthat are not useful for an analysis based on a two-detectorsetup. Finally, all muon trajectories are annihilated at theexit of the outgoing detector. This section discusses the validation of two key aspectswith respect to muon imaging. The first aspect deals withthe energy loss of muons in matter and is discussed inSec. 2.5.1. The second aspect deals with the angular scat-tering of muons after traversing matter with a knownthickness Sec. 2.5.2. In both cases, the referenced thick-ness was specified to 1 mm.
In this section, we compare simulated energy losses ofmuons in relevant target materials - uranium dioxide, poly-ethylene, stainless steel, ductile iron and zirconium alloy -to tabulated (or calculated) values of the mean differentialenergy loss [34,35].The quoted references provide only for two of the listedcompound materials - uranium dioxide and polyethylene -tabulated values. Hence, reference values were calculatedfor the other compounds - stainless steel, ductile iron andzirconium alloy - according to Bragg’s Rule of StoppingPower Additivity [36]: w j = n j · A j (cid:80) k n k · A k (5)d E d x = (cid:88) j w j · d E d x (cid:12)(cid:12)(cid:12)(cid:12) j (6) Here, w j denotes the mass fraction of the material j ,d E /d x | j is the differential energy loss in the material j and d E /d x describes the mean differential energy loss bythe muons in the compound material.Strictly speaking, the simulated energy losses per distance ∆E are not identical to differential energy losses. To in-crease the comparability, we consider for both numbers adistance of 1 mm and cut the lower limit of the consideredkinetic energy range to ensure that the (simulated) meanenergy losses are smaller (or comparable) to five percentof the kinetic energy T , i.e. ∆E (cid:46) . T . The upper limitof the kinetic energy range is set to T = 1 TeV whichequals the maximal kinetic energy of the considered muonprimaries in the present study.The number of simulated events was chosen in a way thatthe relative uncertainties of the mean values ∆E extractedfrom the simulated energy-spectra are below 2 %. The onlyexception is given for polyethylene, for which the limit isincreased to 5 %. In total, between 10 and 5 · eventswere simulated for each kinetic energy and each material.The results are shown in Fig. 5.For the materials stainless steel, ductile iron and zirco-nium alloy, the agreement between simulated and tabu-lated values is very good and the majority of values devi-ate by less than 2 %. Few exceptions were found for kineticenergies close to 1 GeV. It shall be mentioned that due tothe high ∆E dispersion at high projectile energies, themean values are quite sensitive to statistical outliers.The largest deviations were found for polyethylene, forwhich relative deviations close to 7 % were determinedover a broad kinetic energy range.For uranium dioxide, the agreement between T = 80 MeVand T = 10 GeV is very good. However, with increasingkinetic energies a clear trend towards larger deviations ofup to ∼ In this section, we investigate simulated scattering angles θ S of muons after traversing different target materials andcompare the width of the associated distributions to an es-tablished semi-empirical description. Each material had athickness of 1 mm.According to Ref. [37], the root-mean square of the scat-tering angle θ Sp x,y in the x - and y -planes can be describedusing the following formula: σ θ Spx,y = . β · p · c (cid:113) ∆zX · (cid:104) . · ln (cid:16) ∆zX · β (cid:17) (cid:105) (7)Here, β is given by v/c , p is the absolute muon momentum, ∆z is the thickness of the irradiated material and X is theradiation length of the material. The root-mean square homas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks 7 Fig. 5.
Comparison of simulated mean energy losses ∆E perdistance and mean differential energy losses d E /d x (both in[MeV/mm]) of muons in various target materials. The top fig-ure shows simulated mean energy losses (symbols) and com-pares them to tabulated reference values (solid lines). The redcurve (boxes) shows the results for uranium dioxide, the or-ange curve (circles) corresponds to polyethylene, the blue curve(pentagons) corresponds to stainless steel, the green curve (up-pointing triangles) corresponds to ductile iron and the violetcurve (down-pointing triangles) corresponds to zirconium alloy.The lower five figures show for each material the relative de-viations between simulated energy loss per mm and tabulatedmean stopping power values. of the scattering angle θ S in space is given by: σ θ S = √ σ θ Spx,y (8)In case of a compound material, the radiation length X can be calculated using the formula1 X = (cid:88) j w j X ,j (9) Fig. 6.
Comparison of root-mean square (RMS) values of scat-tering angles (in [mrad]) between semi-empirical calculationsand simulated values for muons in various target materials witha thickness of 1 mm. The top figure shows values derived fromthe simulation (symbols) and compares them to calculated val-ues based on a semi-empirical approach (solid lines). The redcurve (boxes) shows the results for uranium dioxide, the or-ange curve (circles) corresponds to polyethylene, the blue curve(pentagons) corresponds to stainless steel, the green curve (up-pointing triangles) corresponds to ductile iron and the violetcurve (down-pointing triangles) corresponds to zirconium al-loy. The lower five figures show for each material the relativedeviations between simulated and calculated values. where w j is the mass fraction of the component j and X ,j is the radiation length of the component j . The ra-diation lengths were either taken directly from Refs. [34,35] or calculated according to Eq. 9. An overview of thevarious radiation lengths X for the relevant materials isgiven in Table 3.Simulations covering a broad range of kinetic energies wereperformed for all relevant materials with 10 events. Withrespect to the energy range, the same restrictions as for Thomas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks
Fig. 7.
The figures in the two upper rows show transmission radiographic images of longitudinal scans of a .) an empty genericmodel ( top row ) as well as b .) a generic model filled with 18 out of 19 possible fuel assemblies ( middle row ) where the centralfuel compartment remained empty. The bottom row shows corresponding difference images, a − b . The color code representsthe ratio of muons that were detected in both detectors over the total number of muons. Different assumptions on the initialmuon directions are considered from left to right and range from a fixed initial muon direction with θ in = 0 ◦ ( left ) to an angulardistribution with respect to θ in of 0 ◦ to 25 ◦ ( right ). The x - and y -coordinates refer to the muon positions at the exit of theincoming detector. the discussion of the energy loss were applied. The simu-lated results are shown and compared to calculated valuesin Fig. 6.The deviations between simulated and semi-empirical σ θ S are usually less than 3 %. Only for UO larger deviationsof about 6 % occur. In all cases, the deviations as a func-tion of the kinetic energy remain rather constant. Overall,the agreement between simulated and calculated values isvery good. Table 3.
Radiation lengths X for materials relevant to thepresent work.Material X [g/cm ] X [cm]Uranium Oxide 6.65 0.6068Polyethylene 44.77 50.31Stainless Steel 13.921 1.808Ductile Iron 14.297 2.014Zirconium Alloy 10.223 1.558 The analysis is split into two main sections. The first sec-tion investigates longitudinal scans of the generic modeland analyses simulated radiographic images. Here, the fo-cus lies on the effects of various angular acceptances on theimage quality. In addition, we compare transmission radio-graphic images with scattering radiographic images, e.g.in terms of image contrast and resolving power. The pre-sentation focuses on qualitative aspects and is describedin more detail in Sec. 3.1.The second section provides a detailed study that focuseson the central fuel assembly. In particular, it investigatesthe suitability of muon scattering radiography to makestatements about the occupancy of individual fuel-rod slotswithin a certain fuel assembly. This section focuses onquantitative aspects and is described in Sec. 3.2 in moredetail. homas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks 9
Fig. 8.
The figures in the two upper rows show scattering transmission images of longitudinal scans of a .) an empty genericmodel ( top row ) as well as b .) a generic model filled with 18 out of 19 possible fuel assemblies ( middle row ) where the centralfuel compartment remained empty. The color code represents for each pixel the median of the associated scattering angledistribution. The bottom row shows corresponding difference images, b − a . Different assumptions on the initial muon directionsare considered from left to right and range from a fixed initial muon direction with θ in = 0 ◦ ( left ) to an angular distribution withrespect to θ in of 0 ◦ to 25 ◦ ( right ). The x - and y -coordinates refer to the muon positions at the exit of the incoming detector. All simulations discussed in this section were performedwith 5 · events. All primary muons were generated ata fixed height ( z ≈ . x - and y -coordinates followed uniform dis-tributions within the limits of − .
25 m ≤ x, y ≤ .
25 m.This area of the initial flux is sufficient to cover the fullcross-section surface of the generic model.The absolute muon momenta p were treated as describedin Sec 2.2, while different scenarios were simulated with re-spect to the initial muon direction. As a first scenario (ref-erence case), we assumed mono-directional initial muons,i.e. (cid:126)p = − p · ˆ e z ( p >
0) and θ in = 0 ◦ . In addition, we re-stricted the incoming muon spectrum to various angularranges with respect to θ in . In particular, we consideredangular distributions of 0 ◦ to 1 ◦ , 0 ◦ to 5 ◦ and 0 ◦ to 25 ◦ .Simulations were performed for two geometries of the ge-neric model, which differed with respect to the occupancyof the fuel compartments with fuel assemblies. In the firstcase, all fuel compartments were empty (no fuel assem- blies). In the second case, all fuel compartments with theexception of the central one were occupied with fuel as-semblies.Projection images were generated using the transmissionradiographic as well as the scattering radiographic ap-proach.In case of the transmission radiographic analysis, the ma-jor imaging information is the ratio of muons reaching theoutgoing detector over the number of muons crossing aspecific pixel of the incoming detector. The pixel size wasspecified as (1 ×
1) cm .In case of the scattering radiographic analysis, the leadingimaging information is the effective scattering angle θ eff with respect to the initial direction, which is calculatedevent-by-event according to: θ eff = arctan (cid:114)(cid:80) i = x,y (cid:16) ∆ i + ∆ z · d i d z (cid:17) z in − z out (10) The basis is given by the position information providedby the incoming ( x in , y in , z in ) and outgoing ( x out , y out , z out ) detectors as well as the normalized muon direction (cid:126)d = ( d x , d y , d z ) provided by the incoming detector. Theprojected distances ∆ i ( i = x, y, z ) are given by i out − i in .Information on positions and directions was processed with-out any attempts to mimic resolution effects. Again, thepixel size was defined as (1 ×
1) cm . Transmission Radiographic Images
Fig. 7 shows transmission radiographic images for the twogeometries and the different angular distributions with re-spect to θ in as well as associated difference images.It can be easily seen that the image quality in terms ofresolution decreases with increasing angular acceptancewith respect to θ in . A good indicator is given in termsof the absorber rods that can be easily identified for theangular acceptance of 0 ◦ ≤ θ in ≤ ◦ . This, however, isnot possible for an angular acceptance of 0 ◦ ≤ θ in ≤ ◦ , at least not with the number of simulated events.A similar picture can be drawn for larger structures suchas fuel assemblies: For the angular acceptance of 0 ◦ ≤ θ in ≤ ◦ , the transmission radiographic image based onthe simulated statistics does not allow for a reliable con-clusion on the occupancy of the central fuel compartmentwith a fuel assembly.It is obvious that difference images between the two ge-ometries benefit from an improved contrast that allowsidentifying structural differences (or similarities) more eas-ily. This, for instance, holds with respect to the occupancyof the central fuel compartment. Even for the angular ac-ceptance of 0 ◦ ≤ θ in ≤ ◦ the difference image providesweak evidence allowing the conclusion that for both ge-ometries the occupancies of the central fuel compartmentwere identical. Scattering Radiographic Images
Fig. 8 shows scattering radiographic images for the twogeometries and the various angular acceptances as well ascorresponding difference images.Similar to the transmission radiographic images, the im-age quality deteriorates significantly with the increasingangular acceptance with respect to θ in . However, the im-age quality is much better and smaller structures can beresolved. For example, the individual walls of the fuel com-partments can be easily identified for the angular accep-tance of 0 ◦ ≤ θ in ≤ ◦ and - with limitations - also for theangular acceptance of 0 ◦ ≤ θ in ≤ ◦ . Unfortunately, alsothe scattering radiographic images would not allow for areliable statement on the occupancy of the central fuelcompartment for an angular acceptance of 0 ◦ ≤ θ in ≤ ◦ based on the number of simulated events.The improved resolution compared to the transmission ra-diographic images is also reflected in the difference images.With respect to the angular acceptance of 0 ◦ ≤ θ in ≤ ◦ ,at least the difference plot provides clear evidence that the occupancies of the central fuel compartment are identicalfor both geometries. General Aspects
The blurring of the structures with increasing angular ac-ceptance can be easily understood in terms of the longi-tudinal extension of the generic model.In general, the simulated results show the advantages ofthe scattering radiographic over the transmission radio-graphic approach. Here, the improved resolving power isthe most prominent indicator. Unlike most radiographicdetectors, tomographic detection systems would be ableto apply cut conditions based on the incoming muon flightdirections which would allow us to realize different angu-lar acceptances with respect to θ in .In summary, the simulations provide evidence that evenwithout any reconstruction efforts to generate tomographicimages it would be possible to make conclusions on theoccupancy of specific fuel compartments with fuel assem-blies within a reasonable amount of time. For example,in case of an angular acceptance of 0 ◦ ≤ θ in ≤ ◦ andbased on reasonable assumptions (see Sec. 3.2), the num-ber of simulated events would correspond to a measuringtime of about 40 hours. This increases to ∼
190 hours and ∼
930 hours in case of 0 ◦ ≤ θ in ≤ ◦ and 0 ◦ ≤ θ in ≤ ◦ ,respectively.So far, these results cannot be extrapolated to smallerstructures such as individual fuel rods. Qualitatively, itcan be expected that for such a level of detail a narrowangular acceptance as well as a much larger muon flux perarea would be required. The following section provides amore detailed discussion of this aspect. In this section, we investigate if muon scattering radiog-raphy can be used to make reliable statements about thecompleteness of individual fuel assemblies. In particular,we investigate whether individual missing fuel rods in anotherwise complete fuel assembly can be detected. Thisinvestigation takes into account a few variable boundaryconditions such as the angular acceptance with respect to θ in as well as the number of events, which most often canbe used to provide reasonable estimates on the requiredirradiation time.All simulations were performed with up to 10 events. Thetreatment of the absolute muon momenta was identicalto the former section 3.1. The considered angular accep-tances with respect to θ in of the primary muons rangedfrom 0 ◦ ≤ θ in ≤ . ◦ to 0 ◦ ≤ θ in ≤ ◦ . As in the for-mer section, we also considered mono-directional incom-ing muons with θ in = 0 ◦ to provide a reference scenario.The initial positions within the x, y -plane were restrictedto − .
35 m ≤ x, y ≤ .
35 m and the initial z -coordinatewas fixed at z ≈ . homas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks 11 the generic model. The first geometry ensured that allfuel compartments of the model were occupied with fuelassemblies and for each fuel assembly all individual slotpositions were occupied according to the nominative lay-out, see Fig. 3. The second geometry differs from the firstone only by three vacated fuel-rod positions along the di-agonal (first, fifth and ninth position - see Fig. 3) withinthe central fuel assembly. The first relevant fuel rod I ( x id = y id = 1) is nominatively placed in the upper leftcorner of the fuel assembly and is surrounded by three fuelrods and the walls of the fuel compartment. The secondrelevant fuel rod II ( x id = y id = 5) is surrounded by twocontrol rods and six fuel rods, while the third relevant fuelrod III ( x id = y id = 9) is surrounded by fuel rods on allsides.Scattering radiographic images were generated using abinning as indicated in Fig. 9. Each bin covers (6 . × .
36) mm and is shifted in a way that for the central fuelassembly the center positions of the bins coincide with thenominative center positions within the ( x, y )-plane of thevarious fuel and control rods.As in the last section, the effective scattering angle θ eff is calculated event by event according to Eq. 10. For eachpixel we obtain a histogram that shows the absolute fre-quencies of the effective scattering angles θ eff . Represen-tative examples of such absolute frequency distributionsare shown in Fig. 10.These frequency distributions are then normalized foreach pixel ( i, j ) into a normalized probability distributionfunction (PDF) ρ i,j ( θ eff ): (cid:90) π/ dθ (cid:48) eff ρ i,j ( θ (cid:48) eff ) = 1In a final step, we use ρ i,j ( θ eff ) to calculate for each pixel i, j the cumulative distribution function (CDF) F Θ i,j ( θ eff ), Fig. 9.
Illustration of the binning used for the analysis inSec. 3.2. The green disks represent the cross sections of thevarious rods in the central fuel assembly. The grid indicatesthe limits of the bins within the ( x, y )-plane. i.e.: F Θ i,j ( θ eff ) = (cid:90) θ dθ (cid:48) eff ρ i,j ( θ (cid:48) eff )This procedure is repeated for each simulation.For illustration, Fig. 11 shows for both geometries the PDF ρ II as well as the CDF F Θ II corresponding to the pixelof the slot position with x id = y id = 5 within the cen-tral fuel assembly. The analysis focuses on a pixel-basedcomparison between the two geometries described above.By computing the local distances between the empiricalCDFs, the derived statistical measure can be used for thepair-wise comparison of images [38,39]. In other words,we compare for each pixel i, j the CDFs F occ Θ i,j ( θ eff ) and F vac Θ i,j ( θ eff ). Here, F occ Θ i,j ( θ eff ) corresponds to the geometryfor which all slot positions within the fuel assemblies areoccupied, while F vac Θ i,j ( θ eff ) corresponds to the geometry forwhich the fuel-rod slot positions within the central fuel as-sembly are empty.We consider for each pixel ( i, j ) in the ( x, y )-plane thetwo-sample Kolmogorov-Smirnov test, D i,j = sup θ eff | F occ Θ i,j ( θ eff ) − F vac Θ i,j ( θ eff ) | , (11) Fig. 10.
Absolute frequency distributions of the effective scat-tering angle θ eff for different pixels and different angular accep-tances. The red and black curves correspond to the pixel of theslot position (5,5) of the central fuel assembly. The black curverepresents the case where this particular slot position is emptywhile the red curve corresponds to the case where the slotposition is filled with a fuel rod. The blue curve correspondsto a pixel between two neighbouring fuel assemblies. The topfigure shows results for a fixed angular acceptance ( θ in = 0 ◦ )while the bottom figure shows results for an angular acceptanceof 0 ◦ ≤ θ in ≤ ◦ . All distributions correspond to simulationswith 10 events.2 Thomas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks Fig. 11.
Probability density functions ρ II ( θ ) ( top ) as well ascumulative probability functions F Θ II ( θ ) ( bottom ) for the pixelcorresponding to the slot position with x id = y id = 5 withinthe central fuel assembly for two different geometries. The redcurve represents the geometry with a complete fuel assembly(all positions are occupied according to the nominative layout),while the black curve corresponds to the geometry where thespecific slot position represented by this pixel is empty. Thedistributions represent results of a simulation with 10 eventsassuming mono-directional muons with θ in = 0 ◦ . which allows us to make quantitative statements on theagreement between F occ Θ i,j ( θ eff ) and F vac Θ i,j ( θ eff ).The null hypothesis - i.e. F occ Θ i,j ( θ eff ) and F vac Θ i,j ( θ eff ) de-scribe identical distributions - is rejected at a statisticalsignificance level α if the test statistic satisfies D i,j > c ( α ) (cid:115) n i,j + m i,j n i,j · m i,j c ( α ) = (cid:112) − . · ln ( α/ n i,j ( m i,j ) describes the number of entries in the( i, j ) pixel of the relevant vacated (occupied) spectrum.The condition can be reformulated as:˜ D i,j ≡ D i,j c ( α ) (cid:114) n i,j · m i,j n i,j + m i,j > α of 0.1 in the follow-ing.Fig. 12 shows a heatmap of ˜ D i,j based on a simulationwith 10 events assuming mono-directional muons with θ in = 0 ◦ . One can clearly identify areas for which theKolmogorov-Smirnov test statistic indicates the signifi-cant deviations. These areas coincide perfectly with areas Fig. 12.
A heat map showing the significance ratio ˜ D in the( x, y )-plane. One can easily identify the locations of the threevacated fuel-rod slot positions along the diagonal. The spec-trum represents simulations with 10 events assuming mono-directional muons. See text for details. for which the two geometries differ, i.e. with respect tothe occupation of the slot positions x id = y id = 1 ( I ), x id = y id = 5 ( II ) and x id = y id = 9 ( III ) within thecentral fuel assembly.Fig. 13 summarizes ˜ D i,j for each of these three particularpixels and for different angular acceptances and quanti-fies the evolution as a function of simulated events. ˜ D I corresponds to the pixel of x id = y id = 1, while ˜ D II and ˜ D III correspond to the pixels of x id = y id = 5 and x id = y id = 9.It is obvious that for all three pixels the significance ratios˜ D decrease in general with an increasing angular accep-tance. The only exception is given in terms of the angularacceptances 0 ◦ ≤ θ in ≤ . ◦ and 0 ◦ ≤ θ in ≤ ◦ , for whichcomparable significance ratios are observed. For all threepixels one can observe a clear trend towards smaller slopesof progression with an increasing angular acceptance andincreasing number of simulated events. It is interestingto note that there are significant differences between thethree pixels with respect to the trends with increasingevent numbers and the achieved significance ratios. Theseeffects are stronger for pixel II ( III ) compared to I . Withrespect to pixel I and based on the maximum number ofevents (10 ), the significant ratio ˜ D I exceeds one only upto an angular acceptance of 0 ◦ ≤ θ in ≤ . ◦ and requiresat least 6 · simulated events in case of the latter. Adifferent picture can be drawn for pixel II . Not only dowe observe that less events (4 · events) are required for˜ D II to exceed one for 0 ◦ ≤ θ in ≤ . ◦ , we also observe sig-nificant deviations with respect to 0 ◦ ≤ θ in ≤ . ◦ . Forthe latter, at least 5 · simulations events are requiredfor ˜ D II to exceed one. In case of pixel III , we observethat ˜ D III also exceeds one for 0 ◦ ≤ θ in ≤ . ◦ , requiring homas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks 13 Fig. 13.
Significance ratios ˜ D i,j as a function of angular ac-ceptance with respect to θ in and simulated events for the threepixels of interest. The top figure shows the results of D I , while D II ( D III ) is shown in the middle ( bottom ) figure . at least 8 . · simulated events. The significant devia-tions between the individual pixels may be related to thelarger numbers of neighbouring fuel rods in case of pixels II and III compared to pixel I .For a specific simulation with known momentum distribu-tion and angular acceptance, the measurement time canbe estimated according to the following formula: ∆t [s] = simulated events A [cm ] · I µ [muons / cm / s] · I θ in · I p · (cid:15) (13)Here, A is the area of the initial ( x, y )-plane, I µ is the in-tegrated muon flux at sea level ( ∼ / cm / min), I θ in is the share within the angular acceptance with respectto θ in , I p is the share of the considered momentum dis-tribution with respect to the full momentum distributionand (cid:15) is the efficiency of the detector system. For a certainangular acceptance, I θ in can be estimated according to: I θ in = (cid:82) θ in dθ (cid:48) cos θ (cid:48) (cid:82) π/ dθ (cid:48) cos θ (cid:48) (14)Values of I θ in for various angular acceptances are listed inTab. 4. In a similar way, a simplified estimate of I p ( ≈ . I p = (cid:82) p ( T =1 TeV) p ( T =1 GeV) dp (cid:48) I ( p (cid:48) µ , (cid:82) ∞ dp (cid:48) I ( p (cid:48) µ ,
0) (15) With respect to 0 ◦ ≤ θ in ≤ . ◦ , ˜ D I exceeds 1.0 for 6 · events. Based on the above estimates, this event numbercorresponds to a measurement time of ∼ . · and 5 · ) are required for the pixelsII and III, which leads to estimated measurement times of ∼ . ∼ . In this work we investigated the theoretical feasibility todetect individual missing fuel rods in an otherwise fullyloaded cask within a reasonable amount of time using cos-mic muons. We used simulations based on the
Geant4 toolkit and a generic model based on the CASTOR ® V/19 (see Fig. 2) loaded with 18x18-24 PWR fuel assem-blies (see Fig. 3). The detectors above and below a verti-cal (standing) model were mimicked by two rectangularplanes with vanishing thickness and an area of 9 m . Thegap between detectors and the model was assumed to be10 cm. The muon characteristics are described in section2.2. The tool can be run with a realistic muon spectrumbut also allows for performing simulations assuming mono-energetic and mono-directional muons. For validation pur-poses, the stopping powers (see Fig. 5) and scattering an-gles (see Fig. 6) of the relevant target materials have beeninvestigated and compared against established referencevalues from the literature. We found good agreement be-tween simulated and literature/analytical values.We simulated longitudinal scans that covered the full crosssection of the model (full-model simulation). In addition,we put a special emphasis on a single fuel assembly withmissing fuel rods.The focus of the full-model simulation was on the effectof angular acceptance criteria on the resolution. We com-pared mono-directional muons ( θ in = 0 ◦ ) with differentangular acceptance ranges with respect to θ in : 0 ◦ ≤ θ in ≤ ◦ , 0 ◦ ≤ θ in ≤ ◦ and 0 ◦ ≤ θ in ≤ ◦ . The integrated fluxwas kept constant with 5 · muons, which correspondsto a radiation time of approximately 40 hours in case of0 ◦ ≤ θ in ≤ ◦ . We performed two analyses and comparedthe empty model with a partially loaded one and the re-sulting difference between the two radiographic images.The partially loaded model was lacking the central fuelassembly. In the first analysis (transmission radiographicscans) we compared the ratio of muons in the lower andupper detectors. This radiographic analysis clearly indi-cated the decreasing resolution with increasing angularacceptance with respect to θ in of the incoming muons (seeFig. 7). The central fuel assembly could still be identifiedas missing based on the difference plot of the acceptancecriteria 0 ◦ ≤ θ in ≤ ◦ . In addition, we repeated the Table 4.
Estimations for I θ in for various angular acceptances.[0 ◦ , . ◦ ] [0 ◦ , ◦ ] [0 ◦ , ◦ ] [0 ◦ , ◦ ] [0 ◦ , ◦ ] I θ in [%] 1.11 2.22 4.44 11.08 52.164 Thomas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks analysis with a focus on the effective scattering angles ofthe muons to receive scattering radiographic images. Theimage quality was much better compared to the trans-mission radiographic analysis and more details could beidentified (see Fig. 8). Again, for the acceptance criteriaof 0 ◦ ≤ θ in ≤ ◦ the difference plot can be used to iden-tify the central fuel assembly as missing.We further investigated the capability of muon scatteringradiography to provide reliable statements on the com-pleteness of a loaded fuel assembly, assuming up to 10 events and angular acceptance criteria of 0 ◦ ≤ θ in ≤ . ◦ , 0 ◦ ≤ θ in ≤ . ◦ , 0 ◦ ≤ θ in ≤ . ◦ , 0 ◦ ≤ θ in ≤ . ◦ , 0 ◦ ≤ θ in ≤ . ◦ and 0 ◦ ≤ θ in ≤ . ◦ in additionto mono-directional muons with θ in = 0 ◦ . Focussing onthe central fuel assembly we investigated a fully loadedmodel and one where the central assembly was missingthree fuel rods on the diagonal (see Fig. 3). All three miss-ing rods were clearly identifiable after the simulation of10 events assuming mono-directional muons with θ in =0 ◦ (see Fig. 12). The significance ratio ˜ D based on theKolmogorov-Smirnov test statistic of each fuel rod (andthus the significance of the contrast of a missing rod) de-pends on the relative position of the missing fuel rod inthe assembly, the angular acceptance criteria of incomingmuons and the number of simulated events (see Fig. 13).The latter can be transformed accordingly into radiationor measuring times. Assuming significance ratios ˜ D i,j > ◦ ≤ θ in ≤ . ◦ .In summary, we have shown that the muon scattering ra-diography is capable of reliably visualizing the inside ofa loaded and sealed model at a resolution scale of a sin-gle fuel rod. The assumptions we made led to timescaleswhich seem feasible compared to the duration of the drystorage of spent nuclear fuel. In the next section we discusshow future work might even shorten the theoretically esti-mated measuring time and how image processing methodsmay be used to gain more detailed insights into the cask’sinterior for the detection of individual missing fuel rods. The present work will serve as a starting point for furtherresearch activities that may evolve into several indepen-dent aspects.The first aspect relates to the developed simulation toolitself. We intend to include the discussion of transversalscans, for which the detectors are located on the sides ofthe generic model. The simulation can also be extendedto additional geometries which do not have to be limitedto storage casks.The second aspect relates to the validation of the simu-lated data and comparisons to other simulations. We al-ready addressed validation aspects in the present workthat concerned the slowing-down process of the muons aswell as the angular scattering. A logical next step wouldbe given by a comparison of simulated to experimental results. For this it might be reasonable to start with lesscomplex geometries or larger objects-of-interest which willrequire less time, both with respect to the computationand the experimental measurement. To take into accountthe uncertainty due to a particular choice of the simu-lation tool, it would be valuable to compare the resultsfrom different simulation tools for identical geometries forbenchmark purposes.A third aspect relates to sensitivity analyses. The presentresults indicate that the significance depends on the rela-tive fuel rod position and, hence, the immediate surround-ings of the considered rod. So far, we have arbitrarily se-lected three fuel rods and it may be worth the effort torepeat the analyses systematically, including the effects ofrods in the immediate surroundings. The significance in-formation of the second analysis part is derived from twopredominantly identical geometries, which deviated onlyby the occupation of three fuel rod slots. It would be in-sightful to investigate the effects of additional differencessuch as slight misalignments.A fourth aspect addresses the extension to tomographicimages which allows using the complete information ofscattered (or absorbed) muons within one visualizationprocessed by various potential statistical muon tomogra-phy reconstruction algorithms as discussed in [40]. Foradaptively comparing muon scattering (or absorption) im-ages of a cask with the purpose of automatically detectingchanges within the interior of the cask, the image process-ing strategy needs to be extended towards tomographicimage reconstruction inherently designed for change detec-tion or in combination with further image analysis meth-ods.
This research was partially funded by the Federal Ministryfor the Environment, Nature Conservation and NuclearSafety in Germany (BMU) under Contract 4720E03366.
References
1. K.-P. Ziock, G. Caffrey, A. Lebrun, L. Forman, P.Vanier and J. Wharton,
IEEE Nuclear Science SymposiumConference Record, 2005, Fajardo, 2005, pp. 1163-1167 ,doi:10.1109/NSSMIC.2005.15964572. V. Brdar, P. Huber and J. Kopp, Phys. Rev. Appl. , 2331(2017), doi:10.1103/PhysRevApplied.8.0540503. G. Bonomi, P. Checchia, M. D’Errico, D. Pagano andG. Saracino, Prog. Part. Nucl. Phys. , 103768 (2020),doi:10.1016/j.ppnp.2020.1037684. P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp.Phys. 2020, 083C01 (2020), doi:10.1093/ptep/ptaa1045. M.P. De Pascale et al. , J. Geophys. Res. A , 3501 (1993),doi:10.1029/92JA026726. P.K.F. Grieder, Cosmic Rays at Earth (Elsevier Science,Amsterdam, 2001)7. L.W. Alvarez et al. , Science , 832-839 (1970),doi:10.1126/science.167.3919.832homas Braunroth et al.: Muon Radiography to Visualise Individual Fuel Rods in Sealed Casks 158. H.K.M. Tanaka, T. Kusagaya and H. Shinohara, Nat. Com-mun. , 3381 (2014), doi:10.1038/ncomms4381 (2014)9. N. Lesparre, D. Gilbert, J. Marteau, Y. D´eclais, D. Carboneand E. Galichet, Geophys. J. Int. , 1348-1361 (2010),doi:10.1111/j.1365-246X.2010.04790.x10. K. Morishima et al. , Nature , 386-390 (2017),10.1038/nature2464711. H.K.M. Tanaka, K. Nagamine, S.N. Nakamura and K.Ishida, Nucl. Instrum. Methods Phys. Res. A , 164-172(2005), doi:10.1016/j.nima.2005.08.09912. K. Borozdin et al. , Nature , 277 (2003),doi:10.1038/422277a13. A. Clarkson et al. , J. Instrum. , P03020 (2015),doi:10.1088/1748-0221/10/03/P0302014. Decision Sciences .15. A. Zenoni. (IEEE, Lisbon, 2015),doi:10.1109/ANIMMA.2015.746554216. J.M. Durham et al. , Phys. Rev. Appl. , 044013 (2018),doi:10.1103/PhysRevApplied.9.04401317. G. Jonkmans, V.N.P. Anghel, C. Jewett andM. Thompson, Ann. Nucl. Energy , 267 (2013),doi:10.1016/j.anucene.2012.09.01118. A. Clarkson et al. , Nucl. Instrum. Methods Phys. Res. A , 64-73 (2014), doi:10.1016/j.nima.2014.02.01919. S. Chatzidakis, M. Alamaniotis and L. H. Tsoukalas,Trans. Am. Nucl. Soc. , 369 (2014), https://ans.org/pubs/transactions/a_36379
20. F. Ambrosino et al. , J. Instrum. , T06005 (2015),doi:10.1088/1748-0221/10/06/T0600521. D. Poulson, J.M. Durham, E. Guardincerri, C.L. Mor-ris, J.D. Bacon, K. Plaud-Ramos, D. Morley and A.A.Hecht, Nucl. Instrum. Methods Phys. Res. A , 48 (2017),doi:10.1016/j.nima.2016.10.04022. D. Poulson, J. Bacon, M. Durham, E. Guardincerri, C. L.Morris and H.R. Trellue, Phil. Trans. Roy. Soc. A , 2137(2018), doi:10.1098/rsta.2018.005223. S. Agostinelli et al. , Nucl. Instrum. Methods Phys. Res. A , 250-303 (2003), doi:10.1016/S0168-9002(03)01368-824. J. Allison et al. , IEEE Trans. Nucl. Sci. , 270-278 (2006),doi:10.1109/TNS.2006.86982625. J. Allison et al. , Nucl. Instrum. Methods Phys. Res. A ,186(2016), doi:10.1016/j.nima.2016.06.12526. R. Brun and F. Rademakers, Nucl. Instrum. Meth-ods Phys. Res. A , 81-86 (1997), doi:10.1016/S0168-9002(97)00048-X27. See also ”ROOT” [software], Release v6.20/04, 01/04/202028. GNS Geselllschaft f¨ur Nuklear-Serivce mbH, Product InfoCastor ® V/1929. Bundesamt f¨ur Strahlenschutz (BfS),
Radioaktive Frachtenunterwegs - Atomtransporte und Sicherheit (brochure, 2000)30. CMS Collaboration, Phys. Lett. B , 83 (2010),doi:10.1016/j.physletb.2010.07.03331. D. Reyna, arXiv:hep-ph/0604145v232. E.V. Bugaev, A. Misaki, V.A. Naumov, T. Sinegovskaya,S.I. Sinegovsky, and N. Takahashi, Phys. Rev. D , 054001(1998), doi:10.1103/PhysRevD.58.05400133. J.W. Eaton, D. Bateman, S. Hauberg and R. Wehbring,GNU Octave version 5.2.0 manual: a high-level interac-tive language for numerical computations.
34. D.E. Groom, N.V. Mokhov and S.I. Strig-anov, At. Data Nucl. Data Tables , 183 (2001),doi:10.1006/adnd.2001.086135. https://pdg.lbl.gov/2020/AtomicNuclearProperties/
36. W.H. Bragg and R. Kleeman, Philos. Mag. , 318-340(1905), doi:10.1080/1478644050946337837. G.R. Lynch and O.I. Dahl, Nucl. Instrum. Methods Phys.Res. B , 6 (1991), doi:10.1016/0168-583X(91)95671-Y38. E. Demidenko, In: Lagan´a A., Gavrilova M.L., KumarV., Mun Y., Tan C.J.K., Gervasi O. (eds) ComputationalScience and Its Applications – ICCSA 2004. ICCSA 2004.Lecture Notes in Computer Science, vol 3046. (Springer-Verlag Berlin, Heidelberg, 2004), doi:10.1007/978-3-540-24768-5 10039. Y. Tang, L. Zhang and X. Huang, Int. J. Remote Sens. ,5719 (2011), doi:10.1080/01431161.2010.50726340. S. Riggi et. al.et. al.