3D imaging of a nuclear reactor using muography measurements

Abstract

The inspection of very large or thick structures represents one of the biggest challenges for nondestructive techniques. For such objects, a particularly powerful technique is muography, which makes use of free, natural cosmic-ray muons. Among other applications, this technique has been applied to provide two-dimensional (2D) images of nuclear reactors, pyramids, or volcanos. Recently, 3D algorithms developed for medical applications were adapted to the special case of muon imaging to derive density maps. The main difficulties relate to the size of the object and to the limited number of available projections. Here, we report on the first 3D imaging of a whole nuclear reactor, obtained without any prior information on its structure and using the largest set of muographic projections ever made in this field.

3D imaging of a whole nuclear reactor is achieved using a non-destructive technique called muon radiography.

INTRODUCTION

Imaging methods using artificial sources face several limitations, such as low penetration (x-ray and neutron) or interpretation issues (radar). On the other hand, the muon imaging, or muography, can reveal the inside of the deepest structures and provide direct images like a penetrating photography. This technique is now successfully applied to various domains such as volcanology or homeland security (1). After the 2011 tsunami in Japan, it was used to check for the internal structure of the Fukushima reactor (2). Its spectacular penetration power was also recently illustrated through the discovery of several voids in Khufu’s Pyramid (3). Although this type of muography is by nature a two-dimensional (2D) imaging, 3D reconstruction can be accessed by combining several projections like in medical imaging (4–6). A proof of concept was recently obtained using only simulated data and a quite homogeneous object with several voids (7). The goal of the current work is to demonstrate the 3D capabilities with real data and a very large and complex object.

The G2 reactor located in the CEA site of Marcoule is one of the first French nuclear reactors and is shown in Fig. 1. It was built in the 1950s and worked between 1958 and 1980. It used natural uranium as fuel, a graphite moderator, and carbon dioxide cooling. A first dismantling process took place from 1986 to 1996 where the cooling systems were removed. Since 1996, the reactor has been regularly inspected, waiting for its final decommissioning. The graphite moderator forms a nearly cubic structure of about 9-m side with 1200 horizontal pipes for the fuel. Its upper part has a smaller section as can be seen in Fig. 1D. The moderator itself is contained in a 34-m-long cylindrical concrete block with a diameter of 20 m. The cylinder lies on a concrete base with an open area below the reactor. The G2 environment thus permits the installation of instruments below and around the reactor, inside a large metallic hall.

Fig. 1. The G2 reactor.

(A and B) Photographs from the North side and below the concrete cylinder, (C) CAD full view, and (D to F) cross sections along the three directions.

In this work, we present a full 3D reconstruction of this reactor using only muon data. We combined 27 projections from four telescopes to derive the density map without any prior information on the reactor internal structure. As will be described in the next section, many substructures can be revealed in the obtained tomographic slices. Beyond the special case of the G2 reactor, this is probably the most complex and large object ever reconstructed in 3D using muons.

RESULTS

2D muography images

The search for cavities or local defects in previous experiments and projects did not require a full agreement between data and simulations over the whole instrument acceptance. The situation is naturally different for a 3D, global reconstruction, and special attention was paid to the performance of the track reconstruction algorithm, including the demultiplexing process. In particular, this demultiplexing failed more often for large-angle tracks, artificially reconstructing them at low angles. This feature yielded a complex angular-dependent efficiency accompanied by a few percent impurity at low angles. For this reason, an algorithm was specifically developed in this study, providing a gain on the efficiency ranging from 10% for perpendicular tracks up to 40% at 45° (see details in the Supplementary Materials and fig. S1). A comparison between open-sky measurements and a simulation suggested that the new track reconstruction efficiency varied by less than 10% within the whole angular acceptance. The open-sky muon rate obtained in data reached 25 to 27 Hz depending on the atmospheric pressure, while the simulated one yielded 25 Hz. Although these numbers cannot provide a precise telescope efficiency, they suggest that this efficiency should be quite high, presumably well above 80%. The 2D muography raw images were then obtained by plotting the distributions of tan(θ_x) versus tan(θ_y), θ_x, and θ_y being the reconstructed muon angles in the x-z and y-z planes, respectively, with z being the axis perpendicular to the telescopes and x and y being the coordinates measured by each detector. After requiring track quality cuts and angles below 45°, the overall muon statistics yielded about 370 millions.

As described in (7), these raw (flux) images should first be converted into an absorption map by dividing them with open-sky distributions. These distributions can be obtained from dedicated acquisitions with the telescopes (one for each zenithal orientation) but are time-consuming measurements. We instead used a Monte Carlo simulation to generate open-sky pseudo-data. Muons were generated according to the Guan parametrization (see Material and Methods). The telescope acceptance was fully simulated by requiring the generated muons to cross at least three detection planes. The spatial resolution of the detector was also taken into account by smearing the true muon position using an angular-dependent parametrization of the resolution. The small, angular-dependent efficiency variation of the telescope was neglected. For each telescope position, a minimum of three times the real data statistics was generated to avoid fluctuations due to the simulation itself. Some of the resulting absorption maps are shown in comparison with the photographs in Fig. 2 (B to M). A small excess of data is observed very close to the horizon and probably originates from low-energy muons or electrons, but this does not affect the 3D reconstruction of the reactor.

Fig. 2. Experimental setup around the G2 reactor.

(A) Positions of the telescopes under and around G2 (indicated by full circles). Small letters indicate the positions corresponding to photographs and muography images provided in the next subfigures. (B to D) Photographs taken from Einstein-P1, Bose-P2, and Dirac-P2, E to M: muography images normalized by the open-sky simulation (i.e., absorption maps). (E), (F), and (G) correspond to the positions of (B), (C), and (D), respectively.

3D reconstruction

The obtained absorption factor in each pixel of the 2D map provides information on the amount of matter in the corresponding direction. More precisely, the absorption can be converted into an opacity, defined as the integral of the density over the muon path length. This conversion requires the knowledge of the muon energy spectrum (which itself depends on the muon angle) and of the energy loss in matter. It thus relies on a set of Monte Carlo simulations to parametrize the absorption-opacity dependence as a function of the zenith angle (7). After converting the data into opacity maps, the SART-based (Simultaneous Algebraic Reconstruction Technique) tomography reconstruction algorithm is applied: (i) The volume to image is first split in cubic pixels (or voxels). In our case, the volume is a parallelepiped containing the whole reactor, i.e., of size 40 m by 30 m by 35 m. Different binnings were tried, the default one corresponding to 25-cm side voxels; (ii) each opacity map is binned in a number of independent measurements. In addition to the [tan(θ_x), tan(θ_y)] variables used to form the 2D maps, the muons can also be binned according to their x and y positions in the telescope. This additional binning can be of interest in the case where the size of the 3D voxels becomes close to the telescope active area, which is the case in our study. All in all, each telescope projection provides an opacity vector of size N(x) × N(y) × N[tan(θ_x)] × N[tan(θ_y)], where N(u) is the number of bins along u. Given the statistics in each projection and requiring at least a few hundreds of muons in each bin, the maximum binning of these maps is of the order of 40,000 (for example, 1 × 1 × 200 × 200 or 2 × 2 × 100 × 100); (iii) the system to be solved can then be written

$$\mathbf{A}\overrightarrow{\mathrm{\rho }}=\overrightarrow{\mathbf{O}}$$

where $\overrightarrow{\mathrm{\rho }}$ represents the (unknown) density in each voxel, $\overrightarrow{\mathbf{O}}$ is the opacity vector, and A is a distance matrix whose element (k,l) contains the mean distance traveled in the lth voxel by muons from the kth opacity measurement. This distance matrix is calculated once the voxels and the data binning have been fixed. The system is solved iteratively by the SART algorithm by computing a density variation in each voxel. All the technical details can be found in (7).

Obviously, many parameters can be varied or tuned in this framework, such as the voxel size, the measurement binning, the number of iterations, or the initialization of the density vector. With the available statistics and requiring at least a few tens of muons in each voxel, the analysis can go down to 25-cm side voxel and images binned in 1 × 1 × 200 × 200 independent measurements. The number of matrix elements in the system is then 27 × 160 × 120 × 140 × 1 × 1 × 200 × 200 ~ 2900 billions. Around 10,000 iterations were enough to converge to a stable result. The initialization choice appeared to be unimportant, and no prior knowledge was therefore introduced, with all elements of the density vector set to zero before the first iteration.

The performance of the tomographic reconstruction is illustrated in Fig. 3 in comparison with the computer-aided design (CAD) model. The plots show the density distribution obtained for the slice between 15 and 20 m high with different parameters. The visible structures are detailed in Fig. 3A, which uses all the available statistics with 25-cm voxels and 1 × 1 × 200 × 200 measurements. The highest density on the right side (1) corresponds to the charging block made of thousands of metallic pipes. The graphite cube is well visible on the center (2), surrounded by the concrete reactor structure (3) with its semispherical extremities. On the left extremity, lower densities indicate the positions of the cooling pipes (4) close to the discharging block structures (5). These pipes go through the reactor with two vertical sections leading to stronger underdensities (6) on both sides of the charging block. Last, some cable caps appear on the right of the reactor (7). The shape and dimensions of all these elements correspond very well with the corresponding slice of the CAD model (see Fig. 3D). The reconstructed density is close to 2 g/cm³ both for concrete and graphite, a value well compatible with the densities of these materials. The (horizontal) spatial resolution is roughly estimated to be of the order of a few tens of centimeters from the smallest visible details, namely, the 80-cm-diameter cooling pipes. Better values can certainly be obtained locally in future measurements. In particular, a few-centimeter resolution is expected on the graphite cube dimensions, by placing telescopes all along its vertical edges.

Fig. 3. Illustration of the reactor tomography with the slice between 15 and 20 m above the ground.

(A) Reconstructed density using the 27 positions, with 25-cm side voxel and muography images binned in 1 × 1 × 200 × 200 independent measurements. The color scale shows the density in g/cm³, with a max density at 2.5 g/cm³ to increase the contrast. (B) Same as (A), with a 1 × 1 × 100 × 100 measurement binning. (C) Same as (B), with a DART-like procedure (see text). (D) Corresponding slice from the CAD model. (E) Same as (B), using only 3 days of data in each projection. (F) Same as (A), using only the five most outer positions.

As a comparison, the same distribution obtained with 1 × 1 × 100 × 100 measurement bins is shown in Fig. 3B, with essentially the same resolution. Figure 3C is obtained by applying a threshold during the SART, after 3000 and 6000 iterations: All the voxels with a density lower than 1 g/cm³ are reset to 0. This procedure, commonly used in discrete versions of the SART (or DART for Discrete Algebraic Reconstruction Technique) enhances the density contrasts. In addition to the parameters mentioned above, the effect of other parameters can be of high interest to design such an experiment, like the acquisition time or the number of projections. Figure 3E shows, in particular, the same density distribution obtained as for Fig. 3B but using only 3 days of data in each projection. The main structures of the reactor are well visible. This time corresponds to 81 days of accumulated data and 20 days of data taking with the four telescopes. The importance of the projection number is illustrated in Fig. 3F, obtained with the five most outer telescope positions (Dirac-P1, Dirac-P2, Bose-P1, Bose-P2, and Einstein-P1). Despite this very small number, it provides a fair reconstruction of the reactor shape, although with a very fuzzy graphite cube and charging blocks. Although outside the scope, the question of the optimization appears to us as a quite interesting problem to tackle for future studies: for example, given an object and a measurement time, how to optimally choose the telescope positions, the number of projections, and the acquisition time in each of them.

The 3D reconstruction reveals additional details of the reactor structure (see Fig. 4 with 1-m-thick slices). The slice in Fig. 4A stands below the graphite cube and shows the concrete base of the reactor. The slice in Fig. 4B shows the lower part of the graphite cube. Compared to Fig. 4A, the y dimension is larger and spherical shells are more pronounced as expected. The lower part of the graphite cube is also visible with the discharging block on its left. On the slice in Fig. 4C, the y dimension is maximal, the plane intersecting the middle of the cylinder. All the details described in Fig. 3A are visible. The slice in Fig. 4D is taken close to the top of the graphite structure where its y dimension is smaller as can be seen. The reactor extension and external shape are very similar to the slice in Fig. 4A. The slice in Fig. 4E is taken just above the reactor cylinder. The concrete hat visible in Fig. 1 is well reconstructed. Although this hat is an external structure, it is worth emphasizing that none of the telescope positions are in visual contact with it, this hat being only seen through the reactor. Another interesting feature of this hat is its asymmetric, octagonal shape, the north-south walls (along x) being much shorter than the y ones. This asymmetry is well visible on the reconstruction. The slice Fig. 4F provides an x-z view of the reconstruction, close to the reactor y center. Because all the projections are taken from the floor (z = 0), the z distribution is more fuzzy and exhibits a global density gradient. Still, the spherical shells and the concrete hat are visible. As presented in the Supplementary Materials, a video further explores the structure of the reconstructed density field by means of several complementary 3D visualization techniques.

Fig. 4. Some tomographic slices obtained from the 3D reconstruction of the reactor, revealing several details of the structure.

(A to E) x-y slices at different heights. (F) x-z slices close to the y axis. See text for more details.

DISCUSSION

The 3D structure of a whole nuclear reactor was obtained using a purely noninvasive and nondestructive method. Despite its complexity and large dimensions, the reactor could be imaged in a relatively short time, with reasonably good-quality reconstruction being achieved with only a few days in each projection. Although anticipated from the simulation, the analysis confirmed that only a limited number of projections are enough to visualize and localize the main elements of the reactor. Last but not least, no prior knowledge on the reactor internal and external structures nor composition was given to the SART algorithm, proving its outstanding capabilities. These conclusions open up new perspectives for the inspection and monitoring of nuclear sites over their entire operating lifetime as in their decommissioning phase, thus contributing to nuclear safety. In addition to the work presented here, some efforts are thus also put on the commercialization of these instruments, with a common laboratory created with an industrial partner.

MATERIALS AND METHODS

Experimental setup

Each of the four muon telescopes used for this study consists of four planes of micropattern gaseous detectors called Micromegas, with an active area of 50 cm by 50 cm. The 2D readout planes are multiplexed (8) so that each coordinate can be read with a single 61-channel connector. The first three telescopes (called Fermi, Dirac, and Bose) are equipped with Micromegas having 1037 strips per coordinate (17 strips per channel) and are therefore identical to the detectors used in (3). The fourth telescope (called Einstein) is made of the new generation of multiplexed detectors, with only 732 strips (12 per channel). With a spatial resolution below 200 μm for perpendicular tracks, all the telescopes yield an angular resolution of the order of 1 mrad. During all the measurements, they were continuously flushed in parallel with a 0.7-liter/hour flow of the so-called T2K gas mixture (Ar-iC₄H₁₀-CF₄, 95-2-3) (9). The trigger was formed from the coincidence of at least two detectors using the self-triggering mode of the Dream-based electronics (10), and the mini-PC performed a preliminary, online muon reconstruction to estimate the mean amplitude in each detector coordinate. Every 5 min, these amplitudes were compared to a target value to adjust the high voltages of each detector. Changes of environmental conditions such as temperature or atmospheric pressure modify the gas density and, thus, the amplification gain of the detectors. Without this online adjustment, the mean signal amplitude can vary by a factor 3 or more. The feedback can thus yield more than 10 V in a single day. It was first implemented for the ScanPyramids campaigns (3) and ensured a gain stability within less than 5% despite temperature, pressure, and humidity fluctuations.

The Fermi telescope was first installed below the reactor in February 2020 and took several months of data in two different positions. The deployment of the other telescopes was slowed down by the pandemic, with an installation of Dirac and Bose in December 2020 and of Einstein in July 2021. Here, we focus on data recorded from March 2021, yielding 370 millions of reconstructed muons for a cumulative time of 1100 detector days and 27 projections. Their positions are indicated in Fig. 2A and given in table S1 with their orientations. Table S1 also shows the duration of the data taking in each position.

Positions of the telescopes

The positions or the telescopes were accurately measured with both a measuring tape and a laser distance meter, reaching centimeter accuracy. The zenith angles were measured with an inclinometer of 0.1° precision, while the azimuth angles were obtained from the laser measurements with respect to the reactor (north-south) axis. All these values are summarized in table S1. X corresponds to the main axis of the reactor cylinder (north-south); Y corresponds to the horizontal axis perpendicular to X (east-west), and Z is the vertical. The reference corresponds to the reactor center on the ground floor (X = Y = Z = 0). α, β, and γ are the Euler angles, the latter being the zenith angle (as an example, Fermi-P4 points to the zenith).

Muon parametrization

The muon flux parametrization is an important ingredient of the 3D reconstruction, as it appears both for open-sky simulations and for the determination of the absorption-opacity dependence. While several such parametrizations were proposed over the past decades, they were often derived for high-energy applications or for a specific angular range. In our case, however, most of the recorded muons have low/medium initial energies (from less than 1 to a few tens of GeV) and cover all the zenith angles. A comparison of six different parametrization was recently performed with data from the same telescopes obtained within the ScanPyramids project, and the results will be published soon. The best parametrization was found to be the Guan model

$$\frac{dI}{dE}=0.14\times {\left[\frac{E}{\mathrm{GeV}}(1+\frac{3.64\mathrm{GeV}}{E{(\cos {\mathrm{\theta }}^{\mathrm{\prime }})}^{1.29}})\right]}^{-2.7}\times [\frac{1}{1+\frac{1.1E\cos {\mathrm{\theta }}^{\mathrm{\prime }}}{115\mathrm{GeV}}}+\frac{0.054}{1+\frac{1.1E\cos {\mathrm{\theta }}^{\mathrm{\prime }}}{850\mathrm{GeV}}}]$$

where I is the differential flux, E is the muon energy in GeV, and θ′ is related to the zenith angle θ through

$$\cos {\mathrm{\theta }}^{\mathrm{\prime }}=\sqrt{\frac{{(\cos \mathrm{\theta })}^2+{0.103}^2-0.068{(\cos \mathrm{\theta })}^{0.959}+0.041{(\cos \mathrm{\theta })}^{0.817}}{0.983}}$$

It is worth mentioning that this model initially comes from an older parametrization and was modified to better reproduce muon data at large zenith angle and lower energies.

3D visualization software

Interactive 3D visualization of the G2 reconstructed density field is performed by means of the Saclay Data visualization (SDvision) software, deployed in the IDL “Interactive Data Language” platform (11). Originally developed for the visualization of astrophysical data, it benefits from the extensive scientific libraries offered by the IDL environment, as well as the high-performance visualization techniques that it provides: hardware-accelerated rendering techniques through “Object Graphics” interfaces to OpenGL and multithreaded usage of multiple-core processors. The SDvision visualization software consists to date in 115,000 lines of code addressing the issues of strategic importance in the field of data visualization: visualization of scalar fields, vector fields, and clouds of points.

Scalar fields, such as the G2 nuclear reactor reconstructed density field, are visualized by means of three complementary techniques: (i) ray-casting volume rendering, a CPU-intensive technique that propagates rays through the data volume under scrutiny and builds up on contributions from crossed cells or voxels, thus requiring a completely new computation at every change in the viewpoint. This technique benefits from several compositing functions to account for the value of a voxel intercepted by a ray: The most basic function is the maximum intensity projection where the color of each pixel on the viewing plane is determined by the voxel with the highest density value along the ray, the color of the voxel being obtained against a lookup table that provides the three RGB colors and opacity functions. A more sophisticated compositing function is the alpha blending, where a recursive equation assumes that the color tables have been premultiplied by the opacity table to obtain semitransparent volumes. This technique is preferred where multiple layers of structures can hide each other. Last, the ray-casting algorithm can be engaged to produce RGBA renderings where red, green, blue, and alpha channels are associated to four different physical fields. The ray-casting algorithm is multithreaded, exploiting all available computing cores on shared-memory computing systems. (ii) Multiple isosurface reconstruction is used to display surfaces of constant value taken by the scalar field. The reconstruction is performed by a procedure known as the Marching-cube algorithm. The resulting surface can be visualized as a Gouraud-shaded polygon or as a simple wireframe polyline object. The interactive visualization of an isosurface polygon benefits from the hardware acceleration by the graphics card. (iii) Volume-slicing is used to map a texture on a simple slice of the volume under scrutiny, at any position and orientation. For all these three techniques, multiple clipping planes along any arbitrary directions can be applied to cutout parts of the data volume to isolate regions of interest.

Supplementary Materials

This PDF file includes:

Materials and Methods

Fig. S1

Table S1

Other Supplementary Material for this manuscript includes the following:

Movie S1