🔬 applied physics

$μ$ TRec: A Muon Trajectory Reconstruction Algorithm for Enhanced Scattering Tomography

This paper introduces the $\mu$ TRec algorithm, a Bayesian-based muon trajectory reconstruction method that significantly outperforms traditional PoCA techniques in cosmic ray muon scattering tomography by achieving superior resolution, signal-to-noise ratios, and detection capabilities for identifying missing fuel assemblies in dry storage casks.

Original authors: Reshma Ughade, Stylianos Chatzidakis

Published 2026-03-09

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CC BY 4.0

Original authors: Reshma Ughade, Stylianos Chatzidakis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to see what's inside a massive, thick, steel-and-concrete safe without opening it. You can't use a flashlight (X-rays) because the walls are too thick, and you don't have a giant particle accelerator to shoot protons at it.

Instead, you decide to use cosmic rays. These are tiny, ghost-like particles called muons that rain down on Earth from space 24/7. They are like invisible bullets that can punch right through mountains and steel.

Here is the problem: When these muon bullets hit the heavy stuff inside the safe (like nuclear fuel), they don't just go straight through. They get bumped around, zigzagging like a pinball hitting bumpers. This is called scattering. The heavier the material, the more the muon gets bumped.

The Old Way: The "Guess the Midpoint" Game

For a long time, scientists used a method called PoCA (Point of Closest Approach) to figure out where the muons got bumped.

Think of it like this: You see a car enter a foggy forest and exit on the other side at a slightly different angle. The old method assumes the car hit one single tree right in the middle of the forest and bounced off. It draws a straight line from the entrance, finds the closest point to the exit line, and says, "The tree must be there."

The flaw: In reality, the car didn't hit one tree; it hit hundreds of tiny twigs and branches all along the way. The old method is too simple. It misses the details, gets blurry, and often needs to wait for millions of cars (muons) to pass through before it can guess what's inside.

The New Way: µTRec (The "Curved Path" Detective)

This paper introduces a new algorithm called µTRec (Muon Trajectory Reconstruction).

Instead of guessing a single bump, µTRec acts like a super-smart detective who knows the laws of physics. It doesn't assume the muon hit one thing. Instead, it calculates a smooth, curved path that the muon likely took as it wiggled through the material.

The Analogy: Imagine you are watching a runner on a track.
- The Old Method (PoCA): You see them start at point A and finish at point B. You assume they ran in a straight line and just tripped once in the middle.
- The New Method (µTRec): You know the runner is tired and the track is muddy. You use a statistical model to draw the actual wavy, curvy path they took, accounting for every little stumble and turn.

By tracing this curved path, µTRec can map out exactly where the heavy materials are, creating a much sharper picture.

What Did They Test?

The researchers tested this on a Dry Storage Cask—a giant, heavy barrel used to store used nuclear fuel. These barrels are incredibly dense and hard to see through.

They simulated four scenarios:

Full Barrel: Everything is there.
Missing a Row: A whole column of fuel is gone.
Missing One: Just one fuel stick is missing.
Missing Half: Half of one fuel stick is gone.

The Results: Why It Matters

The results were like night and day:

Seeing the Invisible: With the old method, if you only had a few muons passing by, the image was so blurry you couldn't tell if a piece was missing. With µTRec, they could spot a missing half-fuel stick even with very few muons.
The "Steel Canister" Detail: The cask has a thin steel wall (2.5 cm thick) inside. The old method couldn't even see this wall. µTRec could clearly outline it, like seeing the rim of a cup inside a box.
Speed and Efficiency: µTRec got clear results with 100 times less data than the old method needed. It's like getting a high-definition photo with a cheap camera instead of waiting for a super-computer to process a blurry one.

The Bottom Line

This new algorithm, µTRec, is a game-changer for nuclear safety. It allows us to "see" inside the most heavily shielded nuclear containers with much greater clarity and speed.

Instead of guessing where the heavy stuff is by looking at a single bump, µTRec traces the entire wiggly journey of the particle. It turns a blurry, fuzzy guess into a sharp, high-definition map, ensuring we can detect even the smallest missing pieces of nuclear fuel without ever having to open the safe.

1. Problem Statement

Muon tomography is a non-invasive imaging technique used to visualize dense, large-scale structures (such as spent nuclear fuel dry storage casks) by tracking cosmic ray muons. As muons traverse matter, they undergo Multiple Coulomb Scattering (MCS), deviating from their original path. The degree of scattering correlates with the material's atomic number ( $Z$ ), allowing for material identification.

However, traditional reconstruction algorithms face significant limitations:

Straight-Line Path (SLP): Assumes muons travel in a straight line, ignoring scattering entirely.
Point of Closest Approach (PoCA): Assumes a single scattering event occurs at the midpoint between the extrapolated incoming and outgoing trajectories.
- Limitations: PoCA fails to model the cumulative, stochastic nature of MCS. It discards muon events where the extrapolated paths do not intersect within the reconstruction volume, reducing usable data. It also suffers from poor spatial resolution and image blurring, particularly in complex geometries or with low muon fluxes.

The paper addresses the need for an algorithm that accurately models the curved trajectory of muons, incorporating realistic MCS and energy loss effects to improve image resolution and detection sensitivity, especially for identifying subtle anomalies like missing fuel assemblies in dry storage casks.

2. Methodology

The authors propose a novel algorithm called Muon Trajectory Reconstruction (µTRec).

A. Theoretical Framework

Bayesian Statistical Approach: Unlike PoCA's geometric single-point assumption, µTRec uses a Bayesian framework with Gaussian approximations to estimate the most probable path of a muon.
Generalized Muon Trajectory Estimation (GMTE):
- The algorithm decomposes the 3D trajectory into 2D projections ( $x-z$ and $y-z$ planes).
- It models the muon state (transverse position $y$ and scattering angle $\theta$ ) as a bivariate Gaussian distribution with a zero mean.
- The probability of a muon's state at an intermediate depth is calculated based on the measured entry and exit states, maximizing the likelihood function.
Physics Integration:
- MCS Modeling: Uses Lynch and Dahl approximations to calculate scattering variances ( $\sigma^2_y, \sigma^2_\theta$ ) based on radiation length ( $X_0$ ) and material properties.
- Energy Loss: Incorporates the Continuous Slowing Down Approximation (CSDA). The algorithm accounts for momentum loss ( $p(z) = p_0 - az$ ) as the muon traverses the material, which is critical for accurate covariance matrix calculation in high-Z materials.
Trajectory Output: The result is a continuous, curved trajectory that asymptotically aligns with the measured incoming and outgoing vectors, rather than a single straight line or a single scattering point.

B. Simulation Setup

Target: A VSC-24 dry storage cask containing 24 spent nuclear fuel assemblies, enclosed in a steel canister and surrounded by concrete shielding.
Scenarios: Four configurations were simulated:
1. Fully loaded.
2. One column of assemblies missing.
3. One single assembly missing.
4. Half of one assembly missing.
Muon Sources:
- Mono-energetic/Unidirectional: 5 GeV muons (idealized).
- Poly-energetic/Multidirectional: Realistic spectrum (1–60 GeV) with zenith angles 0°–90°, generated using a dedicated "Muon Generator."
Detectors: Four ideal scintillation detectors (two upstream, two downstream) with 30 cm spacing.
Flux Levels: Simulations were run with $10^5$ and $10^6$ muons.
Voxel Sizes: 1 cm and 5 cm.

3. Key Contributions

Novel Algorithm (µTRec): Introduction of a physics-based, Bayesian reconstruction method that models muon paths as continuous curves, explicitly accounting for cumulative MCS and energy loss.
Superior Low-Flux Performance: Demonstrated the ability to detect subtle anomalies (e.g., a half-missing fuel assembly) with as few as $10^5$ muons, a regime where PoCA fails completely.
High-Resolution Capability: Achieved reliable reconstruction with voxel sizes as small as 1 cm, enabling the visualization of fine structural details like the 2.5 cm thick steel canister, which is blurred or invisible in PoCA reconstructions.
Quantitative Validation: Provided rigorous metrics (SNR, CNR, Detection Power) comparing µTRec against the industry-standard PoCA method under realistic conditions.

4. Results

The study compared µTRec against PoCA across various flux levels and voxel sizes.

Mono-energetic Simulations:
- µTRec successfully identified all four scenarios (including the half-missing assembly) with high spatial precision.
- PoCA struggled to distinguish subtle anomalies and produced blurred images.
Realistic (Poly-energetic) Simulations:
- At $10^5$ muons: µTRec detected missing columns and single missing assemblies. PoCA could only detect the missing column and failed on single/half-missing cases.
- At $10^6$ muons (1 cm voxels): µTRec resolved all scenarios, including the specific half of the missing assembly and the steel canister outline. PoCA failed to provide sufficient signal strength for confident identification of subtle defects.
- At $10^6$ muons (5 cm voxels): Increasing voxel size improved PoCA's signal-to-noise ratio, allowing it to detect missing columns and single assemblies. However, it still could not resolve the half-missing assembly or the steel canister. µTRec maintained high fidelity and superior detection capabilities even at this coarser resolution.

Quantitative Improvements (Case: One Missing Assembly, $10^6$ muons, 5 cm voxel)

Compared to PoCA, µTRec achieved:

Signal-to-Noise Ratio (SNR): +122%
Contrast-to-Noise Ratio (CNR): +35%
Detection Power (DP): +201%

At lower flux ( $10^5$ muons), the improvements were even more drastic (e.g., DP improvement of ~2500%), highlighting µTRec's robustness in data-scarce environments.

5. Significance

Nuclear Security & Safety: The ability to reliably detect missing or damaged fuel assemblies in dry storage casks without opening them is critical for nuclear waste management and non-proliferation. µTRec reduces the time required for data collection (by requiring fewer muons) and increases confidence in anomaly detection.
Algorithmic Advancement: µTRec moves beyond the "single scattering" approximation that has limited muon tomography for decades. By treating the trajectory as a statistical curve influenced by energy loss, it provides a more physically accurate representation of muon transport.
Practical Applicability: The algorithm's ability to work with realistic, poly-energetic cosmic ray spectra and its superior performance at low fluxes make it highly suitable for field deployment where data acquisition time is a constraint.

Conclusion: The µTRec algorithm represents a significant leap forward in muon tomography, offering higher resolution, better material differentiation, and the ability to detect subtle structural anomalies in heavily shielded nuclear infrastructure compared to conventional methods like PoCA.