# Linear/Gaussian approximation of millimetric spatial uncertainties in particle detectors: A showcase about muon tomography based on GEANT4 simulations

Ahmet Ilker Topuz

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Ahmet Ilker Topuz. Linear/Gaussian approximation of millimetric spatial uncertainties in particle detectors: A showcase about muon tomography based on GEANT4 simulations. 2025. hal-05087477v2

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# Linear/Gaussian approximation of millimetric spatial uncertainties in particle detectors: A showcase about muon tomography based on GEANT4 simulations

Ahmet Ilker Topuzª

Manipal Centre for Natural Sciences, Centre of Excellence, Manipal Academy of Higher Education, Manipal 576104, Indiaª
aitopuz@zohomail.eu

## Abstract

The precise characterization of the spatial uncertainties is a cornerstone for the reliable particle tracking in the modern detector systems. In this study, two approaches, i.e. the linear tracking perturbation and the Gaussian tracking perturbation, which model and simulate the linear and Gaussian spatial uncertainties within the context of particle physics experiments are presented, respectively. Their application in muon tomography through the GEANT4 simulations are demonstrated by investigating how millimetric spatial uncertainties affect the distribution of the scattering angle as well as the variation of the scattering angle with respect to the kinetic energy. This approach provides a systematic and reproducible framework for evaluating the impact of different uncertainty models on the performance of tracking algorithms with all the code and simulation tools made publicly available on GitHub.

**Keywords:** Particle detectors; Muon tomography; Detector resolution; Spatial uncertainty; Monte Carlo simulations; GEANT4

## 1 Introduction

The accurate tracking of the charged particles is a fundamental requirement in the experimental particle physics, underpinning the key measurements in the detector systems [1,2]. The precision of these measurements is inherently limited by the spatial uncertainties, which can originate from the intrinsic detector resolution, the electronic noise, and the environmental effects. The realistic modeling of such uncertainties is essential for the development and validation of the robust tracking algorithms and for the interpretation of the experimental results.

To address this need, two dedicated strategies, i.e. linear tracking perturbation [3] and Gaussian tracking perturbation [4], that enable the simulation of the spatial uncertainties [2] are introduced. These algorithms are designed to be modular and efficient by supporting the integration into the existing Monte Carlo frameworks such as GEANT4 [5]. The linear tracking perturbation allows the users to impose the deterministic as well as interval-based deviations on the spatial variables, whereas the Gaussian tracking perturbation introduces the stochastic fluctuations following a normal distribution by reflecting the statistical nature of many physical processes.

In this study, the application of these tools in the context of the muon scattering tomography [6-9] is showcased by simulating the passage of the cosmic ray muons through a planar detector geometry and quantifying the resulting spatial uncertainties. By systematically comparing the linear model of uncertainty with the Gaussian method of perturbation, insights into their respective impacts on the scattering angle and the detector resolution are provided. All
code and simulation schemes discussed herein are made publicly available by supporting transparency and reproducibility for the future studies [3, 4]. This study is organized as follows. Section 2 describes the methodology that is followed for the corresponding algorithms, i.e. the linear perturbation method and the Gaussian perturbation method. Section 3 states the simulation properties in this study, while section 4 compares the linear spatial tracking uncertainty with the Gaussian spatial tracking uncertainty. Finally, the conclusions are drawn in section 5.

## 2 Methodology

In this work, the spatial uncertainties in the particle tracking are modeled and implemented via two distinct approaches: the linear tracking perturbation and the Gaussian tracking perturbation. Both these methods are designed as the modular extensions compatible with the GEANT4 simulation framework by facilitating their integration into the particle transport and detector response studies.

The linear tracking perturbation introduces a deterministic and bounded spatial deviation to the simulated hit positions of a particle as it traverses a detector layer. For each detector hit, the true position $(x, z)$ is perturbed by a uniformly distributed random variable within a fixed interval $[-a, a]$ where $a$ is the maximum allowed deviation (typically in the range of 0.1 to 1 mm) [1, 2]. The perturbed coordinates are computed as [3]

$$x = -a + 2a \times \text{G4UniformRand}(\cdot) + x \text{ and } z = -a + 2a \times \text{G4UniformRand}(\cdot) + z \quad (1)$$

where G4UniformRand() generates a random number between 0 and 1. This approach allows the user to simulate the worst-case misalignment or the systematic uncertainties by providing a conservative estimate of their impact on the spatial variables.

The Gaussian tracking perturbation models the spatial uncertainty as a stochastic fluctuation, which is statistically more representative of the intrinsic detector resolution or the random noise. Here, the perturbation for each coordinate is drawn from a normal distribution with zero mean and standard deviation $\sigma$ [4]

$$x = x + \text{G4RandGauss::shoot}(0, \sigma) \text{ and } z = z + \text{G4RandGauss::shoot}(0, \sigma) \quad (2)$$

where G4RandGauss::shoot(0, σ) returns a random variable from $N(0, σ^2)$. The value of σ is chosen in accordance with the expected detector resolution, typically ranging from 0.01 to 0.1 mm in the high-precision systems [1, 2]. The linear and Gaussian tracking perturbation methods are tested over the scattering angle in a tomographic setup dedicated to the muon scattering tomography.
<figure><img src="image_2.png" /><figcaption>Figure 1: Simulation layout in order to test the linear as well as Gaussian tracking perturbation for scattering angle in muon scattering tomography.</figcaption></figure>

In the current study, the scattering angle of a muon means the three-dimensional positive angular difference between the direction of the entering muon through the target volume and the direction of the same exiting muon from the same target volume, and this angular deviation is caused by the interactions that stochastically occur between the propagating muons and the target material. As described in Fig. 1, the computation of the scattering angle requires the construction of two independent vectors by utilizing exactly four muon hit locations in the detector layers where the first vector is the difference between the hits locations in the second top detector layer and the third top detector layer, while the subtraction of the hit position in the first bottom plastic scintillator from the hit position in the second bottom plastic scintillator yields the latter vector.

The definition of these two vectors brings forth the scattering angle denoted by $\theta$, and the scattering angle of a muon crossing the target volume is obtained by using these two vectors as follows [10–12]

$$ \theta = \arccos \left( \frac{\vec{v}_1 \cdot \vec{v}_2}{|v_1| |v_2|} \right) \qquad (3) $$

Since a substantial number of muons reach the target volume, the average profile of the scattering angle at a certain energy is quantified by averaging the previously determined scattering angles over $N$ number of the non-absorbed/non-decayed muons as written in

$$ \bar{\theta} = \frac{1}{N} \sum_{i=1}^{N} \theta_i \qquad (4) $$

where its standard deviation is

$$ \delta\theta = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (\theta_i - \bar{\theta})^2} \qquad (5) $$
## 3 Simulation scheme

To perform the aforementioned analysis, the geometrical scheme is depicted in Fig. 1, and it is shown that the plastic scintillators are separated by a distance of 10 cm, whereas the distance between these two hodoscope sections is 100 cm. Furthermore, the dimensions of the detector layers are 100 × 0.4 × 100 cm³. Concerning the target volume of 30 × 30 × 30 cm³ that is made of stainless steel, the target material is held at the center of the tomographic setup.

By fulfilling the geometrical properties of the tomographic system as well as the target volume, the Monte Carlo simulations are conducted via the GEANT4 code in order to register the hit positions in the plastic scintillators. The simulation parameters are listed in Table 1, and the dimension of the simulation box is 100 × 170 × 100 cm³ where the Cartesian components are situated symmetrically in the interval of (-50 cm, 50 cm), (-85 cm, 85 cm), and (-50 cm, 50 cm), respectively as indicated in Fig. 1. A narrow planar multi-energetic mono-directional beam that is generated at ([-0.5, 0.5] cm, 85 cm, [-0.5, 0.5] cm) via G4ParticleGun is used, and the generated muons are propagating in the vertically downward direction as shown by the black arrow in Fig. 1, i.e. from the top edge of the simulation box through the bottom edge.

<table><caption>Table 1: Simulation properties.</caption><thead><tr><th>Particle</th><th>μ<sup>-</sup></th></tr></thead><tbody><tr><td>Beam direction</td><td>Vertical</td></tr><tr><td>Momentum direction</td><td>(0, -1, 0)</td></tr><tr><td>Source geometry</td><td>Planar</td></tr><tr><td>Initial position (cm)</td><td>([-0.5, 0.5], 85, [-0.5, 0.5])</td></tr><tr><td>Number of particles</td><td>10<sup>5</sup></td></tr><tr><td>Energy interval (GeV)</td><td>[0, 8]</td></tr><tr><td>Energy cut-off (GeV)</td><td>0.1</td></tr><tr><td>Bin step length (GeV)</td><td>0.5</td></tr><tr><td>Energy distribution</td><td>Uniform</td></tr><tr><td>Material database</td><td>G4/NIST</td></tr><tr><td>Reference physics list</td><td>FTFP_BERT</td></tr></tbody></table>

In order to maximize the encounter between the incoming muons and the target volume and to minimize the probability of muon absorption in the top detector layers, a uniform energy distribution with an energy cut-off of 0.1 GeV is used, recalling the numerical advantages [13]. 10⁵ is the total number of created μ⁻ in each simulation. The reference physics list utilized in this work is FTFP_BERT, and all of the materials in the simulation geometry are defined in accordance with the GEANT4/NIST material database.

G4Step maintains the muon tracking, and a Python script is used to post-process the registered hit locations. This involves first calculating the scattering angle for each non-absorbed/non-decayed muon, then marching with a step of 0.5 GeV to divide the uniform energy spectrum bounded by 0 and 8 GeV into 16 bins. The central point in the energy sub-interval is then labeled for each energy bin.
## 4 Simulation outcomes

<figure><img src="image_3.png" /><figcaption>Figure 2: (a) Distribution of the scattering angles for stainless steel with a step length of 1 mrad over the energy interval between 0.1 and 8 GeV by using linear spatial tracking uncertainty (b) Variation of the average scattering angle with respect to the energy bins with a bin length of 0.5 GeV over the energy interval between 0.1 and 8 GeV by recalling the standard deviations listed in Table 2.</figcaption></figure>

After the definition of the simulation properties in section 3, this section is dedicated to the simulation outcomes of the detector uncertainties for the scattering angle in the muon scattering tomography by using Eq. (3). While Fig. 2(a) shows the distribution of the scattering angle for stainless steel with a volume of 30 × 30 × 30 cm³ through the linear tracking perturbation, Fig. 2(a) depicts the variation of the average scattering angle in terms of the kinetic energy by using the same strategy. It is observed that the mode of the distributions in Fig. 2(a) increases when the spatial perturbations rise. Furthermore, Fig. 2(b) demonstrates that the average scattering angle of the cosmic ray muons with a higher energy increases if the linear perturbation augments. This variation is also tabulated in Table 2, and the increase in the average scattering angle is more visible when comparing those of cosmic ray muons at the higher energies according to Table 2. For example, when the kinetic energy is 7.75 GeV, the average scattering angle for a linear perturbation of ±1 mm is explicitly greater than that of 0.1 mm, and this trend is remarkable between 0.1 and 1 mm.
<table><caption>Table 2: Average scattering angles of stainless steel with a volume of 30 × 30 × 30 cm³ and their corresponding standard deviations over the energy interval between 0.1 and 8 GeV by using linear tracking perturbation.</caption><thead><tr><th>E [GeV]</th><th>θ<sub>Non-perturbed</sub><sup>Linear</sup> ± δθ [mrad]</th><th>θ<sub>Non-perturbed</sub><sup>Linear</sup> ± δθ [mrad]</th><th>θ<sub>Linear</sub><sup>±0.1 mm</sup> ± δθ [mrad]</th><th>θ<sub>Linear</sub><sup>±0.2 mm</sup> ± δθ [mrad]</th><th>θ<sub>Linear</sub><sup>±0.4 mm</sup> ± δθ [mrad]</th><th>θ<sub>Linear</sub><sup>±0.6 mm</sup> ± δθ [mrad]</th><th>θ<sub>Linear</sub><sup>±0.8 mm</sup> ± δθ [mrad]</th><th>θ<sub>Linear</sub><sup>±1.0 mm</sup> ± δθ [mrad]</th></tr></thead><tbody><tr><td>0.25</td><td>268.835±150.968</td><td>268.795±150.972</td><td>268.762±150.978</td><td>268.716±150.998</td><td>268.697±151.029</td><td>268.703±151.074</td><td>268.736±151.130</td><td>268.736±151.130</td></tr><tr><td>0.75</td><td>128.856±79.373</td><td>128.855±79.384</td><td>128.871±79.401</td><td>128.950±79.447</td><td>129.095±79.507</td><td>129.303±79.584</td><td>129.572±79.683</td><td>129.572±79.683</td></tr><tr><td>1.25</td><td>67.376±37.760</td><td>67.393±37.785</td><td>67.440±37.823</td><td>67.623±37.936</td><td>67.925±38.093</td><td>68.348±38.290</td><td>68.887±38.528</td><td>68.887±38.528</td></tr><tr><td>1.75</td><td>47.417±26.669</td><td>47.438±26.663</td><td>47.499±26.678</td><td>47.744±26.768</td><td>48.158±26.921</td><td>48.737±27.137</td><td>49.472±27.419</td><td>49.472±27.419</td></tr><tr><td>2.25</td><td>36.361±19.958</td><td>36.415±19.952</td><td>36.521±19.972</td><td>36.886±20.095</td><td>37.447±20.328</td><td>38.201±20.662</td><td>39.135±21.090</td><td>39.135±21.090</td></tr><tr><td>2.75</td><td>28.939±16.023</td><td>28.975±16.057</td><td>29.079±16.120</td><td>29.491±16.320</td><td>30.165±16.621</td><td>31.077±17.025</td><td>32.210±17.523</td><td>32.210±17.523</td></tr><tr><td>3.25</td><td>24.708±13.447</td><td>24.751±13.472</td><td>24.873±13.535</td><td>25.348±13.772</td><td>26.119±14.145</td><td>27.160±14.646</td><td>28.445±15.256</td><td>28.445±15.256</td></tr><tr><td>3.75</td><td>21.368±11.787</td><td>21.401±11.798</td><td>21.524±11.852</td><td>22.037±12.073</td><td>22.883±12.446</td><td>24.023±12.961</td><td>25.417±13.604</td><td>25.417±13.604</td></tr><tr><td>4.25</td><td>18.714±10.383</td><td>18.767±10.417</td><td>18.923±10.495</td><td>19.537±10.771</td><td>20.521±11.200</td><td>21.819±11.772</td><td>23.384±12.460</td><td>23.384±12.460</td></tr><tr><td>4.75</td><td>16.761±9.143</td><td>16.813±9.163</td><td>16.980±9.239</td><td>17.649±9.545</td><td>18.719±10.044</td><td>20.120±10.720</td><td>21.795±11.528</td><td>21.795±11.528</td></tr><tr><td>5.25</td><td>14.987±8.209</td><td>15.057±8.240</td><td>15.262±8.325</td><td>16.040±8.672</td><td>17.257±9.220</td><td>18.821±9.945</td><td>20.667±10.787</td><td>20.667±10.787</td></tr><tr><td>5.75</td><td>13.791±7.691</td><td>13.834±7.737</td><td>14.023±7.836</td><td>14.810±8.200</td><td>16.066±8.760</td><td>17.693±9.479</td><td>19.601±10.320</td><td>19.601±10.320</td></tr><tr><td>6.25</td><td>12.645±6.885</td><td>12.726±6.944</td><td>12.962±7.071</td><td>13.872±7.494</td><td>15.263±8.127</td><td>17.010±8.944</td><td>19.016±9.895</td><td>19.016±9.895</td></tr><tr><td>6.75</td><td>11.738±6.424</td><td>11.815±6.475</td><td>12.055±6.600</td><td>12.996±7.036</td><td>14.435±7.694</td><td>16.234±8.535</td><td>18.291±9.504</td><td>18.291±9.504</td></tr><tr><td>7.25</td><td>10.924±6.057</td><td>11.027±6.120</td><td>11.305±6.261</td><td>12.345±6.733</td><td>13.883±7.448</td><td>15.771±8.348</td><td>17.898±9.381</td><td>17.898±9.381</td></tr><tr><td>7.75</td><td>10.186±5.714</td><td>10.303±5.769</td><td>10.607±5.906</td><td>11.699±6.415</td><td>13.300±7.172</td><td>15.252±8.102</td><td>17.442±9.147</td><td>17.442±9.147</td></tr></tbody></table>

When the case of the Gaussian tracking perturbation is investigated, Fig. 3(a) delineates the distribution of the scattering angle for stainless steel via this strategy, and Fig. 3(b) illustrates the variation of the average scattering angle with respect to the kinetic energy by using the same strategy.

<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Distribution of the scattering angles for stainless steel with a step length of 1 mrad over the energy interval between 0.1 and 8 GeV by using Gaussian spatial tracking uncertainty (b) Variation of the average scattering angle with respect to the energy bins with a bin length of 0.5 GeV over the energy interval between 0.1 and 8 GeV by recalling the standard deviations listed in Table 3.</figcaption></figure>

It is seen that the mode of the distributions in Fig. 3(a) increases when the spatial perturbations elevates. Fig. 3(b) shows that the average scattering angle of the cosmic ray muons with a higher energy increases if the linear perturbation furthermore augments.
<table><caption>Table 3: Average scattering angles of stainless steel and their corresponding standard deviations over the energy interval between 0.1 and 8 GeV by using Gaussian tracking perturbation.</caption><thead><tr><th>E [GeV]</th><th>δ<sub>Gaussian</sub><br>Non-perturbed ± δθ [mrad]</th><th>δ<sub>Gaussian</sub><br>± δθ [mrad]</th><th>δ<sub>Gaussian</sub><br>± δθ [mrad]</th><th>δ<sub>Gaussian</sub><br>± δθ [mrad]</th><th>δ<sub>Gaussian</sub><br>± δθ [mrad]</th><th>δ<sub>Gaussian</sub><br>± δθ [mrad]</th><th>δ<sub>Gaussian</sub><br>± δθ [mrad]</th><th>δ<sub>Gaussian</sub><br>± δθ [mrad]</th></tr></thead><tbody><tr><td>0.25</td><td>268.835±150.968</td><td>268.777±150.970</td><td>268.739±150.979</td><td>268.722±151.025</td><td>268.783±151.107</td><td>268.925±151.221</td><td>269.147±151.367</td><td>269.147±151.367</td></tr><tr><td>0.75</td><td>128.856±79.373</td><td>128.857±79.393</td><td>128.906±79.428</td><td>129.152±79.531</td><td>129.585±79.694</td><td>130.204±79.914</td><td>131.005±80.192</td><td>131.005±80.192</td></tr><tr><td>1.25</td><td>67.376±37.760</td><td>67.423±37.825</td><td>67.558±37.929</td><td>68.098±38.237</td><td>68.999±38.659</td><td>70.233±39.218</td><td>71.774±39.923</td><td>71.774±39.923</td></tr><tr><td>1.75</td><td>47.417±26.669</td><td>47.489±26.672</td><td>47.683±26.736</td><td>48.445±27.015</td><td>49.675±27.507</td><td>51.330±28.221</td><td>53.381±29.117</td><td>53.381±29.117</td></tr><tr><td>2.25</td><td>36.361±19.958</td><td>36.480±19.963</td><td>36.754±20.049</td><td>37.752±20.455</td><td>39.312±21.155</td><td>41.373±22.112</td><td>43.864±23.296</td><td>43.864±23.296</td></tr><tr><td>2.75</td><td>28.939±16.023</td><td>29.042±16.093</td><td>29.350±16.242</td><td>30.555±16.773</td><td>32.448±17.619</td><td>34.910±18.750</td><td>37.825±20.127</td><td>37.825±20.127</td></tr><tr><td>3.25</td><td>24.708±13.447</td><td>24.837±13.505</td><td>25.202±13.675</td><td>26.599±14.327</td><td>28.762±15.359</td><td>31.521±16.724</td><td>34.735±18.344</td><td>34.735±18.344</td></tr><tr><td>3.75</td><td>21.368±11.787</td><td>21.480±11.832</td><td>21.861±12.001</td><td>23.365±12.678</td><td>25.690±13.777</td><td>28.634±15.208</td><td>32.029±16.881</td><td>32.029±16.881</td></tr><tr><td>4.25</td><td>18.714±10.383</td><td>18.874±10.466</td><td>19.333±10.685</td><td>21.058±11.493</td><td>23.648±12.709</td><td>26.857±14.225</td><td>30.470±15.987</td><td>30.470±15.987</td></tr><tr><td>4.75</td><td>16.761±9.143</td><td>16.931±9.213</td><td>17.441±9.451</td><td>19.349±10.384</td><td>22.155±11.823</td><td>25.572±13.593</td><td>29.383±15.590</td><td>29.383±15.590</td></tr><tr><td>5.25</td><td>14.987±8.209</td><td>15.196±8.294</td><td>15.784±8.558</td><td>17.914±9.586</td><td>20.970±11.110</td><td>24.613±12.950</td><td>28.607±15.013</td><td>28.607±15.013</td></tr><tr><td>5.75</td><td>13.791±7.691</td><td>13.952±7.799</td><td>14.532±8.078</td><td>16.724±9.110</td><td>19.896±10.596</td><td>23.627±12.427</td><td>27.685±14.485</td><td>27.685±14.485</td></tr><tr><td>6.25</td><td>12.645±6.885</td><td>12.875±7.036</td><td>13.558±7.374</td><td>16.002±8.535</td><td>19.364±10.221</td><td>23.247±12.219</td><td>27.431±14.396</td><td>27.431±14.396</td></tr><tr><td>6.75</td><td>11.738±6.424</td><td>11.968±6.559</td><td>12.672±6.898</td><td>15.178±8.113</td><td>18.608±9.842</td><td>22.558±11.834</td><td>26.792±13.987</td><td>26.792±13.987</td></tr><tr><td>7.25</td><td>10.924±6.057</td><td>11.210±6.215</td><td>11.994±6.598</td><td>14.669±7.927</td><td>18.229±9.778</td><td>22.255±11.904</td><td>26.546±14.161</td><td>26.546±14.161</td></tr><tr><td>7.75</td><td>10.186±5.714</td><td>10.512±5.862</td><td>11.358±6.257</td><td>14.174±7.637</td><td>17.868±9.503</td><td>22.004±11.618</td><td>26.368±13.880</td><td>26.368±13.880</td></tr></tbody></table>

The augmentation in the mode of the scattering angle distribution for the Gaussian tracking perturbation is higher compared to that of the linear perturbation. Moreover, when the perturbation on the spatial coordinates is augmented, the increase in the average scattering angle at the higher energies is also higher than that of the linear perturbation method. This change is also listed in Table 3, and the rise in the average scattering angle is more remarkable when comparing those of cosmic ray muons at the higher energies according to Table 3.

## 5 Conclusion

In this work, the impact of the millimetric spatial uncertainties on the measurement of the muon scattering angles in the tomographic setups is systematically investigated by using two distinct perturbation models: linear and Gaussian tracking perturbations. These models are implemented within the GEANT4 simulation framework, and their effects are analyzed across a wide range of muon energies. It is observed that the simulations through both approaches show an increase in the average scattering angle as the magnitude of spatial uncertainty is increased. This effect is found to be more pronounced at higher muon energies, and the Gaussian perturbation model generally results in a greater rise in the average scattering angle compared to the linear model for the equivalent perturbation scales. The systematic framework and the public availability of the codes and simulation tools are expected to provide a transparent and reproducible basis for the future studies by aiming at evaluating or mitigating the spatial uncertainty effects in the particle tracking. Overall, the necessity of carefully modeling spatial uncertainties in the particle detector simulations is highlighted. It is demonstrated that the choice of perturbation model can significantly influence the interpretation of tomographic and tracking results. These findings are anticipated to inform both the design of the future experiments and the development of more robust tracking algorithms in particle physics and the related fields.

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