# Geometric Simplifications of Natural Caves in Ray-Tracing-Based Propagation Modelling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Ray-Tracing Limits in Irregular Environments

## 3. Environment Simplification

## 4. Simulation Setup and Measurement Campaign

#### 4.1. Caves Physical Properties

#### 4.2. Measurements

#### 4.3. Signal3D Ray Tracer

#### 4.4. Simulation Parameters

## 5. Results and Discussion

#### 5.1. Accuracy of Simplified Models

^{2}average triangle size performs equally well as the grid of 1610 facets and 0.16 m

^{2}average triangle size of the vertex clustering simplification.

#### 5.2. Computational Time Gains

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Signal3D screen snapshots of (

**a**) Skirwith Cave and (

**b**) the KMC Roof Tunnel; both caves are located in the UK.

**Figure 2.**Measured vs. modelled path loss for the two caves at the three selected frequencies; ray tracing and modal theory path loss is compared against the measured values. Modal theory results shown here were reported in [9].

**Figure 3.**The accuracy of ray tracing is evaluated against the measurements in a number of simplified geometrical descriptions of the KMC Roof Tunnel using (

**a**) vertex clustering and (

**b**) edge removal simplifications.

**Figure 4.**The accuracy of ray tracing is evaluated against the measurements in a number of simplified geometrical descriptions of Skirwith Cave using (

**a**) vertex clustering and (

**b**) edge removal simplifications. The missing values in the case of vertex clustering correspond to impassable geometries.

**Figure 5.**Reduction in ray-tracing simulation time in the KMC Roof Tunnel with (

**a**) vertex clustering and (

**b**) edge removal simplifications; the reduction is practically independent of transmission frequency.

**Figure 6.**The reduction in simulation time of the simplified Skirwith Cave geometries in case of (

**a**) vertex clustering is stopped early as the cave becomes impassable, whereas the simulation time of the simplified models of Skirwith Cave using (

**b**) the edge removal algorithm can be halved.

Frequency | Tx Power | Antenna Gain | Antenna Length |
---|---|---|---|

(GHz) | (dBm) | (dBi) | (mm) |

1.2969 | 29 | 5.5 | 340 |

2.3209 | 30 | 9 | 625 |

5.8020 | 22 | 11 | 700 |

Frequency (GHz) | $\mathit{\sigma}$ (KMC Roof) | $\mathit{\sigma}$ (Skirwith) | ||
---|---|---|---|---|

Modal ${}^{1}$ | Ray | Modal ${}^{1}$ | Ray | |

1.2969 | 4.54 | 4.69 | 7.11 | 9.15 |

2.3209 | 9.52 | 6.24 | 9.02 | 6.81 |

5.8020 | 7.89 | 3.40 | 5.74 | 7.77 |

^{1}Values reported in [9].

Cave | Simplification Algorithm | No. of Triangles | Avg. Triangle Size (cm ^{2}) | Simp. VLevel (%) |
---|---|---|---|---|

KMC Roof | None | 53,540 | 53 | 0.0 |

Vertex clustering | 26,686 | 106 | 50.2 | |

13,376 | 209 | 75.0 | ||

6769 | 406 | 87.4 | ||

3285 | 816 | 93.9 | ||

1610 | 1624 | 97.0 | ||

838 | 3025 | 98.4 | ||

614 | 4023 | 98.9 | ||

376 | 6198 | 99.3 | ||

Edge removal | 26,685 | 107 | 50.2 | |

13,375 | 213 | 75.0 | ||

6769 | 420 | 87.4 | ||

3285 | 865 | 93.9 | ||

1610 | 1757 | 97.0 | ||

838 | 3341 | 98.4 | ||

613 | 4519 | 98.9 | ||

312 | 8633 | 99.4 | ||

211 | 12,988 | 99.6 | ||

Skirwith | None | 54,427 | 60 | 0.0 |

Vertex clustering | 27,400 | 119 | 49.7 | |

13,625 | 239 | 75.0 | ||

6905 | 464 | 87.3 | ||

3400 | 905 | 93.8 | ||

Edge removal | 27,399 | 120 | 49.7 | |

13,625 | 241 | 74.6 | ||

6904 | 476 | 87.3 | ||

3400 | 964 | 93.8 | ||

1646 | 1972 | 97.0 | ||

829 | 3831 | 98.5 | ||

428 | 7241 | 99.2 | ||

320 | 9752 | 99.4 |

Simp. Algorithm | Vertex Clustering | Edge Removal | ||||
---|---|---|---|---|---|---|

Frequency (GHz) | 1.2969 | 2.3209 | 5.802 | 1.2969 | 2.3209 | 5.802 |

No. of Triangles | Err Std | Err Std | ||||

53,540 | 4.69 | 6.24 | 3.40 | 4.69 | 6.24 | 3.40 |

26,686 | 5.07 | 6.27 | 3.84 | 4.41 | 6.13 | 3.34 |

13,376 | 5.49 | 6.26 | 3.85 | 4.47 | 6.71 | 3.46 |

6769 | 5.17 | 6.28 | 3.98 | 5.28 | 6.84 | 3.28 |

3285 | 4.41 | 6.58 | 3.79 | 5.12 | 6.18 | 4.25 |

1610 | 5.71 | 8.27 | 3.75 | 5.23 | 6.74 | 4.09 |

838 | 19.97 | 22.50 | 15.46 | 5.39 | 6.72 | 3.57 |

614 | 15.78 | 18.21 | 11.65 | 5.65 | 7.30 | 4.32 |

376 | 57.23 | 55.83 | 56.63 | |||

312 | 7.27 | 7.82 | 4.99 | |||

211 | 12.53 | 14.58 | 8.45 |

Simp. Algorithm | Vertex Clustering | Edge Removal | ||||
---|---|---|---|---|---|---|

Frequency (GHz) | 1.2969 | 2.3209 | 5.802 | 1.2969 | 2.3209 | 5.802 |

No. of Triangles | Err Std | Err Std | ||||

54,427 | 9.15 | 6.81 | 7.77 | 9.15 | 6.81 | 7.77 |

27,400 | 9.59 | 6.88 | 7.96 | 8.35 | 6.38 | 7.92 |

13,625 | 10.01 | 5.74 | 5.81 | 8.93 | 6.51 | 7.57 |

6905 | 9.58 | 5.74 | 5.86 | 9.13 | 7.01 | 8.85 |

3400 | 10.04 | 5.74 | 5.20 | 8.10 | 7.03 | 8.93 |

1646 | 8.69 | 6.56 | 7.21 | |||

829 | 7.59 | 8.68 | 11.53 | |||

428 | 5.30 | 7.62 | 11.14 | |||

321 | 7.44 | 9.03 | 12.58 |

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**MDPI and ACS Style**

Novak, R.; Hrovat, A.; Bedford, M.D.; Javornik, T. Geometric Simplifications of Natural Caves in Ray-Tracing-Based Propagation Modelling. *Electronics* **2021**, *10*, 2914.
https://doi.org/10.3390/electronics10232914

**AMA Style**

Novak R, Hrovat A, Bedford MD, Javornik T. Geometric Simplifications of Natural Caves in Ray-Tracing-Based Propagation Modelling. *Electronics*. 2021; 10(23):2914.
https://doi.org/10.3390/electronics10232914

**Chicago/Turabian Style**

Novak, Roman, Andrej Hrovat, Michael D. Bedford, and Tomaž Javornik. 2021. "Geometric Simplifications of Natural Caves in Ray-Tracing-Based Propagation Modelling" *Electronics* 10, no. 23: 2914.
https://doi.org/10.3390/electronics10232914