# Cross-Correlation Wavelet-Domain-Based Particle Swarm Optimization for Lightning Mapping

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Database

#### 2.2. Lightning Mapping Optimization

#### 2.3. Objective Function

#### 2.4. Optimization Constraints and PSO Parameters

## 3. Results and Discussions

#### 3.1. Optimal Selection of PSO Parameters Using Simulated Data

#### 3.2. Evaluations on Real Lightning Data

#### Evaluation of the Lightning Event Extraction Method

#### 3.3. Lightning Mapping Optimization

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The general structure of the proposed framework for ITF lightning maps optimization, based on PSO algorithm.

**Figure 4.**PSO performance for estimating lightning maps. (

**a**) The PSO-convergence curves as a function of number of iterations and fitness value. (

**b**) An expanded view of (

**a**).

**Figure 5.**PSO algorithm: (

**a**) simulated lightning map plotted in elevation vs. azimuth; (

**b**) expanded view of the simulated lightning maps of (

**a**).

**Figure 6.**Regression performance for estimating lightning map at 100 iterations: (

**a**) elevation regression line; (

**b**) azimuth regression line.

**Figure 7.**Lightning mapping regression performance at 200 iterations: (

**a**) elevation regression performance; (

**b**) azimuth regression performance.

**Figure 10.**Interferometer data for NBE1: (

**a**) lightning map plotted in elevation vs. azimuth; (

**b**) expanded view of lightning map of (

**a**); (

**c**) the breakdown of each colored circle-marker denotes the elevation angles (altitude) vs. time.

**Figure 11.**Interferometer data for NBE2: (

**a**) lightning map plotted in elevation vs. azimuth; (

**b**) expanded view of lightning map of (

**a**); (

**c**) the breakdown of each colored circle-marker denotes the elevation angles (altitude) vs. time.

**Figure 12.**Interferometer data for NBE3: (

**a**) lightning map plotted in elevation vs. azimuth; (

**b**) expanded view of lightning map of (

**a**); (

**c**) the breakdown of each colored circle-marker denotes the elevation angles (altitude) vs. time.

Parameters | Description |
---|---|

${W}_{max}$ | 0.9 |

${W}_{min}$ | 0.4 |

Acceleration coefficients ${c}_{1}$ and ${c}_{2}$ | 2 |

Initial window size (Ws) | $W{s}_{upper}$ = 512, $W{s}_{lower}$ = 100, initial $Ws=$ 256 |

No of population | 2 |

Number of iterations | 100, 200, 300, and 500 |

Cross-correlation wavelet domain | CCWD |

**Table 2.**The performance of CCWD-PSO technique in terms of Euclidean distance. The comparison involves several cross-correlation-related parameters. The sampling parameter indicates the interpolation ratio of the cross-correlation output signal. Two interpolation methods were used, i.e., linear and cubic.

Parameters | Sampling Ratio | Iteration Sets | |||||||
---|---|---|---|---|---|---|---|---|---|

100 Iteration | 200 Iteration | 300 Iteration | 500 Iteration | ||||||

Linear | Cubic | Linear | Cubic | Linear | Cubic | Linear | Cubic | ||

Window size (Ws) | 4 | 282.71 | 446.33 | 290.41 | 446 | 282.55 | 425.56 | 282.46 | 419.72 |

8 | 457.50 | 446.82 | 544.86 | 451.55 | 466.08 | 452.43 | 466.63 | 451.24 | |

Objective function | 4 | 0.6243 | 0.7702 | 0.6328 | 0.7944 | 0.6243 | 0.7913 | 0.6243 | 0.7923 |

8 | 0.7829 | 0.7271 | 0.7940 | 0.7273 | 0.7836 | 0.7325 | 0.7836 | 0.7273 |

**Table 3.**Breakdown characteristics for all the NBEs (NBE1, NBE2, and NBE3) of a tuning window size ${W}_{s}=282$.

NBE | Higher Altitude | Elevation | Captured time | OV | Ws | Length | Time Window | Lightning Event Extraction Method | ||
---|---|---|---|---|---|---|---|---|---|---|

Event Start | Event End | Segment Length | ||||||||

+NBE1 | (15°~66°) | (25°~32°) | 4 December 2017, UTC+8 03:01:42 | 32 | 282 | 4500 | 18 µs | 4596 | 7164 | 2568 |

+NBE2 | (15°~80°) | (22°~29°) | 4 December 2017, UTC+8 03:21:13 | 32 | 282 | 5000 | 20 µs | 4695 | 7877 | 3218 |

+NBE3 | (15°~80°) | (23°~30.5°) | 13 December 2017, UTC+8 5:08:16 | 32 | 282 | 4500 | 19 µs | 4520 | 6840 | 2320 |

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**MDPI and ACS Style**

Alammari, A.; Alkahtani, A.A.; Ahmad, M.R.; Aljanad, A.; Noman, F.; Kawasaki, Z.
Cross-Correlation Wavelet-Domain-Based Particle Swarm Optimization for Lightning Mapping. *Appl. Sci.* **2021**, *11*, 8634.
https://doi.org/10.3390/app11188634

**AMA Style**

Alammari A, Alkahtani AA, Ahmad MR, Aljanad A, Noman F, Kawasaki Z.
Cross-Correlation Wavelet-Domain-Based Particle Swarm Optimization for Lightning Mapping. *Applied Sciences*. 2021; 11(18):8634.
https://doi.org/10.3390/app11188634

**Chicago/Turabian Style**

Alammari, Ammar, Ammar Ahmed Alkahtani, Mohd Riduan Ahmad, Ahmed Aljanad, Fuad Noman, and Zen Kawasaki.
2021. "Cross-Correlation Wavelet-Domain-Based Particle Swarm Optimization for Lightning Mapping" *Applied Sciences* 11, no. 18: 8634.
https://doi.org/10.3390/app11188634