# 3D Lightning Location Method Based on Range Difference Space Projection

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## Abstract

**:**

## 1. Introduction

## 2. Model of TDOA Lightning Detection Sensor Network

^{10}). Thus, we must find some new ways to solve the lightning location problem.

## 3. Geographic Space Projection

#### 3.1. Geographic Space Projection

- Step 1.
- Partition the solution region into a group of grids by the grid step and assign a representative for each of them.
- Step 2.
- Select one representative, calculate its’ TDOA $\mathsf{\Delta}{\tilde{t}}_{i}$ according to Equation (4). Compare the calculated TDOA $\mathsf{\Delta}{\tilde{t}}_{i}$ with the observed TDOA $\mathsf{\Delta}{t}_{i}$; if the absolute difference is less than the designated threshold ${\mathsf{\eta}}_{0}$, $\left|\mathsf{\Delta}{\tilde{t}}_{i}-\mathsf{\Delta}{t}_{i}\right|<{\mathsf{\eta}}_{0}$, the representative gets a vote, if else do nothing.
- Step 3.
- Change the receiver one by one and repeat step 2 until all receivers are processed.
- Step 4.
- Change the representative one by one and repeat steps 2 and 3 until the whole solution domain is processed.
- Step 5.
- Select the representatives whose value of vote is greater than the designated threshold ${\mathsf{\eta}}_{1}$. The coordinates of those representatives are the estimated lightning sources’ position.

#### 3.2. Equivalence Cell

**Property**

**1.**

**,**and ${\mathsf{\alpha}}_{3}$ are not co-planar. Thus, the equivalence cell is the linear transform of a cube, which is a parallel hexahedron and is shown in Figure 5. The eight vertices of the hexahedron can be calculated as:

## 4. RD Space Projection

- Step 1.
- Select three receivers whose triangle covers as many receivers as possible.
- Step 2.
- Calculate the representatives in the RD space according to Equation (16) and solve the non-linear Equation (13) to obtain representatives’ counterparts in the geographic space.
- Step 3.
- Calculate the RDs using the representatives’ counterparts via Equation (4) and perform the geographic space projection discussed in Section 3.1.
- Step 4.
- If the votes of the representatives are greater than the threshold, store the positions of them. Set a new solution domain with some specific size of a 3D cube and center at those positions. Change the grid step to a smaller one. Repeat step 1 to step 4 of the geographic space projection by using the new solution domain and the new grid step until the accuracy of the position result satisfies the system requirement. Then, the representatives’ positions whose votes are greater than the threshold are the estimated lightning sources’ positions.

## 5. Performance Analysis

#### 5.1. Space-Invariant Feature of the RD Space

#### 5.2. Positioning Performance Analysis

- Step 1:
- Select one grid, record the geographic position, and substitute the position in the Equation (4) to calculate the TDOA.
- Step 2:
- Add some gauss noise with standard deviation 100 ns in TDOA calculated in Step 1 to simulate the real observed measurement $\mathsf{\Delta}{t}_{i}$ (according to the GPS measurement error).
- Step 3:
- Use the RD space projection algorithm based on the hierarchical strategy discussed in Section 3.1 and Section 4.
- Step 4:
- Extract the cells whose voting values are greater than the threshold. Store those cells’ position as the estimated lightning source’s positions and the corresponding onset time.
- Step 5:
- Repeat step 2 to step 4 1000 times, calculate the root mean square error (RMSE) in x-y and z between the simulated lightning source position and the estimated source position. The x-y RMSE and the z RMSE were calculated according to$${\mathrm{RMSE}}_{k}^{xy}=\sqrt{\frac{1}{\mathrm{J}}{{\displaystyle \sum}}_{j=1}^{\mathrm{J}}\left({\left({x}_{k}^{j}-{\widehat{x}}_{k}^{j}\right)}^{2}+{\left({y}_{k}^{j}-{\widehat{y}}_{k}^{j}\right)}^{2}\right)},$$$${\mathrm{RMSE}}_{k}^{z}=\sqrt{\frac{1}{\mathrm{J}}{{\displaystyle \sum}}_{j=1}^{\mathrm{J}}{\left({z}_{k}^{j}-{\widehat{z}}_{k}^{j}\right)}^{2}},$$$${\mathrm{RMSE}}_{k}=\sqrt{{\left({\mathrm{RMSE}}_{k}^{xy}\right)}^{2}+{\left({\mathrm{RMSE}}_{k}^{z}\right)}^{2}},$$
- Step 6:
- Repeat step 1 to step 5 $\mathrm{K}$ times, until all surveillance region grids are processed, store all ${\mathrm{RMSE}}_{k}^{xy}$, ${\mathrm{RMSE}}_{k}^{z}$, and ${\mathrm{RMSE}}_{k}$.

^{3}and centered at [100, 100, 6] km. Figure 11 plots the positioning result of the RD space algorithm. From them, we find that the algorithm can position the two lightning sources correctly.

## 6. Conclusions

- Based on the GPS principle, the receiver network can position the lightning flash in 3D space. Different from most lightning location networks, it obtained the position results by optimizing the goodness of fit, and the projection strategy projects the TDOAs into the RD space to accumulate the echoes’ energy and overcomes the problem of how to determine all combinatorial TDOAs due to a common discharge.
- It is difficult to partition the geographic space soundly because of the space-variant feature of the geographic space in face of a vast surveillance region. On the contrary, it is an easy task in the RD space due to its’ space-invariant feature. Thus, the proposed algorithm adopts the grid traverse algorithm to traverse the geographic space by traversing the RD space.
- Increasing the number of receivers can promote the anti-noise performance but cannot improve the positioning precision. The location accuracy is limited by the level of the inherent time uncertainty, the layout, and the size of the receiver network. Due to the measurement time uncertainty, we cannot improve the location accuracy by increasing the receivers or decreasing the grid step further under the fixed size of the receiver network.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 2.**Illustration of the back projection algorithm. A and B are echoes of the lightning flash, F is the false alarm.

**Figure 6.**(

**a**) Geometry of the receiver network with the lightning flash at [0, 0, 6] km; (

**b**) in the RD space, the equivalence cell is a point; (

**c**) in the geographic space, the equivalence cell is almost a point.

**Figure 7.**(

**a**) Geometry of the receiver network with the lightning flash at [100, 120, 6] km; (

**b**) in the RD space, the equivalence cell is a point; (

**c**) in the geographic space, the equivalence cell diffuses significantly.

**Figure 8.**(

**a**) Geometry of the receiver network with the lightning flash at [300, 200, 6] km; (

**b**) in the RD space, the equivalence cell is a point; (

**c**) in the geographic space, the equivalence cell diffuses in a mass region.

**Figure 9.**(

**a**) ${\mathrm{RMSE}}_{k}^{xy}$ under five receivers; (

**b**) ${\mathrm{RMSE}}_{k}^{z}$ under five receivers; (

**c**) ${\mathrm{RMSE}}_{k}^{xy}$ under 10 receivers; (

**d**) ${\mathrm{RMSE}}_{k}^{z}$ under 10 receivers; (

**e**) ${\mathrm{RMSE}}_{k}^{xy}$ under 30 receivers; (

**f**) ${\mathrm{RMSE}}_{k}^{z}$ under 30 receivers. The location errors under the 100 × 100 × 100 m grid step. The triangles indicate the receivers’ locations. The white triangle shows locations at the origin, and the three red triangles form a triangle that covers the others, whose color is green.

**Figure 10.**(

**a**) ${\mathrm{RMSE}}_{k}^{xy}$ under five receivers; (

**b**) ${\mathrm{RMSE}}_{k}^{z}$ under five receivers; (

**c**) ${\mathrm{RMSE}}_{k}^{xy}$ under 10 receivers; (

**d**) ${\mathrm{RMSE}}_{k}^{z}$ under 10 receivers; (

**e**) ${\mathrm{RMSE}}_{k}^{xy}$ under 30 receivers; (

**f**) ${\mathrm{RMSE}}_{k}^{z}$ under 30 receivers. The location errors under the 10 × 10 × 10 m grid step. The white triangle shows locations at the origin, and the three red triangles form a triangle that covers the others, whose color is green.

**Table 1.**Maximum RMSE under different grid step and number of receivers inside the receiver network.

Grid Step(m) | 10,000 × 10,000 × 1000 | 1000 × 1000 × 1000 | 100 × 100 × 100 | 10 × 10 × 10 | ||
---|---|---|---|---|---|---|

RMSE(m) | ||||||

${\mathrm{RMSE}}_{\mathrm{max}}^{xy}$ | 5 | 4382 | 665 | 70 | 38 | |

10 | 3893 | 410 | 63 | 36 | ||

20 | 3067 | 298 | 54 | 32 | ||

30 | 2782 | 156 | 43 | 31 | ||

${\mathrm{RMSE}}_{\mathrm{max}}^{z}$ | 5 | 8674 | 836 | 109 | 39 | |

10 | 7632 | 540 | 92 | 38 | ||

20 | 6756 | 341 | 75 | 33 | ||

30 | 6014 | 263 | 65 | 31 |

**Table 2.**Maximum RMSE under different grid step and number of receivers outside the receiver network.

Grid Step(m) | 10,000 × 10,000 × 1000 | 1000 × 1000 × 1000 | 100 × 100 × 100 | 10 × 10 × 10 | ||
---|---|---|---|---|---|---|

RMSE(m) | ||||||

${\mathrm{RMSE}}_{\mathrm{max}}^{xy}$ | 5 | 7774 | 2431 | 208 | 52 | |

10 | 7725 | 1860 | 152 | 55 | ||

20 | 7177 | 1338 | 135 | 58 | ||

30 | 6958 | 927 | 83 | 56 | ||

${\mathrm{RMSE}}_{\mathrm{max}}^{z}$ | 5 | 16,454 | 13,432 | 4336 | 443 | |

10 | 16,230 | 8972 | 2998 | 487 | ||

20 | 15,886 | 4673 | 1931 | 474 | ||

30 | 15,582 | 2782 | 1817 | 471 |

Searching Level | Searching Grids | Grid Step (km) | Searching Domain (km) | Processing Time (s) |
---|---|---|---|---|

First level | 41 × 41 × 10 | 10 × 10 × 1 | 410 × 410 × 10 | 0.4667 |

Second level | 20 × 20 × 20 | 1 × 1 × 1 | 20 × 20 × 20 | 0.1098 |

Third level | 20 × 20 × 20 | 0.1 × 0.1 × 0.1 | 2 × 2 × 2 | 0.1153 |

Fourth level | 20 × 20 × 20 | 0.01 × 0.01 × 0.01 | 0.2 × 0.2 × 0.2 | 0.1285 |

Total processing time | 0.8203 |

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**MDPI and ACS Style**

Fan, L.; Zhou, C.
3D Lightning Location Method Based on Range Difference Space Projection. *Remote Sens.* **2022**, *14*, 1003.
https://doi.org/10.3390/rs14041003

**AMA Style**

Fan L, Zhou C.
3D Lightning Location Method Based on Range Difference Space Projection. *Remote Sensing*. 2022; 14(4):1003.
https://doi.org/10.3390/rs14041003

**Chicago/Turabian Style**

Fan, Ling, and Changhai Zhou.
2022. "3D Lightning Location Method Based on Range Difference Space Projection" *Remote Sensing* 14, no. 4: 1003.
https://doi.org/10.3390/rs14041003