# Combination of InSAR with a Depression Angle Model for 3D Deformation Monitoring in Mining Areas

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Areas and Data Source

## 3. Methodology

#### 3.1. Basic Theories of InSAR Technology and PIM

_{A}and t

_{B}(t

_{B}> t

_{A}). Then, the phase value at pixel x can be expressed as the following:

_{A}and t

_{B}relative to the initial time; λ is the wavelength; $\Delta {\phi}_{i}^{topo}(x)$ is the residual phase derived from DEM inaccuracy; and $\Delta {\phi}_{i}^{res}(x)$ represents the error phase. After flattening and terrain removal, Formula (2) can be converted to the form as follows:

_{A}to t

_{B}. Formula (3) can be converted into matrix form as follows:

_{max}monitored by InSAR technology is related to the radar wavelength $\lambda $ and SAR image resolution $\mu $. The expression can be written as ${d}_{\mathrm{max}}=\lambda /(2\mu )$. For Sentinel-1, the resolution is 20 m × 20 m after multi-look, and so the corresponding d

_{max}is 1.4 × 10

^{−3}. However, during the practical applications, InSAR is affected by issues, such as spatiotemporal, decoherence, and orbital error, which reduce the maximum deformation gradient that can be monitored by InSAR. Baran et al. [36] have attempted to solve this problem using relationship influence of coherence on the deformation gradient of InSAR technology, which has been presented as follows [36]:

_{max}is the maximum deformation gradient that can be monitored by InSAR and $\gamma $ is the coherence. For Sentinel-1, when the coherence coefficient is 0.3, the D

_{max}= 0, that is, when the coherence coefficient is less than 0.3, InSAR can no longer monitor the effective deformation. If there are n interferometric pairs in the monitoring period, the maximum deformation gradient to be monitored in this period is $n{D}_{\mathrm{max}}$.

^{o}(x) is the subsidence value of the points whose abscissa on the strike main section is x when the strike is fully mined, w

^{o}(y) is the subsidence value of the points whose abscissa on the strike main section is y when the strike is fully mined, and the character behind the semicolon in the right bracket of the equation represents the parameters. For example, t

_{1}in w(y; t

_{1}) represents the corresponding parameters of the mountain up or mountain down boundary; u

^{o}(x) is the horizontal movement value of the points whose abscissa on the strike main section is x when the strike is fully mined. Moreover, u

^{o}(y) is the horizontal movement value of the points whose abscissa on the strike main section is y when the strike is fully mined; r is the main influence radius, and b is the horizontal movement coefficient. Additionally, l and L are the calculating length of the strike and dip, respectively, and $\phi $ is the angle of counterclockwise rotation in the positive direction of the x-axis.

#### 3.2. Research Methods

#### 3.2.1. Depression Angle Model of Surface Displacement Vector

_{1}and b

_{2}to describe the surface subsidence in the uphill and downhill directions, respectively. According to the analysis of Wang Junbao, the model has good applicability when the working face is short and the mining is insufficient [38]. If the working face is long, its accuracy needs further discussion. Assuming that there is a working face with a length and width of ${L}_{1}\times {L}_{2}$, and the main influence radius r is 150 m, the half lengths of the basin are 300 m and 1000 m, respectively (at this time, the strike main section has reached the ultra-full mining). The obtained subsidence basin according to Formula (11) is shown in Figure 5.

_{1}is 1000 m more than 300 m of 2r, it is shown from the PIM that the main section of the strike has achieved super-full mining, and the surface subsidence in this direction has a flat bottom, which is consistent with the actual situation, as shown in Figure 6. Meanwhile, Wang Junbao’s [38] basin model is funnel-shaped, and the maximum subsidence value is in the center of the basin, which is inconsistent with the subsidence basin of the full mining working face with a flat bottom This shows that this model may only apply to the case where the strike and dip main sections are insufficient for mining.

#### 3.2.2. Method for Calculating the Surface Displacement Vector Combined InSAR with the Depression Angle Model

**d**in the LOS direction is the deformation monitored by InSAR. The expression is shown in Formula (18):

## 4. Results

#### 4.1. Application Analysis of the Depression Angle Model

^{2}is 0.9981, and the root mean square error (RMSE) is 2.9°, indicating that the model has good applicability. As the working face is a nearly horizontal coal seam, the maximum subsidence point is located at the center of the basin, and the depression angle of the displacement vector of this point is 90°. The depression angle on both sides of the working face is smaller, revealing that the farther away from the mining area, the closer the surface point moves horizontally.

#### 4.2. InSAR Deformation Monitoring in the Mining Area

**d**is opposite to sign of

**d**.

#### 4.3. 3D Deformation Monitoring Based on PIM and InSAR Combined with Depression Angle Model

**d**(that is, the two vectors have the same direction corresponding to 70 mm sinking). Thus, 70 mm is taken as the critical threshold value of the settlement. When the subsidence value is less than 70 mm, the result of InSAR combined with the depression angle model is taken as the final value; otherwise, the PIM value is taken as the final value. The fusion effect of the two is shown in Figure 16.

#### 4.4. Accuracy Analysis of Vertical Deformation

_{1}to L

_{43}. The monitoring points on the dip line are shown in Figure 14. The goal is to further verify the monitoring accuracy of these methods. L

_{1}~L

_{7}and L

_{35}~L

_{43}, where the InSAR technology can detect effective deformation in this region, are located in the edge area on both sides of the dip line with a settlement of less than 70 mm. L

_{29}~L

_{34}are located in the weighted fusion area between 70 mm and 610 mm where the monitoring results of the InSAR technology and PIM are unreliable. The regions from L

_{8}to L

_{33}are only the predicted results of PIM, which are not part of the analysis. Table 2 presents the error comparisons.

- (1)
- The edge settlement area: The PIM converges too fast, and the predicted result is zero (Figure 20c,d). The difference between the InSAR-monitored results and the measured values is large, especially the InSAR-monitored results in Figure 20d, which shows that the ground surface is uplifted. The settlement after the weighted fusion of L
_{35}~L_{43}in Figure 20a and L_{1}~L_{7}in Figure 20d (the edge area is equivalent to the settlement calculated by InSAR+ depression angle model) is highly consistent with the measured leveling value, with the RMSE equals 10 mm, which has a monitoring accuracy that is much higher than the 65 mm of InSAR and the 61 mm of PIM, and the monitoring accuracy is increased by 85% and 84%, respectively. - (2)
- The weighted fusion area: The predicted result of the PIM increases from almost zero of L34 toward the center of the mining area, and it is gradually consistent with the measured value (Figure 20c). InSAR has begun losing surface deformation information gradually in this area as a result, and the subsidence calculated by InSAR+ depression angle model can only obtain part of the settlement information, which is less than the leveling value (Figure 20a). Through weighted fusion, the monitoring settlement accuracy is improved compared with the PIM, and the RMSE is reduced from 133 mm to 80 mm.
- (3)
- The overall results (edge settlement area and weighted fusion area): The overall monitoring accuracy RMSE of the edge settlement area and weighted fusion area is 42 mm, which is 66% and 44% higher than that of InSAR and PIM, respectively.

## 5. Discussion

## 6. Conclusions

- (1)
- The depression angle model of the displacement vector based on PIM is more consistent with the actual ground movement, which has a high fitting accuracy with the depression angles calculated from the measured values, in which R
^{2}is 0.9981 and the RMSE is 2.9°. This shows that the depression angle model converges gently in the edge region, which is conducive to the inversion of edge subsidence. - (2)
- The subsidence basins monitored by InSAR generally show a certain degree of skewness, and even some regions show surface uplift, which is difficult to reflect in the real surface deformation. The horizontal movement direction field of the mining area surface based on the PIM is consistent with the characteristics of the horizontal displacement of the surface. Compared with the depression angle field of the displacement vector and the surface subsidence information obtained by the method, the 3D deformation information of the whole basin is obtained, which makes up for the shortage of using the PIM or InSAR technology alone.
- (3)
- For the subsidence basin obtained with the proposed method, the subsidence in the edge area is obtained by the depression angle model combined with InSAR, and the RMSE tested by the measured value is 10 mm. The monitoring accuracy is 85% and 84% higher than that of InSAR and PIM, respectively. The RMSE of the weighted fusion area is 80 mm, and the monitoring accuracy improved by 63% and 40%, respectively. The overall RMSE of the two areas is 42 mm, and the monitoring accuracy improved by 66% and 44%, respectively. This shows that the proposed method can obtain more accurate surface subsidence information around the mining area, and the overall subsidence is more consistent with the actual situation.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 11.**Time-series accumulated settlement of InSAR: (

**a**) 12 April 2018, (

**b**) 18 May 2018, (

**c**) 23 June 2018, (

**d**) 29 July 2018, (

**e**) 22 August 2018, (

**f**) 27 September 2018, (

**g**) 18 February 2019, and (

**h**) 13 May 2019.

**Figure 13.**Some deformation field components: (

**a**) horizontal movement direction field, (

**b**) horizontal movement field, and (

**c**) depression angle field.

**Figure 15.**The 3D deformation maps; (

**a**) 3D LOS deformation; and (

**b**) 3D deformation of proposed model.

**Figure 20.**Comparison of monitoring results. (

**A**) Monitoring results of dip main section. (

**a**–

**c**) Comparisons of monitoring results from L

_{29}to L

_{43}by different methods. (

**d**) Comparisons of monitoring results from L

_{1}to L

_{7}.

**Figure 23.**Comparison of deformations under different r. (

**a**) The open-off cut area; (

**b**) the edge area.

Basic Parameters | Sentinel-1A |
---|---|

Orbit | Sun synchronous orbit |

Orbital altitude/km | 693 |

Orbit inclination/(°) | 98.18 |

Angle of incidence | 18.3°~46.8° |

Revisit period/d | 12 |

Imaging mode | IW |

Band | C |

Central incidence angle/(°) | 38.92 |

Cartographic resolution/m | 20 × 20 |

Methods | Edge Subsidence Area (L_{1}~L_{7}, L_{35}~L_{43}) RMSE/mm | Weighted Fusion Area (L_{29}~L_{34}) RMSE/mm | Overall Results (L_{1}~L_{7}, L_{35}~L_{43}, and L_{29}~L_{34}) RMSE/mm |

InSAR | 65 | 214 | 124 |

InSAR+ depression angle model | 10 | 180 | 113 |

PIM | 61 | 133 | 75 |

weighted fusion | 10 | 80 | 42 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Dai, H.; Yan, Y.; Liu, J.; Ren, J.
Combination of InSAR with a Depression Angle Model for 3D Deformation Monitoring in Mining Areas. *Remote Sens.* **2023**, *15*, 1834.
https://doi.org/10.3390/rs15071834

**AMA Style**

Wang Z, Dai H, Yan Y, Liu J, Ren J.
Combination of InSAR with a Depression Angle Model for 3D Deformation Monitoring in Mining Areas. *Remote Sensing*. 2023; 15(7):1834.
https://doi.org/10.3390/rs15071834

**Chicago/Turabian Style**

Wang, Zhihong, Huayang Dai, Yueguan Yan, Jibo Liu, and Jintong Ren.
2023. "Combination of InSAR with a Depression Angle Model for 3D Deformation Monitoring in Mining Areas" *Remote Sensing* 15, no. 7: 1834.
https://doi.org/10.3390/rs15071834