# A Novel Method of Monitoring Surface Subsidence Law Based on Probability Integral Model Combined with Active and Passive Remote Sensing Data

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

#### 2.2. Probability Integration Method

_{0}is the maximum subsidence of the subsidence basin (where m is the average mining thickness, q is the subsidence factor, α is the coal seam dip), erf is the error function, r is the mining influence radius (where H

_{0}is the average coal seam burial depth, β is the mining influence angle), l is the coal seam strike length, and L is the coal seam dip length. After the coal seam is mined out, the original stress balance in the overburden is damaged. When the length and width of the void area exceed 0.2–0.5 times of the average mining depth, the movement, deformation, and damage of the rock mass around the working face gradually extend to the surface. This will eventually lead to discontinuous deformation of the surface, cave-in areas, landslides, and ground cracks, thereby damaging buildings, affecting the normal operation of roads and railways, affecting the phreatic layer, and inducing geological disasters and ecological damage [33,34,35,36]. The failure mechanism and evolution of overburdened rock mass are shown in Figure 2.

_{0}, which is the starting distance. When mining continues to points B, C, and D, subsidence happens at points 1, 2, and 3 on the ground in front of the working face due to the influence of mining. L

_{i}is the influence distance, W

_{i}is the subsidence curve, and ω

_{i}is the leading influence angle. The surface undergoes deformation in different directions when affected by coal mining. In Figure 2b, the red curve indicates subsidence, the green curve inclination, and the pink curve horizontal movement. The 2S201 working face is 1264 m long and 272 m wide, the subsidence factor is 0.79, the coal seam burial depth is 3.26 m, and the mining influence angle (tanβ) is 2.4.

#### 2.3. Geometry Principle

_{1}and S

_{2}are two different shooting positions of the same sensor, R

_{1}and R

_{2}are the distance from the SAR sensor to the target point, θ is the incident angle, B is the baseline, α is the angle between the baseline and the horizontal direction, and Δr is the LOS deformation of the target point. The interferometric phase of surface deformation is calculated as follows: the interferometric phase ϕ indicates that the same target P is deformed to the point P’ at two different acquisition times of the satellites S

_{1}and S

_{2}. The interferometric phase ϕ is influenced by several factors and is calculated as follows [37]. In this experiment, two SAR images taken before and after the deformation and high precision DEM were used for interferometry. During the process of calculating, DEM was projected to the ground coordinate system of the base image. DEM was shifted into topography phrase ϕ

_{Topo}after the projection, as shown in Equation (3).

_{Topo}: phases influenced by topography; ϕ

_{Flat}: phase due to reference surface factors; ϕ

_{Def}

_{o}: phase due to surface deformation factors along the line of sight (LOS); ϕ

_{Atm}: phase due to atmospheric delay factors; ϕ

_{Noise}: phase due to noise factor. In Equation (3): λ is the wavelength of the radar satellite, R

_{1}are the distances of the satellite when it passes the target P, sinθ

_{0}comes from the sensor of incident angle, ${B}_{\perp}^{0}$ represents the vertical baseline of projection and h represents the ground elevation. In Equation (4), ${B}_{\parallel}^{0}$ is the projection of the base line from the ground point to the satellite line at the ellipsoidal reference point. In Equation (5), Δr represents deformations.

_{1}and s

_{2}are standard deviations of fusion DEM; l is the correlation coefficient of fusion image error; In Equation (7), h

_{r}is the DEM data after fusion while h

_{y1}and h

_{y2}are the two DEM data before fusion.

#### 2.4. UAV Subsidence Monitoring

_{0},y

_{0}) are the principal point coordinates of the image; f is the image principal distance; (X,Y,Z) are the ground coordinates of the object;(X

_{S},Y

_{S},Z

_{S})are the coordinates of the camera station in the ground coordinate system; (φ,ω,κ) are the outer azimuth angles of the image; (a

_{i},b

_{i},c

_{i}) are the directional cosine represented by the outer azimuth angles. The flow chart for the encryption of aerial triangulation of UAV images is shown in Figure 5.

#### 2.5. Data Fusion Method

_{0}) with x

_{0}as the center is estimated. The linear estimator can be used for estimation. The equation is as follows:

_{α}is the weight coefficient of Z

_{α}representing the weight of α(1, 2, 3, …, n) deformation values Z

_{α}to the estimated ${Z}_{K}^{*}$. In the interpolation calculation using the ordinary kriging method, the kriging equation set was listed to solve for the full coefficients of λ

_{α}. Secondly, the kriging variance was listed. The equation is as follows:

_{i}) is the monitoring value at subsidence point i; Z(x

_{i}+ h) is the monitoring value with a distance of h away from Z(x

_{i}) spatially; N(h) denotes the number of sample data point pairs with a spatial separation distance of h. Using a spherical model, 15 search points are set, influence range is set to 20 m, search radius is set to 20 m, and empirical values are taken as other parameters. The flow chart of the fusion method is shown in Figure 6.

## 3. Results

#### 3.1. Data Fusion Result

#### 3.2. UAV and InSAR Result

## 4. Discussion

#### 4.1. Comparative Analysis of Data from InSAR, UAV, and GNSS

#### 4.2. Analysis of Observation Method and Subsidence Law

## 5. 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.**Location of the study area in WangJiata, Inner Mongolia Autonomous Region, China and photographs of building crack, collapse pit and ground crack caused by coal mining activity; 2S201 is the monitoring working surface.

**Figure 2.**(

**a**) Leading impact during the advancing process of the working face; (

**b**) Relationship between surface movement basin and main section.

**Figure 4.**(

**a**) ALOS DEM; (

**b**) Fused DEM; 2S201 and 2S202 are working faces, the data is of production.

**Figure 7.**Example of fusion methods. (

**a**) Interpolation and superposition subsidence map by SBASInSAR, DInSAR and UAV; (

**b**) subsidence map by the fusion method; The date marked in red is the mining time; 2S201 and 2S202 are working faces; the data is of production.

**Figure 8.**(

**a**–

**c**) are respectively the settlement maps of UAV time series; 2S201 and 2S202 are working faces; the data is of production.

**Figure 11.**(

**a**–

**c**) are respectively the accumulated settlement maps of UAV time series; 2S201 and 2S202 are working faces; the data is of production.

**Figure 12.**Dynamic cumulative curve of the 2S201 working face, locations of working face during advancement in strike direction, and 3D subsidence map. (

**a**) is the time-series cumulative subsidence curve of 2S201 working face; In (

**b**), the red box indicates the working face, strike direction means the north, dip direction means the east-west, The corresponding scale is the coordinate value.

**Figure 13.**Cumulative subsidence obtained by three methods: (

**a**) cumulative subsidence by InSAR; (

**b**) cumulative subsidence by UAV; (

**c**) cumulative subsidence by fusion method; Z1 to Z26 are GNSS monitoring point; 2S201 and 2S202 are working faces; the data is of production.

**Figure 14.**Time series subsidence rate graph. The black vertical axis represents the time node mining distance; the red vertical axis represents the time node average advancement rate; and the blue axis represents the time node average subsidence rate.

No. | UAV | Camera | Course Overlap% | Lateral Overlap% | Row Height | Collection Date |
---|---|---|---|---|---|---|

1 | Trimble UX5 | SONY A5100 | 80 | 80 | 23 | 9 June 18 |

2 | 4 September 18 | |||||

3 | 16 October 18 | |||||

4 | 16 April 19 |

No. | Product | Beam Model | Polarization | Resolution/(m) | Acquisition Date | Pixel Center | Mean Incident Angle (°) |
---|---|---|---|---|---|---|---|

(Rng × Az) | Lat-Lng (°) | ||||||

1 | SLC | Wide Multi-look Fine | HH | 2.6 × 2.4 | 9 June 18 | 39.5841–110.5944 | 35.2230 |

2 | 27 July 18 | 39.5852–110.5950 | 35.2232 | ||||

3 | 20 August 18 | 39.5851–110.5961 | 35.2224 | ||||

4 | 24 November 18 | 39.591–110.5977 | 35.2128 | ||||

5 | 11 January 19 | 39.5892–110.5969 | 35.2129 | ||||

6 | 4 February 19 | 39.5627–110.5903 | 35.2124 | ||||

7 | 28 February 19 | 39.5729–110.5952 | 35.2165 | ||||

8 | 24 March 19 | 39.5899–110.5995 | 35.2207 | ||||

9 | 17 April 19 | 39.5880–110.5955 | 35.2223 |

No. | InSAR (m) | UAV (m) | Fusion (m) | GNSS (m) | InSAR/GNSS (m) | UAV/GNSS (m) | Fusion/GNSS (m) |
---|---|---|---|---|---|---|---|

1 | −0.051 | −0.042 | −0.187 | −0.174 | −0.123 | −0.132 | 0.013 |

2 | −0.062 | −1.158 | −1.409 | −1.304 | −1.242 | −0.146 | 0.105 |

3 | −0.106 | −1.297 | −1.495 | −1.388 | −1.282 | −0.091 | 0.107 |

4 | −0.152 | −2.52 | −2.628 | −2.542 | −2.39 | −0.022 | 0.086 |

5 | −0.098 | −0.096 | −0.154 | −0.111 | −0.013 | −0.015 | 0.043 |

6 | −0.113 | −0.475 | −0.561 | −0.507 | −0.394 | −0.032 | 0.054 |

7 | −0.034 | −0.091 | −0.156 | −0.077 | −0.043 | 0.014 | 0.079 |

8 | −0.131 | −0.01 | −0.284 | −0.164 | −0.033 | −0.154 | 0.12 |

9 | −0.178 | −2.131 | −2.529 | −2.323 | −2.145 | −0.192 | 0.206 |

10 | −0.150 | −2.646 | −2.724 | −2.668 | −2.518 | −0.022 | 0.056 |

11 | −0.047 | −1.817 | −1.925 | −1.857 | −1.81 | −0.04 | 0.068 |

12 | −0.087 | −1.235 | −1.291 | −1.146 | −1.059 | 0.089 | 0.145 |

Medium Error | 1.426 | 0.099 | 0.103 |

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**MDPI and ACS Style**

Wang, R.; Wu, K.; He, Q.; He, Y.; Gu, Y.; Wu, S.
A Novel Method of Monitoring Surface Subsidence Law Based on Probability Integral Model Combined with Active and Passive Remote Sensing Data. *Remote Sens.* **2022**, *14*, 299.
https://doi.org/10.3390/rs14020299

**AMA Style**

Wang R, Wu K, He Q, He Y, Gu Y, Wu S.
A Novel Method of Monitoring Surface Subsidence Law Based on Probability Integral Model Combined with Active and Passive Remote Sensing Data. *Remote Sensing*. 2022; 14(2):299.
https://doi.org/10.3390/rs14020299

**Chicago/Turabian Style**

Wang, Rui, Kan Wu, Qimin He, Yibo He, Yuanyuan Gu, and Shuang Wu.
2022. "A Novel Method of Monitoring Surface Subsidence Law Based on Probability Integral Model Combined with Active and Passive Remote Sensing Data" *Remote Sensing* 14, no. 2: 299.
https://doi.org/10.3390/rs14020299