Method of Predicting Dynamic Deformation of Mining Areas Based on Synthetic Aperture Radar Interferometry (InSAR) Time Series Boltzmann Function

Abstract

The movement and deformation of rock strata and the ground surface is a dynamic deformation process that occurs as underground mining progresses. Therefore, the dynamic prediction of three-dimensional surface deformation caused by underground mining is of great significance for assessing potential geological disasters. Synthetic aperture radar interferometry (InSAR) has been introduced into the field of mine deformation monitoring as a new mapping technology, but it is affected by many factors, and it cannot monitor the surface deformation value over the entire mining period, making it impossible to accurately predict the spatiotemporal evolution characteristics of the surface. To overcome this limitation, we propose a new dynamic prediction method (InSAR-DIB) based on a combination of InSAR and an improved Boltzmann (IB) function model. Theoretically, the InSAR-DIB model can use information on small dynamic deformation during mining to obtain surface prediction parameters and further realize a dynamic prediction of the surface. The method was applied to the 1613 (1) working face in the Huainan mining area. The results showed that the estimated mean error of the predicted surface deformation during mining was between 80.2 and 112.5 mm, and the estimated accuracy met the requirements for mining subsidence monitoring. The relevant research results are of great significance, and they support expanding the application of InSAR in mining areas with large deformation gradients.

1. Introduction

Surface subsidence caused by underground coal mining is a global problem [1,2,3,4,5]. After coal mining, the original rock stress of the overlying rock strata is redistributed and eventually reaches a new balance through bending, collapse, and spalling, which eventually leads to a series of geological disasters (surface subsidence, damage to buildings, landslides, etc.). Therefore, predicting the surface deformation caused by underground mining in advance can not only help evaluate damage to surface buildings but also provide data support for improving the upper limits of underground coal mining [6,7,8,9].

In the traditional mining subsidence research process, two monitoring lines are usually established on the surface of the first mining face in the mining area. Traditional surveying and mapping technologies (such as geometric leveling and GPS-RTK technology) are used to obtain the three-dimensional deformation of the surface. The parameters of the prediction model are inverted and combined with underground mining information and prediction models. Then, the surface deformation likely to be caused by subsequent mining is predicted based on the obtained parameters [10,11,12]. The disadvantages of this approach are high monitoring costs, labor intensity, and the difficulty of meeting the requirements of high temporal and spatial monitoring resolution in mining areas. InSAR technology has the advantages of low cost, short monitoring cycles, and large coverage. It provides an opportunity to overcome the shortcomings of traditional monitoring technology. However, due to drastic surface deformation in mining areas, it is prone to decoherence and struggles to monitor large deformation areas [13,14,15,16].

Underground mining usually involves inadequate, sufficient, and super-sufficient dynamic mining processes, so the surface deformation it causes is a complex, gradual, and nonlinear spatiotemporal evolution process. Most existing prediction models were established under the assumption of stable surface settlement conditions. Since surface stability requires a certain amount of time to establish, if the prediction parameters under stable settlement conditions are directly used to predict the deformation during mining, the surface deformation value will be overestimated to a certain extent. In order to overcome these defects, the most common solution at present is to multiply a time function on the basis of the prediction model to correct it. Experts and scholars have proposed different time functions to express the surface deformation variable. One is the Knothe time function and its improved model [17,18,19,20,21] and the other is a multi-parameter function such as the Sroka-Schober, B-normal, Weibull, EnKF methods, Hook, Gompertz, arc tangent, and logistic [22,23,24,25,26,27] functions. However, some of the above improved functions still have shortcomings, such as too many parameters and parameter selection difficulties, which reduce the model’s estimated accuracy to a certain extent. Based on this, relevant scholars have integrated the probability integral method prediction model (PIM) with the dynamic prediction function and combined it with InSAR-measured data [28,29,30,31,32] to achieve the prediction of dynamic deformation over a large area of a mining area, which has greatly expanded the application field of InSAR.

However, the above methods have the following two main problems. First, the accuracy of a dynamic prediction model based on InSAR technology depends on the accuracy of InSAR monitoring and the coverage of monitoring points. However, the large gradient deformation in the center of the mining basin often causes the InSAR monitoring technology to lose coherence, resulting in an inability to obtain effective monitoring points or an underestimation of the deformation. Second, the traditional dynamic prediction model achieves the purpose of dynamic prediction by multiplying the prediction model by a time function. The accuracy of this method depends on the accuracy of the time function and its parameter selection. Therefore, it is difficult to make a dynamic prediction of the surface without prior deformation data from the mining area. In view of the above problems, a surface dynamic prediction model that takes into account changes in surface prediction parameters was constructed, and it was combined with effective InSAR data at the edges, thereby establishing a dynamic prediction method for the three-dimensional deformation of the surface of a mining area based on InSAR data describing the edges of the surface of the moving basin.

The remainder of this article is arranged as follows. The second part mainly introduces the construction method of the InSAR-DIB prediction model. The third part uses simulation experiments to verify the feasibility of the model. The fourth part conducts experimental verification in the Huainan mining area, and the fifth section discusses it. The sixth section draws conclusions.

2. Methods

2.1. IB Prediction Model

The rock movement and surface deformation of the thick loose layer in a mining area after coal seam mining differ from those observed in general geological mining conditions. This is mainly reflected in the following aspects: First, the maximum subsidence value of the working face is larger under the conditions of full or over-full mining in strike and dip. Second, the surface movement changes dramatically, the recession period is longer, the boundary converges slowly, and the range of surface movement is larger [33,34]. The accuracy of the currently commonly used PIM is generally poor at the edges. Based on this, scholars in the field have proposed introducing the Boltzmann function and the improved Boltzmann function prediction model into the field of mining subsidence, and these functions have been applied effectively in the Huainan mining area [35,36]. The main principles of the improved Boltzmann function prediction model are described in this manuscript.

Consider the three-dimensional situation shown in Figure 1. The horizontal projections of the coal seam coordinate systems tO₁s and xOy coincide. If we consider a unit B(s,t) with a width of ds and a length of dt, a subsidence prediction formula for the main section of the surface movement basin can be determined by combining the Boltzmann prediction models of two different important influence radii in a certain proportion:

$$W(x)=W_0\left({\displaystyle \frac{1-P}{\exp (-x/R_1)+1}}+{\displaystyle \frac{P}{\exp (-x/R_2)+1}}\right)$$

(1)

where the maximum surface subsidence value W₀ is

$$W_0=mq\cos \alpha$$

(2)

R₁ and R₂ are two important influencing radii, P is the proportional coefficient, q is the subsidence factor, m is the thickness of the coal seam, and α is the dip angle of the coal seam. According to the derivation principle of the PIM, it can be deduced that the subsidence value dW(x,y) caused by any point A(x,y) on the surface is

$$dW\left(x,y\right)=W_e^1\left(x,y\right)\cdot (1-P)+W_e^2\left(x,y\right)\cdot P$$

(3)

where

$$W_e^1\left(x,y\right)={\displaystyle \frac{1}{R_1^2}}{\displaystyle \frac{\exp \left(-(x-s)/R_1-(y-t)/R_1\right)}{{\left(1+\exp \left(-(x-s)/R_1\right)\right)}^2{\left(1+\exp \left(-(y-t)/R_1\right)\right)}^2}}$$

(4)

$$W_e^2\left(x,y\right)={\displaystyle \frac{1}{R_2^2}}{\displaystyle \frac{\exp \left(-(x-s)/R_2-(y-t)/R_2\right)}{{\left(1+\exp \left(-(x-s)/R_2\right)\right)}^2{\left(1+\exp \left(-(y-t)/R_2\right)\right)}^2}}$$

(5)

W_e¹ and W_e² are the ground subsidence values caused by two different important influencing radii.

As shown in Figure 1, if the mining range is O₁CDE, the length of O₁C is D₃, and the length of CD is D₁, then the subsidence value of any point A(x,y) on the surface caused by the entire mining effort is

$$\begin{array}{l}W(x,y)=W_0{\displaystyle {\int }_0^{D_3}{\displaystyle {\int }_0^{D_1}dW\left(x,y\right)}}dtds=W_0{\displaystyle {\int }_0^{D_3}{\displaystyle {\int }_0^{D_1}\frac{1}{R_1^2}\frac{\exp \left(-(x-s)/R_1-(y-t)/R_1\right)(1-P)}{{\left(1+\exp \left(-(x-s)/R_1\right)\right)}^2{\left(1+\exp \left(-(y-t)/R_1\right)\right)}^2}}}+\\ {\displaystyle \frac{1}{R_2^2}}{\displaystyle \frac{\exp \left(-(x-s)/R_2-(y-t)/R_2\right)P}{{\left(1+\exp \left(-(x-s)/R_2\right)\right)}^2{\left(1+\exp \left(-(y-t)/R_2\right)\right)}^2}}dtds={\displaystyle \frac{1}{W_0}}{W}^0\left(x\right)W^0\left(y\right)\end{array}$$

(6)

where

$$W^0\left(x\right)=W_0\left(\begin{array}{l}\left(1-P\right)\left({\displaystyle \frac{1}{\exp (-x/R_1)+1}}-{\displaystyle \frac{1}{\exp (-(x-l_3)/R_1)+1}}\right)+\\ P\left({\displaystyle \frac{1}{\exp (-x/R_2)+1}}-{\displaystyle \frac{1}{\exp (-(x-l_3)/R_2)+1}}\right)\end{array}\right)$$

(7)

$$W^0\left(y\right)=W_0\left(\begin{array}{l}\left(1-P\right)\left({\displaystyle \frac{1}{\exp (-y/R_1)+1}}-{\displaystyle \frac{1}{\exp (-(y-l_1)/R_1)+1}}\right)+\\ P\left({\displaystyle \frac{1}{\exp (-y/R_2)+1}}-{\displaystyle \frac{1}{\exp (-(y-l_1)/R_2)+1}}\right)\end{array}\right)$$

(8)

where l₃ is the calculated length of the working face strike, l₃ = D₃ − S₃ − S₄; l₁ is the calculated length of the working face inclination, l₁ = (D₁ − S₁ − S₂) sin(θ₀ + α)/sin(θ₀); S₁, S₂, S₃, and S₄ are the offset distances of the downhill turning point, the uphill turning point, the open-mining turning point, and the stop-mining line turning point, respectively; and θ₀ is the mining impact propagation angle.

According to the relationship between subsidence and horizontal movement, the horizontal movement value of the main section of the surface movement basin is:

$$U(x)=b{W}_0{\displaystyle \frac{\exp (-x/R_1)}{{\left(1+\exp (-x/R_1)\right)}^2}}\left(1-P\right)+b{W}_0{\displaystyle \frac{\exp (-x/R_2)}{{\left(1+\exp (-x/R_2)\right)}^2}}P$$

(9)

In the above formula, b is the horizontal movement coefficient.

The horizontal movement expression of any point along direction φ can then be obtained as follows:

$$U\left(x,y,\phi \right)={\displaystyle \frac{1}{W_0}}\left[U^0\left(x\right)W^0\left(y\right)\cos \phi +W^0\left(x\right)U^0\left(y\right)\sin \phi \right]$$

(10)

where

$$U^0\left(x\right)=U\left(x\right)-U\left(x-l_3\right)$$

(11)

$$U^0\left(y\right)=\left[U\left(y\right)+W\left(y\right)\cot {\theta }_0\right]-\left[U\left(y-l_1\right)+W\left(y-l_1\right)\cot {\theta }_0\right]$$

(12)

The inclination i(x,y,φ), curvature k(x,y,φ), and horizontal deformation ℇ(x,y,φ) of the surface point A(x,y) along the φ direction can be obtained using similar methods. No further derivation is shown here.

In the model, q, θ₀, S₁, S₂, S₃, S₄, and b have the same meanings as those defined via the PIM, and the parameters are similar. In the new model, R₁ and R₂ are important influencing radii, with the main influence angle tangents being tanβ₁ = H₀/4.13R₁ and tanβ₂ = H₀/4.13R₂. H₀ is the coal seam mining depth. Therefore, there are ten parameters in the new model: q, P, b, θ, S₁, S₂, S₃, S₄, tanβ₁, and tanβ₂.

2.2. Construction of the DIB Prediction Model

Underground working face mining usually involves dynamic processes such as insufficient mining, sufficient mining, and even super-full mining. The traditional dynamic prediction method uses a method of multiplying the dynamic time function to perform dynamic prediction without considering changes in the prediction parameters, which is the reason for its poor prediction accuracy. Through research, it has been found that under insufficient mining conditions, the subsidence factor shows an obvious S-shaped growth form. A change in the mining degree and the inflection point offset causes obvious changes, but the change pattern is not obvious, and the other parameters do not change much.

Based on the actual situation, the following relationship is proposed to describe the relationship between the subsidence rate q, and the subsidence factor q under insufficient mining conditions:

$$q^{\prime }=f_c\left(D_3/H_0\right)·q$$

(13)

$$f_c\left(D_3/H\right)=1-\exp \left[-C_0\left(D_3/H_0\right)\right]$$

(14)

where f_c (D₃/H₀) is the correction function of the subsidence factor and C₀ is the model parameter related to the geological mining conditions.

After constructing a dynamic function model of the subsidence factor in the IB prediction model that changes with the advancement of the underground working face, (that is, Formula (14)), it can be used to replace all q in the IB model. The subsidence factor q is only related to the maximum subsidence value under full mining conditions. Therefore, after replacing q in the IB prediction model during mining with q^, in Formula (2), the maximum subsidence value W₀′ is

$${W^{\prime }}_0=m·q^{\prime }\cos \alpha =m·f_c\left(D_3/H_0\right)·q\cos \alpha$$

(15)

Combined with the prediction model constructed above, all W₀ in the three-dimensional deformation expression in the prediction model are replaced by the above formulas so as to construct a dynamic prediction model based on the change in the subsidence rate. It is known from the above that compared to the IB prediction model, the DIB prediction model has one more parameter, C₀, so the model has a total of 11 parameters: P_DIB = [q, P, b, θ₀, S₁, S₂, S₃, S₄, tanβ₁, tanβ₂, C₀].

2.3. InSAR-DIB Dynamic Prediction Model and Parameter Solution

Before using the DIB prediction model for surface prediction, it is necessary to determine its parameters. Generally, the surface deformation values obtained via traditional geodetic methods (leveling and GPS-RTK measurement) are used to invert the prediction parameters. This method not only requires significant effort and material resources but also has low efficiency overall. InSAR has attracted widespread attention from mining surveying scholars due to its high precision, all-day surface measurement, and low cost (especially with the launch of free data-source radar satellites). This paper combines DIB with InSAR technology to develop a surface prediction model suitable for Huainan’s thick loose layer mining areas, referred to as “InSAR-DIB.” The idea of the model is described below.

When the SAR satellite passes over a surface moving basin, it is assumed that the cumulative LOS value of any pixel i in the moving basin is r_iLOS (which can be obtained by accumulating the deformation in the LOS direction in each period). The subsidence value W_i, the horizontal movement U_iSN in the north direction, and the horizontal movement U_iEW in the east direction of any pixel i are predicted based on the IB prediction model and parameter system. According to the SAR’s imaging geometry principle, the predicted LOS value r^,_iLOS of any pixel i along the LOS direction can be expressed as

$$r_{iLOS}^{\prime }=-U_{iSN}\sin {\theta }_i\cos ({\alpha }_i-{\displaystyle \frac{3\pi }{2}})-U_{iEW}\sin {\theta }_i\sin ({\alpha }_i-{\displaystyle \frac{3\pi }{2}})+W_i\cos {\theta }_i$$

(16)

where θ_i is the incidence angle of the radar and α_i is the heading angle of the satellites.

Among these, W_i, U_iSN, and U_iEW can be calculated according to the mining subsidence prediction model (taking the rectangular shape of the working face as an example):

$$\left\{\begin{array}{l}{W}_i=W(x,y)={\displaystyle \frac{1}{W_0}}{W}^0\left(x\right)W^0\left(y\right)\hfill \\ U_{iSN}={\displaystyle \frac{1}{W_0}}\left[U^0\left(x\right)W^0\left(y\right)\cos {\phi }_{SN}+W^0\left(x\right)U^0\left(y\right)\sin {\phi }_{SN}\right]\hfill \\ U_{iEW}={\displaystyle \frac{1}{W_0}}\left[U^0\left(x\right)W^0\left(y\right)\cos {\phi }_{EW}+W^0\left(x\right)U^0\left(y\right)\sin {\phi }_{EW}\right]\hfill \end{array}\right.$$

(17)

where φ_SN is the angle between the working face direction and the north direction in a counterclockwise direction and φ_EW is the angle between the working face direction and the east–west direction in a counterclockwise direction.

The estimated LOS moving residual v_i of any pixel in the SAR image is defined as

$$v_i=r_{iLOS}^{\prime }-r_{iLOS}$$

(18)

The parameters of the prediction model can be obtained through a multi-population genetic algorithm (MPGA) [36], and the corresponding parameter cost function is

$$F(X)={\displaystyle \sum _{i=1}^n{v}_i}$$

(19)

The essence of an MPGA parameter search is to find a set of predicted parameters X so that the absolute sum of the differences between the predicted and measured values of the LOS movement of all of the pixels involved in the parameter search is minimized, that is, F(X) = min.

However, due to the inherent defects of InSAR technology and the characteristics of mining subsidence movement and deformation, most of the data are incoherent. InSAR technology cannot monitor the cumulative surface deformation value, but instead monitors the deformation between two periods (the cumulative settlement from the initial sinking moment is unknown), and it can often only obtain the settlement value at the edges. From the description in the previous section, it can be seen that the DIB prediction model constructed in this paper can better fit the settlement information of the edges, so the use of edge deformation information can also fit well in the middle of the basin.

Assume that we have obtained the surface deformation information from time t_j to t_j+1, and assuming that the characteristic parameters of the goaf at time t_j are B_gtj = [m, H₀, α, D_3tj, D₁], the characteristic parameters of the goaf at time t_j+1 are B_gtj+1 = [m, H₀, α, D_3tj+1, D₁], and the LOS between the two periods is Δd_LOS. From the previous section, we can see that during the mining process, the DIB model can be used for dynamic prediction at any time, that is, the parameters of the DIB model at times t_j and t_j+1 are unchanged.

The cumulative deformation value expected from the LOS direction of pixel i, representing the surface point t_j at any time after mining, can be expressed as follows:

$$r_{iLOS{t}_j}^{\prime }=-U_{iSN{t}_j}\sin {\theta }_i\cos ({\alpha }_i-{\displaystyle \frac{3\pi }{2}})-U_{iEW{t}_j}\sin {\theta }_i\sin ({\alpha }_i-{\displaystyle \frac{3\pi }{2}})+W_{i{t}_j}\cos {\theta }_i$$

(20)

Then, the cumulative deformation value expected from the LOS direction of pixel i, representing the surface point t_j+1 at any time after mining, can be expressed as follows:

$$r_{iLOS{t}_{j+1}}^{\prime }=-U_{iSN{t}_{j+1}}\sin {\theta }_i\cos ({\alpha }_i-{\displaystyle \frac{3\pi }{2}})-U_{iEW{t}_{j+1}}\sin {\theta }_i\sin ({\alpha }_i-{\displaystyle \frac{3\pi }{2}})+W_{i{t}_{j+1}}\cos {\theta }_i$$

(21)

Δd′_LOS predicted between the two periods can be expressed as follows:

$$\Delta {d^{\prime }}_{LOS}=r_{iLOS{t}_{j+1}}^{\prime }-r_{iLOS{t}_j}^{\prime }$$

(22)

Therefore, the parameter cost function of the DIB prediction model can be constructed as follows:

$$F\left(X\right)={\displaystyle \sum \left(\Delta {d^{\prime }}_{LOS}-\Delta d_{LOS}\right)}$$

(23)

By using a multi-population genetic algorithm, a set of X can be obtained by minimizing the above formula, and the obtained X value can be used for the next step of the dynamic prediction.

Based on the above ideas, a technical roadmap for InSAR-DIB parameter inversion is proposed, as shown in Figure 2. A parameter-finding program was compiled based on a multi-population genetic algorithm.

3. Simulation Experiment

3.1. Simulated Working Face Overview

The simulated working face inclination angle was 5°, the mining height was 3 m, the mining depth was 500 m, the average advancement speed of the working face was 6 m/day, the designed working face strike length was 1000 m, the inclined width was 200 m, and the dip azimuth was 0°. The engineering experiment shows that the characteristics of the mining subsidence in Huainan conform to the DIB prediction model. The predicted parameters of mining subsidence in the simulated working face were: q = 0.9, P = 0.4, b = 0.3, θ = 89, S₁ = S₂ = S₃ = S₄ = 10, tanβ₁ = 0.6, tanβ₂ = 2.2, and C₀ = 2.8. The simulated SAR satellite revisit period was 12 days, the radar incident angle was θ_i = 44.494, the spatial resolution was 10 × 10 m, the radar wavelength was 31 mm, and the satellite’s line of sight azimuth was α_i = 350.596°. This simulation involved four excavation processes at the working face, simulating the cumulative deformation values of the surface LOS when the working face advanced from 12 to 36 days, 36 to 60 days, 60 to 84 days, and 84 to 108 days, respectively. The results are shown in Figure 3.

3.2. Experimental Design and Results Analysis

In order to verify the feasibility of the InSAR-DIB dynamic prediction model based on the edge correction model, the simulation experiment first used the MPGA algorithm to invert the edge settlement information of the two InSAR monitoring pairs obtained from 12 to 36 days and 36 to 60 days to obtain the parameters of the DIB prediction model. Then, the obtained DIB prediction model parameters were used to make a prediction comparison with the LOS deformation values obtained from 84 and 108 days to verify the accuracy of the prediction model.

According to the LOS deformation values obtained from the simulation, according to the layout of conventional surface movement deformation observation stations (strike lines are numbered L1~L181, and dip observation lines are numbered S1~S101), the edge areas with LOS deformation values of less than 60 mm at intervals of 12~36 and 36~60 days were extracted (12~36 days (L1~L21, L68~L97), 36~60 days (L1~L31, L80~L97)). According to the principles of the DIB prediction model, the MPGA algorithm was used to obtain the DIB prediction model parameters through inversion. The comparison between the inverted and design parameters is shown in

Table 1. Comparison of the simulated and measured parameters.

Parameter	q	P	b	θ	S1	S2	S3	S4	tanβ1	tanβ2	C0
Design value	0.9	0.4	0.3	89	10	10	10	10	1.1	4.1	3
Fitted value	0.883	0.396	0.296	87.29	10.07	9.247	10.32	10.52	1.099	3.933	3.080
Absolute error	0.017	0.004	0.004	1.707	0.078	0.753	0.320	0.525	0.001	0.167	0.080
Relative error	1.936	0.968	1.301	1.918	0.784	7.533	3.203	5.251	0.135	4.080	2.671

, and the fitting effect diagram is shown in Figure 4.

It can be seen from Table 1 that the absolute errors of the 11 parameters range from 0.004 to 1.707, of which the largest is the absolute error corresponding to the mining influence angle, which is 1.707. However, its sensitivity to the results was relatively low, and the relative errors corresponding to the remaining parameters were between 0.004 and 0.753. The relative errors corresponding to the 11 parameters were between 0.135 and 7.533%, of which the relative errors corresponding to the four inflection point offsets were relatively large. In the conventional parameter determination process, multiple groups of different inflection point offsets can often obtain similar results, so the combination of inflection point offsets has little effect on the overall parameter determination. From the comparison chart of the measured values and the fitting graph (Figure 4), it can be seen that the LOS edge fitting values and the measured values had basically the same trend. The mean error of the fitting from the 12th to the 36th day was 0.026 mm, and the absolute error of the fitting was between 0 and 0.086 mm. The mean error of the fitting from the 36th to the 60th day was 0.316 mm, and the absolute error of the fitting was between 0.05 and 1.02 mm. Good fitting effects were achieved.

According to the obtained DIB prediction parameters, the surface LOS deformation, subsidence, and horizontal movement on the 84th and 108th days were predicted, and the deformation values of the strike and dip lines were extracted. A comparison between the predicted values and the measured values is shown in Figure 5 and Figure 6.

From Figure 5 and Figure 6, it can be predicted that the LOS deformation and the measured LOS deformation will be in good agreement. The statistics show that the mean error of the LOS deformation (strike line + dip line) on the 84th day was 1.58 mm, with an absolute error of 0 to 5.12 mm. The mean error of the subsidence fitting was 10.35 mm, with an absolute error of 0 to 24.24 mm. The mean error of the east–west horizontal movement fitting was 6.18 mm, with an absolute error of 0 to 27.77 mm. The mean error of the north–south horizontal movement fitting was 1.63 mm, with an absolute error of 0 to 3.82 mm. The mean error of the LOS deformation (strike line + dip line) fitting on the 108th day was 8.77 mm, and the absolute error was between 0 and 30.42 mm. The mean error of the subsidence fitting was 10.76 mm, and the absolute error was between 0 and 24.46 mm. The mean error of the east–west horizontal movement fitting was 6.17 mm, and the absolute error was between 0 and 27.77 mm. The mean error of the north–south horizontal movement fitting was 4.38 mm, and the absolute error was between 0 and 16.36 mm.

Based on the above analysis, the InSAR-DIB prediction model based on the edge correction model established in this paper can be used to predict the three-dimensional deformation of mining subsidence with high predictive accuracy, thus verifying the feasibility of the model.

4. Project Cases

4.1. Overview of the Study Area

The 1613 (1) working face of the Huainan Guqiao Mine adopts the full caving method, with an average mining height of 2.9 m, a working face strike length of 1528 m, a dip width of 251 m, an average mining rate of 5.56 m/d, an average coal seam dip of 3°, a nearly horizontal coal seam, an average mining depth of 668 m, and an average loose layer thickness of 420 m. The layout of the monitoring points of the working face is shown in Figure 7. The surface movement observation time of the 1613 (1) working face is from 2 May 2017 to 7 January 2019. The area above the working face is mainly farmland.

4.2. Data Processing and Results Analysis

Radar imaging of the study area uses Sentinel 1A data, which cover the entire study area. Considering the surface vegetation above the working surface and the large deformation gradient leading to decoherence, a total of six images with a time baseline of 12 days from 4 November 2017 to 5 January 2018 were selected for the experiment. The main parameters of Sentinel imaging are shown in

Table 2. Main parameters of the test area images.

Interference Pair	Master Image	Auxiliary Image	Spatial Baseline/m	Incidence Angle/°	Orbit Type	Path
1	4 November 2017	16 November 2017	−22.290	39.617	Ascending	142/142
2	16 November 2017	28 November 2017	−66.940	39.616	Ascending	142/142
3	10 December 2017	22 December 2017	88.316	39.615	Ascending	142/142
4	22 December 2017	3 January 2018	15.096	39.615	Ascending	142/142

.

Using four pairs of primary and secondary images, differential interferometry was performed using D-InSAR technology to obtain the settlement of each interference pair. Then, the settlement from 4 November 2017 to 28 November 2017 and from 10 December 2017 to 3 January 2018 was obtained by superposition of the deformation variables, as shown in Figure 8 and Figure 9. According to the mining progress map of the 1613 (1) working face, the corresponding geometric parameters of the goaf could be obtained when the impact was acquired, as shown in

Table 3. Geometric characteristic parameters of the goaf.

Parameters	m (m)	H0 (m)	α (°)	D3 (m)	D1 (m)
4 November 2017	2.9	668	3	1230	251
28 November 2017	2.9	668	3	1363	251
10 December 2017	2.9	668	3	1427	251
3 January 2018	2.9	668	3	1528	251

.

The subsidence value information at the edges was extracted as the parameter data. The subsidence values between 4 November 2017 and 28 November 2017 were derived from points Q1~Q27, and the subsidence values between 10 December 2017 and 3 January 2018 were derived from points T1~T26 (see Figure 8 and Figure 9). The dynamic prediction parameters were q = 0.91, P = 0.75, b = 0.31, θ = 86.81°, S₁ = 41.89 S₂ = −58.10 S₃ = 49.68 S₄ = 52.45, tanβ₁ = 1.70, tanβ₂ = 5.71, and C₀ = 1.28. The comparison between the LOS fitting value and the measured value at the boundary is shown in Figure 10 and Figure 11.

It can be seen from Figure 10 and Figure 11 that the absolute error of the parameter fitting from 4 November 2017 to 28 November 2017 was between 0.20 and 6.20 mm, and the mean error of the fitting was 2.95 mm. The absolute error of the parameter fitting from 10 December 2017 to 3 January 2018 was between 0.10 and 9.10 mm, and the mean error of the fitting was 3.64 mm. The subsidence values monitored by InSAR were basically consistent with the values predicted by InSAR-DIB.

Based on the obtained dynamic prediction parameters, the comparison of the surface subsidence values of the dip line during the mining (level monitoring) of the 1613 (1) working face is shown in Figure 12. The fitting error of the four periods was between 80.2 and 112.5 mm. The prediction accuracy met the needs of mining subsidence, realizing the transformation of short-term D-InSAR monitoring of the mining area surface to three-dimensional deformation dynamic monitoring and further verifying the reliability and scientific validity of the three-dimensional deformation dynamic monitoring method of mining subsidence based on D-InSAR technology.

5. Discussion

5.1. Boundary Convergence Performance Analysis of the IB Model

The premise of the boundary parameter improvement scheme is that the model is expected to have higher convergence performance at the boundary. A model verification analysis was carried out on the 1222 (1) working face of the Zhujidong Coal Mine in Huainan. (The detailed geological and mining conditions of the working face are detailed in reference [36]). A total of 110 monitoring points was selected from the working face from 19 March 2017 to 18 September 2018 for analysis. The PIM parameters of the observation station were obtained using a multi-population genetic algorithm: q = 0.86, tanβ = 1.9, b = 0.39, θ = 89.0, S₁ = 21.0, S₂ = 16.0, S₃ = 24.0, and S₄ = 21.0. The estimated parameters of the improved model were: q = 0.83, P = 0.64, b = 0.43, θ = 87.0, S₁ = 10.8, S₂ = −2.8, S₃ = 16.2, S₄ = −12.5, tanβ₁ = 0.85, and tanβ₂ = 2.5. The final measured subsidence value was compared to the predicted value, as shown in Figure 13.

As can be seen from Figure 13, the IB prediction model made up for the shortcomings of the PIM, in which the boundary converges too quickly. The fitting error between the measured and predicted values of the IB prediction model was 25.4 mm, and the fitting error between the predicted and measured subsidence values of the PIM was 32.6 mm. In comparison, the overall fitting accuracy of the IB prediction model improved by 28.3%. For the edge area of the mining subsidence basin, the fitting errors were 27.6 and 29.7 mm, respectively. The accuracy of the IB prediction model in the edge area of the subsidence basin improved by 7.6%. It can be seen that the IB prediction model fits the process of mining subsidence surface deformation well.

5.2. Analysis of the Main Controlling Factors of the IB Prediction Model

It is generally believed that the parameters obtained by fitting the accumulated deformation of the surface are called surface prediction parameters. Actual experiments have shown that from the moment of mining to the moment when the surface stops moving, the surface prediction parameters are not static but change with the change in the degree of mining. The premise of constructing the DIB prediction model is that the subsidence factor is the main controlling factor of the prediction model, and during the mining process, the other prediction parameters are less variable, while the subsidence factor is more variable. Below, we took the Guqiao South 1414 (1) working face [36] as an example to explore the changing law of surface prediction parameters during mining.

According to the principles of the IB prediction model, the displacement deformation values of any point i on the ground surface at time t_i are $W_{t_i}$ and $U_{t_i}$. If the measured subsidence value of the working face at this time is $W_{s{t}_i}$ and the horizontal displacement value is $U_{s{t}_i}$, then the parameter fitness function at this time is

$$F=1/{\displaystyle \sum \left({\left(W_{s{t}_i}-W_{t_i}\right)}^2+{\left(U_{s{t}_i}-U_{t_i}\right)}^2\right)},$$

(24)

In order to obtain the changing law of the IB prediction model with the degree of mining, according to the MPGA parameter-seeking principle and based on the geological mining conditions at the time t_i and the obtained surface movement and deformation, the dynamic IB prediction parameters corresponding to each period can be obtained.

A total of seven data periods during the advancement of the 1414 (1) working face were selected for parameter calculation (mainly the measured data of the 1414 (1) working face strike direction), and the specific parameter values are shown in

Table 4. Inversion results of the dynamic parameters of the 1414 (1) working face with single-phase data.

Time/d	q	P	θ	S1	S2	S3	S4	tanβ1	tanβ2
0.047	0.05	0.45	86.4	−30.8	−12.0	0.2	0.9	1.51	3.31
0.079	0.17	0.46	88.7	−1.6	−32.6	0.1	0.1	1.59	3.31
0.250	0.65	0.57	86.9	−33.9	−30.5	18.4	45.1	1.58	3.38
0.347	0.72	0.55	87.9	−38.3	−39.9	35.3	23.8	1.59	3.23
0.539	0.72	0.52	87.0	−39.9	−21.7	37.8	23.6	1.57	3.50
0.805	0.85	0.43	88.6	−47.7	−49.9	30.9	36.1	1.43	3.29
1	0.89	0.45	88.2	−77.7	−38.7	69.7	52.4	1.47	3.20

. In order to avoid the influence of accidental errors on the parameter calculation results, each period of data was inverted 20 times, and the average value of the 20 times was selected as the dynamic parameter of this period.

From the data in Table 4, as far as q is concerned, q gradually increased with the change in time, and when the mining time approached the complete mining time (343 days), the dynamic IB predicted a parameter value that was actually close to the actual IB predicted parameter. For tanβ₁ and tanβ₂, the parameter changed little with the change in time. θ did not change much with the change in time and tended to be stable. As far as the inflection point offset value is concerned, the parameter value changed little and irregularly with the change in time, while it changed greatly during the actual experiment, and the parameter stability was poor. In order to more intuitively show the change law of the dynamic predicted parameters over time, the change law of the main parameters with the mining degree D₃/H₀ (strike length/mining depth) was plotted into a graph, as shown in Figure 14.

It can be seen from Figure 14 that with the increase in D₃/H₀, the subsidence rate changed the most, the change pattern of the inflection point offset was not obvious, and the changes in the other parameters were small. Through regression analysis, it was found that the regression relationship between the subsidence rate q’ and the subsidence coefficient q was as follows:

$$q^{\prime }\left(D_3/H_0\right)=q\left(1-\exp \left(-C_1\ast \left(D_3/H_0\right)\right)\right)$$

(25)

Through regression analysis, the subsidence rate function of the 1414 (1) working face was obtained as Formula (26). The fitting is shown in Figure 15. It can be seen from the figure that the fitting was good.

$$q^{\prime }\left(D_3/H_0\right)=0.90\left(1-\exp \left(-3.99\left(D_3/H_0\right)\right)\right)$$

(26)

6. Conclusions

Based on the law of changes in predicted parameters with the degree of mining, this study constructs a dynamic prediction model based on changes in the surface’s predicted parameters. Our study also provides a method of obtaining the parameters of the dynamic prediction model based on the relative and absolute deformation values. On the basis of this study, combined with InSAR data, an InSAR-DIB dynamic prediction model based on the edge correction model is constructed which effectively expands the application scope of D-InSAR technology in mining areas. When applied to the Huainan mining area, the predicted mean error of the InSAR-DIB prediction model was between 80.2 and 112.5 mm, and the accuracy can meet the needs of mining subsidence monitoring.

The experimental areas selected in this paper are all located in the Huainan thick loose layer mining area. Whether they are applicable to other geological mining conditions such as mountainous areas, thick bedrock, and thin loose layer areas needs further verification.