A method for predicting mining subsidence throughout the whole cycle by constrained nonlinear optimization

Abstract

The application provides a mining subsidence whole cycle prediction method of constrained nonlinear optimization, relates to the field of mine surveying and mapping and mining subsidence static prediction, and is used for solving the problems of insufficient dynamic description capability of a traditional subsidence model and poor stability of parameter inversion. A three-dimensional dynamic subsidence prediction model is constructed by combining a dynamic probability integral model DPIM with an exponential time function ETF, and a constrained nonlinear optimization method is introduced for parameter inversion, so that the stability and precision of solution are improved while the physical rationality of parameters is ensured. The method can effectively describe the dynamic evolution of the whole subsidence process, realize three-dimensional prediction of surface subsidence and horizontal movement, and has strong stability and engineering applicability.

Description

Technical Field

[0001] This invention relates to the field of mine surveying and static prediction of mining subsidence, and in particular to a method for full-cycle prediction of mining subsidence using constrained nonlinear optimization. Background Technology

[0002] With the continuous development and utilization of coal resources, the intensity of mining in mining areas is constantly increasing. The problems of surface subsidence and horizontal deformation caused by underground mining are becoming increasingly prominent, urgently requiring a technical means to accurately predict surface deformation during mining, thereby providing a basis for safe production, engineering protection, and subsidence disaster prevention in mining areas. Currently, the probability integral method is mainly used for mining subsidence prediction. This method, based on the theory of stochastic media, calculates the subsidence by dividing the underground mining area into several small units and superimposing the influence of each unit on the surface. It has advantages such as few parameters, clear physical meaning, and high computational efficiency, and is therefore widely used in engineering. However, this method is usually used to calculate the final subsidence result after mining is completed, and is a static prediction method, which cannot reflect the dynamic evolution of surface subsidence over time.

[0003] Existing technologies primarily achieve dynamic prediction by introducing time functions. For example, one type of method uses the Knothe time function to describe the settlement process. This method is simple in form but has shortcomings in characterizing complex nonlinear settlement changes. Another type of method uses power-law time functions or Weibull time functions, improving the fitting ability of the settlement development process by introducing exponential or distributed parameters. In addition, there are methods that use piecewise time functions to improve fitting accuracy through staged modeling. These methods achieve a certain degree of description of settlement changes over time, but still rely on the parameter accuracy of the probabilistic integral model, and the parameters usually need to be obtained through inversion. For the parameter inversion problem, existing technologies mainly use gradient-based optimization methods and various intelligent optimization algorithms, including wolf pack algorithms, whale optimization algorithms, hybrid frog leaping algorithms, and invading weed algorithms. These methods obtain parameter solutions by continuously adjusting the model parameters to make the calculation results approximate the measured data, but they generally suffer from problems such as sensitivity to initial values, easy getting trapped in local optima, and insufficient convergence stability. Especially in the case of multi-parameter coupling, the parameter correlation is strong, which further reduces the inversion accuracy.

[0004] Furthermore, with the development of monitoring technologies such as InSAR, the ability to acquire surface deformation data has been continuously enhanced. Some methods attempt to combine monitoring data with probabilistic integral models to achieve dynamic subsidence prediction. However, most of these methods can only perform calculations at discrete time points, making it difficult to achieve continuous dynamic prediction throughout the entire mining process. At the same time, existing methods mainly focus on predicting vertical subsidence, with relatively insufficient ability to dynamically predict east-west and north-south horizontal deformation. Moreover, the coupling between the time function and the probabilistic integral model is still not tight enough, affecting the overall prediction results.

[0005] Therefore, there is an urgent need for a technical method that can achieve a unified temporal and spatial expression based on the probability integral model, improve the stability of parameter inversion, and simultaneously perform dynamic prediction of the entire process of settlement and horizontal deformation. Summary of the Invention

[0006] The purpose of this invention is to provide a method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization. By combining a probability integral model with a power-law time function and introducing an interior-point method for parameter inversion, continuous dynamic prediction of the entire mining subsidence process can be achieved.

[0007] To achieve the above objectives, this invention provides a method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization, comprising the following steps: S1: Combine the probability integral model with the power exponential time function to construct the ETF-DPIM model; S2: The parameter inversion of the ETF-DPIM model is performed using the interior-point method constrained nonlinear optimization. S3: A phased parameter update mechanism is adopted to characterize the subsidence evolution pattern during the mining process based on the mining progress.

[0008] Preferably, in the probability integral model of S1, the settlement of any surface point (x,y) is considered as n The superposition of the effects of each already mined unit: ; In the formula, m For coal seam thickness, q Here, α is the subsidence coefficient, and α is the coal seam dip angle. C x and C y These represent the settlement components in the strike and dip directions, respectively. The horizontal deformation at any point (x, y) on the Earth's surface is represented as: ; In the formula, and The angle between the projected coordinate system of the working surface and the east-west and north-south directions is not specified. D x and D y These are the horizontal displacement components along the strike and dip directions.

[0009] Preferably, the exponential time function of S1 is: ; In the formula, c and k The coefficients are time function coefficients. tThis represents the duration from the start of mining to the current moment. In S2, a probability integral model is combined with a power-law time function to establish an ETF-DPIM dynamic prediction model. At any given time, the subsidence of the surface point (x,y) is considered as the superposition effect of all mined units in space and time. ; The dynamic horizontal deformation of the surface point (x,y) in the east-west and north-south directions is also obtained by introducing a time function and spatiotemporally superimposing the horizontal displacement components of each mining unit: .

[0010] Preferably, based on the settlement function of the surface point (x,y) and the east-west and north-south dynamic horizontal deformation functions of the surface point (x,y) in the established ETF-DPIM dynamic prediction model, an objective function for the difference between measured and simulated values ​​is constructed, expressed as: ; In the formula, N The number of ground observation points. T The number of observation periods; , , These are the three-dimensional deformation components calculated from the model; , , The corresponding measured values ​​are θ = [q, b, tanβ, θ, s1, s2, s3, s4, c, k], which is the parameter vector to be inverted; x lb With x up These are the upper and lower bounds of the parameters; Introducing a logarithmic barrier function transforms the constrained problem into an unconstrained optimization problem, with the augmented objective function being: ; In the formula, α k This is the step size factor, determined by the line search method; and These are the first-order gradient and second-order Hessian matrix of the augmented objective function, respectively. As iteration progresses, the barrier parameter... Gradually decrease, when As time progresses, the augmented objective function gradually approximates the original constrained optimization problem, and the iterative solution converges to the optimal parameter solution.

[0011] Preferably, the formulas for calculating the strike and dip direction settlement components in the probability integral model are as follows: ; In the formula, r As the main influencing radius, r 1 andr 2 These are the main influence radii for the downhill and uphill directions, respectively; L and l The length is calculated based on the dip and strike of the working face, specifically as follows: ; ; ; In the formula, H is the sampling depth, and tanβ is the tangent of the main influence angle; D1 and D3 s1, s2, s3, and s4 are the dip length and strike length of the working face, respectively; s1, s2, s3, and s4 are the inflection point offsets of the downhill, uphill, left boundary, and right boundary, respectively; θ is the angle of propagation of mining influence.

[0012] Preferably, the expressions for the horizontal displacement components along the strike and dip directions in the probability integral model are as follows: ; In the formula, b This is the horizontal shift coefficient. b1 and b2 These are the horizontal movement coefficients for downhill and uphill directions, respectively.

[0013] Preferably, in the probabilistic integral model, the mined area is discretized into several rectangular micro-element units, and the subsidence and horizontal deformation of the surface by each unit are calculated by the probabilistic integral function. The results of all units are superimposed to obtain the deformation distribution at any point on the surface.

[0014] Therefore, this invention employs a constrained nonlinear optimization method for predicting mining subsidence throughout its entire lifecycle. By constructing an ETF-DPIM dynamic prediction model and coupling a probability integral model with a power-law time function, a three-dimensional spatiotemporal dynamic representation of the entire mining subsidence process is achieved. The introduction of interior-point constrained nonlinear optimization for parameter inversion improves the convergence stability and accuracy of multi-parameter solutions. Simultaneously, a phased parameter update mechanism effectively characterizes the subsidence evolution during mining operations, and robust modeling strategies reduce the impact of anomalous data, significantly enhancing the accuracy and engineering applicability of mining surface subsidence prediction. Attached Figure Description

[0015] Figure 1 This is a schematic diagram illustrating the dynamic mining process and the discrete integration unit in an embodiment of the present invention; Figure 2 This is the parameter inversion process of the ETF-DPIM model based on INT in this embodiment of the invention; Figure 3 This is a schematic diagram of the spatial distribution of the working surface and monitoring points in an embodiment of the present invention; Figure 4This is a graph showing the fitting results of the settlement time function at typical monitoring points in an embodiment of the present invention; Figure 5 These are comparison charts of simulated and measured settlement values ​​at monitoring points during various mining stages in this embodiment of the invention. (a) shows the comparison chart at time node 70 days; (b) shows the comparison chart at time node 135 days; (c) shows the comparison chart at time node 213 days; (d) shows the comparison chart at time node 276 days; (e) shows the comparison chart at time node 333 days; and (f) shows the comparison chart at time node 446 days. Figure 6 This is a comparison diagram of the PIM simulated subsidence field and the ETF-DPIM predicted subsidence field at a time node of 135 days in the comparative example of this invention; where (a) is the PIM simulated subsidence field at a time node of 135 days; and (b) is the ETF-DPIM predicted subsidence field at a time node of 135 days. Figure 7 This is a comparison diagram of the PIM simulated subsidence field and the ETF-DPIM predicted subsidence field at time node 213 days in the comparative example of this invention; where (a) is the PIM simulated subsidence field at time node 213 days; and (b) is the ETF-DPIM predicted subsidence field at time node 213 days. Figure 8 This is a comparison diagram of the PIM simulated subsidence field and the ETF-DPIM predicted subsidence field at a time node of 246 days in the comparative example of this invention; where (a) is the PIM simulated subsidence field at a time node of 246 days; and (b) is the ETF-DPIM predicted subsidence field at a time node of 246 days. Figure 9 This is a comparison diagram of the PIM simulated subsidence field and the ETF-DPIM predicted subsidence field at a time node of 333 days in the comparative example of this invention; where (a) is the PIM simulated subsidence field at a time node of 333 days; and (b) is the ETF-DPIM predicted subsidence field at a time node of 333 days. Figure 10 This is a comparison diagram of the PIM simulated subsidence field and the ETF-DPIM predicted subsidence field at a time node of 446 days in the comparative example of this invention; where (a) is the PIM simulated subsidence field at a time node of 446 days; and (b) is the ETF-DPIM predicted subsidence field at a time node of 446 days. Figure 11 This invention is a full-cycle settlement simulation diagram of four monitoring points in a private facility. Detailed Implementation

[0016] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0017] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0018] Example 1 like Figure 1 As shown, this invention provides a method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization, comprising the following steps: S1: Combine the probability integral model with the power exponential time function to construct the ETF-DPIM model; In the probabilistic integral model of S1, the mined area is discretized into several rectangular micro-element units. The subsidence and horizontal deformation of each unit on the surface are calculated using a probabilistic integral function. The results from all units are then superimposed to obtain the deformation distribution at any point on the surface. The subsidence at any surface point (x, y) is considered as... n The superposition of the effects of each already mined unit: ; In the formula, m For coal seam thickness, q Here, α is the subsidence coefficient, and α is the coal seam dip angle. C x and C y These represent the settlement components in the strike and dip directions, respectively. The horizontal deformation at any point (x, y) on the Earth's surface is represented as: ; In the formula, and The angle between the projected coordinate system of the working surface and the east-west and north-south directions is not specified. D x and D y These are the horizontal displacement components along the strike and dip directions.

[0019] The formulas for calculating the strike and dip direction settlement components in the probability integral model are as follows: ; In the formula, r As the main influencing radius, r 1 and r 2 These are the main influence radii for the downhill and uphill directions, respectively; L and l The length is calculated based on the dip and strike of the working face, specifically as follows: ; ; ; In the formula, H is the sampling depth, and tanβ is the tangent of the main influence angle; D 1 and D 3 represents the dip length and strike length of the working face, respectively; s1, s2, s3, and s4 represent the inflection point offsets of the downhill, uphill, left boundary, and right boundary, respectively; θ represents the angle of propagation of mining influence.

[0020] The expressions for the horizontal displacement components along the strike and dip directions in the probability integral model are as follows: ; In the formula, b This is the horizontal shift coefficient. b1 and b2 These are the horizontal movement coefficients for downhill and uphill directions, respectively.

[0021] The exponential time function of S1 is: ; In the formula, c and k The coefficients are time function coefficients. t This represents the duration from the start of mining to the current moment. S2: The parameter inversion of the ETF-DPIM model is performed using the interior-point method constrained nonlinear optimization. By combining a probability integral model with a power-law time function, an ETF-DPIM dynamic prediction model is established. At any given time, the subsidence of the surface point (x, y) is considered as the superposition effect of all mined units in space and time. ; The dynamic horizontal deformation of the surface point (x,y) in the east-west and north-south directions is also obtained by introducing a time function and spatiotemporally superimposing the horizontal displacement components of each mining unit: .

[0022] Based on the settlement function of surface point (x,y) and the east-west and north-south dynamic horizontal deformation functions of surface point (x,y) in the established ETF-DPIM dynamic prediction model, an objective function for the difference between measured and simulated values ​​is constructed, expressed as: ; In the formula, N The number of ground observation points. T The number of observation periods; , , These are the three-dimensional deformation components calculated from the model; , , The corresponding measured values ​​are θ = [q, b, tanβ, θ, s1, s2, s3, s4, c, k], which is the parameter vector to be inverted; x lb With x up These are the upper and lower bounds of the parameters; Introducing a logarithmic barrier function transforms the constrained problem into an unconstrained optimization problem, with the augmented objective function being: ; In the formula, α k This is the step size factor, determined by the line search method; and These are the first-order gradient and second-order Hessian matrix of the augmented objective function, respectively. As iteration progresses, the barrier parameter... Gradually decrease, when As the augmented objective function gradually approximates the original constrained optimization problem, the iterative solution converges to the optimal parameter solution. The process of model construction and parameter inversion is as follows: Figure 2 As shown.

[0023] To verify the applicability and prediction accuracy of the ETF-DPIM model under actual mining conditions, a specific working face was selected as the research object. This working face employs the total caving method for roof management, with a mining cycle of 466 days. The working face has a strike length of 1231m, a dip length of 270m, an average coal seam thickness of 2.5m, an average mining depth of 760m, and a dip angle of approximately 8°. The working face consists of one strike line and one dip line, with a total of 119 monitoring points deployed, including 54 on the strike line and 65 on the dip line, spatially distributed as shown in the diagram. Figure 3 As shown.

[0024] Based on 14 phases of measured settlement data from the mining area, the settlement time series of each monitoring point was extracted, and the power-law time function parameters corresponding to the 119 monitoring points were retrieved using the POS algorithm. c and k The results are shown in Table 1. (Table 1 shows the results for each monitoring point.) c The value range is 0.01–0.031. k The values ​​range from 1.16 to 93.62. The root mean square error (RMSE) is between 0.6 and 46.7 mm, mainly concentrated between 0 and 30 mm, with an average of 9.1 mm, indicating a high overall fitting accuracy.

[0025] Table 1 shows the time function parameters (c, k) obtained from the inversion of 119 monitoring points.

[0026] Substituting some parameters from Table 1 into the power-law time function model, the settlement process at each monitoring point at different stages is simulated as follows: Figure 4As shown, the fitted curve is basically consistent with the measured value, and can well characterize the nonlinear variation of surface subsidence over time.

[0027] S3: A phased parameter update mechanism is adopted to characterize the subsidence evolution pattern during the mining process based on the mining progress.

[0028] Based on the mining progress, the mining process of the working face was divided into six stages, with time nodes of 70 days, 135 days, 213 days, 276 days, 333 days, and 446 days, corresponding to retreat lengths of 180 m, 403 m, 678 m, 862 m, 1027 m, and 1231 m, respectively. In each stage, based on the surface subsidence data from the monitoring points, the INF algorithm was used to invert some parameters of the probability integral model, including b, tanβ, θ, s1, s2, s3, s4, and the stage-specific time function parameter q. The parameter search range is set based on engineering experience: b∈[0.1,0.8], tanβ∈[1,3], θ∈[80,90], s1∈[-300,300], s2∈[-300,300], s3∈[-300,300], s4∈[-300,300], and the value range of the stage parameter q is q∈[0.1,0.3], q∈[0.2,0.5], q∈[0.3,0.8], q∈[0.5,1.0], q∈[0.6,1.1], q∈[0.8,1.3].

[0029] The inversion results are shown in Table 2. The time function parameter q gradually increases with the progress of mining, reflecting the transition of the overlying strata from insufficient mining to full mining. The main influencing angles, tangent tanβ and propagation angle θ, show relatively small changes, indicating relatively stable geological conditions. The boundary parameters S1-S4 fluctuate significantly, greatly influenced by geology and mining methods. The fitting error is small in the initial stage (RMSE 5.8 mm in the 70-day stage), gradually increasing to 105 mm in the 446-day stage, but still within the acceptable range for engineering.

[0030] Table 2 shows the PIM parameter inversion results and error statistics for different mining stages.

[0031] To further verify the reliability of the inversion parameters, the parameters obtained in Table 2 were substituted into the probability integral model PIM to simulate the settlement at each monitoring point. The simulation results are as follows: Figure 5Figures (a), (b), (c), (d), (e), and (f) are shown. The first 54 monitoring points represent the strike line, and the latter 65 represent the dip line. The results show that the simulated settlement values ​​are consistent with the overall trend of the measured data: the strike line monitoring points experienced settlement first, with the maximum settlement gradually increasing to approximately 2500 mm as mining progressed; the dip line monitoring points also gradually experienced settlement as mining progressed, with the maximum settlement also approximately 2500 mm. The simulation results for the central area of ​​the subsidence basin matched the measured values ​​well, while the boundary areas still showed some deviation due to complex geological conditions and mining disturbances.

[0032] Substituting the PIM parameters obtained from each stage of inversion into the model, the settlement field of the mining area at 135 days, 213 days, 246 days, 333 days, and 446 days was visualized, as shown below. Figure 6 (a) Figure 7 (a) Figure 8 (a) Figure 9 (a) Figure 10 As shown in (a) above. Based on this, the time function parameters obtained by inversion from 54 monitoring points along the route are... c and k The final stage PIM parameters [1.19, 0.45, 1.78, 87.5, -210, 274, 63, 154] are used to construct the ETF-DPIM model. Specifically, the monitoring points along the strike of the mining area are divided into 54 sub-regions. First, the settlement value at each spatial location is calculated using a static probability integral model, then multiplied by the corresponding time function to obtain the dynamic settlement prediction results for 0–446 days. Simultaneously, a linear feathering method is used to spatially smooth adjacent sub-regions to ensure the continuity and stability of the spatiotemporal coupling prediction results, as shown below. Figure 6 (b) Figure 7 (b) Figure 8 (b) Figure 9 (b) Figure 10 As shown in (b) of the diagram.

[0033] Accuracy analysis of the ETE-DPIM model was conducted. Monitoring points MLA017, MS037, MLA056, and MCORS02 in the mining area were selected. Daily settlement values ​​from 0 to 446 days were calculated using the ETE-DPIM model, and then overlaid with measured settlement data for visualization analysis. The analysis results are as follows: Figure 11As shown in the figure. The results show that the RMSE of the simulation results at each monitoring point are 22.36 mm, 56.55 mm, 103.41 mm and 42.83 mm, respectively. The trend of the model prediction curve is basically consistent with that of the measured data, and the errors are all within the acceptable range. This further verifies the accuracy and practicality of the ETF-DPIM model in the dynamic prediction of mining subsidence in mining areas.

[0034] Therefore, this invention employs a constrained nonlinear optimization method for predicting mining subsidence throughout its entire lifecycle. By combining a probability integral model with an improved power-law time function and using the interior-point method for parameter inversion, the constructed ETF-DPIM model can effectively simulate vertical subsidence and north-south and east-west horizontal deformation throughout the entire mining process. The model's prediction results show high consistency with simulated working face tests and field monitoring data, demonstrating the feasibility and reliability of this model in three-dimensional dynamic subsidence prediction.

[0035] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization, characterized in that, Includes the following steps: S1: Combine the probability integral model with the power exponential time function to construct the ETF-DPIM model; S2: The parameter inversion of the ETF-DPIM model is performed using the interior-point method constrained nonlinear optimization. S3: A phased parameter update mechanism is adopted to characterize the subsidence evolution pattern during the mining process based on the mining progress.

2. The method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization according to claim 1, characterized in that, In the probability integral model of S1, the settlement at any surface point (x,y) is considered as... n The superposition of the effects of each already mined unit: ； In the formula, m For coal seam thickness, q α is the subsidence coefficient, and α is the coal seam dip angle; C x and C y These represent the settlement components in the strike and dip directions, respectively. The horizontal deformation at any point (x, y) on the Earth's surface is represented as: ； In the formula, and The angle between the projected coordinate system of the working surface and the east-west and north-south directions is not specified. D x and D y These are the horizontal displacement components along the strike and dip directions.

3. The method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization according to claim 2, characterized in that, The exponential time function of S1 is: ； In the formula, c and k The coefficients are time function coefficients. t This represents the duration from the start of mining to the current moment. By combining a probability integral model with a power-law time function, an ETF-DPIM dynamic prediction model is established. At any given time, the subsidence of the surface point (x, y) is considered as the superposition effect of all mined units in space and time. ； The dynamic horizontal deformation of the surface point (x,y) in the east-west and north-south directions is also obtained by introducing a time function and spatiotemporally superimposing the horizontal displacement components of each mining unit: 。 4. The method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization according to claim 3, characterized in that, In S2, based on the settlement function of the surface point (x,y) and the east-west and north-south dynamic horizontal deformation functions of the surface point (x,y) in the established ETF-DPIM dynamic prediction model, an objective function for the difference between measured and simulated values ​​is constructed, expressed as: ； In the formula, N The number of ground observation points. T The number of observation periods; , , These are the three-dimensional deformation components calculated from the model; , , These are the corresponding measured values; δ=[q,b,tanβ,θ,s1,s2,s3,s4,c,k] is the parameter vector to be inverted; x lb With x up These are the upper and lower bounds of the parameters; Introducing a logarithmic barrier function transforms the constrained problem into an unconstrained optimization problem, with the augmented objective function being: ； In the formula, α k This is the step size factor, determined by the line search method; and These are the first-order gradient and second-order Hessian matrix of the augmented objective function, respectively. As iteration progresses, the barrier parameter... Gradually decrease, when As time progresses, the augmented objective function gradually approximates the original constrained optimization problem, and the iterative solution converges to the optimal parameter solution.

5. The method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization according to claim 2, characterized in that, The formulas for calculating the strike and dip direction settlement components in the probability integral model are as follows: ； In the formula, r As the main influencing radius, r 1 and r 2 These are the main influence radii for the downhill and uphill directions, respectively; L and l The length is calculated based on the dip and strike of the working face, specifically as follows: ； ； ； In the formula, H is the sampling depth, and tanβ is the tangent of the main influence angle; D1 and D3 s1, s2, s3, and s4 are the dip length and strike length of the working face, respectively; s1, s2, s3, and s4 are the inflection point offsets of the downhill, uphill, left boundary, and right boundary, respectively; θ is the angle of propagation of mining influence.

6. The method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization according to claim 2, characterized in that, The expressions for the horizontal displacement components along the strike and dip directions in the probability integral model are as follows: ； In the formula, b This is the horizontal shift coefficient. b1 and b2 These are the horizontal movement coefficients for downhill and uphill directions, respectively.

7. The method for predicting mining subsidence throughout its entire lifecycle using constrained nonlinear optimization according to claim 1, characterized in that, In the probabilistic integral model, the mined area is discretized into several rectangular micro-element units, and the subsidence and horizontal deformation of the surface by each unit are calculated by the probabilistic integral function. The results of all units are superimposed to obtain the deformation distribution at any point on the surface.

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