顿志元,王文唱,顿志林,等.基于反正切时间函数的采空区地表剩余变形预计[J].河南理工大学学报(自然科学版)，doi: 10.16186/j.cnki.1673-9787.2025020013.

DUN Z Y, WANG C C, DUN Z L, et al． Prediction of residual surface deformation in goaf areas using an arctangent time function[J]. Journal of Henan Polytechnic University( Natural Science) ， doi: 10.16186/j.cnki.1673-9787.2025020013.

基于反正切时间函数的采空区地表剩余变形预计(网络首发)

顿志元1，2，王文唱3，顿志林2，邹友峰4，任连伟2，陈绍杰5

1.焦作大学 土木建筑工程学院,河南 焦作 454003；2.河南理工大学 土木工程学院,河南 焦作 454003；3.河南理工大学 安全科学与工程学院,河南 焦作 454003；4.河南理工大学 测绘与国土信息工程学院,河南 焦作 454003；5.山东科技大学 能源与矿业工程学院,山东 青岛 266590

摘要: 目的 为掌握采空区地表移动变形演化规律，实现地表剩余变形的精细化、全域预测，本文开展基于反正切时间函数的采空区地表剩余变形预计方法研究。 方法 引入反正切函数并进行优化，建立用于动态预计的反正切时间函数，分析该函数在沉降预计中的时空特征，探讨参数变化对反正切时间函数的地表沉降、沉降速度、沉降加速度曲线形态的影响规律；基于模拟退火算法的随机扰动和概率性求解策略，给出通过该算法提取函数参数最优值的方法；结合概率积分法，提出可预计采空区地表任意点剩余沉降计算方法；以钱家营矿某工作面最大沉降点（23号）为研究对象，引入Knothe，Logistic，MMF时间函数模型进行对比分析。结果 结果表明：反正切时间函数预计曲线与实测值相吻合，其均方根误差（root mean square error，RMSE）、平均绝对误差（mean absolute error，MAE）和平均绝对百分比误差（mean absolute percentage error，MAPE）分别为50.11，40.23和0.17，在对比模型中3项指标值均最低；其可决系数（R2）为0.992 6，在对比模型中最接近1，评价指标值表明采用反正切时间函数进行地表沉降预计误差相对较小，能够有效用于采空区地表剩余变形预计。不同时刻，工作面地表剩余沉降沿开采方向逐渐增大，并在停采线附近地表达到最大值，这与实际工作面开采过程吻合。结论 本文模型在地表移动变形预计中具有良好的适用性，可为采空区剩余变形的精准预测和稳定性判别提供参考。

关键词: 采空区地表；剩余变形；时间函数；概率积分法；动态预计

doi: 10.16186/j.cnki.1673-9787.2025020013.

基金项目: 国家自然科学基金联合基金重点项目（U23A20600）；河南省科技攻关项目（252102320335）

收稿日期：2025-02-22

修回日期：2025-07-30

网络首发日期：2026-10-16

Prediction of residual surface deformation in goaf areas using an arctangent time function

Dun Zhiyuan1,2, Wang Wenchang3, Dun Zhilin2, Zou Youfeng4, Ren Lianwei2, Chen Shaojie5

1.School of Civil and Architectural Engineering,Jiaozuo University,Jiaozuo 454003,Henan,China；2.School of Civil Engineering,Henan Polytechnic University,Jiaozuo 454003,Henan,China；3.College of Safety Science and Engineering,Henan Polytechnic University,Jiaozuo 454003,Henan,China；4.School of Surveying and Land Information Engineering,Henan Polytechnic University,Jiaozuo 454003,Henan,China；5.College of Energy and Mining Engineering,Shandong University of Science and Technology,Qingdao 266590,Shandong,China

Abstract: Objectives A method for predicting residual surface deformation in goaf areas using an arctangent time function was investigated to characterize the evolution of surface movement and deformation and to enable refined， full-area predictions. Methods An arctangent time function for dynamic prediction was developed by introducing and optimizing an arctangent function. Its spatiotemporal characteristics in subsidence prediction were analyzed， and the effects of parameter variations on surface subsidence， subsidence velocity， and subsidence acceleration were examined. An approach for identifying optimal function parameters was proposed using the stochastic perturbation and probabilistic search strategies of the simulated annealing algorithm. A method for predicting residual subsidence at arbitrary surface points in goaf areas was also developed by combining the function with the probability integral method. Results In an engineering case study， measurements at Point 23， the point of maximum subsidence in a mining panel at Qianjiaying Coal Mine， were compared with predictions from the Knothe， Logistic， and MMF time-function models. The arctangent time function provided the closest fit to the field measurements. Its root mean square error （RMSE）， mean absolute error （MAE）， and mean absolute percentage error （MAPE） were 50.11 mm， 40.23 mm， and 0.17， respectively， the lowest among the models compared. Its coefficient of determination （R⟡） was 0.992 6， the value closest to 1 among the models. Residual subsidence predictions at different times showed that residual deformation increased along the mining direction and reached its maximum near the stopping line， consistent with the mining process. Conclusions The proposed model was shown to be feasible for predicting surface movement and deformation in goaf areas. Accurate predictions of residual deformation can support goaf stability assessment.

Keywords: goaf surface; residual deformation; time function; probability integral method; dynamic prediction