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.

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

Received： 2025-02-22

Revised： 2025-07-30

Online： 2026-10-16

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

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