Background The surface subsidence induced by coal seam mining represents a continuous dynamic evolution process. Constructing a full-time perdition model for surface subsidence in goafs holds significant theoretical and practical values for the rational utilization of land resources, the safety guarantee of engineering construction, and the transformation and development of resource-based cities.

Methods To accurately predict the surface subsidence in goafs during its decline and residual periods, this study proposed a prediction model based on the error function (Erf). First, the evolution characteristics of Erf were analyzed, verifying that the subsidence, as well as its velocity and acceleration, predicted using the model conformed to the inherent laws of surface subsidence in goafs. Second, a model for determining the parameters of the Erf function was developed based on the expanding window method (EMW). Then, the parameter solutions of the model were verified and corrected using time functions such as Logistic, Usher, and Erf. These efforts help accurately determine the surface subsidence during its decline and residual periods using monitoring data from the initial and active subsidence periods. Finally, field monitoring data from mining face 1176E in the Qianjiaying Coal Mine, Hebei Province, and mining face S3-13 in the Changcun Coal Mine, Shanxi Province, were used for verification. The maximum surface subsidence was predicted using non-parametric and parametric methods individually.

Results and Conclusions The non-parametric and parametric prediction methods generally exhibited roughly consistent error levels, with errors trending downward over time and a maximum prediction error of merely 4.74%. The errors of the non-parametric prediction method arose primarily from theoretical calculation-induced deviations and the premature convergence of the time functions. In contrast, the parametric prediction method yielded smaller errors, which were primarily determined by the inherent functional characteristics of the Erf-based prediction model. Through the theoretical deductions of the subsidence in the decline and residual periods, a prediction method for the subsidence basin model in these two periods was proposed. Using this method, 3D subsidence models were constructed, ultimately achieving the accurate prediction of surface subsidence in goafs during the two periods. The results of this study provide a new technical pathway and theoretical support for the full-time prediction of surface subsidence in goafs.