## 2 Pressure Inversion Model for Gob Areas Driven by Optical Fiber Monitoring Data
### The Inversion Method for Pressure Distribution in the Gob Area  
The redistribution of geostatic loads and the evolving stress regime in the overburden sequence during extraction operations directly induces ground pressure phenomena. After coal seam extraction, two major systems are formed, i.e., the gob area and the surrounding rock, which are both opposed and unified. The goaf and the surrounding rock mass are interdependent and form a complex, dynamic system. As the mining progresses, the stresses and deformations in this system undergo continuous adjustment, leading to the manifestation of mining pressure in the form of roof falls, floor heave, rib spalling, and other geomechanical phenomena. The height of the falling zone is an important parameter of goaf strain; the height of the falling zone is generally 4-6 times the mining height, and the height of the fracture zone is 10-15 times the mining height. The expression for the height of the caved zone \(H_{c}\) is as follows:  
\[H_{C}=\frac{h_{M}-S_{S}}{b-1},S_{S}\leq S_{\text{Smax}} \tag{11}\]  
where \(h_{M}\) represents the mining height, \(S_{S}\) is the flexural deformation of the overlying formation above the mined-out void, \(S_{\text{Smax}}\) represents the critical subsidence threshold, and \(b\) denotes the bulking factor of the collapsed lithological units. The deformation magnitude within the goaf zone correlates with both the lateral offset from the mining face and the geomechanical properties of the surrounding strata. The expression for the strain \(\varepsilon_{x}\) in the gob area at a distance \(X\) from the coal wall is:  
\[\varepsilon_{x}=\frac{S_{x}}{H_{c}+h_{M}} \tag{12}\]  
Substituting Equation (11) into Equation (12) yields:  
\[\varepsilon_{x}=\frac{S_{x}}{h_{M}b-0.05h_{M}{}^{1.2}} \tag{13}\]  
where \(S_{x}\) represents the deformation of the caved gauge at a distance \(X\) from the coal wall in the gob area. The sinking of the critical layer is shown in Figure 3.  
During the longwall mining process, as the working face advances from the setup room, the sequential extraction of resources creates a goaf zone, which leads to the progressive subsidence of the adjacent rock strata. The expansion index of the fragmented lithological units is unable to fully compensate for the void created by the excavation. This results in bed separation between the immediate and primary roof horizons, leading to a complete redistribution of the overburden load onto the abutment zones. As the working face continues to advance, the immediate roof continues to collapse and fill the gob area, with the caved immediate roof eventually coming into contact with the main roof. At this stage, the majority of the weight from the overlying rock strata is still supported by the surrounding rock formations. As the mining operation progresses further, the scope of roof  
Figure 3: Critical layer subsidence pattern. collapse and failure gradually extends upwards towards the overlying rock strata. The caved gangue, or waste material, is continuously compressed until it becomes compacted, with the weight it bears from the overlying rock strata correspondingly increasing. The geostatic pressure, or the pressure exerted by the weight of the rock formations, transitions from being primarily supported by the peripheral support pillars to alternative load-bearing mechanisms, such as the compacted caved material and the surrounding rock formations. This complex sequence of events, involving the creation of the goaf zone, bed separation, roof collapse, and the redistribution of geostatic pressure, is a critical aspect of the longwall mining process and must be carefully managed to ensure the safety and efficiency of the mining operation.