## 2 Pressure Inversion Model for Gob Areas Driven by Optical Fiber Monitoring Data
### The Inversion Method for Pressure Distribution in the Gob Area
Therefore, the expression for the stress in the gob area at a distance \(X\) is:  
\[\sigma_{x}=\frac{E_{0}\varepsilon_{x}}{1-(\frac{\varepsilon_{x}}{\varepsilon_ {M}})} \tag{14}\]  
where \(E_{0}\) denotes the initial deformation modulus, while \(\varepsilon_{M}\) represents the ultimate strain of the caved goaf material under roof pressure loading. The \(\varepsilon_{M}\) coefficient is derived through:  
\[\varepsilon_{M}=\frac{b-1}{b} \tag{15}\]  
As the longwall mining face advances, the abutment pressure within the goaf material undergoes progressive reconsolidation and compaction. The length of the compaction zone extends synchronously with the advancement of the face, as the caved and fractured rock material is gradually compressed under the increasing overburden pressure [26; 27; 28]. The in situ stress regime of the reconsolidated goaf material gradually re-establishes to approximate the pre-mining geostatic pressure conditions. This means that at equivalent subsurface horizons within the goaf zone, the vertical stress component tends to maintain a state of equilibrium, excluding any significant tectonic influences. Notably, there is a significant reduction in the stress levels in the direction of the gob area around the active mining face. Conversely, the load-bearing demands in front of the coal face increase, leading to the formation of stress concentration in this zone. This redistribution of the in situ stresses is a critical factor in the longwall mining process. The increased load in front of the coal face can lead to ground control challenges, such as excessive roof sagging, rib spalling, or even pillar failure, if not properly managed. Likewise, the reduced stress levels within the gob area can result in unstable caving behavior and potential surface subsidence issues. Continuous monitoring of the stress changes within the goaf and the surrounding rock mass is essential to optimize the longwall mining design and operational parameters. Advanced numerical modeling techniques, in combination with field instrumentation data, can provide valuable insights into the evolving stress regime and guide the implementation of effective ground control measures. The successful management of the stress redistribution within the goaf and the coal face abutment zones is a key aspect of safe and efficient longwall mining operations. It requires a comprehensive understanding of rock mechanics principles, as well as the application of advanced monitoring and data analysis tools to proactively identify and mitigate potential geotechnical risks. By substituting Equations (1)-(3) into Equation (14), we can obtain the strain in the gob area monitored by optical fibers before the key stratum breaks. Similarly, by substituting Equation (10) into Equation (14), we can obtain the strain in the gob area monitored by optical fibers after the key stratum breaks. The specific expressions are as follows:\[\left\{ \begin{array}{ll}\sigma_{x}=\dfrac{E_{0}\bigg{[}L(x)+\sum\limits_{i=1}^{N} \varepsilon_{i}\times(1-\cos\theta)\bigg{]}(b-1)}{b\bigg{[}h_{M}-\bigg{(}\bigg{[} L(x)+\sum\limits_{i=1}^{N}\varepsilon_{i}\times(1-\cos\theta)\bigg{]}\bigg{)} \bigg{]}-0.05{h_{M}}^{1.2}}&\text{Before fracture of the key strata}\\ \\ \sigma_{x}=\dfrac{E_{0}\bigg{(}S_{x}^{\prime}=x_{N}\sqrt{2\sum\limits_{N=1}^{N =x/l}\varepsilon_{N}}\bigg{)}(b-1)}{b\bigg{[}h_{M}-\bigg{(}S_{x}^{\prime}=x_{N} \sqrt{2\sum\limits_{N=1}^{N=x/l}\varepsilon_{N}}\bigg{)}\bigg{]}-0.05{h_{M}}^{ 1.2}}&\text{After the fracture of the key strata}\end{array} \right. \tag{16}\]  
From a theoretical perspective, this research innovatively establishes a goaf pressure inversion model driven by distributed optical fiber monitoring data, uniquely coupling the key stratum fracture mechanics model with the subsidence trajectory function model. Specifically, before the key stratum fracture, the fiber bending deformation characteristics are analyzed based on a simply supported beam model, and the stratum subsidence is obtained through strain integration calculation.