## 2 Pressure Inversion Model for Gob Areas Driven by Optical Fiber Monitoring Data  
### Distributed Optical Fiber Representation and Calculation Method for Subsidence of Key Strata  
The distributed optical fiber sensing (DOFS) methodology for strata deformation monitoring utilizes microbending responses of embedded sensors subjected to mining-induced displacements [25]. The mechanical behavior can be conceptualized as a simply supported beam model, where sensing elements respond to continuous uniform loading from super-incumbent formations. Based on sensor-formation interaction mechanics, the analytical framework for stratum subsidence quantification encompasses two distinct phases: pre-failure and post-failure regimes. In the pre-failure phase, robust interfacial coupling enables direct strain integration computation. Post-failure analysis initially requires fracture zone localization, followed by the integration of geometric deformation parameters to achieve comprehensive subsidence characterization through distributed sensing data. (1) Before fracture of the key strata. Before the fracture of the key strata, the strain data measured by distributed optical fibers can be used to deduce the bending deflection of the key strata. The center of the bending deformation model of the key strata is at the point of maximum deflection in the bending segment, and the deformation of the bending segment assumes an arcuate shape. This deformation characteristic allows the deformation pattern of the optical fibers to be modeled as an arc, thereby enabling a quantitative description of the geometric relationship; the simplified model is shown in Figure 1. The relationship expression for the bending deflection \(S_{x}\) (m) of the rock strata is:  
\[S_{x}=r(1-\cos\theta) \tag{1}\]where \(r\) (m) is the radius of curvature and \(\theta\) (dimensionless) is the arc angle, and its value is limited by the configuration of the embedded sensor. The relationship between them is expressed as:  
\[\left\{ \begin{array}{l}\theta=\frac{L_{d}}{2r}\\ r=\frac{L}{2\sin\theta}\end{array} \right. \tag{2}\]  
where \(L\) denotes the initial embedded length of the optical fiber and \(L_{d}\) represents the deformed length of the optical fiber after axial tension. \(L_{d}\) can be derived by summing the strains at each sampling point of the optical fiber, and its expression is:  
\[L_{d}=L+\sum_{i=1}^{N=-}\frac{L}{\epsilon_{i}} \tag{3}\]  
where \(N\) represents the number of sampling points for the optical fiber embedded in the key strata, \(i\) is the set spatial sampling interval, and \(\epsilon_{i}\) denotes the strain at each sampling point. After the fracture of the key strata. Utilizing rock mechanics fundamentals, the flexural moment of the critical rock layer, which approximates a basic support structure, can be obtained, thereby allowing the stress distribution in the optical fibers to be deduced. When the deformation of the key strata is significant, a longitudinal fissure may appear at a certain location, exposing the optical fibers embedded within it. At this point, the two fractured sections of the rock strata exert tensile forces on the optical fibers at the fissure, leading to a sudden change in stress at that location. At the discontinuity interface, the combined effect of shearing forces and rock-fiber contact friction generates localized stress concentrations \(F_{a}\) and \(F_{b}\) along the rupture plane. The resultant loads can be resolved into orthogonal components: normal forces \(F_{av}\) and \(F_{bv}\), coupled with lateral forces \(F_{all}\) and \(F_{bl}\). Among them, under the action of the horizontal forces \(F_{ah}\) and \(F_{bh}\), compressive stress concentrations form in the optical fibers. For equilibrium preservation, the sensing elements develop counteracting forces \(F_{tu}\) and \(F_{th}\) within the intact competent formation flanking the failure zone.