## 4 Lateral Constraint Inversion Theory
### Optimization of Lateral and Vertical Weighting Factors  
To determine optimal lateral and vertical weighting factors, a fault valley model featuring high- and low-resistivity discontinuities in the lateral direction was developed to mimic water-bearing and non-water-bearing goat conditions based on the local geological context. The corresponding TEM response was computed using unstructured time-domain finite element 3D forward modeling. A tetrahedral mesh was employed for spatial discretization, with localized refinement in the source, receiver, and anomaly regions, resulting in a total of 322,983 cells. To target coal seam goafs, the background resistivity of the fault valley model was set to 200 \(\Omega\)-m, referencing the resistivity values of coal seams provided in Table 1. The transmitter consisted of a 5-turn square loop measuring 50 m \(\times\) 50 m, with an emission current of 5 A. Measurements were taken within a range of (\(-\)400 m, 400 m) with a spacing of 20 m between measurement points. Model I: The schematic diagram of Model I is shown in Figure 4. This model depicts a structure where the middle layer exhibits low resistivity. The resistivity of this layer is 10 \(\Omega\)-m. The top layer is horizontal and located at a depth of 30 m, while the bottom layer changes direction horizontally. The maximum buried depth is 80 m, with the middle layer gradually thinning towards both sides. The minimum buried depth is 50 m. Refer to Figure 4 for a schematic representation of this model. The survey points are set to one, with lateral and vertical weighting factors both set to 0. Figure 5 illustrates the outcomes of single-point inversion using adaptive regularization. A comparison with the model schematic in Figure 4 reveals that the inversion outcomes can mirror the theoretical model's shape. However, the inversion results exhibit overall continuity in both lateral and vertical dimensions, with the possibility of abrupt resistivity changes at specific survey points. Figure 4: Schematic diagram of fault valley model with low resistivity in middle layer. Figure 5: Single-point inversion results. To assess the impact of constraints on inversion outcomes, we employed the control variable method to calculate the inversion results using lateral and vertical constraint matrices. Initially, the vertical weighting factor was set to 0, while the lateral weighting factors were set to 0.1, 0.5, and 5. The corresponding inversion outcomes are depicted in Figure 6. The application of lateral constraints notably improved the lateral continuity of the inverted formation, enabling a more accurate representation of the formation's resistivity variations in the lateral direction. Notably, both the upper and lower interfaces of the low-resistivity structure within the middle layer were delineated. A comparative analysis between Figure 6a,b revealed that the latter exhibited enhanced smoothness, superior lateral continuity of the low-resistivity layer, and clearer depiction of inversion details. As the lateral weighting factor increased to 5, the inversion results exhibited pronounced lateral enhancement, with a tendency towards averaging the characteristics of inclined planes and a reduction in inversion details compared to Figure 6b. While the smoothness and lateral continuity of the inversion outcomes improved progressively with increasing lateral weighting factor, excessively high values led to the suppression of details pertaining to the inclined surface of the low-resistivity layer. Therefore, it is advisable to avoid excessively large lateral weighting factors in practical applications. In this study, a lateral weighting factor of 0.5 proved optimal, striking a balance between lateral continuity and detailed representation of the low-resistivity layer. Figure 6: Inversion results of low-resistivity fault valley model with different lateral weighting factors. (**a**) Lateral weighting factor of 0.1; (**b**) lateral weighting factor of 0.5; (**c**) lateral weighting factor of 5.