## 3 Examples  
### Synthetic Example  
To assess the effectiveness and practicality of the proposed inversion method, a model test was conducted using data from well A, located in a basin in eastern China. Figure 4 shows the planar position of the well. Initially, key parameters such as the mineral content, pore characteristics, and anisotropy were thoroughly analyzed. Based on these parameters, a TTI petrophysical model tailored for the specific geological setting of the study area was constructed. This model was then used to compute the fracture parameter curve for the well. Next, a synthetic seismic record was generated by convolving a Ricker wavelet with a central frequency of 30 Hz and a reflection coefficient. To simulate real-world conditions and align with the methodology of [PERSON] (2022) [11], noise with a signal-to-noise ratio of two was added to the synthetic data, mimicking the interference typically observed in actual seismic recordings. Finally, the results obtained from the proposed inversion method were compared to those from a traditional inversion approach. This comparison allowed for a detailed evaluation of the advantages of the new method, particularly in terms of its ability to enhance the stability and accuracy of the inversion results. Figure 5 presents the model test results for both the proposed inversion method and the traditional method under noise-free conditions. As shown, both methods yielded accurate inversion results without noise. Figure 6 further compares the accuracy of the two methods, with errors remaining within a reasonable range and the results appearing stable for both approaches. However, when noise is introduced, the performance diverges. Figure 7 illustrates the inversion results at a signal-to-noise ratio of two, while Figure 8 compares the associated errors. Under noisy conditions, the traditional method shows a significant decline in its fit with the logging curve, particularly for the fracture weakness parameter inversion, leading to increased errors. In contrast, the proposed method demon  
Figure 4: The location of well A.  
movements in the Yanshan region, particularly since the Paleogene, have triggered frequent fractures, with the main basal faults significantly influencing fracture distribution. This has led to the development of a complex network of inclined fractures and numerous smaller fractures within the carbonate rocks, enhancing the reservoir quality and making it a high-potential target. To improve the seismic data quality, the data in the work area were angle-stacked to increase the signal-to-noise ratio. Angle selection was based on the coverage frequency, with angles that exhibited high signal strength and extensive coverage being prioritized. The final selected incident angles were \(12^{\circ}\), \(20^{\circ}\), and \(28^{\circ}\), and the azimuth angles were \(32.5^{\circ}\), \(77.5^{\circ}\), \(122.5^{\circ}\), and \(167.5^{\circ}\). The processed seismic data are shown in Figure 9.  
Figure 9: The seismic data (**a**) and the angle of incidence are equal to \(12^{\circ}\). (**b**) The angle of incidence is equal to \(20^{\circ}\). (**c**) The angle of incidence is equal to \(28^{\circ}\). In the data inversion process, the initial model for the parameters to be inverted was established using logging data, followed by seismic data inversion through the damping least squares method. Figure 10 presents the inversion results, which align well with the actual logging data. Specifically, the inversion results mirror the trends in the logging data for high values, which correspond to high inversion values, and low values, which correspond to low inversion values. This confirms the reliability of the inversion process. Furthermore, the inversion results were sliced at the target layer, as shown in Figure 11. These slices indicate that the inversion results for the target horizon at well A exhibit high values, supporting their consistency with logging data interpretation.