## 6 Discussion  
### Analysis  
Our numerical experiments demonstrated the effectiveness and robustness of the proposed FAFWI method. By utilizing the first-arrival time information and dynamically selecting seismic traces with minimum phase differences, the FAFWI-PIRNN effectively reduced the dependence on the initial velocity models. A comparison of the inversion results with conventional FWI methods in the Marmousi and 2007 BP TTI velocity models demonstrated the capability of the proposed method to navigate complex subsurface structures and avoid the local minimum traps faced by traditional FWI.  
FAFWI demonstrated dual utilization of first-arrival and reflection wave information. Unlike traditional first-arrival FWI methods, which rely solely on the earliest arriving waves, our method integrates reflection wave information through a first-arrival difference selection mechanism. This integration improves the fidelity of velocity model reconstruction and ensures more comprehensive seismic information utilization, thereby enhancing the overall resolution and accuracy of the inversion results. The dynamic selection of seismic traces based on first-arrival delays ensured that only traces with reliable first-arrival and reflection wave information participated in the inversion process, thereby improving the stability and robustness of the method. Furthermore, we enhanced the FAFWI performance by incorporating a Physics-Informed Recurrent Neural Network (PIRNN). The PIRNN framework embeds physical wave propagation processes into the recurrent neural network, facilitates the integration of physical constraints during network training, and fully utilizes the spatiotemporal field gradients recorded during forward propagation for back propagation, providing nonlinear solutions for full-waveform inversion and ensuring that the results conform to geophysical principles. We comprehensively evaluated the per  
Figure 16: Inversion results of the FAFWI method using the initial models of four constant velocities. (**a**) Initial model for 1800 m/s, (**b**) 2100 m/s, (**c**) 2300 m/s, and (**d**) 2500 m/s.  
Figure 15: Initial models for four constant velocities. (**a**) Velocity of 1800 m/s, (**b**) 2100 m/s, (**c**) 2300 m/s, and (**d**) 2500 m/s.  
formance of the method based on four aspects: effectiveness, applicability, noise resistance, and initial model sensitivity.  
When compared with other strategies aimed at reducing initial model dependency or similar issues, the FAFWI-PIRNN method presents several key distinctions. Multiscale FWI [4], optimal transport [9], and adaptive waveform inversion [7] primarily focus on reshaping the optimization problem or the data itself to be less prone to cycle-skipping. In contrast, the FAFWI-PIRNN method dynamically selects seismic traces based on first-arrival constraints and inputs the full information of these traces into a physically consistent PIRNN. This differs from solely using traveltime tomography to build an initial model [29]. In the realm of deep learning, unlike data-driven FWI methods [13,14], our approach maintains strong physical constraints via the PIRNN while leveraging first-arrival information for data selection, rather than relying predominantly on learning from large datasets or network-induced regularization