## 2 Problem Analysis and Theory
### Mitigating Nonlinearity Using First-Arrival Constraints  
To address the cycle-skipping problem in FWI waveform matching, we introduced first-arrival constraints to mitigate this nonlinearity. Taking P-waves as an example, the first arrival is the earliest seismic wave to reach the receiver during wave propagation. Generally, the first arrival time occurs near the point of maximum energy in the seismic trace. Figure 2a shows an observed seismic record from a near-offset shot in a Vertical Seismic Profiling (VSP) system with dimensions \(\mathrm{T}\times\mathrm{N}\), where \(\mathrm{T}\) represents the sampling time and \(\mathrm{N}\) represents the number of receivers. Figure 2b shows the synthetic seismic record from the initial model for the same near-offset shot. Performing FWI in this situation would lead to the cycle-skipping problem (Figure 1), causing the model updates to become trapped in the local optima. Therefore, these traces should be temporarily excluded from the loss calculation, whereas traces that satisfy the condition of first-arrival time differences of less than half a period should be included in the loss calculation and participate in model updates.  
Figure 2: Illustration of first arrivals in a VSP system. The red box shows the direct waves in the VSP system and indicates the first arrival; the green arrow indicates portions where there are large differences in first arrivals. As indicated by the green arrow, there is a clear time delay exceeding half the period between the observed and synthetic seismic traces in the first arrivals. (**a**) Observed seismic record. (**b**) Synthetic seismic record.  
The First Arrival Constraint Strategy, in the initial stages of inversion, primarily focuses on selecting seismic traces where the first-arrival time difference (\(\Delta t\)) is less than or equal to half a period. These selected traces are then used to construct and optimize the objective function, ensuring accuracy and reliability in the inversion process. This strategy avoids the cycle-skipping phenomenon in FWI, thereby guiding the model parameters to iterate and update in the correct direction. Notably, this method does not directly use first arrivals for inversion; rather, it screens observed and synthetic seismic traces based on the constraint \(\Delta t<\mathrm{T}/2\). The selected seismic traces remain complete, retaining both the first arrival and subsequent reflection information. Taking traces 5 and 50 from Figure 2 as examples, Figure 3a,b display seismic records that satisfy and fail to meet the time threshold condition, respectively. \(\Delta t\) is quantified through calculating cross-correlation functions (Figure 3c,d). For case (a), because \(\Delta t\) is less than half a period and the observed record contains rich wavefield information (including first arrivals and subsequent reflections), high-quality waveform matching can occur. In contrast, in case (b) where \(\Delta t\) exceeds half a period, including these traces in the inversion calculation can lead to cycle-skipping problems and local minima; therefore, they are temporarily excluded during the initial stages of inversion. As the model parameters were iterated and optimized, the number of seismic records satisfying the first arrival time difference threshold condition gradually increased. This approach provides richer subsurface medium structural information and leads to a continuous decrease in the sum of absolute first-arrival time differences across all trace gathers, ultimately converging to a globally optimal solution.