## 2 Problem Analysis and Theory
### Causes of Nonlinearity in FWI  
A FWI is associated with an inherently highly nonlinear optimization problem. This nonlinearity renders the FWI solution highly sensitive to the initial model [31]. Specifically, when the initial model deviates significantly from the true model, the inversion process is prone to becoming trapped in the local minima, leading to results that diverge from the true solution. Figure 1[3] demonstrates the cause of this phenomenon, where (b) represents the observed data of a seismic trace, and (a) and (c) represent the synthetic data of the corresponding trace generated through forward modeling using the initial model. In (a), the time delay \(\Delta t\) between the n-th cycle of the signal and the n-th cycle of the observed data in (b) exceeds half a period (T/2). In this case, the FWI matches the \((\text{n}+1)\)-th cycle of the signal in (a) with the n-th cycle of the observed data in (b), causing the model update to proceed in the wrong direction. Further iterations lead to error accumulation, ultimately resulting in an incorrect model. In contrast, in (c), the time delay \(\Delta t\) between the n-th cycle of the signal and the n-th cycle of the observed data in (b) is less than T/2, allowing FWI to perform correct matching and model updates. Therefore, combining with Equation (2), assuming the time delay between the observed data \(\mathbf{d}_{\text{obs}}(t)\) and the synthetic data \(\mathbf{d}_{\text{cal}}(t,\text{m})\) is \(\Delta t\), the nonlinearity problem of FWI can be described as follows:  
\[\mathbf{d}_{\text{cal}}(t,\text{m})\approx\mathbf{d}_{\text{obs}}(t+\Delta t), \tag{5}\]  
where \(\Delta t\) is the time delay in seismic records caused by model errors. When \(\Delta t<\text{T}/2\), the waveforms of observed and synthetic data are aligned in time, and the objective function has a global minimum at \(\Delta t=0\) (Figure 1c). The gradient points in the correct direction of the time delay enabled FWI to perform accurate matching and model updates. When \(\Delta t>\frac{\text{T}}{2}\), multiple local minima appeared in the parameter space of the objective function \(J_{\text{D}}(\text{m})\), such as at \(\Delta t=\text{T}\), \(\frac{3}{2}\text{T}\), 2T, and so on (Figure 1a). The gradient pointing in the incorrect direction causes FWI to incorrectly match \(\mathbf{d}_{\text{cal}}(t+nT,\text{m})\) with \(\mathbf{d}_{\text{obs}}(t)\). This  
Figure 1: Generation of nonlinearity in FWI. (**a**) For a seismic trace signal with \(\Delta t>\text{T}/2\), the waveform matching between the observed data and the synthetic data is incorrect, and the gradient points in the incorrect direction of the time delay. This phenomenon causes FWI to erroneously match the \((n+1)\)-th cycle of the signal with the n-th cycle of the observed data, leading to error accumulation. (**b**) Observed seismic trace signal. (**c**) For a seismic trace signal with \(\Delta t<\text{T}/2\), the waveforms of the observed data and the synthetic data are aligned in time, enabling FWI to correctly match the n-th cycle of the signal and thus update the model parameters accurately. phenomenon is known as cycle-skipping. After skipping the cycle, the gradient function corresponding to Equation (3) becomes the following:  
\[\frac{\partial f_{D}}{\partial\mathbf{m}}=-2\sum_{s=1}^{N_{\mathrm{s}}}\int_{0}^{ T}\left(\frac{\partial\mathbf{d}_{\mathrm{cal}}(t+\Delta t,m)}{\partial \mathbf{m}}\right)^{\top}(\mathbf{d}_{\mathrm{obs}}(t)-\mathbf{d}_{\mathrm{cal} }(t+\Delta t,m))dt, \tag{6}\]  
where \(\mathbf{d}_{\mathrm{obs}}(t)\) represents the observed data and \(\mathbf{t}\) is the time variable. The synthetic data are denoted as \(\mathbf{d}_{\mathrm{cal}}(t+nT,\mathbf{m})\), where \(\Delta t\) represents the time delay and \(\mathbf{m}\) represents the model parameters. The partial derivative of synthetic data with respect to the model parameters is expressed as \(\frac{\partial\mathbf{d}_{\mathrm{cal}}(t+\Delta t,\mathbf{m})}{\partial \mathbf{m}}\).