## 1 Introduction  
Seismic exploration plays a pivotal role in various geoscientific and resource exploration applications, offering invaluable insights into the Earth's subsurface structures. Modern seismic surveys typically involve generating controlled seismic energy at or near the surface and recording the resulting wavefields with an array of receivers. The recorded data subsequently undergo rigorous processing to enhance signal quality and prepare them for seismic inversion. Seismic inversion aims to deduce quantitative models of subsurface physical parameters, such as seismic velocity, from these surface observations. Among the array of inversion techniques, Full-Waveform Inversion (FWI) has garnered significant attention due to its potential for yielding high-resolution subsurface images. FWI is a method for obtaining detailed subsurface medium parameters and was first proposed by [PERSON] [1]. The primary goal of this inversion is to minimize data misfit between observed and synthetic seismic waveforms to derive an optimal subsurface medium parameter model [2]. However, FWI typically requires a reasonably accurate initial estimate of the subsurface parameters. A starting model that is close to the true subsurface velocity structure is a prerequisite for a successful FWI [3]. This prerequisite is necessary because FWI data generation is largely based on the wave equation, where the medium parameters are integral to the equation [4]. When the initial medium parameters deviate significantly from the true subsurface structure, the inversion tends to converge to local optima, leading to inaccurate results. Therefore, the initial model is a critical factor that directly influences inversion results, and constraints must be applied during FWI to reduce its dependency. In traditional FWI, there are two approaches for processing the initial model. The first step involves estimating the initial model using the observed data. For example, conventional tomography methods and travel time information can generate relatively accurate initial models for FWI [5], followed by full-waveform inversion. Although this approach has been applied successfully, it relies on low-frequency information to ensure the accuracy of the initial model. When this information is missing, it is challenging to provide a sufficiently accurate initial FWI model. The second approach focuses on improving the inversion process to reduce dependency on the initial model. For example, multiscale methods [6] divide the FWI problem by frequency, starting with lower frequencies to capture large-scale subsurface structures and then transitioning to higher frequencies for finer details. Although this method mitigates nonlinearity to an extent, it depends on low-frequency information found within the data, which is challenging to obtain through field surveys. In addition, improper frequency-band partitioning can affect the resolution of the model. Another method involves expanding the optimization search space [7] to avoid local minima by incorporating the wave equation constraint as a regularization term in the objective function. This approach significantly broadens the search space and reduces dependency on the initial model. However, relaxing the physical constraints of the wave equation may lead to results that do not fully satisfy physical laws, thereby compromising accuracy. Similarly, adding regularization terms, such as total variation regularization [8], can constrain the solution space and reduce dependency on the initial model. However, this approach may lead to over smoothing and requires careful parameter tuning. Another approach modifies the FWI objective function by focusing on the observed and synthetic data. Traditional FWI minimizes the squared difference between the observed and synthetic data, but the oscillatory nature of seismic data results in numerous local minima. Alternative methods include using matched filters [9] for trace-by-trace matching and employing optimal transport distances [10; 11] or envelope-based misfits [12; 13].