Synergizing Multigrid Algorithms with Vision Transformer: A Novel Approach to Enhance the Seismic Foundation Model

Due to the emergency and homogenization of Artificial Intelligence (AI) technology development, transformer-based foundation models have revolutionized scientific applications, such as drug discovery, materials research, and astronomy. However, seismic data presents unique characteristics that require specialized processing techniques for pretraining foundation models in seismic contexts with high- and low-frequency features playing crucial roles. Existing vision transformers (ViTs) with sequential tokenization ignore the intrinsic pattern and fail to grasp both the high- and low-frequency seismic information efficiently and effectively. This work introduces a novel adaptive two-grid foundation model training strategy (ADATG) with Hilbert encoding specifically tailored for seismogram data, leveraging the hierarchical structures inherent in seismic data. Specifically, our approach employs spectrum decomposition to separate high- and low-frequency components and utilizes hierarchical Hilbert encoding to represent the data effectively. Moreover, observing the frequency principle observed in ViTs, we propose an adaptive training strategy that initially emphasizes coarse-level information and then progressively refines the model's focus on fine-level features. Our extensive experiments demonstrate the effectiveness and efficiency of our training methods. This research highlights the importance of data encoding and training strategies informed by the distinct characteristics of high- and low-frequency features in seismic images, ultimately contributing to the enhancement of visual seismic foundation models pretraining.