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Carbon capture and storage is a crucial technology for mitigating climate change by reducing carbon dioxide (CO2) emissions through secure geological storage. Seismic monitoring plays a crucial role in ensuring the integrity of CO2 storage sites by detecting potential leakage, tracking plume migration, and assessing induced seismicity. In our study, we integrate Fourier Neural Operators in the branch network with a temporal encoding trunk network. The Fourier-based DeepONet referred as FDON, was trained on the Kimberlina-CO2 dataset, to learn the mapping from multi-source seismic wavefields to time-evolving velocity structures for CO2 plume monitoring. Experimental results demonstrated that Fourier-based DeepONet achieves better performance than two benchmark models with CNN-based DeepONet and standard Fourier Neural Operator (FNO). More importantly, the developed model demonstrated resolution invariant and noise-robust behaviors. These advantages are mainly attributed to spectral domain learning mechanism of the Fourier Blocks, which enables cost reduction in seismic acquisition and processing.