Spatial distribution and physical characteristics of decomposed sunspot wave modes

Sunspot oscillations consist of multiple wave modes, making it challenging to isolate individual physical processes. To decompose these multi-modal oscillations, we apply empirical mode decomposition (EMD) to Doppler velocity data obtained with the Fast Imaging Solar Spectrograph. By avoiding arbitrary frequency filtering, EMD resolves the oscillations into four distinct modes with unique periodicities and spatial distributions: c_1 in the umbra (periods of ∼ 2 min); c_2 in the umbra and penumbra (2.5-4 min); c_3 in the outer penumbra (4-6 min); and c_4 in the superpenumbra (∼ 10 min). We find that all modes, including the high-frequency 1-minute oscillations, coexist in the superpenumbral fibrils and represent potential candidates for transverse waves. Analysis of the c_1 and c_2 modes reveals the coexistence of umbra-trapped body waves and running penumbral waves within the umbra. Meanwhile, the c_3 and c_4 modes demonstrate that sunspot waves undergo significant nonlinear evolution as they propagate through the stratified atmosphere. Our results indicate that EMD effectively classifies dispersive oscillations into distinct groups based on their intrinsic timescales and underlying physical natures.