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The Tonga volcano erupted on 15 January 2022, at 04:15:45 UTC, which significantly influenced the atmosphere and space environment, at the same time, an unprecedented opportunity to monitor ionospheric anomalies is provided by its powerful eruption. In current studies of traveling ionospheric disturbance (TID) triggered by the 2022 Tonga volcanic eruption, the particular phenomenon of ionospheric disturbances in various parts of the world has not been reasonably explained, and the vertical ionospheric disturbances are still not effectively detected. In this paper, we calculate the high-precision slant total electron content (STEC) from more than 3000 ground-based GPS stations distributed around the world, then we obtain the radio occultation (RO) data from near-field COSMIC-2 profiles and investigate the horizontal TID and the vertical ionospheric disturbances by the singular spectrum analysis (SSA). Horizontal TID propagation captured by GPS STEC results indicates that acoustic-gravity waves dominate the energy input at the beginning of the ionospheric disturbance with an approximate speed of 1050 m/s initially. With the dissipation of the shock energy, lamb waves become a dominant mode of ionospheric disturbances, moving at a more stable speed of about 326 m/s to a range of 16,000 km beyond the far-field. Local characteristics are evident during the disturbance, such as the ionospheric conjugation in Australia and the rapid decay of TID in Europe. The shock-Lamb-tsunami waves' multi-fluctuation coupling is recorded successively from the COSMIC-2 RO observation data. The shock and Lamb waves can perturb the whole ionospheric altitude. In contrast, the disturbance caused by tsunami waves is much smaller than that of acoustic-gravity waves and Lamb waves. In addition, influenced by the magnetic field, the propagation speed of TID induced by Lamb waves is higher towards the northern hemisphere than towards the southern hemisphere. [ABSTRACT FROM AUTHOR]