We demonstrated an over 30 W all-fiber supercontinuum source based on a four-mode graded-index multimode fiber (GIMF) pumped by around 5 picoseconds laser pulse. Following the evolution of the supercontinuum, it is reasonable to conclude that the discrete spectral peaks in the range 350-700 nm were generated owing to the geometric parametric instability effect and quasi-phase matched harmonic generation, whereas the radiation of dispersive waves, stimulated Raman scattering, and relative effects of optical soliton caused spectral broadening ranging from 700 nm to 1750 nm (and even to 2400 nm). Furthermore, the properties of the supercontinua generated in the four-mode GIMF and the 50/125 standard GIMF were experimentally compared. This com-parison implied that the wavelengths radiated by the geometric parametric instability differed owing to different dispersion values. And the supercontinuum generated by the four-mode GIMF had a flatter spectral profile and a wider wavelength range.