We present experimental and theoretical results on white-light generation in the filamentation of a high-power femtosecond laser pulse in water and atmospheric air. We have shown that the high spatio-temporal localization of the light field in the filament, which enables the supercontinuum generation is sustained due to the dynamic transformation of the light field on the whole transverse scale of the beam, including its edges. We found that the sources of the supercontinuum blue wing are in the rings, surrounding the filament, as well as at the back of the pulse, where shock-wave formation enhanced by self-steepening takes place. We report on the first observation and demonstration of the interference of the supercontinuum spectral components arising in the course of multiple filamentation in a terawatt laser pulse. We demonstrate that the conversion efficiency of an initially narrow laser pulse spectrum into the supercontinuum depends on the length of the filament with high intensity gradients and can be increased by introducing an initial chirp.