Abstract: In quasi-linear approximation,the electron "transit-time acceleration" by compressional ULF turbulence near the geosynchronous orbit is studied.The frequency range of compressional ULF turbulence is from 2 mHz to 15 mHz.When compressional ULF waves resonate with the background electrons(E<30 keV) and substorm injection electrons(30-300 keV),the electrons with higher-energy increase while the lower-energy electrons decrease,which showed that resonant electrons are accelerated effectively by compressional ULF turbulence.The efficiency of the electron acceleration depends on the character of ULF waves,the larger the amplitude of ULF waves are,the higher the acceleration efficiency is,and the bigger the spectral index is,the lower the acceleration efficiency is.In addition,the more substorm injection electrons are,the more relativistic electrons produced by ‘Transit-time acceleration’ are.Since substorms can offer substorm injection electrons,therefore large flux enhancement events of relativistic electrons(E ≥ 1 MeV) always occur during substorm time.For magnetic storms that are composed of a series of substorms,extremely large flux enhancement events of relativistic electrons can thus occur.