  Trans-ionospheric pulse pairs are the most powerful natural radio signals on
the Earth and associated with lightning. They have been discovered for two
decades by satellites, but their origin still remains elusive. Here we
attribute these radio signals to relativistic electrons produced by
cloud-to-ground lightning. When these electrons strike the ground, radio bursts
are emitted towards space in a narrow cone. This model naturally explains the
interval, duration, polarization, coherence and bimodal feature of the pulse
pairs. Based on electron parameters inferred from x-ray observation of
lightning, the calculated signal intensity agrees with the measurement of
satellites. Our results are useful to develop global warning system of storms
and hurricane based on GPS satellites.
