  Flavour oscillations of sub-GeV atmospheric neutrinos and antineutrinos,
traversing different distances inside the Earth, are a promising source of
information on the leptonic CP phase $\delta$. In that energy range, the
oscillations are very fast, far beyond the resolution of modern neutrino
detectors. However, the necessary averaging over the experimentally typical
energy and azimuthal angle bins does not wash out the CP violation effects. In
this paper we derive very accurate analytic compact expressions for the
averaged oscillations probabilities. Assuming spherically symmetric Earth, the
averaged oscillation probabilities are described in terms of two analytically
calculable effective parameters. Based on those expressions, we estimate
maximal magnitude of CP-violation effects in such measurements and propose
optimal observables best suited to determine the value of the CP phase in the
PMNS mixing matrix.
