  Knowledge of the composition of the Earth's interior is highly relevant to
many geophysical and geochemical problems. Neutrino oscillations are modified
in a non-trivial way by the matter effects and can provide valuable and unique
information not only on the density but also on the chemical and isotopic
composition of the deep regions of the planet. In this paper, we re-examine the
possibility of performing an oscillation tomography of the Earth with
atmospheric neutrinos and antineutrinos to obtain information on the
composition and density of the outer core and the mantle, complementary to that
obtained by geophysical methods. Particular attention is paid to the D$^{\prime
\prime}$ layer just above the core-mantle boundary and to the water (hydrogen)
content in the mantle transition zone. Our analysis is based on a Monte-Carlo
simulation of the energy and azimuthal angle distribution of $\mu$-like events
generated by neutrinos. Taking as reference a model of the Earth consisting of
55 concentric layers with constant densities determined from the PREM, we
evaluate the effect on the number of events due to changes in the composition
and density of the outer core and the mantle. To examine the capacity of a
detector like ORCA to resolve such variations, we construct regions in planes
of two of these quantities where the statistical significance of the
discrepancies between the reference and the modified Earth are less than
$1\sigma$. The variations are implemented in such a way that the constraint
imposed by both the total mass of the Earth and its moment of inertia are
verified.
