  We present detailed computations of the vertical structure of an accretion
disc illuminated by hard X-ray radiation with the code {\sc titan-noar}
suitable for Compton thick media. The energy generated via accretion is
dissipated partially in the cold disc as well as in the X-ray source. We study
the differences between the case where the X-ray source is in the form of a
lamp post above the accretion disc and the case of a heavy corona. We consider
radiative heating via Comptonization together with heating via photo-absorption
on numerous heavy elements as carbon, oxygen, silicon, iron. The transfer in
lines is precisely calculated. A better description of the heating/cooling
through the inclusion of line transfer, a correct description of the
temperature in the deeper layers, a correct description of the entire disc
vertical structure, as well as the study of the possible coronal pressure
effect, constitute an improvement in comparison to previous works. We show that
exact calculations of hydrostatic equilibrium and determination of the disc
thickness has a crucial impact on the optical depth of the hot illuminated
zone. We assume a moderate illumination where the viscous flux equals the X-ray
radiation flux. A highly ionized skin is created in the lamp post model, with
the outgoing spectrum containing many emission lines and ionization edges in
emission or absorption in the soft X-ray domain, as well as an iron line at
$\sim 7 $ keV consisting of a blend of low ionization line from the deepest
layers and hydrogen and helium like resonance line from the upper layers, and
almost no absorption edge, contrary to the case of a slab of constant density.A
full heavy corona completely suppresses the highly ionized zone on the top of
the accretion disc and in such case the spectrum is featureless.
