  We perform an orthogonal basis tight binding fit to an LAPW calculation of
paramagnetic Na$_x$CoO$_2$ for several dopings. The optimal position of the
apical oxygen at each doping is resolved, revealing a non-trivial dependence of
the band structure and Fermi surface on oxygen height. We find that the small
e$_{g'}$ hole pockets are preserved throughout all investigated dopings and
discuss some possible reasons for the lack of experimental evidence for these
Fermi sheets.
