  We complete our study of pulsars' non-uniform surface temperature and of its
effects on their soft X-ray thermal emission. Our previous work had shown that,
due to gravitational lensing, dipolar fields cannot reproduce the strong
pulsations observed in Vela, Geminga, PSR 0656+14, and PSR 1055-52. Assuming a
standard neutron star mass of 1.4 Msol, we show here that the inclusion of a
quadrupolar component, if it is suitably oriented, is sufficient to increase
substantially the pulsed fraction, Pf, up to, or above, the observed values if
the stellar radius is 13 km or even 10 km. For models with a radius of 7 km the
maximum pulsed fraction obtainable with (isotropic) blackbody emission is of
the order of 15% for orthogonal rotators (Vela, Geminga and PSR 1055-52) and
only 5% for an inclined rotator as PSR 0656+14. Given the observed values this
indicates that the neutron stars in Geminga and PSR 0656+14 have radii
significantly larger than 7 km and, given the very specific quadrupole
components required to increase Pf, even radii of the order of 10 km may be
unlikely in all four cases.
  We confirm our previous finding that the pulsed fraction always increases
with photon energy, below about 1 keV, when blackbody emission is used and show
that it is due to the hardenning of the blackbody spectrum with increasing
temperature. The observed decrease of pulsed fraction may thus suggest that the
emitted spectrum softens with increasing temperature.
  Finally, we apply our model to reassess the magnetic field effect on the
outer boundary condition used in neutron star cooling models and show that, in
contradistinction to several previous claims, it is very small and most
probably results in a slight reduction of the heat flow through the envelope.
