  PSR J0108$-$1431 is an old pulsar where the X-ray emission is expected to
have a thermal component from the polar cap and a non-thermal component from
the magnetosphere. Although the phase-integrated spectra are fit best with a
single non-thermal component modeled with a power-law (PL) of photon index
$\Gamma=2.9$, the X-ray pulse profiles do show the presence of phase-separated
thermal and non-thermal components. The spectrum extracted from half the
rotational phase away from the X-ray peak fits well with either a single
blackbody (BB) or a neutron star atmosphere (NA) model, whereas, the spectrum
from the rest of the phase range is dominated by a PL. From Bayesian analysis,
the estimated BB area is smaller than the expected polar cap area for a dipolar
magnetic field with a probability of 86% whereas the area estimate from the NA
model is larger with a probability of 80%. Due to the ambiguity in the thermal
emission model, the polar cap area cannot be reliably estimated and hence
cannot be used to understand the nature of the surface magnetic field. Instead,
we can infer the presence of multipolar magnetic field from the misalignment
between the pulsar's thermal X-ray peak and the radio emission peak. For
J0108$-$1431, we estimated a phase-offset $\Delta\phi > 0.1$ between the
thermal polar cap emission peak and the radio emission peak and argue that this
is best explained by the presence of a multipolar surface magnetic field.
