  The excursion set of a $C^2$ smooth random field carries relevant information
in its various geometric measures. From a computational viewpoint, one never
has access to the continuous observation of the excursion set, but rather to
observations at discrete points in space. It has been reported that for
specific regular lattices of points in dimensions 2 and 3, the usual estimate
of the surface area of the excursions remains biased even when the lattice
becomes dense in the domain of observation. In the present work, under the key
assumptions of stationarity and isotropy, we demonstrate that this limiting
bias is invariant to the locations of the observation points. Indeed, we
identify an explicit formula for the bias, showing that it only depends on the
spatial dimension $d$. This enables us to define an unbiased estimator for the
surface area of excursion sets that are approximated by general tessellations
of polytopes in $\mathbb{R}^d$, including Poisson-Voronoi tessellations. We
also establish a joint central limit theorem for the surface area and volume
estimates of excursion sets observed over hypercubic lattices.
