  We investigate Lipschitz-Killing curvatures for excursion sets of random
fields on $\mathbb R^2$ under small spatial-invariant random perturbations. An
expansion formula for mean curvatures is derived when the magnitude of the
perturbation vanishes, which recovers the Gaussian Kinematic Formula at the
limit by contiguity of the model. We develop an asymptotic study of the
perturbed excursion area behaviour that leads to a quantitative non-Gaussian
limit theorem, in Wasserstein distance, for fixed small perturbations and
growing domain. When letting both the perturbation vanish and the domain grow,
a standard Central Limit Theorem follows. Taking advantage of these results, we
propose an estimator for the perturbation which turns out to be asymptotically
normal and unbiased, allowing to make inference through sparse information on
the field.
