  This paper studies Gaussian random fields with Mat\'ern covariance function with smooth parameter $\nu>2$. Two cases of parameter spaces, the Euclidean space and $N$-dimensional sphere, are considered. For such smooth Gaussian fields, we have derived the explicit formulae for the expected Euler characteristic of the excursion set, the expected number and height distribution of critical points. The results are valuable for approximating the excursion probability and computing p-values for peak inference. 
Smooth Mat\'ern Gaussian Random Fields: Euler Characteristic, Expected   Number and Height Distribution of Critical Points