The expected Euler characteristic approximation to excursion probabilities of smooth Gaussian random fields with general variance functions

Consider a centered smooth Gaussian random field {X(t), t is an element of T} with a general (nonconstant) variance function. In this work, we demonstrate that as u -> infinity, the excursion probability P{sup(t is an element of T) X(t) >= u} can be accurately approximated by E{chi(A(u))} such that the error decays at a super-exponential rate. Here, A(u) = {t is an element of T : X(t)>= u} represents the excursion set above u, and E{chi(A(u))} is the expectation of its Euler characteristic chi(A(u)). This result substantiates the expected Euler characteristic heuristic for a broad class of smooth Gaussian random fields with diverse covariance structures. In addition, we employ the Laplace method to derive explicit approximations to the excursion probabilities.