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Consider a centered smooth Gaussian random field $\{X(t), t\in T \}$ with ageneral (nonconstant) variance function. In this work, we demonstrate that as$u \to \infty$, the excursion probability $\mathbb{P}\{\sup_{t\in T} X(t) \gequ\}$ can be accurately approximated by $\mathbb{E}\{\chi(A_u)\}$ such that theerror decays at a super-exponential rate. Here, $A_u = \{t\in T: X(t)\geq u\}$represents the excursion set above $u$, and $\mathbb{E}\{\chi(A_u)\}$ is theexpectation of its Euler characteristic $\chi(A_u)$. This result substantiatesthe expected Euler characteristic heuristic for a broad class of smoothGaussian random fields with diverse covariance structures. In addition, weemploy the Laplace method to derive explicit approximations to the excursionprobabilities.

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