  Consider a centered smooth Gaussian random field $\{X(t), t\in T \}$ with a
general (nonconstant) variance function. In this work, we demonstrate that as
$u \to \infty$, the excursion probability $\mathbb{P}\{\sup_{t\in T} X(t) \geq
u\}$ can be accurately approximated by $\mathbb{E}\{\chi(A_u)\}$ such that the
error 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 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.
