  This contribution establishes exact tail asymptotics of
$\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous
Gaussian random fields $X$ on a bounded convex set
$\mathbf{E}\subset\mathbb{R}^2$, with variance function that attains its
maximum on a segment on $\mathbf{E}$. These findings extend the classical
results for homogeneous Gaussian random fields and Gaussian random fields with
unique maximum point of the variance. Applications of our result include the
derivation of the exact tail asymptotics of the Shepp statistics for stationary
Gaussian processes, Brownian bridge and fractional Brownian motion as well as
the exact tail asymptotic expansion for the maximum loss and span of stationary
Gaussian processes.
