Probability for all points on a path in a gaussian random field

Question

I have a gaussian random field $u(x)$, $x \in R^2$ , with a covariance function $C(d)$

and I need to calculate probability

$P[ u(x) < u_0, \forall x \in S]$, where $S$ is a straight segment in $R^2$ space.

Is there a direct approach for me to do this without sampling? i haven't been able to find by googling , apologize if too obvious