  We study the local scaling properties of driven interfaces in disordered
media modeled by the Edwards-Wilkinson equation with quenched noise. We find
that, due to the super-rough character of the interface close to the depinning
transition, the local width exhibits an anomalous temporal scaling. It does not
saturate at a characteristic time $t_s(l) \sim l^z$ as expected, but suffers an
anomalous temporal crossover to a different time regime $t^{\beta_*}$, where
$\beta_* \simeq 0.21 $. This is associated with the existence of a local
roughness exponent $\alpha_{loc} \simeq 1$ that differs from the global one
$\alpha \simeq 1.2$. The relevance of the typical size of pinned sites regions
near the critical point is investigated and the definition of the critical
depinning is dicussed.
