  We study the critical behavior of a driven interface in a medium with random
pinning forces by analyzing spatial and temporal correlations in a lattice
model recently proposed by Sneppen [Phys. Rev. Lett. {\bf 69}, 3539 (1992)].
The static and dynamic behavior of the model is related to the properties of
directed percolation. We show that, due to the interplay of local and global
growth rules, the usual method of dynamical scaling has to be modified. We
separate the local from the global part of the dynamics by defining a train of
causal growth events, or "avalanche", which can be ascribed a well-defined
dynamical exponent $z_{loc} = 1 + \zeta_c \simeq 1.63$ where $\zeta_c$ is the
roughness exponent of the interface. We observe that the avalanche size
distribution obeys a power-law decay with an exponent $\kappa \simeq 1.25$.
