  The roughening of interfaces moving in inhomogeneous media is investigated by
numerical integration of the phenomenological stochastic differential equation
proposed by Kardar, Parisi, and Zhang [Phys. Rev. Lett. 56, 889, (1986)] with
quenched noise (QKPZ). We express the evolution equations for the mean height
and the roughness into two contributions: the local and the lateral one. We
compare this two contributions with the ones obtained for two directed
percolation deppining models (DPD): the Tang and Leschhorn model [Phys. Rev A
45, R8309 (1992)] and the Buldyrev et al. model [Phys. Rev. A 45, R8313 (1992)]
by Braunstein al. [J. Phys. A 32, 1801 (1999); Phys. Rev. E 59, 4243 (1999)].
Even these models have being classified in the same universality class that the
QKPZ the contributions to the growing mechanisms are quite different. The
lateral contribution in the DPD models, leads to an increasing of the roughness
near the criticality while in the QKPZ equation this contribution always
flattens the roughness. These results suggest that the QKPZ equation does not
describe properly the DPD models even when the exponents derived from this
equation are similar to the one obtained from simulations of these models.
