  We study the spatiotemporal dynamics of random spatially distributed
noninfinitesimal perturbations in one-dimensional chaotic extended systems. We
find that an initial perturbation of finite size $\epsilon_0$ grows in time
obeying the tangent space dynamic equations (Lyapunov vectors) up to a
characteristic time $t_{\times}(\epsilon_0) \sim b - (1/\lambda_{max}) \ln
(\epsilon_0)$, where $\lambda_{max}$ is the largest Lyapunov exponent and $b$
is a constant. For times $t < t_{\times}$ perturbations exhibit spatial
correlations up to a typical distance $\xi \sim t^z$. For times larger than
$t_{\times}$ finite perturbations are no longer described by tangent space
equations, memory of spatial correlations is progressively destroyed and
perturbations become spatiotemporal white noise. We are able to explain these
results by mapping the problem to the Kardar-Parisi-Zhang universality class of
surface growth.
