We demonstrate that in the continuous limit the etching mechanism yields the Kardar-Parisi-Zhang (KPZ) equation in a ()-dimensional space. We show that the parameters , associated with the surface tension, and , associated with the nonlinear term of the KPZ equation, are not phenomenological, but rather they stem from a new probability distribution function. The Galilean invariance is recovered independently of , and we illustrate this via very precise numerical simulations. We obtain firsthand the coupling parameter as a function of the probabilities. In addition, we strengthen the argument that there is no upper critical limit for the KPZ equation.