(i) Spontaneous Growth
Spontaneous growth defines the occurrence of random urbanization of land. In the cellular automaton framework this means that any non-urbanized cell on the lattice has a certain (small) probability of becoming urbanized in any time step. Thus, whether a given cell U(i,j,t) at coordinate (i,j) at time t will be urbanized at time t+1 can be expressed by

(1) U(i,j,t+1) = f1[ dispersion_coefficient , slope_coefficient , U(i,j,t), random ],

where the parameter dispersion_coefficient ( diffusion_coefficient in previous literature (Clarke, Hoppen, Gaydos 1996)) determines the (small) spontaneous, global urbanization probability, and the slope_coefficient parameter determines the weighted probability of the local slope. The stochasticity of the process is indicated by random. If the cell is already urbanized or excluded from urbanization, it will not change, and therefore the ability to transition also depends on the cell’s own current value.

	Spontaneous Growth: F(dispersion_coefficient, slope_coefficient) { for (p < dispersion_value) { select pixel location (i,j) at random if ((i,j) is available for urbanization) { (i,j) = urban New Spreading Center Growth } } } end spontaneous growth
Spontaneous growth example and pseudo code.