Breeding (Toth and Kalnay, 1997)
At the initial time, a small random perturbation is introduced in the
reference state (or control state), so that a
perturbed state is created. The perturbation is then the vector
difference between the perturbed state and the control state. Both the control
state and the perturbed state undergo a nonlinear (model) evolution. As an
effect of the sensitive dependence of the state evolution on the initial
condition (characteristic of nonlinear, chaotic systems), the components of the
perturbation on the unstable directions grow, while other components decrease.
After a short time interval the pertubation is renormalized so that its
amplitude is reduced to the initial small value, without changing its direction.
This interval may be composed of several time steps, however it has to be short
enough, so that the perturbation evolution can be considered linear. Then the
time evolution of both (perturbed and control) states proceeds for another
interval until the next renormalization time. After several renormalization
steps, following the non-linear trajectory of the control state, the
perturbation approximately assumes the structure of a linear combination of the
most unstable modes.
