Bred vector

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The growth rates of bred vectors in the Lorenz system. Red indicates the fastest-growing bred vectors while blue the slowest.

In applied mathematics, bred vectors are perturbations, related to Lyapunov vectors, that capture fast-growing dynamical instabilities of the solution of a numerical model. They are used, for example, as initial perturbations for ensemble forecasting in numerical weather prediction, they were introduced by Zoltan Toth and Eugenia Kalnay.[1]


Bred vectors are created by adding initially random perturbations to a nonlinear model; the control (unperturbed) and the perturbed models are integrated in time, and periodically the control solution is subtracted from the perturbed solution. This difference is the bred vector; the vector is scaled to be the same size as the initial perturbation, and is then added back to the control to create the new perturbed initial condition. After a short transient period, this "breeding" process creates bred vectors dominated by the naturally fastest-growing instabilities of the evolving control solution.


  1. ^ Toth, Zoltan; Kalnay, Eugenia (December 1993). "Ensemble Forecasting at NMC: The Generation of Perturbations". Bulletin of the American Meteorological Society. 74 (12): 2317–2330. Bibcode:1993BAMS...74.2317T. doi:10.1175/1520-0477(1993)074<2317:EFANTG>2.0.CO;2.
  • Kalnay, E. (2003). Atmospheric Modeling, Data Assimilation and Predictability. Cambridge: Cambridge University Press. ISBN 978-0-521-79629-3.
  • Glickman, T. S., ed. (2000). Glossary of Meteorology (Second ed.). Boston, Massachusetts: American Meteorological Society.