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Characterization of the spatio-temporal evolution of ensembles of initial perturbations

The evolution of ensembles of initial perturbations plays an important role in ensemble
forecasting since these perturbations will determine if the ensemble of forecasts is
predictive and represents correctly the uncertainty existing in the prediction. An ensemble
is predictive if the real variable evolves as one more member of the ensemble
of forecasts. In spatiotemporal chaotic systems, the perturbations grow forming spatial
patterns that affect to the evolution of the ensemble of forecasts. We present a novel
approach to characterize and graphically represent in a 2D diagram the spatiotemporal
evolution of an ensemble of perturbations in spatiotemporal chaotic systems. Previous
studies have characterized how perturbations grow and correlate in time without considering
the spatial counterpart. The novel approach, based on the logarithm of the
perturbations, shows how both the temporal and the spatial components are essential
in the dynamics of the perturbations. The spatial mean and variance of the logarithm
of the perturbations totally describe the spatiotemporal growth of the perturbations.
Both a toy model (Lorenz 96) and numerical weather prediction systems are used to
illustrate the methodology. The 2D diagram allows uncovering some basic features of
the spatio-temporal dynamics of both ensembles of perturbations.

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