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Prediction of the Probable Errors of Predictions

Philip D. ThompsonNational Center for Atmospheric Research, Boulder, CO 80307

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Abstract

We propose here a method of “stochastic-dynamic” prediction that is computationally more efficient than integration of the full set of “second-moment” equations. This gain is achieved by omitting covariances between modes in different interacting triads, and by expressing intratriad covariances in terms of error variances, via the conditions for invariance of products of invariants. The resulting evolution equations for the error variances of all modal amplitudes constitute a closed system involving only those error variances.

To test the accuracy of this method, we have compared the predicted error variances with those calculated directly from an ensemble of 100 individual predictions, starting from an ensemble of 100 initial states containing random errors. These agree very well up to about the doubling time of total rms error, but later diverge as the effects of indirect interactions accumulate.

Abstract

We propose here a method of “stochastic-dynamic” prediction that is computationally more efficient than integration of the full set of “second-moment” equations. This gain is achieved by omitting covariances between modes in different interacting triads, and by expressing intratriad covariances in terms of error variances, via the conditions for invariance of products of invariants. The resulting evolution equations for the error variances of all modal amplitudes constitute a closed system involving only those error variances.

To test the accuracy of this method, we have compared the predicted error variances with those calculated directly from an ensemble of 100 individual predictions, starting from an ensemble of 100 initial states containing random errors. These agree very well up to about the doubling time of total rms error, but later diverge as the effects of indirect interactions accumulate.

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