Empirical Correction of a Dynamical Model. Part I: Fundamental Issues

The possibility of empirically correcting a nonlinear dynamical model is examined. The empirical correction is constructed by fitting a first-order Markov model to the forecast errors using initial conditions as predictors. The dynamical operator of the Markov model can then be subtracted from the original forecast model to correct the forecast. The procedure is based on an earlier work by Leith, suitably modified for practical applications. The effects of analysis errors and finite difference approximations are analyzed. It is shown that uncorrelated analysis errors produce spurious terms in the empirical correction operator that act to damp eddy variance at a rate inversely proportional to the lead time used to estimate the Markov model. The finite difference approximation is appropriate as long as the forecast errors grow linearly with lead time. These restrictions can be interpreted as setting lower and upper limits on the lead time, respectively, and can be checked without knowing the true state exactly. The procedure is sensitive to sampling errors, but this problem was not explored. These formal conclusions are tested on a nonlinear quasigeostrophic model in which model error is created by changing the model parameters. A “systematic error” correction, which does not depend on state and is held constant throughout the integration, does not improve the forecast skill of the model because the contrived error is primarily state dependent. However, an empirical correction, which depends on instantaneous state, improves the forecast skill of the model by 10 days, and further iterations extend the skill to the limit imposed by observation error (about 20 days in this example).