Empirical correction of a dynamical model. Part I: Fundamental issues

The possibility of empirically correcting a nonlinear dynamical model is ex amined. The empirical correction is constructed by fitting a first-order Ma rkov model to the forecast errors using initial conditions as predictors. T he dynamical operator of the Markov model can then be subtracted from the o riginal forecast model to correct the forecast. The procedure is based an a n earlier work by Leith, suitably modified for practical applications. The effects of analysis errors and finite difference approximations are analyze d. 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 b e checked without knowing the true state exactly. The procedure is sensitiv e to sampling errors, but this problem was not explored. These formal conclusions are tested on a nonlinear quasigeostrophic model i n which model error is created by changing the model parameters. A "systema tic error" correction, which does not depend on state and is herd constant throughout the integration, does not improve the forecast skin of the model because the contrived error is primarily state dependent. However, an empi rical correction, which depends on instantaneous state, improves the foreca st skill of the model by 10 days, and further iterations extend the skill t o the limit imposed by observation error (about 20 days in this example).