Observational error structures and the value of advanced assimilation techniques

The ability of four-dimensional variational (4DVAR) assimilation of data to reduce various observational error structures in a quasigeostrophic model is studied. It is found that 4DVAR with assimilation periods on the order o f a week is very efficient at reducing error in phase space directions that have not amplified in the past, that is, those phase space directions that do not lie on the unstable manifold of the system. This is particularly tr ue for observational errors that project in rapidly growing singular vector phase space directions. In general, long period 4DVAR changes the forecast error growth rates to ra tes similar to the leading Lyapunov exponents for the system. However, erro r structures that grow significantly faster than the leading Lyapunov vecto r and are not readily reduced by long period 4DVAR can be constructed by do ing a singular vector decomposition in the subspace of growing backward Lya punov vectors. This procedure is an approximation to calculating the singul ar vectors using an appropriate analysis error covariance metric for the as similation technique. 4DVAR acting on observational errors constructed in t his manner yields forecast error growth a factor of 5 larger than that of t he leading Lyapunov vector over a 4-day forecast period. The addition of model error places limits on the application of long assimi lation period 4DVAR. Model error adds a background level of error to the as similated solution that cannot be reduced, and also limits how far into the past the assimilation period can be extended. These effects combine to red uce the quality of the optimal assimilated state that can obtained by apply ing 4DVAR. However, model error does not diminish the ability of long assim ilation period 4DVAR to reduce rapidly growing singular vector-type error c omponents. Since long assimilation periods can potentially produce large an alysis errors if model error exists, the relative benefit of extending the assimilation period to reduce forecast error growth rates must be weighed i n a given situation.