Discretization error and signal/error correlation in atmospheric data assimilation : (II). The effect of unresolved scales

Abstract

This paper is the 2nd of a 2-part study of the effects of forward interpolation error and model discretization error on atmospheric data assimilation. Utilizing the generalized Kalman filter developed in part (I), that investigation is here extended to the case where not all scales in the signal are resolvable by the analysis mesh (although they may all be sampled by the observation network). The error in the unresolved scales is thus able to affect the forward interpolation and model discretization errors and the signal/error correlations considered in part (I). In some circumstances, the results are found to be qualitatively similar to those observed when all scales were resolved, but at a much higher error level. This strengthens the corresponding results of part (I) by extending them to the situation where there are unresolved scales. The most important findings of this type are the following. (a) Models, however imperfect, reduce analysis error. (b) The neglect of signal/error correlation can result in a serious underestimate of analysis error. (c) While model truncation error has a blue spectrum, proper inclusion of signal/error correlation has the effect of reddening the spectrum. Other important findings of this paper are the following. (i) The error in the unresolved scales cannot be eliminated by data assimilation. Moreover, the existence of unresolved scales not only increases the magnitude of the forward interpolation and model discretization errors due to errors in these scales, but also produces error in the resolved scales. (ii) While simple models can provide useful information about the mechanisms of analysis error growth and decay, the coupling of these models with very special observation networks (e.g., coincident with the analysis mesh) may yield results which are misleading with respect to more realistic models and networks. (iii) For imperfect models, both the magnitude of the analysis error and the forecast-error correlation structure are much more sensitive to model resolution than to observation network density.