Four-dimensional variational assimilation and predictability in a quasi-geostrophic model

Abstract

Four-dimensional variational assimilation (4DVAR) of noisy observations in a multi-layer quasi-geostrophic model is studied, in both the perfect and imperfect model settings. Within the perfect model setting, the quality of the assimilated state improves significantly when the assimilation period is extended more than one week into the past. Specifically, when observations are supplied every 6 h, the squared error in the assimilated state at the end of the assimilation time period (the present) saturates at a value two orders of magnitude smaller than the imposed observational error for an assimilation period of 10 days. Further, this reduction in error occurs not only in measures explicitly minimized by 4DVAR, but for all standard measures of error. For realistic levels of observational error, the extension of forecast lead times is large, exceeding 15 days for global forecasts when the assimilation period is 10 days. This holds even for weather regime transitions, which are shown to be predictable at lead times of 10 days. The use of long assimilation periods extends forecast lead times approximately 5 days over the case when assimilation periods are on the order of one day. The structure of the analysis error when long assimilation period 4DVAR is applied is examined. This error is primarily concentrated in the midlatitude storm tracks. The reduction in analysis error is increasingly efficient at small scales as the assimilation period is increased; consequently, for long assimilation periods the analysis error projects strongly into the subspace of the leading Lyapunov vectors. The performance of 4DVAR in an imperfect model setting is also examined, and is found to depend upon the growth rate of the model errors. For rapidly growing model errors, extension of the assimilation period beyond about 1-2 days results in a degradation in the quality of the assimilated state as well as in the forecast quality. However, for model error growth rates similar to the growth rates of the leading Lyapunov vectors of the system, improvements in the assimilated state similar to those found for the perfect model are obtained. As such, it is estimated that assimilation times of 3-5 days for current levels of model error may improve the quality of assimilated states and forecasts in an operational setting.