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

Four-dimensional variational assimilation (4DVAR) of noisy observations in a multi-layerquasi-geostrophic model is studied, in both the perfect and imperfect model settings. Withinthe perfect model setting, the quality of the assimilated state improves significantly when theassimilation period is extended more than one week into the past. Specifically, when observationsare supplied every 6 h, the squared error in the assimilated state at the end of the assimilationtime period (the present) saturates at a value two orders of magnitude smaller than theimposed observational error for an assimilation period of 10 days. Further, this reduction inerror occurs not only in measures explicitly minimized by 4DVAR, but for all standard measuresof 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 evenfor weather regime transitions, which are shown to be predictable at lead times of 10 days. Theuse of long assimilation periods extends forecast lead times approximately 5 days over the casewhen assimilation periods are on the order of one day. The structure of the analysis error whenlong assimilation period 4DVAR is applied is examined. This error is primarily concentratedin the midlatitude storm tracks. The reduction in analysis error is increasingly efficient at smallscales as the assimilation period is increased; consequently, for long assimilation periods theanalysis error projects strongly into the subspace of the leading Lyapunov vectors. The performanceof 4DVAR in an imperfect model setting is also examined, and is found to depend uponthe growth rate of the model errors. For rapidly growing model errors, extension of the assimilationperiod beyond about 1–2 days results in a degradation in the quality of the assimilatedstate as well as in the forecast quality. However, for model error growth rates similar to thegrowth rates of the leading Lyapunov vectors of the system, improvements in the assimilatedstate similar to those found for the perfect model are obtained. As such, it is estimated thatassimilation times of 3–5 days for current levels of model error may improve the quality ofassimilated states and forecasts in an operational setting.