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4D-Var calculates a forecast model trajectory that best fits the available observations to within the observational error over a period of time. The observational error includes allowance for the finite resolution of the model. Since the model cannot represent the atmosphere exactly, a trajectory cannot be fitted to observations for a period longer than 12 hours without making corrections to it. The expectation is that the best forecast will be achieved by constructing a best fit over a long time interval while making the smallest possible corrections, and then extrapolating forward in time. This exploits the maximum amount of observed information given that only limited observations are available in any one six hour period.

Fitting a model trajectory to observations is very computationally demanding. The only feasible method is to assume that an accurate background forecast is available and that an adequate fit to observations can be made by adding small increments to it. The problem can then be written as minimising a cost function which measures the departure of the corrected forecast from the observations and from the background forecast, weighted by the errors in each.

The iteration strategy is to approximate the evolution of increments to the forecast model, which is nonlinear, by a linear perturbation forecast model. It is then possible to minimise the cost function as a function of the initial increments. Optimising this procedure requires careful preconditioning, and interleaving of nonlinear iterations with linear iterations. The nonlinearity comes both from the forecast model itself and from the nonlinear relation between many observed quantities, such as satellite radiances, and model variables.

4D-Var is used at many operational centres as well as the Met Office. However, future computers will have an increasingly parallel architecture, and ensemble methods, which can fully exploit this, will become more attractive. It is therefore necessary to establish the full potential of 4D-Var in order to see whether it will remain the preferred operational method in the future.

Key Aims

• To determine whether 4D-Var should remain the operational method of data assimilation in the medium to long-term
• To improve the meteorological performance of the operational 4D-Var system, as measured by the resulting forecast accuracy
• To improve the computational efficiency of 4D-Var
• To extend 4D-Var to the convective scale
• To improve the representation of prior information in 4D-Var

Current Projects

• Optimisation: preconditioning of the minimisation using Hessian eigenvectors; optimising the iteration strategy over inner and outer loops; optimising the parameters controlling the expensive perturbation forecast and adjoint integrations.
• Allowing for important nonlinearities while retaining much of the efficiency of linear methods: nonlinear model/spin-up handled via outer-loop; nonlinear relationship between errors and best estimate via nonlinear transforms, nonlinear observed variables and non-Gaussian errors via nonlinear observation operators.
• Model error: allowing for model error by an appropriate regularisation of the inverse problem solved in each data assimilation cycle; explicit representation of parts of the model error; use of longer windows with increments calculated on subwindows.
• Training data for estimating background error: use of MOGREPS, use of ensembles of analyses; use of analysis increments.
• Covariance model-dynamics: use of potential vorticity; inclusion of separate variables for the boundary layer; methods of estimating vertical motion correlated with other variables.
• Covariance model-spatial representation: optimisation of representation in spectral and vertical mode space; use of models giving more information in physical space.
• 4D-Var physics: choice of control variable for moisture; use of total water control variable; exploitation of new physics developments in the Unified Model; extension of the climatological covariance model to include moisture; incorporation of additional physics in the perturbation forecast model.
• Convective scale: incorporation of large-scale information; choice of analysis parameters for the convective scale; training data to calibrate covariances.