Generalized inversion of a global numerical weather prediction model, II: Analysis and implementation

Summary This is a sequel to Bennett, Chua and Leslie (1996), concerning weak-constraint, four-dimensional variational assimilation of reprocessed cloud-track wind observations (Velden, 1992) into a global, primitive-equation numerical weather prediction model. The assimilation is performed by solving the Euler-Lagrange equations associated with the variational principle. Bennett et al. (1996) assimilate 2436 scalar wind components into their model over a 24-hour interval, yielding a substantially improved estimate of the state of the atmosphere at the end of the interval. This improvement is still in evidence in forecasts for the next 48 hours.The model and variational equations are nonlinear, but are solved as sequence of linear equations. It is shown here that each linear solution is precisely equivalent to optimal or statistical interpolation using a background error covariance derived from the linearized dynamics, from the forcing error covariance, and from the initial error covariance. Bennett et al. (1996) control small-scale flow divergence using divergence dissipation (Talagrand, 1972). It is shown here that this approach is virtually equivalent to including a penalty, for the gradient of divergence, in the variational principle. The linearized variational equations are solved in terms of the representer functions for the wind observations. Diagonalizing the representer matrix yields rotation vectors. The rotated representers are the array modes of the entire system of the model, prior covariances and observations. The modes are the observable degrees of freedom of the atmosphere. Several leading array modes are presented here. Finally, appendices discuss a number of technical implementation issues: time convolutions, convergence in the presence of planetary shear instability, and preconditioning the essential inverse problem.With 9 Figures