Optimal nonlinear objective analysis

Abstract Some practical problems in objective analysis for numerical weather prediction are best solved by using nonlinear analysis equations. They include the existence of nonlinear prior constraints on the analysis, and the use of observations which are nonlinearly related to the analysis parameters, or which have non‐Gaussian error distributions. Bayesian methods are used to derive equations for the optimal (maximum likelihood) nonlinear analysis. It is shown how to incorporate a strong constraint that the four‐dimensional evolution of the analysis should be consistent with a NWP model, by reduction of the control variable to the space‐dimensioned initial field for the model. The iterative solution of the nonlinear analysis equation then involves the integration of the NWP model, and its adjoint. The behaviour of the nonlinear equations is demonstrated with a simple one‐dimensional shallow‐water model. It is shown that time‐tendency information, and indirect observations such as wind speed, or the movement of a tracer, can be used in the analysis. The resulting forecasts are better than those made from an analysis from a traditional analysis‐forecast cycle. The nonlinear method is shown to be capable of ‘moving’ a discontinuity similar to a front, to fit observations defining its position, thus giving an analysis with more detail than would be expected from the spatial resolution of the observations. The incorporation of additional nonlinear constraints, such as that used in initialization, is demonstrated. The method can be used to effectively reject observations with gross errors, by specifying a non‐Gaussian error distribution. However, this generates multiple minima which complicate the search for the best analysis, so the complex decision‐taking algorithms associated with other methods of quality control are not avoided. The convergence properties of iterative methods of solution, and approximations to the ideal equations, are studied, in order to provide some indication as to whether the nonlinear effects might be allowed for in a practical analysis scheme.