Application of Exact Newton Optimisation to the Maximum Likelihood Ensemble Filter

The Newton method is used for optimisation in the maximum likelihood ensemble filter (MLEF) to improve analysis convergence and accuracy. The proposed method is compared against the original method using the conjugate gradient (CG) method preconditioned by the Hessian for optimisation. The mechanisms of the two minimisation methods are illustrated with optimisation for the Booth and Rosenbrock functions. Comparisons are then made in simple data assimilation experiments. In the assimilation of a single wind speed, the Newton method is affected by the gradient and Hessian approximated by the forecast ensemble but the gradient norm decreases geometrically. The CG method is terminated at the first step unless the ensemble perturbation matrix in the observation space is fixed. In the cycled experiments using a Korteweg–de Vries–Burgers equation model with a quadratic observation operator, the Newton method and the preconditioned CG method with gradients updated during iterations yield an analysis with comparable accuracy, but the CG with the fixed gradient is found to produce an analysis that leads to unstable forecast. When the number of Newton iterations is limited to one, the solutions remain suboptimal, significantly destabilising the model. The experimental results indicate that the Newton method is a promising alternative to the CG method with a line search for optimisation in MLEF.