A POD reduced order 4-D VAR adaptive mesh ocean modelling approach
F. Fang, C.C. Pain , I.M. Navon, M.D. Piggott , G.J. Gorman, P. Allison, A.J.H. Goddard
A novel Proper Orthogonal Decomposition (POD) inverse model, developed for a mesh adaptive ocean model (the Imperial College Ocean Model, ICOM) is presented here. The new POD model is validated using the Munk gyre ow test case, where it inverts for initial conditions. The optimised velocity fields exhibit overall good agreement with those generated by the full model. The correlation between the inverted modelled and the true velocity is 80%-98% over the majority of the domain. Error estimation (including the projection error, subspace integration error and the error introduced by the controls) was used to judge quality of reduced adaptive mesh models. The cost function (consisting of the misfit between the inverted modelled and true velocity values spatially and temporally) is reduced by 50% of its original value, and further by 25% after the POD bases are updated. In this study, the reduced adjoint model is derived directly from the discretised reduced forward model. The whole optimization procedure is undertaken completely in reduced space. Computational cost for the 4-D Var data assimilation is significantly reduced (here a decrease of 70% in the test case) by decreasing the dimensional size of the control space, in both the forward and adjoint models. Computational efficiency is further enhanced (by a factor of N, here, N is the number of times to run the reduced models) since both the reduced forward and adjoint models are constructed by a series of time-independent sub-matrices. These sub-matrices are calculated prior to running the reduced models. The reduced forward and adjoint models can thus be run repeatedly with negligible computational costs. An adaptive POD 4-D Var is employed to update the POD bases as minimization advances and loses control, thus adaptive updating of the POD bases is necessary (here, when the value of cost function cannot be decreased by more than 10 ..3 between the consecutive iterations). It is noted that the adaptive POD 4D-Var is not always effective. An appropriate choice of initial guess controls can help to achieve a reasonable minimum location during the optimization procedure. Previously developed numerical approaches  are employed to accurately represent the geostrophic balance and improve the efficiency of the POD simulation.