4D variational data assimilation problems in geophysical fluids consist in solving nonlinear least square problems.The incremental 4D-Var method is a popular technique to tackle high dimensional problems as it is equivalent to applying a truncated Gauss-Newton iteration. The development of efficient numerical techniques, like the restricted preconditioned conjugate gradient (RPCG) method, allow to solve the embedded quadratic optimization subproblem (also called inner loop) in observation space. This approach becomes computationally attractive when the number of observations is much smaller than the dimension of the control space (for instance the dimension of the initial state vector). However, the amount of observations to assimilate can still be large even in this favorable case.
In order to reduce the computing costs of the Incremental 4D-Var in observation space, we proposed an observation-thinning strategy that exploits an adaptive structure of the observations in the spirit of multigrid techniques. The thinned observation set is defined using a hierarchy of observations, from coarsest to finest level. Starting from the coarsest set of observations, observations from the next level will be included in the observation set according to the influence they have on the solution, as measured by an estimate on the solution variation between two consecutive levels. Numerical experiments performed in toy models highlighted the benefits of this approach considering both the reductions in the computing costs/time and the amount of assimilated observations. Based on this multilevel approach, we investigate the introduction of multigrid techniques in the resolution of the optimization problem in order to speed up the convergence of the dual space iterative solver. Such techniques aim at exploiting the smoothing properties of iterative solvers by introducing coarse grid correction steps that can efficiently remove the large scale components of the error.