LAROCHE, Stéphane et GAUTHIER, Pierre
(1998).
« A validation of the incremental formulation of 4D variational data assimilation in a nonlinear barotropic flow ».
Tellus A, 50(5), pp. 557-572.
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Résumé
In order to meet current operational limitations, the incremental approach is being used to
reduce the computational cost of 4D variational data assimilation (4D-Var). In the incremental
4D-Var, the tangent linear (TLM) and adjoint of a simplified lower-resolution model are used
to describe the time evolution of increments around a trajectory defined by a complete fullresolution
model. For nonlinear problems, the trajectory needs to be updated regularly by
integrating the full-resolution model during the minimization. These are referred to as outer
iterations (or updates) by opposition to inner iterations done with the simpler TLM and adjoint
models to minimize a local quadratic approximation to the actual cost function. In this study,
the role of the inner and outer iterations is investigated in relation to the convergence properties
as well as to the interactions between the large (resolved by both models) and small scale
components of the flow. A 2D barotropic non-divergent model on a b-plane is used at two
different resolutions to define the complete and simpler models. Our results show that it is
necessary to have a minimal number of updates of the trajectory for the incremental 4D-Var
to converge reasonably well. To assess the impact of restricting the gradient to its large scale
components, experiments are carried out with a so-called truncated 4D-Var in which the complete
model is used to compute the gradient which is truncated afterwards to retain only those
components used in the incremental 4D-Var. A comparison between the truncated and incremental
4D-Var shows that the large-scale components of the gradient are well approximated
by the lower resolution model. With frequent updates to the trajectory, the incremental 4D-Var
converges to an analysis which is close to that obtained with the truncated 4D-Var. This
conclusion is verified when perfect observations with a complete spatial and temporal coverage
are used or when they are restricted to be available at a coarser resolution (in space and time)
than that of the model. Finally, unbiased observational error was introduced and the results
showed that at some point, the minimization is overfitting the observations and degrades the
analysis. In this context, a criterion related to the level of observational noise is found to
determine when to stop the minimization when the complete 4D-Var is used. This criterion
does not hold however for the incremental and truncated 4D-Var, thereby indicating that it
may be very difficult to establish in a more realistic context when the error is biased and the
model itself is introducing a biased error. The analysis and forecasts from the incremental
4D-Var compare well to those from a full-resolution 4D-Var and are more accurate than those
obtained from a low-resolution 4D-Var that uses only the simplified model.