A new hybrid formulation for the background error covariance in the IFS: evaluation

A new hybrid formulation for the background errors in the European Centre for Medium-Range Weather Forecasts (ECMWF) 4D-Var was proposed in Massart (2018) and tested in simple configurations. In the following document, we evaluate this new formulation in a more realistic configuration. In order to perform the evaluation, the new formulation had to be adapted to the current implementation of ECMWF
4D-Var. The 4D-Var analysis is the result of an incremental 4D-Var that successively minimises linear versions of the cost function (outer loop iteration). In ECMWF incremental 4D-Var implementation, the resolution of  the tangent-linear and adjoint versions of the model and observation operators used in the analysis increases with each outer loop. We demonstrate that the new hybrid formulation can be used together with this implementation of the incremental 4D-Var, but one has to consider carefully the change of resolution between outer loops.

We designed three experiments to carry out the evaluation. The first experiment is based on static background errors and referred to as Bs. The second experiment is based on the operational formulation of the background errors as implemented in CY45R1 and referred to as Bo. The last experiment is based on the new hybrid formulation with an hybrid weight of 50 %, and is referred to as a -Bh. They are all designed so that the experiments have the same background error variances on average.
The evaluation is based on a comparison between the a -Bh experiment and the two other at the horizontal resolution of TCo 399, with 3 outer loops, and over a winter season, from November 2017 to February 2018 (4 months). The comparison against the static formulation aims at assessing the impact of flow-dependent background errors. The comparison against CY45R1 operational formulation aims at assessing if the new formulation could complement the current one and help improving the forecast scores.

The main result from the evaluation is that the analysis of the a -Bh experiment is spatially smoother that the analysis from the two other experiments even if, for this experiment, the fields of the atmospheric model variables have more energy in the small scales (from the wavenumber around 100). We deduce that this feature accelerates the convergence of the minimisation of the a -Bh experiment. This experiment requires between 1 and 2 less iterations than for the Bs experiment, and between 5 and 7 less iterations than for the Bo experiment. We also deduce that the smoothness of the analysis from the a -Bh experiment is likely to be the reason why the first-guess derived from it is on average closer to the observations than the one derived from the two other experiments, by 0.5% compared to Bs and by 1.3% compared to Bo. For the satellite sounding data, the Bo experiment performs better and the main difference is for the ATMS instrument for which the Bo first-guess is closer to the observations by an average of 3.3% compared to a -Bh first-guess. This difference may explain why the forecast (compared against its own analysis or observations) from the a -Bh experiment is the best, but only up to day 3 to 5. Then, the medium-range forecast from the Bo experiment tends to be the best.

Overall, the results show that the new hybrid formulation of the background errors provides better forecast scores than when using static background errors. Yet, it currently fails to compete with the current formulation for medium-range forecasts. Nonetheless, the results also show that the current formulation could be improved. For example, the wind forecast is better for all forecast ranges in the southern hemisphere and tropics when it is derived from the a -Bh experiment. The relative humidity forecast fromthe a -Bh experiment is also in general the best. These results encourage us to further develop the new hybrid formulation.