ANNEX 1: Data assimilation and data quality

The aim of the analysis (User Guide to ECMWF Products, 1995) is to provide the general circulation numerical model with the most realistic state of the atmosphere. For so doing the ECMWF uses a finite number of observations and a first guess provided by the same forecasting model (generally the last six hours prevision) or by climatology. The analysis must modify this first guess integrating in the best way the information coming from the observations which, in general, do not describe perfectly the atmospheric state. Different quality controls are performed on the observations in order to select a realistic set of data. The assimilation of the first guess and the observations are made using an algorithm named optimal interpolation. If A is the analyzed parameter, for each grid point one defines:

 

$$a_k^{\rm a}=a_k^{\rm g}+\sum_i\lambda _i\left( y_i^{\rm o}-y_i^{\rm g}\right)$$ (7)

where

• $a_{k}^{\rm a}$ is the value of A that we want to know in the k grid point;
• $a_{k}^{\rm g}$ is the value of the parameter provided by the first guess in the k grid point;
• $\lambda _{i}$ is the averaged weight associated to the quality of the observation (kind of measurement and kind of observed parameter), the distance between the analysis grid point and the observations grid point and the quality of the first guess;
• $y_{i}^{\rm o}$ is the observed value of the parameter Y (Y can be A or, more generally, a parameter linearly correlated to A);
• $y_{i}^{\rm g}$ is the value of Y provided by the first guess.

The optimal interpolation method defines $\lambda _{i}$ in the following way:

$$\lambda _i=\frac{\sigma _{\rm g}^2}{\sigma _{o,i}^2+\sigma _{\rm g}^2}$$ (8)

which transforms the Eq. (7) into

$$a_k^{\rm a}=a_k^{\rm g}+\sum_i\frac{\sigma _{\rm g}^2}{\sigm... ...,i}^2+\sigma _{\rm g}^2}\left( y_i^{\rm o}-y_i^{\rm g}\right)$$ (9)

where $\sigma _{o,i}^2$ is the variance of observational errors, $\sigma _{\rm g}^2$ is the variance of the first guess error resulting from the total uncertainty in the analysis. When an observation is unreliable, $\sigma _{o,i}^2$ is large and $a_{k}^{\rm a}$ takes the value of $a_{k}^{\rm g}$. In this case there is a little impact on the first guess. When the observation is assumed to be accurate, $\sigma _{o,i}^2$ is small and $a_{k}^{\rm a}$ almost takes the value $y_{i}^{\rm o}$.

This analysis, named uninitialized analysis, can sometimes generate, fictitious gravity waves during the simulations. To avoid this, they must be initialized in order to be adapted to the dynamics of the model. This more elaborate analysis is named initialized analysis.

In an operational schedule, the ECMWF produces global analysis and predictions using data collected during time. The predictions are permanently corrected by the observations that are injected in the algorithm of the optimal interpolation. This feed-back system of assimilation data is named Analysis Assimilation.

Acknowledgements

This work is supported by contract (Technical Report UNI-17400-0004) from ESO. The authors are grateful to M.Sarazin for his constant and dedicated presence in this feasibility study. They wish to thank the GMME team of the Centre National de Recherche Meteorologique for their scientific collaboration and logistical help. In particular, they gratefully acknowledge the many helpful discussions with J. Stein, P. Jabouille, J.L. Redelsperger and J.P. Lafore. The authors thank C. Coulman for his helpful comments and revision of the draft.