Data assimilation algorithms

 Data assimilation algorithms and error covariance estimation

Data assimilation is a methodology based on statistics, where the goal is to make a numerical prediction model as close as possible to a true state that is partially observed. In meteorology, this mostly amounts to estimate the initial states (called analyses) of ARPEGE, ALADIN and AROME models from atmospheric observations through the use of various techniques.

Most of the activities undertaken at GMAP contrive to develop variational assimilation techniques, also called 3D-Var and 4D-Var. In particular, an important area of research is the estimation and tuning of the error statistics, including the covariances of estimation errors of the background states.

These background states are provided by short-term forecasts, and they are used as first guesses by the analyses. The estimation of their error spatial covariances relies on ensemble assimilation methods. They consist in simulating system errors with random perturbations, as well as their temporal evolution through successive analysis and forecast steps. This kind of technique allows the space and time dynamics of errors to be simulated, which depends on the weather situation and on the observation network.

It is thus possible, for instance, to represent the relatively large uncertainties associated to intense weather events such as mid-latitude storms and tropical cyclones.

In this context, spatial filtering techniques of error variances (which reflect expected amplitudes of errors) were developed (Raynaud et al 2009, Berre et Desroziers 2010). They seek to keep the robust part of the signal provided by a small size ensemble (with 6 members typically), by spatially filtering out the sampling noise which affects these estimations.

Assimilation of observations in numerical weather prediction systems relies also on modelling of forecast error spatial correlations. These correlations allow spatial structures to be determined from to pointwise observations.

A new model of these correlations is now used operationally, in order to take into account their flow dependence (Varella et al 2011). This approach is based on a wavelet representation (which is in between gridpoint and spectral space representations) of geographical variations of correlations. It relies also on the 6 members of the ensemble data assimilation system, which are sampled in a sliding way over the last 4 current days. This provides a set of 96 perturbed forecasts, from which error correlations can be estimated in a robust way.

Figure 1 shows geographical variations of horizontal correlation length-scales, estimated for the period 24-27 February 2010. Short length-scales are observed in the vicinity of low pressure areas in the Eastern part of USA and over Northern Atlantic and Europe. This allows small scale structures that are observed in these regions to be better described.

Horizontal length-scales of forecast error correlations for wind near 500 hPa (iso-colors, in km), averaged over the period from 24 February 2010 at 00 UTC to 27 February 2010 at 18 UTC, and superimposed with the 500 hPa geopotential field (isolines, in dam) valid on 26/02/2010 at 00H. The length-scale of a local correlation function is a measure of its spatial extension.

In the context of the implementation of a new computer, it is planned in the future to increase the size of the global ensemble. This will allow variability of error structures from one day to the other to be better represented, and global ensemble predictions to be better initialized.

It is also planned to implement an ensemble forecasting system at convective scale, which could also rely itself on an ensemble assimilation using the AROME model.

In the framework of the HyMeX programme, studying Mediterranean meteorological situations with heavy precipitations, an experiment has been conducted with a size of the ensemble one order of magnitude greater than the size used in operations and with also an ensemble assimilation at convective scale, with the AROME and AROME-WMED models.

This "big ensemble" is used both to describe the flow-dependent aspects of the background error covariance matrix at the convective scale. It may be used as a reference in the comparison of different objective filtering techniques of the sampling noise (Ménétrier et. al, 2013). These methods could be used with smaller size ensembles affordable in operations, but also to prescribe objective localization lengthscales.

Work has started on the spatial filtering of background error variance fields by adapting to the limited area and to the non-Gaussian case strategies initially developed for the global model ARPEGE. In order to represent the correlations, a new model based on spatial deformations has been proposed (Michel 2012). It presents some numerical and scientific advantages, especially to preserve variances.

The figure shows humidity background error standard-deviations at model level 50 (around 950 hPa), for the reference ensemble (left panel) and a very small size ensemble (6 members, middle panel), with a very strong sampling noise. The reference map enables to objectively evaluate the spatial filtering technique as applied in the right panel.

Validating these covariance estimations is another important research topic. This relies on the use of diagnostics based on departures from observations (Desroziers et al, 2009). These departures provide independent information on forecast errors, provided that the observation error contribution is filtered out.

Several applications have thus contributed to improve numerical weather forecasting. Research activities are supported by grants from INSU/LEFE (CNRS), ANR and RTRA.

 References

  • Berre, L. and G. Desroziers, 2010: Filtering of Background Error Variances and Correlations by Local Spatial Averaging: A Review. Mon. Wea. Rev., 138, 3693–3720.
  • Desroziers, G., L. Berre, V. Chabot and B. Chapnik, 2009: A Posteriori Diagnostics in an Ensemble of Perturbed Analyses. Mon. Wea. Rev., 137, 3420–3436.
  • Ménétrier, B., T. Montmerle, L. Berre and Y. Michel, 2014: Estimation and diagnosis of heterogeneous flowdependent background error covariances at convective scale using either large or small ensembles. Q.J.R. Meteorol. Soc., 140, 2050–2061.
  • Michel, Y., 2012: Estimating deformations of random processes for correlation modelling in a limited area model . Q.J.R. Meteorol. Soc., 139, 534–547.
  • Raynaud, L., L. Berre and G. Desroziers, 2009: Objective filtering of ensemble-based background-error variances. Q.J.R. Meteorol. Soc., 135, 1177–1199.
  • Varella, H., L. Berre and G. Desroziers, 2011: Diagnostic and impact studies of a wavelet formulation of background-error correlations in a global model. Q.J.R. Meteorol. Soc., 137, 1369–1379.