We have started to study the representation of background error covariances in a wavelet basis. The purpose is to allow for some geographical variation in the covariances while still keeping the covariance matrix B relatively simple.

Currently, B is taken as a diagonal matrix in spectral space, there are non-diagonal elements only for the covariances between different model levels. The result is that the (co-)variance has no geographical variation.

Wavelets are a whole class of function bases that occupy the region between grid space and spectral space. They are partially localized in both grid and spectral space. In the usual theory, the wavelet basis is formed by dilatations and translations of one single "mother wavelet". By dilating this function by (usually) a factor 2 we get a new function that is localized at the same place but analyses our data at a different scale. One can think of such a basis as a set of bases, each living on a subgrid of different scale and describing the data at this scale. So the largest and smallest scale wavelets on a 128x128 domain live on 2x2 and 64x64 grids respectively. It is hoped that an (almost) diagonal B matrix in wavelet space may combine the advantages of spectral space with some geographical variations.

We have chosen to work with the so-called Meyer wavelets. These have compact support in spectral space (i.e. the spectrum of every basis function is strictly limited) but are not as well localized in gridpoint space. For a spectral model like ALADIN this seems the most logical choice.

Most wavelets only scale by 2, restricting the domain sizes to powers of 2. Therefore we started our work using data from ALADIN-Morocco (Raouindi, 2001), which has a domain size of 128x128. We represented temperature errors in wavelet space and found that a simple diagonal approach indeed represents some of the geographical variations in background error covariances.

The restriction to powers of 2 has since then been solved by generalizing the definition of Meyer wavelets to other scales (Deckmyn, 2002). This means that any domain size can now be analysed with such wavelets.

However, there remains a big problem. A restriction to diagonal elements in the wavelet basis gives a strong bias to points that are close to those of the large-scale grid. For example, when we take the B matrix in wavelet space and transform it back to gridpoint space, the variance is concentrated around a few points, especially at high model levels. We have no solution yet for this problem.

References:

Raouindi M., 2001, Etude et représentation spectrales de la variabilité latitudinale des covariances spatiales des erreurs de prévision sur une aire limitée.

Deckmyn A., 2002, Orthogonal Wavelet Transforms with Variable Scale Factor, in preparation.