# Vincent Chabot, Météo-France

## Diagnosis and normalization of wavelet-induced background error variances

##### Coauthors

L. Berre, G. Desroziers#### Abstract:

A wavelet block-diagonal approach can be used in order to specify 3D background error covariances from ensemble data.
At Météo-France, the correlation matrix is represented in wavelet space while the standard deviation fields are computed in grid point space.
However, in practice, the modelled wavelet correlation matrix appears to deviate from an exact correlation matrix because the wavelet modelling modifies variances in grid point space .
It is here demonstrated how resulting variances in grid point space can be expressed and diagnosed from the correlation matrix in wavelet space.
It is shown in particular that these induced grid point variances can be seen as resulting from the application of a scale-dependent spatial filter to wavelet variance fields.
In the context of correlation modelling, these formal results can be used for computing normalization coefficients in an accurate and efficient way, in order to ensure that diagonal elements of the resulting correlation matrix are effectively equal to one.
The links between these normalization coefficients and correlation length scales are also illustrated and discussed.