Estimating deformations of random processes for correlation modelling in a limited area model



Covariance models are widely used in data assimilation, especially when a variational formulation is chosen. Autocovariances for each variable are generally split into variance and correlation parts. Despite recent progress, the correlations are still often modelled as space-stationary. A novel approach is introduced in this paper, where background error correlations are approximated by the spatial deformation of a stationary correlation model. The statistical calibration of this model involves estimating the deformation, its inverse and the stationary correlations. Leveraging previous work in the computer vision community, the problem is first described in two dimensions. Directional wavelets, also called warplets, are able to measure the energy of a random process at different scales and orientations. The texture gradient equation relates the wavelet analysis of a deformed process to local metric changes. From a few realizations, it is therefore possible to estimate local variations of the Jacobian of the deformation. This paper shows that a deformation can be recovered from the deformation gradient by solving partial differential equations of elliptic nature with inhomogeneous coefficients. The choice of boundary conditions is discussed. The method is then applied to the modelling of background error correlations in a limited area model. Qualitative intercomparison with other horizontal correlation models frequently used in atmospheric data assimilation, such as the diffusion equation or the diagonal assumption in a wavelet basis, suggests that the approach is also able to represent anisotropy to a fair degree. Copyright © 2012 Royal Meteorological Society