Modelling static 3-D spatial background error covariances – the effect of vertical and horizontal transform order
A major difference in the formulation of the univariate part of static background error covariance models for use in global operational 4DVAR arises from the order in which the horizontal and vertical transforms are applied. This is because the atmosphere is non-separable with large horizontal scales generally tied to large vertical scales and small horizontal scales tied to small vertical scales. Also horizontal length scales increase dramatically as one enters the stratosphere. A study is presented which evaluates the strengths and weaknesses of each approach with the Met Office Unified Model.
It is shown that if the vertical transform is applied as a function of horizontal wavenumber then the horizontal globally-averaged variance and the homogenous, isotropic length scale on each model level for each control variable of the training data is preserved by the covariance model. In addition the wind variance and associated length scales are preserved as the scheme preserves the variances and length scales of horizontal derivatives. If the vertical transform is applied in physical space, it is possible to make it a function of latitude at the cost of not preserving the variances and length scales of the horizontal derivatives.
Summer and winter global 4DVAR trials have been run with both background error covariance models. A clear benefit is seen in the fit to observations when the vertical transform is in spectral space and is a function of total horizontal wavenumber.