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A major difference in the formulation of the univariate part ofstatic background error covariance models for use in global operational 4DVARarises from the order in which the horizontal and vertical transforms areapplied. This is because the atmosphere is non-separable with largehorizontal scales generally tied to large vertical scales and smallhorizontal scales tied to small vertical scales. Also horizontal lengthscales increase dramatically as one enters the stratosphere. A study ispresented which evaluates the strengths and weaknesses of each approach withthe Met Office Unified Model.  It is shown that if the vertical transform is applied as a function of horizontal wavenumber then thehorizontal globally-averaged variance and the homogenous, isotropic lengthscale on each model level for each control variable of the training data ispreserved by the covariance model. In addition the wind variance andassociated length scales are preserved as the scheme preserves the variancesand length scales of horizontal derivatives. If the vertical transform isapplied in physical space, it is possible to make it a function of latitudeat the cost of not preserving the variances and length scales of thehorizontal derivatives.  Summer and winter global 4DVAR trials havebeen run with both background error covariance models. A clear benefit isseen in the fit to observations when the vertical transform is in spectralspace and is a function of total horizontal wavenumber

Modelling static 3-D spatial background error covariances – the effect of vertical and horizontal transform order