Different approaches to model error formulation in 4D-Var: a study with high-resolution advection schemes

Abstract

A B S T R A C T All numerical models are imperfect. Weak constraint variational data assimilation (VDA), which provides a treatment of the modelling errors, is studied; building on the approach of Vidard et al. (Tellus, 56A, pp. 177-188, 2004). The evolution of model error (ME) is modelled using ordinary differential equations, which involve a scalar parameter. These approaches were tested using different high-resolution advection schemes. The first set of experiments were constructed to see if it is possible to account for (numerical) discretization error within such a framework. In other set of experiments, a systematic source of modelling error was introduced by deliberately specifying an incorrect value for the Coriolis parameter in the model. Results with observational state at half of the model state resolution, are also presented. We also discuss a method of estimating the scalar parameter in the ME through VDA. In all cases, the inclusion of ME provides reduction in forecasting errors. Also, our experiments indicate that different settings of the model (e.g. using different high-resolution advection schemes) would need different ME formulation. Results presented in this paper could be used to formulate sophisticated ME forms to account for systematic errors in higher dimensional models with complex advection schemes.

The effect of numerical model error on data assimilation

“…As a result, given a fixed value for σ 2 o and either N or L, it is possible to choose a value for L and N respectively, that minimises the error in the analysis vector due to numerical model error and observation errors. The result for N in some way works towards answering the question posed by Akella et al [15] as to whether increasing the number of discretisation points would continue to decrease the effects of discretisation errors on the results of 4D-Var. In this instance, we have shown that when considering numerical model and observation errors in strong constraint 4D-Var, increasing the number of discretisation points past the optimal value of N when considering a full sets of observations, would result in an increase in the error in the analysis vector.…”

“…Examining expression (15), we see that the aliasing error inM has a shifted b-periodic nature. Raising the matrixM to the power l −[l] b results inM being raised to a power which is an integer multiple…”

“…However, there is still the problem of gathering prior knowledge of the model error to constrain the initial error in the minimisation problem [18]. Akella et al [15] investigate the impact of choosing growing, constant and decaying linear forms for the deterministic errors.…”

“…Weak constraint 4D-Var uses the model as a weak constraint for the minimisation process where an alternative formulation for the numerical model is chosen to account for model errors [14][15][16]. The model equations are augmented with a model error term, usually assumed to be a Gaussian random variable.…”

Construction of non-diagonal background error covariance matrices for global chemical data assimilation

“…An alternative approach constructs autoregressive models of background errors based on the short-term linearized model dynamics (Constantinescu et al, 2007a). Multivariate, multidimensional background error covariance matrices that maintain the geostrophic and hydrostatic balance have been proposed in Akella and Navon (2009).…”