Effects of sample density on the assimilation performance of an explicit four-dimensional variational data assimilation method

Abstract

The concepts of sample sphere radius and sample density are proposed in this paper to help illustrate that different vector transformations result in diverse sample density with the same sample ensemble, which finally affects their assimilation performance. Several numerical experiments using a onedimensional (1-D) soil water equation and synthetic observations are conducted to evaluate this new theory in land data assimilation.POD/SVD-E4DVAR, data assimilation, sample densityThe four-dimensional variational data assimilation (4DVar) method [1,2] and the ensemble Kalman filter [3][4][5][6] (EnKF) are probably the two most popular assimilation methods in the data assimilation community. They both have their own attractive features. Essentially, the 4DVAR method transforms the data assimilation problem into an optimization one, whose advantages mainly embody on the followings: 1) the physical model provides a strong dynamical constraint; 2) it has the ability to assimilate the observational data at multiple times. However, the control variables (or initial states) are expressed implicitly in the cost function. In order to compute the gradient of the cost function with respect to the control variables, one has to integrate the adjoint model, whose development and maintenance require significant resources. Moreover, the background error covariance applied in the 4DVAR is usually flow-independent, which is obviously inappropriate for forward integration of the weather or climate models. On the other hand, EnKF has become an increasingly popular method because of its simple conceptual formulation and relative of implementation. For example, it requires no derivation of a tangent linear operator or adjoint equations, and no integration backward in time. By forecasting the statistical characteristics, the EnKF can provide flowdependent error estimates of the background errors using the Monte-Carol method. Nevertheless, there are still some deficiencies in the EnKF: the usual EnKF can neither assimilate the observational data at multiple times nor take the physical model as a strong dynamical constraint over an assimilation time window. Although many encouraging research results of the EnKF have been obtained in either global data assimilation or regional mescoscale data assimilation, not much evidence indicates that the EnKF outperforms variational data assimilation system in operational applications. Since variational data assimilation has been practically proved to be a very successful technique in operational numerical weather prediction, application of ensemble based background error covariance statistics to variatioanl data assimilation should be a good choice, which can extract the flow-dependent error covariance from ensemble forecasts to the analysis [7,8] . Lorenc [7] proposed that the control variable was preconditioned upon perturbation of background ensemble forecast so that the background error in variational system is flow-dependent. Buehnerm [8] adopted a similar hybrid ensemble 3DVar (En3DVar) scheme, and...

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