An explicit four-dimensional variational data assimilation method

Abstract

A new data assimilation method called the explicit four-dimensional variational (4DVAR) method is proposed. In this method, the singular value decomposition (SVD) is used to construct the orthogonal basis vectors from a forecast ensemble in a 4D space. The basis vectors represent not only the spatial structure of the analysis variables but also the temporal evolution. After the analysis variables are expressed by a truncated expansion of the basis vectors in the 4D space, the control variables in the cost function appear explicitly, so that the adjoint model, which is used to derive the gradient of cost function with respect to the control variables, is no longer needed. The new technique significantly simplifies the data assimilation process. The advantage of the proposed method is demonstrated by several experiments using a shallow water numerical model and the results are compared with those of the conventional 4DVAR. It is shown that when the observation points are very dense, the conventional 4DVAR is better than the proposed method. However, when the observation points are sparse, the proposed method performs better. The sensitivity of the proposed method with respect to errors in the observations and the numerical model is lower than that of the conventional method. data assimilation, four-dimensional variation, explicit method, singular value decomposition, shallow water equationThe four-dimensional variational data assimilation (4DVAR) has been a very successful technique and used in operational numerical weather prediction (NWP) of some weather forecast centers [1,2] . In this method the optimal estimate of initial condition of a forecast model is obtained by fitting the forecasts to observations within a time window. The attractive features of 4DVAR include: (1) the full-model is set as a strong dynamical constraint, and (2) it has the ability to assimilate the data at multiple time. However, the control variables (initial state) are expressed implicitly in the cost function. In order to compute gradient of the cost function with respect to the control variables, one has to integrate the adjoint model of the forecast model. But coding the adjoint for the 4DVAR and maintaining the adjoint, updated with the model upgrading, are extremely labor-intensive, especially when the forecast model is nonlinear and the model physics contain parameterized discontinuities [3,4] . Some researchers try to avoid integrating the adjoint model or reducing the expensive computation [5][6][7] . But the linear or adjoint model is still required in the methods mentioned above. So the three-dimensional variational data assimilation (3DVAR) becomes the common practice in many numerical weather forecast centers. The 3DVAR can be considered as a simplification of the 4DVAR, but it has lost the two advantages of the 4DVAR mentioned above. In general, the analysis results rely heavily on the information from the background field. However, as we know, in 3DVAR the background error covariance matrix is usually simplified and not flo...

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

“…Tian and Xie [11] proposed a new explicit 4DVAR method by merging the Monte Carlo method and the proper orthogonal decomposition (POD) technique into 4DVAR in order to transform an implicit optimization problem into an explicit one. Qiu et al [9,10] proposed another new method for 4DVAR using the singular value decomposition (SVD) technique based on the theory of the atmospheric attractors. Like the I4DVAR, the POD-E4DVAR also needs to choose an assimilation time window.…”

“…Therefore, the assimilation efficiency would be the best. X X X δ = − is adopted in Qiu et al [9,10] . From eq.…”

“…These methods generally focus on how to produce the flow-dependent background error variance statistics by ensemble forecasts; however, the tangent linear operator or adjoint model is still necessary in these methods. Recently, Qiu et al [9,10] proposed a new method (referred to as SVD-E4VAR) for the 4DVar using a singular value decomposition (SVD) technique based on the theory of the atmospheric attractors to simplify the assimilation process considerably. Tian and Xie [11] developed another explicit 4DVAR method (referred to as POD-E4DVAR) by merging the Monte-Carol method and proper orthogonal decomposition (POD) [12,13] technique.…”

Evaluating the Role of the EOF Analysis in 4DEnVar Methods

“…X en,b , the ensemble perturbations, can also be regarded as a linear transformation from the original K-dimensional space to one spanned by X en,b . This helps avoid the mathematics conceptual difficulty in viewing them as basis vectors, since the columns of X en,b are linearly dependent and the B-matrix has rank at most K − 1 [40]. Yet constructing P y in 4DEnVar uses ensemble perturbations in four-dimensional space, whereas the ETKF uses ensemble perturbations in three dimensions.…”

“…Zhao et al developed a DRP-4DVar assimilation system based on MM5, and successfully assimilated the simulated sea level pressure observations to improve the typhoon-track forecasts in the observing system simulation experiments (OSSEs) [39]. However, previous studies paid little attention to the EOF-, SVD-, or POD-based techniques and their related parameters (e.g., truncation number) in the 4DEnVar method family [20,24,25,40]. Investigating and understanding the role of EOF is of great importance for the study of these 4DEnVars since the basis vectors expressing the analysis variables in the 4D space are constructed by the technique.…”