[1] An ensemble-based four-dimensional variational data assimilation (4DVar) method is proposed to fit the model field to 4-D observations in an increment form in the analysis step of data assimilation. The fitting is similar to that in the 4DVar but the analysis increment is expressed by a linear combination of the leading singular vectors extracted from an ensemble of 4-D perturbation solutions, so the fitting is computationally very efficient and does not require any adjoint integration. In the cost function used for the fitting, the background error covariance matrix is constructed implicitly by the perturbation solutions (through their representative singular vectors) similarly to that in the ensemble Kalman filter, but the perturbation solutions are not updated by the analysis into the next assimilation cycle, so the analysis is simpler and more efficient than that in the ensemble Kalman filter. The potential merits of the method are demonstrated by three sets of observing system simulation experiments performed with a shallow-water equation model. The method is shown to be robust even when the model is imperfect and the observations are incomplete.Citation: Qiu, C., A. Shao, Q. Xu, and L. Wei (2007), Fitting model fields to observations by using singular value decomposition: An ensemble-based 4DVar approach,

[1] Doppler radar observations with high spatial and temporal resolution can effectively improve the description of small-scale structures in the initial condition and enhance the mesoscale and microscale model skills of numerical weather prediction (NWP). In this paper, Doppler radar radial velocity and reflectivity are simultaneously assimilated into a weather research and forecasting (WRF) model by a proper orthogonal-decompositionbased ensemble, three-dimensional variational assimilation method (referred to as PODEn3DVar), which therefore forms the PODEn3DVar-based radar assimilation system (referred to as WRF-PODEn3DVar). The main advantages of WRF-PODEn3DVar over the standard WRF-3DVar are that (1) the PODEn3DVar provides flow-dependent covariances through the evolving ensemble of short-range forecasts, and (2) the PODEn3DVar analysis can be obtained directly without an iterative process, which significantly simplifies the assimilation. Results from real data assimilation experiments with the WRF model show that WRF-PODEn3DVar simulation yields better rainfall forecasting than radar retrieval, and radar retrieval is better than the standard WRF-3DVar assimilation, probably because of the flow-dependence character embedded in the WRF-PODEn3DVar.Citation: Pan, X., X. Tian, X. Li, Z. Xie, A. Shao, and C. Lu (2012), Assimilating Doppler radar radial velocity and reflectivity observations in the weather research and forecasting model by a proper orthogonal-decomposition-based ensemble, three-dimensional variational assimilation method,

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...

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