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 ex-pressed 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 func-tion with respect to the control variables, is no longer needed. The new technique significantly simpli-fies 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.
A three-dimensional variational method is proposed to simultaneously retrieve the 3-D atmospheric temperature and moisture profiles from satellite radiance measurements. To include both vertical structure and the horizontal patterns of the atmospheric temperature and moisture, an EOF technique is used to decompose the temperature and moisture field in a 3-D space. A number of numerical simulations are conducted and they demonstrate that the 3-D method is less sensitive to the observation errors compared to the 1-D method. When the observation error is more than 2.0 K, to get the best results, the truncation number for the EOF＇s expansion have to be restricted to 2 in the 1-D method, while it can be set as large as 40 in a 3-D method. This results in the truncation error being reduced and the retrieval accuracy being improved in the 3-D method. Compared to the 1-D method, the rms errors of the 3-D method are reduced by 48% and 36% for the temperature and moisture retrievals, respectively. Using the real satellite measured brightness temperatures at 0557 UTC 31 July 2002, the temperature and moisture profiles are retrieved over a region （20°-45°N, 100°- 125°E） and compared with 37 collocated radiosonde observations. The results show that the retrieval accuracy with a 3-D method is significantly higher than those with the 1-D method.
