Assimilation of in-situ observed and remotely sensed ocean data (velocity, temperature, and salinity) into numerical model is of great importance in oceanic and climatic research. Usually, both in-situ observed and remotely sensed data are noisy and sparse, and cannot be directly assimilated into numerical models. Based on the fact that any field (temperature, salinity, or velocity) can be decomposed into generalized Fourier series, it is then represented by linear combination of the products of basis functions (or called modes) and corresponding Fourier coefficients. If a rectangular closed ocean basin is considered, the basis functions are sinusoidal functions. If a realistic ocean basin is considered, the basis functions are the eigen-values of the three-dimensional Laplace operator with real topography. Major benefits of using the OSD method are that the boundary conditions for the ocean variables (temperature, salinity, velocity) are always satisfied, and no a-priori knowledge is needed for the variables such as the error covariance matrices.

After the decomposition, the three-dimensional field is represented by a set of Fourier coefficients. This method has three components: (1) determination of the basis functions, (2) optimal mode truncation, and (3) determination of the Fourier coefficients. Determination of basis functions is to solve the eigen-value problem. Chu et al. (2003a, b) developed a theory to obtain the basis functions with open boundaries. The basis functions only depend on the geometry of the ocean basin and the boundary condition. This is to say, the basis functions can be pre-determined before the data analysis. For data without error, the more the modes, the more the accuracy of the processed field. For data with error, this rule of the thumb is no longer true. Inclusion of high-order modes leads to increasing error. The Vapnik variational principal is used to determine the optimal mode truncation. After the mode truncation, optimal field estimation is to solve a set of a linear algebraic equation of the Fourier coefficients. This algebraic equation is usually ill-posed. The rotation method (Chu et al., 2004) is developed to change the matrix of the algebraic equation from ill-posed to well-posed such that a realistic set of the Fourier coefficients are obtained. In this paper, many examples are given to show the capability of the OSD method analyzing the data from Lagrangian and Eulerian measurements such as Texas-Louisiana shelf circulation from surface drifters, Monterey Bay surface circulation from CODAR, global surface circulation from OSCAR, and mid-depth (~1000 m) circulation from Argo floats.