Melnichenko O. Reconstruction and analysis of coastal ocean currents using Helmholtz decomposition.

A new approach is developed to reconstruct a two-dimensional flow from noisy and sparse data in an open domain using a two-scalar spectral representation. According to the Helmholtz theorem the horizontal velocity field is defined in terms of stream function and velocity potential. The reconstruction procedure involves estimating each field of stream function and velocity potential as a linear combination of basis functions using least-square-misfit criterion. The basis functions are the eigenfunctions of Laplacian operator for a given geometry of the ocean domain and homogeneous boundary conditions. An appropriate regression models are developed to estimate unknown amplitudes of basis functions from irregular disposed velocity observations. The optimal number of basis functions used in regression model is determined through a cost function, which is a trade-off between the likelihood of the regression model and a penalty for making a model not too complex for a given data. The capability of the reconstruction scheme is demonstrated by numerous tests and several practical applications.