We developed a new approach to reconstruct a three-dimensional incompressible flow from noisy data in an open domain using a two-scalar (toroidal and poloidal) spectral representation. This approach includes: (a) a boundary extension method to determine normal and tangential velocities at open boundary, (b) establishment of homogeneous open boundary conditions for the two potentials with a spatially varying coefficient kappa, (c) spectral expansion of kappa, (d) calculation of basis functions for each of the scalar potentials , (e) determination of coefficients in the spectral decomposition of both velocity and kappa using linear or nonlinear regressions.

The basis functions are the eigenfunctions of the Laplacian operator with homogeneous mixed boundary conditions and depend on spatially varying parameter kappa at the open boundary. A cost function used for poor data statistics is introduced to determine the optimal spectral truncation. An optimization scheme with iteration and regularization is proposed to obtain unique and stable solutions. We demonstrate the capability of our method through reconstructing a 3D circulation in the northern shelf of the Gulf of Mexico from the drifter buoy data.