Obtaining a complete picture of ocean state has historically been hampered by sparse sampling, both spatial and temporal. A continuous picture of important surface currents contiguous to our coasts is now being realized by shore-based HF surface wave Doppler radars. Their real-time maps every hour reveal the dynamic processes that impact 95% of our maritime economy; their inputs to numerical models offer the promise of accurate coastal forecasts in the near future. Two or more such radars must view the same region of the sea to construct total velocity vector maps. All too often this proves impossible. When only a single radar observes a region, the radial-component maps do not immediately reveal the total flows. And there are many regions blocked to even single-radar coverage.

We describe success with a new methodology that that uses the Navier-Stokes (NS) equations to estimate total velocity fields from radial maps only. The motivation was operation from a single offshore oil platform where a pair of radars is not possible. The pressure-gradient, acceleration, and Coriolis terms are retained in the two coupled NS equations for the polar-system-based surface components. Two approaches on the problem have produced correct results against simulated data, where input fields are known. In the first, the Lagrangian representation of the NS equations is converted to the well-known nonlinear Eulerian representation; the latter are solved with the finite-element code, PDE2D. In the second, the linear Lagrangian system is retained and solved via step-by-step time integration; the velocities of the moving Lagrangian parcels of fluid are transformed after each time step back to the desired fixed Eulerian grid by simple 2-D interpolation. Both methods recover the missing azimuthal fields when only radial fields are available each hour. Examples are shown. Next we plan to apply these methods to actual HF radar radial data soon to be obtained from an offshore rig.