Mass Conserving Interpolation of Velocity Data using Radial Basis Functions

Oceanographic data sets are often sparse, with measurements from widely spaced vessel tracks or moorings. To assist with interpretation of the measurements spatial interpolation is commonly used to span gaps between measurements to give more complete spatial patterns of oceanographic variables such as salinity, temperature or velocity. Thin Plate Spline Radial Basis Functions are used to extract tidal current patterns from noisy moving vessel Acoustic Doppler Current Profiler data. This project develops a spatial interpolator specifically for velocity measurements. The components of the velocity vector are typically interpolated independently of each other, however they are related by physics. This project develops a velocity interpolator which ensures the interpolated velocity components satisfy mass continuity, i.e. are divergence free. The divergence free interpolator is shown to improve the ability to resolve features, such as eddies, whose spatial scale is comparable to the spacing between velocity measurements.

A Divergence Free Spatial Interpolator for Large Sparse Velocity Data Sets

Ross Vennell and Rick Beatson (Dept. Mathematics University of Canterbury)

Vennell, R., and R. Beatson, A Divergence Free Spatial Interpolator for Large Sparse Velocity Data Sets,  Journal of Geophysical Research (3.1),  114, C10024, doi:10.1029/2008JC004973 (2009)

Abstract

A 2D divergence free interpolator is presented which is demonstrated to give more realistic values betweensparsevelocity data than interpolating velocity componentsindependently. The interpolator enforces physicaldependence between the velocity components by ensuring mass is conserved. Tests using data for a synthetic eddy show that independent interpolation of components can give unrealistic velocities between widely spaced data. However divergence free interpolation can reproduce the eddy almost as well with sparse data as with dense data.The 2D divergence free interpolator can be used to interpolate geostrophic velocities or the horizontal transport of high frequency flows such as tidal currents. Tests with data from the Local Dynamics Experiment moorings and from moving vessel ADCP measurements of tidal flow show the divergence free interpolator is better able to reproduce data left out of the interpolation than interpolating velocity components independently. The divergence free interpolator is easily generalized to spatial interpolation of 3D velocity data. An improved smoothing interpolator is presented which is capable of fitting large noisy data sets with relatively few parameters. The combination of imposing the physics, that is the divergence free constraint, and the economy of parameters, makes the interpolation of sparse data much more resistant to following the noise.

Moving Vessel ADCP Measurement of Tidal Streamfunction using Radial Basis Functions

Ross Vennell and Rick Beatson (Dept. Mathematics University of Canterbury)

Moving vessel acoustic Doppler current profiler measurement of tidal stream function using radial basis functions, J. Geophys. Res., 111, C09002, doi:10.1029/2005JC003321(2006), PDF preprint

Abstract

Acoustic Doppler Current Profiler measurements from moving vessels are being used to give detailed observations of the spatial patterns of tidal flows, as well as patterns of vorticity and dynamical terms. Developments in Radial Basis Function interpolation theory are demonstrated to significantly improve the quality of the tidal velocity field extracted from the measurements using thin plate splines. These include placing centers at data locations, enforcing ``side conditions'' on the solution and using higher order splines. For tidal flows with a scale less than a few kilometers, the differentiability of RBFs can be exploited to fit the streamfunction directly to the measurements. This ensures the observed tidal velocity field satisfies mass continuity. Enforcing mass continuity is demonstrated to significantly improve the ability of the splines to interpolate across wide gaps between vessel tracks. Extraordinarily detailed ADCP measurements of the tidal streamfunction within Bluff Harbour, NZ, reveal the stagnation streamline which separates a 200m wide flood tidal jet from an associated 400m diameter eddy. The stagnation streamline clearly shows that the eddy gains its fluid and vorticity from inshore of the jet's vorticity maximum. The eddy forms at 04:45 hours before high tide, grows in spiral fashion, and becomes isolated from the jet 02:30 hours before high tide, after which its vorticity rapidly decays.

Tidal Animation here...