Splines and vector splines

The thrust of this report concerns spline theory and some of the background to spline theory and follows the development in (Wahba, 1991). We also review methods for determining hyper-parameters, such as the smoothing parameter, by Generalised Cross Validation. Splines have an advantage over Gaussian Process based procedures in that we can readily impose atmospherically sensible smoothness constraints and maintain computational efficiency. Vector splines enable us to penalise gradients of vorticity and divergence in wind fields. Two similar techniques are summarised and improvements based on robust error functions and restricted numbers of basis functions given. A final, brief discussion of the application of vector splines to the problem of scatterometer data assimilation highlights the problems of ambiguous solutions.