Spatial interpolation of large climate data sets using bivariate thin plate smoothing splines

Thin plate smoothing splines are widely used to spatially interpolate surface climate, however, their application to large data sets is limited by computational efficiency. Standard analytic calculation of thin plate smoothing splines requires O(n³) operations, where n is the number of data points, making routine computation infeasible for data sets with more than around 2000 data points. An O(N) iterative procedure for calculating finite element approximations to bivariate minimum generalised cross validation (GCV) thin plate smoothing splines operations was developed, where N is the number of grid points. The key contribution of the method lies in the incorporation of an automatic procedure for optimising smoothness to minimise GCV. The minimum GCV criterion is commonly used to optimise thin plate smoothing spline fits to climate data. The method discretises the bivariate thin plate smoothing spline equations using hierarchical biquadratic B-splines, and uses a nested grid multigrid procedure to solve the system. To optimise smoothness, a double iteration is incorporated, whereby the estimate of the spline solution and the estimate of the optimal smoothing parameter are updated simultaneously. When the method was tested on temperature data from the African and Australian continents, accurate approximations to analytic solutions were obtained.