Multivariate-Spline and Scale-Specific Solution for Variational Analyses

A recipe for a cubic B-spline-based solution for multivariate variational formulation of a data analysis and assimilation problem is provided. To represent a signal whose smallest wavelength is L, the spline scale must be at most L\2, or approximately the Nyquist wavelength. This spline scale defines the computational grid, which tends to be coarser than the typical grid required for finite-difference discretization and hence offers a significant advantage in computational efficiency. The geostrophy–thin-plate model is introduced and applied to a set of analysis problems to demonstrate the effectiveness of the solution technique.