A biharmonic spline interpolation is an interpolation of irregularly spaced two-dimensional data points. The interpolating surface is a linear combination of Green functions centered at each data point. The amplitudes of the Green functions are found by solving a linear system of equations.
The surface s(x) is expressed as
|s(x)=Σj=1n wj g(x, xj),|
where n is the number of data points xj = (xj, yj), g(x, xj) is Green's function and wj is the weight of data point j. The weights wj are determined by requiring that the surface s(x) passes exactly through the n data points, i.e.
|s(xi)=Σj=1n wj g(xi, xj), i = 1, 2, ..., n.|
This yields an n×n square linear system of equations which can be solved for the wj.
For twodimensional data Green's function is:
|g(xi, xj) = |xi - xj|2 (ln |xi - xj| - 1.0).|
See Sandwell (1987) and Deng & Tang (2011) for more information.
© djmw, September 15, 2017