
A biharmonic spline interpolation is an interpolation of irregularly spaced twodimensional 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)=Σ_{j=1}^{n} w_{j} g(x, x_{j}), 
where n is the number of data points x_{j} = (x_{j}, y_{j}), g(x, x_{j}) is Green's function and w_{j} is the weight of data point j. The weights w_{j} are determined by requiring that the surface s(x) passes exactly through the n data points, i.e.
s(x_{i})=Σ_{j=1}^{n} w_{j} g(x_{i}, x_{j}), i = 1, 2, ..., n. 
This yields an n×n square linear system of equations which can be solved for the w_{j}.
For twodimensional data Green's function is:
g(x_{i}, x_{j}) = x_{i}  x_{j}^{2} (ln x_{i}  x_{j}  1.0). 
See Sandwell (1987) and Deng & Tang (2011) for more information.
© djmw, September 15, 2017