The thin-plate spline method fits a spline function to the observations.
The fitted function agrees with the observation values at the
observation points.
The coefficients of the functional fit can be stored for later use.
For n observation points, this requires the solution of n simulatneous
equations and the inversion
of an nxn matrix.
As a result, a spline fit is limited to a reasonable number of points.
In Spherekit, if more than 500 points are entered, the domain is broken
up into overlapping
regions, and a surface fit is performed on each region.
Points lying in more than one region are assigned a value that is a
weighted average of the multiple estimates.

The thin plate spline function can be any of the following forms:

##
f(d)= d^{2}log d

f(d)= d^{2}log d^{2}

f(d)= d^{2}(log d - 1)

where d is the spherical distance.

### References

Franke (1982), Harder and Desmarias (1972), and Sandwell (1987).