Default Linear Variogram
The default linear variogram takes the form

where is the unknown nugget effect, and is the unknown slope. We need two defining equations to solve for these two unknown parameters.
According to theory, the expected value of the sample variance is the average value of the variogram between all possible pairs of sample locations (Barnes, 1991); this yields one equation. The second equation is generated by equating the experimental sample variogram for nearest neighbors to the modeled variogram. Thus, we can write

where
= average distance to the nearest neighbor
= average intersample separation distance
= one half the averaged squared difference between nearest neighbors
= sample variance
By solving the two equations for the two unknown parameters, and checking for unreasonable values, we get the final formulae used in Surfer:
