Once we calculate an sample variogram, it is standard practice to smooth the empirical semivariogram by fitting a parametric model to it.

The nugget, the sill and the range parameters are often used to describe variograms:


The following five parametric model are the most commonly used (Gelfand at al. 2010):

\[ \gamma(h, \theta) = \begin{cases} \theta_1 (\frac{3h}{2\theta_2}- \frac{h^3}{2\theta_2^3}) & \text{for } 0\le h \le \theta_2 \\ \theta_1 & \text{for } h > \theta_2 \end{cases}\]

\[\gamma(h, \theta)= \theta_1\{1- \exp(-h/\theta_2)\}\]

\[\gamma(h, \theta)= \theta_1\{1- \exp(-h^2/\theta_2^2)\}\]

\[\gamma(h, \theta)= \theta_1\left( 1- \frac{(h/\theta_2)^\nu\kappa_\nu(h/\theta_2))}{2^{\nu-1}\Gamma(\nu)}\right)\]

where \(\kappa_\nu(\cdot)\) is the modified Bessel function of the second kind of order \(\nu\).