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:

*nugget*: The random error process indicated by the height of the jump of the semivariogram at the discontinuity at the origin.

*sill*: The limit of the variogram tending to infinity lag distances.

*range*: The distance in which the difference of the variogram from the sill becomes negligible. In models with a fixed sill, it is the distance at which this is first reached; for models with an asymptotic sill, it is conventionally taken to be the distance when the semivariance first reaches 95% of the sill.

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

*Spherical*

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

*Exponential*

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

*Gaussian*

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

*MatÃ©rn*

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

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