Semivariogram ($\gamma (h)$)

For an intrinsically stationary random field, the semivariogram is $\gamma (h)=\frac{1}{2}\mathrm{Var}[Z(s+h)-Z(s)]$. It is even and vanishes at zero. With constant mean, a second-order stationary field has $\gamma (h)=C(0)-C(h)$. An isotropic model writes it as a function of nonnegative distance.

For a distance bin $B$, the empirical semivariogram is $\hat{\gamma }(B)=(2|B|{)}^{-1}{\sum }_{(i,j)\in B}(z_i-z_j{)}^2$. A nonconstant drift adds half its squared pairwise differences to the expected raw semivariogram. Residual semivariograms estimate spatial dependence after adjusting for the drift.

The exact range is a separation beyond which the semivariogram reaches its sill. A Gaussian semivariogram with nonzero structured variance approaches its sill asymptotically and therefore has infinite exact range. Software range parameters may instead specify a scale.

A practical range specifies a distance at which a chosen fraction of the structured sill is reached. For the Gaussian semivariogram ${\sigma }^2(1-e^{-(h/a{)}^2})$, the 95-percent practical range is $a\sqrt{\log 20}$, obtained by solving $e^{-(h/a{)}^2}=0.05$.

The sill is the large-distance level of a semivariogram when that limit exists. For covariance tending to zero, the sill is the total marginal variance; a nugget plus a structured component has total sill ${\tau }^2+{\sigma }^2$. A nonzero constant covariance component is invisible to the semivariogram.

The nugget is the discontinuity of a distance semivariogram at zero. Independent location-specific noise of variance ${\tau }^2$ contributes covariance ${\tau }^2$ at zero displacement and zero covariance at nonzero displacement, so its semivariogram is ${\tau }^2{1}_{\{h\ne 0\}}$.

If $W(s)$ is stationary, then $Z(s)=W(s)-W(0)$ has exactly the same increments and semivariogram, but $\mathrm{Var}Z(s)=2{\gamma }_W(s)$ is generally nonconstant. Thus existence of a stationary covariance model for a semivariogram does not imply stationarity of every process with that semivariogram. Adding an independent random constant also leaves the semivariogram unchanged while changing the covariogram.

For $\varphi >0$ and nonnegative variances, $\gamma (h)={\tau }^2{1}_{\{h\ne 0\}}+{\sigma }^2(1-e^{-{\varphi }^2\Vert h{\Vert }^2})$ admits covariance $C(h)={\sigma }^2{e}^{-{\varphi }^2\Vert h{\Vert }^2}+{\tau }^2{1}_{\{h=0\}}$. The Gaussian kernel is positive definite because it is the Fourier transform of the density of a Gaussian distribution. Adding independent white noise supplies the nugget effect. Its scale is $a=1/\varphi$. For ${\sigma }^2>0$, its exact range of a semivariogram is infinite. If ${\sigma }^2=0$, it is a pure nugget effect with no nontrivial structured range.