In geostatistics the spatial correlation is analyzed by the variogram instead of a correlogram or covariogram (Bivand et al. 2008). If the random function \(Z(s)\), is *intrinsic stationary*, which means it has a constant mean \((E(Z(s)) = \mu)\), then the variogram is defined as

\[2\gamma(\mathbf h) = \mathbf E[(Z(\mathbf s_i)-Z(\mathbf s_i+h))^2]\]

where \(Z(\mathbf s_i)\) is the value of a target variable at some sampled location and \(Z(\mathbf s_i+h))\) is the value of the neighbor at distance \(\mathbf s_i+h\). In other words, the covariance between values of \(Z(s)\) at any two locations depends on only their relative locations or, equivalently, on their spatial separation distance, \(h\), also denoted as *spatial lag*. The quantity \(2\gamma(s)\) is known as the **variogram** and the function \(\gamma(h)\) is called the **semivariogram** which is simply written as

\[\gamma(\mathbf h) = \frac{1}{2}\mathbf E[(Z(\mathbf s_i)-Z(\mathbf s_i+h))^2]\text{.}\]

Suppose that there are \(n\) point observations, this yields \(n \cdot (n-1)/2\) pairs for which a semivariance can be calculated (Hengl 2007).

When data locations are irregularly spaced, there is generally little to no replication of lags among the data locations. To obtain quasi-replication of lags, we partition the lag space into lag classes or *bins*. Then, \(N_h\) is the number of lags that fall into the bin \(h_j\). The more bins that are used, the smaller they are and the better the lags are approximated by \(h_j\), but the fewer the number of observed lags belonging to \(h_j\). One popular rule of thumb is to require \(N_h\) to be at least 30 and to require the length of \(h_j\) to be less than half the maximum lag length among data locations (Gelfand at al. 2010).

If we assume *isotropy*, which is the direction independence of the semivariance, the variogram can be estimated from \(N_h\) sample data pairs \(z(s_i), z(s_i + h)\) for a number of distances (or distance intervals) \(\hat h_j\) by

\[\hat \gamma(\hat h_j) = \frac{1}{2N_h}\sum_{i=1}^{N_h}(Z(s_i)-Z(s_i+h))^2, \quad \forall h \in \hat h_j\]

This estimate is called the *sample variogram* or *experimental variogram*.