Summarizing the GRF’s correlation structure

• In an intrinsically stationary GRF, the variance of \(Z(s_i) - Z(s_j)\) is a function of the distance \(h = s_i - s_j\) (see before).
• By definition:
  • variogram = \(2 \cdot \gamma(h) = Var[Z(s + h) - Z(s)] = E[(Z(s + h) - Z(s))^2]\)
  • semivariogram = \(\gamma(h) = \frac{1}{2} \cdot Var[Z(s + h) - Z(s)] = \frac{1}{2} \cdot E[(Z(s + h) - Z(s))^2]\)
• Estimated by the empirical variogram, a tool to evaluate the presence of spatial correlation in data:

\[2 \cdot \hat{\gamma}(h) = \frac{1}{|N(h)|} \cdot \displaystyle\sum_{N(h)}(Z(s_i) - Z(s_j))^2\]