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:

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