Gaussian random fields: properties

GRFs can be:

• strictly stationary: if $Z(\cdot)$ is invariant to shifts: constant distribution
• weakly stationary (2nd order stationarity):
  • process has constant mean $E[Z(s)]$
  • covariances depend only on differences between locations: $C(h) = Cov(Z(s), Z(s + h))$
    • if ‘differences between locations’ only involve distance, not direction, then the process is isotropic (opposite: anisotropic).
• intrinsically stationary:
  • weakly stationary
  • the variogram $Var[Z(s_i) - Z(s_j)]$ follows from the distance $s_i - s_j$.