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 \(s_i - s_j\).