Covariance functions of GRFs

• Matérn family of correlation functions is often used:

\[Corr(h) = \frac{1}{2^{\nu-1} \Gamma(\nu)} \cdot \left(\frac{h}{\phi}\right)^\nu \cdot K_\nu\left(\frac{h}{\phi}\right)\]

• Some special cases for smoothing parameter \(\nu\):
  • \(\nu = 0.5 \implies Corr(h) = e^{-h/\phi}\)
  • \(\nu \to \infty \implies Corr(h) = e^{-(h/\phi)^2}\)