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 $\nu$:
  • $\nu = 0.5 \implies Corr(h) = e^{-h/\phi}$
  • $\nu \to \infty \implies Corr(h) = e^{-(h/\phi)^2}$