# Parametrization of the Matérn Covariance Function

For geostatistics problems, I am used to working with the following parametrization for the Matérn covariance function.

For a stationary and isotropic Gaussian random field $$X(\boldsymbol{s})$$, $$\boldsymbol{s} \in \mathbb{R}^d$$, we have

\begin{align*} \text{Cov}\left[X(\boldsymbol{s}_1), X(\boldsymbol{s}_2)\right] &= r(||h||), \text{ such that } ||h|| \text{ is the Euclidian distance between } \boldsymbol{s}_1 \text{ and } \boldsymbol{s}_2\\ &=\frac{\sigma^2}{2^{\nu - 1}\Gamma(\nu)} (\kappa \cdot ||h||)^{\nu} K_{\nu}(\kappa \cdot ||h||) \end{align*} where $$K_{\nu}(\cdot)$$ is a modified Bessel functions of the $$2^{\text{nd}}$$ order. Additionally,

• $$r(0) = \sigma^2$$ is the variance.
• $$\nu > 0$$ is the smoothness parameter (a Gaussian process with this covariance function is $$\lceil\nu\rceil - 1$$ differentiable).
• $$\kappa > 0$$ determines the practical correlation range; in particular, $$\rho = \frac{\sqrt{8\nu}}{\kappa}$$ is the distance at which the spatial correlation is close to $$0.1$$.

However, using the RandomFields in R, according to its documentation (p.337), the RMmatern() uses the following parameterizationn: \begin{align*} r(||h||) = \frac{1}{2^{\nu-1}\Gamma(\nu)}(\sqrt{2\nu}\cdot||h||)^{\nu}K_{\nu}(\sqrt{2\nu}\cdot||h||), \end{align*} where

• $$\nu > 0$$ is said to be the smoothness parameter (no problem here).

Moreover, the function has the following arguments var, scale, ....

Regarding this parametrization, I have two questions:

1. What does the scale parameter mean? How can I relate it to my original parameterization?
2. Is the parameter var the same as $$\sigma^2$$ in my original parameterization?

There's a hyperlink in the explanation to the documentation for RMmodel, where var and scale parameters are explained. If $$r=\phi(h)$$ is the correlation model, the covariance model is $$C = \mathtt{var}\times\phi(h/\mathtt{scale})$$ That is, var is $$\sigma^2$$ and scale is $$\sqrt{2\nu}/\kappa$$
• The general principle is OK but the expression for scale is ambiguous since it depends on the parameterisation, the OP having two parameterisations involving a parameter $\nu$. This is for the first parameterisation.