# How do nugget interactions work in Gaussian Processes/Kriging?

How do nuggets and nugget interactions fit into the variogram framework? I am especially interested in the case where there is more than one distance term being used (e.g. space and time): where you would potentially want to include a nugget interaction term. How does this term get incorporated into the variogram formula?

I feel I understand how it works for the single distance case of $$\alpha \exp(-h/\beta)$$ as the covariance function:

If there is no nugget, then the semi-variogram $$\gamma$$ is:

$$\begin{split} \gamma(h) & = K(0) - K(h) \\ & = \alpha - \alpha \exp(-h/\beta) \end{split}$$

If there is a nugget, then it is:

$$\begin{split} \gamma(h) & = K(0) - K(h) \\ & = (\sigma + \alpha) - \alpha \exp(-h/\beta) \\ & = \sigma + \alpha\left(1 - \exp(-h/\beta)\right) \end{split}$$

But when there are two distance terms (and we for example assume a distance metric based variogram) it's not clear to me how the nuggets incorporate into the variogram formula. All I can suppose is that it would be:

$$\begin{split} \gamma(h,u) & = K(0,0) - K(h,u) \\ & = (\sigma_h I_{h=0} + \sigma_u I_{u=0} + \sigma_{hu} I_{h=0, u=0} + \alpha) - \alpha \exp\left(-\sqrt{\left(\frac{h}{\beta_h}\right)^2+\left(\frac{u}{\beta_u}\right)^2}\right) \\ \end{split}$$