Considering intensity as a stochastic variable

\(\mu_s\) is fit using a log link and a linear predictor \(\eta_s\):

\(log(\mu_s) = \eta_s =\) fixed effects + spatial random effect (GRF) + unstructured random effect

A model (process) where a response variable can be expressed as a Gaussian process using a log link, is called a Log-Gaussian Cox Process.