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.