Suppose I have been given some process $Y$. Let $Y(s,t)$ denote the value of process at location $s$ and time $t$. For my experiment, I consider a model described as - $$Y(s,t) = \mu(s) + M(t;\beta(s)) + X(s,t) + \gamma(s,t) $$
Here, $\mu(s)$ represents site specific mean $M(t;\beta(s))$ is a large scale temporal model with site specific parameters $\beta(s)$. $X(s,t)$ is a short scale dynamical process. $\gamma(s,t)$ models noise.
I have following questions -
Firstly, $\mu(s)$ is the site specific mean of what? What are the quantities that we are averaging over this space $s$ ?
Secondly, I can't understand what does $ \beta$ and $X$ exactly denote?
All this is given in a paper which I am currently studying - Hierarchical Bayesian space-time models. Here is the link to that paper - http://www.stat.missouri.edu/~wikle/WikleBerlinerCressie1998.pdf