Hi Nathan, thanks for that info. From my understanding of the paper their analysis is only doing simple things in their models of $C(x)$ and $W(x,t)$, primarily
1. The spatial average of $C(x)$ is 0.
2. For each fixed value of the other variable, the spatial averages and temporal averages of $W(x,t)$ are 0.
3. There are constraints on how quickly $C(x)$ and $W(x,t)$ can vary based between nearby observation stations, based partly upon the degree of correlation between observations and upon spatial difference and an estimate of station reliability.
So I think they're doing some basic stuff, but not the kind of deeper structure in the model that you're talking about.