Say I have a number of time series, e.g. a number of temperature records from various stations in a region. I want to get a single temperature record for the whole region with which I could describe aspects of the regional climate. The intuitive approach might be to simply take the average of all stations at each timestep, but my statistical spider-sense (which I'm definitely not well in touch with yet) tells me that this might not be so easy. In particular, I imagine that averaging over the entire region will remove some of the interesting temperature extremes, and I might have problems with dependence between close stations.
What other problems might I face if I tried a strategy like this, and are there ways to overcome them, or more sensible methods of combining this kind of data?
Note: Answers can be more general than the spatial example I've provided.