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I have a number of weather stations, and I know their positions. I would like to interpolate measurements from them for various other positions as a weighted average of these stations, but of course I need weights for this. I am thinking the most logical choice here from a physics point of view will be weighting by the inverse of the distance to the station, but I haven't quite convinced myself that that is right, and am not sure if I should maybe be using the distance squared, or maybe something in between.

Any comments? Are there any other reasonable alternatives I should be considering?

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It sounds like you're describing optimal interpolation (AKA Gauss-Markov analysis, objective analysis, probably others too). This is a solid intro, and this is a powerpoint on the subject. It's a hard process to summarize quickly, but roughly speaking, you're on the right track.

Optimal interpolation is fairly common in meteorology and other environmental sciences, but I'm not convinced it's really the best tool for the job. Any statisticians want to take a crack at it?

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