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?