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I would like to ask a question to experts on spatial statistics:

I have a gridded Digital Elevation Model (DEM) which provides terrain elevation (z) at cartesian coordinates (x,y). Together with the DEM I know the RMS of the elevations. As far as I understand the RMS can be applied to any point in the DEM and the standard deviation of the elevations is stationary.

Now, if I take two points from the DEM, the elevations at these points are not independent, but are auto-correlated if the distance between the points is below a certain range. I can build a variogram from the DEM elevation data which provides the function of elevation variation as a function of distance. For distances beyond the range the elevation variation is independent and just the mean of all elevations in the terrain.

However, what I really want to know is the uncertainty of the elevation at different close distances. I would assume that the uncertainty is equal to the RMS at large range, but apporaches zero if the distance apporaches zero. What I need is the function vor values between zero-distance and the range. I would just apply the variogram function of the elevation to the uncertainty and scale it from the mean elevation to the RMS, but I guess this is a pretty much simplified approach.

Any thoughts and comments are welcome

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Because there may be some confusion in the question, let's address that first. (I'm not saying you're the one confused :-). Typically, an "RMS" is used to assess a "typical" error based solely on the $(x,y)$ location. By employing a variogram, you are implicitly proposing using values of the DEM to predict its values at nearby locations. But what is the point of that (except perhaps when the DEM has some gaps in its interior)? "Applying the variogram" won't work; we're talking about the kriging algorithm here (aka the BLUP). – whuber♦ Feb 15 '12 at 20:51

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