Semivaraince "upside down"

[TimHorstkotte]

Hej all,

I'm doing an analysis of snow hardness in dependence of distance to trees and tree diameter. I therefore constructed a "forest coordinate system" with distance to trees as x-axis, tree diameter as y-axis. I now would like to create a surface showing how snow hardness changes if you a) move further away from trees and b) tree diameter increases. As you can imagine, being close to a tree (e.g 40 cm) can result (depending on canopy structure) in either hard snow (where snow drops out of the canopy) or less hard snow (no such event), while far away (e.g. 5 m) there is no such canopy-impact (snow will be soft everywhere).
In a nutshell, things close to each other (i.e. close to a tree) are highly variable (soft or hard), while things far away (from trees) tend to be similar (soft everywhere)! This is the opposite expressed by usual autocorrelation - can this be handled in a kriging process? I could not manage to model a semivariogram where the line follows a decrease in variation with increasing distance.
Any suggestions how to deal with this - or is it better to abandon this quest as "rubbish"? 

Thanks for your help!

[EricKrause]

The biggest problem I see is that you're trying to model two different processes: one for areas outside of the canopy area, and one for areas within the canopy area.  Kriging assumes that there is only one underlying process, so it isn't going to work.

My first thought is analyze the two processes independently.  First, use kriging to predict the hardness for all areas that are not under a tree canopy.  Second, build some kind of linear model for the areas under a canopy.  You could use distance from the tree and the tree diameter as predictor variables for the snow hardness.  With some work, you might be able to use the Ordinary Least Squares tool in the Spatial Statistics toolbox to perform the second analysis. 

There are some problems with that approach (like assuming the two processes are independent, and mixing a spatial model with a nonspatial model), but that's the only technique that comes to mind.  If I think of anything else, I'll let you know.

[JeffreyEvans]

Eric is spot on. You are convolving the spatial process and making some erroneous assumptions about independence. This is not a problem that you can answer with an interpolation approach. I would highly recommend you look into mixed effects models. This will allow you to explore if there are any random effects in your data, which is quite likely, and control for them. You can then use the model to make a surface estimate of the random field. Because the spatial process in your covariates is somewhat convolved, you will really need to think about model specification and what the results really mean. I do not see any alternatives that could be implemented in ArcGIS.

[TimHorstkotte]

Hej Eric and Jeffrey,

thanks for your help. I surely will try to do some "mixed effect modelling" and see if I can figure out how to deal with the "random" impact of canopies. The modeling in ArcGIS seemed a nice way for me to illustrate the effects I described, but if my assumptions are flawed in the light of data independence, I think of something else.

Thanks again and all the best,

Tim