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I am trying to Krige in ArcGIS from point samples of a chemical concentration. Towards the center, most of the points are larger, and on the periphery, all of the points are 0. The generated raster looks good in the center and towards the edges, but beyond the outside points of zero, the raster values trend back up. I set the major range of the semivariogram to be 50, and the trending upwards is beyond this range of influence for any of the points that are greater than 0. So it seems that Arc is setting an expectation somewhere around the average of all the points. Is there are way to set the expectation to 0 in ArcGIS. If not, what is a simple way to Krige in R or QGIS where you can set the expectation and the parameters of the semivariogram?

Here's a screenshot:

Krigging

See how near the periphery, especially upper left and lower right corners, the value trends back up, when it should be zero.

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    This is how kriging is supposed to behave: in areas where all the supports lie on one side, it is extrapolating; in the limit (beyond the variogram range), it will predict that everything equals the average value in the search window. Extrapolation is a notoriously unreliable thing to do. (The proof of that is to display the kriging variance map: it will become large around the edges to reflect one component of this unreliability.) Therefore, regardless of any other modeling decisions you make, you should mask out the results beyond a small buffer of the convex hull of the data supports. – whuber Jul 15 '14 at 21:57
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If you haven't already found a solution, you might try using Simple kriging and set the mean to zero. Reference: http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/Understanding_simple_kriging/003100000040000000/

I've used simple kriging with a specified mean of zero for interpolating a surface of residuals (local measurements minus estimated regional map). This allowed me to take advantage of the regionally estimated surface where I had little data, but update it locally where data suggested a modification. While your example doesn't involve interpolation of residuals, your expectation is that the ambient concentration is zero. This approach may degrade your cross-validation mean error and RMSE since the mean value of all your points will be higher than zero, but you can still check for a small standardized error and that the standard error is close to the RMSE, as suggested here: http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/Using_validation_to_assess_models/00310000005v000000/

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