The offset correlation, a novel quality measure for planning geochemical surveys of the soil by kriging

The following is a summary of the paper by Murray Lark, which was nominated for the 2013 Best Pedometrics Paper.

One of the useful things about geostatistical prediction is that if you know the variogram of a soil property then you can compute the mean-squared error of the kriging prediction for any location relative to some hypothetical sampling grid.  This means that you can find a sample grid that will allow you to map the property with adequate precision and to avoid over-sampling.  In 1981 Alex McBratney, Richard Webster and Trevor Burgess wrote a paper in Computers and Geosciences where they pointed this out and described a computer program to do it.  This is a simple but elegant approach, and should appeal to the practitioner.

This method has been useful in practice, but it sometimes runs into problems of communication. We have experienced these problems in dealings with government, management, colleagues, farmers and advisors. Even if the manager or official who makes the decision on funding for a survey understands variances the mean square error of predictions is not always useful for planning general baseline surveys with many possible end-users.

The geochemist, or indeed the farmer or other environmental manager, is very aware of the existence of spatial variation. In our experience this sometimes makes them sceptical of the kriged map.  “Aha,” they say, “but if this sample point had been 100m away in the next field then the pH would have been much lower because they never lime that one.”  Our proposed criterion builds on this entirely sensible intuition.

Consider a region sampled on a 500-m square grid.  We collect the data, analyse them, and produce a map.  Now, what would happen if another team, using all the same methods and equipment, sampled at exactly the same intensity, but with their points all 250 m north and 250 m east of the original grid?  The new map will not be identical, but just how different will it be?  How sensitive, in short, is our overall procedure (including the grid spacing) to an arbitrary shift in the origin of the sample grid?  It seems reasonable to propose that a robust sampling scheme to map a spatial variable should not be sensitive to this offset. How sensitive it is will depend on the spatial variability and the grid spacing.

Just as with the kriging variance one can compute, from the variogram alone, the correlation between predicted values on two maps made with the same grid density but a half-grid offset of the origin.  We call this the offset correlation.  It is a bounded measure of the consistency of the map under arbitrary shifts of the origin, potentially easier to explain intuitively to the data user than is a variance.  In our paper we show some hypothetical and real examples of the offset correlation both for ordinary kriging and factorial kriging, considering geochemical data from the East of England and comparing the designs of two national-scale soil sampling schemes from the UK.

Lark, R.M., Lapworth, D.J., 2013. The offset correlation, a novel quality measure for planning geochemical surveys of the soil by kriging. Geoderma 197–198, 27–35.

The full paper in pdf is made available here

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