Kriging, in it's native state, does not ensure positivity of the weights or the estimates. The methodology does not 'know' that such and such a variable (eg concentrations in ppm or permeability in md) has to be positive. For the most part this is a good thing. Consider the 'picture' below. We are trying to estimate elevation on the top of the 'hill' using the 6 data points - marked with a * - that are on the flanks. A reasonable estimate would be given by the + (If the diagram gets screwed up - then the + is at a slightly higher elevation that any of the data - as we expectsince we are estimating the top of the hill)

+

* *

* *

* *

Now kriging does this by assigning weights to all 6 points - as you suggest the nearer ones to the point to be estimated will have high positive weighst and in this case the furthest will have negative weights. The weights need to be negative in this case to get the estimate at the top of the hill higher than any of the data points. You can see this - because the highest possible estimate that you can get using positive weights only is equal to the highest data point (when a weight of one is applied to it and zero to all the other points)

So, to enable kriging to get estimates that are higher than the maximum data point (or lower than the minimum) you need to have negative weights. It is the variogram that determines just how large those negative weights are to be (based on the degree of continuity of the variable at hand). If you really dislike your negative estimates you could change your variogram slightly (Add a small nugget effect / Reduce the range of the variogram /Don't use Gaussian models or other variogram with quadratic behavior at the origin. These are 3 methods that will usually help to improve matters for you). If you object to modifing your variogram you could try 'positive kriging'. There were a couple of papers by Olivier Dubrule on this subject in the mid 80's in Mathematical Geology (there may be more recent stuff by others - I don't know - and I don't have the exact reference to Olivier's papers). However this is fairly heavy duty stuff from a computer resource perspective - so unless it is a real concern or they become too large I would be tempted to live with the small negative estimates and just correct them to zero.

Best Regards

Colin Daly

p.s. I have just 'grabbed' some references for this stuff from the web at Melanie Wall's site http://www.biostat.umn.edu/~melanie/ - I neither endorse nor condemn any of them as I don't know them (with the exception of Barnes - which I can't remember but I think predate the Dubrule papers )

a.. Herzfeld, U.C. (1987) "A Note on Programs Performing Kriging with Nonnegative Weights" Mathematical Geology Vol 21 391-393.

b.. Szidarovsky, F., Baafi, E. Y., and Kim, Y.C., (1987) "Kriging Without Negative Weights" Mathematical Geology Vol 19 549-559.

c.. Baafi, E.Y., and Szidarovsky, F. (1986) "On nonegative weights of linear kriging estimation" Mining Engineering 437-442.

d.. Barnes, R.J. and Johnson, T.B. (1984) "Positive Kriging" Geostatistics for Natural Resources Characterization, Part 1 eds. G. Verly et al. 231-244.

----- Original Message -----

From: Colin Badenhorst

To: ai-geostats@...

Sent: Tuesday, August 07, 2001 1:30 PM

Subject: AI-GEOSTATS: Negative Kriging Weights & Estimates

I have recently carried out ordinary kriging for a ore reserve estimation exercise (using GSLIB), and noted that a very few of the grade estimates are negative (always a very small number e.g. 0.002 ppm). I have been able to trace this back to negative kriging weights, and would like some confirmation of my understanding of how this occurs.

My understanding is that samples lying close to the block centroids being estimated recieve a high weighting, and samples further away recieve a lower weighting. However, if the sample search neighbourhood is very large, and since the sum of the weights must equal 1, the samples lying furtherest away the centroid/s are assigned a very small negative weight, in order for the closer samples to maintain their higher weighting, and for the sum of the weights to equal 1.

Is my understanding of this "compensation" correct? Why wouldn't the weights for the furtherest samples be calculated by subtracting the weighting of the closer samples from 1, instead of compensating using negative weights afterwards?

Colin

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