Hi Colin,
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 )
Herzfeld, U.C. (1987) "A Note on Programs Performing Kriging with
Nonnegative Weights" Mathematical Geology Vol 21 391-393.
Szidarovsky, F., Baafi, E. Y., and Kim, Y.C., (1987) "Kriging Without
Negative Weights" Mathematical Geology Vol 19 549-559.
Baafi, E.Y., and Szidarovsky, F. (1986) "On nonegative weights of linear
kriging estimation" Mining Engineering 437-442.
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: [EMAIL PROTECTED]
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 c
