AI-GEOSTATS: Summary of effect of variogram parameter uncertainty on OK
- Hi all:
In intrinsic geostatistics, spatial continuity is described by a model
(usually the 'theoretical' variogram). Its parameters are estimated by
fitting the model to the moment-based Matheron non-parametric
('empirical') variogram (with several methods available for estimation).
Despite being estimated, the model variogram is treated as known in
intrinsic geostatistics, i.e. the covariance matrix of variogram
parameters is ignored. The assumedly known (but actually estimated)
model is then used in kriging. I asked then what was the feeling of
people in this list about the effect of model-variogram parameter
uncertainty on kriging, in particular OK.
My original question:
Dear list members:
Do you expect that the uncertainty related to estimating variogram
parameters have a strong effect on spatial prediction via ordinary
kriging? It is my impression that usually the analist disregard this
source of variance and proceeds as if these parameters from the
variogram were known. This might be justified since normally the sample
size (pairs of observations) is quite high when fitting the variogram
via weighted least squares, for instance. Any opinion on this and/or
references would be much welcome.
Additional point raised:
Dear list members,
Following Ruben Roa's email, I would like to add that I did not
find any clear criterion to what is the minimum number of data
pairs that should be used in each lag and this would certainly
affect the variogram uncertainty. I know that a minimum of 30 data
pairs is used frequently but my impression is that this is a "rule
of thumb". I guess this rule originated from the central limit
theorem (?). I found that the lack of a clear criterion might have
a big effect on the estimation of the Variogram especially in cases
where the data points are non regular and in cases where a non
parametric variogram might be considered.
I would gratefully appreciate any comments on this as well.
Response to point raised by Arie:
I do not know if it is a rule of thumb. But it also depends on the
variability of your data values. I think it is quite subjective how much
pairs you will use.
But you should also consider some more robust variogram estimators as
or standardized variograms to see if
there are no artificial structures or erratic structures caused by
values or outliers or even trends...
There is a paper which is quite interesting:
Srivastava, R.M.; Parker, H.M., 1989:Robust measures of spatial
continuity. In: Armstrong, M., 1989 (editor):Geostatistics. Vol.I.
Dordrecht, pp. 295-308.
Response to my original question:
The variogram parameters you select will influence the final model. In
my experience, however, the number of neighboring samples can have a
greater influence than the variogram parameters. I have attached a
paper describing the selection of the number of samples to use when
kriging. This is also described in my book on "Geostatistical Error
Management". You can also visit my website at http://www.gemdqos.com
Response to my original question:
You might want to look at a recent book by M. Stein, Some Theory about
kriging, Springer Publishers
Donald E. Myers
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