[ai-geostats] Standardized Semivariograms
Thanks to all who kindly took the time to reply to my question.
I believe your replies will further help to be more careful when using geostatistical software or more simply about the terminology to use when writing papers.
I will add to the replies sent directly to the list the following one received from Donald Myers:
"First of all the sample variogram (whether "standardized", relative or even "robust") is a statistic, i.e., an estimator of some of the values of the variogram (they are not functions in the same sense that the variogram is). I say values because the variogram itself is a function and is not uniquely determined by knowing or estimating values for some distances/directions. If one assumes that the variogram type is known, e.g., spherical, gaussian, etc then it would be sufficient to know/estimate the parameters of the variogram to uniquely determine the variogram. However the sample variogram does not directly estimate the parameters. Indirectly it is possible to estimate the parameters by using for example weighted least squares but there are at least two different weighting schemes that are often used (and sometimes combined). These include inversely with respect to the number of pairs and inversely with respect to the variance of the squared differences in the distance class.
Note that none of the different forms of the sample variogram are unique, the user (or the software programmer) must decide on at least the following: (1) distance tolerance for each lag class, (2) maximum number of lag distances to compute/plot, (3) principal direction, (4) angle tolerance. Even with an omni-directional sample variogram the apparent "information" provided by the sample variogram can change rather drammatically dependent on the choices for the first two.
There are various reasons why one might "standardize" the sample variogram but mostly they are not related to theory. If you look at the ordinary or universal kriging equations you will see that you can divide both sides of the equations (other than the unbiasedness constraints) by the "sill" of the variogram (assuming then that there is a sill) without changing the resulting kriging weights (the Lagrange multiplier(s) will change however and hence the kriging variance will appear to change).
I would agree that it would be nice to "standardize" the use of the terminology but I doubt that will happen, at least in terms of software. There is still the split over "semivariogram" vs "variogram" although most authors in the proceedings of the 1988 Geostatistics conference had already switched to "variogram", including G. Matheron.
Earlier versions of SURFER used the term "kriging" to mean ordinary kriging with a linear variogram model (since in that case it was not necessary to estimate and fit the model, the "slope" only affects the kriging variance and not the weights). That is, the software did not allow the user to estimate/fit the variogram and allowed an unknowing user to believe that it was not necessary. "
Thanks again to all those who reacted to my mail.
Gregoire Dubois (Ph.D.)
JRC - European Commission
IES - Emissions and Health Unit
Radioactivity Environmental Monitoring group
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21020 Ispra (VA)
Tel. +39 (0)332 78 6360
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"The views expressed are purely those of the writer and may not in any circumstances be regarded as stating an official position of the European Commission."