here is a summary of the replies I got to my question on standardised versus
general relative semivariograms. The original question is posted at the end of
this mail. The least I can say is that there is not much work written about it
even if these are frequently used in environmental sciences !
Thanks a lot to Digby Millikan, Nicolas Jeannee, Ulrich Leopold and Yetta
Jager for their kind help.
Digby Millikan suggests the following reading:
M. David "Geostatistical Ore Reserve Estimation" discusses the General
Relative Variogram which was developed to demonstrate the existence of the
proportional effect which he found to exist in ore bodies. He generated
separate variograms of low and high grade regions and found that when
compensated by the mean where found to be identical, showing a direct relation
between correlation and mean.
In the text are graphs of standard deviation vs. mean scatter plots.
While the proportional effect displayed variograms for different regions,
Multiple Indicator Kriging is used where different variabilities for
different grade ranges co-exist in the same regions. Variograms are generated
for separate grade ranges which are then used in the Multiple Indicator
Kriging algorithm to produce the estimates.
Nicolas Jeannee, who works like me on pollutants which typically have highly
skewed distributions, underlines that a problem with these semivariograms is
their explicit estimation of mean and variance, which are non robust
statistics in the case of positively skewed variables, and require
He suggests the use of the weighted variograms presented in
Rivoirard (2000). Weighted Variograms. In: �Geostats 2000�,
W. Kleingeld and D. Krige (Eds.).
Proocedings available on cd-rom only for the moment..
Nicolas and Ulrich Leopold suggested another reference:
Srivastava R. M & H. M. Parker. 1989. Robust measures of spatial continuity.
In: Geostatistics, M. Armstrong (Editor). Kluwer, Dordrecht.
Vol. I, pages 295-308
While my posting was focusing more on the use of these semivariograms for
descriptive analyses, Yetta Jager anticipated already further questions:
�I can see why one would want to use weights inversely related to variance to
account for differences in variance (heteroscedasticity) in estimating
variogram parameters, but I'm not clear how a variogram that is standardized
to means fits into the kriging model. How would you be able to estimate at
points with unknown values if the autocorrelation with neighbors depended on
the unknown estimate?
In my experience with indicator kriging, the closer you get to the median, the
higher the variance (sill) is (p=0.5==>var=0.25), and the more room there is
to observe spatial autocorrelation as a difference between nugget and sill.
As you get to the extremes, the sill drops and it is hard to detect
autocorrelation. I suppose in theory it shouldn't matter, but in practice
that's what I've observed."
Thanks again for all the replies !
>can anyone point the relative advantages and drawbacks of
>- standardised semivariograms (See Pannatier, 1996: Variowin. Software
>for spatial data analysis in 2 D, page 39) where gamma(h) is divided by the
>variance for each lag;
>- general relative semivariogram (Isaaks & Srivastava, 1989: An introduction
>to applied geostatistics, page 164) where gamma(h) is standardised by the
>of each lag.
>I'm currently analysing the spatial structure of radioactive deposition for
>different levels with the help of indicators. Standardised & general
>semivariograms describe very well the structures while the semivariogram is
>not really appropriate for such a highly
>skewed variable. For low values of the chosen thresholds, the standardised
>semivariograms shows me a stronger spatial correlation compared to the
>For higher threshold values, the opposite situation appears.
>Would this mean that low values show strong fluctuations but that the mean
>value remains quite constant in space while high levels of radioactivity
>less fluctuations but the mean values change more in space ?. Has anyone
>experienced similar observations with other variables ?
>Apparently, there has not been much published on these functions, even if
>these are frequently used.
Institute of Mineralogy and Petrography
Dept. of Earth Sciences
University of Lausanne
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