[ai-geostats] Definition of standardized variograms
Michel David coined the term "relative semi-variogram"
back in the 70s for what I think you mean by general
relative -- that is, each lag is divided by the square
of the mean of the samples used at that lag.
Gary Raymond proposed the pairwise relative soon
after. I used the type you are describing where the
whole semi-variogram is divided by the same
mean-squared in my 1979 paper (Does Geostatistics
Work) because I was analysing a line of samples where
all samples are used at every lag.
The term "standardised" in general statistics usually
means dividing through by the variance or standard
deviation (not a mean). This is the first time I have
seen it in context with a semi-variogram. Seen with no
other information, I would have taken this to imply
standardised to total sill of 1. This would mean
dividing by the variance, not the mean-squared.
Relative semi-variograms help you avoid the
proportional effect if you are trying to calculate a
semi-variogram on positively skewed data. Noel Cressie
wrote a paper in Mathematical Geology (early 90s?)
which showed that the David relative semi-variogram
was topologically equivalent to using logarithms. You
data does not have to be lognormal to do this.
Computationally, taking logarithms is faster and more
stable than relative semi-variograms. Probably why
most people don't bother. Gary Raymond provides
software for the pair-wise and Geostat Systems will
have relative semi-variograms. Don't know of any free
- Hi Gregoire,
I agree with you regarding the merits of the standardized semivariogram as implemented
in variowin software. In one of my last studies, the rescaling by the lag variance helped
correcting the preferential sampling of wells with high arsenic levels, leading to a
susbtantial decrease in random fluctuations of the experimental semivariograms.
While the general relative semivariogram approximates the lag variance by the square
of the lag mean, the standardized semivariogram uses the actual lag variance, hence
makes less assumptions.
Regarding the terminology, I guess we should used a term like "lag-standardized"
to distinguish the global and lag-specific standardization or rescaling of semivariogram
From: Gregoire Dubois [mailto:gregoire.dubois@...]
Sent: Tue 4/5/2005 9:48 AM
Subject: [ai-geostats] Definition of standardize variograms
While playing around with different software, I encounter different definitions for standardized variograms.
Surfer (which is using the terminology of Variowin), uses the term "standardized semivariogram" for variograms obtained by dividing the semivariance by the lag variance, while GS+ uses the total variance. While the function obtained in GS+ is only a matter of rescaling variograms, allowing so various variograms to be compared, those proposed in Surfer have the same pupose as the local, pairwise and/or general relative variograms (see Isaaks & Srivastava, page 163-170), that is to reduce the influence of local means. Interestingly enough, one may note that very few software propose relative variograms while I, very personally, consider these functions as essential for detecting spatial structures of many environmental variables.
I have thus here two questions about the use of standardized/relative variogram:
1) What is the correct terminology or definition for standardized variograms? (I personally do not like very much the use of "standardized" when the standardisation is only applied to each lag...)
2) The general relative variogram (lag divided by the mean of the lag) has properties that are very similar to the "standardized" variogram (lag divided by the variance of the lag) but both functions differ. How shall one decide what to use and what are the relative properties of these functions?
Thank you in advance for any feedback.
PS: a few points here good be added to Tom Mueller's FAQ on Geostatistical Software Conventions.
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
Fax. +39 (0)332 78 5466
Email: gregoire.dubois@... <mailto:gregoire.dubois@...>
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