## GEOSTATS: Summary 2: decomposition of nugget effect

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• Hello again, here is a summary of the replies I received to my second question (on the decomposition of the nugget effect). Thanks again a lot to all who
Message 1 of 1 , Sep 6, 1905
Hello again,

here is a summary of the replies I received to my second question (on the
decomposition of the nugget effect). Thanks again a lot to all who
contributed.

2nd question:

Andy Long (aelon@...), Donald Myers (myers@...) and
Denis Marcotte (Denis.Marcotte@...) have underlined the
differences between the theoretical definition of Kriging and its
implementation in most software. Kriging is an exact interpolator since it
always will return the value of the site if evaluated at a site. Kriging makes
no allowance for how the nugget is modelled. Kriging is a JUMP interpolator
in the case of a nugget, which means that the surface will not be continuous
but still exact.

This is clear from the simple kriging equations:

|v(1,1) v(1,2) ... v(1,N)| |lambda_1| = |v(1,x)|
|v(2,1) v(2,2) ... v(2,N)| |lambda_2| = |v(2,x)|
| | | | = | |
|v(N,1) v(N,2) ... v(N,N)| |lambda_N| = |v(N,x)|

Now if x happens to be one of the points 1,...,N, say i, then the solution of
this system is simply lambda_i=1, all other lambda=0, which means that the
data location i will be given all the weight, and will return thus z_i - exact
interpolation

Donald Myers (myers@...) also writes that "The usual versions of
the ordinary, universal kriging equations do not separate the nugget into two
parts and hence if the nugget term is non-zero there will be a jump
discontinuity at the data locations (it will still be an exact interpolator).
However one can modify the kriging equations slightly, if written in variogram
form this means that the diagonal entries are non-zero (the values are the
value of the error variance), the remainder of the nugget term appears in the
variogram values off-diagonal. You will not see this modified version in much
of the geostatistical literature. Cressie discusses it in his book and also in
a paper that appeared in the American Statistician.

You will also find the equations in the following papers

1994, Myers,D.E., Statistical Methods for Interpolation of Spatial Data.
J. Applied Science and Computations 1, 283-318

1994, Myers,D.E., Spatial Interpolation: An Overview. Geoderma 62, 17-28"

Denis Marcotte (Denis.Marcotte@...) writes

The difference between both form of estimates appear only at a
sample point. For every other point (even an epsilon away), you will
get exactly the same estimates. This is possibly one of the reason why
the distinction between EV and MV contributions to the nugget was not
explicitely taken into account in the kriging equations in geostatistical
textbooks before Cressie's book. At a sample point (and of course using this
sample point for the estimation), splitting the nugget in EV and MV will
produce different estimates if programmed properly (it is then nothing else
than a special case of factorial kriging). The reason why the estimates differ
is that with Cressies'equation you are estimating Y(xi), not
Z(xi)=Y(xi)+e(xi) like with other kriging programs.

Another reference given by E. Pebesma is

Ronald Christensen Linear Models for Multivariate, Time Series and Spatial
Data (Springer Texts inStatistics)
Hardcover (January 1991)
Springer Verlag; ISBN: 038797413X

Jeff Myers (jeff_myers@...) also underlines the following:
"The micro variance MV should be highly correlated with the Fundamental
Error (FE) of sampling, and to some degree the grouping and segregation
error (GE). For particulate materials (soils, etc.), the FE can be
extremely high (> 1000% relative error). This is true for environmental
contaminants such as explosives and PCBs or mining variables such as
precious metals. the United States Environmental Protection Agency has
recognized this and now has guidance documents (SW-846 Chapter Nine, FFFI)
to assist in the evaluation of the FE, based on the work of Pierre Gy.
This problem is discussed in Pitard (1993), Myers (1997), and Gy (1998?).
FE is highly dependent on the sample and subsample support (mass) and
occurs at each sampling step (which means it is additive), so it can can
have a great influence on the nugget effect.

In contrast, typical laboratory measurement errors in the environment are
on the order of 20 to 30%. If the FE is less than 20-30%, it will be hard
to distinguish whether the nugget is a result of sampling or measurement
error. An interesting website which deals with this problem is

http://pubs.acs.org/hotartcl/ac/99/aug/settle.html

"

It is 1 AM here, I'm off to bed and will dream of worlds without
nuggets...

Gregoire

Gregoire Dubois
Section of Earth Sciences
Institute of Mineralogy and Petrography
University of Lausanne
Switzerland

Currently detached in Italy

http://curie.ei.jrc.it/ai-geostats.htm

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