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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