I am dealing with heavy metals in upper soil layers of forests. Some

of my (indicator) variograms behave like a spherical model at the

beginning but in stead of reaching the sill they behave like a

power-model for greater distances. I tried

several power models but they didn't fit the variogram very well.

Just two questions:

1. What does this special behavior of the experimental variogram tell

me about the spatial structure or stationarity?

2. Can I mix" a spherical and and a power model to build up one

nested model?

and here are the answers:

Isobel Clark [drisobelclark@...] wrote:

It sounds like you have a trend in your values.

You should have received your copy of the book by now.

Check out Chapter 7 for full information on fitting

trend surfaces, Chapter 9 on diagnosing trend in the

semi-variogram (Wolfcamp example) and Chapter 12 for

Universal Kriging.

second answer of isobel:

Let me clear up the indicator thing for you first.

An indicator transform changes all values below

the 'cutoff' or discriminator value to 0 and all

values above it to 1. At least, that's what ours

does, other packages do it the other way round --

change all values below cutoff to 1 and all values

above to 0.

You only need to do multiple indicators if you have

(a) a very strange distribution or (b) a mixture of

different populations. See Chapter 12 on indicators.

I would recommend you look at the distribution

(histogram,

probability plot, etc) before resorting to something

as

complicated as a multi-indicator approach. If you are

getting decent semi-variograms from your ordinary

values

you don't need indicator kriging!

As far as your semi-variogram goes, if your 'power

model'

looks like a power less than 1 you probably just have

a

nested Spherical structure which hasn't reached the

later

range of influence yet. Try modelling a second

component

with a range longer than yoru maximum lag distance.

Re-reading your e-mail carefully (sorry, I tend to

skim)

you say your first 4 deciles have this unbounded

nature

and the others are bounded. What this is saying is

that

for low cutoffs your ranges are much longer than for

higher cutoffs. Your high values tend to occur in

smaller more coherent 'blobs' than the lower values

which are diffuse and related to much greater

distances.

This could be multiple populations with the high

values

coming from a more localised concentrated source and

the

rest forming a 'background' phenomenon. Best thing to

do

is simply postplot the indicators for the higher

cutoffs

and see WHERE they congregate. Then try to relate that

to

what you know about the origins of your variable. For

example, are these high human concentrations?

Industrial

areas? Forest? Water? Landfill? and so on. You know

your

data better than I do.

Visual assessment of patterns is still a very valuable

tool!

answers of

Donald E. Myers

Department of Mathematics

University of Arizona

Tucson, AZ 85721

http://www.u.arizona.edu/~donaldm

The indicator variogram has to be bounded if the random function is

stationary, note not just second order stationary but stationary. An

unbounded sample indicator variogram suggests that the stationarity

condition is not satisfied.

I mentioned before that "stationarity" in the case of the indicator

transform is not the same as second order stationarity which is the

common

assumption underlying geostatistics. "Stationarity" in this case means

that

for any choice of points s1,.....sn and any vector h, the joint

probability distribution of Z(s1),....,Z(sn) is the same as the joint

probability distribution of Z(s1 +h),...., Z(sn + h). That is, all the

joint probability distributions are translation invariant. Unfortunately

one

will never have the right kind of data to test this condition. This

condition also implies that the probability distribution of Z(s) is the

same

as the probability distribution for Z(s+h) for any point s and any

vector h,

this common distribution is usually called the marginal and might be

denoted

as F(z) (no subscript on the F). Then one can show that the indicator

variogram is actually F(z)[1-F(z)] which is always bounded by 0.25.

This

bound applies for any choice of the cutoff "z". Hence it would not be

theoretically possible for the sample indicator variogram to be bounded

for

some cutoffs and unbounded for others, IF the stationarity condition is

satisfied. Unfortunately all of indicator geostatistics is based on this

assumption and unlike the problem with a non-constant mean it would not

be

sufficient to compute some kind of residuals and then do the analysis on

the

residuals.

A point I should have mentioned to be sure there is no confusion, you

made

reference to water quality "indicators". That use of the term indicator

is

not the same as is referred to in discussing indicator variograms.

Dr. Heinz Burger

Freie Universitaet Berlin

- Geoinformatik -

Malteserstr. 74-100

12249 BERLIN, Germany

Tel. (49) 30-838-70561 Fax: (49) 30-775-2075

e-mail: hburger@...-berlin.de

Web-Seite: http://userpage.fu-berlin.de/~hburger/hb

Linear combinations of admissible models produce admissible models.

The problem is: what will you do? For kriging you only need a good model

within the search radius. For an interpretation of the spatial structure

of

your phenomenon you don't need a model-function in every case (except

for comparison). A detailed trend analysis and resiual analysis (incl.

variogram) will give more information about local and regional spatial

structures. The variogram is only one tool among others.

Edzer J. Pebesma [e.pebesma@...]

>

It may be non-stationary.

> Hello list,

>

> I am dealing with heavy metals in upper soil layers of forests. Some

> of my (indicator) variograms behave like a spherical model at the

> beginning but in stead of reaching the sill they behave like a

> power-model for greater distances. I tried

> several power models but they didn't fit the variogram very well.

> Just two questions:

>

> 1. What does this special behavior of the experimental variogram tell

> me about the spatial structure or stationarity?

> 2. Can I ?mix" a spherical and and a power model to build up one

Yes, most common software, like GSLIB, Variowin, gstat, ...

> nested model?

>

Yetta Jager

Environmental Sciences Division

Oak Ridge National Laboratory

P.O. Box 2008, MS 6036

Oak Ridge, TN 37831-6036

U.S.A.

OFFICE: 865/574-8143

FAX: 865/576-8543

Work email: jagerhi@...

In my opinion, it suggests that one might be better off finding a

covariate

to explain the drift and removing it before modeling the autocorrelation

of

residuals. See our TM report on my website with Pattern-Plus in the

title.

Marco Alfaro [malfaro@...]

You cannot use the power model in the case of indicator variograms.

The variogram g(h) of a random set (indicator 0 or 1) has the following

relationships (not accomplished by the power variogram):

g(h) <= 0.5

g(2h) <= 2g(h)

thanks to all

Lenz

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