AI-GEOSTATS: summary variogram modelling with nested structures
- this was my question(s)
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
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
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
probability plot, etc) before resorting to something
complicated as a multi-indicator approach. If you are
getting decent semi-variograms from your ordinary
you don't need indicator kriging!
As far as your semi-variogram goes, if your 'power
looks like a power less than 1 you probably just have
nested Spherical structure which hasn't reached the
range of influence yet. Try modelling a second
with a range longer than yoru maximum lag distance.
Re-reading your e-mail carefully (sorry, I tend to
you say your first 4 deciles have this unbounded
and the others are bounded. What this is saying is
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
This could be multiple populations with the high
coming from a more localised concentrated source and
rest forming a 'background' phenomenon. Best thing to
is simply postplot the indicators for the higher
and see WHERE they congregate. Then try to relate that
what you know about the origins of your variable. For
example, are these high human concentrations?
areas? Forest? Water? Landfill? and so on. You know
data better than I do.
Visual assessment of patterns is still a very valuable
Donald E. Myers
Department of Mathematics
University of Arizona
Tucson, AZ 85721
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
assumption underlying geostatistics. "Stationarity" in this case means
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
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
as the probability distribution for Z(s+h) for any point s and any
this common distribution is usually called the marginal and might be
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.
bound applies for any choice of the cutoff "z". Hence it would not be
theoretically possible for the sample indicator variogram to be bounded
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
sufficient to compute some kind of residuals and then do the analysis on
A point I should have mentioned to be sure there is no confusion, you
reference to water quality "indicators". That use of the term indicator
not the same as is referred to in discussing indicator variograms.
Dr. Heinz Burger
Freie Universitaet Berlin
- Geoinformatik -
12249 BERLIN, Germany
Tel. (49) 30-838-70561 Fax: (49) 30-775-2075
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
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 oneYes, most common software, like GSLIB, Variowin, gstat, ...
> nested model?
Environmental Sciences Division
Oak Ridge National Laboratory
P.O. Box 2008, MS 6036
Oak Ridge, TN 37831-6036
Work email: jagerhi@...
In my opinion, it suggests that one might be better off finding a
to explain the drift and removing it before modeling the autocorrelation
residuals. See our TM report on my website with Pattern-Plus in the
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
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