1261Re: GEOSTATS: Linear Model of Coregionalization
- Jun 2, 1999Hi Gerhard,
You are perfectly right that there has been a lot of
confusion about the modeling of direct and cross variograms
in the literature.
As explained in my book (p. 123) and illustrated in
Goovaerts (1994), the key point in modeling coregionalization
is to ensure that the matrix of auto and cross covariance
models is positive semi-definite for any possible lag
distance and direction.
You have essentially 2 options:
1. use the Linear model of coregionalization that allows
an easy check of the permissibility of your model.
The price to pay is the requirement that all direct
and cross variograms share the same set of basic structures.
Thus if you use your program that doesn't allow nested structures,
you have to fit the same model with the same range to the three
Note that: a) Variables that are well cross-correlated are likely
to show similar patterns of spatial variability.
What's the point of modeling the coregionalization
of variables that are weakly correlated since
it won't bring any benefit in cokriging (goovaerts, 1998).
b) If you don't restrict yourself to single-structure
models, you get much more flexibility because
there is no need for the direct and cross semivariograms to
include all the basic structures, see Fig. 1.16 in my book.
2. Model independently the 3 variograms and hope that in the subsequent
cokriging all matrices will be positive definite. Many people who
modeled independently their variograms reported negative kriging
variances and numerical instabilities. I remember that some authors
proposed to perturb (slightly!) any troublesome cokriging matrix
to make it positive definite, but I wouldn't recommend this practice.
In summary, my advice is to stay with the well proven and yet reasonably
flexible linear model of coregionalization. You may want to check the
following sources for programs that fit nested models of
1. Morisette J. 1997. Examples using SAS to fit the model of linear
coregionalization. Computer and Geosciences, 23:(3) 317-323.
for public-domain programs to perform coregionalization analysis
Goovaerts, P., 1994. On a controversial method for modeling a
coregionalization. Math. Geol. 26, 197--204.
Goovaerts, P., 1998. Ordinary cokriging revisited.
Math. Geol. 30, 21--42.
| \ / | Pierre Goovaerts
|_ \ / _| Assistant professor
__|________\/________|__ Dept of Civil & Environmental Engineering
| | The University of Michigan
| M I C H I G A N | EWRE Building, Room 117
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Fax: (734) 763-2275
On Tue, 1 Jun 1999, gerhard wrote:
> Dear all,
> I am looking for the "ultimate" truth about the Linear Model of
> I want to model the variograms of variable A and B, and the
> cross-variogram of variables A-B. The variogram and cross-
> variogram models cannot be built independently from one
> another (Issaks & Srivastava, 1989; Goovaerts, 1997).
> This is where the Linear Model of Coregionalization comes
> into play: any basic model that is included in the cross-
> variogram model must be included in all the variogram
> models (ibid.).
> I have a program for calculating variogram and cross-
> variogram models that can handle only one basic model for
> each (cross-) variogram, i.e. no nested structures are allowed.
> From the above description of the Linear Model of
> Coregionalization does it now follow that
> Question 1:
> if I use e.g. a spherical model for the cross-variogram of
> variables A-B, I have to use a spherical model for both, the
> variogram of variable A and the variogram of variable B?
> And further:
> Question 2:
> The ranges of both variogram models and the range of the cross-
> variogram model have to be identical?
> Question 3:
> What are the consequences of violating the Linear Model of
> Coregionalization? Some people seem to think that it is o.k.
> to ignore the Linear Model of Coregionalization as long as
> they appear to get reasonable results from their analysis.
> The reason for my confusion comes from the fact that I have
> seen both cases in the literature:
> people using the same variogram models and the same ranges
> for their variograms and cross-variograms, and
> people (non-geostatisticans?) using different models (e.g.
> an exponential model for variogram A, a spherical model
> for variogram B, and a Gaussian model for the cross-
> variogram A-B) with different ranges.
> I would really appreciate any comments that could
> shed some light on this (for me) confusing topic!
> Thanks for your help and have a nice day!
> | Gerhard Hunner |
> | Ph.D. candidate |
> | Geostatistics, GIS, Remote Sensing |
> | GIS and Remote Sensing Program |
> | Department of Forest Sciences |
> | Colorado State University |
> | Fort Collins, CO 80523 |
> | USA |
> | Tel.: (970) 221-1826 |
> | Fax: (970) 491-6754 |
> | Email: gerhard@... |
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