optimization schemes (annealing, GAs) whereby one

can flexibly model the structure of variates and co-variates

without having to worry about LMC. At the cost of CPU time,

of course.

Syed

Pierre Goovaerts <goovaert@...> on 06/03/99 01:03:41 AM

To: gerhard <gerhard@...>

cc: ai-geostats@...

Subject: Re: GEOSTATS: Linear Model of Coregionalization

Hi 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

variograms.

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

coregionalization:

1. Morisette J. 1997. Examples using SAS to fit the model of linear

coregionalization. Computer and Geosciences, 23:(3) 317-323.

2. http://www.agro.ucl.ac.be/biom/recherche/projets/agromet/

for public-domain programs to perform coregionalization analysis

and cokriging.

References

---------

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.

Cheers,

Pierre

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

________ ________

| \ / | 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

|________________________| Ann Arbor, Michigan, 48109-2125, U.S.A

_| |_\ /_| |_

| |\ /| | E-mail: goovaert@...

|________| \/ |________| Phone: (734) 936-0141

Fax: (734) 763-2275

http://www-personal.engin.umich.edu/~goovaert/

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

On Tue, 1 Jun 1999, gerhard wrote:

> Dear all,

>

> I am looking for the "ultimate" truth about the Linear Model of

> Coregionalization!

>

> 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

>

> --

>

>

>

>

> ******************************************************************

> | 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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