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RES: GEOSTATS: Linear Model of Coregionalization

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  • Jean-Louis Gelot
    A great software to do it right: ISATIS from Center of Geostat Fontainebleau. Cheers Jean-Louis Gelot CGG do Brazil Geosciences Development Manager Rua Conde
    Message 1 of 1 , Jun 2, 1999
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      A great software to do it right: ISATIS from Center of Geostat Fontainebleau.

      Cheers


      Jean-Louis Gelot CGG do Brazil
      Geosciences Development Manager Rua Conde De Lages 44,piso 12
      South America Rio De Janeiro 20241-080
      tel: (55)21-852-5989 Brazil
      fax: (55)21-232-3440
      email: jlgelot@...

      For information on all CGG activities: www.cgg.com
      For information on STRATIMAGIC: www.flagshipgeo.com
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      ----- Mensagem original -----
      De: Pierre Goovaerts [SMTP:goovaert@...]
      Enviada em: Wednesday, June 02, 1999 1:04 PM
      Para: gerhard
      Cc: ai-geostats@...
      Assunto: 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
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      <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>


      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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