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AI-GEOSTATS: The relation between variogram and covariogram (Anisotropic)

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  • jack webster
    Hi all list members Suppose that g denote the semi-variogram and C denote the Covariogram functions. According to Cressie(1993, P. 67) the relation is g(h)
    Message 1 of 4 , May 13, 2002
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      Hi all list members
      Suppose that g denote the semi-variogram and C
      denote the Covariogram functions. According to
      Cressie(1993, P. 67) the relation is
      g(h) = C(0) - C(h)
      where h is the distance vector.In the isotropic case
      C(0) = total sill = c0 + c
      I have an Anisotropic Variogram in which c is varying
      with direction.Therefore my question is what is C(0)
      in this case?

      =====


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    • jack webster
      Hi Dear doctor Abedini I have constructed a more general variogram model, in which all of the parameters are varying with direction. My model more general than
      Message 2 of 4 , May 13, 2002
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        Hi Dear doctor Abedini
        I have constructed a more general variogram model, in
        which all of the parameters are varying with
        direction. My model more general than range anisotropy
        and sill anisotropy. Since I want to simulate from
        this model ,with cholesky decomposition method, I need
        the covariogram model, which is respected to my
        variogram model. Therefore I need the relation between
        variogram and covariogram. In fact I need to specify
        C(0).

        Thank you: W.

        --- "Dr. M.J. Abedini from Civil Eng."
        <abedini@...> wrote:
        >
        >
        > Mr. Webster
        >
        > First of all, C in that equation is covariance which
        > is totally different
        > from covariogram.
        >
        > As for c(0), first you need to specify which type of
        > anisotropy you have.
        > In geometric anisotropy, range is direction
        > dependent and sill [c(0)] is
        > constant while in zonal anisotropy, the sill is
        > direction dependent and
        > range is constant. You might have a combination of
        > geometric and zonal
        > anisotropy for which both sill and range are
        > direction dependent. You may
        > even have nugget effect anisotropy for which negget
        > is direction
        > dependent.
        >
        > I will create a list of papers on modeling
        > anisotropy and will forward it
        > to you shortly.
        >
        > Thanks
        > Abedini
        >
        > On Mon, 13 May 2002, jack webster wrote:
        >
        > > Hi all list members
        > > Suppose that g denote the semi-variogram and C
        > > denote the Covariogram functions. According to
        > > Cressie(1993, P. 67) the relation is
        > > g(h) = C(0) - C(h)
        > > where h is the distance vector.In the isotropic
        > case
        > > C(0) = total sill = c0 +
        > c
        > > I have an Anisotropic Variogram in which c is
        > varying
        > > with direction.Therefore my question is what is
        > C(0)
        > > in this case?
        > >
        > > =====
        > >
        > >
        > > __________________________________________________
        > > Do You Yahoo!?
        > > LAUNCH - Your Yahoo! Music Experience
        > > http://launch.yahoo.com
        > >
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        > ai-geostats@...
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        > >
        >


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      • Gerald van den Boogaart
        Dear Jack Webster, The mapping of the variogram to the covariogram is not one to one. When c(h) is a covariogram having g(h) as variogram, then c(h)+k for
        Message 3 of 4 , May 14, 2002
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          Dear Jack Webster,

          The mapping of the variogram to the covariogram is not one to one.
          When c(h) is a covariogram having g(h) as variogram, then c(h)+k for any
          positive real number k has the same variogram. Thus for your simulations it is appropriate to
          use c(h)=k-g(h) for any k larger than (or equal to) the maximum (or supreme)
          of your directional sills.

          Best regards
          Gerald v.d. Boogaart



          jack webster wrote:
          >
          > Hi Dear doctor Abedini
          > I have constructed a more general variogram model, in
          > which all of the parameters are varying with
          > direction. My model more general than range anisotropy
          > and sill anisotropy. Since I want to simulate from
          > this model ,with cholesky decomposition method, I need
          > the covariogram model, which is respected to my
          > variogram model. Therefore I need the relation between
          > variogram and covariogram. In fact I need to specify
          > C(0).

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        • jack webster
          Hi Dear doctor Myers Thank you for replying I have constructed a more general variogram model, in which all of the parameters are varying with direction. My
          Message 4 of 4 , May 14, 2002
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            Hi Dear doctor Myers
            Thank you for replying
            I have constructed a more general variogram model, in
            which all of the parameters are varying with
            direction. My model more general than range anisotropy
            and sill anisotropy. Since I want to simulate from
            this model ,with cholesky decomposition method, I need
            the covariogram model, which is respected to my
            variogram model. Therefore I need the relation between
            variogram and covariogram. In fact I need to specify
            C(0) in term of the parameters of my model ( My model
            has 6 parameters c01,c02( Nuggets) ,c1,c2 (partial
            sill) and a1,a2 (Ranges) ), because when it be known
            C(h) = C(0) - g(h)

            Thank you again: Webster

            --- "Donald E. Myers" <myers@...> wrote:
            > Two observations
            >
            > 1. The relationship between the variogram and the
            > covariance function is
            > only valid in the case of second order stationarity
            > (presumably you are
            > making the assumption of second order stationarity)
            >
            > 2. You said you had an "Anisotropic variogram", do
            > you mean you have a
            > model for a variogram with anisotropy or do you mean
            > that the
            > directional sample variograms indicate a change in
            > the sill with
            > direction? The apparent sill indicated by a sample
            > variogram
            > (directional or not) will combine the nugget effect
            > component and also
            > the sill due to the non-nugget part of the variogram
            > model, in addition
            > the nugget effect component will be separately
            > evident on the graph of
            > the sample variogram. If you check the various
            > geostatistical software
            > packages you will see that they only allow for a
            > geometric anisotropy in
            > the variogram model, this means that the RANGE of
            > the variogram changes
            > with respect to direction, the software does not
            > allow the sill to
            > change with respect to direction. To allow the sill
            > to change you would
            > need a "zonal" aniostropy, i.e., a non-geometric
            > anisotropy and the
            > problem is constructing a valid variogram model with
            > this kind of
            > anisotropy.
            >
            > A. Journel and I showed that one method that had
            > been used leads to
            > semi-definite models, i.e., non-invertible
            > coefficient matrices for the
            > * kriging system 1990, D .E. Myers and A. Journel,
            > Variograms with Zonal
            > Anisotropies and Non-Invertible Kriging Systems.
            > Math. Geology 22, 779-785
            >
            >
            >
            > Donald E. Myers
            > http://www.u.arizona.edu/~donaldm
            >
            > jack webster wrote:
            >
            > >Hi all list members
            > >Suppose that g denote the semi-variogram and C
            > >denote the Covariogram functions. According to
            > >Cressie(1993, P. 67) the relation is
            > > g(h) = C(0) - C(h)
            > >where h is the distance vector.In the isotropic
            > case
            > > C(0) = total sill = c0 +
            > c
            > >I have an Anisotropic Variogram in which c is
            > varying
            > >with direction.Therefore my question is what is
            > C(0)
            > >in this case?
            > >
            > >=====
            > >
            > >
            > >__________________________________________________
            > >Do You Yahoo!?
            > >LAUNCH - Your Yahoo! Music Experience
            > >http://launch.yahoo.com
            > >
            > >--
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            > ai-geostats@...
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            > remember to post a summary of any useful responses
            > to your questions.
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            > ai-geostats" followed by "end" on the next line in
            > the message body. DO NOT SEND Subscribe/Unsubscribe
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            > http://www.ai-geostats.org
            > >
            > >
            >
            >


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