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

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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)
Message 1 of 4 , May 13, 2002
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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• 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
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?
> >
> > =====
> >
> >
> > __________________________________________________
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> > LAUNCH - Your Yahoo! Music Experience
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> >
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>

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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
Message 3 of 4 , May 14, 2002
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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• 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
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
> >
> >--
> >* To post a message to the list, send it to
> ai-geostats@...
> >* As a general service to the users, please
> remember to post a summary of any useful responses
> to your questions.
> >* To unsubscribe, send an email to
> majordomo@... with no subject and "unsubscribe
> ai-geostats" followed by "end" on the next line in
> the message body. DO NOT SEND Subscribe/Unsubscribe
> requests to the list
> >* Support to the list is provided at
> http://www.ai-geostats.org
> >
> >
>
>

__________________________________________________
Do You Yahoo!?
LAUNCH - Your Yahoo! Music Experience
http://launch.yahoo.com

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