Hello,

I'd like to add a few comments to Pierre's remarks on automatic variogram

modeling. Variogram modeling is traditionally done by manually modeling

individual directional sample variograms and then somehow interpolating

between all the directional models to obtain the full model in either 2 or 3

dimensional space. Remember, that the variogram model must be able to return

Gamma(h) for any separation vector. For example, in 3D space this means the

model must be able to return Gamma(h) for any lag distance h given any

azimuth alpha, and any dip angle phi.

Most geostatisticians find the job of combining more than 4 or 5 directional

models into one full 3D positive definite model a bit more than they can

handle. I actually witnessed an enthusiastic geostatistician calculate and

model something like 50 directional sample variograms. He then laid out 50

hard copies of the directional models on several large drafting and map

tables and spent several days trying to put them all together to form one

consistent model in 3D space. Finally, in disgust he selected 5 directional

models that more or less corresponded with his prior intuition -- modeled

them and discarded the remainder. The orientations and ratios of the

anisotropies (there were several - one for each structure) represented by

the 50 directional models were more than he could make sense of. The result

was an over simplified model.

Such problems are common to the mining industry. Many mining data sets have

plenty of samples, sometimes as many as 50,000 or more. It turns out that

mother nature has generally also done a pretty good job of messing up the

spatial continuity of the regionalized variable(s). So it is not uncommon to

find that as many as three nested structures may be required to capture the

messed up spatial continuity with each structure characterized by a unique

anisotropy ratio and orientation.

Experience has shown that it is definitely an advantage to be able to model

all 50 directional sample variograms simultaneously using an automatic

fitter. Computers are very good at searching out and fitting data in 4

dimensional space (length, width, depth, and Gamma). The automatically fit

model can be a very accurate representation of the underlying spatial

continuity, particularly when the anisotropy ratios are relatively severe.

But of course, the model must be critically reviewed and judged acceptable

in the final analysis.

Many mining companies use automatic variogram modeling routinely; Placer,

Amec, BHP, Barrick, Newmont, CVRD, and Hecla Mining to name a few. So I

think it can be said from experience that in general, the advantages of

automatic variogram modeling outweigh the disadvantages. The proof is in the

pudding.

More information on automatic variogram modeling can be found at

www.isaaks.com

----- Original Message -----

From: "Pierre Goovaerts" <goovaert@...>

To: <m.pawley@...>

Cc: <ai-geostats@...>

Sent: Sunday, April 04, 2004 6:51 PM

Subject: Re: AI-GEOSTATS: Automated Variogram modelling

> Hello,

>

> The issue of automatic versus manual modeling of semivariogram

> has been the subject of much debate in the past.

> In my graduate class, I used to ask the students to model their

> experimental semivariograms first manually (i.e. bye eye), then using

> non-linear regression. The resulting models were then used in kriging

> and cross-validation allowed them to assess the prediction

> performances of both types of models. Most were surprised to find out

> that manually fitted semivariograms could lead to more accurate

> predictions than automatically fitted ones. The take-home lesson

> was that the modeling of the semivariogram is usually a preliminary step

> towards prediction or simulation, and influence partially their results.

>

> Automatic semivariogram modeling is useful to model complex anisotropies

> as long as the experimental semivariograms are reasonably well defined and

> also when multiple semivariograms need to be modeled (i.e. indicator

> kriging). In addition, working now for a software R&D company and

> developing new applications of geostatistics to health science, I have

> to keep in mind that most users migth not have the necessary background to

> compute and model semivariograms. The challenge is then to find a

> procedure to achieve meaningful fits without asking much from the user...

>

> The issue of automatic versus manual modeling is particularly important

> when data are sparse, making the semivariogram erratic... Then the

> modeling procedure is more than a mere exercice of fitting a curve to

> experimental values. It aims at creating a model for the spatial

> variability of the phenomenon under study and it relies greatly

> on ancillary information (e.g. magnitude of nugget effect, directions of

> anisotropy) typically provided by expert knowledge.

>

>

> Pierre

>

>

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

<><>

>

> Dr. Pierre Goovaerts

> President of PGeostat, LLC

> Chief Scientist with Biomedware Inc.

> 710 Ridgemont Lane

> Ann Arbor, Michigan, 48103-1535, U.S.A.

>

> E-mail: goovaert@...

> Phone: (734) 668-9900

> Fax: (734) 668-7788

> http://alumni.engin.umich.edu/~goovaert/

>

>

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

<><>

>

> On Mon, 5 Apr 2004 m.pawley@... wrote:

>

> > Hi all,

> > I have a couple of questions for the list.

> >

> > I understand that most theoretical variograms are fit by eye, and I was

interested in

> > gauging the usefulness of automated (purely data-driven) estimation for

theoretical variograms.

> >

> > i.e. Would it be useful to practitioners to be able to fit to be able to

fit something like a

> > 'constrained spline' as the theoretical variogram function to give your

kriging results?

> > (the spline could be constrained to be positive-semi-definite)

> >

> >

> > 1. Is this something that has been examined in detail in the past?

> > 2. If not - would it be something that geostatisticians would find

useful?

> >

> >

> > Any thoughts and references on this matter would be most welcome.

> >

> > Many thanks in advance,

> > Matthew Pawley

> >

> >

>

> --

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