- O boy, I wish my world included the kind of data which would allow modelling of anisotropy on a 10m scale! I am full of envy.Isobelhttp://www.stokos.demon.co.uk
wrote:*Edward Isaaks <ed@...>*Hello List

FYI, a few comments related to the ongoing discussion re Geostats Scam.

Stephen Henley makes some valid points on the shortcomings of geostatistics.

In particular, I have also been troubled by the application of models

"invariant under spatial translation" to real world data.

"The proper selection of data" for estimation is considered by many to be

the fundamental mantra of ore resource estimation. Typically, the selection

of data for variography and the estimation of block model grades is

controlled through manually interpreted models of lithology, alteration,

grade shells, structural domains and so on. Although these models are

practical at the scale of the deposit, they are not practical at local

scales where local scale is defined by distances as short as 10 m, over

which abrupt changes in the direction of geologic trends such as grade,

fault direction, fracture patterns, and rock type contacts are observed.

Obviously, it is not practical to control the selection of data at this

scale using manually interpreted models.

However, the good news is that the proper selection of data for kriging can

be achieved at a local scale by aligning an anisotropic search ellipsoid

with the local geologic trend(s) on a block by block basis.

The idea is simple. Before each block is estimated, the anisotropy ratios of

the local search neighborhood are adjusted and the axes aligned with local

trends in the data. The method has come to be known as local anisotropy

kriging or LAK. The results are remarkable. I have an example where LAK

applied to grade control out performs ordinary kriging by reducing dilution

and ore loss. You can read more about this relatively new implementation of

an old idea by visiting www.isaaks.com and clicking on "Geo Docs".

I would also add a note regarding popular geostatistical misconceptions, and

there are several. For example, Krige, Deutsch, Vann and many others have

published papers admonishing conditional bias. However, in mining

applications (where geostatistics has its roots) conditional bias is

actually irrelevant, unless the estimates are used for grade control. See

"The Kriging Oxymoron" at www.isaaks.com "Geo Docs" for a peer reviewed

paper on the subject.

I recently read a paper by J Vann, S Jackson, and O Bertoli (2003) that

actually proposes a method for designing the kriging search neighborhood

based on minimizing conditional bias. Horrifying - do they not realize that

such practice actually increases the estimation error of the predicted

tonnes and grade above cutoff? This paper is probably the worst (best?)

example I have seen of a faulty misconception in 25 years of ore resource

assessment. One can almost understand why geostatistics might be labeled a

scam.

However, I'm not sure I agree with Stephen where he appears to suggest that

the "vast array of methods and an array of intensely mathematical published

papers" are somewhat responsible for providing the means to deliver

"whatever the client wants". The difference between a professional and an

amateur practitioner is knowing which is the correct tool for the job and

how to use it properly. I'd argue that a packed toolbox is not the problem

but rather, the inexperienced or dishonest practitioner is the problem.

And finally, I have done some research on the subject of computing "weighted

variances" since a number of "weighted variance" estimators can be found in

the literature including Mr. Merks' version. Each of the estimators I found

provided a different estimate of the weighted variance and to make matters

worse, not one was shown to be a valid statistical estimator -- they were

simply stated without derivation(see footnote). However, the good news is

that with careful work and with help from Colin Daly and Don Myers, I now

have the mathematical derivation of an unbiased estimator for the population

variance given a sample of N (iid) observations with associated weights.

Now, it turns out that in spite of all the huffing and puffing by our

colleague Mr. Merks, the "weighted variance" estimator that he so loudly

champions is biased under the iid model! Perhaps Mr. Merks' time could have

been better spent looking for an unbiased variance estimator rather than

stalking the "lost variance". :-)

A copy of this work will be made available to the list following

publication.

----------------------------------------------------------------------------

--------------------

Derivation -- A logical or mathematical process indicating through a

sequence of statements that a result such as a theorem or a formula

necessarily follows from the initial assumptions.

Edward Isaaks

Reference

J Vann, S Jackson and O Bertoli, (2003), "Quantitative Kriging Neighbourhood

Analysis for the Mining

Geologist - A Description of the Method With Worked Case Examples", 5th

International Mining Geology Conference, AusIMM.

* By using the ai-geostats mailing list you agree to follow its rules

( see http://www.ai-geostats.org/help_ai-geostats.htm )

* To unsubscribe to ai-geostats, send the following in the subject or in the body (plain text format) of an email message to sympa@...

Signoff ai-geostats