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.
Isobel
http://www.stokos.demon.co.uk
Edward Isaaks <[EMAIL PROTECTED]> wrote:
Edward Isaaks <[EMAIL PROTECTED]> wrote:
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.
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