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Re: AI-GEOSTATS: re CLT and mixtures

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  • Digby Millikan
    Brian, I am not familiar with the derivation of kriging equations, but of note I know the lognormal population does not follow the central limit theorem, and
    Message 1 of 7 , Dec 30, 2002
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      Brian,

      I am not familiar with the derivation of kriging equations, but of note
      I know the lognormal population does not follow the central limit
      theorem, and it is necessary to transform lognormal populations to normal
      populations before kriging using lognormal kriging. I know the same also
      applies to mixed populations and that mixed populations must be transformed
      to normal populations using hermite polynomials with disjunctive kriging,
      maybe this is for the same reason.

      Best wishes for the New Year everyone.

      Regards Digby Millikan.
      Geolite Mining Systems
      digbym@...


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    • Digby Millikan
      Francis, Have you looked at indirect lognormal corrections and affine corrections for sample sizes. Many kriging softwares use diffrent sample sizes for their
      Message 2 of 7 , Jan 1, 2003
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        Francis,
        Have you looked at indirect lognormal corrections and affine corrections
        for sample
        sizes. Many kriging softwares use diffrent sample sizes for their algorithm
        e.g. a single
        line of zero diameter representing all core sizes, with each block
        discretized to match
        the sample sizes, probably sufficient. Where poor lognormal semivariograms
        are available
        in the past I have used a sichel estimate of the mean of the population then
        cut the grades
        until the arithmetic mean equals the sichel mean and used inverse distance
        modelling
        (White Devil, N.T. Australia). Populations where actually mixed (lognormal
        with a small
        high grade portion) it may have been more useful to provide a mixed
        distribution mean rather
        than a sichel mean which I think a paper has been published on. Unforunately
        it can be
        rare at some undergrounds and such operating environments where several
        mines target
        one mill. However this underground grade modelling was only for financial
        planning,
        as stope design was done entirely from geological interpretation. However at
        open pit
        operations with a single mine and mill operating and more sophisticated
        grade control
        samples are available it is possible to reconcile a resource model to a
        grade control model
        and mill production to within a 0.5% by adjusting variograms and selective
        mining unit
        cut off grades, though I personally have no experience with nugget effect
        adjustments.

        Regards Digby Millikan
        Geolite Mining Systems
        digbym@...




        ----- Original Message -----
        From: "Francis T. Manns" <artesian1@...>
        To: "Isobel Clark" <drisobelclark@...>; "Digby Millikan"
        <digbym@...>
        Cc: <ai-geostats@...>
        Sent: Wednesday, January 01, 2003 8:07 AM
        Subject: RE: AI-GEOSTATS: re CLT and mixtures


        > Dear all,
        >
        > Please excuse my jumping into the dialogue without a proper introduction.
        I
        > just enrolled in the forum in order to search for a collaborator to assist
        > me in understanding what I've done with grade distributions for gold and
        > copper. My recent manuscript on Sadiola was rejected by Canadian
        Institute
        > of Mining Metallurgy and Petroleum [CIM] on the basis that my empirical
        > nugget effect solution needed a mathematical basis. Therefore I would
        like
        > to interest someone, perhaps a grad student somewhere, in pursuing the
        math
        > that eludes me. Would that mathematical proofs be required to have
        > empirical support in earth sciences? How many new mines struggle because
        > the expected grades fall short of predictions, estimates, and
        'calculations'
        > from the geostatistical side?
        >
        > My hypothesis is that in far too many instances, the sample size (weight)
        is
        > too small to represent the end product - the estimate (assay) of the metal
        > content of a tonne of rock or a block of ore. Moreover, the industry
        > standard aliquot of 30 grams is practical for a lab is not practical for
        at
        > least half of the gold deposits in the universe. We're asking 30 grams to
        > represent 1 million grams and then require four or so assays to represent
        > 300 tonnes (e.g., 120 grams to represent to estimate 300 million grams of
        > rock).
        >
        > I use a uniform probability plot to describe the assay distribution. A
        > straight segment on a uniform plot indicates a random number sequence. In
        > most cases, the distributions shows two or more random runs of numbers at
        > the high end of the distribution (these separate runs represent geological
        > loci of crystallization). Though the sample size is small, we at least
        get
        > a collection of random numbers or samples from the deposit. Treating
        these
        > empirically it is possible to correct the arithmetic average of the sample
        > distribution for the sample size limitation.
        >
        > As for conventional geostatistics, it's GIGO. A systematic error or
        errors
        > appear if the sample size is too small for the distribution of metal in
        the
        > rocks. I believe that if we were able to twin all the holes of the garden
        > variety exploration level deposit, we would find that very few holes would
        > actually be twinned, but the mean, standard deviation and standard error
        > would be indistinguishable.
        >
        > I have found that the slope of the straight segment of the uniform
        > probability plot is proportional to the error of the arithmetic average.
        > When one deducts the intercept of the slope at the fiftieth percentile
        from
        > the arithmetic average, one gets a very good estimator of the plant grade
        of
        > the deposit. I would be very happy to correspond with someone who has
        > insight into this phenomenon. It works for a copper distribution that I
        > once had in my possession. The raw data are gone now. I believe the 50th
        > percentile is important on a semi-log plot as it represents the square
        root
        > of the slope "factor", for want of any clearer understanding. I'll add my
        > website, and hope someone could have a glance at what's happening here.
        > It's geologically based, as you can see from my Borneo photograph in the
        > site.
        >
        > http://www.geocities.com/fmanns_artesian/index.html
        >
        > Thank you,
        >
        > Francis Manns, PhD
        > Artesian Geological Research
        > Toronto Ontario
        >
        >
        > -----Original Message-----
        > From: ai-geostats-list@... [mailto:ai-geostats-list@...]On
        > Behalf Of Isobel Clark
        > Sent: December 31, 2002 5:30 AM
        > To: Digby Millikan
        > Cc: ai-geostats@...
        > Subject: Re: AI-GEOSTATS: re CLT and mixtures
        >
        >
        > Digby
        >
        > The original derivation of the kriging equations does
        > not demand any specific distribution. However, it only
        > makes sense if the 'increments' - difference in value
        > - have a fairly stable variance.
        >
        > Kriging works much better IN PRACTICE if those
        > differences are Normal, hence the various
        > transformations offerred such as lognormal kriging,
        > hermitian polynomials, Normal scores, rank transforms
        > and so on.
        >
        > Normalising a mixture of distributions will result in
        > specious answers unless the distributions are mixed in
        > the same proportions all over the study area.
        >
        > Usually a mixed distribution is due to a mixture of
        > real populations. For example, in geology oxide and
        > sulphide samples may show different behaviour. There
        > may be separate phases of deposition.
        >
        > In short, a mixture of distributions generally
        > indicates a violation of the assumption of homogeneity
        > (or, if your prefer, stationarity) which is needed for
        > the proper application of any geostatistics. If
        > possible the populations should be separated and the
        > analysis carried out. If not, you may be able to cope
        > with the data using indicator kriging such as
        > suggested in my 1993 paper. There is also an example
        > in my Cardiff paper of 2000, I think.
        >
        > Isobel
        > http://geocities.com/drisobelclark/resume/Publications.html
        >
        > __________________________________________________
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        >
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      • Digby Millikan
        Francis, On topic in Lognormal-de Wijsian Geostatistics for Ore Evaluation D.G.Krige pp5. points out that the slope of the log-probability plot provides an
        Message 3 of 7 , Jan 2, 2003
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          Francis,
          On topic in "Lognormal-de Wijsian Geostatistics for Ore Evaluation"
          D.G.Krige pp5. points out that the slope of the log-probability plot
          provides an estimate of the variance which Rendu 78 and Storrar 77
          also published information on. The graph paper in the book actually
          has a variance slope scale on it from which you can estimate the
          variance. There is also a graph which gives the realtionship between
          mean value, pay limit, pay value and percentage pay for lognormal
          distributions. I will take a look and see how this relates to the 50th
          percentile. It is interesting that your mill grade was related to the
          50th percentile as this would depend on the pay limit or cutoff grade
          which we call it in Australia which is entirely based on economic
          factors and hence should show no relationship unless at operations
          of similar commodity type and operating costs and revenues.
          Overestimating gold grade used to be due to a reluctance to cut grades
          so operators would hear the figures they wanted to, in regards to
          nugget effect, I am a mining engineer, on what basis do geologists
          select core size? is it e.g. hole depth, mineralization type etc.

          Regards Digby Millikan
          Geolite Mining Systems
          digbym@...
          http://www.users.on.net/digbym


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        • Digby Millikan
          Francis, Having looked at your website further comments regarding the Australian gold mining industry that only 2 out of 35 operations acheived feasibility
          Message 4 of 7 , Jan 2, 2003
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            Francis,
            Having looked at your website further comments regarding the Australian
            gold mining industry
            that only 2 out of 35 operations acheived feasibility head grades, that in
            1990 it was not common
            knowledge among the mining industry, though common knowledge to
            geostatisticians, that gold
            deposits are typically overestimated due to lognormality. This demonstrates
            geostatistics has its
            place in educational institutions and the Australian mining industry.
            Hopefully with "due diligence"
            (1995 Newmont Mining Australia appointed a technical due diligence officer,
            and
            "Basic Linear Geostatistics" 1998 Margaret Armstrong discusses due
            diligence) on the increase
            this is being addressed.
            Is your Sadiola paper available online?

            Regards Digby Millikan
            Geolite Mining Systems
            digbym@...
            http://www.users.on.net/digbym


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