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Re: GEOSTATS: the lognormal in geology

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  • Samantha (Sama) Low Choy
    Dear Peter, Hi, I m a statistician, but I ve read some soil science literature recently. I m not sure what the units of measurement for geologic reserves are,
    Message 1 of 10 , Mar 11, 1997
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      Dear Peter,

      Hi, I'm a statistician, but I've read some soil science literature
      recently. I'm not sure what the units of measurement for geologic
      reserves are, so I will answer with regard to concentrations (about
      which I have read recently). From what I can gather, most data which is
      "concentrations" of some chemical/metal/etc in soil/water/etc naturally
      follows a log-normal distribution. This makes sense as concentrations
      range from very low to very high, and are usually graphed on a log scale,
      due to wide variation in scales. "Experience" or "industry practice" in
      these fields has shown that usually this type of data is normal on the
      logged scale. If you have little data for your variable, it might make
      sense to "assume" that it has similar behaviour to other variables in the
      field. Then again, it might be safer to use non-parametric tests instead,
      or collect more data!

      Hope this is not too vague,
      Samantha.

      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
      Sama Low Choy s.lowchoy@...
      Final-year PhD student in statistics ph: +61 07 864 1114
      School of Mathematics fax: +61 07 864 2310
      Queensland University of Technology, Brisbane, Australia
      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

      On Tue, 11 Mar 1997, Peter Ehlers wrote:

      > Date: Tue, 11 Mar 1997 05:31:34 -0700 (MST)
      > From: Peter Ehlers <ehlers@...>
      > To: ai-geostats@...
      > Subject: GEOSTATS: the lognormal in geology
      >
      >
      > Kind geofolk,
      >
      > The following question was recently put to me. Unfortunately, I'm
      > a statistician, not a geologist. Can anyone provide a brief answer?
      >
      > Question:
      >
      > What is the justification for using the lognormal for modeling
      > geologic reserves when only a few data are available? Many
      > practitioners tell me they do this, but none of them can explain
      > why (other than "it's industry practice").
      >
      >
      > Peter Ehlers
      > Math & Stats, U of C, Calgary AB, Canada T2N 1N4
      > ehlers@...
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    • Eric PIRARD
      ... Hello, I ll try to give you my geological point of view.... Lognormal and normal probability distributions are indeed the most widely used in geology and
      Message 2 of 10 , Mar 12, 1997
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        >
        >Kind geofolk,
        >
        >The following question was recently put to me. Unfortunately, I'm
        >a statistician, not a geologist. Can anyone provide a brief answer?
        >
        >Question:
        >
        >What is the justification for using the lognormal for modeling
        >geologic reserves when only a few data are available? Many
        >practitioners tell me they do this, but none of them can explain
        >why (other than "it's industry practice").

        Hello,
        I'll try to give you my geological point of view....

        Lognormal and normal probability distributions are indeed the most widely
        used in geology and particularly in ore reserve estimation.

        As a very general rule, symetric distributions are generated by major
        elements with high mobility.

        Take iron for example :
        The iron concentration in the earth crust is about 5-6%. You can take any
        piece of rock and you'll always find concentrations gently oscillating
        around this value because iron is mobile and tends to diffuse in the
        environment (homogeneity).

        On the other hand, asymetric distributions are generated by trace elements
        with very limited mobility. The best examples in this case are gold or diamond.
        A gold deposit contains 5 to 10 grams per ton, but this gold is very
        locallly concentrated into pure gold nuggets and will never diffuse
        (heterogeneity)

        Hence, when you sample a gold deposit, the majority of samples will contain
        only a very limited amount of nuggets but exceptionally you may encounter a
        sample with tens of nuggets (and hence a very large grade).

        In conclusion, there is absolutely no justification for using lognormal
        distributions in case of sparse sampling unless one has a geological argument.
        Spitefully it is not possible to predict the distribution law for any
        element in any geological context.

        If the problem you are concerned with is the estimation of a global mean for
        the deposit, you should be careful that procedures designed for lognormal
        distributions like (Sichel's t estimator) perform badly when the real
        distribution deviates form lognormality.... but this is the statistical side
        of the problem.

        Greetings to all geostatisticians on the ai-geostat planet,

        Eric /.
        Prof. Eric PIRARD /.
        Universite de Liege
        Caracterisation des Matieres Minerales Naturelles (MICA)
        Avenue des Tilleuls, 45
        4000 LIEGE
        BELGIUM

        Tel.: +32-4-366.95.28. FAX: +32-4-366.95.20. e-mail : Eric.Pirard@...

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      • Eric LEWIN - IPGP Geochimie
        ... I would not say that ! As a matter of fact, C.J.Allegre and I have committed a paper (Allegre, C.J. and Lewin, E., 1995. Scaling laws and geochemical
        Message 3 of 10 , Mar 13, 1997
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          At 12:03 +0100 12/03/97, Eric PIRARD wrote:

          >In conclusion, there is absolutely no justification for using lognormal
          >distributions in case of sparse sampling unless one has a geological argument.
          >Spitefully it is not possible to predict the distribution law for any
          >element in any geological context.

          I would not say that ! As a matter of fact, C.J.Allegre and I have
          committed a paper (Allegre, C.J. and Lewin, E., 1995. Scaling laws and
          geochemical distributions. Earth and Planetary Science Letters, 132: 1-13)
          which places the log-normal law as one of the "end-members" for "natural"
          geochemical distributions ("natural" as opposed to "due to sampling") for
          geological systems submitted to a series of fractionnation events --like
          the normal distribution does in the case of mixing scenarios--. The
          log-normal law appears quite naturally for instance for incompatible trace
          elements in the case where a geological system is recurrently
          fractionnated, a fractionnation process occuring indistinctively on all
          sub-parts of the system, have they been still enriched or depleted by
          previous fractionnation episodes.

          Note that the importance of this log-normal law has been widely studied by
          L.Ahrens in a tenth of papers, the first of which was published in Nature
          in 1953 (i have not the exact ref. at hand, sorry), the second being:
          Ahrens, L.H., 1954. The lognormal distribution of the elements (a fundamental
          law of geochemistry and its subsidiary). Geochimica et Cosmochimica Acta,
          5: 49-73.
          and the last (as far as i know):
          Ahrens, L.H., 1966. Element distributions in specific igneous rocks - VIII.
          Geochim. Cosmochim. Acta, 30: 109-122.

          Note also that the log-normal law appears in multifractal studies for the
          case of binomial multiplicative processes, which is exactly the
          mathematical basis of our model. And the spatial distribution of some
          mineral ressources has been described as multifractal (Mandelbrot, Scholtz,
          among others...). For instance, recently, Cheng & Agterberg (Math.Geol.
          (1996)28,1-16) have studied the famous DeWijs' example of a log-normal
          distribution (De Wijs, H.J., 1951. Statistics of ore distribution: 1)
          frequency distribution of assay values. Geologie en Mijnbouw, 13: 365-375;
          De Wijs, H.J., 1953. Statistics of ore distribution: 2) theory of binomial
          distribution applied to sampling and engineering problems. Geologie en
          Mijnbouw, 15: 12-24.) as a multifractal distribution...

          At the opposite end of such iterative fractionnation processes, if the
          process of enrichment or of remobilisation is systematically selective, for
          instance by redifferentiating preferentially only the most enriched parts
          of the geologic system, like in chromatographic ones, or if the sampling is
          selective, like mining, a self-affine (Pareto, or fractal) distribution law
          is more liable to be observed.

          One limit of application of our modelling is the validity of the
          fractionnation law which gives a central role to the partition coefficient:
          for instance, major elements may not follow this, simply because of the
          closure problem; compatible ones will not too (they tend to exhibit
          negatively-skwed, because of their buffering.

          In conclusion, i would say that I agree with Eric Pirard's arguments about
          mobility, since somehow element mobility is a result of geochemical
          incompatility. But, contrary to him, I think this is the reason for finding
          naturally this log-normal distribution...

          Now, about Peter Ehlers' concern, with a sparse sampling: GIGO ! Garbage
          in, garbage out...

          This is now netly open to discussion...

          --Eric Lewin



          =====[ Eric LEWIN - lewin@... - IPGP Geochimie ]=====
          = Labo de Geochimie et Cosmochimie / Institut de Physique du Globe =
          ------- Tel:(33|0)1.44.27.24.59 ---- Fax:(33|0)1.44.27.37.52 -------
          > IPGP Geochimie; case 89; 4,place Jussieu; F-75252 Paris cedex 05 <
          = <A HREF="http://www.ipgp.jussieu.fr/">L'I.P.G.Paris</A> =
          |\_ __ ___ TEC: <http://www.ipgp.jussieu.fr/~lewin/> :WIP ___ __ _/|
          ========================== Paris - FRANCE ==========================


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        • Frederic Verhelst
          Dear all, In the discussion about the type of distributions in geology I would like to mention the fact that until now only central-limit types of
          Message 4 of 10 , Mar 14, 1997
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            Dear all,

            In the discussion about the type of distributions in geology I would like to mention the fact that until now only "central-limit" types of distributions were considered. Depending on the goal of the study also "extreme value" distributions might be considered. For instance in the field of the evaluation of diamond deposits it is crucial due to the extreme non-linear relationship between size of one stone and its value.

            More information can be found in the PhD thesis of Jef Caers: "Statistical and geostatistical valuation of diamond deposits" and in his journal papers. More information can be obtained ate the following E-mail address:
            Jef.Caers@...
            and from the following site:
            http://www.kuleuven.ac.be/mining/

            Regards,

            Frederic

            --------
            Frederic Verhelst
            Delft University of Technology, Center for Technical Geoscience
            Faculty of Applied Physics, Laboratory of Seismics and Acoustics
            P.O. Box 5046, NL-2600 GA Delft, The Netherlands

            Tel: + 31 15 2785188
            Fax: + 31 15 2783251
            e-mail: F.Verhelst@...
            Home-page: URL: http://wwwak.tn.tudelft.nl/~fre/--
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          • Harini Nagendra
            What I seem to remember from my reading of distributions is this - if several factors influence a distribution, acting independent of each other and being
            Message 5 of 10 , Mar 16, 1997
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              What I seem to remember from my reading of distributions is this - if
              several factors influence a distribution, acting independent of each
              other and being roughly equal in influence, you get a normal
              distribution. So what then causes lognormals? Is is something like
              several fatcors acting multiplicative? The previous discussions have been
              specifically about geological distributions - I was wondering if there is
              a more general way of descibing this. There are many more things which
              are lognormally distributed in nature - species abundance distributions,
              for eg, are most always lognormal in the tropics. So what is common
              between the 2?

              ------------------------------------------------------------------------
              | Harini Nagendra E-MAIL : harini@... |
              | Center for Ecological PHONE : 91 80 309 2639 (Hostel: LR-94)|
              | Sciences : 91 80 309 2506 (Department) |
              | Indian Institute of Science FAX : 91 80 334 1683 |
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            • Rane Curl
              ... You are correct. The sum of random variables, no matter how they are individually distributed, normalized to finite mean and variance, approaches the
              Message 6 of 10 , Mar 16, 1997
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                On Sun, 16 Mar 1997, Harini Nagendra wrote:

                > What I seem to remember from my reading of distributions is this - if
                > several factors influence a distribution, acting independent of each
                > other and being roughly equal in influence, you get a normal
                > distribution. So what then causes lognormals? Is is something like
                > several fatcors acting multiplicative? The previous discussions have been
                > specifically about geological distributions - I was wondering if there is
                > a more general way of descibing this. There are many more things which
                > are lognormally distributed in nature - species abundance distributions,
                > for eg, are most always lognormal in the tropics. So what is common
                > between the 2?

                You are correct. The sum of random variables, no matter how they are
                individually distributed, normalized to finite mean and variance,
                approaches the Normal distribution. This is the Central Limit Theorem in
                operation. If instead, a random variable is constructed as the product of
                many random variables, the logarithm of the random variables so
                constructed will approach Normal, since it is the sum of the logarithms of
                all the contributing random variables.

                In natural systems, however, the Central Limit Theorem operates up to a
                point, but no natural distributions are exactly normal, or log-normal. For
                example, the normal distribution allows for infinite positive and negative
                values, but no natural process does. The log-normal admits of positive
                infinite values, but that is also impossible. The question is, are you
                able to reject a Null Hypothesis, based on a finite sample, that a
                population is Normal or log-normal? With small samples, the answer is
                usually no.

                The log-normal distribution is a unimodal distribution of positive values,
                which can be said of most natural distributions. In addition, it is
                distinguished from the Normal distribution when variance is large. Therefore
                most non-negative, unimodal, distributions with large variance - and small
                samples - will appear log-normal.

                There are alternatives. For example, the large infinite suite of Gamma
                distributions. They will fit almost any non-negative unimodal data. They
                also arise 'naturally' from physical processes of fragmentation (for
                example). They are, however, harder to work with, than the log-normal.

                --Rane L. Curl
                University of Michigan

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              • Eric PIRARD
                Hi, Many thanks to Eric Lewin for giving a much more argumented point of view on the appearance of normal and lognormal distributions in geology. The idea of
                Message 7 of 10 , Mar 17, 1997
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                  Hi,

                  Many thanks to Eric Lewin for giving a much more argumented point of view on
                  the appearance of normal and lognormal distributions in geology.
                  The idea of fractionnation and mixing scenarii is very elegant.

                  I would however insist on the following.
                  In ore reserve estimation, one is not so much interested in grade
                  distributions of small samples (drill cores) but merely in grade
                  distributions on blocks of thousands cubic metres (SMU= Selective Mining Unit).

                  A change of support often implies :

                  - that the mean grade is preserved.
                  - that the variance is reduced.

                  In case of such a support change, the lognormality of a drill core
                  distribution will probably tend to be "symetrized" at block scale.

                  Can anyone comment on this increased symmetry ?
                  Is there anyone having field data for illustrating this in ore deposits ?

                  Best regards,

                  Eric /.


                  Prof. Eric PIRARD /.
                  Universite de Liege
                  Caracterisation des Matieres Minerales Naturelles (MICA)
                  Avenue des Tilleuls, 45
                  4000 LIEGE
                  BELGIUM

                  Tel.: +32-4-366.95.28. FAX: +32-4-366.95.20. e-mail : Eric.Pirard@...

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                • Syed.R.Syed@EXXON.sprint.com
                  Prof. Eric P s questions refer: 1. The degree to which the PDF approaches symmetry is quite a strong function of the degree of spatial correlation in the
                  Message 8 of 10 , Mar 17, 1997
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                    Prof. Eric P's questions refer:

                    1. The degree to which the PDF approaches symmetry is quite a strong
                    function of the degree of spatial correlation in the underlying data.
                    The more continuous the phenomena, the slower the PDF approaches
                    symmetry. Srivastava's book gives a nice graphical example of this.

                    2. Someone mentioned the use of extreme value statistics. Still not very
                    popular yet, but indicator approaches are an elegant means to quantify
                    the spatial correlation of extreme values, inter alia:

                    a. more robust variogram analysis, i.e., modeling of "outliers"
                    b. distribution-free
                    c. PDF's dependent on neighborhood data
                    d. connectivity of extreme values that are usually averaged out in normal
                    variogram analysis

                    3. Normal (ordinary) kriging of log-transformed variables results in biased
                    predictions. The expectation in log-space becomes the median in
                    normal space. Furthermore, predictions are highly sensitive to even slight
                    changes in the sill values.

                    4. Assuming log-normality is just a decision/choice. It does not necessarily
                    make it the correct one. Checks using simulation methods can help to
                    gauge the reasonableness of the assumption.

                    Regards, Syed
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