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Re: GEOSTATS: the lognormal in geology

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  • Rane Curl
    ... I would venture to say it is because the distribution of geologic reserves is generally unimodal and has large variance, but negative values are
    Message 1 of 10 , Mar 11, 1997
      On Tue, 11 Mar 1997, Peter Ehlers wrote:

      > The following question was recently put to me. Unfortunately, I'm
      > a statistician, not a geologist. Can anyone provide a brief answer?
      >
      > 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").

      I would venture to say it is because the distribution of geologic reserves
      is generally unimodal and has large variance, but negative values are
      impermissible. In addition, log-normal probability paper has been available
      for a long time, and practically anything looks moderately straight on it.

      --Rane L. Curl

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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 2 of 10 , Mar 11, 1997
        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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        >

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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 3 of 10 , Mar 12, 1997
          >
          >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 4 of 10 , Mar 13, 1997
            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 5 of 10 , Mar 14, 1997
              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 6 of 10 , Mar 16, 1997
                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 7 of 10 , Mar 16, 1997
                  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 8 of 10 , Mar 17, 1997
                    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 9 of 10 , Mar 17, 1997
                      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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