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[ai-geostats] Re: Kriging Small Blocks

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  • Isobel Clark
    Jul The warning about kriging small blocks is about small relative to the sampling density. For example, less than about one-third of the grid spacing. The
    Message 1 of 5 , Jul 19, 2004
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      Jul

      The warning about kriging small blocks is about
      "small" relative to the sampling density. For example,
      less than about one-third of the grid spacing.

      The warning is the same as the one about 'point'
      kriging (mapping) The map is too smooth - or, at
      least, a lot smoother than the real surface would be.
      High value areas will be under-estimated and low value
      areas will be over-estimated.

      If your objective in kriging is to obtain general maps
      of an area with an idea of where the highs and lows
      are, then ordinary kriging is sufficient. The over-
      and under- estimations cancel out on average.

      In mining applications, where block kriging
      originated, most applications require a 'cutoff',
      where values below a certain value are not included in
      the 'plan'. In this case, mapping or estimating small
      blocks will result in an over-estimation of 'payable'
      ground and an under-estimation in average value.

      In pollution or environmental applications, the areas
      at risk will be under-estimated as will the true
      toxicity or risk factors.

      There are two major ways round this problem:

      (1) use a non-linear kriging approach such as
      disjunctive kriging or the multivariate gaussian. Ed
      Isaacs and Mohan Srivastava's book is th ebest
      reference for the latter. Rivoirard's book for DK.

      (2) simulation. There are lots of simulation methods
      around, which allow you to 'put back the roughness'
      and get an idea how bad the problem might be. GSLib is
      pretty good on this.

      Isobel
      http://geoecosse.bizland.com/course_brochure.htm

      If, as in mining, you wish to apply some sort





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    • Isobel Clark
      Nicolau I was talking about kriging before cutoff is applied. If the cutoff is applied to the block estimates my comments stand. If you aply the cutoff to your
      Message 2 of 5 , Jul 19, 2004
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        Nicolau

        I was talking about kriging before cutoff is applied.
        If the cutoff is applied to the block estimates my
        comments stand. If you aply the cutoff to your data
        first and then krige, you get the opposite problem,
        because you will over-estimate every value and
        under-estimate the tonnage.

        My point (1) is that, if you wish to avoid conditional
        bias in your kriging, you could consider using a
        non-linear kriging method such as those mentioned. I
        have no experience with either, since I follow a
        different route in the correction of conditional bias
        in mineral resource estimation.

        Isobel
        http://geoecosse.bizland.com/whatsnew.htm



        --- nicolau.barros@... wrote: > Isobel,
        >
        > So for mining purposes can't we just krige before
        > applying the cut-off
        > criteria? I mean, for most mining applications one
        > will prefer to have a
        > more realistic geologic block model and will always
        > have the chance to
        > evaluate his/her panels under the appropriate
        > cut-off criteria, but applying
        > that criteria after estimating small blocks, right?
        >
        > Could you please explain your point in solution (1)
        > below? Thanks for
        > indicating the literature.
        >
        > Thanks
        >
        > Nicolau Barros
        > Engineer
        > Mine Planning and Production Control Department
        > Mineração Rio do Norte S.A.
        > nicolau.barros@...
        > +55 (93) 549 8215
        >
        > Confidencialidade
        > Esse e-mail e possíveis anexos podem possuir
        > informações confidenciais e de
        > interesse somente do destinatário. Portanto, se você
        > recebeu esta mensagem
        > por engano, favor comunicar imediatamente o
        > remetente e deletá-la logo em
        > seguida. Esteja ciente que o uso indevido do
        > conteúdo das informações em
        > questão é estritamente proibido.
        > Confidentiality
        > This message and any possible attached files may
        > contain confidential
        > information and only for interest of the intended
        > recipient. If you have
        > received this message by mistake, please notify the
        > sender and delete the
        > message immediately. Be aware that the unauthorized
        > use of the
        > above-mentioned information is strictly forbidden.
        >
        > -----Mensagem original-----
        > De: Isobel Clark [mailto:drisobelclark@...]
        > Enviada em: segunda-feira, 19 de julho de 2004 05:23
        > Para: Julhendra_Solin@...
        > Cc: ai-geostats@...
        > Assunto: [ai-geostats] Re: Kriging Small Blocks
        >
        > Jul
        >
        > The warning about kriging small blocks is about
        > "small" relative to the sampling density. For
        > example,
        > less than about one-third of the grid spacing.
        >
        > The warning is the same as the one about 'point'
        > kriging (mapping) The map is too smooth - or, at
        > least, a lot smoother than the real surface would
        > be.
        > High value areas will be under-estimated and low
        > value
        > areas will be over-estimated.
        >
        > If your objective in kriging is to obtain general
        > maps
        > of an area with an idea of where the highs and lows
        > are, then ordinary kriging is sufficient. The over-
        > and under- estimations cancel out on average.
        >
        > In mining applications, where block kriging
        > originated, most applications require a 'cutoff',
        > where values below a certain value are not included
        > in
        > the 'plan'. In this case, mapping or estimating
        > small
        > blocks will result in an over-estimation of
        > 'payable'
        > ground and an under-estimation in average value.
        >
        > In pollution or environmental applications, the
        > areas
        > at risk will be under-estimated as will the true
        > toxicity or risk factors.
        >
        > There are two major ways round this problem:
        >
        > (1) use a non-linear kriging approach such as
        > disjunctive kriging or the multivariate gaussian. Ed
        > Isaacs and Mohan Srivastava's book is th ebest
        > reference for the latter. Rivoirard's book for DK.
        >
        > (2) simulation. There are lots of simulation methods
        > around, which allow you to 'put back the roughness'
        > and get an idea how bad the problem might be. GSLib
        > is
        > pretty good on this.
        >
        > Isobel
        > http://geoecosse.bizland.com/course_brochure.htm
        >
        > If, as in mining, you wish to apply some sort
        >
        >
        >
        >
        >
        >
        ___________________________________________________________ALL-NEW
        > Yahoo!
        > Messenger - sooooo many all-new ways to express
        > yourself
        > http://uk.messenger.yahoo.com
        >





        ___________________________________________________________ALL-NEW Yahoo! Messenger - sooooo many all-new ways to express yourself http://uk.messenger.yahoo.com
      • Edward Isaaks
        Hi I think the discussion below is missing some key points: 1. If the individual block estimates are to be used for actual selection at the time of mining,
        Message 3 of 5 , Jul 19, 2004
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          Hi

          I think the discussion below is missing some key points:
          1. If the individual block estimates are to be used for actual selection at
          the time of mining, then conditional bias will impact the predicted
          recoveries and should be minimized.
          2. However, if the block model is to be used for long term mine planning,
          the preparation of production schedules etc. etc., then it is unlikely that
          these same block estimates will be used for selection at the time of mining.
          In this scenario, it is sufficient to know the distribution of block grades
          within a mining period such as annual, semi-annual, or quarterly, etc. and
          conditional bias is irrelevant.
          3. Now here is the rub. One cannot accurately estimate the distribution of
          block grades within a mining period without invoking conditional bias unless
          each block estimate is perfect, e.g., no error!

          If you read Michel Davids, "Geostatistical Ore Reserve Estimation" you will
          find that he also points out this apparent contradiction (page 313 section
          11.3.2) The apparent contradiction is:
          1. If the block grades are conditionally unbiased, then the distribution
          (histogram) of block estimates is necessarily smoothed. Thus, the prediction
          of in situ tones and grade above cutoff is inaccurate (biased)!
          2. If the histogram of estimated block grades yields the correct in situ
          proportions and grades above cutoff (for all cutoff grades), then the block
          estimates are necessarily conditionally biased.

          I often refer to this as the "kriging Oxymoron", and it appears to be very
          poorly understood with in the geostat community. Even Dr. Krige wrongly
          claims that conditional bias should be removed or minimized in a long term
          mine planning model, when in fact it is irrelevant.


          -----Original Message-----
          From: Isobel Clark [mailto:drisobelclark@...]
          Sent: Monday, July 19, 2004 8:50 AM
          To: nicolau.barros@...
          Cc: ai-geostats@...
          Subject: [ai-geostats] Re: Kriging Small Blocks

          Nicolau

          I was talking about kriging before cutoff is applied.
          If the cutoff is applied to the block estimates my
          comments stand. If you aply the cutoff to your data
          first and then krige, you get the opposite problem,
          because you will over-estimate every value and
          under-estimate the tonnage.

          My point (1) is that, if you wish to avoid conditional
          bias in your kriging, you could consider using a
          non-linear kriging method such as those mentioned. I
          have no experience with either, since I follow a
          different route in the correction of conditional bias
          in mineral resource estimation.

          Isobel
          http://geoecosse.bizland.com/whatsnew.htm



          --- nicolau.barros@... wrote: > Isobel,
          >
          > So for mining purposes can't we just krige before
          > applying the cut-off
          > criteria? I mean, for most mining applications one
          > will prefer to have a
          > more realistic geologic block model and will always
          > have the chance to
          > evaluate his/her panels under the appropriate
          > cut-off criteria, but applying
          > that criteria after estimating small blocks, right?
          >
          > Could you please explain your point in solution (1)
          > below? Thanks for
          > indicating the literature.
          >
          > Thanks
          >
          > Nicolau Barros
          > Engineer
          > Mine Planning and Production Control Department
          > Mineração Rio do Norte S.A.
          > nicolau.barros@...
          > +55 (93) 549 8215
          >
          > Confidencialidade
          > Esse e-mail e possíveis anexos podem possuir
          > informações confidenciais e de
          > interesse somente do destinatário. Portanto, se você
          > recebeu esta mensagem
          > por engano, favor comunicar imediatamente o
          > remetente e deletá-la logo em
          > seguida. Esteja ciente que o uso indevido do
          > conteúdo das informações em
          > questão é estritamente proibido.
          > Confidentiality
          > This message and any possible attached files may
          > contain confidential
          > information and only for interest of the intended
          > recipient. If you have
          > received this message by mistake, please notify the
          > sender and delete the
          > message immediately. Be aware that the unauthorized
          > use of the
          > above-mentioned information is strictly forbidden.
          >
          > -----Mensagem original-----
          > De: Isobel Clark [mailto:drisobelclark@...]
          > Enviada em: segunda-feira, 19 de julho de 2004 05:23
          > Para: Julhendra_Solin@...
          > Cc: ai-geostats@...
          > Assunto: [ai-geostats] Re: Kriging Small Blocks
          >
          > Jul
          >
          > The warning about kriging small blocks is about
          > "small" relative to the sampling density. For
          > example,
          > less than about one-third of the grid spacing.
          >
          > The warning is the same as the one about 'point'
          > kriging (mapping) The map is too smooth - or, at
          > least, a lot smoother than the real surface would
          > be.
          > High value areas will be under-estimated and low
          > value
          > areas will be over-estimated.
          >
          > If your objective in kriging is to obtain general
          > maps
          > of an area with an idea of where the highs and lows
          > are, then ordinary kriging is sufficient. The over-
          > and under- estimations cancel out on average.
          >
          > In mining applications, where block kriging
          > originated, most applications require a 'cutoff',
          > where values below a certain value are not included
          > in
          > the 'plan'. In this case, mapping or estimating
          > small
          > blocks will result in an over-estimation of
          > 'payable'
          > ground and an under-estimation in average value.
          >
          > In pollution or environmental applications, the
          > areas
          > at risk will be under-estimated as will the true
          > toxicity or risk factors.
          >
          > There are two major ways round this problem:
          >
          > (1) use a non-linear kriging approach such as
          > disjunctive kriging or the multivariate gaussian. Ed
          > Isaacs and Mohan Srivastava's book is th ebest
          > reference for the latter. Rivoirard's book for DK.
          >
          > (2) simulation. There are lots of simulation methods
          > around, which allow you to 'put back the roughness'
          > and get an idea how bad the problem might be. GSLib
          > is
          > pretty good on this.
          >
          > Isobel
          > http://geoecosse.bizland.com/course_brochure.htm
          >
          > If, as in mining, you wish to apply some sort
          >
          >
          >
          >
          >
          >
          ___________________________________________________________ALL-NEW
          > Yahoo!
          > Messenger - sooooo many all-new ways to express
          > yourself
          > http://uk.messenger.yahoo.com
          >





          ___________________________________________________________ALL-NEW Yahoo!
          Messenger - sooooo many all-new ways to express yourself
          http://uk.messenger.yahoo.com
        • Isobel Clark
          Ed I would differ from your explanation on one point. If you are merely declaring a mineral resource, i.e. mineral in the ground, then the conditional bias may
          Message 4 of 5 , Jul 19, 2004
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            Ed

            I would differ from your explanation on one point.

            If you are merely declaring a mineral resource, i.e.
            mineral in the ground, then the conditional bias may
            not be relevant at the "pre feasibility" stage.

            However, as soon as you introduce any economic or
            technical parameters which entail selection, the
            conditional bias makes its appearance.

            In every project I have worked on, from
            pre-feasibility onwards, I have been asked for a
            grade/tonnage calculation - no matter how hand-waving
            it may be. The grade/tonnage curve will be affected by
            the conditional bias. By how much has to be assessed
            at the time. Most of Chapter 3 in Practical
            Geostatistics 1979 is devoted to working out what the
            (theoretical) global grade tonnage curve looks like
            when you adjust for the variance reduction. Even this
            will differ from the curve derived from the kriged
            estimates, no matter what size the block.

            The problem is even greater for environmental
            applications, especially toxic level risks. A 'global
            view' - i.e. a map - will not identify the true peaks
            because of the conditional bias. The fact that the
            overall average is unbiassed is irrelevant when trying
            to identify an area which is likely to be lethal.

            So, there is no contradiction. Conditional bias is
            unimportant (or irrelevant) until you apply some
            selection criterion. Yes, we agree. However, selection
            criteria can be relevant at very early stages of a
            project. It depends on your objective.

            Isobel
            http://uk.geocities.com/drisobelclark/practica.htm for
            free downloads of Practical Geostatistics 1979

            PS: sorry I mis-spelled your name, I know it drives me
            nuts when people call me 'Clarke' ;-)





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          • Edward Isaaks
            Hi Isabel If you go to www.isaaks.com and click on Otherstuff , you will find an example where the block estimates are conditionally biased (rather severely),
            Message 5 of 5 , Jul 19, 2004
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              Hi Isabel

              If you go to www.isaaks.com and click on "Otherstuff", you will find an
              example where the block estimates are conditionally biased (rather
              severely), but the grade tonnage curves are right on the money. Perhaps this
              will help clear the confusion. Ed

              -----Original Message-----
              From: Isobel Clark [mailto:drisobelclark@...]
              Sent: Monday, July 19, 2004 10:47 AM
              To: Edward Isaaks
              Cc: ai-geostats@...
              Subject: [ai-geostats] Re: Kriging Small Blocks

              Ed

              I would differ from your explanation on one point.

              If you are merely declaring a mineral resource, i.e.
              mineral in the ground, then the conditional bias may
              not be relevant at the "pre feasibility" stage.

              However, as soon as you introduce any economic or
              technical parameters which entail selection, the
              conditional bias makes its appearance.

              In every project I have worked on, from
              pre-feasibility onwards, I have been asked for a
              grade/tonnage calculation - no matter how hand-waving
              it may be. The grade/tonnage curve will be affected by
              the conditional bias. By how much has to be assessed
              at the time. Most of Chapter 3 in Practical
              Geostatistics 1979 is devoted to working out what the
              (theoretical) global grade tonnage curve looks like
              when you adjust for the variance reduction. Even this
              will differ from the curve derived from the kriged
              estimates, no matter what size the block.

              The problem is even greater for environmental
              applications, especially toxic level risks. A 'global
              view' - i.e. a map - will not identify the true peaks
              because of the conditional bias. The fact that the
              overall average is unbiassed is irrelevant when trying
              to identify an area which is likely to be lethal.

              So, there is no contradiction. Conditional bias is
              unimportant (or irrelevant) until you apply some
              selection criterion. Yes, we agree. However, selection
              criteria can be relevant at very early stages of a
              project. It depends on your objective.

              Isobel
              http://uk.geocities.com/drisobelclark/practica.htm for
              free downloads of Practical Geostatistics 1979

              PS: sorry I mis-spelled your name, I know it drives me
              nuts when people call me 'Clarke' ;-)





              ___________________________________________________________ALL-NEW Yahoo!
              Messenger - sooooo many all-new ways to express yourself
              http://uk.messenger.yahoo.com
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