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  • John Butler
    Sender Phone Number: My question relates to the use of geostatistics in a field we call quantitative metallography (in metallurgy). The first paper I have seen
    Message 1 of 1 , Dec 2, 1999
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      Sender Phone Number:
      My question relates to the use of geostatistics in a field we call
      quantitative metallography (in metallurgy).

      The first paper I have seen on this subject was Sampling in Quantitative
      Image Analysis. Examples of the Application of Sampling to Measurements of
      the Contents of Phases in Sinter Types and of Inclusions in Steel Types',
      T.Hersant and D.Jeulin, Rev. Met. Mem. Sci., 1976, Vol. 73, (7/8)pp
      503-517). They use the Variogram to describe the error committed if the
      phase content of X, P(x), measured at point x is attributed to point x+h.,
      i.e.
      2*gamma(h) =E{[P(x+h)- P(x)]^2}..................(1)
      The TAS texture analyser is used to sample the phase type at a series
      points, giving P(x)=1 if x pertains to phase X and P(x)=0 if it does not.
      (NB from stereology a standard result is the equality of point, area and
      volume fractions).
      By algebraic manipulation and using this 'indicator ?' approach equation
      (1) becomes in a semivariogram form :-
      gamma(h)= E{P(x)^2}- E{P(x)*P(x+h)}
      Do they get this because E{[P(x)]^2} = E{[P(x+h)]^2} ?)
      They let C(h) = E{P(x)*P(x+h)}
      This gives gamma(h)=C(0)-C(h)
      where C(h) is the covariance of phase X, i.e. the probability that the two
      points separated by a distance of h both belong to X.

      They then state that the properties of the semivariogram are
      C(0)= volume fraction of phase X, and
      C(infinity)= (volume fraction of phase X)^2
      Can anyone explain how these these limits arise ?
      Does C(0)= Vv because E{[P(x)]^2} = (0)^2*(1- Vv) + (1)^2*(Vv) ?

      However, I cannot see why C(infinity) = (Vv)^2 , the plateau of the
      covariance, could anyone explain this ? This may be simple to someone who
      has attended lectures in this field, is it ?

      They determined the approximate formula for the extension variance (also
      termed estimation variance) equations (equations 12 to 14 in the paper)
      and certain ranges, but is not clear how these arise ? I believe that J.
      Chone also used these ranges (for lineal, areal and volumetric phase
      anlysis) in the papers which he published on this subject whilst at IRSID
      around 1976-1977.

      Finally, is there a less exacting introduction to the subject matter of
      Chapter 9 on The Covariance in J.Serra's book on Mathematical Morphology,
      that would be suitable for a non-mathematician ? Here he presents equations
      which show the difference on going from a global representation of the
      covariance to a local one, equations IX-1 to IX-25. Is there explanation
      available of this and the derivation which is easier to follow ?


      [As an aside, could I point out that I am not a mathematician, but a
      metallurgist who has received mathematical training as an engineering
      school undergraduate (over 20 years ago), so please excuse my ignorance.
      For background reading I have looked at Chapter 9 of 'Image Analysis and
      Mathematical Morphology' by J. Serra on The Covariance, but found it heavy
      going (as stated above I could not follow fully equations IX-1 to IX-25,
      i.e. moving from a global to a local representation of the covariance
      function, which seems to be relevant), and Chapter 2 of 'Mining
      Geostatistics' by Journel and Huijbregts on The Theory of Regionalised
      Variables, which I was able to understand a greater proportion of (useful
      for extension of variance, derivation of the general equation). For an
      introduction to general geostatistics I have read 'Practical Geostatistics'
      by Isobel Clark.]

      Thank you for your attention.


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