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Re: GEOSTATS: interpolating pH values

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  • Jeffrey White
    ... Log transformations are commonly used to provide a more normal population, although many have expressed their view that normality is not a necessary nor
    Message 1 of 6 , Feb 5, 1998
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      On Thu, 5 Feb 1998, Syed Abdul Rahman wrote:

      > One has to reason why logarithmic transforms are used, a common
      > reason is to work with a manageable range of values, and another
      > is to get a stable variogram. Nevertheless, krigings become extremely
      > sensitive to changes in sill values, once the results are transformed
      > to their raw values. Another problem is that such results are
      > biased, what you expect to be the expectation (assuming this is what
      > you're after) would be the median after backtransformation. Therein
      > lies the bias. To be theoretically more "rigorous", go for log-normal
      > kriging.

      Log transformations are commonly used to provide a more normal population,
      although many have expressed their view that normality is not a necessary
      nor inherent assumption in the theory of regionalized variables, but MAY
      be a condition for unbiased estimation. With respect to pH, it is the
      transformation bias of which Syed wrote that is the problem when pH units
      are used directly in any additive process (such as averaging or
      semivariography and kriging), rather than the actual concentrations that
      they represent.

      An illustation: A simple example is mixing equal volumes
      of two solutions of different pH, say pH 6 and pH 8. Simple (and
      erroneous!) averaging of the pH's indicates that the resulting solution
      would have pH 7, while the true pH is about pH 6.3 : pH 6 => hydrogen
      ion concentration of 0.000001 M; pH 8 => hydrogen ion concentration of
      0.00000001 M. Equivolume mixture has hydrogen ion concentration of
      (0.000001 + 0.00000001)/2 = 0.000000505 M, negative log (or pH) of which
      is 6.296.


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    • Rane Curl
      ... I am responding to this as a chemist, and not as expert in kriging. I am also sure that others have dealt with pH values in this context in one way or
      Message 2 of 6 , Feb 5, 1998
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        On Fri, 6 Feb 1998, Inakwu Ominyi Odeh wrote:

        > The transformation bias of pH values as negative log of H+ concentrations being
        > referred to here is important iff the pH range within the immediate domain used
        > for kriging is large. If the range is within a unity, then there is probably
        > not much to be gained by transforming back to H+ concentration before kriging.
        > If, however, the pH range is 2 or more, then theoretically, additivity bias may
        > be introduced as a result of any quantitative interpolation. One should judge
        > each case carefully before embarking on kriging of pH values or their H+ conc.
        > But then other problems may arise as the H+ concentration (in soil) is
        > generally very low and may vary (spatially) exponentially.

        I am responding to this as a chemist, and not as expert in kriging. I am
        also sure that others have dealt with pH values in this context in one way
        or another, but I would raise questions about the procedures being
        recommended and the theoretical foundation for them.

        While [H+] is "additive" as a strong acid with nothing else present, in an
        real environment there are many other solutes present as well as reactive
        minerals. Natural solutions are usually buffered, so mixing two solutions
        of different pH does *not* average the [H+]. I would suggest that the
        objective be looked at more deeply and a transformation of pH for the
        purposes of kriging be chosen on a more realistic basis than additivity of
        [H+].

        There may even be justification for using pH. After all, the *solutions*
        do not get mixed when kriging - just the numbers. pH does represent a real
        thermodynamic quantity, which is the Gibbs Free Energy of formation of
        [H+] in solution and, under some conditions, free energies are additive.
        Other reactions also respond in a multiplicative manner to [H+] (via
        mass-action equilibria), so the pA of (the activities) of those other
        species respond in a generally additive manner to changes in pH.

        I presume that someone, somewhere, has done a much more thorough
        investigation of the meaning of kriging in pH, but I would not dismiss it
        out of hand unless some more fundamental objections can be put forward
        that the "additivity of [H+]" (sometimes, but rarely).

        --Rane L. Curl
        Prof. of Chemical Engineering


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      • Inakwu Ominyi Odeh
        The transformation bias of pH values as negative log of H+ concentrations being referred to here is important iff the pH range within the immediate domain used
        Message 3 of 6 , Feb 6, 1998
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          The transformation bias of pH values as negative log of H+ concentrations being
          referred to here is important iff the pH range within the immediate domain used
          for kriging is large. If the range is within a unity, then there is probably
          not much to be gained by transforming back to H+ concentration before kriging.
          If, however, the pH range is 2 or more, then theoretically, additivity bias may
          be introduced as a result of any quantitative interpolation. One should judge
          each case carefully before embarking on kriging of pH values or their H+ conc.
          But then other problems may arise as the H+ concentration (in soil) is
          generally very low and may vary (spatially) exponentially.

          --
          Dr Inakwu Ominyi Odeh, |
          Dept. of Agric. Chem. & Soil Science |
          The University of Sydney |
          Ross Street A03 NSW 2006 AUSTRALIA |
          Phone +61 (02) 9351 4178 |
          Fax: +61 (02) 9351 5108 |
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