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Re: GEOSTATS: Kriging variance

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  • Pierre Goovaerts
    Dear Edzer, You are perfectly right that, IF the sample histogram is normal, the ensemble of all (conditional) realisations at any unsampled grid node should
    Message 1 of 8 , Dec 7 7:21 AM
      Dear Edzer,

      You are perfectly right that, IF the sample histogram is normal,
      the ensemble of all (conditional) realisations at any unsampled
      grid node should have a normal distribution with for mean the simple
      kriging estimate and with variance the simple kriging variance.
      In many situations, however, the sample histogram is non-normal
      and we have to transform the data first.
      As I mentioned in my original e-mail, you could use a lognormal
      transform, but then there is the problem of backtransforming
      the kriging estimate. More importantly, what do you do with
      the lognormal kriging variance??

      Stochastic simulation is more appropriate than kriging whenever
      the sample histogram is non-normal. You proceed as follows:
      1. A normal score transform is first used to normalize the
      histogram, that is each data is replaced by the corresponding
      quantile of the standard normal distribution. Unlike the lognormal
      transform, that kind of graphical transform ensures that the
      resulting histogram is normal, regardless the shape of the
      original histogram.
      2. You perform the simulation (sequential or others) within the
      multiGaussian framework.
      3. You back-transform your simulated normal scores using the
      inverse of the normal score transform.
      As a result, although you use the multiGaussian formalism, the
      distribution of simulated values at each grid node is not normal!

      I agree with you that the incorporation of uncertainty into
      decision-making requires not only a measure of the spread of
      the distribution of possible values but also the values
      themselves. In other words, it might not be worth sampling
      a location with a large uncertainty if it appears that the
      probability of exceeding a regulatory threshold is negligeable
      at that location.

      Pierre


      <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

      ________ ________
      | \ / | Pierre Goovaerts
      |_ \ / _| Assistant professor
      __|________\/________|__ Dept of Civil & Environmental Engineering
      | | The University of Michigan
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      <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>


      On Mon, 7 Dec 1998, Edzer J. Pebesma wrote:

      > Pierre Goovaerts wrote:
      > > Cross validation has also its limitations. For example, it can not
      > > be used to derive the location of additional monitoring sites.
      > > For that type of applications, I would use a simulation approach
      > > whereby a set of realizations, which reproduce the distribution
      > > and pattern of variability of the data, is generated conditionally
      > > to the measurements. For example, the generation of 100 realizations
      > > would provide, at each grid node, a distribution of 100 simulated values,
      > > the spread of which could be used as a measure of uncertainty.
      > > In other words, locate your station where the uncertainty is the
      > > largest (widest distribution of simulated values).
      > >
      > When you're using Gaussian simulation here, you're back at the kriging
      > variance: the ensemble of all (conditional) realisations should have a
      > normal distribution with mean the kriging estimate and with variance the
      > kriging variance.
      >
      > Personally, I often find it hard to believe that kriging variance only can
      > provide a sensible measure for deciding where to place additional sampling.
      > Most often, one will be interested in high values, in low values, or in the
      > decision whether interpolated values are below/above a critical threshold.
      > Then, both kriging estimate and kriging variance should be taken into account,
      > even when you're kriging indicators.
      > --
      > Edzer
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