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GEOSTATS: Block Kriging Log Data and estimation error

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  • William C. Thayer
    I would like to receive feedback on block kriging with log data: I wish to use log-transformed data in block kriging to estimate global means. I had
    Message 1 of 3 , May 6, 1998
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      I would like to receive feedback on block kriging with log data: I wish
      to use log-transformed data in block kriging to estimate global means.
      I had anticipated taking the anti logs of the block values and kriging
      variances and then estimating the global mean by averaging the block
      values. Is this OK? My understanding of block kriging is the average
      covariances between sample points and points within the block to be
      estimated are used to calculate the block kriging weights. Does this
      averaging of the covariances result in problems when back transforming
      the kriged block values? Any suggestions on how to combine block
      variances to form an estimate of the variance (error) of the global
      mean? Thanks,
      Bill



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    • Pierre Goovaerts
      Hello Bill, You cannot backtransform block kriged estimates obtained from log data. Indeed, the block kriged estimate is a linear average of log values, which
      Message 2 of 3 , May 7, 1998
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        Hello Bill,

        You cannot backtransform block kriged estimates
        obtained from log data. Indeed, the block kriged
        estimate is a linear average of log values,
        which is different from the log of a block estimate.
        If you want to work on log data y(u) = log (z(u)), you
        have to use point kriging to estimate the attribute of
        interest at locations discretizing the block, then backtranform
        the point estimates y^*(u) (don't take the antilog but use
        the transform z*(u) = exp [y*(u) + s^2(u)/2], where s^2(u)
        is the kriging variance at location u), and last
        average the point kriging estimates within the block.
        The next problem is the combination of kriging variances,
        which is not trivial.

        I would recommend to either avoid lognormal kriging, or use
        stochastic simulation. The basic idea of simulation consists
        of generating realizations of the spatial distribution of the target
        attribute which reproduce the pattern of spatial variability of point
        measurements. Block simulated values are then computed using linear or
        non-linear averages of the point simulated values inside each block.
        The major advantage of stochastic simulation is that it provides
        a non-parametric measure of the uncertainty attached to the prediction
        of a single block or multiple spatially dependent blocks. In other words,
        you don't have to assume any particular distribution for the prediction
        error.

        Sincerely,

        Pierre Goovaerts


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        ________ ________
        | \ / | Pierre Goovaerts
        |_ \ / _| Assistant professor
        __|________\/________|__ Dept of Civil & Environmental Engineering
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        |________________________| Ann Arbor, Michigan, 48109-2125, U.S.A
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        On Wed, 6 May 1998, William C. Thayer wrote:

        > I would like to receive feedback on block kriging with log data: I wish
        > to use log-transformed data in block kriging to estimate global means.
        > I had anticipated taking the anti logs of the block values and kriging
        > variances and then estimating the global mean by averaging the block
        > values. Is this OK? My understanding of block kriging is the average
        > covariances between sample points and points within the block to be
        > estimated are used to calculate the block kriging weights. Does this
        > averaging of the covariances result in problems when back transforming
        > the kriged block values? Any suggestions on how to combine block
        > variances to form an estimate of the variance (error) of the global
        > mean? Thanks,
        > Bill
        >
        >
        >
        > --
        > *To post a message to the list, send it to ai-geostats@....
        > *As a general service to list users, please remember to post a summary
        > of any useful responses to your questions.
        > *To unsubscribe, send email to majordomo@... with no subject and
        > "unsubscribe ai-geostats" in the message body.
        > DO NOT SEND Subscribe/Unsubscribe requests to the list!
        >

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      • Syed Abdul Rahman
        ... For the indicator approach, yes. The problem with calculating a strongly skewed variable is that variograms tend to be rather unstable, hence the usual
        Message 3 of 3 , May 9, 1998
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          Pierre Goovaerts wrote:

          > I would recommend to either avoid lognormal kriging, or use
          > stochastic simulation. The basic idea of simulation consists
          > of generating realizations of the spatial distribution of the target
          > attribute which reproduce the pattern of spatial variability of point
          > measurements. Block simulated values are then computed using linear or
          > non-linear averages of the point simulated values inside each block.
          > The major advantage of stochastic simulation is that it provides
          > a non-parametric measure of the uncertainty attached to the prediction
          > of a single block or multiple spatially dependent blocks. In other words,
          > you don't have to assume any particular distribution for the prediction
          > error.

          For the indicator approach, yes. The problem with calculating
          a strongly skewed variable is that variograms tend to be
          rather unstable, hence the usual practice of applying some sort
          of transform. Moreover, such transforms would make predictions
          sensitive to changes in sill values for the modeled variograms.
          The other problem (outlined above) is that of deriving biased
          estimates unless a non-linear approach is utilized. Indicator
          approaches have the advantage of pinpointing patterns of
          spatial variability for classes of values, unlike conventional
          geostatistics where a single globally averaged variogram value
          is calculated. Indicator variograms alss tend to be more stable,
          since binary values are used. All of this, while good on
          paper, also implies more work -- indicator coding of soft and
          hard data, division of data into thresholds, less data at the
          high and low ends of the thresholds, etc etc. Practical
          considerations are therefore more numerous, hence wide acceptance
          of such techniques has been slow in coming, which IMHO is true
          at least for oil and gas problems.

          Regards,

          Syed
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