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GEOSTATS: adaptive interpolation

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  • Donald Myers
    A few thoughts 1. Adaptive can mean several different things. The kriging estimator/predictor is already adaptive in several senses. a. It adapts to the
    Message 1 of 1 , Sep 29, 1998
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      A few thoughts

      1. Adaptive can mean several different things. The kriging
      estimator/predictor is already "adaptive" in several senses.

      a. It "adapts" to the spatial pattern of the data locations by
      incorporating the spatial correlation for pairs of data locations. (The
      data locations need not be on a grid).

      b. It "adapts" to the spatial relationship of the location to be estimated
      to the data locations used in the estimation

      c. By using a search neighborhood it makes several adaptations. Note
      however that using search neighborhoods may introduce discontinuities in
      the interpolated surface.

      One of the supposed justifications for using a search neighborhood is
      that for most variogram models the data locations outside the search
      neighborhood will have small or zero weights even if included (this
      obviously depends on the choice of the search neighborhood and the
      variogram model and its parameters) hence there is no substantial change
      by not including them.

      One of the presumed advantages of using a search neighborhood is that the
      constant mean assumption can be slightly weakened, i.e., the mean is only
      assumed constant (or nearly constant) within each search neighborhood
      (separately). See Journel et al (Math Geology) on "Why do we need a trend

      d. By allowing the variogram model to be "adapted" to the data, the
      kriging estimator incorporates several aspects of the data including a
      directional effect in the spatial correlation.

      Most of the above are what might be called pre-adaptations, i.e, they do
      not occur during the actual kriging.

      2. In order that the kriging variance be non-negative (remember the
      kriging variance is the minimized estimation and mimimizing non-negative
      valued functions is very different from mimimizing functions that can take
      on both postive and negative values) and that the kriging system have a
      unique solution it is (essentially) necessary that the variogram be
      conditionally negative definite (or that the covariance be positive
      definite). If the variogram is allowed to change from one search
      neighborhood to another there are two potential problems, (i) do the
      multiple forms collectively determine a valid model, (ii) if the
      neighborhoods overlap then which one does one use in the intersection (for
      consistency they should in fact match on the intersection). One of the
      practical reasons that we commonly use only a small number of standard
      models for variograms is that it is difficult to check whether a given
      function is a valid variogram.

      3. It is well-known that the kriging estimator can be re-written in the
      so-called "dual" form, which shows the relationship to radial basis
      function interpolation. However there is a significant practical
      difference. In the usual form of the kriging estimator the coefficients
      change with the location to be estimated whereas in the dual form the
      coefficients do not depend on the location to be estimated. That is, the
      radial basis form produces an interpolating function whereas the kriging
      estimator only implicitly determines the interpolating function (by giving
      its values one at a time for each location). Hence it is not so simple to
      use a search neighborhood since in that case one must actually confront the
      piecing together functions defined on separate (but usually overlapping)
      regions and hence the discontinuities will become very apparent. (See
      Myers, "Interpolation with Positive Definite Functions", Sciences de la
      Terre, 1987).

      4. If the function to be interpolated is really so variable that one would
      have to consider a locally varying variogram then maybe it is too
      complicated to be represented by second order properties.

      5. Finally one must remember that geostatistics is essentially based on the
      idea that the data is a (non-random) sample from one realization of a
      random function. The variogram or covariance function is a characteristic
      of the random function, not of the data. The sample variogram (or other
      variogram estimator) is a statistic based on the one realization, some form
      of ergodicity and stationarity is needed to use this spatial statistic to
      estimate/model the variogram which is an ensemble statistic. Ideally one
      should have data from multiple realizations at each data location, i.e.,
      replications. In some instances where one has spatial temporal data, it
      might be assumed that there is no temporal dependence and hence data
      collected at multiple time points for a given location might be treated as
      replications. This would allow local fitting/estimation of variograms and
      without invoking (spatial) stationarity.

      This doesn't mean that it might not be possible to have a locally varying
      variogram but it does mean that one should be careful both in the modeling
      step and in using it for kriging because there are hidden implications.

      Donald Myers
      Department of Mathematics
      University of Arizona

      (520) 621-6859


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