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GEOSTATS: SUM: Kriging variance & search neighborhood

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  • Gregoire Dubois
    Dear all, here is a summary of the answers I got to my question about the impact of local search on kriging variance. I got many replies showing me that my
    Message 1 of 1 , Dec 11, 1905
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      Dear all,

      here is a summary of the answers I got to my question about the
      impact of local search on kriging variance.

      I got many replies showing me that my question was still
      unclear or could be understood in different ways.

      My point really was that the kriging variance, which is
      dependent on the covariance model and the data configuration,
      can not properly reflect the variance of the estimates if one uses
      a local search neighborhood since the experimental semivariogram
      is usually modeled for the whole data set.
      When I was asking about the impact of a search neighborhood
      on the kriging variance, I was more thinking in terms of the
      interpretation of the kriging variance.

      I can only recommend to read pages 178-180 of Pierre Goovaerts'
      book "Geostatistics for Natural Resources Evaluation" for what
      concerns the practice of using only the data closest to the
      location where an estimation is made.

      Other references suggested by Victor De Oliveira and
      Mustafa Touati are given hereafter. The first two references
      propose a statistically sound likelihood-based method to choose
      the "neighborhoods" for doing estimation and prediction.

      Vecchi (1992), A new method of prediction for spatial regression models
      with correlated errors, J. of the Royal Statistical Society B, 813-830

      Vecchia (1988), Estimation and model identification for continuous
      spatial processes, J. of the Royal Statistical Society B, 297-312.

      Poids de Krigeage ( A Ph.D. Thesis from the Paris School of Mines).
      (this was done early in the 80th) by Jaques Rivoirard from the
      Centre de Geostatistique


      You will find here under other replies that could be interesting for
      the readers:

      From Edzer J. Pebesma:

      Obviously, when using kriging in a small neighbourhood,
      you only use (and need) the variogram function for the smaller
      distances. The good thing is that for smaller distances the
      variogram is easier to estimate than for longer distances,
      given a single replicate (realization) of a RF. OTOH, even
      for kriging with a global neighbourhood the variogram at the
      smaller distances is more important than at larger distances.

      For ordinary or universal kriging, all data in the neigbourhood
      are used for prediction, also values beyond the correlation
      distance (range): one of the things they are used for is
      (re-)estimating the trend value or parameters at the kriging location.
      A larger neighbourhood will therefor usually lead to smaller
      kriging variances, but the effect may be small after
      taking, say, the 20 or 30 nearest observations. For a pure nugget
      effect things are easy to write out analytically, because
      data configurations don't matter here. For other cases, I would
      just try a couple of neighbourhood sizes, perhaps do some cross
      validations.

      Robert C Reynolds & Yetta Jager remind that the kriging variance
      is affected by the number, distance, position and orientation of data
      used. A limited number of nearest neighbors is frequently used so as
      to keep the kriging system of equations small. Kriging weights
      become small for sample points as one move away from the location
      estimated and have a negligible influence, especially of there is a
      strong spatial correlation. Points beyond the sill contribute in no way
      to the kriged estimate and so can be ignored. Ideally, points should
      reside fairly evenly around (interpolation) the point to be estimated
      rather than tending off to one side (extrapolation) for best results.


      Mats Soderstrom reminds that it might be more efficient in some cases
      to use local variograms instead of variograms made on the whole data set
      and that the VESPER software (See the Software list of AI-GEOSTATS or
      directly http://www.usyd.edu.au/su/agric/acpa/vesper/vesper.html) proposes
      a local approach.


      Last but not least, from Dan Cornford:

      "The answer depends on the range of the process and the
      distance beyond which the sites are and the sites locations - i.e.
      it is not particularly easy to answer. In general a small number
      of surrounding sites is used in kriging for several reasons:

      Computational (matrix inversion scales as n^3, where n is the
      number of sites. Stability (conditioning) of matrix inversion is
      also likely to be improved. Stationarity becomes less of a problem
      (only local needed) - but of course this is conditional on knowing
      the variogram which was estimated locally thus this is a bit of a fix!

      Nowadays it is possible to invert matrices of up to about 1000
      observations in one go - that is there is no need to do the local
      thing - but this is not always a good idea for the reasons given
      above.

      Adding more observations should (I think - it is a while since I have
      looked at this) alweays decrease the kriging variance, but in practice
      the decrease will be tiny".


      Many other replies suggested me to test it by increasing the numbers
      of points in the neighbourhood and to check the kriging variance.



      Thanks a lot also to Nicla Giglioli and Andrew (Nefia ?)
      for their kind replies.

      Have a nice week !

      Gregoire



      Gregoire Dubois
      Section of Earth Sciences
      Institute of Mineralogy and Petrography
      University of Lausanne
      Switzerland

      Currently detached in Italy

      http://curie.ei.jrc.it/ai-geostats.htm

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