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  • Soeren Nymand Lophaven
    Dear list, Some time ago I asked the following question: I am currently working with spatial interpolation of geophysical data. Each observation is associated
    Message 1 of 1 , Feb 14, 2003
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      Dear list,

      Some time ago I asked the following question:

      I am currently working with spatial interpolation of geophysical
      data. Each observation is associated with an individual and known
      standard deviation. How should this infomation be incorporated if I
      want to use ordinary kriging for interpolation ?? My idea was the

      When finding the vector of weights (w) by solving the system of
      linear equations A*w=b, I would exchange the zeros in the diagonal
      of the A-matrix with the individual observation variances. Does this
      sound reasonable ??

      and got the following four very useful answers. Thank you very much !!

      Colin Daly wrote:

      That works if your matrix is made up of covariance terms rather than
      variogram terms. However you should use the variance of the error term
      instead of the standard deviation.

      So in your notation A_ij = C_ij = D - Gamma_ij

      where Gamma_ij are the variogram values, D is the sill of the variogram
      (or larger than the largest variogram value if your variogram does not have
      a sill) and at the diagonal you use C_ii+K_ii where K_ii are the variance
      error terms.

      This will not interpolate your data! It will filter the noise terms (which
      you say that you know the variance of at each point)

      Rubens Caldeira Monteiro wrote:

      I suggest you to incorporate those information as soft data,
      i.e., a data that you have some uncertainties related to them. A good
      approach to do it is using Bayesian Maximum Entropy. This spatiotemporal
      geostatistical method allows you to incorporate hard and soft data and
      even deterministic (e.g., physical equations) and stochastic (e.g.,
      variograms) general knowledge.

      Main references about BME are:
      Christakos, G., P. Bogaert, and M.L. Serre, 2002, Temporal GIS,
      Springer-Verlag, New York, N.Y., 220 p., CD Rom included.

      Christakos, G., 2000, Modern Spatiotemporal Geostatistics, Oxford
      University Press, New York, NY, 304 p. (3rd Reprint, 2001).

      Isobel Clark wrote:

      I presume what you have is a sort of 'analytical
      error' for each sample? That is, the standard
      deviation for two samples at the same location around
      the 'true value' at the same location?

      In this case, you can put the variance down the
      diagonal of your kriging system to obtain optimal
      weights under the uncertainty admitted for your data

      You would need to be careful that the 'analytical
      variance' was not greater than the nugget effect of
      the semi-variogram model.

      The kriging system would be similar to that obtained
      when the sample is not treated as a 'point', but
      rather as a volume. This results in a lower kriging
      variance than using zero on the diagonal, so to
      compensate you should probably add the complete
      'analytical variance' back on to get realistic
      estimation variances.

      There seems to be a lot of confusion in the books (and
      software) about what happens if you have a significant
      replication variance.

      Dimitri D'OR wrote:

      You surely find a proper way of solving your problem in using the
      Bayesian Maximum Entropy (BME) approach. This method is especially
      dedicated to the incorporation of various types of soft (imprecise)
      information as intervals of values, pdf's, etc.

      In your case, if you consider a Gaussian distributed error, you may
      consider your data as soft data of the pdf-type. The best predictor will
      thus be nonlinear, which is not possible if you stay within the class of
      kriging predictors. Moreover, BME does not require hard (accurate) data.
      It is able to make good use of data sets made of only soft information.

      For more information about this method, check the books by G. Christakos,
      and some references on my web page (you will find the address in the
      "people" section of the AI-Geostat Web site). Have also a look at the
      BMElib software proposed in the software section (It's a freeware running
      with Matlab).

      Best regards / Venlig hilsen

      Søren Lophaven
      Master of Science in Engineering | Ph.D. student
      Informatics and Mathematical Modelling | Building 321, Room 011
      Technical University of Denmark | 2800 kgs. Lyngby, Denmark
      E-mail: snl@... | http://www.imm.dtu.dk/~snl
      Telephone: +45 45253419 |

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