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1361Re: GEOSTATS: Interpolation of data with line collection error

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  • Colin Daly
    Sep 6, 1999
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      Marco,

      I missed the main set of responses to your question - but I feel that the
      responses you posted didn't mention the convolution aspect of the problem -
      so I thought that I should add a bit.

      As you have posed your problem the main issue appears to be the convolution
      or smoothing along the direction of sampling. You do not state that there
      are any measurement errors - but I would be surprised if there were none as
      I will explain.

      I have looked at a problem like this in the past - in 1986 I did some work
      for a French Oil company on removing mismatches in 2d seismic lines. Here
      the data was well correlated along the lines - but where the lines crossed
      there was a mismatch. Interpolated images were full of wierd and wonderful
      bulls-eyes.

      I found it useful to go through an analysis as follows. Assume that the
      correct or 'true' data is Z(x). It is this data and its derivatives that you
      wish to find. Sampling is done along lines - so assuming that there are no
      errors - what we observe is a convolution (this may not be true in your
      case - I don't know what the sampling procedure is exactly)

      Y(x) = Int( w(x-y) Z(y) dy) where Int means Integral of and
      w(x-y) is some weighting function along the direction of sampling with
      'bandwidth' equal to the smoothing interval in the sampling process. The
      correlation function for the observed data is related to the true data
      correlation by

      Cov(Y(x),Y(y)) = Int( w(x-z) Cov(Z(z),Z(z')) w(y-z') dz dz') (1)

      It is then possible to Krige Z(x) from the data Y(x). It is not much more
      difficult to estimate derivatives of Z(x) - it simply involves using
      derivatives of the covariance function on the right hand side of the kriging
      equations.

      Note - estimation of the the 'bias' as you call it seems to be to be an
      attempt to calculate w(x) - this can be attempted by trial and error or more
      sophisticated techniques (although it is a bit trickey) - some work on this
      is done by Seguret (see below)

      However note that if this analysis is correct so far - then for small lags
      we might not expect the large differences in correlation that you say as we
      go across lines as opposed to along lines (seems obvious by inspecting
      equation(1) for various configurations of x and y). You suggest that there
      is a large difference which seems to me to indicate that there is something
      else going on- in fact this was the case for the seismic lines problem that
      I mentioned. In that case there were errors within the sampling line (ie a
      constant or a correlated error along the sampling line - associated with the
      sampling itself. This was not there in the other directions). I modelled
      this at the time in a rather heavy handed way by using multi valued random
      functions or some such. However essentially it just means that you have to
      have a correlation term for each line which must be filtered in the estimate
      of Z. Clearly - no off the shelf kriging program will handle it - but it is
      not too difficult to write.

      Some other work that (if my memory is still working ok) seems fairly related
      is the thesis and any related publications by S Seguret from the Centre de
      Geostatistique - which were to do with data recovered by boats - this had
      some extra problems in that there was also a time component (so 24hr
      peroiodicities etc)

      Some references on this - I'm sorry that I don't have access to their full
      names at the moment - but there are copies of them all in the Centre de
      Geostatistique library in Fontainebleau. The librarian Mme F. Poirier is
      very helpful and may be able to provide more detailed information

      (1) Deconvolution: Renard and Jeulin (or Jeulin and Renard) - Deconvolution
      and Kriging (about 1988-1990)

      (2) Seguret S. (about 1990) Thesis and related publications

      (3) Daly C. (1986) Removing Seismic mismatches (Mastere report - this is
      hand written, fairly abstract and in bad French - a last resort!!!)

      Not very clear but hopefully of some use.

      regards

      Colin Daly


      However

      Marco Albani wrote:

      > Hello,
      >
      > I have a geostatistical problem and I was wondering if somebody could
      > point me in the right direction as I haven't got a lot of experience
      > with geostatistic.
      >
      > I am working with elevation data which is collected on a semi-regular
      > grid from air-photo stereocouples.
      >
      > The data is collected along lines, in the same direction, so the
      > measurements are done in sequence (let us say North-South). Because the
      > data represents a continous surface, it is expected to be strongly
      > autocorrelated, but a directional variograms across the collection line
      > shows much higher variance at short lags than the along-collection line
      > direction, which I assume to be due to autocorrelation in measurement
      > errors along the correction lines. This is confirmed by the "corrugated"
      > surface that one obtains interpolating the points through any exact
      > interpolation method.
      >
      > Since my main interest mapping the first and second derivative of the
      > surface, this collection line bias (or autocorrelation of the error) is
      > quite bothersome. My objective is to estimate the collection line bias
      > so to remove it from the data.
      > I was wondering if anyone has encounterd this kind of problem before and
      > has any insight to give.
      >
      > Will post summary.
      >
      > Cheers,
      >
      > Marco
      > --
      > Marco Albani - PhD Candidate, Quantitative Landscape Ecology
      > Department of Forest Sciences, University of British Columbia
      > 3041 - 2424 Main Mall - Vancouver, BC V6T 1Z4 Canada
      > Phone: (604) 822 8295 Fax: (604) 822 9102
      > --
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