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Re: GEOSTATS: Interpolation of data with line collection error

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  • Gilles Bourgault
    ... Hi Marco, An idea will be the filtering of those errors on each line. Factorial kriging can be used to filter out correlated errors. You will need to model
    Message 1 of 3 , Aug 18, 1999
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      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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      Hi Marco,

      An idea will be the filtering of those errors on each line. Factorial kriging can be used
      to filter out correlated errors. You will need to model a variogram for each line. Then,
      you will have to identify the variogram's component that represents the correlated
      noise and filter out that component from your data.

      You may find a bit more information on factorial kriging in GSLIB book. You will have to
      adapt GSLIB FORTRAN programs to do factorial kriging.

      Good Luck,

      Gilles

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    • Colin Daly
      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
      Message 2 of 3 , 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
        > --
        > *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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