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GEOSTATS: intro to splines

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  • Beatrice Eiselt
    Hi there, After checking out kriging I m interested in what seems to be an alternative: splines, so I m looking for a simple introduction to splines, thin
    Message 1 of 4 , Oct 3, 1997
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      Hi there,
      After checking out kriging I'm interested in what seems to be an
      alternative: splines, so I'm looking for a simple introduction to splines,
      thin plate smoothing splines, for the analysis of meteorological data
      (average data) (simple as for example "An Introduction to Applied
      Geostatistics" by Isaaks and Srivastava for kriging). Can anybody suggest
      some references?!

      Thanks,
      Beatrice Eiselt
      _______________________________________________________

      Beatrice Eiselt
      Joint Research Centre of the European Commission
      Institute for Systems, Informatics and Safety
      Joint Research Centre
      21020 ISPRA (VA)
      ITALY

      Tel +339-332-785941
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    • Matthew Kay
      ... Dear Beatrice I have used a form of splines called minimum curvature (MC) A classic reference is: Briggs, I. 1974. Machine contouring using minimum
      Message 2 of 4 , Oct 4, 1997
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        > After checking out kriging I'm interested in what seems to be an
        > alternative: splines, so I'm looking for a simple introduction to splines,
        > thin plate smoothing splines, for the analysis of meteorological data
        > (average data) (simple as for example "An Introduction to Applied
        > Geostatistics" by Isaaks and Srivastava for kriging). Can anybody suggest
        > some references?!

        Dear Beatrice

        I have used a form of splines called minimum curvature (MC)

        A "classic" reference is:

        Briggs, I. 1974. "Machine contouring using minimum curvature" Geophysics,
        39

        An example is presented in:

        Mendonca, C. & Silva, J. (1974). "Interpolation of potential field
        data by equivalent layer and minimum curvature", Geophysics (60)

        However, I prefer kriging over splines/MC, as kriging can incorporate
        anisotropies present in your data.

        I may be wrong, but as far as i know, the only way anisotropies can be
        dealt with in MC, is by varying the search ellipse (an inferior solution).

        It has been shown by Matheron (1980) that a formal equivalence exists
        between kriging and splines, with splines being equivalent to kriging with
        a particular covariance model. Thus MC may give poorer results than
        kriging when your data's covariance (model) is different to the fixed one
        used by MC when there are few data in the area being interpolated.

        In zones where you have good spatial sampling then kriging and MC perform
        equally well.

        I hope this helps

        cheers

        matthew


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      • soroko@netcom.ca
        Hi, The classic reference, as M.Kay pointed out is: Briggs, I. 1974. Machine contouring using minimum curvature Geophysics, 39 And an improvement to the
        Message 3 of 4 , Oct 4, 1997
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          Hi,

          The "classic" reference, as M.Kay pointed out is:

          Briggs, I. 1974. "Machine contouring using minimum curvature" Geophysics,
          39

          And an improvement to the method (gridding with tension to reduce the erroneous inflections) is
          in:

          Smith, W.H.F. and Wessel, P. 1990. "Gridding with continuous curvature splines in tension".
          Geophysics. Vol.55, No.3, p.293-305.

          Fortran Code is offered by Swain in:

          Swain, C.J. (1976) "A Fortran IV Program for Interpolating Irregularly Spaced Data using the
          Difference Equations for Minimun Curvature". Computers and Geoscience. Vol.1, p.321-240.

          And I haven't studied it intensively, but a more recent paper which uses directionalized tension
          as a way to incorporate simple anistropy.

          Mitasova, H. and Mitas, L. (1993). "Interpolation by Regularized Spline with Tension: I. Theory
          and Implementation". Mathematical Geology. Vol.25, No.6.


          I hope this helps!

          Jason Soroko
          University of Ottawa
          soroko@...



          > After checking out kriging I'm interested in what seems to be an
          > alternative: splines, so I'm looking for a simple introduction to splines,
          > thin plate smoothing splines, for the analysis of meteorological data
          > (average data) (simple as for example "An Introduction to Applied
          > Geostatistics" by Isaaks and Srivastava for kriging). Can anybody suggest
          > some references?!

          Matthew Kay responded:

          Dear Beatrice

          I have used a form of splines called minimum curvature (MC)

          A "classic" reference is:

          Briggs, I. 1974. "Machine contouring using minimum curvature" Geophysics,
          39

          An example is presented in:

          Mendonca, C. & Silva, J. (1974). "Interpolation of potential field
          data by equivalent layer and minimum curvature", Geophysics (60)

          However, I prefer kriging over splines/MC, as kriging can incorporate
          anisotropies present in your data.

          I may be wrong, but as far as i know, the only way anisotropies can be
          dealt with in MC, is by varying the search ellipse (an inferior solution).

          It has been shown by Matheron (1980) that a formal equivalence exists
          between kriging and splines, with splines being equivalent to kriging with
          a particular covariance model. Thus MC may give poorer results than
          kriging when your data's covariance (model) is different to the fixed one
          used by MC when there are few data in the area being interpolated.

          In zones where you have good spatial sampling then kriging and MC perform
          equally well.

          I hope this helps

          cheers

          matthew


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        • Syed Abdul Rahman
          A couple of caveats: Before embarking on work using splines, or neural networks, etc, might be worth noting that all work on the same premise, i.e. an
          Message 4 of 4 , Oct 5, 1997
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            A couple of caveats:

            Before embarking on work using splines, or
            neural networks, etc, might be worth noting that
            all work on the same premise, i.e. an underlying
            spatial correlation structure, and an interpolation
            component.

            Thin plate splines can be equivalent to same
            covariance models. Likewise radial basis function
            neural networks, which can be equivalent to
            kriging using a particular covariance model,
            depending on the radial basis function used.

            Unfortunately, splines and RBF networks are
            mostly packaged as black boxes (with some
            particular covariance model). Or, if "customization"
            is allowed, the spatial correlation structure can
            sometimes be modified, within certain limits,
            but it is still not as intuitive as graphical variogram
            modeling. The danger here is that the spatial
            analysis part -- the most important component
            in an interpolation exercise -- is relegated to the
            computer.

            Re: anisotropies:

            1) One can use the earlier suggestion, i.e. elliptical
            or ellipsoidal search neighborhoods when using
            splines or RBF networks, or

            2) Go back to the code, and modify the distance
            measures to include the anisotropy ratio, i.e.
            exagerating distances along directions of minimum
            continuity and compressing them along directions of
            maximum continuity.

            Note that a normal variography phase is still recommended
            to derive the anisotropy ratios in the first place.

            Regards,

            Syed

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