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Re: GEOSTATS: intro to splines

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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 1 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 2 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 3 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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