3-D velocity models — transformation from general to natural splines

  • Petr Bulant
  • Luděk Klimeš
Open Access


The functions describing material parameters and structural interfaces in velocity models are frequently represented by splines. The general cubic splines differ from the natural cubic splines by the boundary conditions at the outermost gridpoints. The general cubic splines have a general curvature at the outermost gridpoints used for interpolation, whereas the natural splines have a zero normal curvature at the outermost gridpoints. It is thus very useful to employ a simple algorithm for the transformation between the general and natural splines. The transformation from the natural to general (bi–) (tri–) cubic splines is straightforward, because the natural splines represent a special case of the general splines. This paper is devoted to the algorithm of transformation from the general to natural (bi–) (tri–) cubic splines. We present the formulae necessary for the transformation together with their derivation.

We illustrate the presented formulae on the example of fitting a 1–D quadratic function by natural cubic splines, and on the example of a velocity model of a layered structure with two 3–D structural interfaces.


velocity model general and natural splines transformation 


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Copyright information

© The Authors 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provided a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Petr Bulant
    • 1
  • Luděk Klimeš
    • 1
  1. 1.Department of Geophysics, Faculty of Mathematics and PhysicsCharles UniversityPraha 2Czech Republic

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