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

**Petr Bulant**
&
**Ludek Klimes**
### Summary

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.

### Keywords

Velocity model, general and natural splines, transformation.

### Whole paper

The reprint is available in
PDF (3628 kB).

*Stud. geophys. geod.*, **63** (2019), 137-146.