Quantitative Stratigraphy, Splining and Geologic Time Scales - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 326

Fig. 9.13 De Boor ( 1978 ,
Fig. 8.1, p. 224) simulated
irregular spacing along x -
axis by selecting 12 points
( solid circles ) from set of
49 regularly spaced
measurements of a variable
( y ) as a function of another
variable ( x ). The optimum
fifth order interpolation
spline (with seven knots)
provides a poor fit except
around the peak (Source:
Agterberg 1990 , Fig. 3.9)

together on the peak than in the valleys. De Boor used this example to illustrate that
poor results may be obtained even if use is made of a method of optimum spline
interpolation in which best locations are computed for ( n

k ) knots of a k -th order

spline. For the example of Fig. 9.13 , k 5 so that seven knots are used. Although

these seven knots have optimum locations along the X -axis, the result is obviously
poor, because the shape of the relatively narrow peak is reflected in non-realistic
oscillations in between the more widely spaced data points in the valleys. De Boor
( 1978 , p. 225) pointed out that using a lower-order spline would help to obtain a
better approximation. In subsequent applications, use is made of cubic splines only
( k

¼

3). Figure 9.14a shows the cubic interpolation spline for the 12 regularly
spaced points of Fig. 9.2 using knots coinciding with data points. Contrary to the
fifth order spline with seven knots, the new result provides a good approximation.
However, deletion of three more points from the valleys (Fig. 9.14b ) begins to give
the relatively poor cubic interpolation spline of Fig. 9.14c which has unrealistic
oscillations in the valleys because all intermediate data points were deleted.
The bottom half of Fig. 9.14 shows results obtained by applying the indirect
method to the situation that led to the worst cubic-spline result for the previous
example (seven data points in Fig. 9.14c ). Figure 9.14d is the cubic interpolation
spline for seven regularly spaced “levels”. Figure 9.14e is a monotonically increas-
ing cubic smoothing spline with a small positive value for SF for the relation
between X and level. Figure 9.14f represents the combination of the curves of
Fig. 9.14d , e . Its approximation to the original pattern for the 49 values of Fig. 9.13
is only relatively poor in the valleys where no data were used for control. Unreal-
istic oscillations could be avoided by the use of the three-step indirect method of
Fig. 9.14d-f .

¼