About principles of spline-extrapolation.

Possible applications of spline mathematics is discussed for situations typical for
geophysical observations when only numerical values of time series of data are known, to build
a physical dynamic model is either impossible or too complicated, unreasonable mainly because
of complexity of geological “scene” on which the events occur. Dipmeter survey, systematic
measurements of varying level and temperature of ground water, radon concentration in wells
are associated traditionally with search for possible precursors of earthquakes, forecasting of
the nearest following value at the sequences of such kind does not presuppose from the outset
knowledge or even existence some dynamic connection between phenomena. As a consequence, an
interpolation on the set of experimental points proves not approximation, in sense of convergence
to something a priori existing, as well as an extrapolation it should be treated as a designing,
modeling of dependence for given sum-total of the points at plane. “Principle of maximum
simplicity”, or Occam’s Razor, may mean minimal distinction of model from polygonal line joining
points, then cubic splines appear as extremals for functional of the second derivative norm. Simple
idea of transferring of properties of the set of the points beyond the bounds of given interval is
being found as an only way: net of knots on specified segment is supplemented by a potentially
predictable point, “prognostic” spline on the augmented net is built, we must ensure minimum of
integral of quadratic deviation depending on ordinate of the add-on point as a parameter. The first
key moment is an expansion of “prognostic” spline in terms of system of fundamental splines,
a way is opened up to simple simultaneous linear equations expressing minimum condition,
analytical solution can be written in explicit form. In the second place, very essential problem is
end conditions for spline interpolating experimental points, as a rule, nothing is known about the
values of the first and, all the more, second derivatives at the moments of regular measurement,
for example, temperature of air. One can make the choice of end values of the first derivatives
not depending on interpreter if it is realized as a result of complementary optimization: the
main spline must “differ to the least degree” from a cubic polynomial. For uniform net and
end conditions of continuity of the third derivative of “prognostic” spline structural units of the
extrapolation algorithm are represented in form of sequence of expansions in terms of coordinates of the specified points, expansion coefficients are available analytically. In wrap-up result ordinate
of the forecasted point does not depend on uniform net spacing, that is very essential applying
to situations of forecasting concerning regular measurements when principal is not a value of
interval between measurements but its sameness
Keywords: interpolation, principles of extrapolation; cubic splines.