Complexity of the Geological Framework and Use of Mathematics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 14

Fig. 1.5 First derivative of hypsometric curve for Earth's surface (After Scheidegger 1963 ); and
hypsometric curve for continents only (Source: Agterberg 1974 , Fig. 11)
borehole. This implies that untransformed Sr content decreases exponentially in
this direction. The data points deviate from the trend line. Their residuals are
irregular but not randomly distributed along the best-fitting straight line. In order
to improve the closeness of fit, a polynomial of a higher degree could be fitted. It
would be shown by statistical inference ( cf . variance analysis, Chap. 2 ) that the fit of
the polynomial of degree 20 is “significantly” better. It is based on solving an
equation with 21 unknown coefficients whereas only two coefficients were needed
to obtain the straight line. It is unlikely that the fluctuations in the second curve are
meaningful. Polynomials of degree 10 or 30 show different patterns and, for lack of
data, one could not say which fit is best.
1.3.2 Elementary Differential Calculus
The purpose of this section and the next one is to show by means of simple examples
how elementary methods of calculus using differentiation and integration can be
applied for the analysis of various contour maps and cross-sections. Of course, these
methods have been implemented in various special-purpose software systems but a
good understanding of what is being done in software applications remains important.
Geophysicists have been interested in studying the surface of the Earth by using
hypsometric curves. A hypsometric curve for an area is a plot of the percentage of
area above a certain height level against height. The diagram on the right side
of Fig. 1.5 is the hypsometric curve for continents. The first derivative of the
hypsometric curve for the Earth's surface is shown on the left side of Fig. 1.5 .
These curves, which were fitted by hand, to observed heights, and their absolute or
relative frequencies can be used to determine quantities such as average height by