Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 371

1 mile (1.6 km) fits in with the patterns for larger radii. This illustrates that
the so-called roll-off effect had not yet set in on for the shapes of contours on the
large-scale map. This example illustrates that measuring fractal dimension can have
value in quantitatively assessing degree of smoothing of geological features on
maps at different scales.
Lovejoy and Schertzer ( 2007 ) have reviewed early history of fractal geometry
pointing out that, early on, several mathematicians and geophysicists had pointed
out problems of differentiability and integrability in connection with some types of
natural phenomena. For example, Perrin ( 1913 ) wrote: “Consider the difficulty in
finding the tangent to a point of the coast of Brittany ... depending on the resolution
of the map the tangent would change. The point is that a map is simply a
conventional drawing in which each line has a tangent. On the contrary, an essential
feature of the coast is that

at each scale we guess the details which prohibit us
from drawing a tangent”. With respect to problems of integrability, Steinhaus
( 1954 ) stated: “The left bank of the Vistula when measured with increased precision
would furnish lengths ten, hundred, or even 1,000 times as great as the length read
off a school map. A statement nearly adequate to reality would be to call most arcs
encountered in nature as not rectifiable.” Among the early pioneers, Lovejoy and
Schertzer ( 2007 ) list Vening Meinesz ( 1951 ) who argued that the power spectrum
P ( k ) of the Earth's topography has the scaling form k β where k is a wave number
(

...

2, which according to Lovejoy and Schertzer (2007 ,
p. 466) is “close to the modern value

¼ 2 n frequency) and β ¼

2.1”. Vening Meinesz ( 1951 ) derived his
scaling model of the Earth's topography as follows.
Prey ( 1922 ) originally had developed the Earth's topography in terms of spher-
ical harmonics up to the 16th order. For the mathematics of spherical functions; see,
e.g., Freeden and Schreiner 2008 . Vening Meinesz ( 1951 ) normalized Prey's
coefficients after making a separation between continents and oceans. He showed
that mean square elevation ( y ) is approximately related to order x of harmonic
according to the power law relation y

β ¼

C · x β where C and ¼ β

are constants. This
kind of relationship is in accordance with the concept that the Earth's topography
can be described as a universal multifractal ( cf . Sect. 12.7 ). Orders of spherical
harmonics are analogous to wave numbers in a periodogram. On the original graph
(Vening Meinesz 1951 , Fig. 3), y multiplied by { n ·( n 1)} was related to

n according to a curve that was approximately horizontal for x > 2. Heiskanen

and Vening Meinesz ( 1958 ) argued that multiplication by { n ·( n 1)} is to be

preferred to multiplication by n 2 but this is a minor refinement only.
In 1957, at the request of Vening Meinesz, the Netherlands Geodetic Commis-
sion completed a new development in spherical harmonics of the Earth's topogra-
phy using improved data, especially on the topography of water-covered areas,
extending Prey's calculations from the 16th to the 31st order. Figure 10.3 shows the
resulting refinement of the 1951 analysis subsequently obtained by Vening Meinesz
( 1964 ). The new curve is subhorizontal for n

> 5, although there may exist an

increase in y for higher orders suggesting that is slightly less than 2, especially for

continental topography. Although the precise value of

β

is not known exactly, it is
clear that the Earth's topography on the continents as well as on the ocean floor
approximately displays the same type of fractal/multifractal behavior.

β