ENLARGED SECTION 
7:3000 
  
  
  
  
  
  
  
  
  
PA 
Fig. 4. Selfsimilarity in Norway. 
This concept may be illustrated in the Norwegian terrain. Figure 4 shows 
two drawings of the same profile at different scales. Both these drawings 
look very similar in their structure and their scale or scale relation can- 
not be derived by mere inspection of the drawings. 
Based on this definition of selfsimilarity, Mandelbrot introduced ran- 
dom functions, called "Fractals", which possess this property and from 
which - under mild restrictions - the terrain models (3), (5) and (8) 
can be deduced. These Fractals are in principle deduced by a yER_summa- 
tion (integration) of independent random variables. Although we are only 
familiar with L = and 2 order summations (cf. the error theory of strip 
triangulation; Vermeir, 1954; Ackermann, 1965), a fractal summation or 
integration of order Y(Y noninteger) may be defined by extending the clas- 
sical definition of the n"! integral to noninteger values n - Y (Holmgreen- 
Riemann-Liouville fractional integral, cf. Levy, 1953) 
20) xw) [x37 ants) (10) 
9 
with dB normally distributed, independent and equally accurate increments 
(white noise) and K(Yy) a constant depending on Y. The constant Y relates 
to a and B by Y = +(8+1)= # a. The concept of Fractals is recently applied 
very intensively to model and classify terrain forms, cf. for example Há- 
kanson, 1978; Shelberg et al 1982; Goodchild 1980, and Mandelbrot 1975. 
€ 
Much more attention for this concept is expected in future. 
= 
In analogy with fractal integration one may define fractal differentiation, 
(Y) er) 
à is dh 
de À Li LT