In the applet below you see 1D realisation of white and
correlated noise with equidistant step in x. Independent random
points uniform distribution on interval (-1, 1) make the white
noise (the blue curve). Correlated random points Vi (the red
curve) are obtained by averaging of white noise in radius Rc
sphere, i.e. kernel Ko is used Vi = ∑j= -Rc,Rc fi+j To get a 2D
fractal noise (mountain) you take an elastic string (see Fig.1),
then a random vertical displacement is applied to its middle
point. The process is repeated recursively to the middle point
of every new segment. The random displacement decreases m times
each iteration (usually m = 2 are used). Using Fourier
transformation for V(r), K(r), f(r) g(k) = ∫ g(r) eikr dr , we
get for V(k) V(k) = K(k) f(k) . I.e. averaging (*) means the
white noise filtration by means of a filter with bandwidth K(k).
The bandwidths for the two used filters are shown in Fig.3 (for
Rc = 1) KG(k) ~ exp[-(Rck)2], Ko ~ sin(Rck)/k. At last 2D
correlated random landscape. To get