Editing Fractal landscape (section)

==Behavior of natural landscapes==
[[Image:FractalLandscape.jpg|thumb|A fractal landscape rendered in [[Terragen]].]]
[[File:Fractal terrain texture.jpg|thumb|Computer generated fractal terrain using [[Perlin noise]] with [[Adobe Photoshop]] and Terragen.]]
[[File:BlueRidgePastures.jpg|thumb|Computer-generated fractal wooded hills using [[Visual Nature Studio]].]]
Whether or not [[natural landscape]]s behave in a generally fractal manner has been the subject of some research. Technically speaking, any surface in three-dimensional space has a [[topological dimension]] of 2, and therefore any [[Fractal_dimension#Fractal_surface_structures|fractal surface]] in three-dimensional space has a [[Hausdorff dimension]] between 2 and 3.<ref>[[#Lewis|Lewis]]</ref> Real landscapes however, have varying behavior at different scales. This means that an attempt to calculate the 'overall' fractal dimension of a real landscape can result in measures of negative fractal dimension, or of fractal dimension above 3. In particular, many studies of natural phenomena, even those commonly thought to exhibit fractal behavior, do not do so over more than a few orders of magnitude. For instance, Richardson's examination of the western coastline of Britain showed fractal behavior of the coastline over only two orders of magnitude.<ref>[[#Richardson|Richardson]]</ref> In general, there is no reason to suppose that the geological processes that shape terrain on large scales (for example [[plate tectonics]]) exhibit the same mathematical behavior as those that shape terrain on smaller scales (for instance, [[soil creep]]).

Real landscapes also have varying statistical behavior from place to place, so for example sandy beaches don't exhibit the same fractal properties as mountain ranges. A [[fractal function]], however, is statistically stationary, meaning that its bulk statistical properties are the same everywhere. Thus, any real approach to modeling landscapes requires the ability to modulate fractal behavior spatially. Additionally, real landscapes have very few natural minima (most of these are lakes), whereas a fractal function has as many minima as maxima, on average. Real landscapes also have features originating with the flow of water and ice over their surface, which simple fractals cannot model.<ref>[[#Musgrave1|Ken Musgrave, 1993]]</ref>

It is because of these considerations that the simple fractal functions are often inappropriate for modeling landscapes. More sophisticated techniques (known as 'multi-fractal' techniques) use different fractal dimensions for different scales, and thus can better model the frequency spectrum behavior of real landscapes<ref>[[#Jense|Joost van Lawick van Pabst et al.]]</ref>