Definition of a self-affine fractal

Up to this point we have considered self-similar fractals; we now turn to self-affine fractals (Mandelbrot, 1985). Topography is an example of both. In the two horizontal directions topography is often self-similar; the ruler method can be applied to a coastline or to a contour on a topographic map to define a fractal dimension. The box-counting method can also be applied to a coastline, and square boxes are used to determine a fractal dimension. These are examples of self-similar fractals. Consider next the elevation of topography. Three examples of elevation h as a function of distance x along linear tracks are given in Figure 7.1. The vertical coordinate is statistically related to the horizontal coordinate but systematically has a smaller magnitude. Vertical cross sections of this type are often examples of self-affine fractals (Dubuc et al., 1989a).

A statistically self-similar fractal is by definition isotropic. In two dimensions defined by x and y coordinates the results do not depend on the geometrical orientation of the x- and y-axes. This principle was illustrated in Figure 2.8, where the box-counting method was introduced. The fractal dimension of a rocky coastline is independent of the orientation of the boxes. A formal definition of a self-similar fractal in a two-dimensional xy-space is that f(rx, ry) is statistically similar to f(x, y) where r is a scaling factor. This result is quantified by applications of the fractal relation (2.6).