Fractal techniques and the surface roughness of talus slopes: Reply

Linear plots of log N against log G, where N is the number of steps of length G to span a transit, are conventionally used as evidence that geomorphic surfaces are self-similar fractals (i.e. the surfaces have a constant fractal dimension). In this study 42 transits on talus slope surfaces in Niagara and Letchworth Gorges, western New York, are investigated to ascertain whether they are self-similar. Log N-log G plots, which r2 values in excess of 0·99 suggest are linear, are found upon more rigorous testing to be curvilinear. It is concluded that the talus slope surfaces are not self-similar, and that log N-log G plots are relatively insensitive to departures from self-similarity. The curvilinearity of the log N-log G plots is explained with the aid of a randomized square-wave model of the talus slope surfaces. This model is used to extend the range of measurement beyond that which was possible in the empirical analysis. The negative of the gradient of the log N -log G relation at a point is the fractal dimension D. Measurements made upon the randomized square-wave model indicate that the relation between D and scale of measurement has an asymmetrical wave shape with a peak (i.e. maximum D) where the scale of measurement is equal to the characteristic scale of roughness. In other words the value of D for a surface is not absolute but depends on the scale of measurement relative to the scale of roughness. Linear regression analysis reveals that at the scale of measurement employed in this study, D is positively correlated with particle size. This is because the values of D fall on the right-hand tail of the wave-shaped relation between D and scale of measurement. Transects (normal to the direction of slope) are found to have higher values of D than profiles (parallel to the direction of slope), and this is explained in terms of particle orientation, shape, and juxtaposition. Because D varies continuously with scale of measurement, there are considerable difficulties in using it to characterize and compare the surface roughness of talus slopes. Generalizing from talus slopes to other ground surfaces, it is evident that to the extent that any natural ground surface has a characteristic scale of roughness, it will depart from self-similarity, and D should be used with caution in quantifying the roughness of the surface. Geomorphologists are therefore urged to be more rigorous in their testing of self-similarity before employing D to characterize surface roughness.

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