Conventional analytical techniques do not cope adequately with multiscale processes. Mandelbrot's concept of fractal dimension, a novel approach to multiscale phenomena, is applied to the problem of coral reef topography. A fractal dimension D = 1.9-2.0 for a contour on the reef slope is obtained. This contrasts strongly with the well established value of D = 1.2-1.3 for coastlines. We speculate that in common with the human lung, another living surface with a high D value, the coral reef slope is maximizing its contact with the surrounding medium.

We agree that the method of computing fractals described by Mark (1984) is both more appropriate and more informative than the method we used (Bradbury and Reichelt, 1983). In this note, we argue that it is the departure from strict self-similarity in real structures that gives them their ecological interest. We support these arguments with the analysis of new finer-resolution data from the previously studied coral reef which suggests that reefs are extremely smooth structures at the scale of coral colonies, and that this smoothness is reduced at both finer a n d coarser scales.David Mark (1984) has voiced what we ourselves have also realized. The method of computing fractals that h e describes (Mandelbrot, 1977;Goodchild, 1980) is both more appropriate and more informative than the method we used (Bradbury and Reichelt, 1983). However, we feel that there is a risk that Mark's discussion may leave some readers with the impression that lack of self-similarity invalidates the whole fractal approach to ecosystem studies. In this note we wish to emphasize that this is not so; it is precisely the lack of self-similarity that is of greatest ecological relevance and interest.Problems with variograms. The variogram approach to fractal estimation assumes that one is dealing with a continuous function W(t) of some independent variable t. In this context, the only possible interpretation of t is 'straight-line' distance along a transect, while W must represent the corresponding distance measured along the reef surface. Fractal estimates are then based on the variance of increments where T = step length used (Berry and Lewis, 1980). The chief practical difficulty is that in the reef transects, our steps with dividers were made along the reef surface, whereas T i n this context represents increments in the straight line distance. A further problem is Ecological interpretation of fractal dimensions. True fractals are abstractions; no real-world phenomenon is self-similar at all scales. Individual biological (and other) processes always operate over a restricted range of scales. However, the geometric properties of the patterns they produce, if extended over all possible scales, would define true fractal curves. Thus estimates of fractal dimension made over the restricted range of scales do reflect the geometric properties of the processes. Peaks and dips in the value D of this 'fractal index' (our term for the curve showing estimates of fractal dimension against scale) indicate shifts in the sources of biological pattern. The fractal index is thus a statistical tool for examining patterns and processes in coral communities.Estimates of D for real processes (whether obtained by the variogram or ratio method) may stray outside the range 1.0 to 2.0 in certain circumstances. Such transgressions should not b e interpreted as evidence that the application of fractal theory is not valid. They might, on the contrary, provide useful information about the phenomenon concerned, for they may indicate limits to the range over ...

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