The duality of fractals: roughness and self-similarity

The fractal dimension (D (HB)) is an interesting metrics because it is supposed to quantify by a single value, scale independence and roughness of ecological objects. However, we show here that those two properties may be quantified by a single dimension only in some specific cases. In general, a non-integer D (HB) quantifies only the roughness, and self-similarity needs to be evidenced or postulated by other means. Second, we revisit some aspects of the practical estimation of D (HB). We recommend the use of madogram instead of variogram for estimations based on geostatistics. We propose a simplification of its estimation for 2D fields and discuss its possible relationship with self-similarity. We finally underline the problem of scale and resolution. Field data recorded during a scientific acoustic survey on the North Sea herring are used for illustrations. The paper concludes on a synthesis of practical recommendations to ecologists when using fractal dimension.