Robustness of fractal dimension estimators for vector talweg network characterization
Résumé
The fractal approach is often used to characterize natural objects. Numerous studies have focused on fractal analysis of river networks in particular. However, only few papers discuss the estimation methods and the uncertainty of the main fractal indicator, the fractal dimension. Firstly, the distinction between infinite mathematical fractal and nature fractal should be taken into account to estimate fractal dimension. Moreover, the networks are most of the time integrated in GIS database and represented by vector object. This type of representation possesses its own properties and we think that the impact on fractal measure should be evaluated. In this context, the work we propose aims at testing the robustness of different fractal dimension estimators for the characterization of vector talweg networks. We focus on the two most popular estimators: a classical estimator for river networks, based on a topological approach with the Horton-Strahler ratios, and the box-counting dimension, based on a geometric approach. A third estimator, the less known correlation dimension, also based on a geometric approach, offers interesting possibility for calculating a stable fractal indicator, in particular in the case of a reduced number of stream-segments. These methods are applied on both virtual (such as Scheidegger network), and actual vector networks. The actual case is a network extracted from a high resolution DTM of the Draix badlands in the French Alps. Three main methodological results can be highlighted: 1- the study of virtual network contributes to the assessment of the estimators relevance, according to the network branching structure; 2- an empirical fractal domain must be determined on the Log-Log curve with an objective method to estimate fractal dimensions that can be compared; 3- the observation of uncertainty of the fractal dimension is necessary for any valid comparison.