Comparison of gliding box and box-counting methods in river network analysis

Abstract. We use multifractal analysis to estimate the Rényi dimensions of river basins by two different partition methods. These methods differ in the way that the Euclidian plane support of the measure is covered, partitioning it by using mutually exclusive boxes or by gliding a box over the plane.

Images of two different drainage basins, for the Ebro and Tajo rivers, located in Spain, were digitalized with a resolution of 0.5 km, giving image sizes of 617×1059 pixels and 515×1059, respectively. Box sizes were chosen as powers of 2, ranging from 2×4 pixels to 512×1024 pixels located within the image, with the purpose of covering the entire network. The resulting measures were plotted versus the logarithmic value of the box area instead of the box size length.

Multifractal Analysis (MFA) using a box counting algorithm was carried out according to the method of moments ranging from −5<q<5, and the Rényi dimensions were calculated from the log/log slope of the probability distribution for the respective moments over the box area. An optimal interval of box sizes was determined by estimating the characteristic length of the river networks and by taking the next higher power of 2 as the smallest box size. The optimized box size for both river networks ranges from 64×128 to 512×1024 pixels and illustrates the multiscaling behaviour of the Ebro and Tajo. By restricting the multifractal analysis to the box size range, good generalized dimension (D_q) spectra were obtained but with very few points and with a low number of boxes for each size due to image size restrictions. The gliding box method was applied to the same box size range, providing more consistent and representative D_q values. The numerical differences between the results, as well as the standard error values, are discussed.