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

Abstract

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.

Why is achieving good ecological outcomes in rivers so difficult?

“…Rivers are well known to exhibit fractal structural properties (e.g. De Bartolo, Veltri & Primavera, 2006; Saa et al. , 2007; Rodriguez‐Iturbe et al.…”

Texture appearance characterization of pre-sliced pork ham images using fractal metrics: Fourier analysis dimension and lacunarity

“…The Fourier power spectrum method conveniently represents the statistical nature of real images by describing them in terms of a fractional Brownian motion model (Dougherty & Henebry, 2001). Lacunarity computation using the gliding box algorithm has the advantage of the large sample size that usually leads to more consistent statistical results (Saa, Gascó, Grau, Antón, & Tarquis, 2007). Furthermore, lacunarity is suitable to describe the spatial distribution of real data sets, because translational invariance can also be a property of non-fractal sets.…”

Fractal characteristics of the main channel of Yellow River and its relation to regional tectonic evolution

“…According to the existing multifractal theory, the characterization of a geometrical multifractal generally requires not a single dimension but a sequence of generalized fractal dimensions (D q ). Saa et al (2007) used multifractal analysis to estimate the generalized fractal dimensions (D q ) of river basins by the box-counting and gliding-box methods. The D q values obtained for Tajo and Ebro river networks images with distinctive spatial arrangement were analyzed by Saa et al (2007).They pointed out that, in terms of modeling, it is important to characterize the multiscale heterogeneity of river networks in a useful way.…”

“…Saa et al (2007) used multifractal analysis to estimate the generalized fractal dimensions (D q ) of river basins by the box-counting and gliding-box methods. The D q values obtained for Tajo and Ebro river networks images with distinctive spatial arrangement were analyzed by Saa et al (2007).They pointed out that, in terms of modeling, it is important to characterize the multiscale heterogeneity of river networks in a useful way. However, the D q values obtained by the box-counting or gliding-box methods are generally unable to describe the spatial variability in the fractal behaviors of river channel geometry.…”