Mathematical characterization of fractal river networks

Carraro and Altermatt1 performed an extensive analysis to study scaling properties of river networks. They concluded that: (1) branching “probability” is better termed as branching “ratio” for real river networks because it is not a probability; (2) branching ratio is a scale-dependent quantity as the value changes across spatial resolutions at which river networks are extracted (expressed as the threshold catchment area AT that initiates channels or pixel size l); (3) Optimal Channel Networks most accurately predicted the metapopulation stability and capacity. However, the supporting ground for these conclusions is either seriously flawed or inconclusive due to the improper use of branching probability, scale invariance, dimensions, and units.

better termed as branching "ratio" for real river networks because it is not a probability; (2) branching ratio 23 is a scaledependent quantity as the value changes across spatial resolutions at which river networks are ex 24 tracted (expressed as the threshold catchment area that initiates channels or pixel size ); (3) OCNs most 25 accurately predicted the metapopulation stability and capacity. However, the supporting ground for these 26 conclusions is either seriously flawed or inconclusive due to the improper use of branching probability, scale 27 invariance, dimensions, and units. 28 First, branching probability is certainly a probability by definition. In previous studies, branching proba 29 bility has been defined as a cumulative probability distribution of link length [km] in real river networks 30 (a "link" represents a river segment from one confluence to another). For example, Terui et al. 6,7 fitted an 31 exponential distribution to link length as ∼ Exp( ) ( is the rate parameter and refers to an individual 32 link) and estimated branching probability as = 1 − exp(− ) ( was set to be 1 [km] in Terui et al. 6,7 ). 33 Thus, [km −1 ] in this example is the probability of including a confluence (or an upstream terminal) per 34 unit river km. Therefore, the author's claim"…it is in fact improper to refer to a probability when analyzing 35 the properties of a realized river network…" is simply a misunderstanding. between and fell exactly on a 1:1 line ( Figure 1A), thus = . By definition, branching probability is 42 a monotonic increasing function of with a possible range of 0 − 1 ( Figure 1B); yet, they are not identical.

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Note that, in the following paragraphs, I use for my analysis to be consistent with the original article.

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Since branching probability is a simple transformation of , the arguments are translatable between the 45 two. at the 50 rivers. Note that and were estimated independently; nevertheless, the data points fell exactly on the theoretical relationship between and (the gray line), confirming the equality between and . Inlet zooms into the observed range of , which is denoted by the vertical broken lines.
Second, the term "scale invariance" was falsely used in their article. To explain this, let me consider an object 47 whose structural property is assessed under observation scale . For example, Mandelbrot 10 studied the 48 length of a coastline as multiples of a ruler with a unit length (the "observation scale"). In this practice, the which a physical quantity is measured, and a unit is a way to assign a number to a particular dimension to 68 make it relative. For a simple straight line, length is a dimension (dimension = 1), and a meter is a unit of 69 length. Throughout the article, the authors used the number of pixels to measure the total river length , 70 the total catchment area , and the threshold catchment area . There is no issue with using the number 71 of pixels as a unit. However, a problem in their article is that they obscured the dimensions of pixels. For 72 example, they made it very unclear that the river length has a dimension of pixel "length," while the total 73 catchment area and the threshold catchment area have a dimension of pixel "area." In particular, the 74 authors incorrectly defined as a dimensionless quantity (Methods) even though its unit is [pixel length −1 ].

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The improper use of units caused more serious problems in the analysis of "real" river networks; the authors 94 log 10 , = log 10 ( ) + log 10 , + (M0) log 10 , = log 10 ( ) + ( ) log 10 , + where is the error term that is properly weighted by Huber's function. Robust regression analysis is appro priate because it is robust to outliers caused by the small sample size (typical for large values). The first 96 model (M0) assumes the "universal" scaling with the single exponent across watersheds; i.e., the branching 97 ratio at all the 50 watersheds follows the same power law with the watershedspecific constant log 10 ( ) 98 ( ( ) is watershed for a data point ). In contrast, the second model (M1) assumes the "localized" scaling 99 with the watershedspecific exponent ( ) . I estimated the evidence ratio of the two models using the ap and assessing the rivers' rank in terms of , one observes that rivers that look more "branching" (i.e., have 108 higher ) than others for a given value can become less "branching" for a different value (Fig. 3)." 109 I also must note that I did not find any significant correlation between watershed area [km 2 ] and branching Codes and data are available at https://github.com/aterui/publicproj_fractalriver.