A Fractal Description of Fluvial Networks in Chile: a Geography not as Crazy as Thought

Chilean geography is highly variable, not only from a climatic and hydrological point 1 of view, but also a morphological one, showing unpredictable natural patterns with marked 2 contrasts throughout the country, for which sometimes it is considered as a "crazy" geography. 3 In this paper we have investigated this apparent disorganized character by exploring the fractal 4 properties of fluvial networks extracted from basins distributed across the continental territory. 5 Analytical and semi-empirical methods were applied, finding striking patterns of organization 6 in the distributions of Horton parameters and the fractal dimension of the drainage networks. 7 Fractal dimension reveals to be quite dependent on the drainage area of each unit, showing clear 8 groupings by tectonic and climatological factors. Such dimension reveals to be an important 9 geomorphic parameter, if not the only one able to capture the real morphology of a fluvial network. 10 From our results and despite the diversity of landforms, hydrological, climatic and tectonic 11 conditions, Chilean’s geography is perhaps not as crazy and disorganized as believed. 12

where R A,ω , R B,ω and R L,ω denote the area, bifurcation and length ratios of the 41 streams, respectively. The parameters L ω , N ω and A ω correspond to the length and 42 number of drains of the sub-catchments of area A ω , where ω = 2, ..., Ω is the order of the 43 streams, whose maximum value is Ω [9]. An striking observation from Horton is that 44 for high enough dense networks, the parameter R q,ω shows almost convergent values. 45 Such observation can be interpreted, a priori, as an intimate connection between fractals 46 and self-similar trees [10,11]. These convergent values can be well represented by the 47 average of each ratio, let's say R q = 1 Ω ∑ Ω ω=1 R q,ω , for q = A, B, L. The ratios R A , R B , R L 48 can be considered characteristic parameters for a given fluvial network. 49 Inspired on Hack's law [12], Mandelbrot  and l the length of the mainstream, respectively. Based on this relation and assuming the 52 constancy of the drainage density across the network, Feder [6] improved this definition 53 proposing the relationship d = 2ln(R L )/ln(R B ). Rosso et al. [13] extended this result 54 suggesting the law d = max(1, 2ln(R L )/ln(R A ), in agreement with the topological 55 minimum path dimension deduced by Liu [14]. One of the problems of previous formulations is the need of connecting them with 58 the fractal dimension of the entire network. La Barbera and Rosso [2] derived a law 59 for this situation, assuming that Horton parameters holds through the whole network 60 across different scales: According to [2], Eq.2 leads to values in the range 1.5 < D 1 < 2.0 (1.67 in average).

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The authors argued about the impossibility of reaching values close to 2, car fluvial 63 networks show decreasing drainage densities for increasing contributing areas. Tarboton 64 et al. [15] states that [2] assumes that mainstreams identify with topological objects of 65 dimension 1. However, many real streams show meandering patterns where d ̸ = 1. 66 Tarboton et al. [15] proposed the following law to estimate the fractal dimension in this 67 situation: Eq.3 to values in the range D 2 < 2 ( [3,15,16]) . This is coherent with observations made 70 at larger scales where it is reasonable to assume that streams drain each point of the 71 basin as pointed by [17]. In an interesting exchange, La Barbera and Rosso [18] refuted 72 the conclusion of Tarboton proposing a modification of Eq.3 as follows: where β = 1 2−d . Also from a theoretical point of view, Liu [14] worked with infinite The tectonic segments of Chilean territory, ordered from north to south, can be also observed.
Although Eqs.2-5 shows to be a practical approach to describe the fractal dimension 79 of stream networks, they requires a huge amount of geographic information to obtain 80 the morphometric parameters. On the other side, there are several limitations of these 81 methods that deserves to be considered. One of them is related with the self-similarity 82 hypothesis. This assumption has been objectively refuted by Kirchner [19]), giving 83 rise to another approaches based on self-affine attributes of networks [17,[20][21][22][23]. This 84 self-affine character naturally arises from the morphological anisotropy of the network 85 and the combination of the different tectonic processes that constrains the diffusion of 86 the streams over the time ( [24] to Rodriguez-Iturbe and Rinaldo [27], the fractal dimension of the network arising from 104 this method can be calculated from the next relationship: In order to compare the morphological characteristics of each network, we have also 106 estimated some geomorphic parameters typically used to describe the characteristics of 107 drainage basins [5]. This is, the shape index F, the circularity index C and the elongation 108 factor E, respectively: where L is a characteristic length of the watershed (usually the longest distance of 110 the basin) and P its perimeter. We also introduce the drainage density (ρ) of the network.

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This parameter is provided by GRASS-GIS and defined as follows: following the criteria proposed by Strahler [28]. These parameters are presented in Table   124 1.
125 Table 1. Morphometric parameters of large-basins ordered from north to south and by tectonics influence. (NPN: Nazca plate-north, NPFS: Nazca plate-flat slab,NPS: Nazca plate-south, AP: Antarctic plate). Here A is the drainage area, i m the mean slope, H m the mean elevation of the unit and Ω the maximum order of each network according to Strahler's hierarchical ordering. Drainage density and geomorphic indexes were also included. (The superscript () * denotes the basins chosen for analysis at sub-basin level).

Tectonics
Basin In order to explore the fractal properties of the basins at a finer scale, a sub-set of characteristics of these sub-networks can be observed in Table 2.
137 Table 2. Morphometric parameters of large basins, with A the drainage area, i m the mean slope, H m the mean elevation of the unit and Ω the maximum order of the network according to Strahler's hierarchical ordering. Geomorphic indexes were also included and the drainage density, as well.

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The basins located between the Copiapo and Aconcagua rivers (CO to AC in Figure   173 2) are quite similar both in their network patterns and their drainage density. They   However, the parameter E shows a significant increase falling into the range 0.51 ≤ E ≤ 215 0.60. All of these indexes shows a dramatic departure from the values reported in Table   216 1. A similar conclusion can be deduced with respect to ρ. Despite the similar order of 217 the watersheds (Ω), the differences between Tables 1-2 are non-negligible.

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In this section the interplay between the fractal dimension and some morphological 246 properties of the networks is explored. From data presented in Table 3, we have built the curves shown in Figure 5. In (a) three regions were drawn to separate the slope regimes from low to high. In (b) two regions were drawn, one for ρ ≤ 1.24 (very coarse drainage networks) and another one, for ρ > 1.24 (coarse drainage networks). Loa, Elqui, Valdivia and Baker basins were indicated to remark the differences between large and sub-networks.  area-dependent parameter. Curiously, this fit has been also obtained from measurements 292 of fractal dimension in urban environments(see [34]).   Taking these elements into account, both growth regimes can be roughly described 323 by the next relationships:

Discussion
where κ 1 , κ 2 , κ 3 are fitting constants. When q = L we obtain (κ 1 , κ 2 , κ 3 ) = (0.06 ± 325 0.02, 0.99 ± 0.1, 1.21 ± 0.02), but for q = A we obtain instead (κ 1 , κ 2 , κ 3 ) = (0.13 ± 326 0.04, 1.18 ± 0.1, 1.60 ± 0.02). Notice that κ 1 is small, but not small enough to neglect the 327 term κ 1 log(A) at all. In any of both regimes, significant contrast arise when considering 328 R L or R A in the determination of ζ and thus, the fractal dimension D k for k = 1, ...4. Such 329 saturation effect observed for very large areas emphasises the idea that fractal dimension 330 of fluvial networks cannot growth indefinitely. There must be a limit for this index 331 according to the full-filling space concept proposed by [24], that is, fractal dimension Then, serious differences are expected when comparing both methods of calculation as 345 shown in Figure 8. This effect was also suggested by [35]. are objects whose behaviour is far from "pure" fractals typically reported in literature.

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Another striking element arises when analysing sub-networks from the North of 361 the country. In this case, dataset shows an important dispersion suggesting the influence 362 of tectonics control in our measurements, particularly when we discuss about this scale-363 invariance property. This competition between tectonics and water-erosive effects have 364 been discussed by [36] and it is inherent to the fractal dimension of a fluvial network.  (Figures 5a-5b). In this last case, however, D F strongly concentrates in the range shows a tendency to group in regimes different from those observed for larger units.

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This departure could be explained because of the detailed morphological information 377 obtained at a finer measurement scale, not possible to recover when analysing the same 378 morphological unit with a larger "ruler".

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Our results also shows that the determination of the fractal dimension is quite sen- for such purpose, that is: whereD i is the average value of the sub-networks extracted from a given large-  exists. Such interplay can be observed in Figure 10. Climatic and tectonics effects on 410 data were also qualitatively indicated in the same graph.   previously by [34] from measurements on urban environments.

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Finally, all the results presented in this paper invite to consider the fractal dimension 472 as a rich geomorphic parameter revealing the self-affine character of fluvial networks as 473 inferred from the the results presented in Table 4. This parameter could definitely help (e.g. Figures 6,9,10). These striking patterns of organisation seems to be camouflaged into 480 each basin, inviting to consider Chilean territory not as so "crazy", nor so unpredictable 481 as believed.

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Chilean landscapes could be taken as a powerful natural laboratory to test the 483 validity of mprphometric and fractal scaling-laws, widely disseminated in the literature 484 (e.g. Eqs. [2][3][4][5]. Most of these findings deserve to be analysed with more detail, but due 485 to the scale of such work it requires to use a methodology different from that reported 486 here. In this context, a multifractal analysis is an interesting tool to conduct this process 487 considering the complex mechanisms involved on the generation and evolution of a 488 fluvial network. This is a very challenging task will definitely be the goal of a next report.