Simple statistical models for river networks

Physical Review Letters, 1993

Optimal channel networks (OCN's) obtained by minimizing the local and global rates of energy expenditure evolve automatically from arbitrary initial conditions to network configurations exhibiting fractal and multifractal statistics indistinguishable from those observed in nature. It is suggested that OCN's are spatial models of self-organized criticality and that natural fractal structures like river networks may arise as a joint consequence of optimality and randomness.

Journal of statistical …, 1998

Two distinct models for self-similar and self-affine river basins are numerically investigated. They yield fractal aggregation patterns following non-trivial power laws in experimentally relevant distributions. Previous numerical estimates on the critical exponents, when existing, are confirmed and superseded. A physical motivation for both models in the present framework is also discussed.

Geomorphology, 2001

We investigate the connection between surface elevation and the growth and scaling of river networks. Three planar Ž . models Scheidegger, Eden, and invasion percolation are first considered. These models develop aggregating networks according to stochastic rules but do not simulate erosion because the network growth is independent of the surface elevation. We show that none of these planar growth models produces scaling results consistent with observations for natural river basins. We then modify the models to include elevation, simulating the effects of fluvial erosion by enforcing the slope-area Ž relationship. The resulting configurations have scaling properties that still depend on the model Scheidegger, Eden, or . invasion percolation but are closer to natural river networks when compared with those from the planar growth rules. We conclude that inclusion of the vertical dimension in these three models is critical for explaining the formation and regularities of fluvial networks. q

Water Resources Research, 1999

The well‐known Norton's laws are empirical observations on how the means of measurements for river networks and basins vary with Horton‐Strahler order. It is now known that these laws are a consequence of an average‐sense self‐similarity in the bifurcation structure of river networks. In this paper we present a reformulation of Horton's laws which generalizes the familiar scaling of first moments, or means, to scaling of entire distributions. We also present extensive data analysis which supports this reformulation and show that this feature is also exhibited by Shreve's well‐known random topology model.

Water Resources Research, 1989

The analysis of large fiver networks obtained from digital elevation models has given insight into the variation of channel slope with scale. Investigators have recently suggested that channel slopes are self-similar with magnitude or area as a scaling parameter. Our data indicates otherwise; in particular, the variance of channel slope is larger than that predicted by simple self-similarity. This suggests multiscaling. The scaling exponent for the standard deviation is approximately half the corresponding exponent in the relationship of the slope mean to magnitude or area. A model for channel slopes based on a point process of elevation drops along the channel is presented. The model reproduces observed multiscaling properties when the density of elevation increments is related to area (or magnitude) as A-0. 1. Zhang, Y., Plotting positions of annual flood extremes considering extraordinary values, Water Resour. Res., 18(4), 859-864, 1982.

Journal of Hydrology, 1996

Informational entropy of river networks, as defined by , proved to be a useful tool in the interpretation of several properties exhibited by natural networks. In this paper, self-similar properties of river networks are taken as the starting point for investigating analogies and differences between natural networks and geometric fractal trees, comparing their variability entropy with parameters of both classes of networks. Attention is directed particularly to relations between entropy and Horton order and entropy and topological diameter of subnetworks. Comparisons of these relations for fractals and natural networks suggest that network entropy can contribute to clarify important points concerning self-similar properties of river networks. Moreover, the estimation of the fractal dimension of branching for natural networks can be considerably improved using the relation between entropy and Horton order throughout the network.

Water Resources Research - WATER RESOUR RES, 1992

This paper studies the fractal structure and properties of stream networks through a comparison between stream networks and loopless random aggregate trees. Five fractal dimensions are recognized for stream networks, namely, topological dimension (dt), minimum path dimension (dm), fractal dimension of stream networks as a whole (df), diffusion dimension (dw), and spectral dimension (ds). Mathematical definitions and derivations of the fractal dimensions are proposed, which provide a theoretical basis for some recognized empirical relationships in drainage basin geomorphology. Relationships among the fractal dimensions are examined, and the average values of the fractal dimensions for natural stream networks are estimated. The hydrological implications of the fractal dimensions of stream networks are also discussed.

A new geographic information system (GIS) numerical framework (NF), called CUENCAS, for flows in river networks is presented. The networks are extracted from digital elevation models (DEMs). The program automatically partitions a basin into hillslopes and channel links that are required to correspond to these features in an actual terrain. To investigate the appropriate DEM resolution for this correspondence, we take a high-resolution DEM at 10-m pixel size, and create DEMs at eight different resolutions in increments of 10 m by averaging. The extracted networks from 10-30 m remain about the same, even though there is a tenfold reduction in the number of pixels. By contrast, the extracted networks show increasing distortions of the original network from 40-90 m DEMs. We show the presence of statistical self-similarity (scaling) in the probability distributions of drainage areas in a Horton-Strahler framework using CUENCAS. The NF for flows takes advantage of the hillslope-link decomposition of an actual terrain and specifies mass and momentum balance equations and physical parameterizations at this scale. These equations are numerically solved. An application of NF is given to test different physical assumptions that produce statistical self-similarity in spatial peak flow statistics in a Horton-Strahler framework.

Water Resources Research, 2007

We analyze the distribution of the distances between tributaries of a given size (or of sizes larger than a given area) draining along either an open boundary or the mainstream of a river network. By proposing a description of the distance separating prescribed merging contributing areas, we also address related variables, like mean (or bankfull) flowrates, channel and riparian areas width, which are derived under a set of reasonable hydrologic assumptions. The importance of such distributions lies in their ecological, hydrologic and geomorphic implications on the spreading of species along the ecological corridor defined by the river network and on the propagation of infections due to water-borne diseases, in particular in view of exact theoretical predictions explicitly using the alongstream distribution of confluences carrying a given flow. Use is made here of real river networks, suitably extracted from digital elevation models, Optimal Channel Networks (OCNs) and exactly solved tree-like constructs like the Peano and the Scheidegger networks. The results obtained redefine theoretically in a coherent and general manner -and verify observationally -the distribution function of the above distances and thus provide the general probabilistic structure of tributaries in river networks. Specifically, we find that the probability of exceedence of the alongstream distance d of tributaries of size larger than a has the explicit form P (≥ d) = exp (−Cd/a H/(1+H) ) where C is a constant that depends on the choice of boundary conditions and H ≤ 1 is the Hurst exponent.

Three freely meandering rivers in the Amazon basin were analyzed for statistical scaling properties and oxbow lake size-frequency distributions. The rivers are the Purus (central Amazon, planform data), Juruá (central Amazon, planform and oxbow lake data), and Madre de Dios (Peruvian Amazon, oxbow lake data). Long reaches were found to be power- law scaling over more than two orders of mag- nitude. These river planforms are self-affine fractals. Oxbow lake data suggest that the lakes are sampled from a skewed hyperbolic (Pareto) size-frequency distribution. To examine the long-term behavior of freely meandering rivers, a deterministic continuum model of meandering rivers has been used for extensive simulations of free meandering mo- tion. The simulation outcomes are consistent with a dynamical state of self-organized criticality, which has the following characteristic behavior: (1) stationary mean sinuosity of the final state; (2) robustness, in the sense that the same final state is reached from any initial conditions; and (3) formation of a spatiotemporal fractal structure. Sensitivity tests showed that this behavior is not affected by valley confinement down to a valley width of 50 w (river width), and by chute cutoffs of mature meanders up to 3 w long, but the average sinuosity value reached in the final state is sensitive to valley width less than 100 w, and chutes longer than 1.5 w. Comparison with empirical data confirmed the validity of the simulations as models of river meandering. All tests found data and simulation results to be in close agreement.