A Note on Fractal Channel Networks
Abstract
This paper studies the relation between the structure of river networks and the features of their geomorphologic hydrologic response. A mathematical formulation of connectivity of a drainage network is proposed to relate contributing areas and the network geometry. In view of the connectivity conjecture, Horton's bifurcation ratio RB tends, for high values of Strahler's order Ω of the basin, to the area ratio RA, and Horton's length ratio RL equals, in the limit, the single‐order contributing area ratio Ra. The relevance of these arguments is examined by reference to data from real basins. Well‐known empirical results from the geomorphological literature (Melton's law, Hack's relation, Moon's conjecture) are viewed as a consequence of connectivity. It is found that in Hortonian networks the time evolution of contributing areas exhibits a multifractal behavior generated by a multiplicative process of parameter 1/RB. The application of the method of the most probable distribution in view of connectivity contributes new inroads toward a general formulation of the geomorphologic unit hydrograph, in particular generalizing its width function formulation. A quantitative example of multifractal hydrologic response of idealized networks based on Peano's construct (for which RB = RA = 4, RL = 2) closes the paper.
Citing Literature
Number of times cited according to CrossRef: 78
- Andrea Rinaldo, Marino Gatto, Ignacio Rodriguez-Iturbe, , River Networks as Ecological Corridors, 10.1017/9781108775014, (2020).
- Cesar R. Castillo, İnci Güneralp, Billy Hales, Burak Güneralp, Scale‐Free Structure of Surface‐Water Connectivity Within a Lowland River‐Floodplain Landscape, Geophysical Research Letters, 10.1029/2020GL088378, 47, 16, (2020).
- Sara Bonetti, Milad Hooshyar, Carlo Camporeale, Amilcare Porporato, Channelization cascade in landscape evolution, Proceedings of the National Academy of Sciences, 10.1073/pnas.1911817117, (201911817), (2020).
- Chengming Li, Wei Wu, Pengda Wu, Yong Yin, Zhaoxin Dai, Selection Method of Dendritic River Networks Based on Hybrid Coding for Topographic Map Generalization, ISPRS International Journal of Geo-Information, 10.3390/ijgi9050316, 9, 5, (316), (2020).
- Matthew Bartos, Branko Kerkez, Hydrograph peak-shaving using a graph-theoretic algorithm for placement of hydraulic control structures, Advances in Water Resources, 10.1016/j.advwatres.2019.03.016, 127, (167-179), (2019).
- Andrea Rinaldo, Marino Gatto, Ignacio Rodriguez-Iturbe, River networks as ecological corridors: A coherent ecohydrological perspective, Advances in Water Resources, 10.1016/j.advwatres.2017.10.005, 112, (27-58), (2018).
- Houda Boudhraâ, Christophe Cudennec, Hervé Andrieu, Mohamed Slimani, Net rainfall estimation by the inversion of a geomorphology-based transfer function and discharge deconvolution, Hydrological Sciences Journal, 10.1080/02626667.2018.1425801, 63, 2, (285-301), (2018).
- Joshua S. Rice, Ryan E. Emanuel, James M. Vose, The influence of watershed characteristics on spatial patterns of trends in annual scale streamflow variability in the continental U.S., Journal of Hydrology, 10.1016/j.jhydrol.2016.07.006, 540, (850-860), (2016).
- Samuele De Bartolo, Francesco Dell'Accio, Giuseppe Frandina, Giovanni Moretti, Stefano Orlandini, Massimo Veltri, Relation between grid, channel, and Peano networks in high‐resolution digital elevation models, Water Resources Research, 10.1002/2015WR018076, 52, 5, (3527-3546), (2016).
- Bruce L. Rhoads, Fluvial geomorphology, Progress in Physical Geography, 10.1177/030913339201600404, 16, 4, (456-477), (2016).
- Jonathan A. Czuba, Efi Foufoula‐Georgiou, Dynamic connectivity in a fluvial network for identifying hotspots of geomorphic change, Water Resources Research, 10.1002/2014WR016139, 51, 3, (1401-1421), (2015).
- Joakim Riml, Anders Wörman, Spatiotemporal decomposition of solute dispersion in watersheds, Water Resources Research, 10.1002/2014WR016385, 51, 4, (2377-2392), (2015).
- A. S. Al-Wagdany, Fractal properties of Indiana basins, Arabian Journal of Geosciences, 10.1007/s12517-014-1470-3, 8, 6, (4139-4146), (2014).
- A. Rinaldo, R. Rigon, J. R. Banavar, A. Maritan, I. Rodriguez-Iturbe, Evolution and selection of river networks: Statics, dynamics, and complexity, Proceedings of the National Academy of Sciences, 10.1073/pnas.1322700111, 111, 7, (2417-2424), (2014).
- Chandana Gangodagamage, Efi Foufoula‐Georgiou, Patrick Belmont, River basin organization around the main stem: Scale invariance in tributary branching and the incremental area function, Journal of Geophysical Research: Earth Surface, 10.1002/2014JF003304, 119, 10, (2174-2193), (2014).
- Joanna Fac-Beneda, Fractal structure of the Kashubian hydrographic system, Journal of Hydrology, 10.1016/j.jhydrol.2013.02.033, 488, (48-54), (2013).
- A. I. Mejia, Scaling of the Network Instantaneous Response Function from Basin Geomorphology and Hydraulic Geometry, Journal of Hydrologic Engineering, 10.1061/(ASCE)HE.1943-5584.0000760, 18, 12, (1786-1789), (2013).
- B Daya Sagar, Terrestrial Surface Characterization, Mathematical Morphology in Geomorphology and GISci, 10.1201/b14933, (123-188), (2013).
- B Daya Sagar, Morphological Shape Decomposition, Mathematical Morphology in Geomorphology and GISci, 10.1201/b14933, (215-260), (2013).
- M. Gatto, L. Mari, E. Bertuzzo, R. Casagrandi, L. Righetto, I. Rodriguez-Iturbe, A. Rinaldo, Generalized reproduction numbers and the prediction of patterns in waterborne disease, Proceedings of the National Academy of Sciences, 10.1073/pnas.1217567109, 109, 48, (19703-19708), (2012).
- Marco Marani, Andrea Rinaldo, Riccardo Rigon, Ignacio Rodriguez‐Iturbe, Geomorphological width functions and the random cascade, Geophysical Research Letters, 10.1029/94GL01933, 21, 19, (2123-2126), (2012).
- Riccardo Rigon, Andrea Rinaldo, Ignacio Rodriguez‐Iturbe, On landscape self‐organization, Journal of Geophysical Research: Solid Earth, 10.1029/93JB03601, 99, B6, (11971-11993), (2012).
- Ignacio Rodriguez‐Iturbe, Andrea Rinaldo, Riccardo Rigon, Rafael L. Bras, Ede Ijjasz‐Vasquez, Alessandro Marani, Fractal structures as least energy patterns: The case of river networks, Geophysical Research Letters, 10.1029/92GL00938, 19, 9, (889-892), (2012).
- Chandana Gangodagamage, Patrick Belmont, Efi Foufoula‐Georgiou, Revisiting scaling laws in river basins: New considerations across hillslope and fluvial regimes, Water Resources Research, 10.1029/2010WR009252, 47, 7, (2011).
- JarosŁaw Jasiewicz, Markus Metz, A new GRASS GIS toolkit for Hortonian analysis of drainage networks, Computers & Geosciences, 10.1016/j.cageo.2011.03.003, 37, 8, (1162-1173), (2011).
- Andrea Rinaldo, Gregor K. Vogel, Riccardo Rigon, Ignacio Rodriguez‐Iturbe, Can One Gauge the Shape of a Basin?, Water Resources Research, 10.1029/94WR03290, 31, 4, (1119-1127), (2010).
- Scott D. Peckham, New Results for Self‐Similar Trees with Applications to River Networks, Water Resources Research, 10.1029/94WR03155, 31, 4, (1023-1029), (2010).
- John D. Snell, Murugesu Sivapalan, On geomorphological dispersion in natural catchments and the geomorphological unit hydrograph, Water Resources Research, 10.1029/94WR00537, 30, 7, (2311-2323), (2010).
- Vladimir I. Nikora, On self‐similarity and self‐affinity of drainage basins, Water Resources Research, 10.1029/93WR02017, 30, 1, (133-137), (2010).
- Vladimir I. Nikora, Victor B. Sapozhnikov, River network fractal geometry and its computer simulation, Water Resources Research, 10.1029/93WR00966, 29, 10, (3569-3575), (2010).
- Keith R. Helmlinger, Praveen Kumar, Efi Foufoula‐Georgiou, On the use of digital elevation model data for Hortonian and fractal analyses of channel networks, Water Resources Research, 10.1029/93WR00545, 29, 8, (2599-2613), (2010).
- Riccardo Rigon, Andrea Rinaldo, Ignacio Rodriguez‐Iturbe, Rafael L. Bras, Ede Ijjasz‐Vasquez, Optimal channel networks: A framework for the study of river basin morphology, Water Resources Research, 10.1029/92WR02985, 29, 6, (1635-1646), (2010).
- Tom Beer, Michael Borgas, Horton's Laws and the fractal nature of streams, Water Resources Research, 10.1029/92WR02731, 29, 5, (1475-1487), (2010).
- Mauro Fiorentino, Pierluigi Claps, Vijay P. Singh, An entropy‐based morphological analysis of river basin networks, Water Resources Research, 10.1029/92WR02332, 29, 4, (1215-1224), (2010).
- Andrea Rinaldo, Ignacio Rodriguez‐Iturbe, Riccardo Rigon, Rafael L. Bras, Ede Ijjasz‐Vasquez, Alessandro Marani, Minimum energy and fractal structures of drainage networks, Water Resources Research, 10.1029/92WR00801, 28, 9, (2183-2195), (2010).
- Samuele De Bartolo, Francesco Dell’Accio, Massimo Veltri, Approximations on the Peano river network: Application of the Horton-Strahler hierarchy to the case of low connections, Physical Review E, 10.1103/PhysRevE.79.026108, 79, 2, (2009).
- Shlomo P. Neuman, Hortonian scaling and effect of cutoffs on statistics of self‐affine river networks, Water Resources Research, 10.1029/2009WR008124, 45, 12, (2009).
- Ignacio Rodriguez‐Iturbe, Rachata Muneepeerakul, Enrico Bertuzzo, Simon A. Levin, Andrea Rinaldo, River networks as ecological corridors: A complex systems perspective for integrating hydrologic, geomorphologic, and ecologic dynamics, Water Resources Research, 10.1029/2008WR007124, 45, 1, (2009).
- Roger Moussa, What controls the width function shape, and can it be used for channel network comparison and regionalization?, Water Resources Research, 10.1029/2007WR006118, 44, 8, (2008).
- C. Spence, J. Hosler, Representation of stores along drainage networks in heterogenous landscapes for runoff modelling, Journal of Hydrology, 10.1016/j.jhydrol.2007.09.035, 347, 3-4, (474-486), (2007).
- Matteo Convertino, Riccardo Rigon, Amos Maritan, Ignacio Rodriguez‐Iturbe, Andrea Rinaldo, Probabilistic structure of the distance between tributaries of given size in river networks, Water Resources Research, 10.1029/2007WR006176, 43, 11, (2007).
- CHRISTOPHE CUDENNEC, On width function-based unit hydrographs deduced from separately random self-similar river networks and rainfall variability: Discussion of “Coding random self-similar river networks and calculating geometric distances: 1. General methodology” and “2. Application to runoff simulations”, Hydrological Sciences Journal, 10.1623/hysj.52.1.230, 52, 1, (230-237), (2007).
- E. Bertuzzo, A. Maritan, M. Gatto, I. Rodriguez‐Iturbe, A. Rinaldo, River networks and ecological corridors: Reactive transport on fractals, migration fronts, hydrochory, Water Resources Research, 10.1029/2006WR005533, 43, 4, (2007).
- Bruno Lashermes, Efi Foufoula‐Georgiou, Area and width functions of river networks: New results on multifractal properties, Water Resources Research, 10.1029/2006WR005329, 43, 9, (2007).
- Peter Molnar, On geometrical scaling of Cayley trees and river networks, Journal of Hydrology, 10.1016/j.jhydrol.2005.02.035, 322, 1-4, (199-210), (2006).
- Daniel Campos, Joaquim Fort, Vicenç Méndez, Transport on fractal river networks: Application to migration fronts, Theoretical Population Biology, 10.1016/j.tpb.2005.09.001, 69, 1, (88-93), (2006).
- Andrea Rinaldo, Jayanth R. Banavar, Amos Maritan, Trees, networks, and hydrology, Water Resources Research, 10.1029/2005WR004108, 42, 6, (2006).
- Lea Tien Tay, B. S. Daya Sagar, Hean Teik Chuah, Allometric relationships between traveltime channel networks, convex hulls, and convexity measures, Water Resources Research, 10.1029/2005WR004092, 42, 6, (2006).
- Kelly K. Caylor, Salvatore Manfreda, Ignacio Rodriguez-Iturbe, On the coupled geomorphological and ecohydrological organization of river basins, Advances in Water Resources, 10.1016/j.advwatres.2004.08.013, 28, 1, (69-86), (2005).
- Daniel Campos, Vicenç Méndez, Reaction-diffusion wave fronts on comblike structures, Physical Review E, 10.1103/PhysRevE.71.051104, 71, 5, (2005).
- L. Chockalingam, B. S. Daya Sagar, Morphometry of network and nonnetwork space of basins, Journal of Geophysical Research: Solid Earth, 10.1029/2005JB003641, 110, B8, (2005).
- B. S. Daya Sagar, L. Chockalingam, Fractal dimension of non‐network space of a catchment basin, Geophysical Research Letters, 10.1029/2004GL019749, 31, 12, (2004).
- BETTY RICHARDS-PECOU, Scale invariance analysis of channel network width function and possible implications for flood behaviour, Hydrological Sciences Journal, 10.1080/02626660209492942, 47, 3, (387-404), (2002).
- Ing-Marie Gren, Georgia Destouni, Raul Tempone, Cost effective policies for alternative distributions of stochastic water pollution, Journal of Environmental Management, 10.1006/jema.2002.0569, 66, 2, (145-157), (2002).
- Thian Yew Gan, Qiang Wang, Michael Seneka, Correlation dimensions of climate subsystems and their geographic variability, Journal of Geophysical Research: Atmospheres, 10.1029/2001JD001268, 107, D23, (ACL 23-1-ACL 23-17), (2002).
- A. Maritan, R. Rigon, J. R. Banavar, A. Rinaldo, Network allometry, Geophysical Research Letters, 10.1029/2001GL014533, 29, 11, (3-1-3-4), (2002).
- Seth A. Veitzer, Vijay K. Gupta, Statistical self-similarity of width function maxima with implications to floods, Advances in Water Resources, 10.1016/S0309-1708(01)00030-6, 24, 9-10, (955-965), (2001).
- Merab Menabde, Seth Veitzer, Vijay Gupta, Murugesu Sivapalan, Tests of peak flow scaling in simulated self-similar river networks, Advances in Water Resources, 10.1016/S0309-1708(01)00043-4, 24, 9-10, (991-999), (2001).
- Brent M. Troutman, Thomas M. Over, River flow mass exponents with fractal channel networks and rainfall, Advances in Water Resources, 10.1016/S0309-1708(01)00031-8, 24, 9-10, (967-989), (2001).
- Achille Giacometti, Local minimal energy landscapes in river networks, Physical Review E, 10.1103/PhysRevE.62.6042, 62, 5, (6042-6051), (2000).
- James P McNamara, Douglas L Kane, Larry D Hinzman, An analysis of an arctic channel network using a digital elevation model, Geomorphology, 10.1016/S0169-555X(99)00017-3, 29, 3-4, (339-353), (1999).
- K. Helming, M. J. M. Römkens, S. N. Prasad, H. Sommer, Erosional development of small scale drainage networks, Process Modelling and Landform Evolution, 10.1007/BFb0009716, (123-145), (1999).
- A Flammini, F Colaiori, Exact analysis of the Peano basin, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/29/21/006, 29, 21, (6701-6708), (1999).
- Roger Moussa, Is the drainage network a fractal Sierpinski space?, Water Resources Research, 10.1029/97WR01899, 33, 10, (2399-2408), (1997).
- Vijay K. Gupta, Sandra L. Castro, Thomas M. Over, On scaling exponents of spatial peak flows from rainfall and river network geometry, Journal of Hydrology, 10.1016/S0022-1694(96)03088-0, 187, 1-2, (81-104), (1996).
- David G. Tarboton, Fractal river networks, Horton's laws and Tokunaga cyclicity, Journal of Hydrology, 10.1016/S0022-1694(96)03089-2, 187, 1-2, (105-117), (1996).
- B.S.Daya Sagar, Fractal relation of a morphological skeleton, Chaos, Solitons & Fractals, 10.1016/S0960-0779(96)00037-9, 7, 11, (1871-1879), (1996).
- Carlos E. Puente, Pablo A. Castillo, On the fractal structure of networks and dividers within a watershed, Journal of Hydrology, 10.1016/S0022-1694(96)03094-6, 187, 1-2, (173-181), (1996).
- Carmelo Agnese, Francesco D'Asaro, Giovanna Grossi, Renzo Rosso, Scaling properties of topologically random channel networks, Journal of Hydrology, 10.1016/S0022-1694(96)03095-8, 187, 1-2, (183-193), (1996).
- Pierluigi Claps, Mauro Fiorentino, Giuseppe Oliveto, Informational entropy of fractal river networks, Journal of Hydrology, 10.1016/S0022-1694(96)03092-2, 187, 1-2, (145-156), (1996).
- Vladimir Nikora, Richard Ibbitt, Ude Shankar, On Channel Network Fractal Properties: A Case of Study of the Hutt River Basin, New Zealand, Water Resources Research, 10.1029/96WR02396, 32, 11, (3375-3384), (1996).
- Pierluigi Claps, Giuseppe Oliveto, Reexamining the determination of the fractal dimension of river networks, Water Resources Research, 10.1029/96WR01942, 32, 10, (3123-3135), (1996).
- Carlos E. Puente, A new approach to hydrologic modeling: derived distributions revisited, Journal of Hydrology, 10.1016/S0022-1694(97)87978-4, 187, 1-2, (65-80), (1996).
- Roger Moussa, Claude Bocquillon, Fractal analyses of tree-like channel networks from digital elevation model data, Journal of Hydrology, 10.1016/S0022-1694(96)03093-4, 187, 1-2, (157-172), (1996).
- Pierluigi Claps, Mauro Fiorentino, Giuseppe Oliveto, The Most Probable Hydrologic Response of Fractal River Networks, Proceedings of the International Conference on Hydrology and Water Resources, New Delhi, India, December 1993, 10.1007/978-94-011-0389-3_14, (191-204), (1996).
- Tao Sun, Paul Meakin, Torstein Jøssang, Minimum energy dissipation model for river basin geometry, Physical Review E, 10.1103/PhysRevE.49.4865, 49, 6, (4865-4872), (1994).
- Tao Sun, Paul Meakin, Torstein Jøssang, A minimum energy dissipation model for drainage basins that explicitly differentiates between channel networks and hillslopes, Physica A: Statistical Mechanics and its Applications, 10.1016/0378-4371(94)00053-0, 210, 1-2, (24-47), (1994).
- A. Rinaldo, I. Rodriguez-Iturbe, R. Rigon, E. Ijjasz-Vasquez, R. L. Bras, Self-organized fractal river networks, Physical Review Letters, 10.1103/PhysRevLett.70.822, 70, 6, (822-825), (1993).
