Abstract
We review four quantitative models of drainage networks. The main characteristic of channel networks is selfsimilarity. But networks are not deterministic, there is natural variability that needs to be taken into account. The first model is simple, it takes into account variability and can be tested. Nevertheless, it does not reproduce observations adequately because self-similarity is not explicitly considered in the model construction. The second model does consider self-similarity in the construction but it does not take into account variability and it cannot be tested. The third model considers both self-similarity and variability. The fourth model defines a firmer theoretical basis butit also needs testing against observations. We conclude stressing the need for a thorough test of the model against observations and analysing implications for hydraulic geometry and hydrologyc similarity. © 2018. Acad. Colomb. Cienc. Ex. Fis. Nat.
References
Barenblatt, G. I. (1996). Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press.
Barenblatt, G. I. (2003). Cambridge University Press.
Dawdy, D. R. (2007). Prediction versus understanding (the 2006 Ven Te Chow lecture). J Hydrol Eng. 12 (1): 1-3.
Dawdy, D. R., Griffis, V. W., Gupta, V. K. (2012). Regional floodfrequency analysis: How we got here and where we are going. J Hydrol Eng. 17 (9): 953-959.
de Vries, H., Becker, T., Eckhardt, B. (1994). Power law distribution of discharge in ideal networks. Water Resour Res. 30 (12): 3541-3543.
Dodds, P. S. and Rothman, D. H. (1999). Unified view of scaling laws for river networks. Phys Rev. 59 (5): 4865.
Eagleson, P. S. (1970). Dynamic Hydrology. McGraw-Hill, New York.
Feller, W. F. (1968). An introduction to probability theory and its applications Vol. 1. Wiley, New York, third edition.
Feder, J. (1968). Fractals. Plenum Press, New York.
Furey, P. R., Gupta, V. K., Troutman, B. M. (2013). A top-down model to generate ensembles of runoff from a large number of hillslopes. Nonlinear Process Geophys. 20 (5): 683-704.
Gibbings, J. C. (2011). Dimensional analysis. Springer.
Gupta, V. (2016). Scaling theory of floods for developing a physical basis of statistical flood frequency relations. In Oxford Research Encyclopedia of Natural Hazard Science. Oxford University Press. Retrieved 16 Nov. 2018, from http://naturalhazardscience.oxfordre.com/ view/10.1093/acrefore/9780199389407.001.0001/acrefore- 9780199389407-e-301
Gupta, V. K. &Mesa, O. J. (2014). Horton laws for hydraulic– geometric variables and their scaling exponents in selfsimilar Tokunaga river networks. Nonlinear Process Geophys. 21 (5): 1007-1025.
Gupta, V. K., Mesa, O. J., Waymire, E. C. (1990). Tree-dependent extreme values: The exponential case. J Appl Probab. 27 (1): 124-133.
Gupta, V. K., Troutman, B. M., Dawdy, D. R. (2007). Towards a nonlinear geophysical theory of floods in river networks: an overview of 20 years of progress. In Tsonis, A. A. and Elsner, J. B., editors, Nonlinear Dynamics in Geosciences, p. 121-151. Springer, New York, NY 10013, USA.
Gupta, V. K. & Waymire, E. C. (1998). Spatial variability and scale invariance in hydrologic regionalization. In Sposito, G., editor, Scale Dependence and Scale Invariance in Hydrology, p. 88-135. Cambridge University Press, London.
Hack, J. T. (1957). Studies of longitudinal stream profiles in Virginia and Maryland. USGS Professional Paper. 294 (B): 1-97.
Horton, R. E. (1945). Erosional development of streams and their drainage basins; hydrophysical approach to quantitative morphology. Geological Society of America Bulletin. 56 (3): 275-370.
Kovchegov, Y. & Zaliapin, I. Horton law in self-similar trees. Fractals. 24 (02): 1650017, 2016.
Kovchegov, Y. & Zaliapin, I. Horton self-similarity of kingman’s coalescent tree. Ann Inst H Poincare Probab Statist. 53 (3): 1069-1107, 08 2017. doi: 10.1214/16-AIHP748. https://doi. org/10.1214/16-AIHP748.
Kovchegov, Y. & Zaliapin, I. Tokunaga self-similarity arises naturally from time invariance. Chaos. 28 (4): 041102, 2018.
La Barbera, P. and Rosso, R. (1989). On the fractal dimension of stream networks. Water Resour Res. 25 (4): 735-741.
Langbein, W. B., et al. (1947). Topographic characteristics of drainage basins. Water Supply Paper 968-C. US Government Printing Office.
Leopold, L. B. (1994). A View of the River. Harvard University Press.
Leopold, L. B., Wolman, M. G., Miller, J. P. (1964). Fluvial Processes in Geomorphology. W. H. Freeman, San Francisco.
Mantilla, R. (2007). Physical basis of statistical scaling in peak flows and stream flow hydrographs for topologic and spatially embedded random self-similiar channel networks. PhD thesis, University of Colorado at Boulder.
Mantilla, R. & Gupta, V. K. (2005). A GIS framework to investigate the process basis for scaling statistics on river networks. IEEE Geosci Remote S. 2 (4): 404-408.
Mantilla, R., Gupta, V. K., Troutman, B. M. (2012). Extending generalized Horton laws to test embedding algorithms for topologic river networks. Geomorphology. 151-152: 13-26.
Mantilla, R., Mesa, O. J., Poveda, G. (2000). Análisis de la ley de Hack en las cuencas hidrográficas de Colombia. Avances en Recursos Hidráulicos. 1 (7): 1-18.
Mantilla, R., Troutman, B. M., Gupta, V. K. (2010). Testing statistical self-similarity in the topology of river networks. J Geophys Res. 115: F03038.
Mcconnell, M. & Gupta, V. K. (2008). A proof of the Horton law of stream numbers for the Tokunaga model of river networks. Fractals. 16 (03): 227-233.
Mesa, O. J. (1986). Analysis of Channel Networks Parameterized by Elevations. PhD thesis, University of Mississippi.
Mesa, O. J. & Gupta, V. K. (1987). On the main channel lengtharea relationship for channel networks. Water Resour Res. 23 (11): 2119-2122.
Mueller, J. E. (1973). Re-evaluation of the relationship of master streams and drainage basins: Reply. Geol Soc Am Bull, 84: 3127-3130.
Peckham, S. (1995a). New results of self-similar trees with applications to river networks. Water Resour Res. 31 (4): 1023-1029.
Peckham, S. (1995b). Self-Similarity in the Three-Dimensional Geometry and Dynamics of Large River Basins. Ph. D. thesis, Univ. of Colo., Boulder.
Peckham, S. & Gupta, V. K. (1999). A reformulation of Horton’s laws for large river networks in terms of statistical selfsimilarity. Water Resour Res. 35 (9): 2763-2777.
Rodríguez-Iturbe, I. & Rinaldo, A. (2001). Fractal River Basins: Chance and Self-Organization. Cambridge University Press.
Shreve, R. L. (1966). Statistical law of stream numbers. J Geol. 74: 17-37.
Shreve, R. L. (1967). Infinite topologically random channel networks. J Geol. 75: 178-186.
Shreve, R. L. (1969). Stream length and basin areas in topologically random channel networks. J Geol. 77: 397-414.
Shreve, R. L. (1974). Variation of mainstream length with basin area in river networks. Water Resour Res. 10 (6): 1167-1177.
Sivapalan, M., Takeuchi, K., Franks, S. W., Gupta, V. K., Karambiri, H., Lakshmi, V., Liang, X., McDonnell, J. J., Mendiondo, E., O’Connell, P. E., et al. (2003). IAHS decade on predictions in ungauged basins (PUB), 20032012: Shaping an exciting future for the hydrological sciences. Hydrolog Sci J. 48 (6): 857-880.
Stirling, J. (1730). The differential method: A Treatise of the Summation and Interpolation of Infinite Series. Methodus Differentialis: sive Tractatus de Summatione et Interpolatione Serierum Infinitarum. Gul. Bowyer, London, English translation by Holliday, J., 1749.
Strahler, A. N. (1952). Hypsometric (area-altitude) analysis of erosional topography. Geol Soc Am Bull. 63 (11): 1117-1142.
Strahler, A. N. (1957). Quantitative analysis of watershed geomorphology. Eos, Transactions American Geophysical Union. 38 (6): 913-920.
Tokunaga, E. (1966). The composition of drainage network in Toyohira River basin and valuation of Horton’s first law (in Japanese with English summary). Geophys Bull Hokkaido Univ. 15: 1-19.
Tokunaga, E. (1978). Consideration on the composition of drainage networks and their evolution. Geogr Rep. 13: 1.
Troutman, B. M. & Karlinger, M. (1998). Stochastic Methods in Hydrology: Rain, Landforms and Floods, chapter Spatial Channel Network Models in Hydrology, p. 85-128. World Sci., River Edge, N. J.
Veitzer, S. & Gupta, V. K. (2000). Random self-similar river networks and derivations of generalized horton laws in terms of statistical simple scaling. Water Resour Res. 36 (4): 1033-1048.
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