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Critical Tokunaga model for river networks.

Yevgeniy Kovchegov1, Ilya Zaliapin2, Efi Foufoula-Georgiou3

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This study mathematically explains Horton's laws in river basins using random self-similar trees. It reveals the origin of scaling laws and fractal dimensions in geomorphology.

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Area of Science:

  • Geomorphology
  • Hydrology
  • Complex Systems

Background:

  • River basin organization and self-similarity have been studied since Horton (1945).
  • The mathematical origins and interrelations of Horton's laws remain unclear.
  • Existing models do not fully capture observed river network scaling properties.

Purpose of the Study:

  • To elucidate the mathematical origin of Horton's laws and related scaling relationships in river basins.
  • To develop a theoretical framework explaining Hack's laws, fractal dimensions, and power-law distributions.
  • To extend hydrologic prediction capabilities to finer resolutions.

Main Methods:

  • Utilized a recently developed theory of random self-similar trees.
  • Introduced a one-parametric family of self-similar critical Tokunaga trees.
  • Analyzed the mathematical relationships between stream attributes and basin structure.

Main Results:

  • Provided a unified mathematical explanation for Horton's laws, Hack's laws, and basin fractal dimensions.
  • Demonstrated that Tokunaga trees can approximate observed river networks with realistic exponents.
  • Identified power-law distributions and relations between distinct attributes.

Conclusions:

  • The theory of random self-similar trees offers a robust framework for understanding river basin organization.
  • The findings provide tools to analyze landscape organization under various hydroclimatic conditions.
  • The study advances scaling relationships for improved hydrologic prediction accuracy.