Critical Tokunaga model for river networks

TL;DR

This paper introduces a new mathematical model of river networks that explains their self-similar structure and scaling laws, connecting various observed empirical laws through a unified theoretical framework.

Contribution

It develops a one-parameter family of self-similar critical Tokunaga trees that unify and extend existing models, clarifying the origin of key hydrological scaling laws.

Findings

The model explains the origin of Horton and Hack's laws.

It reproduces observed river network exponents.

It provides a framework for understanding landscape organization.

Abstract

The hierarchical organization and self-similarity in river basins have been topics of extensive research in hydrology and geomorphology starting with the pioneering work of Horton in 1945. Despite significant theoretical and applied advances however, the mathematical origin of and relation among Horton laws for different stream attributes remain unsettled. Here we capitalize on a recently developed theory of random self-similar trees to introduce a one-parametric family of self-similar critical Tokunaga trees that elucidates the origin of Horton laws, Hack's laws, basin fractal dimension, power-law distributions of link attributes, and power-law relations between distinct attributes. The proposed family includes the celebrated Shreve's random topology model and extends to trees that approximate the observed river networks with realistic exponents. The results offer tools to increase our understanding of landscape organization under different hydroclimatic forcings, and to extend scaling relationships useful for hydrologic prediction to resolutions higher that those observed.