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Cross-relationships among Scaling Indicators for Self-similar River Network Geometry

  • 양수현 (Yang, Soohyun, 대학원 건축사회환경공학과 수자원공학전공)

초록/요약

In nature, irregular and fragmented shapes that classical Euclidean geometry is not applied to are existed. These shapes have been named as ‘fractals’ that was coined by mathematician Mandelbrot. Fractal infinitely exhibits the same repeating pattern at all scales. This is an essential property of fractal, which is called as ‘self-similarity’. Among many fractals in nature, hydrologists and geomorphologists have mainly focused on river networks to quantitatively describe self-similar characteristics of river network geometry. As invaluable findings, it was found that self-similarity of river networks can be detected as mathematical expressions such as log-linear or power-law relationships. In addition to scaling properties, fractal dimensions can also play a role in identifying self-similarity. The term of ‘scaling indicator’ is used to synthetically name those quantitative indicators to represent self-similarity of river networks. Although many studies were conducted to find not only each of scaling indicators but also their relationships, all relationships among scaling indicators have not been revealed yet and neither do physical implications of the relationships. Based on these backgrounds, this study was conducted with three purposes: the first is to revisit Horton’s ratios of which variables are not perfectly consistent, the second is to verify the exponent of relationship between source area and drainage density as a scaling indicator and include it in the existing relationship between Hack’s exponent and the exponent in exceedance probability distribution of upstream area, and the last is to find new relationships among scaling indicators and elucidate their practical meanings. To deal with inconsistent definition of area variable among Horton’s ratios, the concept of eigenarea meaning directly draining area to a stream was introduced. Considering the constant property of lengths of overland flows over stream orders supported by empirical results, it was found that the stream length ratio is nearly equal to the eigenarea ratio. In cases of two other Horton’s ratios, they were linked to the eigenarea ratio introducing the equations fractal dimensions expressed as Horton’s ratios. As a result, classical three Horton’s ratios were expressed as the functions of one term as the eigenarea ratio. Power-law relationship between source area and drainage density was verified as another scale-free phenomenon. Furthermore, the range of its exponent was derived by referring to empirically revealed properties of fractal dimensions in the previous studies. From both theoretical and empirical approaches, it was found that the scaling exponent is almost identical to the exponent in exceedance probability distribution of upstream area. This result contributed to make inter-relationship among scaling exponents to be expanded. To reveal new relationships among scaling indicators and those implications, random-walk model was used to simulate various cases of drainage networks. According to previous studies, random-walk drainage networks are eligible to be surrogate natural river networks in terms of representing self-similarity. Based on scaling properties of random-walk drainage networks found in this study, reasonable relationships between scaling indicators and physically-based indicators were suggested. Topological characteristics (the sinuosity of mainstream, the distribution of streams, and the basin shape) of river networks could be inferred from the new relationships. In conclusion, this study contributes to the adjustment, the expansion, and the discovery of cross-relationships among scaling indicators for river network geometry.

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목차

1 Introduction
2 Revisiting Horton’s Ratios with Eigenarea
2.1 Introduction
2.2 Relation between Eigenarea Ratio and Stream Length Ratio
2.3 Relationships of Eigenarea Ratio to Other Horton’s Ratios
2.4 Conclusions
3 Inter-relationship among Three Scaling Exponents
3.1 Introduction
3.2 Analytical Derivation for the Relationship between ε and η
3.3 Case Study
3.4 Conclusions
4 Scaling Properties of Random-walk Drainage Networks
4.1 Introduction
4.2 Method
4.2.1 Random Walk Model
4.2.2 Simulation Conditions
4.3 Result Analysis and Discussions
4.3.1 Dependent Relationships among Horton’s Ratios
4.3.2 Relationships between Fractal Dimensions and Horton’s Ratios
4.3.3 Relationships between Fractal Dimensions and Scaling Exponents
4.4 Case Study
4.5 Conclusions
5 Summary and Conclusions

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