[Paper Review] Comparing Topology of Engineered and Natural Drainage Networks

Soohyun Yang, Kyungrock Paik+5 more|arXiv (Cornell University)|Jul 16, 2017

Urban Stormwater Management Solutions29 references18 citations

TL;DR

This study compares the topological scaling properties of engineered urban drainage networks (UDNs) and natural river networks using power-law scaling and exceedance probability distributions. It finds that UDNs evolve from linear to power-law scaling with size, exhibit exponential tempering in upstream area distributions, and gradually resemble river networks over time, though with higher power-law exponents and structural heterogeneity due to design constraints.

ABSTRACT

We investigated the scaling and topology of engineered urban drainage networks (UDNs) in two cities, and further examined UDN evolution over decades. UDN scaling was analyzed using two power-law characteristics widely employed for river networks: (1) Hack's law of length ($L$)-area ($A$) scaling [$L \propto A^{h}$], and (2) exceedance probability distribution of upstream contributing area $(δ)$ [$P(A\geq δ) \sim a δ^{-ε}$]. For the smallest UDNs ($<2 \> ext{km}^2$), length-area scales linearly ($h\sim 1$), but power-law scaling emerges as the UDNs grow. While $P(A\geq δ)$ plots for river networks are abruptly truncated, those for UDNs display exponential tempering [$P(A\geq δ) \> ext{=}\> a δ^{-ε}\exp(-cδ)$]. The tempering parameter $c$ decreases as the UDNs grow, implying that the distribution evolves in time to resemble those for river networks. However, the power-law exponent $ε$ for large UDNs tends to be slightly larger than the range reported for river networks. Differences in generative processes and engineering design constraints contribute to observed differences in the evolution of UDNs and river networks, including subnet heterogeneity and non-random branching.

Motivation & Objective

• To analyze scaling laws in engineered urban drainage networks (UDNs) and compare them with natural river networks.
• To investigate how UDN topology evolves over decades in urban areas.
• To identify differences in network generative processes and structural organization between engineered and natural systems.
• To quantify topological deviations in UDNs, such as non-random branching and subnet heterogeneity.

Proposed method

• Applied Hack's law ($L \propto A^h$) to examine length-area scaling in UDNs of varying sizes.
• Used exceedance probability distribution $P(A \geq \delta) \sim a \delta^{-\epsilon}$ to characterize upstream contributing area distributions.
• Modeled tempering in UDN distributions using $P(A \geq \delta) = a \delta^{-\epsilon} \exp(-c\delta)$, with $c$ as the tempering parameter.
• Analyzed UDNs from two cities across multiple decades to assess temporal evolution of topological properties.
• Compared power-law exponents ($h$, $\epsilon$) and tempering parameters ($c$) between UDNs and natural river networks.
• Evaluated network heterogeneity and non-random branching patterns in UDNs due to engineering constraints.

Experimental results

Research questions

• RQ1 How do length-area scaling relationships in engineered urban drainage networks compare to those in natural river networks?
• RQ2 How does the exceedance probability distribution of upstream contributing area differ between engineered and natural drainage systems?
• RQ3 In what ways does the topology of UDNs evolve over time, and how does this evolution compare to natural river networks?
• RQ4 What role do engineering design constraints play in shaping the topological structure of UDNs?
• RQ5 Why do UDNs exhibit different power-law exponents and tempering behaviors compared to natural river networks?

Key findings

• For small UDNs (<2 km²), length-area scaling is linear ($h \sim 1$), but power-law scaling emerges as networks grow larger.
• Exceedance probability distributions in UDNs show exponential tempering ($P(A \geq \delta) = a \delta^{-\epsilon} \exp(-c\delta)$), unlike the abrupt truncation seen in river networks.
• The tempering parameter $c$ decreases with UDN size, indicating a temporal evolution toward a form resembling natural river networks.
• The power-law exponent $\epsilon$ for large UDNs is slightly higher than the typical range observed in natural river networks.
• Network heterogeneity and non-random branching in UDNs are attributed to engineering design constraints and generative processes.
• Despite topological convergence over time, UDNs retain distinct structural signatures due to anthropogenic design influences.