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arXiv · 2009.08014

1D and 2D Flow Routing on a Terrain

Aaron Lowe·Svend C. Svendsen·Pankaj K. Agarwal·Lars Arge
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Abstract

An important problem in terrain analysis is modeling how water flows across a terrain creating floods by forming channels and filling depressions. In this paper we study a number of \emph{flow-query} related problems: Given a terrain $\Sigma$, represented as a triangulated $xy$-monotone surface with $n$ vertices, a rain distribution $R$ which may vary over time, determine how much water is flowing over a given edge as a function of time. We develop internal-memory as well as I/O-efficient algorithms for flow queries. This paper contains four main results: (i) We present an internal-memory algorithm that preprocesses $\Sigma$ into a linear-size data structure that for a (possibly time varying) rain distribution $R$ can return the flow-rate functions of all edges of $\Sigma$ in $O(\rho k+|\phi| \log n)$ time, where $\rho$ is the number of sinks in $\Sigma$, $k$ is the number of times the rain distribution changes, and $|\phi|$ is the total complexity of the flow-rate functions that have non-zero values; (ii) We also present an I/O-efficient algorithm for preprocessing $\Sigma$ into a linear-size data structure so that for a rain distribution $R$, it can compute the flow-rate function of all edges using $O(\text{Sort}(|\phi|))$ I/Os and $O(|\phi| \log |\phi|)$ internal computation time. (iii) $\Sigma$ can be preprocessed into a linear-size data structure so that for a given rain distribution $R$, the flow-rate function of an edge $(q,r) \in \Sigma$ under the single-flow direction (SFD) model can be computed more efficiently. (iv) We present an algorithm for computing the two-dimensional channel along which water flows using Manning's equation; a widely used empirical equation that relates the flow-rate of water in an open channel to the geometry of the channel along with the height of water in the channel.

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Aaron Lowe, Svend C. Svendsen, Pankaj K. Agarwal, Lars Arge. 2020-09-17. 1D and 2D Flow Routing on a Terrain. https://doi.org/10.1145/3397536.3422269

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