Flood-risk analysis on terrains under the multiflow-direction model

Abstract

An important problem in terrain analysis is modeling how water flows across a terrain and creates floods by filling up depressions. In this paper we study a number of flood-risk related problems: Given a terrain Σ, represented as a triangulated xy-monotone surface with n vertices, a rain distribution R and a volume of rain ψ, determine which portions of Σ are flooded. We develop efficient algorithms for flood-risk analysis under the multiflow-directions (MFD) model, in which water at a point can flow along multiple downslope edges to more accurately represent flooding events. We present three main results: First, we present an O(nm)-time algorithm to answer a terrain-flood query: if it rains a volume ψ according to a rain distribution R, determine what regions of Σ will be flooded; here m is the number of sinks in Σ. Second, we present a O(n log n)-time algorithm for preprocessing Σ into a linear-size data structure for answering point-flood queries: given a rain distribution R, a volume of rain ψ falling according to R, and point q ∈ Σ, determine whether q will be flooded. A point-flood query can be answered in O(nk) time, where k is the number of maximal depressions in Σ containing the query point q. Alternately, we can preprocess Σ in O(n log n + nm) time into an O(nm)-size data structure so that a point-flood query can be answered in O(|R|k+k2) time, where |R| is the number of vertices in R with positive rain fall. Finally, we present algorithms for answering a flood-time query: given a rain distribution R and a point q ∈ Σ, determine the volume of rain that must fall before q is flooded. Assuming that the product of two k × k matrices can be computed in O(kω) time, we show that a flood-time query can be answered in O(nk + kω) time. We also give an α-approximation algorithm, for α > 1, that runs in O(nk + k2(log(n + logα)ρ))-time, where ρ is a variable on the terrain which depends on the ratio between depression volumes. We implemented our terrain-flooding algorithm and tested its efficacy and efficiency on real terrains