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 article, 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 and which more accurately represent flooding events. We present three main results: First, we present an O ( n log n )-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. Second, we present a O ( n log n + nm )-time algorithm for preprocessing Σ containing m sinks into a data structure of size O ( nm ) 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 (| R | k + k 2 ) time, where k is the number of maximal depressions in Σ containing the query point q and | R | is the number of vertices in R with positive rainfall. 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 ( k ω ) time, we show that a flood-time query can be answered in O ( nk + k ω ) time. We also give an α-approximation algorithm, for α > 1, which runs in O ( n log n log α ρ)-time, where ρ is a variable on the terrain that depends on the ratio between depression volumes. We implemented our algorithms for computing terrain and point-flood queries as well as approximate flood-time queries. We tested the efficacy and efficiency of these algorithms on three real terrains of different types (urban, suburban, and mountainous.)