Flood-Risk Analysis on Terrains under the Multiflow-Direction Model

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 Sigma, represented as a triangulated xy-monotone surface with n vertices, a rain distribution R and a volume of rain psi, determine which portions of Sigma 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 psi according to a rain distribution R, determine what regions of Sigma will be flooded; here m is the number of sinks in Sigma. Second, we present a O(n logn)-time algorithm for preprocessing Sigma into a linear-size data structure for answering point-flood queries: given a rain distribution R, a volume of rain psi falling according to R, and point q is an element of Sigma, 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 Sigma containing the query point q. Alternately, we can preprocess Sigma in O(n log n + nm) time into an O(nm)-size data structure so that a point-flood query can be answered inO(vertical bar R vertical bar k+k(2)) time, where vertical bar R vertical bar 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 is an element of Sigma, determine the volume of rain that must fall before q is flooded. Assuming that the product of two k x k matrices can be computed in O(k(omega)) time, we show that a flood-time query can be answered in O(nk + k(omega)) time. We also give an alpha-approximation algorithm, for alpha > 1, that runs in O(nk + k(2) (log( n+log(alpha))rho))-time, where rho 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