Abstract
We study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x,y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n logn) time algorithm to compute the minimal and the maximal watershed of a vertex, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time.
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References
Borga, M., Gaume, E., Creutin, J., Marchi, L.: Surveying flash floods: gauging the ungauged extremes. Hydrological Processes 22, 3883–3885 (2008)
Craddock, W., Kirby, E., Harkins, N., Zhang, H., Shi, X., Liu, J.: Rapid fluvial incision along the Yellow River during headward basin integration. Nature Geoscience 3, 209–213 (2010)
Danner, A., Mølhave, T., Yi, K., Agarwal, P.K., Arge, L., Mitásová, H.: TerraStream: from elevation data to watershed hierarchies. In: Proc. 15th ACM Int. Symp. on Geographic Information Systems (ACM-GIS 2007), pp. 212–219 (2007)
de Berg, M., Cheong, O., Haverkort, H., Lim, J.-G., Toma, L.: The complexity of flow on fat terrains and its I/O-efficient computation. Comput. Geom. Theory Appl. 43(4), 331–356 (2010)
de Berg, M., Haverkort, H., Tsirogiannis, C.: Flow on noisy terrains: An experimental evaluation. In: 27th Eur. Worksh. Comput. Geom., pp. 111–114 (2011)
de Berg, M., Haverkort, H., Tsirogiannis, C.: Implicit flow routing on terrains with applications to surface networks and drainage structures. In: Proc. 22nd ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 285–296 (2011)
Fisher, P.F., Tate, N.J.: Causes and consequences of error in digital elevation models. Progress in Physical Geography 30(4), 467–489 (2006)
Gray, C., Evans, W.: Optimistic shortest paths on uncertain terrains. In: Proc. 16th Canad. Conf. on Comput. Geom., pp. 68–71 (2004)
Gray, C., Kammer, F., Löffler, M., Silveira, R.I.: Removing local extrema from imprecise terrains. In: CoRR, abs/1002.2580 (2010)
Hebeler, F., Purves, R.: The influence of elevation uncertainty on derivation of topographic indices. Geomorphology 111(1-2), 4–16 (2009)
Henzinger, M., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. J. Computer and System Sciences 55(1), 3–23 (1997)
Koster, R.D., Mahanama, S.P.P., Livneh, B., Lettenmaier, D.P., Reichle, R.H.: Skill in streamflow forecasts derived from large-scale estimates of soil moisture and snow. Nature Geoscience 3(9), 613–616 (2010)
Lindsay, J., Evans, M.: The influence of elevation error on the morphometrics of channel networks extracted from DEMs and the implications for hydrological modelling. Hydrological Processes 22(11), 1588–1603 (2008)
Liu, Y., Snoeyink, J.: Flooding triangulated terrain. In: Proc. 11th Int. Symp. Spatial Data Handling, Berlin, pp. 137–148 (2005)
Montanari, A.: What do we mean by ‘uncertainty’? The need for a consistent wording about uncertainty assessment in hydrology. Hydr. Proc. 21, 841–845 (2006)
Tarboton, D.: A new method for the determination of flow directions and upslope areas in grid dig. elev. models. Water Resources Research 33(2), 309–319 (1997)
Tetzlaff, D., McDonnell, J., Uhlenbrook, S., McGuire, K., Bogaart, P., Naef, F., Baird, A., Dunn, S., Soulsby, C.: Conceptualizing catchment processes: simply too complex? Hydrological Processes 22, 1727–1730 (2008)
Vrugt, J.A., Diks, C.G.H., Gupta, H.V., Bouten, W., Verstraten, J.M.: Improved treatment of uncertainty in hydrologic modeling: Combining the strengths of global optimization and data assimilation. Water Resources Research 41 (2005)
Wechsler, S.P.: Uncertainties associated with digital elevation models for hydrologic applications: a review. Hydrology and Earth System Sc. 11(4), 1481–1500 (2007)
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Driemel, A., Haverkort, H., Löffler, M., Silveira, R.I. (2011). Flow Computations on Imprecise Terrains. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_30
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DOI: https://doi.org/10.1007/978-3-642-22300-6_30
Publisher Name: Springer, Berlin, Heidelberg
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