1. Flow modeling on grid terrains

2. DEM Representations 324 758 719 324 758 719 324 758 719 324 758 719 TIN Grid Contour lines Sample points

3. Computations on Terrains Reality: Elevation of terrain is a continuous function of two variables h(x,y) Estimate, predict, simulate  Flooding, pollution  Erosion, deposition  Vegetation structure  …. GIS: DEM ( Digital Elevation Model ) is a set of sample points and their heights {  x, y, h xy  } Model and compute indices

4. Modeling Flow on Terrains  What happens when it rains? Predict areas susceptible to floods. Predict location of streams. Compute watersheds.  Flow is modeled using two basic attributes Flow Direction (FD) The direction water flows at a point Flow Accumulation (FA) Total amount of water that flows through a point (if water is distributed according to the flow directions)

5. Flow Direction (FD) on Grids  Water flows downhill follows the gradient  On grids: Approximated using 3x3 neighborhood SFD (Single-Flow Direction): FD points to the steepest downslope neighbor MFD (Multiple-Flow direction): FD points to all downslope neighbors

6. Computing FD  Goal: compute FD for every cell in the grid (FD grid)  Algorithm: Scan the grid For each cell compute SFD/MFD by inspecting 8 neighbor cells  Analysis: O(N) time for a grid of N cells  Is this all? NO! flat areas: Plateas and sinks

7. FD on Flat Areas  …no obvious flow direction  Plateaus Assign flow directions such that each cell flows towards the nearest spill point of the plateau  Sinks Either catch the water inside the sink Or route the water outside the sink using uphill flow directions model steady state of water and remove (fill) sinks by simulating flooding: uniformly pouring water on terrain until steady state is reached Assign uphill flow directions on the original terrain by assigning downhill flow directions on the flooded terrain

8. Flow Accumulation (FA) on Grids FA models water flow through each cell with “uniform rain” Initially one unit of water in each cell Water distributed from each cell to neighbors pointed to by its FD Flow conservation: If several FD, distribute proportionally to height difference Flow accumulation of cell is total flow through it Goal: compute FA for every cell in the grid (FA grid)

9. Computing FA  FD graph: node for each cell (directed) edge from cell a to b if FD of a points to b  FD graph must be acyclic ok on slopes, be careful on plateaus  FD graph depends on the FD method used SFD graph: a tree (or a set of trees) MFD graph: a DAG (or a set of DAGs)

10. Computing FA: Plane Sweeping  Input: flow direction grid FD  Output: flow accumulation grid FA (initialized to 1)  Process cells in topological order. For each cell: Read its flow from FA grid and its direction from FD grid Update flow for downslope neighbors (all neighbors pointed to by cell flow direction)  Correctness One sweep enough  Analysis O(sort) + O(N) time for a grid of N cells  Note: Topological order means decreasing height order (since water flows downhill).

11. DEM and Flow Accumulation [Panama]

14. Flow accumulation with MFD

15. Flow accumulation with SFD

16. Uses Flow direction and flow accumulation are used for:  Computing other hydrological attributes river network moisture indices watersheds and watershed divides  Analysis and prediction of sediment and pollutant movement in landscapes.  Decision support in land management, flood and pollution prevention and disaster management