Flow modeling on grid terrains

DEM Representations

3 2 47 5 87 1 9

3 2 47 5 87 1 9

3 2 47 5 87 1 9

3 2 47 5 87 1 9

TIN

GridContour lines

Sample points

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, hxy}

Model and

compute indices

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)

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

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

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

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)

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)

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).

DEM and Flow Accumulation[Panama]

Flow accumulation with MFD

Flow accumulation with SFD

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