Flow modeling on grid terrains

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Flow modeling on grid terrains

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DEM Representations
TIN
Grid
Contour lines
Sample points
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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
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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)

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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

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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

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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

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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)

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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)

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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).

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DEM and Flow AccumulationPanama
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Flow accumulation with MFD
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Flow accumulation with SFD
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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