Flow modeling on grid terrains

Why GIS?

How it all started..• Duke Environmental researchers:

• computing flow accumulation for Appalachian Mountains took 14 days (with 512MB memory)

– 800km x 800km at 100m resolution ~64 million points

GIS (Geographic Information Systems)• System that handles spatial data

• Visualization, processing, queries, analysis…• Rich area of problems for Computer Science

• Graphics, graph theory, computational geometry, scientific computing…

GIS and the EnvironmentIndispensable tool • Monitoring: keep an eye on the

state of earth systems using satellites and monitoring stations (water, pollution, ecosystems, urban development,…)

• Modeling and simulation: predict consequences of human actions and natural processes

• Analysis and risk assessment: find the problem areas and analyse the possible causes (soil erosion, groundwater pollution,..)

• Planning and decision support:

provide information and tools for better management of resources

Lots of rainDry

Precipitation in Tropical South America

High nitrogen concentrations

Nitrogen in Chesapeake Bay

GIS and the EnvironmentBald Head Island Renourishment

Sediment flow

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

Massive Terrain Data

Remote sensing technology • Massive amounts of

terrain data

• Higher resolutions (1km, 100m, 30m, 10m, 1m,…)

NASA-SRTM • Mission launched in 2001• Acquired data for 80% of

earth at 30m resolution • 5TB

USGS • Most of US at 10m

resolution LIDAR

• 1m res

Example: LIDAR Terrain Data

Massive (irregular) point sets (1-10m resolution) Relatively cheap and easy to collect

Example: Jockey’s ridge (NC coast)

It’s Growing!

Appalachian Mountains

Area if approx. 800 km x 800 km

Sampled at:

• 100m resolution: 64 million points (128MB)

• 30m resolution: 640 (1.2GB)

• 10m resolution: 6400 = 6.4 billion (12GB)

• 1m resolution: 600.4 billion (1.2TB)

Computing on Massive Data GRASS (open source GIS)

• Killed after running for 17 days on a 6700 x 4300 grid (approx 50 MB dataset)

TARDEM (research, U. Utah)• Killed after running for 20 days on a 12000 x 10000 grid

(appox 240 MB dataset)• CPU utilization 5%, 3GB swap file

ArcInfo (ESRI, commercial GIS)• Can handle the 240MB dataset • Doesn’t work for datasets bigger than 2GB

Scalability Problem

We can compute FD and FA using simple O(N)-time algorithms

..but.. for large sets..??

Dataset Size (log)

Scalability Problem: Why? Most (GIS) programs assume data fits in memory

• minimize only CPU computation But.. Massive data does not fit in main memory!

• OS places data on disk and moves data between memory and disk as needed

Disk systems try to amortize large access time by transferring large contiguous blocks of data

When processing massive data disk I/O is the bottleneck, rather than CPU time!

track

magnetic surface

read/write armread/write head

Disks are Slow

“The difference in speed between modern CPU and disk technologies is analogous to the difference in speed in sharpening a pencil using a sharpener on one’s desk or by taking an airplane to the other side of the world and using a sharpener on someone else’s desk.” (D. Comer)

Scalability to Large Data Example: reading an array from disk

• Array size N = 10 elements

• Disk block size = 2 elements

• Memory size = 4 elements (2 blocks)

1 2 10 9 5 6 3 4 8 71 5 2 6 3 8 9 4 7 10

Algorithm 2: Loads 5 blocksAlgorithm 1: Loads 10 blocks

N blocks >> N/B blocks Block size is large (32KB, 64KB) N >> N/B

N = 256 x 106, B = 8000 , 1ms disk access time

N I/Os take 256 x 103 sec = 4266 min = 71 hr

N/B I/Os take 256/8 sec = 32 sec

I/O-model

I/O-operation• Read/write one block of

data from/to disk

I/O-complexity• number of I/O-operations

(I/Os) performed by the algorithm

External memory or I/O-efficient algorithms:

Minimize I/O-complexity

RAM model

CPU-operation

CPU-complexity• Number of CPU-operations

performed by the algorithm

Internal memory algorithms:Minimize CPU-complexity

I/O-Model Parameters

N = # elements in problem instance

B = # elements that fit in disk block

M = # elements that fit in main memory

Fundamental bounds:• Scanning: scan(N) = • Sorting: sort(N) =

D

P

M

Block I/O

)log(BN

BN

BMO

NBN

BN

BN

BM <<< log

In practice block and main memory sizes are big

)(BNO

TerraFlow TerraFlow is our suite of programs for flow routing and flow

accumulation on massive grids [ATV`00,AC&al`02]

Flow routing and flow accumulation modeled as graph problems and solved in optimal I/O bounds

Efficient• 2-1000 times faster on very large grids than existing software

Scalable• 1 billion elements!! (>2GB data)

Flexible • Allows multiple methods flow modeling

http://www.cs.duke.edu/geo*/terraflow

TerraFlow TerraFlow is our suite of programs for flow routing and flow

accumulation on massive grids [ATV`00,AC&al`02]

Flow routing and flow accumulation modeled as graph problems and solved in optimal I/O bounds

Efficient• 2-1000 times faster on very large grids than existing software

Scalable• 1 billion elements!! (>2GB data)

Flexible • Allows multiple methods flow modeling

http://www.cs.duke.edu/geo*/terraflow

TerraFlow

GRASS cannot handle Hawaii dataset (killed after 17 days) TARDEM cannot handle Cumberlands dataset (killed after 20 days)

Significant speedup over

ArcInfo for large datasets• East-Coast

TerraFlow: 8.7 Hours

ArcInfo: 78 Hours

• Washington state

TerraFlow: 63 Hours

ArcInfo: %

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10

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40

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60

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80

90

KaweahPuerto Rico

Sierra Nevada

Hawaii

CumberlandsLower NE East-Coast

MidwestWashington

Running Time (Hours)

TerraFlow 512

TerraFlow 128

ArcInfo 512

ArcInfo 128

ArcInfo

TerraFlow

Massive Data• Massive datasets are being collected everywhere• Storage management software is billion-$ industry

(More) Examples: Phone: AT&T 20TB phone call

database, wireless tracking Consumer: WalMart 70TB

database, buying patterns (supermarket checkout)

WEB: Web crawl of 200M pages and 2000M links, Akamai stores 7 billion clicks per day

Geography: NASA satellites generate 1.2TB per day