Map Algebra and Beyond: 1. Map Algebra for Scalar Fields Xingong Li University of Kansas 3 Nov 2009

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Map Algebra and Beyond: 1. Map Algebra for Scalar Fields Xingong Li University of Kansas 3 Nov 2009. Topics. The conventional map algebra Local operations Focal operations Zonal operations Global operations. Map Algebra.

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### Map Algebra and Beyond:1. Map Algebra for Scalar FieldsXingong LiUniversity of Kansas3 Nov 2009

Topics
• The conventional map algebra
• Local operations
• Focal operations
• Zonal operations
• Global operations
Map Algebra
• Raster layers are manipulated by math-like expression to create new raster layers

5

6

7

6

Precipitation

-

Losses (Evaporation, Infiltration)

=

Runoff

-

3

3

2

4

=

2

3

5

2

Map Algebra Operations
• Tomlin (1990) defines and organizes operations as local, focal, zonal, and globalaccording to the spatial scopeof the operations
• Geographic Information System and Cartographic Modeling, Englewood Cliffs: Prentice Hall, 1990.

Global

Local

Focal

Zonal

Local Operations
• Compute a new raster layer.
• The value for each cell on the output layer is a function of one or more cell values at the same location on the input layer(s).
Local Operations
• Arithmetic operations
• +, -, *, /, Abs, …
• Relational operators
• >, <, …
• Statistic operations
• Min, Max, Mean, Majority, …
• Trigonometric operations
• Sine, Cosine, Tan, Arcsine, Arccosine, …
• Exponential and logarithmic operations
• Sqr, sqrt, exp, exp2, …
Removing Clouds Using a Local Operation
• Two consecutive ocean surface temperature raster layers for the same area (measured at a slightly different time).

Images are from: http://rs.gso.uri.edu/amy/avhrr.html

Oct.

Feb.

Apr.

Jun.

Dec.

Aug.

30-Year (1971-2000) Monthly PRISM Precipitation

How do we define seasonality of precipitation at a single location?

Seasonality at San Francisco

Total = 18.69”

[4.01 3.48 2.69 1.30 0.48 0.11 0.01 0.02 0.19 0.74 1.57 4.09]

x=p*sin(monthAngle)

y=p*cos(monthAngle)

Monthly Precipitation as a Vector Quantity

Each month’s duration is equivalent to a 30° angle

Monthly data are plotted at midpoint: 15°, 45°, 75°, ……

x=p*sin(monthAngle)

y=p*cos(monthAngle)

Seasonality Analysis

Monthly precipitation (in inches):

[4.01 3.48 2.69 1.30 0.48 0.11 0.01 0.02 0.19 0.74 1.57 4.09]

Add all vectors = Resultant vector

Seasonality at San Francisco
• Average monthly precipitation at San Francisco in inches
• [4.01 3.48 2.69 1.30 0.48 0.11 0.01 0.02 0.19 0.74 1.57 4.09]
• Precipitation vectors
• x=1.09, 2.46, 2.59, 1.26, 0.34, 0.03, 0, -0.01, -0.18, -0.71, -1.11, -1.04
• y=3.86, 2.47, 0.74, -0.32, -0.33, -0.11, -0.01, -0.01, -0.05, 0.19, 1.11, 3.95
• Resultant vector
• sx=4.7
• sy=11.5
• Magnitude=12.41
• Direction=22.25
• Time of occurrence
• Direction in month (= January)
• Seasonality index
• 1-magnitude of resultant vector/total precip.
• = 1- (12.41/18.69)
• = 0.34 (larger number means more uniform)

12.41

22.25°

• sy
• Cos(15) * [p01] + Cos(45) * [p02] + Cos(75) * [p03] + Cos(105) * [p04] + Cos(135) * [p05] + Cos(165) * [p06] + Cos(195) * [p07] + Cos(225) * [p08] + Cos(255) * [p09] + Cos(285) * [p10] + Cos(315) * [p11] + Cos(345) * [p12]
• sx
• Sin(15) * [p01] + Sin(45) * [p02] + Sin(75) * [p03] + Sin(105) * [p04] + Sin(135) * [p05] + Sin(165) * [p06] + Sin(195) * [p07] + Sin(225) * [p08] + Sin(255) * [p09] + Sin(285) * [p10] + Sin(315) * [p11] + Sin(345) * [p12]
• Magnitude of resultant vector
• Sqrt(Sqr([sx]) + Sqr([sy]))
• Total precipitation
• [p01] + [p02] + [p03] + [p04] + [p05] + [p06] + [p07] + [p08] + [p09] + [p10] + [p11] + [p12]
• Seasonality
• 1 - ([Mag. Of resultant vector] / [Total Precip])
Seasonality Index

0.34

Large number means more uniform

Small number means more seasonal

Map Algebra Operations
• Operations are grouped as local, focal, zonal,andglobalaccording to the spatial scope of the operations.
Focal Operations
• Compute an output value for each cell as a function of the cells that are within its neighborhood
• Widely used in image processing with different names
• Convolution, filtering, kernel or moving window
• Focal operations are spatial in nature
Neighborhoods
• The simplest and most common neighborhood is a 3 by 3 rectangle window
• Other possible neighborhoods
• a rectangle, a circle, an annulus (a donut) or a wedge
Finding Appropriate Wind Farm Sites
• Wind speed
• Higher elevation higher speed
• Elevation (>= 1000m)
• Aspect
• facing prevailing wind direction
• Wind exposure
• Not blocked by nearby hills in the prevailing wind direction
• Data
• Prevailing wind direction
• 225  to 315
• DEM
• Wedge neighborhood
• 0 degree is East, counterclockwise (135—225)
Wind Exposure Analysis
• Find max elevation in the prevailing wind direction
• FocalMax with a wedge neighborhood
• Find cells not blocked by hills in the neighborhood
• DEM > FocalMax
Map Algebra Operations
• Operations are grouped as local, focal, zonal, andglobalaccording to the spatial scope of the operations.
Zonal Operations
• Compute a new value for each cell as a function of the cell values within a zone containing the cell
• Zone layer
• defines zones
• Value layer
• contains input cell values
Zonal Statistical Operations
• Calculate statistics for each cell by using all the cell values within a zone
• Zonal statistical operations:
• ZonalMean, ZonalMedian, ZonalSum, ZonalMinimum, ZonalMaximum, ZonalRange, ZonalMajority, ZonalVariety, ….
Zonal Statistical Operation Example

Value Layer

Zone Layer

ZonalMax

Output Layer

Outputs of Zonal Operations
• Raster layer
• All the cells within a zone have the same value on the output raster layer
• Table
• Each row in the table contains the statistics for a zone.
• The first column is the value (or ID) of each zone.
• The table can be joined back to the zone layer.

Measurement spatial resolution

p1

p2

a2

a1

a3

a4

p3

p4

Calculating Subwatershed Precipitation Depth

pi—precipitation depth in an hour

ai—portion of the watershed that falls in the ith cell

Map Algebra Operations
• Operations are grouped as local, focal, zonal,andglobalaccording to the spatial scope of the operations.
Global Operations
• Operations that compute an output raster where the value of each output cell is a function of all the cells in the input raster
• Global statistical operations
• Distance operations.
• Euclidean distance
• Cost distance
Distance Operations
• Characterize the relationships between each cell and source cells (usually representing features)
• Distance to nearest source cell
• Direction to nearest source cell
EuclideanDistance Operation
• Calculates the shortest straight distance from each cell to its nearest source cell (EucDistance)
• Assigns each cell the value of its nearest source cell (EucAllocation)
• Calculates the direction from each cell to its nearest source cell (EucDirection)
EucDistance Example

Buffers can be delineated from the distance raster

EucAllocation Example

Voronoi diagram

Thesisen’s polygon

Non-Euclidean Distance (Cost Distance)
• Straight line distance (between A and B) is a type of cost
• Cost could also be measured as time or money spent
• Friction may vary space
• Least cost and least-cost-path

source

friction

CostDistance Operation
• Compute the least accumulative cost from each cell to its least-cost source cell
• Source raster
• Representing features (points, lines, and polygons)
• No-source cells are set to NODATA value
• Friction raster
• Cost encountered while moving in a cell (distance, time, dollars and efforts)
• Unit is: cost per unit distance
• Can have barriers (NODATA cells)
Fungus Invasion
• Fungus spreading depends on the availability of precipitation
• A fungus is introduced at a seaport in January 1
• Questions
• Which area would be affected by July?
• Will the fungus reach Austin by the end of July?
• Fungus travel speed depends on precipitation.
• < 100 mm/month, 0 m/day
• 100 – 200 mm/month, 4000 m/day
• > 200 mm/month, 7000 m/day

Travel speed

The Friction Raster

Unit of the friction raster:

days per unit distance

Friction = 1 / speed

The Least Cost Raster

What do the values mean on the cost raster layer?

The days that the fungus will take to reach a cell

February

Friction Varies in Space & Time
• Precipitation varies both in space and time.
• How could we model the spreading of the fungus now?

April

Sum of Two Cost Surfaces

The least cost between A and B and passes through a cell.

Corridor Analysis

Corridor = accumulative cost < a threshold value

Summary Concepts
• What you have just seen is the basis for the map algebra language in ArcGIS Grid and Spatial Analyst
• Local functions
• Focal functions
• Zonal functions
• Global functions