map algebra 1

8/12/2019 Map Algebra 1

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Map Algebra and Beyond:

1. Map Algebra for Scalar

Fields

Xingong LiUniversity of Kansas

3 Nov 2009


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Topics

The conventional map algebra

Local operations

Focal operationsZonal operations

Global operations


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Map Algebra

Precipitation

-Losses

(Evaporation,

Infiltration)

=Runoff

5 22 3

2 43 3

7 6

5 6

-

=

Raster layers are manipulated by math-like expression tocreate new raster layers


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Map Algebra Operations

Tomlin (1990) defines and organizes operationsas local, focal, zonal, andglobalaccording tothe spatial scopeof the operations

Geographic Information System and Cartographic

Modeling, Englewood Cliffs: Prentice Hall, 1990.

Local ZonalFocal Global


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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 locationon the input

layer(s).


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Local Operations

Arithmetic operations

+, -, *, /, Abs,

Relational operators

>,


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Local Operation--Examples

9 9 7

9 8 5

6 3 0

0 0 2

0 0 1

0 0 0

9 9 9

9 8 6

6 3 0+ =

9 9 79 8 5

6 3 0

0 0 2

0 0 1

0 0 0

N N 3.5

N N 5

N N N

/ =


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Removing Clouds Using a Local

Operation

Two consecutive ocean surface temperature raster layersfor the same area (measured at a slightly different time).

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


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30-Year (1971-2000) Monthly PRISM

Precipitation

Dec.

Oct.

Aug.

Jun.

Apr.

Feb.

How do we define seasonality

of precipitation at a single

location?


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Seasonality at San Francisco

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Average Monthly Rainfall in San Francisco

(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]

Total = 18.69


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Monthly Precipitation as a Vector Quantity

0

1

2

3

45

Average Monthly Rainfall in San

Francisco

(Inches)

1

2

3

4

5

30

210

60

240

90270

120

300

150

330

180

0

x=p*sin(monthAngle)

y=p*cos(monthAngle)

Each months duration is equivalent to a 30 angle

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


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Seasonality Analysis

5

10

15

30

210

60

240

90270

120

300

150

330

180

0

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]

1

2

3

4

5

30

210

60

240

90270

120

300

150

330

180

0

x=p*sin(monthAngle)

y=p*cos(monthAngle) Add all vectors = Resultant vector


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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)

22.25


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Seasonality Analysis: Local functions at

each cell over the whole domain

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])


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Time of Occurrence


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Seasonality Index

Large number means more uniform

Small number means more seasonal

0.34


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Operations are grouped as local, focal,

zonal,andglobal according to thespatial

scopeof the operations.


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Focal Operations

Compute an output value for each cell as a function ofthe cells that are within its neighborhood

Widely used in image processing with different names

Convolution, filtering, kernel or moving window

Focal operations arespatialin nature


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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


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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

225to 315

DEM

Wedge neighborhood

0 degree is East, counterclockwise(135225)


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Wind Exposure Analysis

Find maxelevation in the prevailing wind direction FocalMax with a wedge neighborhood

Find cells not blocked by hills in the neighborhood

DEM > FocalMax


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zonal, andglobal according to thespatial



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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


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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, .


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Zonal Statistical Operation Example

1 1 4 3 3

1 1 4 3 3

2 2 2 3 4

2 1 2 3 4

1 1 4 4 4

1 2 3 4 5

6 7 8 9 1

2 3 4 5 6

7 8 9 1 2

3 4 5 6 7

Zone Layer Value Layer

ZonalMax

Output Layer

8 8 8 9 9

8 8 8 9 99 9 9 9 8

9 8 9 9 8

8 8 8 8 8


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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.


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NEXRAD Cell Precipitation

Measurement spatial resolution


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Subwatershed Precipitation from NEXRAD

Cells Precipitation

model/application resolution


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NEXRAD Subwatershed Precipitation


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Calculating Subwatershed Precipitation Depth

i

ii

a

pa

aaaa

papapapa *****depthprecip.edsubwatersh

4321

44332211

p1 p2

p3 p4

a1a2

a3 a4

meanzonal*

*depthprecip.edsubwatersh

So,.sizecellislayer,edsubwatershrasteraWith

n

p

an

pa

a

pa

aa

ii

i

ii

i

piprecipitation depth in an

hour

aiportion of the watershed that

falls in the ith cell


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zonal,andglobalaccording to thespatial



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Global Operations

Operations that compute an output raster where the valueof each output cell is a function of all the cells in the inputraster

Global statistical operations

Distance operations.

Euclidean distance

Cost distance


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Distance Operations

Characterize the relationships

between each cell and source

cells (usually representing

features)

Distance to nearest source cell

Direction to nearest source cell


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Euclidean Distance Operation

1 1

1

2

Calculates the shortest straight distance from each cell toits 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)

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1

2 1 1 1 1 1

2 2 2 1 1 1

2 2 2 2 2 1


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EucDistance Example

Buffers can be

delineated from

the distanceraster

EucAllocation Example


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EucAllocation Example

Thesisens polygon

Voronoi diagram


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Non-Euclidean Distance (Cost Distance)

Straight line distance (between A and B) is a type ofcost

Cost could also be measured as time or money spent

Friction may vary space

Least cost and least-cost-path


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CostDistance Operation

Compute the least accumulativecost from each cell to its least-cost source cell

Source raster

Representing features (points, lines,and polygons)

No-source cells are set toNODATA value

Friction raster

Cost encountered while moving ina cell (distance, time, dollars andefforts)

Unit is: cost per unit distance

Can have barriers (NODATA cells)

friction

source


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Fungus Invasion

Fungus spreading depends

on the availability of

precipitation

A fungus is introduced at aseaport in January 1

Questions

Which area would be affected

by July?

Will the fungus reach Austin

by the end of July?


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Fungus Spreading Speed

Fungus travel speed depends on precipitation.

< 100 mm/month, 0 m/day

100200 mm/month, 4000 m/day

> 200 mm/month, 7000 m/day


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The Friction Raster

Friction = 1 / speed

Unit of the friction raster:

days per unit distance

Travel speed


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The Least Cost Raster

What do the values mean onthe cost raster layer?

The days that the fungus

will take to reach a cell


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Friction Varies in Space & Time

Precipitation varies both inspaceand time.

How could we model the spreading of the fungus now?

AprilFebruary


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Fungus Invasion


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Fungus Invasion by Month


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Sum of Two Cost Surfaces

The least cost between A and B and

passes through a cell.


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Corridor Analysis

Corridor = accumulative cost < a threshold value


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Summary Concepts

What you have just seen is the basis for the mapalgebra language in ArcGIS Grid and Spatial

Analyst

Local functions

Focal functions

Zonal functions

Global functions

map algebra 1

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