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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Map Algebra Operations
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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Map Algebra Operations
Operations are grouped as local, focal,
zonal, andglobal according to thespatial
scopeof the operations.
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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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Map Algebra Operations
Operations are grouped as local, focal,
zonal,andglobalaccording to thespatial
scopeof the operations.
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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
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