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# Topics - PowerPoint PPT Presentation

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

• The conventional map algebra

• Local operations

• Focal operations

• Zonal operations

• Global operations

• 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

• 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

• 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).

• 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, …

• 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

Feb.

Apr.

Jun.

Dec.

Aug.

30-Year (1971-2000) Monthly PRISM Precipitation

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

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]

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°, ……

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

• 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°

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])

0.34

Large number means more uniform

Small number means more seasonal

• Operations are grouped as local, focal, zonal,andglobalaccording to the spatial scope of the 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

• 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

• 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)

• Find max elevation in the prevailing wind direction

• FocalMax with a wedge neighborhood

• Find cells not blocked by hills in the neighborhood

• DEM > FocalMax

• Operations are grouped as local, focal, zonal, andglobalaccording to the spatial scope of the 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

• 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, ….

Value Layer

Zone Layer

ZonalMax

Output Layer

• 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

Subwatershed Precipitation from NEXRAD Cells Precipitation

model/application 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

• Operations are grouped as local, focal, zonal,andglobalaccording to the spatial scope of the 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

• 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)

Buffers can be delineated from the distance raster

Voronoi diagram

Thesisen’s polygon

• 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

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

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

Friction Varies in Space & Time

• Precipitation varies both in space and time.

• How could we model the spreading of the fungus now?

April

Fungus Invasion

Fungus Invasion by Month

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

Corridor = accumulative cost < a threshold value

• 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