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

2. Topics The conventional map algebra Local operations Focal operations
Zonal operations
Global operations

3. Losses (Evaporation, Infiltration)
Map Algebra
Raster layers are manipulated by math-like expression to create new raster layers
Precipitation -
Losses (Evaporation, Infiltration) =
Runoff 5 2 3 4 7 6 - =

4. Map Algebra Operations
Wednesday, April 19, 2017
Map Algebra Operations
Tomlin (1990) defines and organizes operations as local, focal, zonal, and global according to the spatial scope of the operations
Geographic Information System and Cartographic Modeling, Englewood Cliffs: Prentice Hall, 1990.
Tomlin (1983, 1990) defined and organized operations on raster data model as local, focal and zonal according to the spatial scope of the operations.
Geographic Information System and Cartographic Modeling, Englewood Cliffs: Prentice Hall, 1990.
Menton et al (1991) add global operations.
Local
Focal
Zonal
Global

5. 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).

6. 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, …

7. Local Operation--Examples9 7 8 5 6 3 2 1 9 8 6 3 + = 9 7 8 5 6 3 2 1 N
3.5 5 / =

8. Removing Clouds Using a Local Operation
Wednesday, April 19, 2017
Removing Clouds Using a Local Operation
Two consecutive ocean surface temperature raster layers for the same area (measured at a slightly different time).
Biweekly NDVI green report
Images are from:
Images are from:

9. 30-Year (1971-2000) Monthly PRISM Precipitation
Feb.
Aug.
Apr.
Oct.
How do we define seasonality of precipitation at a single location?
Jun.
Dec.

10. Seasonality at San Francisco
Total = 18.69”
[ ]

11. Monthly Precipitation as a Vector Quantity
x=p*sin(monthAngle)
y=p*cos(monthAngle)
Each month’s duration is equivalent to a 30° angle
Monthly data are plotted at midpoint: 15°, 45°, 75°, ……

12. Seasonality Analysis Add all vectors = Resultant vector
Monthly precipitation (in inches):
[ ]
x=p*sin(monthAngle)
y=p*cos(monthAngle)
Add all vectors = Resultant vector

13. Seasonality at San Francisco
Average monthly precipitation at San Francisco in inches
[ ]
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°

14. Seasonality Analysis: Local functions at each cell over the whole domainsy
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])

15. Time of Occurrence

16. Seasonality Index Large number means more uniform
0.34
Large number means more uniform
Small number means more seasonal

17. Map Algebra Operations
Wednesday, April 19, 2017
Map Algebra Operations
Operations are grouped as local, focal, zonal, and global according to the spatial scope of the operations.
Tomlin (1983, 1990) defined and organized operations on raster data model as local, focal and zonal according to the spatial scope of the operations.
Geographic Information System and Cartographic Modeling, Englewood Cliffs: Prentice Hall, 1990.
Menton et al (1991) add global operations.

18. 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

19. 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

20. 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)

21. Wind Exposure Analysis
Wednesday, April 19, 2017
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
Relaxed conditions
DEM > FocalMean
DEM > FocalMedian
FocalPercentile > 95%
How large should the neighborhood be?
Prevailing wind direction may vary from cell to cell (raster layers)

22. Map Algebra Operations
Wednesday, April 19, 2017
Map Algebra Operations
Operations are grouped as local, focal, zonal, and global according to the spatial scope of the operations.
Tomlin (1983, 1990) defined and organized operations on raster data model as local, focal and zonal according to the spatial scope of the operations.
Geographic Information System and Cartographic Modeling, Englewood Cliffs: Prentice Hall, 1990.
Menton et al (1991) add global operations.

23. 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

24. 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, ….

25. Zonal Statistical Operation Example1 4 3 2 1 2 3 4 5 6 7 8 9
Zone Layer
Value Layer
ZonalMax 8 9
Output Layer

26. 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.

27. NEXRAD Cell Precipitation
Measurement spatial resolution

28. Subwatershed Precipitation from NEXRAD Cells Precipitation
model/application resolution

29. NEXRAD Subwatershed Precipitation

30. Calculating Subwatershed Precipitation Depth
pi—precipitation depth in an hour
ai—portion of the watershed that falls in the ith cell a1 a2 a3 a4

31. Map Algebra Operations
Wednesday, April 19, 2017
Map Algebra Operations
Operations are grouped as local, focal, zonal, and global according to the spatial scope of the operations.
Tomlin (1983, 1990) defined and organized operations on raster data model as local, focal and zonal according to the spatial scope of the operations.
Geographic Information System and Cartographic Modeling, Englewood Cliffs: Prentice Hall, 1990.
Menton et al (1991) add global operations.

32. 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

33. Distance Operations
Characterize the relationships between each cell and source cells (usually representing features)
Distance to nearest source cell
Direction to nearest source cell

34. Euclidean Distance Operation
Wednesday, April 19, 2017
Euclidean Distance 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) 1 2 1 2
Hypotenuse

35. EucDistance Example
Buffers can be delineated from the distance raster

36. EucAllocation Example
Voronoi diagram
Thesisen’s polygon

37. 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

38. CostDistance Operation
Wednesday, April 19, 2017
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)
source
friction
Distance example: Slope distance or vertical distance
Time example: reciprocal of speed

39. 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?

40. Fungus Spreading Speed
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

41. The Friction Raster Unit of the friction raster:
Wednesday, April 19, 2017
The Friction Raster
Travel speed
Unit of the friction raster:
days per unit distance
reciprocal
Friction = 1 / speed

42. 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

43. Friction Varies in Space & Time
Wednesday, April 19, 2017
Friction Varies in Space & Time
Precipitation varies both in space and time.
How could we model the spreading of the fungus now?
February
Sometime fungus won’t move
Some time they move fast and some time they move slowly
April

44. Fungus Invasion

45. Fungus Invasion by Month

46. Sum of Two Cost Surfaces
Wednesday, April 19, 2017
Sum of Two Cost Surfaces
All the cells on the least-cost path have the same cost. The path cells are at the bottom of the sum surface.
The least cost between A and B and passes through a cell.

47. Corridor Analysis Corridor = accumulative cost < a threshold value
Wednesday, April 19, 2017
Corridor Analysis
All the cells on the least-cost path have the same cost. The path cells are at the bottom of the sum surface.
Corridor = accumulative cost < a threshold value

48. 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