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

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

  3. 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

  4. 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

  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--Examples = + = /

  8. 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

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

  10. 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]

  11. 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°, ……

  12. 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

  13. 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°

  14. 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])

  15. Time of Occurrence

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

  17. Map Algebra Operations • Operations are grouped as local, focal, zonal,andglobalaccording to the spatial scope of the 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 • Find max elevation in the prevailing wind direction • FocalMax with a wedge neighborhood • Find cells not blocked by hills in the neighborhood • DEM > FocalMax

  22. Map Algebra Operations • Operations are grouped as local, focal, zonal, andglobalaccording to the spatial scope of the 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 Example Value Layer Zone Layer ZonalMax 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. 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

  31. Map Algebra Operations • Operations are grouped as local, focal, zonal,andglobalaccording to the spatial scope of the 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. 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)

  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. 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)

  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. Travel speed The Friction Raster Unit of the friction raster: days per unit distance 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. February Friction Varies in Space & Time • Precipitation varies both in space and time. • How could we model the spreading of the fungus now? April

  44. Fungus Invasion

  45. Fungus Invasion by Month

  46. Sum of Two Cost Surfaces The least cost between A and B and passes through a cell.

  47. Corridor Analysis 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

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