1. Spatial Analysis

2. Overview DEM derivatives: slope and aspect Neighborhood Operations
Boolean Operations with Raster Layers
Algebraic Operations w/ Raster Layers
Neighborhood Operations
Problems with point location data

3. DEM derivatives: slope and aspect

4. GIS Terrain Surface & Topographic Analyses
Operations on terrain data
terrain data frequently produced using spatial interpolation and/or stereoscopic interpretation of aerial photography
Terrain data models
usually represented using a DEM (digital elevation model)
also sometimes as a TIN (triangulated irregular network)

5. GIS Terrain Surface & Topographic Analyses
Basic terrain surface properties, e.g.:
Slope angle (gradient, steepness)
Slope aspect (direction/orientation)
Combine basic properties to achieve more complex analyses or create models

6. Derived Properties: Slope Angle
Slope angle: Change in elevation per unit horizontal change
i.e., how steep is the slope?, what is its gradient?
units generally are degrees or percent
The incline, or steepness, of a surface. Slope can be measured in degrees from horizontal (0-90), or percent slope (which is the rise divided by the run, multiplied by 100). A slope of 45 degrees equals 100 percent slope. Can be done in vector or raster; with raster, must compensate for minor errors due to grid cell quantization of space
GIS calculates slopes and places them into classes
Not limited to elevation  could look at slope of change in population, precipitation, etc.
TIN: software compares the distance between vertices of each of the TIN facets with horizontal coordinates
Raster: Search eight neighboring cells

8. Calculating slope: Raster
From Demers (2005) Introduction to Geographic Information

9. 100 meter cell size 40
100 70 80 50
Gradient?
Slope angle?

10. Steepness of Slope
Slope is a measure of the steepness of a surface and may be expressed in either degrees or percent of slope. In this example, the red cells show steep areas and the green cells show flat areas.

11. Calculating slope: Applications
Erosion analyses
Landslide vulnerability
Directing land development
Updating soil surveys

12. Derived Properties: Slope Aspect
Slope aspect: Orientation of the line of steepest slope
i.e., what direction does the slope face
units generally degrees from cardinal north
Vector: Each of the facets of a TIN has a specific slope and aspect; can be grouped into classes
Raster: Compare a central cell to its neighbors

13. Calculating aspect: Applications
Relating aspect to others layers such as soils, vegetation
Building wind generators
Land use planning
As an input to moisture index

14. Boolean Operations
Boolean operations of OR & AND correspond to UNION & INTERSECTION, used in vector-based analyses A B A B OR A B
UNION A B
AND A B
INTERSECTION
We can apply these concepts in the raster spatial data model, when two input layers contain true/false or 1/0 data:

15. Boolean Operations with Raster Layers
The AND operation requires that the value of cells in both input layers be equal to 1 for the output to have a value of 1: 1 1
AND =
The OR operation requires that the value of a cells in either input layer be equal to 1 for the output to have a value of 1: 1 1 OR =

16. Algebraic Operations w/ Raster Layers
We can extend this concept from Boolean logic to algebra
Map algebra:
Each cell is a number
Mathematical operations are applied to the layers, calculated on a cell-by-cell basis
The result for each cell is placed in a new layer.
Example: suitability analysis
Multiple input layers determine suitable sites:
The pixels attributes in each layer represent ‘scores’.
Layers are weighted based on their importance.
Layer scores are added on a per-pixel basis.

17. Simple Arithmetic Operations1 + = 2
Summation 1  =
Multiplication 1 + = 3 2
Summation of more than two layers
Near the mall
Near work
Near friend’s house
Good place to live?

18. Raster (Image) Difference5 1 7 6 3 4 2 - = -2 -1 -3
The difference between two layers:
An application of taking the differences between layers is a form of change detection:
Example: you have 2 images of the same forest, 10 years apart. To measure forest growth or cuts, you can take the difference in infrared light between image dates.
More infrared light = more chlorophyll and more greenery
Question: How can the locations where a substantial changes have occurred be identified using the difference layer?

19. Monitoring forest fire
Pre-forest fire
Post-forest fire
Burned area identified from space

20. Raster (Image) Division
Question: Can we perform the following operation? Are there any circumstances where we cannot perform this operation? Why or why not?  =

21. More Complex Operations
Linear Transformation 2 3 5 1 4 + = a b c
This could applied in the context of computing a statistical linear regression (e.g. output y = a*x1 + b*x2 + c*x3) using raster layers
Example: suitability analysis.
Suitable place to live example.
Near the mall – low importance (a=1)
Near work – high importance (b=4)
Near friend’s house – moderate importance (c=3)

22. The Revised Universal Soil Loss Equation
(RUSLE) (Wischemeier and Smith 1978)
A=RKLSCP A: the average soil loss
R: the rainfall-runoff factor
K: soil erodibility factor
S: slope steepness
L: the slope length factor
C: crop management factor
P: the support practice factor

23. Neighborhood Operations
In neighborhood operations, we look at a neighborhood of cells around the cell of interest to arrive at a new value.
We create a new raster layer with these new values.
An input layer
A 3x3 neighborhood
Cell of
Interest
Neighborhoods of any size can be used
3x3 neighborhoods work for all but outer edge cells
Neighborhood operations are called convolution operations.

24. Neighborhood Operations
The neighborhood is often called:
A window
A filter
A kernel
They can be applied to:
raw data (BV’s)
classified data (nominal landcover classes)
A 3x3 neighborhood

25. Neighborhood Operation: Mean Filter
The mean for all pixels in the neighborhood is calculated.
The result is placed in the center cell in the new raster layer.
This operation can be done for all cells, or just some cells. 6 4 1 9 8 3 7 2 5
Input
Layer 2 3 4 5 1 7 6 9 8 2 3 4 5 1 7 6 9 8
Result
Layer 3 4

26. Landsat TM 543 False Color Image of Tarboro, NC
Smoothed Image
Landsat TM 543 False Color Image of Tarboro, NC
Normal Image
Smoothing Filter

27. Neighborhood Operations
Suppose you have a nominal dataset: a landuse classification.
Sometimes classifications are ‘speckled’.
Usually a few misclassified pixels within a tract of correctly-classified landcover
How do we correct this? Filter.
We want to reclassify those pixels as the surrounding landcover type.
What kind of filter do we use for this operation?

28. Neighborhood Operation: Majority Filter
The majority value (the value that appears most often, also called a mode filter): 2 4 1 3 8 7 5
Input
Layer
Result

29. Neighborhood Operation - Variance
We may want to know the variability in nearby landcover for each raster pixel:
To find cultivated areas - usually less variability than natural areas
To find where areas where eco-zones meet
The variance of a 3x3 filter on, for instance, an NIR (near infra red) satellite image band will help find such areas. 6 4 1 9 8 3 7 2 5
Result
Layer
Input
2.75
5.75

30. The Mean Operation Revisited
In the mean operation, each cell in the neighborhood is used in the same way:
1/9 6 4 1 9 8 3 7 2 5
Input
Layer
Result

31. Edge Enhancement Cells can be treated differently within a kernel:
This is an edge enhancement filter (discussed below). -1 9 6 4 1 9 8 3 7 2 5
Input
Layer
Result -7
-26

32. Edge Enhancement Filter-1 9
Why is this an edge enhancement filter?
It enhances edges. Ha!
Let’s look at the kernel’s behavior at and away from edges:
Away from edge (in areas with uniform landcover)
At edges (between areas with differing landcover)

33. Edge Enhancement Filter-1 9 20 19 21 5 4 6 21 11 22 30 76 67 66 56
-21
-38
-48 8 9 -6 16
Filter Result:

34. Edge Enhancement Edge enhancement filters sharpen images. Normal Image
Sharpening Filter
Edge enhancement filters sharpen images.

35. Problems with point location data
You have point location data, but you want data for your whole site/region.
Examples:
Air temperature maps created from point data.
Water pollution levels measured at points.
Solution:
Estimate areas with no data.

36. Spatial Interpolation
You have point data (temp or air pollution levels).
You want the values across your full study site.
Spatial interpolation estimates values in areas with no data.
creates a contour map by drawing isolines between the data points, or
creates a raster digital elevation model which has a value for every cell

37. Spatial Interpolation: Inverse Distance Weighting (IDW)
One method of interpolation is inverse distance weighting:
The unknown value at a point is estimated by taking a weighted average of known values
Those known points closer to the unknown point have higher weights.
Those known points farther from the unknown point have lower weights.

38. Sample weighting function
Spatial Interpolation: Inverse Distance Weighting (IDW)
point i
known value zi
distance di
weight wi
unknown value (to be interpolated) at
location x
The estimate of the unknown value is a weighted average
Sample weighting function

39. Issues with IDW
Weighted average estimates are always between the min and max known values.
If the known (sampled) points did not include the minima and maxima (e.g., mountain peaks and valleys), your data will be less extreme than reality
It is thus important to position sample points to include the extremes whenever possible

40. Issues with IDW The dashed line is a hill.
The x’s are sampled elevation points.
The black line is the interpolated (estimated) hill elevation. With IDW, the unknown points tend towards the overall mean.