1. Spatial Analysis

2. Digital Elevation Model (DEM)
DEM is the representation of continuous elevation values over a topographic surface by a regular array of points with 3 coordinates (x,y for location on the surface and z-value for altitude).

3. DEM Derivatives Slope Hillshade Aspect
A number of surfaces can be derived from a DEM: slope, aspect, hillshade, contour lines, viewsheds.
Aspect
DEM Analysis:

4. Slope and aspect
Slope and aspect are calculated at each point in the grid, by comparing the point’s elevation to that of its neighbors
Slope is the incline or steepness of a surface (measured in degrees 0 – 90, or as a percentage of a rise divided by a run)
Aspect is the compass direction that a topographic slope faces usually measured in degrees from north

5. Draping
Hillshading creates a 3-dimensional effect that provides a sense of visual relief, derived using a hypothetical illumination of a surface according to a specified azimuth and altitude for the sun
Draping – a rendering of a 2 dimensional image onto a 3 dimensional surface
Upper image: DEM draped over hillshade
Lower image: Slope draped over hillshade

6. Buffering
Creates a new object consisting of areas within a user-defined distance of an existing object, for example:
To determine areas impacted by a proposed highway
To determine the service area of a proposed hospital
Can be done for both a raster and a vector

7. Buffering
Point
Polyline
Polygon

8. Point-in-polygon transformation
Determine whether a point lies inside or outside a polygon
generalization: assign many points to containing polygons
used to assign crimes to police precincts, voters to voting districts, accidents to reporting counties

9. Point-In-Polygon

10. Map Algebra
A language that allows to transform a raster map or combine two or more raster maps by applying mathematical operations and analytical functions
Local: cell-by-cell operations
Focal: operations performed on a user-defined neighborhood of the focus cell
Zonal: process all cells within a user-defined regions (zones)
Global: the cell values for the output grid can be dependent upon all the cells in the input grid
Map algebra:
Creates new output
Uses combinations of mathematical, logical, and Boolean operations (e.g. +, AND, >=, mean, tan)

11. Map Algebra Example: Sum
What if we add a constant number e.g. 3 to the 1st grid? How would you do that?
The neighborhoods can vary in shape and size, here a 3x3 square neighborhood is used.

12. Map Algebra Example: Sum
Sample zones: red is urban, yellow is rural
Global sum: what about the other cells?

13. Spatial interpolation (Tobler’s First Law of Geogaphy)
The process of using points with known values to estimate values at other points. These points with known values are called known points, control points, sample points, or observations.
Spatial interpolation is a process of intelligent guesswork, in which the investigator (and the GIS) attempt to make a reasonable estimate of the value of a continuous field at places where the field has not actually been measured. The one principle that underlies all spatial interpolation is the Tobler Law.

14. Spatial interpolation
Distance Decay

15. Importance of the Density of Sample Points
Imagine this elevation cross section: If each dashed line represented a sample point, this spacing would miss the major local sources of variation, like the gorge

16. Importance of the Density of Sample Points
If you increase the sampling rate (take samples closer together), the local variation will be more accurately captured

17. Importance of the Density of Sample Points

18. Kriging
Kriging is a spatial interpolation technique that assumes that the spatial variation of an attribute may consist of three components: a spatially correlated component, representing the variation of the regionalized variable; a ‘drift’ or structure, representing a trend; and a random error term.
Developed by Georges Matheron to evaluate new GOLD mines with a limited number of borholes.

19. Density estimation
Spatial interpolation is used to fill the gaps in a field
Density estimation creates a field from discrete objects. The field’s value at any point is an estimate of the density of discrete objects at that point
E.g. estimating a map of population density (a field) from a map of individual people (discrete objects)

20. Kernel Density Surfaces
Each discrete object is replaced by a mathematical function known as a kernel. Kernels are summed to obtain a composite surface of density. The smoothness of the resulting field depends on the width of the kernel.
When the kernel width is too small (in the left case it is ~80km when search area equals 20,000km2) the surface is too rugged and bumpy, and each point generates its own peak. The kernel width in the right case is ~178.5km (when search area equals 100,000km2) and produces much more plausible smooth density surface.
Summary: The density is a function of a distance used to calculate it.
Data shown:
Ozone measurement stations in CA (NOT the measurements of ozone themselves)
ArcGIS Spatial Analyst was used to generate the surfaces.
Search radius: 20K km2
Search radius: 100K km2