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### Spatial Analysis: “Transformations”

Slope and Aspect

- Calculated from a grid of elevations (a digital elevation model)
- Slope and aspect are calculated at each point in the grid, by comparing the point’s elevation to that of its neighbors
- usually its eight neighbors
- but the exact method varies
- in a scientific study, it is important to know exactly what method is used when calculating slope, and exactly how slope is defined

Slope Definitions

- Slope defined as an angle
- … or rise over horizontal run
- … or rise over actual run
- various methods
- important to know how your favorite GIS calculates slope

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Slope and Aspect- measured from an elevation or bathymetry raster
- compare elevations of points in a 3x3 neighborhood
- slope and aspect at one point estimated from elevations of it and surrounding 8 points
- number points row by row, from top left from 1 to 9

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Typical Slope Calculation- tan (slope) = sqrt (b2 + c2)
- b = (z3 + 2z6 + z9 - z1 - 2z4 - z7) / 8D
- c = (z1 + 2z2 + z3 - z7 - 2z8 - z9) / 8D
- b denotes slope in the x direction
- c denotes slope in the y direction
- D is the spacing of points (30 m)
- find the slope that fits best to the 9 elevations
- minimizes the total of squared differences between point elevation and the fitted slope
- weighting four closer neighbors higher

Longley et al., chs. 13 and 14

Transformations

- Create new objects and attributes, based on simple rules
- involving geometric construction or calculation
- may also create new fields, from existing fields or from discrete objects

“ Transformations” - New Data

- new objects and data sets from existing objects anddata sets
- BUFFERING
- POINT IN POLYGON
- POLYGON OVERLAY
- SPATIAL INTERPOLATION
- DENSITY ESTIMATION

Buffering

- buffering takes points, lines, or areas and creates areas
- every location within the resulting area is either:
- in/on the original object
- within the defined buffer

width of the original object

Applications

- find all areas of Siuslaw National Forest beyond 1 mile from a road
- find all households within 1 mile of a proposed new freeway
- and send them notification of proposal
- find all liquor stores within 1 mile of a school
- and notify them of a proposed change in the law

Variants

- vary the object\'s buffer width according to an attribute value
- e.g. noise buffers depending on road traffic volume
- vary the rate of spread according to a friction field
- Thiessen polygons

for point objects

Point-in-Polygon

- Determine whether a given point lies inside or outside a given polygon
- assign a set of points to a set of polygons
- e.g., count numbers of accidents in counties
- e.g., whose property does this phone pole lie in?
- Algorithm
- draw a line from the point to infinity
- count intersections with the polygon boundary
- inside if the count is odd
- outside if the count is even

Point-in-Poly Algorithm

- inside if the count is odd
- outside if the count is even
- what if the point lies on the boundary?

Polygon Overlay

- Create polygons by overlaying existing polygons
- how many polygons are created when two polygons are overlaid?
- Discrete object
- find overlaps between

two polygons

- creates a collection

of polygons

Overlay Issues

- in raster the values in each cell are combined -- binary, rating models??
- major computing workload
- Indexing
- swamped by slivers
- tolerance

Spatial Interpolation & GIS

- to calculate some property of a surface at a given point
- model all the REAL intricacies of a surface
- to provide contours
- highlight general spatial trend of data for decision-making

What’s A Wildlife Manager to Do?

- 23 animals assumed everywhere?
- pitfalls of applying classical statistics to spatial data
- give “spatial” characterization to the mean (23)
- let’s interpolate!

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Spatial Interpolation- need to estimate values at locations where there are no explicit data
- estimates must be determined from surrounding values

Spatial Interpolation - cont.

- We need to be able to modela surface based on the sampled points.
- A predictive numerical model of Z values
- Can be conceptually very simple but it requires an a prioriassumption
- Can the numbers in each successive step be determined by a simple mathematical procedure?

Linear interpolation

- a prioriassumption
- Assigning values between known points
- Create an isarithmic (contour map)

Nonlinear Interpolation

- When things aren\'t so simple
- Can’t assume linearity of features
- Basic types:1. Trend surface analysis

2. Minimum Curvature Spline

3. Inverse Distance Weighted 4. Kriging

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