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

Slope Definitions (cont.)

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

1 2 3

4 5 6

7 8 9

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 1 2 3

4 5 6

7 8 9

Aspect

• tan (aspect) = b/c

• Angle between vertical and direction of

steepest slope

• Measured clockwise

• add 180 to aspect if c is positive, 360 to

aspect if c is negative and b is positive

1 2 3

4 5 6

7 8 9

Spatial Analysis:

“Transformations” 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!

X

X

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 model a surface based on the sampled points.

• A predictive numerical model of Z values

• Can be conceptually very simple but it requires an a priori assumption – Can the numbers in each successive step be

determined by a simple mathematical procedure?

Linear interpolation

• a priori assumption

• Assigning values between known points

• Create an isarithmic (contour map)

Contouring a TIN

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