1. BUFFERING

2. POINT IN POLYGON

3. POLYGON OVERLAY

4. SPATIAL INTERPOLATION

5. DENSITY ESTIMATION

Transformations create new objects and data sets from existing objects and data sets

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

Two versions

discrete object:

for every object, result is a new polygon object

new objects may overlap

field (objects cannot overlap):

every location on the map has one of two values:

inside buffer distance
outside buffer distance

every location on the map has a value of distance to the nearest object

CSISS cookbook and UCI's medical center

Determine whether a given point lies inside or outside a given polygon

a type of spatial join

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?

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

diagram

point must lie in exactly one polygon

where are the California ozone monitoring stations?

and how are they distributed by California habitat?

point can lie in any number of polygons, including zero

Create polygons by overlaying existing polygons

how many polygons are created when two polygons are overlaid?

find overlaps between two polygons

e.g. a property and an easement

creates a collection of polygons

overlay two complete coverages

creates a new coverage

e.g. find all areas that are owned by the Forest Service and classified as wetland

in vector or raster

in raster the values in each cell are combined, e.g. added

Areal interpolation

determining attributes for zones from other non-congruent zones

source zones

attributes are known

target zones

attributes are needed

overlay polygons, measure areas, use as weights

diagram

California example

What is interpolation?

intelligent guesswork

an interval/ratio variable conceived as a field

temperature
soil pH
population density

sampled at observation points

needed:

values at other points
a complete surface

a contour map
a TIN
a raster of point values

inverse-distance weighting (IDW)

Kriging (geostatistics)

estimates are averages of the values at n known points

known values z₁,z₂,...,z_n

unknown value z = Sum over i (w_iz_i) / Sum over i (w_i)

where w is some function of distance, such as:

w = 1/d^k

w = e^-kd

an almost infinite variety of algorithms may be used, variations include:

the nature of the distance function
varying the number of points used
the direction from which they are selected

is the most widely used method

objections to this method arise from the fact that the range of interpolated values is limited by the range of the data

no interpolated value will be outside the observed range of z values

peaks and pits will be missed if they are not sampled

outside the area sampled the surface must flatten to the average value

summary: IDW is popular, easy, but full of problems

ozone concentrations at CA measurement stations

objectives:

1. estimate a complete field, make a map
2. estimate ozone concentrations at other locations

e.g. cities

data sets:

measuring stations and concentrations (point shapefile)
CA outline (polygon shapefile)
DEM (raster)
CA cities (point shapefile)

IDW wizard in Geostatistical Analyst

opening screen defines data source

next screen defines interpolation method

which power of distance? (2)
how many sectors? (4)
how many neighbors in each sector? (10-15)

next screen gives results of cross-validation

results map

things to notice

amount of detail where there is no data

generally smooth surface

highs in LA, S central valley

developed by D.G. Krige as an optimal method of interpolation for use in the mining industry

the rate at which the variance between points changes over space

expressed in the variogram

shows how the average difference between values changes with distance

analysis of the data

then application to interpolation

vertical axis is E(z_i - z_j)²

the average difference in elevation of any two points distance d apart

d (horizontal axis) is distance between i and j

most variograms show behavior like the diagram

sill: the upper limit (asymptote)

range: distance at which this limit is reached

nugget: intersection with the y axis

an irregularly spaced sample of points

divide the range of distance into a set of discrete intervals

e.g. 10 intervals between distance 0 and the maximum distance

for every pair of points, compute distance and the squared difference in z values

assign each pair to one of the distance ranges

accumulate total variance in each range

compute the average variance in each distance range

plot this value at the midpoint distance of each range

fit one of a standard set of curve shapes to the points

"model" the variogram

variogram is used to estimate distance weights for interpolation

weights are selected so that the estimates are:

unbiased (if used repeatedly, Kriging would give the correct result on average)

minimum variance (variation between repeated estimates is minimum)

problems with this method:

when the number of data points is large this technique is computationally very intensive

the estimation of the variogram is not simple, no one technique is best

results from this technique can never be absolute

example

selection of method

simple Kriging

co-Kriging includes a correlated variable
indicator Kriging is for binary data

analysis of the variogram

fitting a model
directional effects

how many neighbors?

cross-validation

things to notice

similar pattern

less detail in remote areas
smoother

rebounds to the mean at the edge

better cross-validation

Suppose you had a map of discrete objects and wanted to calculate their density

density of population

density of cases of a disease

density of roads in an area

density would form a field

density estimation is one way of creating a field from a set of discrete objects

count the number of points in every cell of a raster

measure the length of lines, e.g. roads

result depends on cell size

result is very noisy, erratic

think of each point being replaced by a pile of sand of constant shape

add the piles to create a surface

example kernel

width of the kernel determines the smoothness of the surface

density of ozone measuring stations

using Spatial Analyst

kernel is too small (radius of 16 km)

kernel radius 150 km

what's the difference?