Statistical Surfaces

• Any geographic entity that can be thought of as containing a Z value for each X,Y location– topographic elevation being the most obvious

example– but can be any numerically measureble attribute

that varies continuously over space, such as temperature and population density (interval/ratio data)

Surfaces

• Statistical surface

• Continuous

• Discrete

Statistical Surfaces

• Two types of surfaces:– data are not countable (i.e. temperature) and

geographic entity is conceptualized as a field – punctiform: data are composed of individuals

whose distribution can be modeled as a field (population density)

Statistical Surfaces

• Surface from punctiform data

Distribution of trees

2

43

3

2

2

3

4

2

3

3

4

4

3

6

2

2

3

1

1

2

2

3

3

2

Find # of trees w/in the neighborhood of each grid cell

Point data Density surface

Statistical Surfaces

• Storage of surface data in GIS

– raster grid– TIN– isarithms (e.g. contours for topographic

elevation)– lattice

Statistical Surfaces

• Isarithm

10

2030

40

50

60

7080

60

Statistical Surfaces• Lattice: a set of points with associated Z values

Regular Irregular

Statistical Surfaces• Interpolation

– estimating the values of locations for which there is no data using the known data values of nearby locations

• Extrapolation– estimating the values of locations outside the

range of available data using the values of known data

We will be talking about point interpolation

Statistical Surfaces

Estimating a point here: interpolation

Sample data

Statistical Surfaces

Estimating a point here: interpolation

Estimating a point here: extrapolation

Statistical Surfaces• Interpolation: Linear interpolation

Elevation profile

Sample elevation data

A

B

If

A = 8 feet and

B = 4 feet

then

C = (8 + 4) / 2 = 6 feetC

Statistical Surfaces• Interpolation: Nonlinear interpolation

Elevation profile

Sample elevation data

A

B

C

Often results in a more realistic interpolation but estimating missing data values is more complex

Statistical Surfaces• Interpolation: Global

– use all known sample points to estimate a value at an unsampled location

Use entire data set to estimate value

Statistical Surfaces• Interpolation: Local

– use a neighborhood of sample points to estimate a value at an unsampled location

Use local neighborhood data to estimate value, i.e. closest n number of points, or within a given search radius

Statistical Surfaces• Interpolation: Distance Weighted (Inverse Distance Weighted - IDW)

– the weight (influence) of a neighboring data value is inversely proportional to the square of its distance from the location of the estimated value

4

3

2

100

160

200

Statistical Surfaces• Interpolation: IDW

4

3

2

100

160

200

100 x 1 = 100 160 x 1.8 = 288 200 x 4 = 800

1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500

.0625 / .0625 = 1 .1111 / .0625 = 1.8 .2500 / .0625 = 4

Weights Adjusted Weights

100 +288 + 800 = 1188

1188 / 6.8 = 175

Statistical Surfaces

• Interpolation: 1st degree Trend Surface– global method– multiple regression (predicting z elevation with x and y location

– conceptually a plane of best fit passing through a cloud of sample data points

– does not necessarily pass through each original sample data point

Statistical Surfaces• Interpolation: 1st degree Trend Surface

x

yz

x

y

In two dimensions In three dimensions

Statistical Surfaces• Interpolation: Spline and higher degree

trend surface– local– fits a mathematical function to a neighborhood

of sample data points– a ‘curved’ surface– surface passes through all original sample data

points

Statistical Surfaces• Interpolation: Spline and higher degree trend surface

x

yz

x

y

In two dimensions In three dimensions

Statistical Surfaces

• Interpolation: kriging– common for geologic applications– addresses both global variation (i.e. the drift or

trend present in the entire sample data set) and local variation (over what distance do sample data points ‘influence’ one another)

– provides a measure of error