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Areal Interpolation and Spatial ConvolutionAreal Interpolation and Spatial Convolution

Michael F. GoodchildUniversity of California

Santa Barbara

Areal interpolationAreal interpolation

Given a set of non-overlapping, space-exhausting source zones– census tracts– with attributes

Estimate attribute values for a second set of non-overlapping, space-exhausting target zones– that cut the source zones

Attribute typesAttribute typesSpatially intensive– mean value of a field over a zone– average income, population density, percent

married– mean, density, proportion

Spatially extensive– population count– integral of a density field– volume under the surface

Thou shalt not map a spatially extensive variable– or mix the two types in a single linear model

Approaches (1)Approaches (1)

1. Replace source zones by centroids2. Interpolate a continuous field on a

dense raster (spatially intensive)3. Re-aggregate cells to the target zones

Approaches (2)Approaches (2)

Area weighting (spatially extensive)

∑∑=i

iji

ijij aaST

1 target zone4 source zones

AB

C

D

10% of A

15% of B

5% of C

50% of D

PopTARGET = 0.10 PopA + 0.15 PopB + 0.05 PopC + 0.50 PopD

Simple variantsSimple variants

Spatially intensive variableTarget zones homogeneous– OLS solution for target zone densities

Control zones homogeneous

Approaches (3)Approaches (3)

Pycnophylactic interpolation– Tobler, JASA ~1980

Find a field z– represented on a raster– spatially intensive– integral over each source area matches the input

(spatially extensive) attribute– the field is maximally smooth– imposed boundary condition

• outside = 0• outside equal

Tobler's algorithmTobler's algorithm

1. Apportion each source zone's spatially extensive attribute among overlapping cells

2. Replace each cell's value by the mean of its neighbors

3. Renormalize each source zone's sum to given source value

4. Repeat from (1) until stable

A geostatistical approachA geostatistical approach

Definition– Support = Domain informed by each datum or

unknown value

Assumption of underlying point support field– Actual value unknown– Viewed as realizations of stationary Random

Field (RF) model– Parametrized by a mean and covariance

function

{ }( ),z D∈x x

( ){ } ,ZE Z m= ∀x x ( ) ( ){ } ( ), ' 'ZCov Z Z C= −x x x x

Framework propertiesFramework properties

General: can handle integrated measurements over arbitrary domainsSimple: utilizes standard geostatistical theory with minor modificationsComprehensive: can handle alternative types of point covariance modelsConsistent: guarantees reproduction of data at larger scales (mass preserving)Providing uncertainty assessment: regarding target predictions

A simple exampleA simple exampleTarget zone configurationTarget zone configurationSource zone configuration and data valuesSource zone configuration and data values

Results: No spatial correlationResults: No spatial correlation

Pure nugget effect

Unit sill

Spherical semivariogram modelSpherical semivariogram model

Spherical model

Unit sillRange = 10

Gaussian semivariogram modelGaussian semivariogram model

Gaussian model

Unit sillRange = 20

Exponential variogram modelExponential variogram model

Anisotropic exponential model

Unit sillRange = 50 for -45°Range = 10 for 45

Example resultsExample results

“Geographic effect” more pronounced for smoother variogram modelsPrediction uncertainty is a function of target areaSum of product of source values and source areas equals sum of product of target predictions and target areas

( ) ( )∑∑==

⋅=⋅K

1kkk

P

1ppp sszttz

Spatial convolutionSpatial convolution

A point in the context of its surroundings– how to characterize the surroundings?

Containing polygon– varying size and shape– no control over scale– arbitrary– points near the edge

Convolution functionConvolution function

Centered on the pointDecreasing with distanceNegative exponential

0)( xxbexw −−=

Convolution functionConvolution function

Rasterize the layer– e.g. population

Sum over the raster– cell value times weight

Divide by the sum of the weights