areal interpolation and spatial convolution - csiss · areal interpolation and spatial convolution...
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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