1. Accuracy, Support, and Interoperability
Michael F. Goodchild
University of California
Santa Barbara

2. The traditional view
Every object has a true position and set of attributes
with enough time and resources we could build a perfect geographic database in any thematic domain
It is the responsibility of the appropriate government agency to construct and disseminate the database

3. A contemporary view
There will be many potential sources of data on any theme
varying in format
varying in the meaning of terms
varying in all dimensions of quality
positional accuracy, attribute accuracy, logical consistency, lineage, currency
varying in spatial support
sampling design

4. Sampling a field f(x) Accuracy of measurement of f
point sample
weather data
linear transect
bathymetry
reporting zone
social data
averaging proportions/ratios
integrating densities
Accuracy of measurement of f
f* = f + δf
Accuracy of measurement of x

5. point sample
reporting zone sample
transect sample

6. For example… f(x) = elevation DEM DLG Spot heights
a raster of point samples
DLG
digitized isolines
Spot heights
irregularly spaced point samples
Triangular mesh (TIN), elevation polygons, splines, means over cells, finite elements,…

7. Support The objects used to characterize a field points, lines, areas
possibly overlapping
though not normally in a single dataset

8. A traditional solution
Force everything to a common support
downscaling?
DLG to raster point sample
congruent with DEM
contour-to-grid (CTOG) interpolation
Spot heights to raster point sample
Weighted average
weights should vary spatially

9. A spatial web solution Query/analysis environment DEM data DLG data
Spot height data

10. The areal interpolation problem
Given attributes of a set of reporting zones
e.g. counties
Estimate attributes of a set of incompatible reporting zones
e.g. watersheds
e.g. to integrate county data with watershed data

11. 1 target zone
4 source zones A B C D
10% of A
15% of B
5% of C
50% of D
PopTARGET = PopA PopB PopC PopD

12. Newer methods co-Kriging Pycnophylactic interpolation hard data
point observations of variable z
sparse but accurate
soft data
point observations of a covariate y
dense but inaccurate or imperfectly correlated
elevation field used to help interpolate a temperature field
Pycnophylactic interpolation
hard data are areal, e.g. population count
interpolate a field of density
ensure integral over areas equals hard data

13. Organizing the methods
Assumptions about characteristics of fields
homogeneous over zones
smoothly varying
Dependence on scale and region
A comprehensive solution
offered as a remotely invokable Grid service

14. The nominal case c(x) every point in the plane assigned to a class
the area-class map
maps of land cover, land use, vegetation class, habitat, ownership, county name

16. Soils Classification Schemes
Petén – Simmons (1959). USDA.
Only data set with soil attributes for Drainage and Fertility
Mexico – Modified FAO/UNESCO (1969)
Belize – Wright (1959). British Honduras Land Use Survey Team.

17. Empirical comparison Areas of overlap Areas of adjacency
Aij = area that is Class i on Map A and Class j on Map B
permute classes to obtain diagonal or block-diagonal matrix
Areas of adjacency
Lij = length of boundary that is Class i on Map A and Class j on Map B

19. Drainage
Finished product

20. Fertility
Finished product

21. Positional uncertainty
The fundamental item of geographic information <x,z>
uncertainty in x
Geographic location
absolute
stored at point, object, data set level
return absolute position

22. Measurement of position
Position measured
x = f(m)
Position interpolated
between measured locations
surveyed straight lines
registered images
The inverse function
m = f -1(x)

28. Errors of position Location distorted by a vector field
x' = x + (x)
(x) varies smoothly
Database with objects of mixed lineage
different vector fields for each group of objects
lineage may not be apparent
e.g. not all houses share same lineage

29. Absolute and relative error
Two points x1, x2
perfect correlation of errors, (x1) = (x2)
no error in distance
zero correlation of errors
maximum error in distance
Absolute error for a single location
measured by (x)
Relative error for pairs of locations
value depends on error correlations

31. Implications Most GIS operations involve more than one point
e.g. distance, area measurement, optimum routing
knowledge of error correlations is essential if error is to be propagated into products
joint distributions are needed
statistics such as the confusion matrix provide only marginal distributions

32. The inverse f -1 An error is discovered in x
error at x1 is correlated with error at x2
both errors are attributed to some erroneous measurement m
to determine the effects of correcting x1 on the value of x2 it is necessary to know f and its inverse f -1

33. Definitions Coordinate-based GIS Measurement-based GIS
locations represented by x
f, f -1 and m are lost during database creation
Measurement-based GIS
f and m available
x may be determined on the fly
f -1 may be available

34. Partial correction
The ability to propagate the effects of correcting one location to others
preserving the shapes of buildings and other objects
avoiding sharp displacements in roads and other linear features
Partial correction is impossible in coordinate-based GIS
major expense for large databases

35. Updating a street database through transactions

36. The geodetic model Equator, Poles, Greenwich
Sparse, high-accuracy points
First-order network
Dense, lower-accuracy points
Second-order network
Interpolated positions of even lower accuracy
Locations at each level inherit the errors of their parents

38. Formalizing measurement-based GIS
Structured as a hierarchy
levels indexed by i
locations at level i denoted by x(i)
locations at level (i+1) derived through equations of the form x(i+1) = f(m,x(i))
locations at level 0 anchor the tree
locations established independently (GPS but not DGPS) are at level 0

39. An example A utility database
Pipe's location is measured at 3 ft from a property boundary
m = {3.0,L}
property at level 3, pipe at level 4
Property location is later revised or resurveyed
new m = {2.9,L}
effects are propagated to dependent object

40. Beyond the geodetic model
National database of major highways
100m uncertainty in position
sufficient for agency
relative accuracies likely higher, e.g. highways are comparatively straight, no sudden 100m offsets
Local agency database
1m accuracy required
two trees with different anchors

42. Merging trees Link with a pseudo-measurement displacement of 0
standard error of 100m
revisions of the more accurate anchor can now be inherited by the less accurate tree
but will normally be inconsequential

44. Conclusions Almost universal adoption of coordinate-based GIS
assumes it is possible to know location exactly
design precision greatly exceeds actual accuracy
in practice exact location is not knowable
attempts at partial correction lead to unacceptable topological and geometrical distortions

45. Measurement-based GIS
Retains measurements and derivation functions
may obtain absolute locations on the fly
Supports incremental update and correction
Supports merger of databases with different inheritance hierarchies
Legacy GIS designs are not optimal