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

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

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

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

point sample reporting zone sample transect sample

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,…

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

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

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

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

1 target zone 4 source zones A B 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

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

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

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

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.

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

Drainage Finished product

Fertility Finished product

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

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)

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

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

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

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

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

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

Updating a street database through transactions

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

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

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

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

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

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

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