Field arrangements of blue and white cells exhibiting extreme negative spatial autocorrelation (B) a dispersed arrangement (C) spatial independence (D) spatial clustering (E) extreme positive spatial autocorrelation (Source: Goodchild 1986 CATMOG, GeoBooks, Norwich)

A Sierpinski carpet at two levels of resolution (A) coarse scale (B) finer scale In general, measures of spatial and temporal autocorrelation are scale dependent

Individual rocks may resemble the forms of larger structures, such as rock outcrops or eroded coastlines (© PauloFerreira/iStockphoto)

Spatial sample designs simple random sampling (B) stratified sampling (C) stratified random sampling; (D) stratified sampling with random variation in grid spacing (E) clustered sampling (F) transect sampling (G) contour sampling

An example of physical terrain in which differential sampling would be advisable in order to construct a representation of elevation (Source: M. Langford, University of Glamorgan)

(© Rachel Turk/iStockphoto) We require different ways of interpolating between points, as well as different sample designs, for representing mountains and forested hillsides

The attenuating effect of distance linear distance decay, wij 5 (B) negative power distance decay, wij 5 (C) negative exponential distance decay, wij 5 exp(2bdij).

The creation of isopleth maps point attribute values (B) user-defined classes (C) interpolation of class boundary between points (D) addition and labeling of other class boundaries (E) use of hue to enhance perception of trends (After Kraak and Ormeling 2003: 134)

Choropleth maps a spatially extensive variable, total population (B) a related but spatially intensive variable, population density Many cartographers would argue that (A) is misleading and that spatially extensive variables should always be converted to spatially intensive form (as densities, ratios, or proportions) before being displayed as choropleth maps. (Courtesy Daryl Lloyd)

Map showing 10-, 20-, and 30-minute travel times to a doctor’s surgery in South London. (Courtesy Daniel Lewis; travel time data © Transport for London; map data courtesy Open Streetmap www.openstreetmap.org)

Situations in which a scientist might want to measure spatial autocorrelation (A) point data (wells with attributes stored in a spreadsheet: linear extent of image 0.6 km) (Source: Courtesy ESRI)

(B) line data (accident rates in the southwestern Ontario provincial highway network)

(C) area data (percentage of population that are old age pensioners in Southeast England) (Courtesy Daryl Lloyd)

(D) volume data (elevation and volume of buildings in London: east to the top of the image; image extent 1.6 km north-south, 1.5 km east-west) (Courtesy Duncan Smith)

A simple mosaic of zones

(Courtesy Peter Haggett)

The relationship between recorded length (L) and step length (r)

Part of the Maine coastline (© Anna Omeltchenko.iStockphoto)

The coastline of Maine, at three levels of recursion the base curve of the coastline (B) approximation using 100-km steps (C) 50-km step approximation; and (D) 25-km step approximation.

 Lewis Fry Richardson (Photo from wikipedia: http: 12/16/2009 120//en.wikipedia.org/wiki/File:Lewis_Fry_Richardson.png)