MAPPING URBAN GROWTH AND DEVELOPMENT AS CONTINUOUS FIELDS IN SPACE AND TIME
Authors: Small Christopher Sennen 
Published: June 2015

<p>Cities are often depicted as discrete entities – both on maps and in analysis. While the individual components of cities may be discrete (e.g. people, buildings, firms), the functional entity of a city can be difficult to define as a discrete object. Administrative boundaries of cities are discrete but in many ways they are functionally irrelevant to the processes that occur within the city and with its surrounding communities.  Attempts to classify cities as discrete spatial objects generally fail to produce useful depictions because the definitions on which the classifications are based are often arbitrary. Two persistent obstacles to discrete classification of urban extent are the lack of a consistent definition and the scale dependence of the most easily measurable quantities (e.g land cover) on which the definitions are based. As an alternative to discrete definition, cities can be depicted as parts of a continuum. Depiction of a city as part of a continuum can accommodate both form and function.  In terms of form, a city might be considered a local maximum of density of some component, or combination of components (e.g. population density, building density, or road density) that varies continuously in space and time. In terms of function, a city might be considered a local maximum of activity, or combination of activities (e.g. economic output, innovation or information exchange). Depicting cities as entities within continuous fields offers at least two advantages over discrete classification; flexibility and information content.  Discrete classifications trade information for simplicity. The two modes of viewing geography in Google Earth (map &amp; satellite) provide an example of this trade-off.  The map view is generally simpler to interpret but it contains far less information than the satellite view. However, continuous fields can still provide a basis for discrete classification when simplicity of depiction is needed. Continuous fields may be segmented into discrete components by imposing thresholds. For example, a city might be defined as the area with population density above some threshold. The ability to impose different thresholds on a single continuous field offers the flexibility to accommodate different definitions in situations where there is no consensus on a single definition. For example, a city might be considered alternately as a place with population density greater than 100 persons/km<sup>2</sup>, 1000 persons/km<sup>2</sup> or 10,000 persons/km<sup>2</sup>. One important asymmetry between discrete and continuous depictions is the ability of continuous fields to represent abrupt changes and boundaries and the inability of discrete depictions to represent gradual changes and gradients. The objective of this paper is to illustrate some benefits of continuous fields for depiction of urban growth and development. In this context, urban growth refers to expansion in space, either vertical or horizontal. Development refers to progressive changes in form or function that lead to improved living standards.  Continuous field depictions are illustrated using remotely sensed imagery – but the underlying concepts are generally applicable to other measurable quantities like population density or economic activity. Continuous field depictions can be extended to represent change by mapping differences in time.  Examples are provided for multiple quantities (land cover type and night light brightness), measured by multiple sensors (Landsat, DMSP-OLS and VIIRS) at multiple times (1990, 2010, 2012).  Some characteristics of urban growth and development are illustrated with continuous field depictions of large cities and their surrounding regions from the rapidly developing BRIC countries; Brazil, Russia, India and China. In addition to multi-scale, multi-sensor, multi-temporal depictions of urban areas as continuous fields, the utility of discretization with multiple thresholds is illustrated by comparing city size distributions obtained from night lights.<strong><em></em></strong></p>