Density, Fragmentation and Compactness Metrics

The area metrics defined in the previous section give rise to two measures of average density:

• built-up area density: the ratio of the population within the administrative area of the city and the area of its built-up pixels;
• city footprint density: the ratio of the population within the administrative area of the city and the area of its city footprint.

Five metrics were used to measure different aspects of fragmentation:

• openness index: the average share of open space in the walking distance circle around each built-up pixel in the city;
• city footprint ratio: the ratio of the city footprint and the built-up area in the city;
• infill: all new development that occurred between two time periods within all the open spaces in the city footprint of the earlier period excluding exterior open space;
• extension: all new development that occurred between two time periods in contiguous clusters that contained exterior open space in the earlier period and that were not infill.
• leapfrog: all new development that occurred between two time periods entirely outside the exterior open space of the earlier period.

The Figure on the left below shows a small area on the northern fringe of Bandung in 1991, distinguishing between the different types of open spaces and built-up areas. The figure on the right introduces the infill, extension and leapfrog that took place between 1991 and 2001.

Bandung, 1991: Infill, Extension and Leapfrog in a Small Area on the Urban Fringe

Compactness metrics were calculated to determine to what extent the urban footprint approximates a circle. The most relevant metrics for measuring the compactness of cities are the Proximity Index and the Cohesion Index defined by Angel et al (2010). The proximity index measures the relative closeness of all locations in the city to its center. The cohesion index measures the relative closeness of all locations in the city to each other. To calculate these indices we first define the Equal Area Circle: a circle with an area equal to that of the City Footprint centered at the city's center, identified as the location of its City Hall (CBD).

The equal area circle for Bandung in 1991 is shown in the figure below.

Bandung, 1991: The Equal-area Circle of the City Footprint

The two compactness indices used in the Atlas are defined as follows:

• The Proximity Index is the ratio of the average distance from all points in the equal-area circle to its center and the average distance to the city center from all points in the city footprint.
• The Cohesion Index is the ratio of the average distance among all points in an equal-area circle and the average distance among all points in the city footprint.

Both indices vary between 0 and 1, with higher values corresponding to city footprints that are closer in shape to the circle. The value of Proximity Index for Bandung in 1991 was 0.87 and the value of the Cohesion Index was 0.82.

For every city in the global sample of 120 cities, the right-hand pages of the atlas display values for all the metrics defined here. The values are displayed both in table form and in graph form. The graphs compare the metrics for the city to both the regional average and the global average.

For every city in the global historical sample of 30 cities, we only have data for urban land cover and density and these are displayed in both table form and graph form. The graphs show the change in population, urban land cover, density, and density change over time.