Using Space/Time Transformations
to Map Urbanization in

Lee De Cola

U.S. Geological Survey
521 National Center
Reston VA 20192

*A revised version of a paper that originally appeared in the Technical Papers of the 1997 Convention of ACSM/ASPRS, Vol. 5 Auto-Carto 13, pp. 56-65.

During the past year researchers at the U.S. Geological Survey have been using historical maps and digital data for a 2° × 2° (168-km × 220-km) area of the Baltimore/Washington region to produce a dynamic database that shows growth of the transportation system and built-up area for 30-meter grid cells for several years between 1792 and 1992. This paper presents results from the development of a Mathematica package that spatially generalizes and temporally interpolates these data to produce a smoothly varying urban intensity surface that shows important features of the 200-year urban process. The boxcount fractal dimension of a power-2 grid pyramid was used to determine the most appropriate level of spatial generalization. Temporal interpolation was then used to predict urban intensity for 4320-m cells for 10-year periods from 1800 to 1990. These estimations were spatially interpolated to produce a 1080-m grid field that is animated as a surface and as an isopleth (contour) map (see USGS 1997 for the Internet address of the animation). This technique can be used to experiment with future growth scenarios for the region, to map other kinds of land cover change, and even to visualize quite different spatial processes, such as habitat fragmentation due to climate change.

Baltimore-Washington study area
Figure 1. The study area in the Boston-Washington megalopolis.

In 1994 a team of U.S. Geological Survey (USGS) and academic researchers produced an animation of the growth of the San Francisco/Sacramento region using a temporal database extracted from historical maps, USGS topographic maps, digital data, and Landsat imagery (Gaydos and Acevedo 1995). Publicly televised videotapes of this work received sufficient attention to support a larger team that had planned to work on the development the Boston/Washington megalopolis (Gottmann 1990) (The current research involves staff from USGS, National Air and Space Administration, the Smithsonian Institution, and University of Maryland Baltimore County.) Resource and time constraints, however, limited efforts to the southern part of the region shown in figure 1 (Crawford-Tilley et al 1996, Clark et al. 1996). The animation of urbanization in this region is based on a 5122-cell grid data structure that represents whether or not a given 270-meter cell is built-up in each of 8 base years (figure 2). This raster was interpolated for intervening years, but still represents a binary condition for each of the grid cells. Throughout this work there was interest in how we might analyze the intensity of development, perhaps by sacrificing spatial resolution for temporal and measurement resolution (table 1). Because the urban phenomenon (cartographic feature) is self-organized, complex, and probably also critical (Bak 1996), it is reasonable to suppose that scaling properties would assist in this transformation (Quattrochi and Goodchild 1997).

Table 1. Dimensions of the data

Consider therefore a location in space x at a given time t and spatial resolution level l for which a measurement f is made; call this measurement fl(xt). For example, in the present case we are interested in whether or not a given grid cell of a certain size is built-up (covered by buildings, has a dense road network, etc.). In this simplest case we have a binary function fl(xt) = {1 if xt is built-up, 0 otherwise}. Assume at the finest scale level l = 0 that this measurement is reliable-but what can be said of the phenomenon at other spatial scales? Table 2 shows how a 10-level power-2 image pyramid can be built upon the 0-level data in the present case. One (not necessarily obvious) way to examine data at coarser scales is simply fboxl+1(xt) = {0 if all fl(xt) = 0, 1 otherwise}, i.e. the value of a higher-level l + 1 cell will be "on" if any of the lower-level l cells is on. This is called a box-covering algorithm because a high-level box is needed to cover 1 or more lower-level boxes (De Cola 1997).

1992 level 0
Figure 2. Level 0 grid for 1992.

Consider the 0-level image of figure 2, which contains 41,183 built-up cells, as reported in the last row of table 3, which presents the box counts for each level and each of the raw data years The table shows at the next highest level l = 1 that 14,892 cells are necessary to cover these cells. This number is 45% larger than the (41,183 / 4 =) 10,296 level-1 cells that would be necessary if all the level-0 cells were spatially compact. The excess number is due to spatial complexity of the urban phenomenon, which has fractal dimension D < 2, where D = 2 would be the dimension of say a perfect disk (for a comprehensive discussion of the fractal nature of cities see Batty 1995).

Table 2. Characteristics of the image pyramid
pyramid table

The 0-level row of the table 3 illustrates that for at least 200 years there has been some urbanization in the region (A fit of a linear model to the 0-level data yields ln[f0(xt)] = 40 + 0.026 t which predicts a y-intercept at about the year 1575). The table cells that are shaded represent completely covered pyramid levels, showing how in later years the windows rapidly become saturated. This happens at l = 8 in 1792 and by level 6 in 1972 and later. One way to avoid this saturation is to expand the extent of the study area, and this indeed is underway. But another problem with this analysis is that traditional maps (1772-1850) produced to widely varying cartographic styles, are being analyzed along with carefully standardized USGS maps (1900-1953) and satellite imagery (1972-1992). Nevertheless-and this is another advantage of multiscale analysis-at coarser scales the difference among these disparate data sources diminishes.

Table 3. Box Counts for each year and level
boxcount table

The box counts in table 3 can be used to compute the fractal dimension of the built-up area for each year. For example, figure 3 shows the regression line estimating log2[fl(x1953)] = 0.89 1.51 l for 1953, which yields a fractal dimension of D1953 = 1.51 and an R2 = .99 (Falconer 1990).

Estimation of fractal dimension
Figure 3. Fractal dimension estimation 1953.

The box counts for each level and each year are used to compute the 8 values of Dt, the fractal dimensions for each of the data years, shown in figure 4. There is a continuing debate in urban studies about how regions develop. One school argues that so-called "primate" metropolitan regions continue to grow from a point to a centralized but spreading metropolitan pole. But another school envisions a dispersed metropolis that may eventually completely disperse, returning to a collection of isolated points (Alonso 1980, De Cola 1985). Figure 3 certainly shows the early stages of this process; we can only speculate about whether Dt will eventually decline, although its rate of increase seems to be leveling off. This scenario suggests the possibility of future dispersion in which the urban complex not only breaks up into dispersed centers but even perhaps returns to the low-dimension post-industrial "village" system similar to that of the 18th century.

Time series of fractal dimensions
Figure 4. Boxcount fractal dimensions 1792-1992.

Each of the fractal dimensions Dt for the data years is a linear estimate of the behavior of the box counts over the scale levels. Yet the fit is not perfect, as figure 3 shows for 1953; there is a similar pattern of parabolic residuals among all the years. In general the middle scale levels l = 4 and 5 have higher residuals, suggesting that at about the 6-km scale the urban area has its most compact representation. But the box count aggregation algorithm, which yields 0/1 values, cannot be used to generalize the data. Another way to aggregate grid data is to sum lower-level values using fsuml+1(xt) = fl(xt) where the aggregation is over subwindows of 4 cells each. The algorithm fsum is like a mean filter that aggregates subregions into a higher-level region whose value is the average of lower-level elements. The generalized animation is therefore based on the level-4 generalization, which gives for each of 322 = 1024 cells of size 4320-m an 8-bit dynamic range of [0, 256] (see table 2). Figure 5 shows what happens to the 1992 data for 5 successive levels of aggregation. The lower-level images allow us to focus on the individual features of the region, while the higher-level images highlight the unified nature of the BaltWash metropolis.

3 levels and 2 interpolations
Figure 5. 3 levels from the image pyramid with 2 interpolations.

Let l = 4 and consider the central-cell x = (col, row) = (16, 16) for each of the t = 1,…,8 data years. The values of fsum4(xt) for this cell are shown in figure 6 and (as did Dt in figure 4) these points suggest a logistic curve, which can be estimated with an interpolation (prediction) function fsump that predicts fsum for any year and not just the 8 data years. Figure 6 shows {fsump4(xt): x = (16,16), t [1750 to 2000]}. When this function is used at level-4 we only get 322 predictions. This is how we obtain a

central cell interpolation
Figure 6. Actual and interpolated values for cell (16,16).

The unique temporal interpolation functions for each of the (322 =) 1024 level-4 cells can be arrayed into a Mathematica table that provides a grid of predictions for any year in the study period. A sample for 1990 is shown in figure 7, taken from the animation (USGS 1997). The data have been spatially linearly interpolated to level 1 (540 meters) to provide a smooth surface for visualization (for a alternative approaches to the interpolation problem see Tobler 1979 and Bracken and Martin 1989). The image, which is one frame of a 20-period animation, illustrates the polycentric nature of the Baltimore/Washington urban process. The animation shows reveals a self-organizing system that has been growing along the Northeast U.S. transportation corridor. During the past 200 years urban leadership has shifted between the two centers at least three times, and since World War II there has arisen a polycentric post-industrial system whose fractal dimension has been growing logistically and may be leveling off.

Another way to visualize the growth process is isopleths or contours, which emphasize the geographic location of urbanization. figure 8 shows not only the 2 urban centers in 1992, but such other features as the edge cities of Frederick, Annapolis, and La Plata, MD as well as Potomac Mills, VA. The picture also highlights the linear nature of the whole system, oriented along Interstate 95, which continues from Boston to Miami.

6 selected years of surfaces
Figure 7. 3-d plot of estimated built-up areas for selected years.

Naturally we are interested in the future of the region, and the analysis suggests approaches. (A logistic curve fitted to the 0-level data in table 2 yields f0(xt) = 55800 [1 + Exp(2.09 - 0.0469(t - 1923))]-1, which has a maximum growth rate of 2.1% in 1923 (Haggett, Cliff and Frey 1977:238)). This expression has an asymptotic value of 55,800 pixels, which is only about 20% of the window at level-0.

labeled contour plot 1990
Figure 8. Contour plot of estimated built-up areas 1990.

The analysis of the last 3 data years (1972, 1982, 1992) was based on Landsat imagery, and the growth both of the fractal dimension Dt (figure 4) as well as of one of the generalized cells fsum4(xt) (figure 6) show a linear growth trend. The growth rate for 1972-1992 is mapped in figure 9; darker shades show faster growth-up to 2% per year. Recent metropolitan development displays the doughnut patterns typical of U.S. cities (Whyte 1968). The Baltimore growth ring is broken by Patapsco Bay and the Washington ring by a Potomac River "greenbelt" that would clearly be the fastest growing edge city were the river bridged from Sugarland Run VA to Seneca Creek MD. It is interesting how strongly topography still influences the development of this region.

growth 1972-92
Figure 9. Contour plot of growth rates, 1972-1992.

The research presented here is part of a 118-year history of the use of USGS core skills in the physical, and-more recently-human and biological sciences to understand human-induced land transformations. These efforts exhibit not only institutional expertise but also rich historical databases that can be used to understand spatial processes, to forecast change, and help to shape future policy. The dimensions highlighted in table 1 suggest new directions for this research. First, the analysis can profit from a broader spatial view, expanding to Megalopolitan and even world urbanization. Second temporal extrapolation and deeper "data mining" will help planners envision the future of the region-as well as its distant past. Third, more features (shoreline, land cover, climate) need to be studied and animated. A central theoretical and policy problem highlighted by this work therefore is the development of rigorous, informative, and visually effective transformations of data along and among spatial, temporal, and phenomenological scales.


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