A satellite picture of adjacent cities clearly illustrates the analytical dynamics of urban growth

Waldo R. TOBLER

Mappemonde

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VISUAL EVIDENCE FOR URBAN POTENTIAL FIELDSWaldo R. TOBLER*M1/1991APPEONDEA satellite picture of adjacent citiesclearly illustrates the analytical dynamics ofurban growth.• DALLAS • DYNAMICS • FORT WORTH • SA-TELLITE PICTURE • URBAN GROWTHUne image satellite de deux villes voi-sines met clairement en évidence la dynamique«analytique» de la croissance urbaine.• CROISSANCE URBAINE • DALLAS • DYNA-MIQUE • FORT WORT • PHOTO SATELLITE Una imagen por satélite de dos ciu-dades vecinas evidencia claramente la dinámi-ca «analítica» del crecimiento urbano.• CRECIMIENTO URBANO • DALLAS • DINÀMICA• FORT WORTH • FOTOGRAFÍA SATELITARIAFor over a hundred years attempts have been made to apply to thesocial sciences concepts expropriated from physics. The dangerinherent in reasoning from analogy has always caused these effortsto be regarded with some suspicion. The gravity model of humaninteraction, however, is one model which seems to have gainedwidespread acceptance, mainly because it describes aggregatehuman behavior quite well (1).Most of the evidence used to calibrate the social gravity model hascome from census enumerations, the time and space scales ofhuman interactions being such that direct sensory observations aredifficult. The satellite photograph (fig. 1), showing two terrestrialcities, it is suggested, provides direct pictorial confirmation of theexistence of potential functions in the field of human geography.In terms of potential theory, if a source of strenght k1 is located at z= a of the complex plane, and another strength of k2 is located at z= - a, then the complex potential is:Ω (z) = k1 ln (z + a) + k2 ln (z - a)and this is readily separated into the equipotential and stream func-tions (2). The general configuration shown in the satellite photo-graph appears to conform to these equations. In particular, thedeveloping suburbanization between the two cities appears to out-line the stream function (a = 25 km, k1 = 679,684, the Dallas popu-lation in 1960; k2 = 356,268, the Fort Worth population in 1960).The second figure shows this somewhat more schematically, withthe city areas approximated by circles of appropriate size (3).There remains the possibility that the example is spurious or fortu-itous; time sequences or examples from twin cities in other parts ofthe world should illuminate this possibility. In particular, it is sug-gested that further high resolution photographic exploration of thisunusual planet may reveal other phenomena of equally great inter-est, including further instances similar to that reported here.(1) Cf. J. Stewart and W. Warntz, Journal of regional Science, vol. 1,1968, pp. 99-123; Gunnar Olsson, Distance and Human Interaction,Philadelphia, Regional Science Research Institute, 1965; and Robert E.Nunley, Living Maps of the field Plotter: Analog Simulation of SelectedGeographic Phenomena, Washington, D.C.: Association of AmericanGeographers, Commission on College Geography, Technical Paper n° 4,1971.(2) Cf. M. Spiegel, Theory and Problems of Complex Variables, Schaum,1968, p. 248.46W. Tobler Mappemonde 1/91ABSTRACT RÉSUMÉ RESUMEN* University of California, Santa Barbara, California.47W. Tobler Mappemonde 1/912. Schematic diagram Schematic diagram of a portion of the expected stream function, if Fort Worth and Dallas induce a potential field, for comparison with the satellite photograph (fig. 1).1. Apollo VI photograph Photograph AS6-2-1462, showing the built up areas of Fort Worth (32° 45’ N, 97° 20’ W) and Dallas (32° 47’ N, 96° 48’ W), Texas, U.S.A. 

Visual evidence for urban potential fields

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Visual evidence for urban potential fields

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