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distance decay

  The attenuation of a pattern or process with distance. Distance was one of the \'fundamental spatial concepts\' identified by Nystuen (in Berry and Marble, 1968) and the importance of distance decay (sometimes called a distance lapse rate) was enshrined in Tobler\'s famous \'first law of geography: everything is related to everything else, but near things are more related than distant things\' (Tobler, 1970). The empirical significance of this had of course been recognized in the early formulations of social physics, but only achieved wider formal significance within geography with the emergence of the search for general theorems of spatial organization. Underlying many of the classical models of spatial structure, for example, the central place models of Christaller and Lösch and the diffusion models of Hägerstrand, are assumptions about spatial interaction which, in the typical gravity model form, postulate a definite inverse \'distance effect\', which is capable of a series of mathematical expressions (see the figure). These various transformations have such a powerful effect on the lapse rate that Olsson (1980) argued that the identification of a distance decay \'may reveal as much about the language I am talking in as it does about the phenomena I am talking about\'. But in any event the lapse rate is evidently not independent of the geometry of the system within which interaction takes place, and in some locational models this is partially recognized through a parallel discussion of the accessibility of points arrayed on a movement surface (or network) around some hypothetical centre; for example, the von Thünen model of agricultural land use or the density gradients of conventional urban land-use models. Because of these logical connections Bunge (1962) represented interaction and geometry as \'the inseparable duals of geographic theory\'; but the matter clearly does not end there, because such interdependence poses formidable interpretative difficulties (e.g., see Curry, 1972; Cliff et al., 1975, 1976). Hence, while distance-decay curves can be identified empirically it is by no means clear how far their form depends on the model structures used to replicate them; nor to what extent their parameters can be given substantive meaning. (DG)

{img src=show_image.php?name=bkhumgeofig21.gif }

distance decay Distance decay curves and transformations (Taylor, 1971)

References Berry, B.J.L. and Marble, D.F., eds, 1968: Spatial analysis: a reader in statistical geography. Englewood Cliffs, NJ: Prentice-Hall. Bunge, W. 1962: Theoretical geography. Lund: C.W.K. Gleerup. Cliff, A., Martin, R.L., and Ord, J.K. 1975 and 1976: Map pattern and friction of distance parameters. Regional Studies 9: 285-8 and 10: 341-2. Curry, L. 1972: A spatial analysis of gravity flows. Regional Studies 6: 131-47. Olsson, G. 1980: Birds in eggs/eggs in bird. London: Pion; New York: Methuen. Taylor, P.J. 1971: Distance transformation and distance decay functions. Geographical Analysis 3: 221-38. Tobler, W. 1970: A computer movie. Economic Geography 46: 234-40.

Suggested Reading Olsson (1980), ch. 13; Sheppard, E.S. 1984: The distance decay gravity model debate. In G.L. Gaile and C.J. Willmott, eds, Spatial statistics and models. Dordrecht: D. Reidel, 367-88.



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