Distance Decay

Distance decay is a major cornerstone of spatial-temporal analysis and, hence, geography. Essentially, it states that the nearer two locales are, the greater is the expected attraction/interaction between the two and, conversely, the greater the distance is between two locales, the less the expected interaction. This seemingly commonsense notion has been restated as a formulation of spatial autocorrelation by the geographer Waldo Tobler. It has come to be known as Tobler's “first law of geography,” which states that “everything is related to everything else, but near things are more related than distant things.”

In its simplest form, the distance decay function can be portrayed graphically as a concave curve descending from the y axis, representing the intensity of a phenomenon, and extending along the x axis, representing the distance from some origin. Examples of this phenomenon can be found in virtually all subfields of geography and at all scales for a wide range of travel interactions. The universality of distance decay necessarily limits the discussion here to travel and human geography. In human geography, the distance decay effect has been noted in the movement of international tourists; travel to local coffee shops in Japan; use of mental health clinics in New York City and Baltimore; behavior of a serial killer in Baton Rouge, Louisiana; and forest land use patterns in the northeastern Philippines. An example of the distance decay function for a series of everyday activities is presented in Figure 1.

Distance decay functions are quantitatively rooted in the family of gravity models based on Isaac Newton's fundamental law of attraction. The term distance decay characterizes how the attraction between two bodies decreases as the distance between them increases. The distance decay concept can be traced to the principle of least action, first espoused by Pierre Louis Moreau de Maupertuis (1698–1759), who first gave the notion verbal expression:

It is difficult for any single discipline to “claim” Maupertuis for he wrote in physics, mathematics, metaphysics, and biology. He was the force behind expeditions to Lapland and Peru, traveling to the former to measure the difference in Earth's radius and demonstrate that Earth was an oblate spheroid rather than ovate (egg shaped).

The concept of distance decay formulated by Maupertuis, although somewhat deterministic today, does capture much of the essence of mechanics. In applying it to physics, Maupertuis suggested that the quantity to be minimized was the product of the duration (time) of movement within a system and the vis viva (living force), twice what we now call the kinetic energy of the system.

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