# Distance decay

Distance decay, also known as the Gravity Model or the Inverse Square Law, is the tendency of a spatial relationship between one place and another to weaken as the distance between them increases. According to Waldo R. Tobler's First law of geography, "Everything is related to everything else, but near things are more related than distant things." An example of this type of relationship is the likelihood that residents will use a facility such as a park: the further they live from the park, the less likely they are to go there. Friction of Distance, the costs associated with moving over space, is the primary underlying cause of distance decay, because there will be higher costs associated with interacting between more distant locations, which will therefore be avoided. With the advent of faster and cheaper transportation and communications, these costs have reduced, and the distance decay effect has diminished in many fields in which it was formerly crucial.

## Mathematical model

Distance decay can be expressed mathematically as an exponential function or a power function, in which the strength of the given relationship (T) is dependent on the distance (D) between two places (i,j): can be seen below where D is the independent variable and T is the dependent variable, with F as the distance unit of measurement.

Tij = Dij-P

Here, the exponent P can model different patterns of decay. The Inverse Square Law, which describes the effect for physical properties such as gravity and light intensity, uses a power of 2. This is also most common for many geographic applications, which are often called "gravity models" because of the mathematical similarity (not because gravity is a causal force in these applications).

## Applications of Distance Decay

Distance decay models were first introduced to geographic applications during the Quantitative revolution of the 1960s, and have since become common in a number of uses:

• Interpolation of fields, using methods such as inverse distance weighting (IDW).
• Location analysis: modeling the potential use of a facility (e.g., a park, a store, a water source) by individuals distributed over space.
• Spatial interaction: modeling the degree of interaction between entities, such as international trade.

## GIS&T Body of Knowledge

This page is part of topic AM3-5 in the 2006 GIS&T Body of Knowledge.