186 / Geographical Analysis the gravity model and then provided such a basis for the "intervening opportunities" model in [ 141. Notwithstanding, only pragmatic considerations justified the use of the gravity formulation until very recently. Research from two sources [23, 101 has now lent mathematical rigor to the theory of spatial gravity flows, and provided a consistent theoretical rationale for the use of the gravity model.Both approaches estimate the travel distribution by the constrained minimization of a convex function. The nonlinear programming problems proposed by Wilson and by Charnes, Raike, and Bettinger are similar in form, although each embodies different basic assumptions and results in a different estimate of interzonal interactions. Since the two approaches have remarkably divergent histories and derivations, the intimate connections between the two estimation procedures are all the more striking.The implications of the two new approaches to gravity flows will influence the further development of Spatial analysis. It is the intent of this paper to begin to explore some of these implications.
ObjectivesThe paper begins with a concise review of the respective approaches of [ 10, 231 to the gravity model, emphasizing the physical analogs from which they were derived. Incorporating insights contributed by other researchers since the publication of the original results, especially with respect to Wilson's method, the authors hope to have achieved a more lucid presentation of the derivations than has been possible heretofore. Some formal relationships between the two approaches will be demonstrated. The common and complementary features of the two methods point to some useful generalizations. These include a hypothesis generator which may be used as a general method of extending the theory of spatial gravity flows. Partial answers to some questions posed by colleagues and reviewers concerning relationships between mechanical, statistical-mechanic, and statistical theories in geography also emerge. Comments on some of the computational realities of each approach conclude the paper.
The Gravity ModelConsider a region containing n distinct zones. A trip within the region may originate or terminate at any of the n zones. The following variables and constants are understood to apply to a particular fixed time period: 0, =the number of trips originating from zone i, i = 1, . . , , n, D, =the number of trips terminating at zone j , j = 1, , , . , n, K , = the "generalized impedance" to travel between zone i and zone ti, = the number of trips from zone i to zone