1. Overview

One methodology of particular importance to transport geography relates to how to estimate flows between locations, since these flows, known as spatial interactions, enable to evaluate the demand (existing or potential) for transport services.

A spatial interaction is a realized movement of people, freight or information between an origin and a destination. It is a transport demand / supply relationship expressed over a geographical space. Spatial interactions cover a wide variety of movements such as journeys to work, migrations, tourism, the usage of public facilities, the transmission of information or capital, the market areas of retailing activities, international trade and freight distribution.

Economic activities are generating (supply) and attracting (demand) flows. The simple fact that a movement occurs between an origin and a destination underlines that the costs incurred by a spatial interaction are lower than the benefits derived from such an interaction. As such, a commuter is willing to drive one hour because this interaction is linked to an income, while international trade concepts, such as comparative advantages, underline the benefits of specialization and the ensuing generation of trade flows between distant locations. Three interdependent conditions are necessary for a spatial interaction to occur [Ullman, 1956]:

• Complementarity. There must be a supply and a demand between the interacting locations. A residential zone is complementary to an industrial zone because the first is supplying workers while the second is supplying jobs. The same can be said concerning the complementarity between a store and its customers and between an industry and its suppliers (movements of freight).
• Intervening opportunity. There must not be another location that may offer a better alternative as a point of origin or as a point of destination. For instance, in order to have an interaction of a customer to a store, there must not be a closer store that offers a similar array of goods.
• Transferability. Freight, persons or information being transferred must be supported by transport infrastructures, implying that the origin and the destination must be linked. Costs to overcome distance must not be higher than the benefits of related interaction, even if there is complementarity and no alternative opportunity.

Spatial interaction models seek explain spatial flows. As such it is possible to measure flows and predict the consequences of changes in the conditions generating them. When such attributes are known, it is possible for example to better allocate transport resources such as highways, buses, airplanes or ships since they would reflect the transport demand more closely.

2. Origin / Destination Matrices

Each spatial interaction, as an analogy for a set of movements, is composed of an origin / destination pair. Each pair can itself be represented as a cell in a matrix where rows are related to the locations (centroids) of origin, while columns are related to locations (centroids) of destination. Such a matrix is commonly known as an origin / destination matrix, or a spatial interaction matrix.

In the O/D matrix the sum of a row (Ti) represents the total outputs of a location (flows originating from), while the sum of a column (Tj) represents the total inputs (flows bound to) of a location. The summation of inputs is always equals to the summation of outputs. Otherwise, there are movements that are coming from or going to outside the considered system. The sum of inputs or outputs gives the total flows taking place within the system (T). It is also possible to have O/D matrices according to the age group, income, gender, etc. Under such circumstances they are labeled sub-matrices since they account for only a share of the total flows.

In many cases where spatial interactions are relied on for planning and allocation purposes, origin / destination matrices are not available or are incomplete, requiring surveys. With economic development, the addition of new activities and transport infrastructures, spatial interactions have a tendency to change very rapidly as flows adapt to a new spatial structure. The problem is that an origin / destination survey is very expensive in terms of efforts, time and costs. In a complex spatial system such as a region, O/D matrices tend to be quite large. For instance, the consideration of 100 origins and 100 destinations would imply 10,000 separate O/D pairs. In addition, the data gathered by spatial interaction surveys is likely to become obsolete quickly as economic and spatial conditions change. It is therefore important to find a way to estimate as precisely as possible spatial interactions, particularly when empirical data is lacking or is incomplete. A possible solution leans on the use of a spatial interaction model to complement and even supplant empirical observations.

3. the Spatial Interaction Model

The basic assumption concerning many spatial interaction models is that flows are a function of the attributes of the locations of origin, the attributes of the locations of destination and the friction of distance between the concerned origins and the destinations. The general formulation of the spatial interaction model is as follows:

• T_ij : Interaction between location i (origin) and location j (destination). Its units of measurement are varied and can involve people, tons of freight, traffic volume, etc. It also relates to a time period such as interactions by the hour, day, month, or year.
• V_i : Attributes of the location of origin i. Variables often used to express these attributes are socio-economic in nature, such as population, number of jobs available, industrial output or gross domestic product.
• W_j : Attributes of the location of destination j. It uses similar socio-economic variables than the previous attribute.
• S_ij : Attributes of separation between the location of origin i and the location of destination j. Also known as transport friction. Variables often used to express these attributes are distance, transport costs, or travel time.

The attributes of V and W tend to be paired to express complementarity in the best possible way. For instance, measuring commuting flows (work-related movements) between different locations would likely consider a variable such as working age population as V and total employment as W. From this general formulation, three basic types of interaction models can be constructed:

• Gravity model. Measures interactions between all the possible location pairs. The gravity model is covered in more details here.
• Potential model. Measures interactions between one location and every other location.
• Retail model. Measure the boundary of the market areas between two locations competing over the same market.