1. Definition

Accessibility is a key element to transport geography, and to geography in general, since it is a direct expression of mobility either in terms of people, freight or information. Well-developed and efficient transportation systems offer high levels of accessibility (if the impacts of congestion are excluded), while less-developed ones have lower levels of accessibility. Thus accessibility is linked with an array of economic and social opportunities.

Accessibility is defined as the measure of the capacity of a location to be reached by, or to reach different locations. Therefore, the capacity and the arrangement of transport infrastructure are key elements in the determination of accessibility.

All locations are not equal because some are more accessible than others, which implies inequalities. The notion of accessibility consequently relies on two core concepts:

• The first is location where the relativity of space is estimated in relation to transport infrastructures, since they offer the mean to support movements.
• The second is distance, which is derived from the connectivity between locations. Connectivity can only exist when there is a possibility to link two locations through transportation. It expresses the friction of distance and the location which has the least friction relatively to others is likely to be the most accessible. Commonly, distance is expressed in units such as in kilometers or in time, but variables such as cost or energy spent can also be used.

There are two spatial categories applicable to accessibility problems, which are interdependent:

• The first type is known as topological accessibility and is related to measuring accessibility in a system of nodes and paths (a transportation network). It is assumed that accessibility is a measurable attribute significant only to specific elements of a transportation system, such as terminals (airports, ports or subway stations).
• The second type is known as contiguous accessibility and involves measuring accessibility over a surface. Under such conditions, accessibility is a measurable attribute of every location, as space is considered in a contiguous manner.

Last, accessibility is a good indicator of the underlying spatial structure since it takes into consideration location as well as the inequality conferred by distance to other locations.

2. Connectivity and Total Accessibility

The most basic measure of accessibility involves network connectivity where a network is represented as a connectivity matrix (C1), which expresses the connectivity of each node with its adjacent nodes. The number of columns and rows in this matrix is equal to the number of nodes in the network and a value of 1 is given for each cell where this is a connected pair and a value of 0 for each cell where there is an unconnected pair. The summation of this matrix provides a very basic measure of accessibility, also known as the degree of a node:

• C1 = degree of a node.
• c_ij = connectivity between node i and node j (either 1 or 0).
• n = number of nodes.

The connectivity matrix does not take into account all the possible indirect paths between nodes. Under such circumstances, two nodes could have the same degree, but may have different accessibilities. To consider this attribute, the Total accessibility Matrix (T) is used to calculate the total number of paths in a network, which includes direct as well as indirect paths. Its calculation involves the following procedure:

• D = the diameter of the network.

Thus, total accessibility would be a more comprehensive accessibility measure than network connectivity.

3. The Shimbel Index and the Valued Graph

The main focus of measuring accessibility does not necessarily involve measuring the total number of paths between locations, but rather what are the shortest paths between them. Even if several paths between two locations exist, the shortest one is likely to be selected. Consequently, the Shimbel index calculates the minimum number of paths necessary to connect one node with all the nodes in a defined network. The Shimbel accessibility matrix, also known as the D-Matrix, thus includes for each possible node pairs the shortest path.

The Shimbel index and its D-Matrix fail to consider that a topological link between two nodes may involve variable distances. It can thus be expanded to include the notion of distance, where a value is attributed to each link in the network. The valued graph matrix, or L-Matrix, represents such an attempt. It has a very strong similarity with the Shimbel accessibility matrix and the only difference lies that instead of showing the minimal path in each cell, it provides the minimal distance between each node of the network.

4. Geographic and Potential Accessibility

From the accessibility measure developed so far, it is possible to derive two simple and highly practical measures, defined as geographic and potential accessibility. Geographic accessibility considers that the accessibility of a location is the summation of all distances between other locations pondered by the number of locations. The lower its value, the more a location is accessible.

• A(G) = geographical accessibility matrix.
• d_ij = shortest path distance between location i and j.
• n = number of locations.
• L = valued graph matrix.

This measure (A(G)) is an adaptation of the Shimbel Index and the Valued Graph, where the most accessible place has the lowest summation of distances.

Although geographic accessibility can be solved using a spreadsheet (or manually for simpler problems), Geographic Information Systems have proven to be a very useful and flexible tool to measure accessibility, notably over a surface simplified as a matrix (raster representation). This can be done by generating a distance grid for each place and then summing all the grids to form the total summation of distances (Shimbel) grid. The cell having the lowest value is thus the most accessible place.

Potential accessibility is a more complex measure than geographic accessibility, since it includes simultaneously the concept of distance weighted by the attributes of a location. All locations are not equal and thus some are more important than others. Potential accessibility can be measured as follows:

• A(P) = potential accessibility matrix.
• d_ij = distance between place i and j (derived from valued graph matrix).
• P_j = attributes of place j, such as its population, retailing surface, parking space, etc.
• n = number of locations.

The potential accessibility matrix is not transposable since locations do not have the same attributes, which brings the underlying notions of emissiveness and attractiveness:

• Emissiveness is the capacity to leave a location, the sum of the values of a row in the A(P) matrix.
• Attractiveness is the capacity to reach a location, the sum of the values of a column in the A(P) matrix.

Likewise, a Geographic Information System can be used to measure potential accessibility, notably over a surface.