Accessibility indicators

There is no single "accessibility" number. There is a family of indicators, each answering a different question and each making different demands on data. This page puts them side by side.

Notation used throughout: $i$ indexes origins, $j$ destinations, $d_{ij}$ is the travel impedance between them, $A_j$ the attraction (size) of destination $j$, $P_i$ the population at origin $i$, and $f(\cdot)$ a distance decay function.

Choosing an indicator

If your question is	Use	Section
How far is the nearest hospital?	Nearest destination measure	1
How many jobs can be reached within 45 minutes?	Cumulative opportunities	2
How well placed is this location overall?	Potential accessibility	3
Is there enough supply for the demand nearby?	Competition corrected	4
Can this person reach work, school and shops in a day?	Person based	5
What is the welfare effect of this new line?	Utility based	6
How central is this place in the network?	Network centrality

A rule of thumb: use the simplest indicator that can distinguish the alternatives you are evaluating. Complexity that does not change the ranking only costs credibility.

1. Nearest destination measures

$$Acc_i = \min_j ; d_{ij}$$

Travel time or distance to the nearest destination of a given type, optionally to the k-th nearest.

• Reads as: minutes to the nearest GP, metres to the nearest supermarket.
• Strengths: intuitive, no parameters to defend, works with a single destination type, easy to communicate to non specialists.
• Weaknesses: ignores capacity and choice entirely. A village with one small school scores the same as a suburb with five large ones next door.
• Typical use: essential services analysis, spatial equity monitoring, and suitability factors for housing. In our projects in the social domain, distances from dwellings to services are inputs to housing suitability for people with mobility impairments.

2. Cumulative opportunity (contour) measures

$$Acc_i = \sum\limits_{j} A_j \cdot \mathbb{1}[d_{ij} \le \bar{d}]$$

The number of opportunities reachable within a threshold $\bar{d}$, the "isochrone count".

• Reads as: jobs within 45 minutes by car, residents within 30 minutes by public transport.
• Strengths: transparent, no decay parameter to estimate, directly comparable across regions and modes, and the most policy legible of all the indicators.
• Weaknesses: the threshold is a cliff edge. A job at 44 minutes counts fully and one at 46 counts not at all. Results can be sensitive to the threshold, so report at least two.
• Typical use: national accessibility monitors. The inverse form, the travel time needed to reach X people or jobs, is often easier to map.

Related variants used in practice:

• travel time required to reach a fixed number of people or jobs
• average travel time to all populated cells within a fixed radius, or within a time budget
• travel time to the nearest settlement of a given rank in a settlement hierarchy, useful where no administrative zoning is available, as in the worldwide network model

3. Potential (gravity) measures

$$Acc_i = \sum\limits_{j} A_j \cdot f(d_{ij})$$

Every destination contributes, weighted by a decreasing function of impedance. This is the attraction potential formulation described in Accessibility.

• Reads as: a dimensionless index, meaningful in comparison rather than in absolute terms.
• Strengths: no arbitrary cut off, behaviourally better founded, and a smooth response to network changes, which makes it well suited to scenario comparison and to land use models that iterate.
• Weaknesses: requires a decay function and its parameters, which must be estimated or borrowed and then defended. The resulting values are hard to explain to a non technical audience.
• Typical use: attractiveness factors in land use allocation, potential accessibility to jobs.

The choice of $f$ is consequential; see Distance decay functions. The underlying theory, including the doubly constrained forms, is in Spatial Interaction Models.

4. Competition corrected measures

Potential measures count supply but ignore that others compete for it. Competition corrected measures divide each destination's attraction by the demand pressing on it:

$$Acc_i = \sum\limits_{j} \frac{A_j \cdot f(d_{ij})}{D_j}, \qquad D_j = \sum\limits_{k} P_k \cdot f(d_{kj})$$

• Shen (1998) and the two step floating catchment area method (2SFCA, Luo & Wang, 2003) are the standard formulations. 2SFCA with a step function is the special case where $f$ is a cut off.
• Reads as: effective supply per capita, such as GP places per 1,000 residents adjusted for travel.
• Strengths: the right family whenever capacity is genuinely limited, as in healthcare, childcare, school places and sports facilities.
• Weaknesses: needs capacity data per destination, not just location. It doubles the number of routing runs, and is sensitive to the same decay parameters as the potential measure.

Where the question is not how much supply is reachable but where supply should be located, this becomes a facility location problem. See the European network model for a service allocation procedure built on travel time matrices.

5. Person based measures

Derived from Hägerstrand's space time geography: given an individual's fixed commitments, such as being at work from 09:00 to 17:00, and their available modes, which destinations fall inside their space time prism?

• Strengths: the only family that properly handles the temporal and individual components of accessibility, including opening hours, time budgets and car availability.
• Weaknesses: requires individual or synthetic population data and activity schedules. Results are hard to map, because accessibility is a property of a person rather than of a place.
• Typical use: transport equity and social exclusion research.

6. Utility based measures

The expected maximum utility of the available choice set, usually the logsum of a discrete choice model:

$$Acc_i = \frac{1}{\mu} \ln \sum\limits_{j} e^{\mu V_{ij}}$$

• Strengths: grounded in economic theory, and directly convertible into welfare effects and therefore into cost benefit analysis.
• Weaknesses: needs an estimated choice model. The scale is not interpretable on its own, only differences are.
• Typical use: appraisal of infrastructure investments.

Reporting an indicator honestly

Whichever family you choose, state alongside the result:

1. the destination type and attraction measure (is $A_j$ jobs, floor space, or capacity?)
2. the mode, and whether it is multimodal
3. the time of day and day of week used for impedance
4. the threshold or decay parameters, and a sensitivity check on them
5. the spatial unit, and how origins were connected to the network (Zoning and spatial units)

Two studies of the same region that differ on any of these are not comparable, however similar their maps look.

In the GeoDMS

Families 1 to 3 fall out of an ordinary impedance call. The nearest destination measure is a cut off run, or service_area when you also want to know which destination it was. The cumulative and potential measures are aggregations over the resulting Origin Destination matrix (trip table).

The potential measure is also produced directly as $D_i$ by the interaction section of the impedance options, which computes it as a by-product of the flow model described in Spatial Interaction Models. That is usually the cheapest route to it, since the routing happens once.

Family 4 has no single option. The demand side $D_j$ is a second impedance run with the roles of origins and destinations reversed, after which the two results are combined. Note that the interaction model constrains the origin side rather than the destination side, so it does not give you competition correction for free.

Related

• Distance decay functions · Spatial Interaction Models · Network centrality
• Applications, indicators as actually published
• Impedance functions and impedance options on geodms.nl