We are grateful to Zoé Andreyev who translated the whole text with many technical words.
 1 Image et Ville Laboratory, URA 902, CNRS, currently UMR 7011 CNRS.
 2 This study does not refer to the Graph theory, which is very important in the field of transport (...)
1A great deal of research on accessibility has been conducted in Strasbourg1 since the early 1980s. Though these studies were initially based on the research of A. Pred and G. Tornqvist (1973, 1974), their number and the diversity of scales on which they were performed has led researchers to develop an original methodology, whose statistical and cartographic results are both stable and comparable on all levels: international (Western Europe, Eastern Europe, etc.), national (France, South Korea, etc.), regional (e.g Upper Rhine Graben), or on the city level (e.g. Strasbourg, Luxembourg). After a reminder of basic definitions, we will briefly present the different stages of our research, as well as the methods and software we used in our approach2.
 3 Direction à l'Aménagement du Territoire et à l'Action Régionale – Bureau of Physical Planning and (...)
 4 Cauvin C., Reymond H., Enaux C., et al.
2We refer to the two following definitions of accessibility, which we justified in a report sent to DATAR3 in 19924, and in a book published in 1998 (Reymond H. et al.):
3Theoretical and processual definition :
4Accessibility is a permanent construction effort aimed at establishing regular and easy communication (one can also speak of "degree of facility") between places of supply or wealth, socially recognized as such created and maintained by societies aware of their necessity and of the fact that they are not symmetrically distributed in space.
5Operational and structural definition :
6Accessibility is the measurable quantity of spatial distance between the place where one wishes to go and the place where one is, depending on the chosen means of transport. It is the minimal time distance necessary to link together two places thanks to a means of transport, via a given itinerary, and taking into account the notion of comfort.
7These definitions consider places — cities, crossroads, etc.— as the nodes of a network, and as such they must take into account not the administrative divisions, but the functional partitions of the territories in which these places are located. The first definition is essential from a conceptual angle, while the second enables us to conceive and develop databases adapted to a given means of transport. Part 1 presents the underlying principles of the databases; parts 2 and 3 describe the local and global approaches; in the conclusion, we will provide a diagram summing up our approach.
 5 For this presentation, we took into consideration only the time measurements; however, our labora (...)
8The notion of accessibility implies a circumscribed area which includes places considered as points in this space; it is measured in terms of spacetime5, that is time intervals between any two places. In order to obtain comparable and spatialized data, it is necessary to have a set of geographic references (a map) and thematic references (reference distances, in this case orthodromes).
9This base includes spatial and thematic references.
10This is the main map with its georeferenced anchor points; it must include at least an outline, an envelope of the considered area, and the points representing the selected places. Additional illustrations can be added as well: main roads, wooded areas, builtup areas, etc. (see figure 1). This map as a closed shape must be digitalized in vector mode.
Figure 1. Geographical base and digitalisation
The shape and the statistical analysis of the spatial distribution of the points can be verified with the software Geocart.
 6 The orthodromic distance between two points is equal to the length of the arc of the great circle (...)
11The more commonly used distance reference is orthodromes6, the shortest distances between two points on a geoïd's surface, i.e. considering the curvature of the surface of the earth. To measure this distance, one must first determine the geographic coordinates of each of the places, that is their longitudes and latitudes. Then these coordinates must be entered into a file, as a table containing n rows (=places) and 2 columns, one for latitudes, the other for longitudes; this presentation is shown in figure 2.
Figure 2. Table of orthodromes
12Based on this table, one can compute orthodromes thanks to specifically adapted software, such as "Orthodromie".
13The reference distances are computed with the software Orthodromie
14The various distancetime measurements obtained for each means of transport will be compared to these reference distances.
15When building accessibility databases, the capture of information in connection with the underlying hypotheses must follow very strict rules. However, schedules can lead to considerable variations; for this reason, we established a distinction between the construction of databases corresponding to movements governed by timetables and those corresponding to free movement.
 7 The definition and relevance of the stay time are based on the research conducted by the two auth (...)
16The general principle is the following: "Given all possible ways of linking point A to point B, the best access time is the time taken to cover the shortest journey". However, this principle is not sufficient if accessibility is considered within a larger context, that is including the duration of stay7. In that case, other terms must be defined, as shown in figure 3.

Transit time, Ttdr, is the total time spent by a person outside of his/her home between the time of departure from residence (or any other place considered as a point of departure) and time of return:
Ttdr = time of return—time of departure

 8 For the sake of simplicity, we decided to call all points of departure or arrival of a given type (...)
Accesstime : At is the time taken to go somewhere added to the time taken to return. It corresponds to the time needed to travel from one's place of departure to one's destination, and from the destination point back to the departure point. Thus it corresponds to the time spent travelling, excluding the time spent in the place of destination. Access time includes time spent getting to airports, train stations, bus stations8, etc. in both directions, checking in (if necessary), waiting for transfers, and includes nights if the trip includes spending the night in a place other than one's destination.
At = At there+ At back
or
At = Att+ Atb

Traveltime : Tt is the time spent actively travelling, that is the time during which the chosen means of transport is actually moving. It corresponds to access time minus all forms of waiting time.
Tt = Tt there + Tt back = Ttt + Ttb
or
Tt = At — (all waiting times)
Figure 3. Accessibility components

Staytime (duration of stay) This time is more complex, and must be divided into total time of stay and useful stay time.

Total staytime : TST is the amount of time actually spent by a person in his/her place of destination B. It is obtained by calculating the difference between the time of departure from B and the time of arrival in B. But depending on when these times occur during the day, the time of stay can be considered useful or not; for this reason a second notion was defined, that of "useful time of stay".
TST = Tt — At

The useful staytime, UST, is the amount of time during which a person is actually involved in various activities and exchanges. Depending on arrival time, part of the time spent in one's destination can be considered to be not useful in the context of one's scheduled activities. The parameters are defined according to the hypotheses of the study, and the time slots considered useful or not useful can vary.

The time slots, TS, are the exact times during which the useful duration of stay can be determined. For example, a useful duration of stay lasting one hour between 9 and 10 AM will not be considered in the same way as a the same duration of one hour situated between 1 and 2 PM.
17The number of rules for elaborating databases has thus increased; these can be stated as follows:

minimize total transit time, Ttdr

minimize access time there and back, Att, Atb

consequently reduce the difference between access time and travel time, At, Tt

maximize the useful stay time,UST

minimize the difference between total stay time, TST, and useful time UST.
18These rules make it possible to establish priorities when computing the time needed to link each pair of points, but the construction of each database depends on the means of transport and on whether or not it is governed by a schedule.
19In the case of trips made in vehicles which are not tied to schedules (private car, bicycle, etc.), each trip is divided into a sequence of sections (see figure 4), each of them corresponding to a given distance in kilometres and an average speed, weighted according to the type of landscape, crossroads, cities crossed, etc.
Figure 4. Notion of road section
20This information is entered into a table, each row corresponding to a link, and each column corresponding to a section. The access time is the sum of the distances in kilometres weighted by the corresponding speeds. Some softwares provide additional information, but they are often insufficient or incomplete, depending on the level of precision sought or on the hypotheses. The collecting of data can be facilitated by using a GPS.
21These databases are more difficult to establish. Usually, a choice is made concerning the time of departure and arrival. Whether the person is travelling by train, by bus or by airplane, one must determine the shortest possible trip while taking into account constraints linked to departure and arrival times and avoiding too many transfers when there is no direct connection. The database is thus made up of a sequence of times indicating the changes in the "actions". For the time base, different countries have developed various types of software, but a great deal of personal input on the part of the person building the base is still necessary.
22Once these databases have been elaborated, it is necessary to transform the information into accessibility matrices.
23The software Acced is perfectly adapted to this purpose: it instantly converts information into access time from one location to another, the rows and columns of the matrix corresponding to the chosen locations. It can also calculate, depending on need, other times and other matrices: waiting times, transfers, trips etc.
Figure 5. Accesstime matrix
24To be usable, the tables of the bases must be built according to the precise instructions for use provided with the software.
25To establish time matrices, use the software Acced.
26The accessibility matrices can be symmetrical or not, depending on the chosen means of transport; for instance, for trains and planes, they are usually assymetrical. "Intercity" roads can be considered symmetrical, but within cities, roadways are rarely symmetrical: the itineraries from A to B or B to A in a city are rarely symmetrical, due to oneway streets or traffic circles. In some cases it may be necessary to work with a symmetrical matrix. In order to do this, one must first verify the strength of the asymmetry by calculating, for example, an asymmetry index (Tij/Tji) and studying the statistical variation of this index. If the variation is weak, then the matrix can be made symmetrical by calculating the halfsum of this matrix and of its transpose. These computations can be done with Calmat software.
27To obtain asymmetry indices or symmetrize a matrix, use Calmat.
28Once the matrix has been obtained, it is possible to analyse the data, in other words study the accessibility of the space under consideration. There are many ways of processing the data: they are classified here according to two large categories, depending on whether they concern the entire matrix or only one row or column of the matrix. In the first case, we may speak of system accessibility or global accessibility (total matrix), and in the second case of subsystem accessibility or local accessibility (a row or column of the matrix), as shown in figure 6.
Figure 6. System and subsystem accessibility
29Both types of processing have been considered. We will first present the system accessibilitytype of processing.
30Three groups of methods can be considered depending on the elements included in the accessibility matrix. We can summarize the information, transform the available times into frequencies or process the entire matrix (see figure 7); each method brings new, additional information which completes the rest.
Figure 7. System accessibility methods
31On the basis of the accessibility matrix, one calculates indices summing up all the information; one thus obtains three indices, often used in the field of transport (see figure 8):


the potential accessibility index, which is the sum of times in rows or columns, a typical transport index (Lowe, Moryadas, 1975)

the centrality index, the sum of the times in rows or columns, weighted by the double sum of the matrix; this adimensional index is included between 0 and 1.

the dispersion index, double sum of the matrix weighted by the sums of times in rows or columns.
Figure 8. Commonly used accessibility indices
32These indices can be computed with Calmat.
33After the statistical analysis, these indices can then be mapped as isolines (isopleth map) thanks to cartography software or a geographical information system with an interpolation function. The results are easy to understand and provide an initial image of network accessibility.
34However, another mapping system can also be used to visualize potential accessibility: the piezopleth map. In this method, developed by C. Schneider (Cauvin, Schneider, 1989), the different thematic values are not represented through distinct grey areas; the variable is considered as having an impact on the geographic surface in the same way as a force will alter a given material. This material, and consequently the geographic surface, expands or contracts under the impact of this force; thus the surface variations of the map's elementary units express the impact of the phenomenon (see figure 9). This method, which is seldom used, was developed in Strasbourg, along with the programme that goes along with it, called STRUCT; its present version is called Anaplaste.
35In order to obtain a piezopleth map, use the software Anaplaste.
C. Cauvin, 2000
Figure 9. Potential accessibility and piezopleth map
36The second approach reveals a hierarchy of places, in terms of accessibility. The focus is no longer exact time but significant time — or accessibility — spans The times included in the matrix are converted into frequencies; to do this, one first defines a time span, for example every ten minutes or every hour, depending on the type of transport and the area under consideration. Then the number of places accessible from each place, for each time span, is determined. In the new table, the rows indicate places, the columns indicate time spans and the cells show the frequencies (see figure 10). This operation can be performed with Calmat.
Figure 10. Table of accessibility frequency profiles
37To obtain frequency profiles, use Calmat.
38With this table, one can build frequency profiles with the time spans on the abscissa and the frequencies on the ordinate. For the comparison of profiles and the construction of typologies, correspondence analysis is a perfectly adapted tool. This can be done thanks to different data analysis software such as XLSTAT, BMDP, SPSS, ADDAD, or SPAD. The new map — with point symbols— will then reveal the hierarchy. Simple cartography software is sufficient, there is no need for GIS software.
39A last category of methods, which uses all the information contained in the accessibility matrix, takes us still further, enabling us to obtain a functional space linked to a given transport matrix — in other words the space through which we move, and within which the points can be considered either nearby or far away, depending on how fast the distance is covered, compared to their location in the space of reference. Two main steps are necessary to construct this functional space (see figure 11).
40First, one must determine the relative position of the locations in terms of time, that is their functional configuration; next, one must compare this configuration to the space of reference, in order to obtain through adjustment and interpolation the corresponding functional, and comparable, space.
41In order to determine the relative positions of points on the basis of timedistance, one can use methods belonging to the family of multidimensional scaling, or elastic mapping (Müller, 1982), or still trilateration (Tobler, 1977). The basic principles and the general approaches of most of these methods are similar or almost similar (see figure 12).
42Let us consider a matrix of timedistances Δij. First, we choose [or calculate or randomly select] an initial configuration with 2 dimensions, X and Y, and n rows (n places). Next, we also choose a distance function which will express the underlying structure of the studied space. Thanks to this function, we calculate the distances, dij, between the n places of the initial configuration; these are then compared to the distances observed either directly or after regularization of the observed distance matrix Δij thanks to an fit function. We then calculate an index of fit (goodnessoffit function or loss function), called Stress (according to the KYST method ; it is like a similarity index) which varies from 0 to 1. The closer it is to 0, the better the fit, the greater the similarity. If this index is lower then the chosen threshold, the configuration is selected; if it is higher, then the configuration must be improved and the procedure is repeated. The algorithm can be stopped once the index has reached the fixed threshold or when the function has reached its minimum. This configuration can be obtained with various software such as KYST, SSA, SPSS; the software used here is AMDP, which is similar to KYST, developed by Kruskal et al. (1968).
43For the multidimensional scaling analysis, use AMDP
44This configuration only indicates relative positions which are not comparable to geographical positions; for this reason, the configuration must be transformed in order to become comparable.
Figure 11. From time matrix to functional space : main steps
Figure 12. Approach of multidimensional scaling analyses (the KYST method)
45From configuration to functional space
46The functional configuration (W) differs from the reference map (Z) as concerns origin, orientation and the scale of the coordinates, independently of the differences due to the characteristics of the chosen means of transport. To ensure the comparability of the two spaces, it is necessary to apply bidimensional regression as suggested by W. Tobler (1977). There are two steps to this method: first, one adjusts the configuration to the geographic map and next, one performs an interpolation in order to generalize results (see figure 13).
Figure 13. Simplified bidimensional regression
47Fitting makes it possible to obtain an adjusted functional configuration W'; the vectors attached to the studied points originate in the positions located in the space of reference and terminate in their homologous points located in the configuration; the norm of this vector is equal to the movement of the point depending on the type of transport considered.
48The interpolation is generally based on a squared grid. It is alternately performed along the rows and columns, making it possible to violate the monotonicity and subsequently the topology if necessary; an image of the associated functional space of transportation, W'', can then be obtained (see figure 14).
Figure 14. Complete bidimensional regression
49A great many maps can thus be generated, providing numerous results either as concerns the homologous points, or the grid nodes (figures 15 and 16); all can be interpreted in terms of transport.
Figure 15. Results tied to the bidimensional regression
Figure 16. Maps of Luxembourg
50Bidimensional regression, developed by W. Tobler on the basis of W. d'Arcy Thompson's research, can be applied thanks to the Darcy programme, first developed by W. Tobler between 1963 and 1978, then adapted by different persons in Canada and in Strasbourg. The programme used here was entirely rewritten by B. Guérin.
51To apply the bidimensional regression, use the Darcy programme.
52Once these three stages completed, we have at our disposal many results which complement each other, providing a great deal of material for interpretation; local examples provide illustrations for the different sets thus identified.
53Until now our use of the accessibility matrix was based on all the rows and all the columns; on the basis of these global results, we will now look separately at one or the other of these rows or columns. This is a local analysis, in other words a subsystem analysis. Several approaches are possible; three will be retained in this presentation. The first takes into account the raw values of access time, the second underscores the movement of the points and the third highlights the spatial distortion caused by these movements.
54The row or the column corresponding to the studied place is considered as a vector representing the location's accessibility from all other locations of the network. These vectors can be statistically analysed and cartographically represented thanks to lines of equal accessibility (see figure 17). Any cartography or GIS programme that includes isopleth maps can provide this type of easily read map.
Figure 17. Unipolar isoaccessibility
55Another representation aims to emphasize the relative closeness or distance from the original location according to the chosen mode of transport. On one hand, we have the time matrix W, on the other the reference distance matrix Z and the basic map. The double sum of each of the two distance matrices is calculated, and the matrices are made comparable by multiplying the time matrix by the ratio of the double sums, a ratio which in fact expresses the average speed necessary to cover the reference space using any given means of transport:
56To obtain the matrix of weighted times, use Calmat.
57We thus obtain the weighted time matrix, W'. The matrices W' and Z, expressed in metric units, are now comparable. We are aware of the geographic position of the different locations in relationship to the original location thanks to the reference map. In order to determine where these locations are in terms of time, we must draw all the halflines linking the original location to all the other locations within the area under consideration. The displacement vectors starting from the geographical positions are located on these halflines. In order to find the magnitude of the displacement vectors, we must calculate the difference between the matrices W' and Z; the time locations, which are the extreme points of the vectors, are located on the halflines, either in the direction of the original location (moving closer) or in the other direction (moving further), depending on whether the sign of the magnitude is + or — (figure 18).
Figure 18. Isodistance maps
58Many representations can now be based on these new positions, as shown in figures 19 and 20. These can be obtained thanks to the software Isodist.
59To obtain isodistances, use Isodist software.
Figure 19. Isodistances representations
Figure 20a. Examples with Luxembourg
Figure 20b. Examples with Luxembourg
60However, only the modifications of the positions of the points were taken into account; there is no global distortion of the space.
61We will now only take into account the positions of the points obtained through Isodist and those of the homologous points in the reference area. They are directly comparable and the locations corresponding to the travel time can be generalized by interpolation thanks to bidimensional regression, applied only during the second phase. We then obtain an image of the unipolar functional space (figure 21).
62In order to obtain the unipolar functional space, use Darcy.
Figure 21. Obtaining unipolar spaces
63These methods make it possible to illustrate the characteristics of certain places depending on their role in the global typologies.
64The method described in this article makes it possible to determine the accessibility of a given location with any given means of transport; it explains how to obtain access time data (Acced software), how to obtain a global view of accessibility or an illustration of the specific characteristics of a given place. As announced in the introduction, a general diagram enabling the user to follow all or only some of the different steps of this approach, depending on the aim of the study, is provided below (see figure 22).