1Urban sprawl is frequently associated with the idea of an unsuitable development, leading to increasing economic, social and environmental problems. Moreover, its control is difficult because it combines multiple overlapping patterns relaying on numerous traditional urban planning fields. So, considering urban sprawl usually demands to take into account several patterns including numerous sociodemographic indicators, as density or family size. Here, it will only be considered in its morphological aspects, as a combination between built and nonbuilt areas. Thus, urban sprawl process will be defined as the spatial expansion of the built areas of a city through time. Modelling appears a useful means to understand and manage its complexity. The urban area of Belfort (NorthEast of France, approximatively 80000 inhabitants in 1999) seems an interesting area for measuring the sprawl process since the 50’s, and simulate scenarios of expansion towards 2015. The model is developed within a threeyear collaboration with the Agence d’urbanisme du Territoire de Belfort (urban planning agency). Their aim is to collect data and test the model’s operational feature, according to their planning issues. Concerning the urban sprawl and the planning problems, the model must answer to three questions about the future urban growth to help decisionmaking : how much ? where ? what ?
2The aim of this paper is then to present a methodological approach to model the urban sprawl process. It deals with the construction of a global model based on three successive steps. Each step corresponds to a model with a precise goal :

quantification of the urban expansion caused by the urban sprawl process (question "how much ?")

location of the expansion according to the geographic characteristics of the periurban outskirts (question "where ?")

 1 At this time, the third step uses cellular automata, but leaves other ways open to answer the quest (...)
differentiation of the located expansion in order to determine which kind of land use will be generated in each location (question "what ?")1. However, first of all, a solid database is needed in order to apply the models described in each step (fig. 1).
Figure 1 :Three methodological steps to construct a global model
3Constructing such a database will allow to achieve two important goals :

to collect and store data taking into account the sprawl process’ scale

to develop a tool in order to query and analyse data in a spatiotemporal way.
4The construction of the database stresses different questions which have to be answered and validated. The first one is about collecting raw data : what kind of information is needed ? How to get it ? The second one is about storing the data : is there a database structure allowing to organize it in order to process easily and quickly ? At last, we will assess the technical and thematic uses of the database : which criteria should we favour ? Is it possible to find a compromise according to both uses ? Of course, the answers to these questions must be included in a Geographical Information System (GIS) approach, considered as a generic tool for collecting, storing and analyzing spatial data.
5According to the goal of the model, two important facts have to be considered :

 2 The dates correspond to the French censuses.
Urban sprawl has to be considered within a temporal dimension to be understood as a process. The Belfort Agency proposes to study it from the 50’s, considering the rural depopulation (19541968) and the massive periurbanization (19681999) statistically measured by the French statistics institute (INSEE)2 for the Belfort area, and more generally for France ;

Measuring the urban sprawl requires to consider space at the urbanblock scale or at the individual house scale, and not at the community scale as in the available censuses : the land use changes introduced by periurbanization cannot by visualized at a small scale and demand to use precise indicators. To find information corresponding to these major goals, three data sources can usually be used : satellite imagery, aerial photographs and topographic maps.
 3 Actually, the mapped information for 1955 is based on 4 sheets drawn between 1954 and 1957 by the I (...)
6Satellite images, with high (or very high) spatial resolution, appear like one of the most interesting data source (Puissant, 2000). But, in our case, they do not exist before the 70’s and do not allow to consider the early time of the process. Aerial photographs exist since the beginning of the century, but only a few photographic coverages are available on the study area : we have a first coverage in 1954 and a second one in 1992. No coverage is available between these dates. Then, the more complete sources of information for considering urban sprawl through time seem to be the different topographic maps produced at three dates (1955, 1975 and 1995)3 by the French national geographic institute (IGN). These maps cover regularly the study area : one map a twentyyear period. Nevertheless, with only these sources, the thematic choice of the data is strongly determined by its representation on the map. The discrimination of the land use categories related to the urban sprawl process is then a delicate operation : they must be represented on each map in order to be regularly considered for the encoding. According to the map keys, thirteen land use categories only appear systematically for the three dates. For the more recent date (1995), they have been checked and completed by field surveys. The chosen land use categories include natural areas, built areas, and networks (fig. 2) :
 4 The names of some categories are very difficult to translate from French into English : they often (...)
Figure 2 : Land use categories4
The C_{t} row corresponds to the codage in the database. It will be described in section 1.2.
7In order to study the urban sprawl process, we must be able to visualize the land use transformations of urban and periurban areas, and to compare these transformations from one location to another and from one date to another. So, the problem is how to harmonize this disparate raw data for obtaining comparable information in space and time. Using a tessellation appears an interesting solution for collecting and storing the information.
8Tessellation consists in transforming continuous original mapped data into discreet spatial data within regular (squares, hexagons) or irregular polygons (Thiessen polygons). In our case, a regular mesh seems adapted and square cells appear the simplest geometric form to generate it. The result of the tessellation is a lattice. To make it easier, we call it a grid. Such a grid makes it possible to consider the geographic information inside the cells, whatever its date and its source. Through the grid, the three georeferenced maps allow to visualize the land use for each cell at the three dates (1955, 1975 and 1995). A rasterization (vector to raster conversion) could then be an interesting solution. But, technically, considering that the only software used during our collaboration with the Agence d’urbanisme is a vector GIS, this operation must be processed in a vector mode. Then, we have to think about creating vector square cells and using vector GIS structure for the tessellation. We chose to manage spatial information in polygons S (a regular grid of square cells covering continuously the study area) associated with temporal information (stored in a table of attributes C_{t} corresponding to each polygon) :
9S (x, y, C_{t1, }…, C_{tn})
10x,y : geographical coordinates
11n = 3, t_{1} = 55, t_{2} = 75, t_{3} = 95
12After digitising the maps, the grid encoding (space) for each date (time) is automatically done by a spatial request (GIS request). The cells’ geographic location is given by their coordinates (x,y), and their land use for each date is given by their attributes (C_{t}). Each polygon is associated with three attributes, C_{55}, C_{75} and C_{95} corresponding to their land use category in 1955, 1975, and 1995. The consequence of using such a vector structure is that the whole data is stored in a unique GIS layer, while a raster technique would need one layer for each date. The disadvantage is that the structure is heavier than raster and can quickly lead to technical dataprocessing problems.
13Nevertheless, using vector or raster mode, the tessellation stresses the important question of the cells size : the efficiency (and also consistency and quality) of the database depends on it. On the one hand, the smaller the cells are, the more precise and efficient the database will be. But, on the other hand, if the cells are very small, they are numerous and they make the database less easy to consult (this is one of the major problems induced by the vector GIS structure). The database may quickly need too much computing resources and finally it may be difficult to handle in a prospective way. This is why we chose to determine the optimal size of the cells according to the following objectives :

create cells as small as possible (thematic criterion)

reduce their number (technical criterion).
14Four spatial resolutions have been tested on a testfield : 200, 100, 50, and 25 meters corresponding respectively to 4 ha, 1 ha, 0,25 ha and 0,0625 ha for each cell. The fiftymeters resolution seems the more interesting. It reproduces the urban morphology in a correct way and produces 160000 cells. Of course, the test results depend very much on the test field’s distinctive features. We are presently reproducing the same exercise on other test fields, in order to eliminate these features and to validate the choice of a fiftymeters resolution. The first results seem to confirm it. Then, with such a resolution, the amount of cells is important but may still be handled with a GIS set up on a powerful computer. So, this choice is based on a thematic criterion (characterization of the morphological urban area) but also on a technical criterion (construction of a database easy to handle). It appears as a compromise linking the thematic analysis of the sprawl process and the optimization of calculations and simulations speed processes. This choice should lead to construct a tool allowing both to improve urban sprawl knowledge and to run quickly useful simulations for making planning decisions. With thirteen land use categories and a fiftymeters resolution, the grid database allows to quantify the importance of each category for each date within cells of 0,25 ha. For example, it shows with a rather good precision that the built area of Belfort grows constantly (fig. 3) :
Figure 3 : Morphological growth of Belfort
15Processing the database allows then to map the urban morphology through time (fig. 4) and grid mapping appears an interesting method to quantify the builtarea limits and to represent the evolution of the built / nonbuilt ratio. This operation consists of a simple GIS request, searching for the residential and nonresidential built cells, (C_{t} = 4, 5, 6, 7, 8 and 9 for each date, cf. fig. 2).
Figure 4 : Evolution of the urban morphology
16The whole information concerning the land use changes can then be queried through space and time within a unique GIS layer. But, the vector structure which has been used at the beginning in order to harmonize data, offers now an interesting grid to run different models. We can use it to apply successively three models, based on principles as different as gravity or Artificial Distributed Intelligence. Each one will be applied to approach a particular part of the urban sprawl global model.
17If we systematically reproduce the request described in the section 1.3, the quantification of every land use changes is easy. For example, we can search the cells containing forest areas in 1955 (c_{55 }= 2), considered as open spaces in 1975 (c_{75 }= 1) and occupied by individual houses in 1995 (c_{95 }= 4). Such an exercise leads quickly to the construction of two contingency tables, displaying the transformations between 1955 and 1975 and between 1975 and 1995. It consists in counting cells which changed during the two periods and storing the results within two 13*13 tables linking the land use categories at the date t and t+1 (fig. 5) :
Figure 5 : Contingency table
N : number of cells which changed from one category to another between t and t+1
18These land use changes can be visualized through a graphic, showing the complexity of past transitions (fig. 6). Apart from particular cases as highway nodes (C_{t} = 12), there are few transition impossibilities. The nonsymmetry of the two parts indicates that changes have a tendency to stabilize in the second period : the possibilities of transition are less important during 19751995 than during 19551975. To make it easier, it is now possible to reduce this complexity by considering the transitions in relation to their occurrence probability, and not only in relation to their presence (in a binary way). From the frequencies extracted from the database, it is possible to calculate the probability of each cell to move from one category to another, in order to create two transition matrices. The importance of the probabilities shown by the diagonal of the matrices demonstrates that a significant inertia force exists and that it is most important in the second period than in the first one (fig. 6). Although almost all land use transformations are possible, the majority of the cells do not change from 1975 to 1995.
Figure 6 : Graphic of observed transitions through time and transition matrices
(Blank cells correspond to null probabilities)
19From the two transition matrices, it is possible to estimate the transitions corresponding to the Belfort urban morphology towards 2015. This operation requires the use of a transition model, taking into account the stabilization of the transformation probabilities observed over the graphic and the second transition matrix diagonal (19751995). For example, we can test linear regressions and other methods relying on Markov processes (Collins, 1975). The comparison and the interpretation of the results obtained with these models depend strongly on the related hypotheses, but each one predicts precisely the land use categories which will increase or decrease. Then for instance, by summing up the two transition matrices weighted by their effectives on each line (Berchtold, 1998), we can create a homogeneous matrix representing the general trend of changes (fig. 7).
Figure 7 : Generation of a homogeneous matrix of transition and simulation in 2015
20Of course, this method is based on the assumption that the changes follow a trend which is similar through time (19552015), and that it can be determined from the two previous periods (19551975, 19751995). Using this general trend, we can then calculate the probable transitions between 1995 and 2015, and determine the number of cells of each category that will compose the study area in 2015 (fig. 7). This method seems quite interesting to estimate the future land use changes (fig. 8) but does not bring any information about the location of the cells that will be built. This is why to consider the changes, we have to complete this first step with a second one introducing spatial dimension, in order to locate the areas involved in urban sprawl.
Figure 8 : Estimation of the future urban expansion (brought down to 1)
21Different spatial models can be used to locate the n cells that will be built in 2015 (n being defined in the first step). For instance, we can use a potential model (cf. Weber, 1997, 1998). It takes simultaneously into account the complementarities between each cell and the rest, their respective distances, and the intervening opportunities existing on the area, as it usually appears in spatial interaction models (Abler and al., 1972). This way, the potential model assumes that :

urban expansion creates an interaction between new and old urban spaces which can be considered as complementary spaces

urban expansion is based on the minimization of the mathematical distances between the concerned cells

urban expansion favors the best solutions by testing all the possibilities of complementarities and distances.
 5 This calculation is computed in a software developed by Olivier Klein in Image et Ville laboratory, (...)
22In order to better explain, we will take an example using one of the most simple equations of the potential model, and will consider only the open space cells (C_{t} = 1)5 :
P_{i} = potential value of the cell i,
m_{j} = mass value of the cell j,
d_{ij} = distance from i to j
23Calibrating such a model means defining the mass values m corresponding to the land use category of each cell j. Empirically, we will use intuitive values according to the presumed attractiveness of each category and we will try to validate them by running the model on the known past periods : from the initial state 1975, we will try to find the initial state 1995. So, the last map shows the location of the n cells corresponding to the most important potential values for the 19751995 period obtained by modeling (fig. 9). These n cells should correspond to the ones which have been built during the same period. The results are quite relevant : about 25% of the simulated cells are located in the same location as observed cells, and about 90% are located within a radius of 200 meters.
Figure 9 : Locations obtained by the potential model
m values used for calculation (from C_{t}=1 to C_{t}=13) : 0 0 0 3 2 2 2 2 0 2 4 0 0 ; radius of calculation : 300 meters
24A method based on multiplecorrespondence analysis (Lebart and al., 1997) is currently under development to measure the nature of neighbor cells around the cells built between 1955 and 1975, and between 1975 and 1995, within different radii. This should help to determine the mass values m.
 6 The first matrix (initial state) contains only the land use information. That means that to take in (...)
25At this stage, the approach allows to locate the n cells that should change from a nonbuilt category to a built category, but we cannot describe more precisely the changes. For instance, we cannot know if the n (step 1) located (step 2) cells concerned with urban sprawl will be residential, industrial units or facilities. This discrimination remains to be done. The recent works in urban geography, using techniques issued from Artificial Distributed Intelligence, offer a new conceptual framework to approach this kind of problems (Langlois, Phipps, 1997). Here, cellular automata seem an interesting tool to determine the land use category of the cells located in step 2. Cellular automata should assume that the category of the cells is determined by its neighborhood, actually by the categories of the neighbor cells. In fact, the analysis of the neighborhood should allow to define some transition rules that could be applied to the nondefined cells to choose their category. Of course, the biggest problem is the definition of pertinent rules. At first, they can be determined by empirical tests, according to the users’ estimation. Then, a statistical method should be developed to construct a set of rules firmly relying on quantified observations (it could be quite similar to the method developed for the calibration of the potential model, cf. section 2.2). Finally, it will be interesting to integrate administrative information or legislation (that means information not directly concerning the land use) so as to refine the transition rules6.
26By associating these three steps, the methodological approach presented here can be associated to an imbrication of models. The final global model is not yet completed but should correspond to an interesting solution allowing to understand the urban sprawl and the role of the neighborhood (and its attractiveness) in the process. It should allow to run simulations easily. The parameters needed for these simulations can be modified at each step, according to the hypotheses given by the scenarios : modification of the general trend (step 1), of the global attractiveness (step 2), of the neighborhood’s role (step 3). In any case, the use of a grid database developed within GIS allows every possible continuation in modeling. From collecting to encoding spatiotemporal data to calculating complex simulations, grid mapping can be used efficiently at every step.