ABSTRACT: When studying dynamic geographic phenomena such as rural-urban transformation in a raster GIS environment, information between two consecutive snapshot layers may not be available. This paper describes an application of fuzzy logic which enables the completion of unknown information in generated intermediate layers using annual temporal resolution and spatio-temporal interpolation. Change is modeled temporally by performing three possible scenarios with a different duration of rural-urban transition, and spatially by applying two standard GIS methods. This methodology is tested with data from the Montreal Metropolitan area in Quebec, Canada, which cover the 1956 to 1986 period and have a temporal resolution of 10 years. User-friendly modules were developed and incorporated in GRASS4.1 environment in order to simulate the spatio-temporal changes which occur in the study area.
In current GIS raster databases, data are stored in a series of snapshot layers associated with particular instants in time. As a result, information about the change that has occurred in the interval between two consecutive snapshots is not available. When studying dynamic geographic phenomena which happened in the past, it is often impossible to obtain the missing information from data sources such as maps or remote sensing images. One possible solution to the problem consists of performing a temporal interpolation between two consecutive snapshot layers registered in the raster GIS database. An approach has been developed where fuzzy logic was applied to generate intermediate layers in a raster database to study land-use transformations in a rural-urban environment over the last 50 years (Dragicevic and Marceau 1997).
These basic concepts of fuzzy logic are illustrated in Figure 1 and are explained using as an example the transition from rural to urban land use. Given two snapshot layers captured at two time instants [t.sub.1] and [t.sub.2], two basic cases can occur during the transformation in the database. Case I shows that a geographic entity may stay unchanged in both instants [t.sub.1] and [t.sub.2]; thus the generated cells that correspond to the intermediate layers at instants [t.sub.1i] and [t.sub.1j] remain unchanged. In case II, the boundaries of the geographic entity have changed; therefore, some generated cells do not preserve their values since they are engaged in the transition.
[FIGURE 1 OMITTED]
The cell which is in the transition receives values determined by a set of membership functions which describe the current change. Furthermore, all cells known to be in the transition in every intermediate layer will carry the information about the change. This information varies, depending on the grades of membership specified in one set of membership functions. The transition modeling is based on the temporal proximity of the cell to the older or more recent snapshot layer.
For example, as shown in Figure 1, for the intermediate layer at [t.sub.1.i] which is closer in time to the older snapshot at [t.sub.1.0], the lower value of membership grade is assigned to the cell for urban land use, and the higher value of the grade to rural land. On the other hand, for the intermediate layer at [t.sub.1.j] which is temporally closer to the more recent snapshot at [t.sub.2.0], the higher value of membership grade is assigned to urban land use and the lower value to rural land use. Using this approach, other transitions could be similarly considered, such as the transition from forest to agricultural land use, agricultural land use to roads, or from forest to urban land use (Dragicevic and Marceau 1997).
A geographic entity is, however, subject to continuous spatial changes in time. Case III in Figure 1 illustrates a progressive spatial change of a geographic entity from layer to layer until it reaches its ultimate boundaries at the snapshot layer [t.sub.2]. Therefore, the information about change varies from one intermediate layer to another and does not depend only on one set of fuzzy membership functions, but also on the model of spatial propagation of an entity during a period of time.
In this paper, an extension to the approach developed by Dragicevic and Marceau (1997) is proposed in order to solve the problem of missing information about a continuous change that occurs in time. To simplify matters, a rural-to-urban transition is used to illustrate the extended approach.
Various approaches exist for urban expansion simulation (Meaille and Wald 1990; Landis 1994; Batty and Xie 1994; Clarke et al. 1997; Allen 1997), but they are often implemented in temporal extrapolation, i.e., in a prediction processes. These approaches are also based on a large volume of data on urban growth obtained over 100 with a large number of snapshots (Acevedo et al. 1996; Crawford-Tilley et al. 1996; Bell et al. 1995). Consequently, trends in urban growth patterns can be estimated using the probability theory. These prediction models integrate a large number of parameters which may influence urban expansion such as socioeconomic factors, road and hydrographic network, slope, demography or urban morphology. When studying urban expansion which had occurred in the past, the most accurate information may not be available on how the spatial expansion process progressed, only the information stored in the database snapshots.
The objective of this study is to simulate the progressive change of rural-urban transition for which essential information is lacking by means of spatio-temporal interpolation of conditions lacking the essential information about urban growth. This approach is based on fuzzy logic with multiple series of fuzzy membership functions which were developed for each intermediate layer separately. The transition from rural to urban land use is modeled temporally in three possible scenarios of different duration, and spatially by using two standard GIS methods (surface interpolation and buffer operation) for simulating the spatial evolution of a transition. The methodology was tested in North Shore of Montreal's metropolitan area in Quebec, Canada.
The Fuzzy Logic Theory and Its Application in Spatio-Temporal Interpolation
This section describes basic concepts related to fuzzy logic, as developed by Zadeh (1965, 1978), and its applications to spatio-temporal interpolation in a raster GIS database. Any classical set is characterized by an abrupt boundary which could be drawn between two classes of elements: those which fully belong to the set and those that are outside the set and belong to the set's complement. As opposed to the classical set, a fuzzy set consists of its elements and their respective degrees of membership in the set. For example, if T is considered to be a set of n elements [t.sub.i], their respective degrees of membership are [Mu]([t.sub.i]). A fuzzy subset A of T is characterized by a membership function [[Mu].sub.A]:T [right arrow] [0,1], which is used to calculate a degree of membership [[Mu].sub.A](t) between two extreme states: 0 and 1. The value of 0 indicates that the considered element does not belong to the fuzzy subset A; the value of 1 denotes a full membership to the fuzzy set. The closer the value is to 1, the higher is the element's degree of belonging to the set.
Recently, fuzzy reasoning has found its use in the areas of remote sensing and GIS. Some scientists point out that the classical Boolean approach to the representation of geographic phenomena used in current geographic information systems (GIS) is not always appropriate …