Landscape and Urban Planning

Abstract

In recent years, cellular automata (CA) models for urban growth simulation have proliferated because of their simplicity, flexibility and intuitiveness, and particularly because of their ability to incorporate the spatial and temporal dimensions of the processes. Though apparently simple, CA models are capable of modeling complex dynamic systems such as urban systems. Currently, one of the main problems in actually applying CA models to urban planning practice is the choice or design of the most suitable CA model. For this reason, a review of urban CA models applied to real-world cases is provided, along with an analysis of their capabilities and limitations. The review and classification of CA models based on the main characteristics of the models has allowed for the analysis of their strengths and weaknesses. Finally, a discussion of the needs for further research is presented.

1. Introduction

In recent decades, a number of modeling techniques have been developed to better understand and predict urban growth. Early studies of urban processes used transportation and land-use planning models based on gravity theory or optimizing mathematics, but soon evolved into more dynamic spatial models (Berling-Wolff and Wu, 2004a) , such as cellular automata (CA).

CA were first developed in the late 1940s by S. Ulan and J. von Neumann like copies of the same Turing machine placed at each cell of a lattice and connected together. Wolfram (1984) demonstrated that complex natural phenomena can be modeled by CA and, later, laid the foundations for a Theory of Cellular Automata (Wolfram, 2002) , defined as discrete dynamic systems in which local interactions among components generate global changes in space and time. CA simulation was soon applied to physical sciences, natural sciences and mathematics. Tobler (1979) first proposed the application of cellular space models to geographic modeling. In the 1980s, the first theoretical approaches to CA-based models for the simulation of urban expansion appeared (Batty and Xie, 1994; Couclelis, 1985; White and Engelen, 1994) . Itami (1994) reviewed CA theory and its application to the simulation of spatial dynamics, and Batty (2005b) provided an analysis of diverse applications of urban CA.

Conceptual advances in CA research and the development of computing power led to the emergence of the first operational urban CA models applied to real-world urban systems in the 1990s. The ability of CA to simulate urban growth is based on the assumption that past urban development affects future patterns through local interactions among land uses. The interest of CA-based models for urban simulation can be explained in terms of the simplicity, flexibility, intuitiveness and transparency of CA. Additionally, CA can be easily integrated with Geographical Information Systems (Itami, 1994; Wagner, 1997) and, consequently, model at high spatial resolution with computational efficiency. Besides, many authors have shown that the nonlinearity of the iterative process of CA leads to regular fractal patterns, i.e. to regular and ordered spatial patterns that generate similar geometries at different scales. Such fractal structures, derived from complex phenomena, are characteristic of urban developments (Batty, 1991; Batty and Longley, 1994; Longley and Mesev, 2000) .

When actually applying CA-based models to urban planning, choosing the most suitable model from among the many options available (Pinto and Antunes, 2007) is difficult (Li and Yeh, 2002a) . In order to present a structured overview that facilitates the choice of a particular method for a given application problem, an analysis of 33 urban CA models has been performed. The differences among the different approaches are highlighted and a classification of the models is proposed. The objective of this paper is to provide a basis for a literature review of urban CA and for the future development of new advances in this area. To this end, the general characteristics of urban CA are described and the different techniques used in the design of such models are characterized and classified. Such techniques have been summarized in several tables. In addition, the strengths and weaknesses of the different models have been identified from the analysis and discussion of the characteristics of the models.

2. Relaxations Of Ca For Urban Simulation

A CA consists of a discrete cell space, in which states characterize every cell. In urban CA, states can be (i) binary values (urban, nonurban) , (ii) qualitative values that represent different land uses, (iii) quantitative values that represent, for example, population density (Li et al., 2003) , degree of development (Yeh and Li, 2002) or the value of buildings (Cecchini and Rizzi, 2001) , or (iv) a vector of several attributes (Portugali and Benenson, 1995) . The state of each cell depends on its previous state and on the state of its neighboring cells according to a set of transition rules. The 'background conventions' of CA limit their ability to realistically simulate complex geographical phenomena (Couclelis, 1985) . For this reason, adapting CA to urban simulation requires considering the particularities of this phenomenon, which usually entails a relaxation of the original structure of CA in order to introduce more complexity in models (Couclelis, 1997) . The most common modifications include:

(1) Irregular cell space. Formal CA assume a cell space represented by a regular grid usually composed of square cells, although some authors have proposed using hexagonal cells in order to obtain a more homogeneous neighborhood (Iovine et al., 2005) . Additionally, the cell space can consist of a three-dimensional matrix that allows to represent growth in height of urban areas (Semboloni, 2000) . The regular grid space can be modified by using irregular tessellations such as Voronoi polygons (Shi and Pang, 2000) or graphs (O'Sullivan, 2001a,b) . Irregular spatial units may provide a more faithful representation of the objects being modeled. For example, using cadastral parcels instead of regular cells (Stevens and Dragicevic, 2007) provides a representation that comes closer to reality. Yet, using cadastral units complicates the definition of the neighborhood. (2) Non-uniform cell space. In standard CA, the cell space is homogeneous, i.e., all cells are identical and are characterized exclusively by their state. However, land-use change depends largely on other land attributes such as slope, elevation or accessibility. Consequently, the cell space is not uniform, such that some cells are more suitable for certain land uses. (3) Extended neighborhood. In strict CA, the neighborhood must be the same for every cell and must be composed of the geometrically closest set of cells (e.g. Moore and von Neumann neighborhood). In urban systems, this local neighborhood must be extended in order to consider the influence of neighboring cells located at a certain distance. When the neighborhood is extended, a distance-decay effect is usually introduced in the model, such that the effect of a neighboring cell decreases with the increase in distance between both cells. In spaces composed of irregular units, the neighborhood can be defined as the adjacent units, as the units within a specified distance or using the Voronoi spatial model (Shi and Pang, 2000) . (4) Non-stationary neighborhood. The neighborhood space may be defined differently for each cell. Such a relaxation is widely acknowledged (Couclelis, 1985) , but seldom implemented. Models in which each cell receives a weight according to its state and location within the neighborhood (e.g. White and Engelen, 1993) allow for the application of neighborhoods of different sizes and shapes by introducing weights equivalent to zero.

(5) More complex transition rules. The transition rules of a formal CA consider solely the current state of the cell and its neighbors. However, a variety of factors influence urban processes, such as suitability for a land use, accessibility, socioeconomic conditions, or urban planning. Consequently, urban CA are not closed systems, as established by the CA formalism, since urban CA models can consider external factors. The transition rules of urban CA can be designed in many ways and reflect various urban theories, based on microeconomic theories of planning (Webster and Wu, 2001) , centrality and potential models (Polidori and Krafta, 2005) , etc. (6) Non-stationary transition rules. The transition rules of strict CA are static, however the processes that govern land-use change may vary over time and space. Therefore, it may be necessary to adapt the transition rules to the specific characteristics of each area and period. Spatial and temporal variation can be achieved through calibration (Geertman et al., 2007; Li et al., 2008) . The most evident example of transition rules that vary over time are the self-modification rules of SLEUTH (Clarke et al., 1997) . Phipps and Langlois (1997) proposed a system that modified the transition rules at every time step according to changes in configuration and external parameters. (7) Growth constraints. In conventional CA the number of cells that change state is endogenously defined by the application of transition rules. However, urban land demand is usually determined by exogenous social, economic or environmental constraints, such as demographic evolution or urban planning, which constrain overall urban growth. (8) Irregular time steps. There are many urban CA in which different cells may be subject to time steps of different lengths (Stevens and Dragicevic, 2007) . A less frequent relaxation is using variable time steps (Couclelis, 1997) to simulate specific events of different lengths of time. Cecchini and Rizzi (2001) suggested applying two types of rules: structure rules, applied in every iteration, and conjunctures rules, applied only when a specific event was implemented.

3. Analysis Of Urban Ca Models

Early applications of CA to urban dynamics modeling were theoretical models aimed at simulating simple urban structures. Takeyama and Couclelis (1997) defined the language of geo-algebra as the mathematical basis for the development of urban CA within GIS. Theoretical developments were followed by the design of abstract simulation models of urban evolution (Batty, 1998; Xie, 1994, 1997; Batty et al., 1999; Semboloni, 1997) . Such theoretical approaches allowed modelers to test hypotheses of urban theories and simulate general urban forms (Batty, 2005a) . Other theoretical applications of CA to urban modeling are described by Itami (1988) , Portugali and Benenson (1995) , Cecchini (1996) , Phipps and Langlois (1997) , Wu and Webster (1998a) , Webster and Wu (1999a,b) , Liu and Phinn (2003) and Kocabas and Dragicevic (2006a) . These theoretical models laid the foundations for the development of operational urban CA, but were never tested in real cities.

The next step was the application of the above theoretical models to the simulation of real-world urban development processes. The emergence of GIS has contributed to the change from artificial applications to real simulations (Couclelis, 1997) . This paper focuses on models that have been applied to real cities and, in most cases, have been subject to calibration and validation.

Some of the first applications of urban CA to the simulation of real-world cases were carried out by Batty and Xie (1994) and Xie (1996) in Amherst, New York. However, the first widespread empirical application of these models was developed by White et Residential use: the attractiveness of the parcel for residential use is the sum of the scores given for proximity to parks, commercial areas, light industry and heavy industry, and the score given by adjacency to residential land. If the adjacency score is higher than the sum of the other four scores, all the parcels adjacent to currently developed parcels will begin to be developed until the demanded area is fully developed. Commercial use: if the parcel is adjacent to a road, the neighborhood has enough population and the neighborhood is not saturated with other developed commercial properties, the cell is developed. The product of the neighborhood effect and the constraint score determines the additional 'grey' state, which is added to the previous 'grey' state to calculate the 'grey' state. The state of a cell will be: developed if the 'grey' state of the cell is 1, partially developed if the 'grey state' is between 0 and 1, and the previous state is the 'grey state' is 0./In Li and Yeh (2002b) , the additional 'grey' state is the product of the stochastic disturbance term, the neighborhood effect, and a function of the distance from the cell to the 'ideal point', where this distance is based on the attributes in the principal component space./In Yeh and Li (2002) , the additional 'grey' state is calculated by a more complex function which involves the neighborhood effect, a constraint function for the hierarchy of a major center and subcenters, and a nonlinear transformation of different weighted factors./ Yeh and Li (2001) use the same function as Yeh and Li (2002) (1997) , based on the previous models of Engelen (1993, 1997) , which calculated for every cell the potential for transition to different land uses as a function of the neighborhood, the suitability for each land use, an inertia effect, and a stochastic disturbance term. A number of models based on this type of transition potential were applied to the Island of St. Lucia (Engelen et al., 1995) , Cincinnati , the IJmond/Kennemerland region of Holland (Engelen et al., 1999) , The Netherlands (White and Engelen, 2000) , Dublin (Barredo et al., 2003) , Lagos (Barredo et al., 2004) and San Diego (Kocabas and Dragicevic, 2006b) . These studies confirmed the possibility of achieving highly realistic predictions of urban evolution using CA-based models. This type of models were improved by Arai and Akiyama (2004) , who used discriminant analysis to facilitate their calibration, and Caruso et al. (2005) , who incorporated an urban economic model to reinforce their theoretical basis. Another widespread model is SLEUTH, developed by Clarke et al. (1997) to simulate urban dynamics in San Francisco and further applied to other regions in North America (e.g. Clarke and Gaydos, 1998; Yang and Lo, 2003; Herold et al., 2003) , Europe (Silva and Clarke, 2002) , South America (Leao et al., 2004) and Asia (Mahiny and Gholamalifard, 2007) .

The models of Wu (1998b Wu ( , 2002 , Wu and Webster (1998b) and Wu and Martin (2002) focused on the calculation of the development probability for every cell according to a number of factors, such as the neighborhood. The first urban CA models developed by Yeh (2000, 2002b) and Li (2001, 2002) were based on grey cells, which represented the cumulative degree of development, and focused on finding feasible alternatives for the planning of sustainable development. More recently, these authors have focused on defining transition rules using artificial intelligence methods. DINAMICA (Soares-Filho et al., 2002) is a CA-based model originally designed for the simulation of deforestation processes that was later applied to urban processes (Almeida et al., 2003 (Almeida et al., , 2005 .

The analysis of the most relevant characteristics of these and other CA-based models has allowed us to group and characterize the models. The main characteristics of these urban CA are summarized as follows:

(1) Transition rules. The relaxations allowed in CA to more realistically simulate urban growth have led to the emergence of a large variety of rules. To facilitate analysis, transition rules have been classified into six types. Yet, some rules can be included in several groups.

Type-I rules are strictly orthodox transition rules, in the sense that the state of a cell is a function of the cell's current state and the state of its neighbors, which can be implemented by means of simple rules, based exclusively on the number of neighboring cells of each land use (Jenerette and Wu, 2001; Yüzer, 2004) , or of more complex rules (Besussi et al., 1998; Stevens and Dragicevic, 2007; Ward et al., 2000) (Table 1) .

Type-II rules. In type-II rules, the key driver of urban evolution is the transition potential, i.e. the probability that each cell changes to a specific land use, which is a function of the current land use of the cell and its neighbors (CA component), and of other factors that constrain land-use evolution.

White and Engelen were the first to propose the calculation of transition potentials (Engelen et al., 1999; . The models designed by Wu (1998b Wu ( , 2002 , Wu and Martin (2002) and Wu and Webster (1998b) determined the probability of development as a function of a number of factors, such as neighborhood. These factors are not preset, which allows modelers to introduce the most suitable factors for each specific application. The same applies to the models developed by Almeida et al. (2003 Almeida et al. ( , 2005 ), Cheng and Masser (2004) and He et al. (2006 He et al. ( , 2008 . Conversely, the models proposed by Lau and Kam (2005) , Sui and Zeng (2001) and White and Engelen (2000) used preset factors. Yet, the models by White and Engelen (2000) and Lau and Kam (2005) allowed for the incorporation of the desired factors in the calculation of the suitability or of the attribute effect respectively.

The transition potential is usually calculated as the weighted sum or product of a number of factors (Table 2) , among which the effect of neighborhood stands out. In addition, a stochastic disturbance parameter is commonly used to model the uncertainty associated with urban processes. These models may include a series of constraints that take the value 0 if a cell cannot be developed or 1 otherwise. When the transition potential is calculated as a weighted sum, different techniques can be used to determine the weights, such as logistic regression (Sui and Zeng, 2001; Wu, 2002) or multicriteria evaluation (Wu, 1998b; Wu and Webster, 1998b) .

In other models, the transition potential is calculated using more complex functions. For example, Wu (1998b) uses an exponential function justified because the sites with higher scores are more likely to be developed. Such complex functions are usually based on statistical or urban theories. The first group includes the models of Almeida et al. (2003) , in which the transition probability is based on the weights of evidence for different factors; Li and Yeh (2002b) , which uses Principal Components Analysis (PCA) to define the factors of the transition rule; and Lau and Kam (2005) , which uses statistical tools to identify the factors to form the attribute and heterogeneity effects.

The second group includes the models of Caruso et al. (2005) , based on urban economics, which allows, according to the authors, a further understanding of urban processes; He et al. (2008) , which uses a potential model to calculate the urban expansion potential by taking into account the effect of the spatial distribution of capital and population; and Yeh and Li (2002) , which incorporates density and constraint functions to simulate different types of urban forms.

Type-III rules. These transition rules are based on urban shape and form for the reproduction of the spatial patterns of urban growth (Table 3 ). An example of type-III rules is found in SLEUTH, a pattern-extrapolation model that considers four types of urban growth: spontaneous growth, new spreading center growth, edge growth and road-influenced growth. The model proposed by is based on urban shape to the extent that it is based on the assimilation of the spatial appearance of urban expansion to a physical diffusion process. The DINAMICA model can be included in type-III rules because, from among the cells with the highest probabilities, some cells are chosen using two functions, which are aimed at spatially simulating urban growth: the Expander function, which expands or contracts the existing patches and the Patcher function, which generates new patches.

Type-IV rules. Type-IV rules use artificial intelligence methods (Table 4 ) such as neural networks, kernel-based learning methods, or Case-Based Reasoning (CBR), which are learning algorithms aimed at recognizing complex patterns based on data, or data mining methods, which automatically reconstruct explicit transition rules.

Type-V rules. Type-V rules are based on fuzzy logic, which allows the uncertainty of human behavior to be included in the simulation and the definition of transition rules through a natural language (Table 5) .

Type-VI rules. Type-VI rules include transition rules that cannot be grouped into a general method (Table 6 ). These models are based on simple rules, defined through conditional logical operations. By contrast, the model developed by Xie (1996) is more complex insofar as it defines a general scheme on which multiple models can be built.

Table 3

4. Strengths And Weaknesses Of Urban Ca Models

Because all the models have advantages and disadvantages, some of the most relevant aspects of urban CA models have been Table 6 Main characteristics of other urban CA models. Author AGR -annual growth rate for urban land; POP -population growth projection; PLA -urban regulation planning; MOD -model mentioned in the 'other methods' section; OTH -other studies.

Table 7

4.1. Balance Between Realism And Preservation Of Ca Features

The relaxations of the original scheme of CA may lead to the loss of the fundamental characteristics of simplicity and locality, or even to models in which the CA component is no longer the core of the model. Except for the models developed by Besussi et al. (1998) and Jenerette and Wu (2001) , most models show some relaxation. Yet, some models present more important modifications. For example, in the neural network proposed by Li and Yeh (2001) , the CA component is present exclusively in the network input variables pertaining to the distance from a cell to each land use. In this model, even the application of the only concept that is essential to the definition of a model as a CA-based model, namely, the neighborhood concept, is quite vague. Almeida et al. (2003) defined their model as a cell-space model rather than a CA model because the CA component was quite hidden. In Li and Yeh (2004) and Liu et al. (2008a) , the neighborhood function was only one of the nine spatial variables used to define the transition rules, whereas in Li and Liu (2006) and Deadman et al. (1993) , neighborhood was not explicitly addressed.

A possible solution to the problem of achieving a balance between realism and simplicity is the multi-cellular automaton proposed by Cecchini and Rinaldi (1999) , which consists in an ordered sequence of CA sub-models, such that every single CA conforms to the orthodox formulation (Besussi et al., 1998) .

4.2. Flexibility

Flexibility is the ability of the model to adapt to different real-world urban situations and depends on the flexibility of the transition rules, on the factors considered in such rules, on the land uses modeled and on neighborhood adaptability.

With regard to transition rules, the most flexible models are the models that propose a general scheme within which multiple specific models can be defined (e.g. Xie, 1996; Besussi et al., 1998) . The least flexible models are the models that use neural networks, CBR or data mining, in which rule definition and calibration are simultaneous, such that the resulting transition rules are closely adapted to local conditions. Consequently, these models are not suited to define general urban CA, but rather a methodology to define specific models for each situation. Most of the rest of the models show intermediate flexibility because they are based on more or less general rules that are adjusted to every situation through calibration. The adaptability of the model largely depends on calibration.

With regard to the factors included in the transition rules, using strict rules limits the ability of the model to represent different real-world urban situations. For type-II rules, a distinction has been made between rules that include specific factors and more flexible rules that allow for the use of any factor. Models that use artificial intelligence techniques are flexible in terms of the factors considered, as opposed to models based on urban shape and form. In principle, CA-based models based on fuzzy logic allow modelers to consider an indefinite number of factors. However, in practice, problems arise when a large number of factors are included in such models.

With regard to land uses, simulating the evolution of multiple land uses is far more complex than simulating urban growth. However, Dietzel and Clarke (2006) demonstrated that models that include only urban/non-urban data oversimplify the dynamics of the system because of the strong link between the likelihood of urbanization and the type of land use that will be converted to urban. Yet, only 13 out of the 33 models reviewed in this paper consider multiple land uses.

With respect to neighborhood, only a few models provide the possibility of applying different types of neighborhood. Only the model developed by Stevens and Dragicevic (2007) considered adjacent parcels or parcels located at a specified distance depending on each land use. Xie (1996) considered three types of neighborhood, whereas Ward et al. (2000) used a local neighborhood and a wider one. In the models proposed by White and Engelen, the size and shape of the neighborhood can be adjusted, and different neighborhoods can be applied to different uses by setting weights equal to zero from a certain distance, in some cells or for certain uses. In Al-Ahmadi et al. (2009) , neighborhood size is one of the parameters defined during calibration. Other models were tested using several neighborhoods, but in different simulations.

Most models have been tested only for the regions for which they have been designed. Examples of the adaptability of urban CA models to the local characteristics of very different regions can be found only for the most widespread models, among which SLEUTH and the family of models designed by White and Engelen.

4.3. Explanatory Power

There are descriptive and explanatory models. Descriptive models tell us what is happening, but not why. For example, the transition rules of SLEUTH do not allow for the analysis of the causes of the spatial patterns generated. In the neural network of Li and Yeh (2001) , knowing the weight of each variable in the final output and, consequently, explaining the origin of the simulated phenomenon are not possible. Something similar happens in the model of Li and Liu (2006) . However, this limitation does not affect the ability of the model to predict urban growth or to solve 'what if'-type questions.

At the opposite end of a black-box approach are explicit transition rules obtained using data mining techniques or fuzzy inference, which are transparent and easily understood by decision-makers. However, explicit rules have difficulties in representing very complex relationships and the choice of fuzzy functions is subjective and largely influences the results.

An intermediate approach uses models based on mathematical equations that provide information about the causes of the process but are less intuitive than explicit rules, and can manage more complex relationships than explicit rules but show more limitations than artificial intelligence techniques to simulate nonlinear complex systems.

Generally, the main advantage of descriptive models, particularly of models based on informing theories (e.g. Wu and Webster, 1998a) , is their ability to explore and validate hypothetical ideas related to urban dynamics (Torrens and O'Sullivan, 2001 ). Yet, only a few urban CA models applied to real-world processes are based on well-developed theoretical models (e.g. Caruso et al., 2005; He et al., 2008) .

4.4. Data Requirements

Data requirements depend upon the factors considered in the model, so are usually in opposition to flexibility: the only input required for strict rules are land-use maps, whereas in rules of types II, IV and V, input data vary greatly.

4.5. Software Availability

Most of the models reviewed use the geographic information management capabilities of standard GIS, and program the models using general programming languages or macrolanguages. In both cases, modelers need programming knowledge to implement the models, which makes their diffusion dif- Table 8 Overall accuracy and kappa index of the results of various models.

Overall Accuracy (%)

Kappa index Al-Ahmadi et al. (2009) 92-94 Barredo et al. (2003) 0.62-0.93 Barredo et al. (2004) 0.63-0.88 Cheng and Masser (2004) 55 Engelen et al. (1999) 0.88-1 He et al. (2006) 0.73-0.80 He et al. (2008) 0.75-0.86 Lau and Kam (2005) 82.9-99.5 0.71-0.99 Li and Liu (2006) 82-86 0.51-0.53 Li and Yeh (2001) 79 Li and Yeh (2002a) 83 Li and Yeh (2004) 72.4-82 Liu et al. (2008a) 76.8-83.3 0.53-0.64 Liu et al. (2008b) 74.1-79.0 0.48-0.57 Sui and Zeng (2001) 71-83 0.51-0.69 Wu (2002) 72.6-79.5 Wu and Webster (1998b) 68.9-81 Yang et al. (2008) 84.9-87.25 0.68-0.7 ficult and impedes non-expert users to apply such models. Some authors have developed independent software applications, such as AUGH (Besussi et al., 1998) , or iCity (Stevens and Dragicevic, 2007) . Among the few applications available are SLEUTH (http://www.ncgia.ucsb.edu/projects/gig/) and DINAMICA (http://www.csr.ufmg.br/dinamica/).

4.6. Accuracy Of The Results

Overall, the accuracy of these models is good (Table 8) . Nevertheless, the results are not directly comparable because they are largely dependent on the land-use pattern of each area. However, all the models considered the total surface area of the study area and, therefore, the percentage of cells that changed land use with respect to the total number of cells was usually low. By excluding the area that was already developed at the beginning of the simulation, Jantz et al. (2003) obtained an accuracy of 19% and a k value of 0.19. However, when the entire surface area was considered, the accuracy amounted to 93.1% for an area in which the developed area accounted for 22% of total area. Overall accuracy will be less significant for areas in which the developed area is smaller. Consequently, when the evaluation is performed without excluding the area in which land uses remain constant, such an evaluation should include at least the proportion of converted area. Almeida et al. (2008) have solved this problem by using a fuzzy similarity metric to compare patterns.

5. Conclusions

Urban CA models provide a tool with great potential for the development of operational models that generate realistic urban patterns, and contribute to a better understanding of urban dynamics and theories. The main strength of CA-based models is their ability to integrate the modeling of the spatial and temporal dimensions of urban processes. Yet, the main reason for the widespread acceptance of these models is their simplicity. Anyway, the relative simplicity of CA-based models is also their main weakness insofar as flexibility limits the ability of the model to represent real-world phenomena, thus leading to the relaxations mentioned earlier in this paper. As stated in Section 4.1, when modifications are too extensive, it remains in doubt as to whether urban CA actually constitute CA at all. Another shortcoming of urban CA is the lack of a standard method for the definition of transition rules, although this allows modelers to design the most suitable model for each case. The main difficulty is the definition of simple rules that represent the complexity of the processes. The tradeoff between simplicity and flexibility in the design of the transition rules has been discussed in Section 4.2. Other difficulties for the implementation of urban CA models, described in Sections 4.4 and 4.5, include data requirement and the lack of easily configurable and usable software. Because of the above shortchomings, the use of urban CA models is most often limited to academic exercises.

In recent years, many researchers have focused on calibration because calibration is a key aspect to achieve reliable simulations and, therefore, to apply CA-based models to practical cases of urban planning. Recently, some authors have used artificial intelligence techniques to calibrate urban CA models or to define the transition rules of the models. Other authors have focused on the analysis of the effects of the different parameters included in the model, such as neighborhood type and size (Kocabas and Dragicevic, 2006b) , cell size (Dietzel and Clarke, 2004; Jantz and Goetz, 2005; Samat, 2006) , both (Ménard and Marceau, 2005) , land-use classes (Dietzel and Clarke, 2006) or temporal resolution (Liu and Andersson, 2004) . However, the impact of other factors such as cell type or the stochastic component remains unstudied.

Further research in this area would involve the development of new validation methods, based on urban pattern recognition, and the exploration of the CA potential for integration with other conventional geographical and urban theories, which would provide urban CA with a better developed theoretical background. Similarly, by integrating CA-based models with other techniques such as multi-agent systems or transportation models, hybrid models that overcome some deficiencies of CA could be obtained (e.g. Torrens and Benenson, 2005) . Finally, further demonstrations of how such models can help solve practical planning issues are required (e.g. Berling-Wolff and Wu, 2004b; Jantz et al., 2003; Syphard et al., 2005) . Mainly, such models will be not be used to exactly predict a phenomenon, but to interactively simulate different scenarios by modifying the parameters of the model.

SECTION

a D -descriptive; P -predictive; M -multiple land uses; PC -prescriptive. b AGR -annual growth rate for urban land; POP -population growth projection; PLA -urban regulation planning; MOD -model mentioned in the 'other methods' section; OTH -other studies.