Prior to 1950s, the concept of growth of cities was limited to assumption that cities grow in a centric way, growth is concentrated around the central business districts or the central zones ( Wegener, 1994 ). Models like concentric zone model (1925) ( Brown, 2015 ), Sector model ( Hoyt, 1939 ), Gravity model (1946) ( Zipf, 1946 ), multi nuclei model ( Harris & Ullman, 1945 ) were developed based on the assumption and mainly categorized under central place theories based models as shown in Figure 2 . These models mainly focused on spatial interactions where the intensity of two spatially interacting components is closely related to their properties and the distance between them as in gravity model or totally dependent on the attraction of the similar components as in the sector model ( Li & Gong, 2016 ; Batty, 2008 ).

Post 1950s, due to rapid development of interstate highways in USA and similar growth in transport infrastructure in Western Europe led to development of growth along the peripheries of highway routes and transportation networks. New cities started appearing along the highway routes as the number of car users increased. Industries started settling around the routes for better access to transportation. This led urban planners to realize the centric models of urban growth are insufficient to predict the land use and trip and location decisions co-determine each other and impact in the rate and density of land use; therefore transport and land use planning needed to be coordinated ( Hansen, 1959 ; Berling-Wolff & Wu, 2004 ). Figure 3 represents the models based on the location-based theories.

The CA models were first proposed in geographic modelling by Tobler in 1979 and applied in urban growth modelling in the 1980s and 1990s ( Batty & Xie, 1994 ; Couclelis, 1985 ) and have been widely used in the simulation and prediction of urban-land dynamics since then ( Phipps & Langlois, 1997 ; Feng & Qi, 2018 ; Liu & Phinn , 2003). These models are based on the concept that the pixels in raster geographical representation also called cells, interact with each other based-on neighborhood rules and transition based on the rule based or probabilistic statistical transition functions as shown in Figure 4 ( Li & Gong, 2016 ).

( Li & Gong, 2016 ) described the CA model using Equation (1)

$s_{ij}^{t+1}=f\left(s_{ij}^t,{\Omega }_{ij}^t,T\right)$ Equation (1)

where:

$s_{ij}^{t+1}$ is the stage of cell (i, j) at time t + 1

( $s_{ij}^t$) is the state of cell representing the stage time t

${\Omega }_{ij}^t$is state of neighborhood state and transition rules

T is the transition rule by linkage to function f.

The CA models are better at representation of spatial interaction at the pixel level. However, traditional bottom up CA models were not able to capture the macro socio-economic driving factors of growth ( White & Engelen, 2000 ). These models also faced challenges in terms of number of driving factors sensitivity, fixed cellular state, and integration of human decision-making in the change process, thus failing to represent dynamic nature of spatial change ( Santé et al., 2010 ; Liu et al., 2021 ). The development of CA models in urban growth modelling thus follows the trajectory to rectify and address the issues in the traditional CA modelling approaches including changes in cell shape, cell size, neighborhood configuration, transition rules.

Initial CA models were developed in the raster-based GIS platforms where each cell shape representation is a regular rectangular or square lattice ( Yi, Min, & Lei, 2015 ; Stevens, Dragicevic, & Rothley, 2007 ). To address this shortcoming, the patch-based raster cells are proposed ( Chen et al., 2019 ; Wang & Marceau, 2013 ). Patch represents a block or parcel of land as combination of cells with homogenous land use type as in Figure 5 and conversion is on the patch rather than cell by cell basis ( Yang et al., 2020 ). However, patch based simulations still represents the regular and fixed neighborhood, and hence cannot represent heterogeneity of space ( Zhai et al., 2020 ).

Change in spatial heterogeneity was done using vector based cellular automata. The vector-based CA allowed the representation of land use classes as Voronoi polygons, Delaunay triangles, spatial polygons, irregular automata in forms of cadastral land parcels, census parcels and planning zones ( Lu et al., 2020 ; Zhuang et al., 2022 ). This approach also provides links between varying land classes, socio economic information and spatial information which makes the model more realistic and allows simulation in much finer spatial scales improving the simulation accuracy ( Zhu et al., 2021 ).

However, in vector-based CA models the definition of neighborhood becomes complex as the size and shape of cells vary. ( Stevens, Dragicevic, & Rothley, 2007 ) have proposed three alternative solutions to this problem, defining the neighbors of each parcel as: a) adjacent parcels only, b) those parcels that are totally or partially covered by a distance buffer, or c) the area within a buffer. ( Dahal & Chow, 2014 ) proposed another neighborhood definition based on topological relations, proximity and intercepted buffers, and the extended neighborhood, where every other parcel acts as a neighborhood. ( Barreira-González & Barros, 2017 ) proposed the use the spatial metrices to characterize and measure the neighborhood effect in irregular cells.

Moreover, transition rules are the most important aspect of CA models and determine the evolution of a cell. Transitional rules are a set of functions either deterministic or probabilistic based on which a cell changes its state ( Roodposhti, Aryal, & Bryan, 2019 ). Traditional CA models use neighborhood influence rules as the benefactor to determine the transition rule ( Roodposhti, Hewitt, & Bryan, 2020 ). As in Von Neumann or Moore Neighborhoods in raster-based CA, there is a certain combination of number of cells defined that determines the evolution of the target cell also shown in Figure 4 .

To make CA models more efficient in simulation, in addition to neighborhood transition rules, different statistical, probabilistic, machine learning and heuristic methods are introduced to determine the change of cell state/transition rules. Figure 6 shows the classification of the CA models based on used of different methods to define the transition rules. Various statistical methods like Markov Chain ( Feng et al., 2018 ; Aburas et al., 2021 ; Rimal et al., 2018 ; Aburas et al., 2017 ), regression methods ( Long et al., 2014 ; Cao et al., 2020 ), decision trees ( Basse, Charif, & Bódis, 2016 ), fuzzy sets ( Liu, 2012 ) have been used in the determination of transition rules in CA models.

With the evolution of CA models and use of variable influence factors, it was essential that transition rules be determined based on the influence of the driving factors that play a significant role in change of urban geography ( Hewitt, van Delden, & Escobar, 2014 ). This means that a cell will change from non-urban to urban if it meets the required neighborhood transition rule and has higher influence of the driving factors like proximity and accessibility to different biophysical and socioeconomic factors.

Also, to address challenges in addressing correlation in interaction of influencing factors, spatial and temporal heterogeneity of spatial growth; variable machine learning methods like Artificial neural networks ( Li et al., 2013 ), Support vector machines ( Okwuashi & Ndehedehe, 2021 ), and heuristic methods like Genetic Algorithm ( Liu, Feng, & Pontius, 2014 ), Artificial Bee colony( Feng & Tong, 2020 ), grey wolf optimizer ( Cao et al., 2019 ), Particle Swarm Optimization ( Feng et al. 2022 ) are used to mine the transition rules. ( Feng & Tong, 2019 ) used three different methods to quantify spatial heterogeneity of land use classes in the neighborhood cells and used genetic algorithm to mine the transition rules. ( Pinto, Antunes, & Roca, 2017 ) used Particle Swarm method in irregular cell CA to deduce transitional rules. ( Yan et al., 2021 ) used eigenvector spatial filtering techniques in CA modeling to reduce the spatial autocorrelation in driving factors and spatial interactions in urban growth simulation of Suzhou, China. ( Gao et al., 2020 ) used three spatial regression methods to address spatial heterogeneity: a spatial lag CA model (SLM-CA), a spatial error CA model (SEM-CA) and a geographically weighted regression CA model (GWR-CA) for simulating urban growth at Nanjing, China.

In ABM models the pixels/grids are called agents. These agents are free to move in space unlike the cells in CA models and thus, agents can mimic the interaction between humans and environment and make choices and decisions based on this interaction ( Matthews et al., 2007 ). These agents are given certain attributes based on socio economic formulation of the urban system and interact with each other and transition based on the outcome of the interaction with each other, and via the utility function which is assigned based on preference and suitability of agents to choose a grid or pixel (refer to Figure 7 ) ( O’Sullivan et al., 2012 ). The aggregated behavior of individual agents thus defines the structure of the whole system ( Tan et al., 2015 ).

In urban growth modelling, the development of ABM largely constitutes the definition of the agent type, behavior and the decision-making rules that determines the outcomes of their interaction ( Robinson et al., 2012 ). The agents can be defined and allocated based on the socio-economic formulations of the society or the interactions we intend to model between different land user types ( An, 2012 ). This may include individual households, residential developers, government agencies, and so on and can be defined based on social hierarchy of the urban area ( Filatova et al., 2013 ). These predefined agents are defined with certain attributes using statistical or regression methods, that define their status and suitability at a point in time, based on which they interact with other agent types

and select a land based on the outcome of those interactions. The decision between two agent types can be decided based on simple argumentative statements of “if and then”, case-based reasoning, statistical methods like decision trees ( Deadman et al., 2004 ), theories from social sciences like utility theory ( Evans & Kelley, 2004 ) and interactive decisions theory like Game theory ( Tan et al., 2015 ; Kaviari et al., 2019 ).

ABM has also been used in combination with various quantitative methods as a hybrid model as shown in Figure 8 . This has been done to enhance the behavior rules or decision-making rules of agents. ( Xu et al., 2015 ) used ABM with Ant Colony Optimization (ACO) method where ACO was used to define the behavior rules of agents for LULC simulations of Erhai Lake Basin. ( Hashemi Aslani, Omidvar, & Karbassi, 2022 ) used Multi-Layer Perceptron neural network with ABM to predict the future land use transformation of North Ahvaz watershed in Iran. Coelho et al. (2016) used artificial intelligence method call BDI (Belief-Desire-Intention) integrated with ABM, where BDI defines the transitional rules of the ABM for land use transformation studies of Cerrado Biome, Brazil. ( Zhang et al., 2013 ) developed an ABM model integrated with Game theory for the study of urban expansion in Fuyang, China. ( Li et al., 2020 ) used an ABM-Learning model, where learning behavior of human was embedded to study the impact ton land use patterns and simulate the urban growth of Shenzhen, China.

The advantage of ABM is that the agents can move in space, interact with other agents based on feedback loop mechanisms, thus representing impacts of individual decision making and representation of socio-economic factors. Thus, Agent based models in urban domain have been used vastly in simulation of human nature interactions and simulation of factors and actors for urban indicators like pollution ( Wang, Cao, & Zeng, 2020 ; Gurram, Stuart, & Pinjari, 2019 ), water management ( Arasteh & Farjami, 2021 ; Mashhadi Ali, Shafiee, & Berglund, 2017 ), traffic management ( Motieyan & Mesgari, 2018 ; Filomena & Verstegen, 2021 ), agricultural land use change Models, case study for Grain for Green Program in China and wide variety of other social and environmental issues including scenario and policy analysis ( Dai et al., 2020 ). However, ABM lack a clear geo spatial representation in simulation environments and are not easily integrable with other methods and techniques, hence, fall behind in application of urban growth prediction to CA models ( Heppenstall et al., 2012 ; Crooks et al., 2021 ). Also, the data requirements for ABM are large as the agents must be calibrated at the micro level which limits its application on larger scale compared to CA models ( An, 2012 ; Batty et al., 2012 ; Zhang et al., 2013 ).

Research in deep learning methods like Neural Networks boomed in decades of early and mid-2000s for image recognition, classification, and pattern recognition ( Grekousis, 2019 ). Important feature of these methods is that they could incorporate any number of inputs, handle large amounts of nonlinear input data and would train themselves to look into the relationships between different input-output parameters in a nonlinear way ( Li & Yeh, 2001 ). This provided urban growth modelers to resolve model sensitivity to number of data inputs and largely to bypass the autocorrelation between different input factors. Figure 9 shows various neural network frameworks currently in use in urban geography. The different layers are represented by nodes, each node acts on its own to determine the relationship between variables in the connected layer to determine the output and acts as an input for the next layer.

The first implementation of ANN methods in urban growth modelling was done in 2002 via integration of GIS with ANN ( Li & Yeh, 2001 ). Since then, ANN has been used widely to simulate and predict urban growth ( Basse et al., 2014 ). Different ANN models like convolution neural networks, multi layered perceptron neural networks have been used in urban growth modelling area ( Omrani, Tayyebi, & Pijanowski, 2017 ; SİPahİOĞLu & ÇAĞDaŞ, 2022 ). ( Maithani, 2009 ) used Multi-Layer perceptron network to study the Urban growth patterns in Saharanpur, India. ( Alqadhi et al., 2021 ) used MLP to model the future Land Change in semi-arid region of Asir, Saudi Arabia. ( Qiao et al., 2017 ) used multi-layer backpropagating neural network to study the land use change in Nanjing, China.

The advantage of the neural network-based models is that these models can address the non-linearity in relation between variable driving factors and growth, can handle large amounts and discreet input data types which was a challenging issue for CA, ABM models, meanwhile achieve acceptable simulation efficiencies. These models also addressed the issues of representation of spatial heterogeneity and sensitivity of CA, ABM models to data input. This made ANN one of the frequently used models for prediction of urban land use change ( Basse et al., 2014 ). However, due to the black box nature of operations of ANN, it is difficult to describe the transparency in relationship between input variables and quantize the impact of the factors, and thus hard predicting the error propagation in modelling process ( Roodposhti, Aryal, & Bryan, 2019 ).

The basic nature of individual models is that the representation of urban system and growth in simple representation may not describe the broader range of chaotic urban dynamic processes ( Liu et al., 2021 ). Various integration techniques are thus introduced to increase the sensitivity of the models. The models have been integrated to incorporate different transitional rules, different neighborhood configurations, inclusion of varying number and type of driving factors. The CA models are one of the frequently used in integration with other models because of their flexibility, simulation efficiency and representation of geospatial growth ( Aarthi & Gnanappazham, 2018 ; Li & Gong, 2016 ; Aburas et al., 2016 ).

Towards the late 2000s to early 2010s, it was realized that combining CA and ABM models with each other as well as with modern statistical and regression approaches have more capabilities to account the complex phenomena. ( Mozaffaree Pour & Oja, 2021 ) used a CA-ABM-Markov model to address spatial heterogeneity; ( Kumar et al., 2021 ) integrated CA-ABM to analyze impact of socioeconomic factors, spatial neighborhoods, stakeholder choices, and development plans on LULC of Dehradun City; ( Dahal & Chow, 2014 ) integrated ABM with irregular CA to determine if integration of these two models improves the simulation efficiency for San Marcos, Texas; ( Tan et al., 2015 ) integrated CA-ABM coupled with Game theory to simulate urban growth of Wuhan; ( K Agyemang, Silva, & Fox, 2019 ) used CA-ABM model to study geospatial behavior of key urban development actors, including households, real estate developers and government in Accra, Ghana. In such integration, CA represents the spatial component, ABM represents the environment and socio-economic variables and thus complementing the deficiencies of individual models by representing landscape as well as human role in urban growth-related decision-making process.

However, with the rise in number of CA-ABM models and their calibration methods, there were some inefficiencies of these integrated models to represent the temporal dynamicity of driving factors ( Feng et al., 2019 ), spatial heterogeneity ( Feng & Tong, 2020 ), model sensitivities to number of driving factors ( Wang et al., 2011 ), multicollinearity between factors ( Feng & Tong, 2017) . To address this, integration of CA-ANN based models and ABM-ANN, and integration of CA models with different machine learning and heuristic methods was done ( Okwuashi & Ndehedehe, 2021 ). To account for the transparency and black box nature of machine learning, methods like DoT (Dictionary of Trusted Rules), and to increase capability over to ANN, deep belief network (DBN) have been introduced in urban growth modelling ( Zhou et al., 2017 ).

In urban growth domain, the objectives may include prediction of total land area increase, visualizing human-nature interactions and quantifying the impacts of these interactions, addressing the urban issues related in regional scale like formation of slums, high density growth, peri-urban growth, urban agglomeration etc. or scenario-based analysis like analysis of impacts of climate change in an urban area, analysis of impacts of flood in future urban growth, etc.

A single model even with various techniques may not be able to address all the issues within the localized simulation environment ( National Research, 2014 ). ( Liu et al., 2021 ) reviewed and highlighted how current practice of CA models in limited mostly to urban expansion rather than incorporating broader urban dynamics. ( Ren et al., 2019 ) highlighted reasons for poor validation of models and hence the applicability and lack of global standards in the practice. ML models like ANN have shown great potential in urban growth modelling, however, as with other models a clearly defined objective to guide modelling process has limited their applicability or to larger case acceptability. Moreover, there is a gap in developing a clear classification of objectives on development of urban growth models which certainly has limited their integration in decision making processes. Thus, is necessary to define the modelling objective clearly before a model can be selected. A model should not be taken as a comprehensive tool to address all the issues in the local urban region from delineation of growth forms, morphology, direction of growth, influencing factors of growth, incorporation of any available data and simulation efficiency at the end. Necessary tradeoffs are required, and model selection should be done based on what issue or objective we intend to address through the modelling process.

This will help in increasing the applicability of all model types. Example, the early top-down models like Concentric Zone Model, Sector models or multi nuclei models can still be used in simple scenario analysis, non-vital planning, or reconnaissance survey to study the socio-economic distribution of urban area like distribution of households based on income, family size and other characteristics. Balakrishnan and Jarvis (1991) applied the Burgess CCD model to study the socio-economic status and family size of residents in 14 Canadian cities. The Land Use transportation model can be used in Transport Planning, determination of impacts of use of different modes of transport on housing, employment. Also, the land rent models can be applicable for simple analysis of variations of land prices based on distances, distribution of economic groups based on land use and variations. ( Gautrin, 1975 ) used the simple Land rent model to evaluate the impact of Aircraft Noise on Land Rent Prices. ( Park, 2014 ) reviewed the land rent theory in light of Global financial crisis of 2009 to discuss its applicability on current housing market. These top-down approaches are thus simple in application and may provide a reasonable estimate for simple scenario analysis rather than using complicated, data intensive and time-consuming complex models. Use of complex models like CA, ABM, ANN should be done when the objective is to model influence of various factors on various urban form related changes.

Another characteristic of urban growth models is the use of variable amounts of input data. Since CA models can be calibrated and validated at the global scale, they are less data intensive and cost less computational power. However, in ABM the data must be calibrated at the local level on the scale of agents. Thus, depending on the number of agents defined and their attributes and utility function, the data demand at the local level can get very intensive and require higher computational power. Whereas models like ANN are flexible in terms of data requirements and computational power, as they can be used with scarce as well as large volumes of data. Depending on the data volumes, their computational requirements can vary depending.

CA and ABM types of models may be susceptible to large amounts of data in terms of simulation times and computational power required to handle the data. With the emergence of big data, the generic CA and ABM models, thus, may be more unfavorable to ML methods like the ANN. This is supported by development of intense data demanding ML methods like RNN, LSTM, GRU ( Koumetio Tekouabou et al., 2022 ). Also, advanced ML models have great potential in addressing several challenges of urban form including spatial and temporal heterogeneity, non-linearity, however, increase in data demand and computational complexity is a challenge in large scale implementation of ANN based models ( Koumetio Tekouabou et al., 2022 ).

Urban growth is a complex process derived by the combination of biophysical (Geographical and Physical), socio-economic factors, environmental and policy induced factors. The biophysical factors include the proximity to physical infrastructures and characteristics of the geography of urban land like slope, elevation, proximity to road network, health facilities, etc. Socio-economic factors include population density, GDP, household combination and human nature interaction. The environmental factors include proximity to or restricted development to waterbodies, conservation areas, parks, disaster prone areas etc. Policy related factors include policies like zoning, ecological conservation policies, land classification policies, infrastructure development policies that may induce or limit growth in policy implemented areas.

Figure 10 shows that the use of bio physical driving factors is predominant in the last 5 years. Especially proximity and accessibility related factors like distance to roads, city centers, public transportation, slope, elevation, and type of land use more primarily urban land use seem to be more used in delineation of Urban growth. This also provides a picture that the use of CA based models is frequently.

CA models are capable of incorporation of biophysical factors, however, fail to incorporate socio economic factors. Process based models like ABM more easily incorporate the socio-economic factors, however, lag in their spatial representation. This is due to the reason that biophysical factors can be represented effectively as static, whereas socio economic factors are dynamic and the rates of change of such factors are variable with time. This also makes CA models capable of simulation at global scales and thus one model developed can be used in different regions. However, ABM models are generic at the local scales and thus application of the same models at global scales may be inefficient. Hence, individually CA and ABM models cannot comprehensively represent both driving factors during the simulation. However, integrated models like CA-ABM can incorporate

both biophysical as well as socio economic factors.

CA models are also susceptible to the number of driving factors input. Thus, generic CA models or CA models combined with statistical methods cannot incorporate large number of driving factors. It is important that the weights of driving factors be determined firsthand and only factors with influence in growth be input in the modelling platform. However, CA-ANN models or CA-heuristic can address this issue effectively. Also, a random selection of articles between 2018-2022 showed that most of articles used driving factors based on past studies of similar models or have not clearly defined the selection of factors. Only about 27% of articles have used a range of driving factors after either correlation analysis or after determining the quantitative impact of these factors. This provides a grim picture as the growth of an area may be dependent on other factors which may be missed in the modelling process. Thereby highlighting the fact that modelling accuracies are more prioritized rather than actual understanding of reasons of growth.

Urban growth is an evolution of various forms of urban areas and relevant policies that shape these forms. Examples include questions like What policies an urban area has adapted for its residential, industrial development, what kinds of housing density to develop, what policies determine the land class for development, what is the socioeconomic policy of urban area that defines health, education, economic growth, and connectivity and many more. However, as these policies vary based on geographical areas, a single standard cannot be used which makes universal modelling of policy scenarios challenging. It may be imperative to use policy scenarios specific to the area being modelled, however, this has hindered practicing uniform modelling approaches.

As briefed in Section 4, evolution of urban growth models has been more focused on improvising models’ performance rather than application as decision support systems for various planning and policy scenarios. There is a strong literature understanding on various scenario analysis impacting the urban growth process in terms of land use change ( Wang, Murayama, & Morimoto, 2021 ). Authors have suggested development of relevant policies to manage different forms or rates of growth ( Maturana et al., 2021 ; Kushwaha et al., 2021 ; Geng et al., 2022 ; Khan & Sudheer, 2022 ; Kisamba & Li, 2022 ). However, there is still limited understanding on how variable policies to manage these scenarios can impact this change after implementation. Recent review on various urban growth models revealed that only 6% of articles have used policy as a variable of change ( Wang, Derdouri, & Murayama, 2018 ). Figure 10 also supports this fact. Moreover, most of the policy scenarios are focused on bio physical or socio-economic influence related policies ( Cheng & Liu, 2022 ; Spyra et al., 2021 ; Xu et al., 2020 ).

Another policy issue largely ignored in urban growth modelling, is the urban sustainability issues ( Diehl et al., 2020 ). Now, as many urban areas are trying to develop sustainable practices and policies to enhance the sustainability of urban areas, urban growth models have largely bypassed this incorporation due to challenges in modelling environment ( Sakieh et al., 2015 ). Widely used CA based model like SLEUTH is able to model different forms and urban change, allocate restricted areas for development for ecological conservation ( Akyol Alay, Tunçay, & Clarke 2021 ; Eyelade, Clarke, & Ijagbone, 2022 ). However, this model has fixed input layers, which cannot broadly and clearly establish the relation between indicators of sustainability as input and growth as output. Model concepts like CLUE, FLUS have been primarily designed to incorporate various policy scenarios in relation to socio-ecological conservation and protection ( Peng et al., 2020 ; Chasia, Olang, & Sitoki, 2023 ; Liu et al., 2017 ). However, large scale applications targeting urban areas still remain limited. Sustainable practice policies change the way urban area’s function, hence, what is essential is integrating and testing of these sustainable practices as influence factors of growth.

Also, climate change is a global phenomenon and as most urban areas around globe will certainly be impacted by this phenomenon in one way or the other. The primary impact will be in terms of environmental and ecological parameters within the urban boundary like varying rainfall patterns, rise of sea levels, loss of vegetation and varying frequency of other natural hazards like floods, urban heat, fluctuating temperatures. These parameters will in a closed feedback loop impact the future of growth by influencing adjacent policies, socio-economic distribution and hence changes in patterns of urban growth ( Hansen, 2010 ). Data requirements, computational complexity, and other inherent challenges in models will most likely be not that instrumental compared to challenges possessed by climate change in future. Climate change is one of major problems among current day grand challenges in integration of policy scenarios in urban growth models ( Batty, 2011 ). However, the study on impact of climate change on urban growth is very limited ( Liu et al., 2017 ). As the impacts of climate change get more intense, the necessity to understand this pattern becomes more fundamental. Moreover, CO₂ is main driver of climate change, however, study on impacts of carbon emissions due to different forms of urban growth is limited ( Hong, Hui, & Lin, 2022 ). This consequently limits the application of urban growth models to forecast growth due to carbon emission induced urban change.