Unveiling urban area growth dynamics: insights from a comprehensive study of urban area growth curves

ABSTRACT

Amidst rapid global urbanization, comprehending urban area growth (UAG) dynamics is vital for urban planning and sustainability. Despite ample urban development research, the historical developmental patterns of UAG remain underexplored. Long-term UAG exhibits an initial acceleration followed by deceleration, thus, using 200-years UAG data from the U.S., we meticulously compare the S-shaped curve (Logistics and Gompertz models) with other time-series models, and unveil insights into UAG's intricacies by examining how S-shaped models perform across different urban stages. Notably, the Logistics model emerges as the more accurate modeling tool, boasting an RMSE of 0.019, which surpasses the Gompertz model's 0.032. Moreover, the parameters of the S-curve explicitly describe the fundamental properties of UAG, and we unveil the remarkable stability of the maximum growth rate in the S-shaped model through a thorough parameter analysis, which underscores its role as a reliable and universally applicable assessment tool for UAG. Fundamentally, this meticulously analytical research delves deep into the complexities of the UAG curve, robustly confirming the S-shaped trend as the ‘realistic’ consequence of UAG over time. Importantly, this enduring S-shaped trend remains consistent across historical and contemporary contexts. These findings significantly advance our understanding of UAG dynamics for informed urban planning and development.

KEYWORDS:

• Urban area growth
• S-shaped curve
• logistics model
• gompertz model
• maximum growth rate

1. Introduction

Urbanization, a hallmark of contemporary global development, entails the persistent expansion of urban areas (Seto et al. 2017; Zhang et al. 2022). Urban areas (UA) are characterized as land regions marked by high population density, concentrated economic activities, and dense infrastructure and architecture (Zhang et al. 2022). The physical boundary growth of urban area is encapsulated by the term Urban Area Growth (UAG), which signifies the ongoing expansion of urban land, gradually encompassing the surrounding rural and natural areas (Montgomery 2008; Wei and Ewing 2018; Zhou et al. 2018). UAs have long transcended their role as mere spaces for human activity, firmly establishing themselves as the bedrock of sustainable urban development. The UAG not only influences the efficient utilization of land resources but also intricately intertwines with the ecological balance surrounding the city, as well as the population and societal development (Acuto, Parnell, and Seto 2018; Wachsmuth, Cohen, and Angelo 2016). Therefore, diving into the study of UAG evolution serves as an intriguing catalyst for gaining insights into the multidimensional perspective of urban development.

In examining UAG, traditional UAG metrics such as urban expansion rate (Liu, Zhang, and Wang 2016), urban form index (Yang, Qian, and Lau 2013), and land use change (Du et al. 2014) are commonly utilized. However, these metrics often confine themselves to portraying the overarching shifts in total UA (Deng et al. 2009; Weng 2007), employing a discrete approach that fails to capture the continuity and dynamism of UAG. The distinctive feature of the UAG curve lies in its ability to offer a continuous temporal perspective, in sharp contrast to traditional measurement methods (Song et al. 2016). It provides a more precise means of capturing subtle shifts and trends in UAG. This curve not only unveils the growth rate of UAs but also chronicles every moment of UAG, enabling us to observe the rhythm, pace, phased characteristics, and fluctuations in UAG. This type of continuous temporal curve aids in unveiling underlying patterns in UAG, predicting future trends, and simultaneously provides more accurate data support for urban planning and sustainable development. The key to studying UAG lies in selecting appropriate tools to accurately capture its continuous trends, which not only need to describe the overall growth patterns of UAG but also account for variations in its growth rate. Furthermore, we must investigate whether such descriptive methods hold the potential for reliable predictions of future UAG. However, studying UAG curves is not a straightforward task, as they involve complex urban geographical expansion (Bagan and Yamagata 2012; Chen, Qiu, et al. 2020), which is intricately shaped by the interplay of social, economic, and environmental factors (Bettencourt et al. 2007; Bettencourt and West 2010; Verma and Raghubanshi 2018; Yang et al. 2023). The intricacies of these factors further exacerbate the challenges in conducting such research.

The process of UAG often follows a staged pattern. In the early stages of development, urban areas typically experience slow growth, often because the formation of a city takes time. As urban infrastructure gradually improves and populations continue to migrate, the growth of urban areas accelerates. Eventually, the growth rate of urban areas slows down, reaching a relatively stable state. This slowdown may be due to the limited availability of land resources and various internal constraints within the city. Therefore, the process of UAG, characterized by slow initial growth, followed by rapid expansion, and eventual deceleration, can be roughly likened to an S-shaped growth pattern, akin to the urbanization curve (Chen, Ye, and Zhou 2014; Mulligan 2013b). Urbanization typically exhibits an S-shaped pattern, comprising identifiable stages of initiation, acceleration, and terminal. Similarly, UAG follows a comparable three-stage pattern, demonstrating a characteristic resemblance (Chen, Ye, and Zhou 2014; Mulligan 2013a). Utilizing mathematical equations to fit the S-shaped curve of UAG offers a plausible hypothesis concerning the underlying growth logic of urban areas. Nonetheless, this conjecture warrants substantial validation before achieving broad acceptance. The proposed curve holds great potential as a longitudinal quantification tool, particularly suitable for examining UAG in diverse city types across the globe. Considering the pressing issues of current global urban development disparities, researchers seem compelled to reach a consensus and explore a simple yet comprehensive model to considerably enhance our grasp of urban evolution and facilitate comparative studies of UAG in contemporary global cities.

This study provides a comprehensive discussion of the UAG curve. Considering that urban expansion often unfolds over centuries, we utilized the Historical Settlement Data Compilation for the United States (HISDAC) (Leyk and Uhl 2018) to construct a comprehensive UAG trajectory. This dataset covers a time span of 200 years and records the historical UAG patterns across the United States, providing a solid data foundation for studying UAG curves. Building upon this foundation, this study revolves around the core scientific question of how to model and simulate the dynamic evolution of urban area growth (UAG). It focuses on addressing three key challenges related to UAG: (1) What are the general laws governing the evolution of UAG? (2) Which models are suitable for expressing the evolution processes of UAG for different types/stages? (3) How to integrate multiple models to finely represent the evolution of urban area growth processes. Through in-depth model analysis and parameter exploration, we enhanced our understanding of UAG dynamics, revealing the complexity of urban growth. These contributions advance our understanding of UAG patterns, offering valuable insights into the field of urban development, aiding in the development of more informed urban planning strategies, particularly in the context of rapid global urbanization.

2. Conceptual base of urban area growth (UAG)

UAs typically serve as central hubs for economics, industry, innovation, education, culture, and politics (Albino, Berardi, and Dangelico 2015; Grodach, O'Connor, and Gibson 2017). As a result, UAG becomes a complex process influenced by a lot of interconnected factors (Pickett et al. 2011). These factors steer UAG, encompassing phenomena such as rural-to-urban and natural-to-urban migration, among others. Economic factors attract population influx and the need for more urban land by generating jobs and economic activities, which drive UAG to meet growing commercial and industrial demands (Petrov, Lavalle, and Kasanko 2009; Yeh, Yang, and Wang 2015; Yigitcanlar, O’Connor, and Westerman 2008). However, economic factors do not exist in isolation. Government policies related to land use, infrastructure development plans, and taxation policies can alter the dynamics of economic factors, consequently impacting the characteristics of UAG (Liu, Chen, and Gu 2019). Additionally, enhanced infrastructure and technological innovations improve the quality of urban life, increasing attractiveness and further promoting UAG (Jia et al. 2020). These factors interact collectively, forming the dynamic process of UAG. Thus, the essence of UAG is deeply rooted in the intricate interplay of multiple interconnected factors, constituting the fundamental driving force behind it (van Vliet 2019).

UAG typically exhibits a temporal pattern characterized by initial slow growth, followed by rapid expansion, and ultimately tapering growth (Figure 1). In the initial stages, UAG progresses slowly due to limited rural and natural migration into cities, along with underdeveloped economic and infrastructural conditions (Feng, Liu, and Qu 2019). As urban areas offer more job opportunities and economic activities, they attract a surge in population, leading to a phase of rapid growth (Zhao et al. 2021). Government urban planning initiatives and infrastructure improvements enhance urban appeal, further propelling this growth (Jia et al. 2020). However, as urban areas approach market saturation and governmental regulations may be implemented, growth begins to decelerate (Colsaet, Laurans, and Levrel 2018). Ultimately, UAG stabilizes as it reaches a more mature and stable state, influenced by market dynamics, resource availability, and environmental considerations (Li et al. 2019).

Figure 1. The general regulation of UAG curve.

This dynamic progression of UAG often resembles an S-shaped growth curve. In the initiation stage of urbanization, cities tend to concentrate their development in specific regions or city centers (Jiao 2015; Yang et al. 2022), resulting in relatively slower growth rates. As urbanization enters the acceleration stage, cities expand and sprawl into surrounding areas. Moreover, variations in UAG progresses across different regions depend on factors such as socio-economic conditions, policy support, and natural environments (Acuto, Parnell, and Seto 2018). Cities in emerging economies or developing countries may exhibit early-stage curves as they actively seek investment and promote economic growth (Ravallion 2002). On the other hand, cities in developed countries may be in the later stages of the curve, experiencing slower growth as they reach the maturity of urbanization. By discerning various developmental stages, we can pinpoint the current positions of diverse cities on the UAG curve. For instance, London, New York, and Paris may have reached the terminal stage, while Beijing, Mumbai, and Chengdu are in the acceleration stage. Emerging cities in developing nations may be in the initiation stage. It is essential to note that the shapes of UAG curves vary among different cities due to various influencing factors. Consequently, the eventual size and growth rate (i.e. the steepness of the growth curve) of urban areas may differ, providing a robust basis for intercity comparisons (Güneralp et al. 2020). The UAG model serves as a highly flexible tool for investigating distinct developmental patterns in different cities. As more data and research accumulate, we can further delve into the underlying significance of UAG models, enabling better guidance for the scientific growth of cities.

3. Modeling urban area growth

3.1. Conceptual framework of urban area growth curves

UAG models serve as pivotal tools in the fields of urban geography. They not only facilitate a profound understanding of the intricacies of urbanization but also offer insights into the evolution of urban scale and morphology. Rooted in mathematical and statistical principles, these models, through numerical simulations and empirical analyses, aim to capture the expansion and development of urban regions, along with a spectrum of pivotal factors influencing this process. The fundamental concept at the heart of UAG models lies in simulating the UAG process itself. This entails an endeavor to quantitatively describe how cities gradually evolve from relatively modest scales into complex and sprawling urban landscapes. Such modeling approaches aid in unraveling the dynamics of UAG, thereby unveiling the patterns and trends that underlie urbanization.

In this conceptual framework, we will delve into the mathematical underpinnings and operational principles of UAGmodels, and explore how these models can be applied to address the challenges of urban planning and urbanization. The workings of UAG models typically employed mathematical equations or models to describe the growth trends of cities, which a commonly used model is the S-shaped curve model. The core concept of the S-shaped curve model is that urban growth starts relatively slowly, then gradually accelerates under specific conditions, ultimately reaching a saturation point. This curve reflects the distinct phased characteristics of the urbanization process, encompassing initial sluggish development, a rapid growth phase, and eventual saturation.

The application of the S-shaped curve model in urban growth research is extensive. By fitting the S-shaped curve model to actual UAG data, researchers can determine the values of various parameters, thereby gaining a deeper understanding of the city's developmental process. This model is instrumental not only for retrospective analysis but also for predicting future urban growth trends, evaluating the effectiveness of urban planning policies, and comparing growth patterns and characteristics among different cities. The shape of the S-shaped curve is influenced by a range of parameters, including maximum urban area, urban take-off area, urban start time, growth constant, maximum growth rate, and the time at which the maximum growth rate occurs. In the realm of UAG modeling, comprehending the concept and application of the S-shaped curve model is of paramount importance. It furnishes us with a robust tool for analyzing and elucidating the urbanization process, thereby facilitating decision-making in urban planning and urbanization.

This study elucidates the fundamental characteristics of the S-shaped curve and identifies the key parameters that influence its shape. Subsequently, the stability of these parameters was assessed through subsequent experiments. The parameters include maximum urban area (m), urban take-off area (b), growth constant (k), urban start time (c), maximum growth rate (R), and time of maximum growth rate (t(R)). Among these, m, b, k, and c determine the shape of the UAG curve, while G measures the maximum growth rate of urban expansion on the S-curve, representing its steepest segment, and ‘t(R)’ indicates the time when G occurs. Here, we introduce the concept of ‘Maturity’, calculated as the ratio of each city's urban area in 2015 to the logistic model's maximum urban area prediction. This measure is employed to convey the degree to which each city's UAG curve signifies the accomplishment of its urban growth progression.

3.1.1. Maximum urban area

Maximum urban area refers to the upper limit or the peak level of urban area can achieve within a given time frame and a given region. It represents the highest value that the urban area can reach on the S-curve, which indicates the saturation point of urban expansion.

3.1.2. Urban take-off area

Urban take-off area refers to the urban area in time when a city's urban growth starts to accelerate significantly, marking the transition from a relatively slow growth phase to a rapid expansion period.

3.1.3. Growth constant

Growth constant, represented by ‘k’ in the UAG curve formula, is a significant parameter that influences the shape and rate of urban growth over time. From a mathematical perspective, the growth constant controls the steepness of the UAG curve, determining how quickly the urban area expands during different stages of development. Rapidly growing economies, efficient infrastructure, significant population influx, and proactive urban planning can lead to higher growth constants.

3.1.4. Urban start time

Urban start time, represented as ‘c’ in the UAG curve equation, refers to the initial point in time when urban growth begins to accelerate noticeably. It marks the transition from a relatively slow initial phase of urbanization to a more rapid and exponential expansion phase. From a mathematical perspective, the urban start time ‘c’ is a critical parameter that determines the point of inflection on the UAG curve, where the growth rate starts to increase rapidly. It indicates the time at which the city's development shifts from gradual to accelerated growth.

3.1.5. Maximum growth rate

Maximum growth rate, denoted as ‘R’ in the UAG curve equation, refers to the highest growth rate on the curve where urban growth is occurring at its fastest pace. It represents the peak rate of urban expansion during the accelerated phase of development. The maximum growth rate ‘R’ corresponds to the steepest slope on the UAG curve, indicating the period when the city experiences the most rapid increase in urban area over time.

‘R’ is an excellent parameter that characterizes Urban Area Growth (UAG) and possesses a stability not found in other parameters of the UAG curve. It serves as an aggregate parameter for the UAG, reflecting the characteristics of specific UAG trajectories under particular conditions. As a result, it remains unaffected by annual fluctuations or stages in the UAG cycle, distinguishing it from other parameter s. Compared to other parameters for UAG, ‘R’ exhibits greater robustness when dealing with different growth models, limited data availability, and deviations from the ideal S-shaped curve, as we will demonstrate in subsequent experiments. This implies that when uncertainty arises between the Logistics Model and the Gompertz Model in accurately representing UAG, ‘R’ becomes an exceptional factor in depicting the actual UAG.

3.1.6. Time of maximum growth rate

Time of maximum growth rate refers to the point in time when a city experiences its highest rate of urban area expansion during the UAG process. From a mathematical perspective, it corresponds to the time ‘t(R)’ in the UAG curve formula where the growth rate ‘R’ reaches its peak value.

3.2. Urban area growth curves

The primary metric used to study UAG is the UAG rate, which is the ratio of the difference between the current urban area and the initial urban area to the initial urban area (Bren d’Amour et al. 2017; Liu et al. 2021), and similar metrics used in UAG research encompass Urban Expansion Speed, Urban Form Index, Land Use Change. While these metrics provide valuable insights, they exhibit a relatively limited scope, primarily offering insights primarily focused on specific moments in time, which often lack the necessary temporal granularity required to comprehensively capture the nuanced dynamics of both short-term and long-term UAG. Compared to quantifying UAG with traditional metrics, UAG curves can provide a more intuitive representation of the UAG process. The UAG curve serves as a modeling approach for capturing the UAG process, elucidating the continuous and long-term transformations of UAs over time.

The mathematical modeling of the UAG curve allows for the parameterization of urban area growth, leading to a quantifiable model. When selecting suitable mathematical equations to represent the S-shaped UAG curve, it is crucial to prioritize simplicity and clarity while ensuring that the parameters have precise and practical interpretations. An appropriate equation should concisely describe the shape of the UAG curve and provide clear insights into the different stages of urban growth.

Among the various S-shaped curve models, the logistics model (Chen et al. 2014) and the Gompertz model (Cai and Wu 2022) stand out as the most concise and widely applied. Both models effectively capture the essence of the S-shaped curve and have found broad applications in areas such as population statistics (Storper, van Marrewijk, and van Oort 2012), cultural transmission, spatial diffusion (Mulligan, Partridge, and Carruthers 2012), and energy development (Madsen and Hansen 2019). An advantage of these models is their capacity for fitting and estimation using straightforward OLS regression. The logistics model exhibits a unique symmetry on both sides of the inflection point, while the Gompertz model displays non-symmetrical characteristics and allows for prolonged growth after the inflection point. Both the logistics and Gompertz models can be expressed using 3-parameter equations:

3.2.1. Logistics model

(1) $f(t)={\displaystyle \frac{m}{1+{\displaystyle \frac{m-b}{b}{e}^{-k(t-t_0)}}}}$(1)

3.2.2. Gompertz model

(2) $f(t)=m{e}^{\ln {\displaystyle \frac{b}{m}{e}^{-k(t-t_0)}}}$(2) The parameters in the logistics model and the Gompertz model can be mutually related. In these models, $e$ is the natural logarithm, which is about 2.718; $k$ is the growth constant, $c$ is the urban start time, $b$ is the urban take off area, and $m$ is the maximum urban area. When taking derivatives of the logistics model and the Gompertz model, the moment at which the derivative attains its utmost value corresponds to the $R$, with the corresponding time point being the $t(R)$. These two models differ from the traditional logistics model and Gompertz model. We have designed the parameters to ensure that when $t$ equals $t_0$, the model's value is $b$, and as $t$ approaches infinity, the model's value approaches $m$ infinitely.

In addition, for comparison, we also considered a simple UAG alternative model, namely the piecewise linear model. The piecewise linear modelassumes a linear relationship between urban area and time, with a constant increase in area per unit of time. UAG typically involves three phases: slow growth, rapid growth, and decelerated growth, thus a segmented linear model with three segments have been opted. Although the piecewise linear modelis straightforward, it neglects the nonlinear characteristics of the UAG process and cannot accurately capture the acceleration and deceleration phases of urban expansion. The equation of piecewise linear modelis as follow:

3.2.3. Piecewise linear model

(3) $t<t_0:f(t)=k_1\ast t+b$(3) (4) $t>=t_0\mathrm{and}t<t_1:f(t)=k_2\ast (t-t_1)+k_1\ast t_0+b$(4) (5) $t>=t_2:f(t)=k_3\ast (t-t_2)+k_2\ast (t_2-t_1)+k_1\ast t_0+b$(5)

3.3. Evaluation metric for urban area growth curves

To assess the goodness of fit for the three UAG curves to the urban surface growth process, we employed three metrics: R-squared ($R^2$), Adjusted R-squared (Ad-$R^2$), and Root Mean Square Error (RMSE). Their calculation methods are as follows: (6) $R^2=1-{\displaystyle \frac{S{S}_{\mathrm{res}}}{S{S}_{\mathrm{tot}}}}$(6) (7) $\mathrm{Ad}\text{-}{R}^2=1-{\displaystyle \frac{(1-R^2)(n-1)}{(n-k-1)}}$(7) (8) $\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}\sum _{i=1}^N{(f_i^{{}^{\mathrm{\prime }}}-f_i)}^2}}$(8) where $S{S}_{\mathrm{res}}$ is the sum of the squared residuals, $S{S}_{\mathrm{res}}$ is calculated by the differences between the $f_i^{{}^{\mathrm{\prime }}}$ and $f_i$, $S{S}_{\mathrm{tot}}$ is the total sum of squares, which means the total variance in the dependent variable, $f_i^{{}^{\mathrm{\prime }}}$ and $f_i$ refer to the estimated urban area and reference urban area, N and n represents the number of input data, and $k$ represents the represents the number of independent variables in the UAG model.

$R^2$ is commonly used as an indicator to measure the overall goodness of fit of a model, representing the proportion of variance explained in the dependent variable. However, ordinary $R^2$ may overestimate the generalization ability of the model to unknown data. Ad-$R^2$, building upon $R^2$, takes into account the number of independent variables in the model (Miles 2005). In comparison to $R^2$, it is more sensitive to the model's complexity, helping to prevent overfitting. The Ad-$R^2$ offers a more accurate estimate of model fitness, particularly when dealing with complex models involving multiple independent variables, providing a better reflection of the model's fit to the data. RMSE measures the magnitude of errors between model predictions and actual observed values. A smaller RMSE indicates lesser prediction errors, making it valuable for assessing the accuracy of the model's predictive capabilities.

4. Data

4.1. Sampled cities

Presently, the United States possesses a distinctive capacity to furnish an extensive record of Urban Area Growth (UAG) spanning more than a century, a crucial asset bolstering UAG curve research. In pursuit of this, our study zeroes in on a cohort of 258 cities nationwide, delineated by the Core Based Statistical Area Description (CBSA), encompassing cities with populations exceeding 10 thousand (Figure 2). We define the ‘urban area’ as a pixel portrayal of documented built-up regions, delineated by city boundary data. The CBSA categorizes U.S. cities into Metropolitan Statistical Areas and Micropolitan Statistical Areas, reflecting discrete scales and geographical scopes. We opt for the utilization of CBSA, rather than administrative divisions, to delineate and define urban areas in the United States. This choice is driven by CBSA's comprehensive consideration of the overall socioeconomic connections within cities and their surrounding regions, aligning more closely with the complexities of contemporary urbanization. By employing similar measurement standards, CBSA facilitates enhanced comparability among diverse cities, thus capturing the nuances of modern urban development. This categorization facilitates UAG curve fitting and analysis across diverse urban development typologies.

Figure 2. City distributions included in this study. 258 cities were defined and selected based on the Core-Based Statistical Area description.

We categorized the 258 cities into 9 subsets based on common geographical divisions in the United States. Cities within these regions exhibit unique development patterns influenced by independent political, geographical, historical, cultural, and economic factors. The divisions primarily fall into four major regions, each further subdivided into smaller zones. The United States is geographically organized into nine major regions: New England (NE), the Mid-Atlantic (MA), West North Central (WNC), West South Central (WSC), South Atlantic (SA), East South Central (ESC), East North Central (ENC), Mountain (MO), and Pacific (PA) (Figure 3).

Figure 3. The 9 major geographic divisions of the United States.

4.2. Time series urban area growth records

Cities often undergo a process of evolution spanning hundreds of years, from their inception to expansion and eventual maturity. Hence, reconstructing Urban Area Growth (UAG) trajectories often necessitates reliance on digital records spanning over a century. Over the past two centuries, the United States has undergone significant urbanization, a process meticulously documented in the Historical Settlement Data Compilation (HISDAC) dataset, spanning from 1815 to 2015 (Leyk and Uhl 2018).

However, due to significant data gaps, we conducted a meticulous selection process, excluding cities with minimal growth between 1815 and 2015 or those exhibiting pronounced data deficiencies. As per the 2015 Core Based Statistical Area (CSBA) definitions, the United States encompasses a total of 940 cities. Further refinement led to the selection of 258 cities with urban areas exceeding 10 square kilometers in 2015. Subsequently, leveraging the HISDAC-US dataset representing the historical period from 1815 to 2015, UAG curves were fitted for each of these 258 cities, enabling an in-depth study of their UAG processes.

5. Experiments and results

5.1. Fitting the urban area growth curve

We commenced by conducting a comprehensive analysis of 258 cities across the United States, spanning the years 1815–2015, with data points collected at five-year intervals, capturing the UAG curve. Subsequently, employing the method of least squares, we individually fitted these curves using the Logistics model, Gompertz model, and Linear model (Table 1) (Figure 4). We meticulously assessed the efficacy of these three models in capturing the dynamics of UAG curves.

Figure 4. The logistic model (yellow) and the Gompertz model (blue) fitting the S-shaped curve of UAG for the exemplar cities. Gray dots represent historical UAG data.

Table 1. Evaluation of the fitted UAG curves.

City	Logistics model			Gompertz model			Piecewise linear model
	$R^2$	Ad-$R^2$	RMSE	$R^2$	Ad-$R^2$	RMSE	$R^2$	Ad-$R^2$	RMSE
New York	0.997	0.994	0.348	0.998	0.996	0.829	0.843	0.704	5.510
Los Angeles	0.997	0.994	0.110	0.997	0.994	0.287	0.774	0.589	4.721
Mission Viejo	0.999	0.997	0.025	0.997	0.995	0.179	0.630	0.382	3.493
Chattanooga	0.996	0.993	0.031	0.994	0.988	0.047	0.878	0.765	0.153
Washington D.C.	1.000	0.999	0.218	0.997	0.995	0.681	0.668	0.433	7.958
Dallas	0.999	0.998	0.717	0.980	0.959	5.580	0.570	0.308	18.090
Oklahoma	0.998	0.997	0.064	0.998	0.996	0.146	0.749	0.551	2.286
Chicago	0.998	0.995	0.680	0.997	0.994	1.339	0.783	0.604	9.585
Denver	0.996	0.993	0.498	0.994	0.987	0.803	0.648	0.406	7.885
Tulsa	0.999	0.998	0.060	0.999	0.997	0.068	0.756	0.561	1.099
Cincinnati	0.997	0.993	0.269	0.996	0.992	0.415	0.808	0.645	2.649
Orlando	0.999	0.998	0.434	0.998	0.996	0.759	0.576	0.316	10.818
Wichita	0.997	0.994	0.198	0.997	0.995	0.253	0.783	0.604	1.841
Laredo	0.995	0.991	0.038	0.995	0.990	0.046	0.614	0.361	0.070
Dubuque	0.982	0.964	0.016	0.985	0.969	0.019	0.928	0.857	0.095
Salisbury	0.996	0.993	0.062	0.997	0.993	0.075	0.741	0.538	0.591
Omaha	0.993	0.986	0.160	0.990	0.980	0.187	0.866	0.743	1.670
Appleton	0.972	0.943	0.019	0.966	0.931	0.026	0.909	0.822	0.121
Bozeman	0.987	0.974	0.021	0.986	0.972	0.024	0.734	0.528	0.118
Hays	0.972	0.943	0.010	0.977	0.953	0.011	0.777	0.594	0.070

*Ad-$R^2$ stand for Adjust $R^2$_.

*RMSE: 100 $k{m}^2$_.

We conducted an analysis of the adjusted R-squared values for each city's fitted curve, revealing that both the Logistics and Gompertz models exhibit significantly higher goodness of fit compared to the linear model. The average $R^2$ for the Logistics and Gompertz models across 258 cities are 0.993 and 0.991, respectively, with adjusted $R^2$ of 0.985 and 0.983. These values significantly surpass those of the linear model, which stand at 0.759 and 0.595. This observation suggests that urban area growth (UAG) aligns more closely with an S-shaped growth pattern.

In the comparison between the Logistics and Gompertz models, the difference is minor, yet overall, the $R^2$ of Logistics model slightly outperforms that of the Gompertz model. From the perspective of maturity, as maturity increases, the fitting effects of the Logistics model and the Gompertz model become more similar (Madsen and Hansen 2019). However, at lower levels of maturity, the predictive values of the Gompertz model tend to be significantly higher than those of the Logistics model. Furthermore, we proceeded to compare the predictive capabilities of the Logistics and Gompertz models using the last 5 data points. We computed the RMSE for the last 5 data points, resulting in an RMSE of 4.8 $k{m}^2$ for the Logistics model, which outperforms the Gompertz model's RMSE of 9.8 $k{m}^2$. The results indicate that, in terms of UAG prediction, the Logistics model outperforms the Gompertz model overall, implying a stronger alignment of the Logistics model with the growth pattern of UAG.

5.2. Urban area growth curve fitting at different time intervals

By applying the UAG curves, we have demonstrated that the Logistics model and the Gompertz model are more suitable for fitting the S-shaped growth pattern. However, in highly urbanized American cities, the urban growth curve often exhibits a nearly complete S-shaped pattern. Consequently, both of these models can achieve effective fitting results. In this section, we have extracted different phases of the UAG curve for fitting experiments, aiming to further substantiate the applicability of the Logistics model and the Gompertz model.

We have retained the last 3 points to calculate the RMSE. Subsequently, starting from the forth-to-last data point, we retrogressed in time and selected 8, 10, 20, 30, and 35 data points, corresponding to time spans of 40, 50, 100, 150, and 185 years, for further fitting analysis. We conducted experiments with time spans of 40, 50, 100, 150, and 185 years. The choice of an evenly spaced decreasing sequence (50, 100, 150, and 185 years) aimed to determine the optimal time span for effective data fitting, assessing the UAG model's performance relevance to various usage scenarios. the maximum time span of 185 years, rather than 200 years, was chosen because the HISDAC dataset used in our study covers only 200 years, and we needed to reserve the last three years (2005, 2010, 2015) for RMSE calculations. Additionally, we incorporated a 40-year time span into our design, considering that the longest temporal coverage of current remote sensing imagery (e.g. Landsat) is around 40 years. This choice aimed to explore the UAG model's applicability to remote sensing imagery, ensuring its broad and practical usage. To ensure comparability of RMSE across various stages of different cities, we normalized the values of the last 3 points before calculating RMSE. Subsequently, we computed the RMSE based on this normalization.

By contrasting the performances of the Logistics model and the Gompertz model across diverse time spans, it becomes evident that the Logistics model exhibits heightened predictive accuracy and lower RMSE values (Figure 5). Taking a 200-year time span as an example, the Logistics model demonstrates an average relative RMSE of merely 0.019 across 258 cities, while the Gompertz model registers an average RMSE of 0.032. This advantage of the Logistics model in terms of relative RMSE persists across alternate time spans. Across varying time spans, the Logistics model consistently maintains a relative RMSE lower than that of the Gompertz model, with a differential of approximately 0.0179. This implies that the Logistics model displays a slightly superior predictive prowess relative to the Gompertz model across diverse temporal scales (Table 2).

Figure 5. A boxplot depicting the predicted RMSE of 258 cities using the Logistics and Gompertz models under various time spans. The ‘L’ represents the Logistics model, while ‘R’ denotes the Gompertz model.

Table 2. Normalized RMSE of Logistics and Gompertz model at different phases.

City	Normalized RMSE (Logistics)					Normalized RMSE (Gompertz)
	40 years	50 years	100 years	150 years	185 years	40 years	50 years	100 years	150 years	185 years
New York	0.009	0.009	0.015	0.013	0.007	0.011	0.006	0.043	0.035	0.017
Los Angeles	0.006	0.018	0.012	0.010	0.010	0.012	0.017	0.019	0.019	0.010
Mission Viejo	0.011	0.016	0.013	0.013	0.011	0.006	0.008	0.023	0.023	0.009
Chattanooga	0.014	0.014	0.012	0.013	0.013	0.015	0.012	0.045	0.046	0.022
Washington D.C.	0.013	0.013	0.011	0.011	0.010	0.024	0.032	0.075	0.080	0.021
Dallas	0.043	0.037	0.041	0.043	0.018	0.028	0.029	0.058	0.058	0.031
Oklahoma	0.026	0.013	0.018	0.026	0.008	0.013	0.016	0.052	0.050	0.018
Chicago	0.029	0.016	0.043	0.024	0.016	0.034	0.023	0.086	0.080	0.032
Denver	0.057	0.059	0.051	0.049	0.027	0.042	0.034	0.132	0.134	0.043
Tulsa	0.034	0.038	0.029	0.034	0.012	0.041	0.021	0.036	0.035	0.011
Cincinnati	0.072	0.052	0.049	0.032	0.023	0.074	0.057	0.086	0.077	0.036
Orlando	0.021	0.020	0.022	0.023	0.018	0.037	0.050	0.084	0.084	0.041
Wichita	0.095	0.088	0.041	0.034	0.029	0.082	0.073	0.064	0.047	0.037
Laredo	0.070	0.071	0.051	0.063	0.040	0.060	0.056	0.055	0.051	0.053
Dubuque	0.044	0.046	0.112	0.071	0.023	0.050	0.033	0.087	0.075	0.043
Salisbury	0.67	0.062	0.088	0.054	0.042	0.039	0.076	0.101	0.069	0.057
Omaha	0.092	0.090	0.065	0.032	0.029	0.082	0.070	0.044	0.037	0.048
Appleton	0.033	0.034	0.094	0.083	0.032	0.027	0.026	0.115	0.115	0.020
Bozeman	0.156	0.155	0.082	0.089	0.065	0.174	0.181	0.079	0.077	0.040
Hays	0.074	0.086	0.176	0.163	0.045	0.083	0.097	0.207	0.219	0.021

Across various time spans, the Logistics model exhibits higher RMSE stability compared to the Gompertz model. The variance of the fitted values from the Logistics model across different time spans is merely 0.02, whereas the Gompertz model has a variance of 0.031. This signifies that the Logistics model demonstrates enhanced stability and consistency in its predictive outcomes, regardless of the time frame. This further implies that the Logistics model not only aptly forecasts urban growth trends but also maintains greater stability in fitting different types of cities.

5.3. Urban area growth curve fitting at different stage

In our previous analysis, we have already validated the effectiveness of fitting with different time spans. Due to the relatively high maturity of American cities, it is easier to obtain complete UAG curves and achieve effective fitting. However, when it comes to cities at different stages of maturity, such as nascent cities (low maturity), young cities (moderate maturity), and mature cities (high maturity), the process of fitting UAG curves presents highly exploratory variations.

Now, we are shifting our focus to explore the stability of fitting under different levels of maturity. To maintain the consistency of validation, we adopted a consistent approach by using the last three available data points to verify the models. In more specific terms, we extracted the maturity of each city from the logistics fit within a time span of 200 years. Subsequently, we categorized these cities into different maturity levels ranging from 20% to 90%, with intervals of 10%. Next, we utilized data corresponding to these distinct maturity levels to separately apply the logistics and Gompertz models for fitting. We then predicted the urban area growth (UAG) for the years 2005, 2010, and 2015. Following this, we computed the relative RMSE between the predicted values and actual observations (Table 3).

Table 3. Normalized RMSE of Logistics and Gompertz model at different maturity.

City	Normalized RMSE (Logistics)					Normalized RMSE (Gompertz)
	20%	30%	50%	70%	90%	20%	30%	50%	70%	90%
New York	0.996	0.003	0.018	0.032	0.030	0.052	0.071	0.082	0.088	0.038
Los Angeles	0.006	0.006	0.005	0.010	0.009	0.055	0.012	0.058	0.018	0.005
Mission Viejo	0.127	0.085	0.361	0.006	0.012	0.427	0.767	1.468	0.809	0.075
Chattanooga	0.026	0.025	0.009	0.009	0.130	0.051	0.591	0.081	0.206	0.395
Washington D.C.	0.128	0.349	0.195	0.012	0.012	0.125	0.214	0.510	0.243	0.042
Dallas	0.104	0.107	0.082	0.075		0.158	0.270	0.260	0.066
Oklahoma	0.031	0.062	0.068	0.007	0.008	0.246	0.257	0.130	0.203	0.029
Chicago	0.037	0.091	0.110	0.020		0.216	0.251	0.031	0.144
Denver	0.332	0.403	0.539	0.054		0.514	0.160	0.146	0.096
Tulsa	0.016	0.030	0.042	0.043	0.034	0.176	0.204	0.055	0.191	0.035
Cincinnati	0.106	0.174	0.037	0.024		0.302	0.333	0.179	0.078
Orlando	1.310	0.124	0.030	0.023		0.051	0.046	0.082	0.079
Wichita	0.185	0.177	0.094			0.134	0.048	0.042
Laredo	0.255	0.038	0.048			0.421	0.058	0.046
Dubuque	0.053	0.165	0.025			0.190	0.277	0.037
Bemidji	0.223	0.212	0.066			0.364	0.333	0.057
Omaha	0.280	0.030				0.402	0.071
Appleton	0.219	0.144				0.297	0.157
Bozeman	0.367	0.144				0.454	0.090
Hays	0.048	0.114	0.153			0.155	0.221	0.212

Through comparing the performance of the Logistics model and the Gompertz model across different levels of maturity, a trend emerges where both the Logistics and Gompertz models exhibit a gradual reduction in relative RMSE values as urban maturity increases. Specifically, the relative RMSE of the Logistics model decreases from an initial 0.334–0.03, while that of the Gompertz model decreases from 0.409 to 0.071.

In the context of varying levels of maturity spans, both the Logistics model and the Gompertz model exhibit notable differences in their RMSE stability (Figure 6). Our analysis reveals distinct stability patterns between these two models across diverse maturity scenarios. Specifically, when the urban maturity is below 60%, the Gompertz model demonstrates a lower standard deviation of relative RMSE values in various maturity scenarios compared to the Logistics model. On average, the standard deviation of relative RMSE values in each maturity scenario is approximately 0.169 for the Gompertz model, which is lower than the 0.206 of the Logistics model.

Figure 6. A boxplot depicting the predicted RMSE of 258 cities using the Logistics and Gompertz models under various time spans. The ‘L’ represents the Logistics model, while ‘R’ denotes the Gompertz model.

Conversely, a shift occurs when urban maturity surpasses 60%. In this context, the Logistics model displays better stability in terms of relative RMSE across different maturity scenarios than the Gompertz model. The average standard deviation of relative RMSE values in each maturity scenario for the Logistics model is about 0.042, which outperforms the 0.06 of the Gompertz model. This observation underscores the influence of urban maturity on the stability of both models’ predictions. The shift in stability patterns at different maturity thresholds highlights the nuanced dynamics of these models’ performances.

5.4. Parameters stability of urban area growth curves

In the preceding sections, we have evaluated the fitting performance of the Logistics model and the Gompertz model for urban growth curves under various scenarios. Next, we will further compare the differences in curve shapes and parameters between these two models after fitting, aiming to delve into the reasons behind their disparities. We begin by applying the Logistics model and the Gompertz model to fit data across different maturity scenarios, recording their respective parameters. Subsequently, we calculate the relative differences in parameters between the Logistics and Gompertz models for each maturity scenario. This relative difference is computed as the ratio of the corresponding parameters (e.g. parameter ‘k’) between the two models. A ratio closer to 1 indicates a higher similarity in parameters between the models. Through this analysis, we can gain a better understanding of the similarities and differences between the Logistics and Gompertz models.

We conducted an analysis of the relative differences between the Logistics model and the Gompertz model (Figure 7). The average relative differences for parameters ‘b’, ‘m’, ‘c’, ‘k’, ‘R’, and ‘t(R)’ were 0.89, 1.83, 1.02, 0.41, 0.94, and 1.01, respectively. These parameters mainly shape urban growth curves, with the most notable are ‘m’, ‘k’, ‘R’, and ‘t(R)’. The disparity in parameter ‘k’ arises from the distinct mathematical equations of the Logistics model and the Gompertz model. Despite relative differences in the ‘k’ values between the two models, their relative proportions remain relatively stable. Parameter ‘m’ represents the maximum potential urban area a city can reach. Notably, the disparity in parameter ‘m’ is particularly pronounced between the Logistics model (m_L) and the Gompertz model (m_G), with m_G reaching up to 27 times m_L in extreme cases. This discrepancy underscores significant differences in the estimations of the maximum potential size of cities between the two models.

Figure 7. A boxplot depicting the relative differences of fitted parameters between the Logistics and Gompertz models under various maturities. RD represents relative difference, P(Logistics) and P(Gompertz) represents parameters of Logistics and Gompertz, respectively.

The relative differences of the other four parameters, namely ‘b’, ‘c’, ‘R’, and ‘t(R)’, are close to 1, indicating a similar performance of the two models. Among them, parameters ‘b’ and c can be directly obtained from the urban growth curve and have a minor impact on the shapes of both the Logistics and Gompertz models. On the other hand, parameters ‘R’ and ‘t(R)’ respectively represent the maximum growth rate of the urban growth curve and the time at which this maximum growth rate occurs, providing an intuitive reflection of the growth potential and critical moment of urban expansion. The proximity of parameters ‘R’ and ‘t(R)’ in both logistics and Gompertz models suggests a potential for strong parameter robustness (Table 4).

Table 4. Parameters of Logistics model and Gompertz model fitted at 200 years span.

City	Logistics						Gompertz
	k	c	m	b	t(R)	R	k	c	m	b	t(R)	R
New York	0.23	11.85	51.23	1.92	26	2.93	0.11	10.87	62.09	0.54	25	2.53
Los Angeles	0.32	13.81	30.17	0.50	27	2.40	0.19	15.72	32.05	0.11	25	2.24
Mission Viejo	0.43	14.68	10.63	0.01	31	1.15	0.20	14.90	12.03	0.00	30	0.90
Chattanooga	0.33	14.71	2.25	0.03	28	0.19	0.14	17.23	2.74	0.05	27	0.14
Washington D.C.	0.27	14.74	23.36	0.21	32	1.59	0.13	14.97	29.43	0.01	31	1.41
Dallas	0.28	14.75	48.57	0.19	35	3.36	0.11	17.55	73.95	0.03	36	3.05
Oklahoma	0.26	14.65	8.90	0.10	32	0.57	0.12	17.17	11.62	0.05	32	0.51
Chicago	0.19	9.36	48.48	0.85	31	2.29	0.08	12.37	69.71	0.75	31	2.03
Denver	0.19	13.62	30.00	0.32	37	1.44	0.06	16.40	62.86	0.24	43	1.49
Tulsa	0.28	14.25	5.46	0.05	31	0.38	0.14	17.09	6.60	0.02	30	0.34
Cincinnati	0.16	11.56	15.26	0.47	33	0.62	0.06	13.92	28.55	0.56	38	0.60
Orlando	0.28	14.38	29.79	0.10	35	2.06	0.11	16.70	46.99	0.01	36	1.88
Wichita	0.17	14.11	11.33	0.20	38	0.48	0.05	15.93	38.94	0.21	52	0.65
Laredo	0.20	14.14	1.78	0.01	39	0.09	0.05	15.74	9.70	0.01	58	0.17
Dubuque	0.11	15.53	1.07	0.10	35	0.03	0.04	16.53	2.53	0.11	48	0.03
Salisbury	0.18	15.96	2.69	0.05	38	0.12	0.05	16.68	9.25	0.04	52	0.16
Omaha	0.13	12.81	12.50	0.28	42	0.41	0.03	15.85	98.41	0.39	59	0.86
Appleton	0.10	14.93	1.04	0.08	39	0.03	0.02	16.51	7.73	0.09	59	0.05
Bozeman	0.16	15.49	0.71	0.01	41	0.03	0.03	16.39	13.78	0.01	59	0.09
Hays	0.16	15.01	0.39	0.01	39	0.02	0.05	16.00	0.90	0.01	47	0.02

Upon further observation of these 6 parameters across different levels of maturity, notable differences were observed in parameters ‘k’ and ‘m’ (Figure 8). Interestingly, the relationship between parameters ‘k’ from both models exhibited relative stability (p-value less than 0.01), indicating a certain level of convergence. However, the correlation between parameters ‘m’ of the Logistics model and the Gompertz model was not significant, suggesting a lack of apparent association in their variations. Specifically, we observed a wider dispersion in parameter ‘m’ for the Gompertz model, potentially leading to instances of overestimation. This may imply that the Gompertz model's estimations of the maximum potential size of cities could carry significant uncertainty in certain scenarios.

Figure 8. Differences between parameters of Logistic and Gompertz models.

The impact of maturity on parameters ‘b’, ‘c’, ‘R’, and ‘t(R)’ is relatively minor, especially for parameter R. When maturity is below 20%, parameters b and c can also be directly obtained from existing urban growth data, whereas parameter ‘R’ requires computation through fitting the Logistics and Gompertz models. In such cases, the parameters ‘R’ from both the Logistics and Gompertz models exhibit a strong correlation and nearly identical values. This indicates the stability and universality of parameter ‘R’, as it remains consistent across different models and data scenarios. Thus, it can be considered a highly stable and universally applicable assessment parameter, holding significant significance for urban growth (UAG) evaluation.

Although there is a relatively high correlation between parameters ‘t(R)’ of the Logistics and Gompertz models (correlation coefficient of 0.62), the degree of correlation is relatively low. This suggests that parameter ‘t(R)’ is influenced by model selection and may exhibit variations between different models. Hence, compared to parameter R, parameter ‘t(R)’ might be less suitable as a universal assessment parameter, potentially having a more limited scope of applicability.

6. Discussion

6.1. The applicability of the S-curve to the UAG patterns

UAG is intricately shaped by multifaceted factors, particularly on a microscale within short time frames (Chauvin et al. 2017; Chen, Li, et al. 2020). However, considering longer temporal contexts, macro-level influences tend to steer the trajectory of urban growth, often adopting an S-shaped trajectory (Liu et al. 2018). This pattern entails gradual initial growth, followed by accelerated expansion, culminating in saturation. In capturing intricate UAG patterns, the Logistics and Gompertz models exhibit clear superiority over the linear model (Mulligan 2013b). The adjusted $R^2$ of the piecewise linear model, at 0.595, indicates that it can explain approximately 59% of the variability in UAG. The piecewise linear model can relatively well describe the growth trend of UAG but may not fully capture all of its variations. In contrast, both the Logistics and Gompertz models have adjusted $R^2$ exceeding 0.98. This signifies that these two can almost perfectly account for the variability in UAG, signifying their remarkable precision in depicting UAG trends. The significantly higher adjusted R-squared values for both the Logistics and Gompertz models underscore their efficacy in describing the S-shaped growth trajectory of urban areas. This observation aligns with the expectation that UAG is more accurately represented by an S-shaped curve.

S-shaped curve has proven to be a robust framework for depicting the trajectory of urban area expansion, while associated parameters such as Maximum Growth Rate, Maximum Urban Area, Urban Take-off Area, and Urban Start Time exhibit disparities in their intuitive clarity. Maximum Growth Rate and Maximum Urban Area within the S-shaped curve model boast explicit and straightforward interpretations, representing the pinnacle growth rate and ultimate size achievable in urban area expansion, respectively. However, Urban Take-off Area and Urban Start Time, integral to the urban development process, present challenges in terms of their elusive nature, necessitating consideration of factors such as population dynamics, economic activities, infrastructure development, and government-implemented urban planning policies for precise quantification (Mulligan 2013b). Given the intricacy of these parameters, our study employed fitting with logistics and Gompertz models to obtain quantified values. Through this mathematical modeling process, our objective was to objectively derive values for Urban Take-off Area and Urban Start Time, grounded in the specific conditions and features observed in the development of each city. It is crucial to emphasize that while the results obtained through model fitting may not adhere to conventional quantitative standards, these values are derived from observed growth patterns in each city. Future research directions should focus on refining these models and potentially introducing additional factors to enhance the accuracy and reliability of quantifying Urban Take-off Area and Urban Start Time.

The Logistics and Gompertz models effectively characterize the UAG, and their comparison reveals subtle distinctions in their performance. The Logistics model consistently outperforms the Gompertz model in terms of R-squared values and root mean square error (RMSE) for various time spans. In fitting UAG, the Logistics model demonstrates greater stability compared to the Gompertz model, particularly across various temporal scales. Although both models converge when urban development approaches saturation, disparities arise at different maturity levels, notably in lower maturity scenarios (Satoh 2021). It's worth noting that the Gompertz model tends to overestimate urban growth in less mature situations (Trappey and Wu 2008). In contrast, the Logistics model provides estimates that align more closely with actual circumstances and developmental trends (Dhar and Bhattacharya 2018). Additionally, given the current trend towards compact urban development, the Logistics model serves as a more suitable estimation tool, even when applied to shorter UAG datasets (Bengisu and Nekhili 2006). The significance lies in validating the successful reconstruction of urban UAG processes using the Logistics model, even with as little as 40 years of data. This insight proves valuable for research into urban area growth based on remote sensing data, underscoring the robustness of the Logistics model in capturing expansion patterns under data constraints.

6.2. Applicability of the UAG model under different development scenarios

The assessment of the universal UAG model's applicability across different urban expansion scenarios is crucial. To address this, we evaluated whether cities undergoing expansion in diverse development contexts exhibit S-shaped growth patterns consistent with the UAG model. Utilizing the geographical divisions in the United States, we categorized the sample cities into nine groups based on their unique development patterns shaped by historical, cultural, and economic factors. For instance, New England relies on robust industrial and educational sectors, while the Mid-Atlantic, represented by New York City, functions as a global financial hub. The Midwest, with regions like Midwest Northeast and Midwest Northwest, showcases early industrialization, particularly in automobile manufacturing and steel. Lastly, the Pacific region emphasizes high-tech industries, the film industry, and international trade.

In our study, we conducted a comprehensive assessment of the fitting performance across various geographical regions for all cities (Figure 9). Overall, despite the diverse developmental patterns and scenarios exhibited by these cities, they collectively demonstrate a trend indicative of S-shaped growth, suggesting the applicability of the UAG model to cities in different developmental contexts. Spatially, subtle variations in the fitting performance of the UAG model were observed across different regions. Specifically, in highly developed regions such as the Pacific, Atlantic Central, New England, and the Midwest Southwest, cities tend to conform more closely to the S-shaped growth pattern. This observation can be attributed to the diverse and advanced economic structures in these areas, encompassing high-tech industries, financial services, and cultural and creative sectors. Additionally, these regions typically possess outstanding educational systems and innovative ecosystems, attracting highly skilled talents and fostering innovation and technological development (Chauvin et al. 2017). The diversity of driving factors in these areas contributes to a more comprehensive and steady urban growth, aligning with the inherent growth patterns of cities. In contrast, regions heavily reliant on agriculture and resources, such as the Mountain region and the Northwest Central Midwest, may experience less stable urban growth due to the singular nature of these driving factors.

Figure 9. Precision evaluation of Logistic and Gompertz models based on geographical regions: a. Urban areas in different regions in 2015, b. Maturity of cities in various regions, c. R² values for Logistic models across regions, d. R² values for Gompertz models across regions, e. Root Mean Squared Error (RMSE) for Logistic models across regions, f. RMSE for Gompertz models across regions.

Our in-depth evaluation of the correlation between urban maturity and the fitting accuracy of the UAG model (Figure 10) reveals that maturity serves as a representative measure of urban development. In general, higher urban maturity corresponds to better fitting accuracy. Cities with elevated maturity levels are predominantly located in the Pacific, Atlantic Central, New England, and the Midwest Southwest – regions characterized by high economic levels and well-developed industries. Conversely, cities with lower maturity levels are concentrated in the Mountain region and the Northwest Central Midwest, where industries are relatively singular, resulting in slightly lower fitting accuracy compared to their highly mature counterparts. This discrepancy may be attributed to the instability of urban driving factors caused by the singular nature of industrial structures, leading to slower and occasionally stagnant urban development. This instability manifests in the growth curves as erratic increases, ultimately impacting the fitting accuracy.

Figure 10. Correlation distribution of Maturity and model accuracy for Logistic and Gompertz models across different regions.

In conclusion, this study robustly validates the alignment of the UAG model with diverse urban growth trajectories shaped by varying development patterns. The evaluation, based on geographical divisions in the United States, consistently demonstrates an S-shaped growth trend across cities, emphasizing the model's adaptability to different developmental contexts. Particularly, regions characterized by well-established economic structures exhibit a more pronounced adherence to the S-shaped growth pattern. In contrast, regions heavily dependent on agriculture and resources may encounter less stable growth, emphasizing the need for heightened attention when applying the UAG model to such city types.

6.3. S-curve for reconstructing spatial historical UAG or forecasting future UAG

The value of the S-curve in studying urban area growth lies in its ability to reconstruct the entire temporal sequence of UAG. This implies that it can not only be used for predicting future UAG but also for retracing the historical processes of UAG. This characteristic provides robust support for contemporary UAG research. Currently, conventional UAG research heavily relies on remote sensing imagery, a method that originated in the 1970s (Huang et al. 2021). However, by that time, global urban development had already matured significantly, especially in developed countries, where the urbanization process was largely complete (Reba and Seto 2020). Data like the long-term 200-year historical records of UAG in the United States, as provided by HISDAC, are exceedingly rare. Consequently, obtaining data for researching the historical development of UAG has faced substantial limitations. In contrast, the S-curve offers a significant tool for addressing this issue. Its application enables researchers to capture the status of UAG at various time points, thereby reconstructing the entire historical process.

To investigate the application of the S-curve in reconstructing the UAG process, we conducted experiments combining the S-curve with cellular automata (Figure 11). We selected five cities at different stages of development: (a) Washington D.C., (b) New York, (c) Orlando, (d) Salisbury, and (e) Wichita. Using UAG data from 1970 to 2000, we successfully fitted the entire UAG process for these cities, achieving a close match between the fitted and observed UAG. Subsequently, we employed cellular automata to spatialize the UAs for 2015 and 1950 based on the fitted UAG, with an average accuracy of 86.3%. This experiment holds three primary significances. Firstly, it deepens our understanding of the significance of the S-curve. It demonstrates that despite the macro-historical process of UAG being influenced by various complex factors, it still adheres to some fundamental growth principles, similar to other types of growth processes. Secondly, we provide a universal method for spatiotemporal simulation of UAG. By combining the S-curve with cellular automata, effective modeling of the spatiotemporal aspects of UAG can be achieved. Lastly, our choice of using data from 1970 to 2000 to fit UAG provides a robust foundation for future spatiotemporal reconstructions of global UAG. Existing global UAG records primarily begin after 1970. Our experiments demonstrate that even for different types of cities, using satellite UAG data from this period can successfully facilitate the spatiotemporal reconstruction of UAG, offering a reasonable opportunity for historical research on global urban UAG.

Figure 11. Combining Logistics modeling with cellular automata to reconstruct historical spatial Urban Area Growth (UAG) or predict future UAG for five distinct regions: (a) Washington D.C., (b) New York, (c) Orlando, (d) Salisbury, and (e) Wichita. Here, ‘Obs’ denotes the observed real UAG, while ‘Sim’ represents the simulated UAG for the corresponding year obtained through the Logistics modeling approach. Grey points represent all the observed real UAG data, while the red points constitute the subset used for fitting the S-curve.

6.4. Assessing the potential application through parameter ‘R’

Within the exploration of UAG models, a resilient parameter, denoted as ‘R’ emerged as a pivotal metric for evaluating disparate urban developmental trajectories. In contrast to the conspicuous disparities and volatility displayed by various parameters across the Logistics and Gompertz model, parameter ‘R’ exhibits remarkable stability (Dhar and Bhattacharya 2018; Satoh 2021). Contrasted with ‘k’, the advantage of ‘R’ lies not only in its heightened stability but also in its more precise physical interpretation, which is that ‘R’ directly signifies the maximum growth rate of the UAG curve (Iyer et al. 2015; van Sluisveld et al. 2015). ‘R’ remains stable regardless of whether UAG has already been completed or is nearing completion (i.e. UAG saturation), or is in the middle stages of growth, such as near the inflection point of the S-curve. This relational framework becomes instrumental in gauging whether a city's expansion has attained the zenith of its growth rate, thus affording an appraisal of both present and future developmental trajectories.

We designed a quantitative experiment to discuss the significance of ‘R’. We fitted the growth rate changes for 258 cities in this study and compared them with ‘R’, evaluating the growth status of these 258 cities in the ‘current’ year (2015) (Figure 9). First, we fitted the growth rate curves of the Logistic and Gompertz models. Then, based on the relationship between the growth rate fitting curves and the positions of their corresponding ‘R’ values, we classified the cities into three categories: Stall Growth, Stable Growth, and Rapid Growth.

• Stalled growth: The ‘R’ values from Logistics and Gompertz models occurs after the latest point in time within UAG, and the growth rate has dropped below 20% of ‘R’. The maturity level generally exceeds 90%.

• Stable growth: The growth rate is higher than 20% of ‘R’, and the ‘R’ values from Logistics and Gompertz models appear shortly after the growth rate exceeds this threshold. At this stage, the maturity level typically surpasses 50%.

• Rapid growth: The timing of the ‘R’ in the Logistics and Gompertz models exhibits certain discrepancies, or both ‘R’ appear after the latest point in time within UAG. At this stage, the maturity level typically falls below 50%.

From the fitting results, it can be observed that in most cases, the values of ‘R’ in the Logistic and Gompertz models are quite consistent, especially when maturity is above 50%. The results revealed that among these 258, 109 cities were in the Stalled growth, and 86 cities were in the Stable growth phase, mainly concentrated on the U.S. East Coast, West Coast, Great Lakes region, and the Southeast, which are highly developed areas with advanced urbanization. Additionally, 63 cities were in the Rapid growth phase, primarily situated in the U.S. Midwest and West. This classification suggests that a significant number of U.S. cities have largely completed their urban growth, indicating a high level of urban development in the United States (Figure 12).

Figure 12. To classify the developmental stages of 258 cities based on the parameter ‘R’. a. The 258 cities are categorized into three groups: Stall Growth, Stable Growth, and Rapid Growth. b. The growth rate curves for cities, with yellow and blue representing the velocity fits (derivatives) of the Logistics and Gompertz models, respectively. The dashed line represents the timeline for the year 2015.

The significance of parameter ‘R’ resides in its role as a relatively steadfast gauge of urban developmental states, impervious to the influences of diverse model equations (Comin, Hobijn, and Rovito 2008). This integration lessens model disparities, bolstering research reliability (Creutzig et al. 2017; Martino 2003). The evaluation of UAG using ‘R’ demonstrates a rational assessment approach that does not rely on either the Logistics or Gompertz models individually but instead integrates both models to assess different growth stages, thereby enhancing the credibility of the results. This confers upon researchers the capacity to embrace a more comprehensive comprehension of urban expansion dynamics, transcending sole reliance on model fitting efficacy. Furthermore, this study provides additional evidence supporting the theoretical framework and the advantages of S-curve-based UAG models, which demonstrates that this framework allows us to quantitatively investigate and assess UAG growth models from various perspectives. Through the monitoring of its fluctuations, researchers can prognosticate the vector of urban growth, ascertain the necessity for adjustments in developmental strategies, and discern latent bottlenecks that might arise.

7. Conclusion

In summary, the perplexing aspect is that the field of urban research has yet to converge on a singular or a few simple models to address the monumental transformations underway in numerous nations and their regions as they swiftly transition from rural to urban settings. This article proposes the adoption of urbanization curves as a succinct method of description, and strongly advocates for the utilization of the Logistics model as an analytical tool to depict the diverse urbanization processes occurring across the globe. In practice, within the context of current urbanization, particularly in emerging economies, employing both straightforward time series models and more intricate urban evolution approaches could yield substantial dividends. Furthermore, the validation of our conclusions would necessitate further applied research, encompassing a multitude of additional observational outcomes, while potentially considering various factors such as the geographic locations of distinct nations. These insights hold significant implications for urban planning, policy formulation, and remote sensing studies, enriching our understanding of the complexities inherent in urban expansion and development. The study's methodological rigor and the robustness of the findings position it as a valuable contribution to the field of urban studies, with potential applications in guiding sustainable urban development practices worldwide.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (No. 42330103) and (No. 42271469), and also supported by the National Key Research and Development Program of China (No.2021YFE0117100 and No. 2022YFF1301102).