1
A Diffusion-based Macro Model for Regional Urban Expansion
in the Case of the Eastern Part of China
Yohei SATO * and Lin LI*
*
Dept. of Biological & Environmental Engineering, Graduate School of Agricultural & Life Sciences
The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-8657, Japan
Abstract: This year, we made a progress on how to find driving factors in general means. As known to us, it
would be difficult to find such factors and the relationship between the factors and urban growth because there is
no universal approach to determine the driving factors. Based on pervious experiments, we summarized a
general method to derive driving factors from a land cover change. In this method, it would be available to
deduce the driving factors from potential factors based on easily accessible land cover data. In addition, we
implement zoning factor in the diffusion-based model. By assuming normal distribution of population, we took
Xian (in China) as the case study city to make some experiments. This study shows that Xian city can achieve
higher economic growth at cost of using less urban land in the future under current allocation of functional zones
and zoning would be available way to balance economic growth and urban land growth.
Keywords: urban expansion, population distribution, zoning, land cover
1. Background
1.1 Main work in the previous years (1998-2000)
Main work to have been done is listed as the following:
1) Determining the case study area
2) Preparing and processing remote sensing data—TM images
3) Setting up a basic theoretical model for urban expansion—Diffusion-based Model
4) Designing a method for distributing the population of a county—Population Allocating
5) Collecting data(statistical data and geographical data)
6) Matching images with a geographical map
7) Implementing the Diffusion-based Model
8) Implementing the Population Allocating
9) Making experiments of simulation of urban expansion
1.2 Main work this year (2000-2001)
This year, we made a deeper research on the diffusion-based model
1) Summarizing the method to derive driving factors from an urban expansion on a regional scale
2) Preparing data (statistical data and geographical data) for examining suitability of this model to the
zoning factor
3) Implementing the zoning factor in this diffusion-based model in a city (Xian city, China)
4) Building a simple DEMO on the Internet
2. Scheme for Deriving Driving Factors from a Land Cover Change
Although practical work about this aspect was done last year, we made a valuable theoretical study based on the
work and summarized a general scheme for finding driving factors for urban expansion on a regional scale. We
may have knowledge on the general category of the driving factors, but we cannot know those concrete factors
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for land use changes because the factors driving urban expansion vary with regions.
Since the urban expansion is similar to a diffusion process in these four aspects:
a. flowing in single way: non-urban to urban,
b. an urban area becomes bigger and b igger,
c. an expansion happens on the edges of urban land, and
d. expansion is continuous in terms of year; we can use the concept of basic diffusion process to describe an
urban expansion in the case study area. The diffusion-based model is represented by
p ij (t+1) = p ij (t) + ρ×∆2 p ij (t)
(1)
( with initializing: p ij (1) = τ×p ij (0) )
where p is population density, p arameter ρ represents environmental (or physical) impacts on the expansion(such
as roads); p arameter τ represents the integrated influence of socio-economic factors (driving factors) on an urban
expansion.
However, it would be a little hasty to determine what value of t means before simulation. There would be
two ways to determine the meaning of t through sample data analysis : temporal sampling and spatial sampling.
(1) Temporal Sampling
On the condition that t means the same things for an area during a certain period, we collect a series of data about
urban growth (land cover change) at some interval of this period. Through the diffusion process, we can obtain a
series of simulations by adjusting the values of t so that each simulation is close to actual case respectively.
Those data can be filled in a table (Table 1).
By some means of a data analysis among the values and the potential driving factors, we can make certain
what the t means in terms of social and/or economical indices.
Table 1 data in Temporal Sampling
Time
τ
Potential
driving
factors
intervals
values
GDP
Secondary
Industry
…
Certain
t1
τ1
a1
b1
c1
…
Period
t2
τ2
a2
b2
c2
…
(T)
…
…
…
…
…
…
tn
τn
an
bn
cn
…
Table 2 data in Spatial Sampling
Spatial
τ
Potential
driving
factors
units
values
GDP
Secondary
Industry
…
u1
τ1
a1
b1
c1
…
whole
u2
τ2
a2
b2
c2
…
region
…
…
…
…
…
…
un
τn
an
bn
cn
…
(2) Spatial Sampling
This way is similar to the above temporal sampling except that we divide whole region into a set of spatial units
instead of dividing a period into intervals. The factors influencing urban growth should be kept the same and the
spatial units are homogeneous. However, size of the units may different. Also, we simulate urban expansion in
each unit and adjust the value of t so that the simulation for each unit is close to an actual case respectively. We
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have Table 2 similar to table 1 and do some data analysis.
Usually on modeling urban dynamics including social behaviors, an often-met problem in practice is about
data. In some area, there is very limited data available especially on land use, even some data exist but cannot be
accessible due to some reasons. However, it is quite convenient to access to land cover data by remote sensing. In
above tables, only land cover data are used. It is convenient for us to apply these approaches extensively. These
two ways can boil down to a scheme for deriving driving factors for urban growth from its land cover changes
(Figure 1).
Adjusting parameter τ
Diffusion
Land
cover
Model
Real land
cover
Potential
driving
factors
Future
land
cover
Simulating
Figure 1 Scheme for deriving driving factors
Regarding to our case study area, we used spatial sampling approach to derive driving factors. The case
study area was divided into 5 cities and 19 counties so that there are 24 basic units in total. We used data in 1990
and in 1995. Through the experiments made last year, we find an approximated relationship between t and
socio-economic factors as fo llowing:
τ = k×(t×a)
(k is a constant)
t : output value of township and village industry
a: average land area
This formula reveals that the degree of urban expansion in a county was related to its output value of
township and village industry and limited by its land resource. Since we know that total land resource is
unchanged and the evolution of township and village industry varies at different periods, we would be certain
that township and village industry is a driving factor for urban expansion.
3. Implementing the Zoning Factor in this Diffusion-based Model
In order to examine some features of the model, we implement the zoning factor in modeling urban growth of a
city. We choose Xian--capital city of Shangxi Province, China as the case study city. It was the capital of many
Dynasties (from Dang Dynasty). Although the city has long history (about 3100-year), it is not located in the
developed area in China and it is on the initial stage of modernization.
3.1 Case city-Xian
3.1.1 General
The city of Xian (shown in Figure 2) consists of 6 communities and the total population in 1997 was about 2.86
millions, among which 1.597 million (about 56%) are located in the three communities (Xincheng, Beilin and
Lianghu) that mostly center the city. Population density in city center (the rectangle area in the middle) is much
higher than that in other area. The density in the center arrives at 25577/km2 within the rectangle area fenced by
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the wall, and the density in the others ranges from 1000/km2 to 3000/km2 . It is obvious that population density
decays rapidly with distance from the city center. Because the density in the center is too high (10-20 times that
in other area), the city government tries to make some policy and plan to decrease the density according to the
city planning.
Figure 2 Xian City Land-use Map
Figure 3 Functional zones in Xian
3.1.2 Functional zones
A city may form some structures or different functional zones in the process of its development. Those zones
play different roles in different economic situations or different periods. The construction on those zones may
evolve with economy and represents levels of development of the city.
In Xian city, four main zones were used to examine the influence of these functional zones on the spatial
pattern of urban growth. Figure 3 shows these four zones: protected, economic, high-tech and textile. Urban
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development will take different speeds on these zones at different scenarios.
1) Protected zone: Since Xian once was the capital of China in several dynasties and there exist a lot of historic
sites (archaic tombs of empires) which would be invaluable for Chinese culture and also an alluring tourist
resource. These zones are protected from being constructed to preserve these resources.
2) Economic zone: This region is used for developing and launching enterprises of industry for this city.
3) High-tech zone: It is used for various activities related to the latest and high technology such as computer,
communication, information and so forth.
4) Textile zone: It is mainly used for textile industry.
5) Ordinary zone: other area than above four zones.
3.2 Basic principle
3.2.1 Distribution of population
In general, if we go through a city, we may find a declining pattern of population density from the city center of
the city to its edge. To describe such pattern, “economists tempt to use so-called negative exponential function,
which was supported to some extent by empirical studies.” (F Wang, 1998) .
Population density p(r) at distance r from the center of the city declines according to the negative
exponential:
p ( r ) = K × exp( − λ × r )
(2)
Figure 4 Negative exponential curve
Intuitively, this function produces a very sharp peak (shown in Figure 4) for population distribution over a
space that would not be suitable for our case study. Whether we chose the negative exponent or other functions,
the most important criterion is that the chosen function should obey the pattern “density decays with distance”.
Instead, we chose the function producing a relatively flatted peak. Thus, this formula (2) can be extended to the
following form that inherits the feature of density decay and preserves the similar trend of population density
change:
K
r2
p (r) =
exp( − 2 )
2σ
σ 2π
where K is a constant,
p ' (r) =
(3)
1
r2
exp( − 2 ) is the form of normal distribution N(0, σ2 ). In this
2σ
σ 2π
sense, the formula (3) is called the normal-distribution-based population density. Based on this function, we can
easily find the meaning of the constant.
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∞
∞
K
−∞
σ 2π
∫ p( r ) = ∫
−∞
exp( −
r2
2σ
2
∞
1
−∞
σ 2π
)=K ∫
exp( −
r2
2σ 2
)=K
Figure 5 Normal distribution curve
So the constant K refers to the total population. According to normal distribution (Figure 5), we have:
P{-2σ<p<=2σ}=0.9544
,
P{-3σ<p<=3σ}=0.9974
(4)
That means that only less than 5% of the population lie outside 2 standard deviations from the city center and
about 0.26% outside 3 standard deviations far from the center.
3.2.2 Zoning factor
These zones can be described as frictions to urban flow. For example, in a protected zone, urban expansion will
be blocked; it would be faster for an urban area to extend in an economic zone than in the other zones. So we use
parameter r to represent the roles of different zones in the process of urban expansion.
3.3 Implementation
3.3.1 Normal distribution on urban area
Since we only know the total population, we must set a value for the standard deviation value in order to
calculate population distribution. Suppose the data size represents all area of this city. The urban area of this city
is not regularly shaped and is not continuous over the two -dimensional space. Only the urban area is allocated by
people or population density. For the simplicity of description, pixel is used as length unit. The shape of the city
is like rectangle with about 330 pixels in length and about 160 pixels in width. The size of the city is about 300
48000
≈ 124 . The city center is about 70×48; and comparing with
π
total size, the center is rather an area than a point. This is one main reason why we chose the normal distribution
rather than the negative exponential function. The extent of the city is supposed to be covered by this
image--namely 550×520. According to the characteristics of the normal distribution; the value for the standard
×160=48000, and the average radius is
deviation should be between 86 and 91. In that case, we here assume that the standard deviation at the beginning
is 90 (pixels). We allocate 2.86 millions people on this geographical extent (550×520 pixels) by the normal
distribution with standard deviation s=90. Figure 6 shows the distribution over this space. If the city is regular
shaped (a circle) and sprawl all the extent, the distribution will look like Figure 6(a). Figure 6(b) shows that the
normal distribution is constrained by the actual urban area.
3.3.2 Implementing sprawl process with CA
A cellular automata is a discrete dynamical system1). Space, time, and the states of the system are processed in
discrete way. Space consists of regular cells that can have any one of a finite number of states. The states of the
cells are updated according to a local rule. That is, the state of a cell at a given time depends only on its own state
one time step previously, and the states of its nearby neighbors at the previous time step. All cells on the lattice
are updated synchronously. Thus the state of the whole space advances in discrete time steps and the overall
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pattern is determined by or can be attributed to the changes of local states 2),3),4),5),6),7),8),9).
Based on the formula and the methods provided in above sections, it is found that computational features
are the followings (except initial state):
1) Locality: density change of each cell or grid solely depends upon the value of the cell and of its nearest
neighbors.
2) Homogeneity: density of a cell can be calculated by the unified formula and transition of a cell from non-urban
to urban can be formatted by the same rule.
In order to implement simulating process, cellular automata (CA) is chosen as an implementing tool.
There are two main reasons for doing so.
Figure 6 Distribution of population over urban space
1) The description of state changes is quite simple and understandable and land change from non-urban to urban
can be easily represented by local rules in cellular automata;
2) By using CA, it would be easy to implement our model as general simulating tool. For examp le, we can
introduce other distribution functions to imitate various possible situations.
Suppose we have a case area that is divided into n×m small square cells, n is the number of rows and m the
columns. [i,j] denotes the location of the cell at ith row and jth column, c[i,j] state of the cell and p[i,j] density of
the cell. A cell has only two state denoted by 0 and 1 representing non-urban and urban respectively. State can be
changed in single direction from 0 to 1, that is that density of a cell can only increase.
3.3.3 Calculation of density
Density increase of a cell is attributed to density of its 8-neighbors and only those cells with higher density will
contribute to its density since it is assumed that density keeps up in a diffusion process. Considering that 8
neighbors have different spatial connections, the weights are assigned as the following:
w[i-1,j] = w[i+1,j] = w[i,j-1] = w[i,j+1] =0.2 ,
w[i-1,j-1] = w[i-1,j+1] = w[i+1,j-1] = w[i+1,j+1] =0.05 ;
∑w =1.
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Thus, double difference is substituted by increment ∆p
∆p = w[i-1,j] × (p[i-1,j] − p[i,j]) + w[i+1,j] × (p[i+1,j] −p[i,j]) +
w[i,j-1] × (p[i,j-1] − p[i,j]) + w[i,j+1] × (p[i,j+1] − p[i,j]) +
w[i-1,j-1] × (p[i-1,j-1] − p[i,j]) + w[i-1,j+1] × (p[i-1,j+1] − p[i,j]) +
w[i+1,j-1] × (p[i+1,j-1] − p[i,j]) + w[i+1,j+1] × (p[i+1,j+1] − p[i,j])
Density of a cell at time t+1 is calculated by
p ( t+1) [i, j ] = p (t ) [i, j ] + ρ[i, j] × ∆p (t ) [i, j ]
3.3.4 Rule of state transition:
K
(i − i0 ) 2 + ( j − j 0 ) 2
c [ i, j ] = 1 when p [i , j ] ≥
exp( −
)
2σ 2
σ 2π
( t)
(t )
where [i0 ,j0 ] is the location of city center.
3.4 Experiments and analysis
3.4.1 Scenarios
City growth is always tied with economic growth depending upon the development in agriculture, industry, and
technology. Agriculture cannot support high economic increases and traditional industry can hardly support
sustainable economic increases. New technology, informatics as well as service can arrive at very high increase
of economy. According to agenda of future development in China, economic growth should be about 7%-10%
overall. Since city economy plays leading role in regional economic development, economic growth would be
larger than the average growth rates. In such view, three growth rates of city economy (GDP) are assumed up to
2040, that are 6%, 10% and 14%, respectively called low growth, middle growth and high growth. Table 3 shows
the structure of three industries composing GDP.
Rate
Scenarios
1997
Low
2040
Middle
High
Table 3 Scenarios of economic growth
GDP
(108 Yun)
Primary
Secondary
14.3%
(500.55)
≈6%
(621 7.13)
≈10%
(30201.16)
≈14%
(139696.53)
9.06%
(51.33)
5%
(418.32)
(7%)
941.62
(10%)
3092.12
16.5%
(209.70)
7%
(3846.81)
(10%)
(13658.62)
(14%)
58682.28
Tertiary
13.4%
(239.52)
5%
(1952.00)
(10.2%)
(15600.92)
(14.4%)
77922.13
3.4.2 Experiments
Urban growth is lead by an economic increase of a city and its extent on different functional zones will vary with
different economical situations. In city economy, Secondary and tertiary industries dominate city development
so that urban growth on zones related to these industries is faster than that on the other zones, that will lead to a
different spatial pattern of the city in the future.
Denote ρ P ,ρE ,ρH ,ρ T and ρo as Protected, Economic, High-tech, Textile zone and Ordinary area respectively.
Figure 7 shows two simulated cases of urban growth in 2040 with different diffusion coefficient r for those
functional zones. The case (a) shows the spatial pattern of urban growth in 2040 when economic growth occurs
mostly on the economic zone and the city will extend largely to north. The case (b) shows the spatial pattern
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when economic growth of high technology dominates the economic increase in the city so that the city will
extend largely to south. Urban land on protected zone has little increase in case (a) or no any increase in case (b).
The spatial patterns may reveal an influence of a different kind of economies on the city development as well as
a social and economic evolution in the surrounded regions. The la rger of the coefficient r, the easier the urban
grows.
Figure 7 Influences of zones on pattern of urban growth
We assume that urban growth on functional zones is proportional to the growth rate of industries. According
to the explanation of concepts of these 3 industries in statistic yearbook, the first industry mainly means
agriculture; the secondary mainly means traditional industry and the tertiary includes information industry
related to high technology. The structure of GDP in functional zones is supposed in Table 4. Thus, the coefficient
r can be set in following Table 5 that also includes the case “Ordinary” meaning all functional zones degenerate
to ordinary area. Simulated urban growths are shown in Figure 8.
Table 4 Supposed structure of GDP in functional zones
Protected
0.1×ρ o
Economic
1.0×Secondary
High-tech
1.0×Tertiary
Textile
0.5×First+0.5×Secondary
Table 5 Coefficient r for different Scenarios
Scenarios
ρP
ρE
ρH
ρT
ρo
Low
Middle
0.0
0.1
0.84
1.43
0.6
1.45
0.78
1.2
0.6
1.0
High
0.14
1.96
2.03
1.68
1.4
Ordinary
1.0
1.0
1.0
1.0
1.0
(In each scenario except "Ordinary", basic ρ for zone other than the 4 zones is supposed equal to total GDG
growth×10, those ρ for the 4 zones are calculated by its scale to lowest growth rate, then the value is multiplied
by ρ o . For example at low growth, ρ E =0.6×(7%/5%)=0.6×(1.4) =0.84,
ρ T =0.6×((7%+5%)2/5%)=0.6×(1.2) =0.78 )
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Figure 8 Simulated results for different cases
New increments of land on zones are collected in Table 6.
Table 6 New increased land on zones
1997
2 Low
0 Middle
4 High
0 Ordinary
Protected
1328
0
0
0
975
Economic
9754
6503
6734
6508
3497
Hightech
11210
3262
5788
5944
3197
(unit: pixels)
Textile
Others
Increment
4084
2420
1982
1852
1654
61164
21916
20227
16698
27109
(%)
39.0%
39.7%
35.4%
41.6%
3.4.3 Analysis
Usually, high economic growth will lead to larger extent of urban growth. However, through those experiments,
we found that less urban land would be used for this city at high economic growth under the restriction of normal
distribution of population. The main reason would be that Xian is at its initial stage on the way to
industrialization and industry enterprises, business and residence are all concentrate on or near to the city center.
Although economic and high-tech zones, which mainly support city economy, are located around edge of the city,
the shape of the city is east-west longer than north-south and these two zones lie in north and south respectively,
where are near to the center. The places nearer to the center can hold more people according to the normal
distribution. When urban growth on these zones is higher than that on the other zones, places with high density
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increase faster so that less land would be used for the same number of people.
Population density also has a relationship with the story of buildings and locations. High buildings can hold
more people than low buildings in the same area, but it would be much expensive to build them. Construction
near the center costs more than that of being far from the center. In that sense, only high economic growth can
support such pattern of urban growth. In the period of low economic growth, economic effects are paid much
more attention to than the other factors; places far from the center and low building are constructed at the first
priority so that more lands will be occupied for holding the same size of population.
In comparison with the ordinary case without zoning, a city arranged for different functional zones may use
less urban land for its economic development. The case study shows that zoning in city planning would be
available way to balance economic growth and urban land growth. These simulations also showed that this city
could be successful in supporting its high economic growth and population growth aiming to use less land.
4. Conclusion
Since urban growth is a very complicated process, finding driving factors is vital to forecast or estimate future
urban growth. It is a valid way to derive the driving factors from potential elements related to social and
economic indexes by means of Temporal Sampling or Spatial Sampling as shown in our case study.
Although assumptions such normal distribution in the case study of Xian may not fit to the reality,
simulations can provide valuable information for examining relationship between economic situations and urban
growth. Implementing this model can provide planners or decision-makers with an interactive tool to survey
different possible effects based on different assumptions in the future for strategic planning and
decision-support.
As for the simulated results for various scenarios in the case study of Xian, they may not contain the real
one in the future. It is impossible to make those data absolutely be the same as the reality due to complexity of
such a process. However, relative data in different scenarios can clearly draw the patterns of urban growth. In
this study, the most important conclusion is that Xian city can achieve high economic growth at cost of using less
urban land although we cannot certain how much urban land will precisely increase.
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