1 
 
Geographical Modeling: from Characteristic Scale to 
Scaling 
Yanguang Chen 
(Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 
100871, P.R.China. E-mail: chenyg@pku.edu.cn) 
 
Abstract: Geographical research was successfully quantified through the quantitative revolution of 
geography. However, the succeeding theorization of geography encountered insurmountable 
difficulties. The largest obstacle of geography’s theorization lies in scale-free distributions of 
geographical phenomena which exist everywhere. The first paradigm of scientific research is 
mathematical theory. The key of a quantitative measurement and mathematical modeling is to find 
a valid characteristic scale. Unfortunately, for many geographical systems, there is no characteristic 
scale. In this case, the method of scaling should be employed to make a spatial measurement and 
carry out mathematical modeling. The basic idea of scaling is to find a power exponent using the 
double logarithmic linear relation between a variable scale and the corresponding measurement 
results. The exponent is a characteristic parameter which follows a scaleful distribution and can be 
used to characterize the scale-free phenomena. The importance of the scaling analysis in geography 
is becoming more and more evident for scientists. 
 
Key words: Geographical modeling; Spatial analysis; Characteristic scale; Scaling; Scale-free 
distribution 
 
1 Introduction 
The development level of a subject depends heavily on the methods which are applied to this 
subject. The scientific method include two basic paradigms: one is the mathematization of the world 
picture, and the other, experience and experiment (Henry, 2002). Sixty year ago, Einstain (1953) 
once pointed out: “The development of Western science has been based on two great achievements, 

2 
 
the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers，and 
the discovery of the possibility of finding out causal relationships by systematic experiment (at the 
Renaissance).” (see Crombie, 1963, page 142) Formal logical system includes mathematics and is 
related to the relationships between symbols (Bunge, 1962). A consensus of the scientific 
community is that the basic methods of science are mathematical theory and controlled systematic 
experiment. However, geographical systems especially human geographical systems are generally 
uncontrollable so that it is impossible to implement regular systematic experiments for geographical 
research. The experiment method is always replaced by experience of geographers. On the other 
hand, the application of traditional mathematical tools to geography is limited because of spatio-
temporal complexity of geographical systems. 
Mathematical theory is the first paradigm for scientific research. Mathematical methods have two 
basic functions in a discipline: one is to construct postulates and make models for developing 
theories, and the other is to possess and analyze experimental data or observational data. However, 
in quantitative revolution, most geographers preferred the second function to the first function, 
despite the fact that the first function is more important than the second function for development 
of geography (Moss, 1985). In fact, there are three obstacles to mathematical modeling for 
geographical systems, including spatial dimension, time lag, and interaction. All these difficult 
problems are related with scale dependence. The conventional mathematical tools such as calculous, 
linear algebra, probability theory and statistics can be applied to geographical analysis, but the 
conclusions are always inconsistent with the facts: the predicted values of models are often 
significantly different from the observed values, or the causalities cannot be efficiently revealed by 
the mathematical modeling. Some models, including the gravity model and allometric model, are 
useful in practice, but the model parameters cannot be interpreted with Euclidean geometry. 
Because mathematical theory and systematic experiment cannot be efficiently applied to the 
traditional geographical research, geography once became of exceptional discipline that is different 
from the standard science (Schaefer, 1953). The biggest conundrum of geography development lies 
in mathematization of human geographical patterns and processes. Quantitative revolution resulted 
in quantification of human geography, which was regarded as becoming a real science (Stimson, 
2008). Geography made great achievements in quantitative analysis and spatial modeling. However, 
the succeeding theorization of human geography is not really successful and quantitative human 

3 
 
geography seems to turn a full circle (Johnston, 2008; Philo et al, 1998). The consequence of failure 
of geographical theorization is serious, which led to nihilistic mood in geographical circles. 
Atkinson and Martin (2000) once said, “Why, as geographers, would we want to throw away the 
geography?” (page 2) Earlier, Hurst (1985) once declared that “geography has neither existence nor 
future”. In fact, the puzzles of geography boil down to two aspects. First, scale invariance. 
Traditional mathematical methods are based on characteristic scales, but geographical distributions 
are always free of characteristic scales; Second, irreducibility. Modern science is based on 
reductionism (Gallagher and Appenzeller, 1999; Waldrop, 1992), but geographical systems are 
actually irreducible. This paper is devoted to investigating the first aspect. In section 2, the key of 
mathematical modeling for geographical systems is clarified. In section 3, the new idea of 
geographical modeling based on scaling is illustrated. In section 4, the obstacles of geographical 
theorization and the solutions to these problems are discussed. Finally, the article is concluded with 
brief summary of this work. 
2 The key of mathematical modeling 
2.1 Characteristic scales 
The key of key of quantitative measurement and mathematical modeling is to find characteristic 
scales, including characteristic length and characteristic parameters. Lord Kelvin once pointed out: 
“When you can measure what you are speaking about, and express it in numbers, you know 
something about it; but when you cannot measure it, when you cannot express it in numbers, your 
knowledge is of a meager and unsatisfactory kind.” (see Taylor, 1977, page 37) However, not all 
measurements and numbers are valid for geographical analysis. If you want to measure a 
phenomenon or a system and express it in numbers, you must find the characteristic length, which 
represents the key scale of calculation or quantitative analysis. For example, for a circle, the 
characteristic length is its radius; for a square, the side length is its characteristic length; for a matrix, 
the eigenvalues give the characteristic lengths in different directions; for a statistical distribution, 
the average value is associated with the characteristic length. For a geometrical object such as circle, 
if we know the characteristic length, i.e., radius, we can know other geometrical information 
including the circumference and the area within the circle. For a statistical distribution, if we know 

4 
 
the average value, we can compute the standard deviation and covariance, and the structure of 
probability become clear. A statistical inference is always made by means of probability structure 
comprising average value, standard deviation, and covariance. 
Characteristic scales are basis of mathematical modeling and quantitative analysis by 
conventional methods. Generally speaking, a good mathematical model of a geographical system 
includes three levels associated with three scales: the basic level based on the characteristic scale, 
the macro level based on the global/large scale of geographical environment, and the micro level 
based on the local/small scale, which can be expressed with some of coefficients (Chen, 2008; Hao, 
1986). An efficient and valid quantitative analysis is always based on the characteristic scale or the 
characteristic parameter of a model associated with the characteristic scale of the corresponding 
geographical system. For example, the rate parameter of Clark’s model of urban population density 
is just the reciprocal of the characteristic radius (Chen, 2008a; Takayasu, 1990), and the Moran’s 
index of spatial autocorrelation proved to be a characteristic root of a spatial correlation matrix 
based on the spatial weight matrix (Chen, 2013a). However, in many cases, we cannot find valid 
numbers indicating characteristic length for a geographical system, especially for a human 
geographical system. For instance, for the rank-size distribution of cities which follow Zipf’s law, 
we can never find a valid mean to represent the characteristic length. Of course, we can calculate an 
average value, but the mean value depends on sampling results and sample sizes. We cannot find 
the fixed average value for the city-size distribution within a geographical region. Especially, the 
mean value does not represent the characteristic of the most elements in a sample. The calculated 
average value barely has no practical significance for us to explain the rank-size distribution of cities 
and predict urban development. This suggests an important concept: scale-free distributions. There 
is no typical size for a city in any country (Buchanan, 2000). It is meaningless to find the optimal 
city size for a regional system, but in theory, there is an optimal city size distribution (Chen, 2008a). 
In short, there are two types of geographical phenomena, which correspond to two kinds of 
probability distributions. One is the scaleful distribution, which bears a characteristic scale; and the 
other is the scale-free distribution, which bears no characteristic scale (Table 1). The two sorts of 
distributions indicates two types of geographical space: scaleful geo-space and scale-free geo-space. 
The geographical phenomena satisfying a scaleful distribution can be treated with the conventional 
mathematical methods such as the higher mathematics. However, the geographical phenomena 

5 
 
satisfying a scale-free distribution cannot be dealt with using the conventional mathematical tools 
(Figure 1). It needs the method of scaling analysis such as fractal geometry, allometric theory, and 
complex network theory (Batty, 2008; Batty, 2013; West, 2017). 
 
Table 1 A comparison between scaleful distribution example and scale-free distribution example 
Order 
or step 
Scaleful distribution: measurement 
result is independent of 
measuringscale 
Scale-free distribution: measurement 
result depends on measuringscale 
Scale x 
Measurement y0 
Scale x 
Measurement f(x) 
1 
1000 
15 
1000 
9 
2 
100 
15 
100 
16 
3 
10 
15 
10 
28 
4 
1 
15 
1 
50 
 
 
Figure 1. Two types of geographical phenomena corresponding to two types of probability 
distribution 
2.2 Two types of spatial distributions 
After quantitative revolution, geography evolved from a descriptive discipline into a science of 
spatial distributions. In the process of quantitative analysis, a spatial distribution of a geographical 
phenomenon can be converted into a statistical distribution using mathematical language and 
Geographical 
distributions 
Scaleful 
distributions 
Conventional 
mathematical methods 
Singular mathematical 
methods 
Higher 
mathematics, etc 
Fractal 
geometry, etc 
Scale-free 
distributions 
e.g. Gaussian distribution 
e.g. Pareto distribution 

6 
 
observational data. As stated above, statistical distributions fall into two types: scaleful distributions 
and scale-free distribution (Table 2). The typical scaleful distribution is normal distribution, which 
is often termed Gaussian distribution (Figure 2(a)). This is a very simple statistical distribution 
2
2
(
)
2
1
( )
2
x
f x
e






,                               (1) 
where x refers to the scale and f(x) to the probability density, μ denotes the average value of x, and 
σ is standard deviation of x. The mean value μ indicates the characteristic length of the statistical 
distribution, and both the mean value μ and the standard deviation σ define the probabilistic structure 
of the distribution. The representative geographical model is the normal function urban population 
density distribution (Dacey, 1970; Sherratt, 1960; Tanner, 1961). The typical scale-free distribution 
is the Pareto distribution (Figure 2(b)). This is a power-law distribution familiar to many scientistis: 
( )
b
f x
Cx

,                                  (2) 
in which C refers to the proportionality coefficient, and b to the scaling exponent. For Pareto 
distribution, the average value of x is invalid for directly quantitative analysis because it depends on 
the sample size. The urban traffic model of Smeed (1963) indicates an inverse power law 
distribution of road density around a city. 
 
Table 2 A comparison between simple distributions and complex distributions 
Item 
Scaleful distribution 
Scale-free distribution 
Typical case 
Gaussian distribution 
Pareto distribution 
Basic property 
With characteristic scale 
Without characteristic scale 
Probability 
structure 
Can be defined with mean, standard 
deviation, and covariance 
Cannot be defined with mean, 
standard deviation, and covariance 
Quantitative 
criterion 
Characteristic length 
Scaling exponent 
Probability 
curve 
Unimodal curve 
Long-tailed curve 
Geographical 
system 
Simple systems 
Complex systems 
Example 
Urban 
population 
density 
distribution 
City rank-size distribution 
 

7 
 
 
  a. Gaussian distribution                      b. Pareto distribution 
Figure 2. The distribution with characteristic scale and the distribution without characteristic 
scale 
(Note: The first one is the normal distribution with an average value of 15 and a standard deviation of 2. The 
second one is the power-law distribution with a scaling exponent of 1.2.) 
 
If the probability density takes on a unimodal curve, it suggests a distribution with a characteristic 
scale. However, not all the scaleful distributions take on unimodal curves. For example, the 
exponential distribution takes on one-side decay curve with a thin tail rather than a single peak curve, 
but it is a scaleful distribution because the exponential distribution has a valid average value. The 
negative exponential function is as below: 
0
/
0
( )
x x
f x
f e

,                                (3) 
where f0 refers to the constant coefficient, and x0 to the scale parameter associated with a average 
value (Chen, 2008b). The well-known urban density model proposed by Clark (1951) is just a 
negative exponential function. Equation (3) is actually a density distribution function defined in 1-
dimensional space. The curve based on equation (3) has no single peak. However, we can derive a 
unimodal curve defined in a 2-dimensional space from the 1-dimensional density distribution 
function (Chen, 2020). Compared with the power-law distribution, exponential distribution is 
simple (Barabasi and Bonabeau, 2003), while compared with the exponential distribution, the 
normal distribution is very simple (Goldenfeld and Kadanoff, 1999). In short, if a statistical 
distribution bears an average value indicating a characteristic scale, it is simple; while if a 
distribution has no valid mean indicative of characteristic scale, it is a complex distribution. Simple 
0.0
0.1
0.2
0.3
0
10
20
30
f(x)
x
0.0
0.1
0.2
0.3
0.4
0
10
20
30
f(x)
x

8 
 
statistical distributions suggest simple geographical spatial or size distributions, while complex 
statistical distributions correspond to complex geographical spatial or size distributions. The simple 
distributions can be modeled with traditional mathematical methods. However, the complex 
distributions cannot be efficiently analyzed with higher mathematics. 
3 New approach to mathematical modeling 
3.1 Scaling and mathematical modeling 
As indicated above, the key to quantitative measurement and mathematical modeling is the 
characteristic scale. If and only if we find the characteristic scale for a system through observational 
data, we can make an efficient mathematical model. For examples, for the normal distribution, 
equation (1), the characteristic scale is x=μ; for the exponential distribution, equation (3), the 
characteristic scale is x=x0. However, for a geographical system such as a city or a network of cities, 
we cannot find the characteristic scale in most cases. In other words, no characteristic length can be 
measured and expressed in numbers. For instance, for the power-law distribution, equation (2), no 
characteristic scale can found for x. This suggests that the traditional mathematical tools including 
calculous, linear algebra, probability theory and statistics cannot be directly and efficiently applied 
to modeling the systems without characteristic scales. In this instance, we need new notion of 
mathematical modeling for geographical systems and the characteristic scale should be replaced 
with scaling.  
In scaling analysis, characteristic lengths are often replaced by scaling exponents. In order to 
show how to substitute scaling for characteristic scale, let’s see a very simple example 
d
A
r


,                                   (4) 
where r is scale variable, A is the corresponding measurement, d is a power exponent. If d=2, we 
will have a formula for the area of a circle, A=πr2, in which π≈3.1416. The radius r can be determined 
and it is just the characteristic length, which can give the area and perimeter of the circle. On the 
other hand, the power exponent d=2 is known and shows no useful information for the geometrical 
object. However, for urban form, the situation is different, and equation (4) should be rewritten as 
0
D
A
A r

,                                   (5) 

9 
 
in which the coefficient A0 is not necessarily equal to 3.1416, and D is not equal to 2. In both theory 
and empirical work, we cannot find a certain radius for a city. Changing the radius r, we will have 
different urban area and perimeter. In other words, the urban measurements depend on the scale 
which is adopted. In this case, the radius cannot act as the characteristic length for the urban studies, 
but the power exponent can be used as a characteristic parameter, which is termed fractal dimension 
of urban form (Batty and Longley, 1994; Frankhauser, 1994; Frankhauser, 1998). Based on the 2-
dimensional digital maps, the fractal dimension values come between 0 and 2. A low value (D<1) 
and high value (D→2) rarely appear, and most results are near 1.7 (Batty, 1991; Batty and Longley, 
1994; Chen, 2010a). This suggest that a city’s radius has no characteristic length (no valid fixed 
radius), but the scaling exponent based on the variable radius is constant and reflects spatial 
characteristics. The dual relationships between characteristic scale and scaling can be expressed as 
follows 
2
0
,  is  a characteristic length
( )
( )
,  is  a  characteristic parameter
d
D
A
r
r
A r
r
A r
A r
D









,          (6) 
This shows the similarities and differences between Euclidean geometry and fractal geometry, each 
going it own way. 
Nowadays, fractal geometry, allometric theory, and complex network theory have broken a new 
path to mathematical modeling of geographical systems. All these theories can be integrated into a 
new framework by the scaling concept (Batty, 2008; Chen, 2013b). We can compute the fractal 
parameters, or the scaling exponents, which represent new spatial characteristic quantities and 
reflect spatial characteristics of geographical systems from new perspective. In other words, a fractal 
has no characteristic length, but its fractional dimension has characteristic length. Analogously, a 
complex geographical system has no characteristic length, but its scaling exponents have 
characteristic length. 
3.2 Fractal geometry for geographical spatial analysis 
Among all the theory or mathematical methods based on scaling idea, fractal geometry is the 
most important one for geographical analysis. First, fractal geometry is a mathematical tool, which 
belongs to the first paradigm of scientific methods. The first paradigm is the most significant 
paradigm for scientific research. Second, fractal modeling can combine both numbers and patterns 

10 
 
in the best way. It makes a good link between observational data and geographical patterns. Third, 
fractal measurement depends heavily on fractional dimension. The fractal dimension is just a spatial 
parameter and is powerful for geo-spatial analysis. A fractal includes three elements: form, chance, 
and dimension (Mandelbrot, 1977; Mandelbrot, 1982). These elements correspond to patterns, 
processes, and information of geographical systems (Table 3). 
 
Table 3 The relationships between fractals and geographical research 
Fractal 
Geographical systems 
Geographical analysis 
Form 
Geographical patterns 
Spatial distribution 
Chance 
Geographical process 
Dynamical evolution 
Dimension 
Geographical information 
Spatio-temporal analysis 
 
The famous difficult problems of mathematical modeling include spatial variables, time lag, and 
interaction. First, spatial variables always result in dimension puzzle of quantitative analysis. The 
well-known economist Arthur (1992) once said: “Only we don’t usually consider the spatial 
dimension in economics much, so that makes economics a lot simpler.” (see Waldrop, 1992, page 
141) Unfortunately, geographers cannot evade and must face the problem of spatial dimension. 
Second, time lag always leads to nonlinearity. Where there is a time lag, there is a responding delay 
indicating nonlinear problem, which, generally speaking, cannot be solved using traditional 
mathematical methods (Chen, 2009). Third, interaction between multiple objects leads to complex 
dynamics. Interaction is one of the scientific difficult problems in the 20th century (Liu et al, 2003). 
All these conundrums are associated with scale dependence. Scale dependence suggests scale-free 
distributions, in which no characteristic length can be found directly by spatial measurements. 
Fractal geometry is one of the important and useful tools for solving the three problems 
abovementioned. That is, fractal geometry can be used to deal with dimensional relations, nonlinear 
processes, and scale-free distributions. In this sense, fractal geometry provides an efficient tool for 
geographical spatial modeling. 
4 Geography: on the threshold of theoretical revolution? 
4.1 The shortcomings of higher mathematics 
In the conventional scientific research, the basic tools of mathematical modeling are the higher 

11 
 
mathematics, including calculus, linear algebra, and probability theory and statistics. However, 
these mathematical methods cannot meet the needs of geographical research effectively. In fact, the 
higher mathematical methods are often incompatible with geographical analyses in nature (Table 4). 
First, calculus theory is based on regular geometry. However, geographical patterns need irregular 
geometry. Second, linear algebra is based on linear superposition principle. However, geographical 
process bears nonlinearity, which violates superposition property. Third, probability theory and 
statistics are based on the distributions with characteristic scales. However, geographical 
distributions are always free of scale. That is to say, geographical distributions are of scale 
invariance and follow scaling laws. Because of these insufficiencies of the traditional mathematical 
methods, the ‘theoretical revolution’ of geography after its ‘quantitative revolution’ is unsuccessful. 
In other words, geography has succeeded in quantification, but failed to achieve its goal of 
theorization (Philo et al, 1998). 
 
Table 4 The inconsistency between the higher mathematical methods and geographical analyses 
Mathematics 
Mathematical base 
Geographical 
phenomenon 
Calculus 
Regular geometry 
Irregular patterns 
Linear algebra 
Linear 
superposition 
principle 
Nonlinear processes 
Probability 
theory 
and 
statistics 
Scaleful distributions 
Scale-free distributions 
 
Many geographical phenomena cannot be described with the traditional higher mathematics. 
Applying a mathematical method based on characteristic scale to the geographical patterns and 
processes without characteristic scales always results in problems of oversimplification. This 
phenomenon indicates the spherical chicken syndrome (Lederman and Teresi, 1993; Kaye, 1989), 
or even what is called the “‘gene for’ syndrome” (Gallagher and Appenzeller, 1999). Recent years, 
a number of new mathematical tools have emerged, including fractal geometry, allometric scaling, 
complex network, renormalization group, wavelet analysis, artificial neural network, genetic 
algorithm, and chaotic dynamics. Among all these new mathematical methods based on the idea 
from scaling, fractal geometry is the most important and powerful tool for geographical analysis. 
The reasons are as below: First, fractal geometry can be employed to characterize the irregular 

12 
 
patterns of geographical systems. Second, fractal geometry can be used to explore the nonlinear 
processes of geographical evolution. Third, fractal geometry can be adopted to describe the scale-
free distributions of geographical phenomena. In short, fractal geometry is the most effective tool 
to explore nonlinearity, singularity and irregularity. This geometric tool can remedy the defects of 
the old higher mathematics where geographical research is concerned. 
4.2 Reinterpreting traditional geographical models using new ideas 
For geography, the achievements of quantitative revolution are very great. However, due to the 
confusion between the concept of characteristic scales and that of scaling, the value of many 
achievements has been buried for a long time. The essence of a theory rests with models, especially 
mathematical models (Holland, 1998). Geographers presented many important theories and made 
significant models during the period of quantitative revolution (e.g. see Haggett et al, 1977). 
Unfortunately, it used to be hard to develop further parts of these theories and models, some of 
which were even abandoned for a time not because they are not good in practice, but because they 
could not be interpreted with the notions of traditional mathematics. However, the inexplainable 
theories and models can be explained using the concepts from fractals and scaling. In a sense, a 
number of theoretical puzzles in human geography are dimensional conundrums, which cannot be 
solved with Euclidean geometry. The simple and typical examples are the law of allometric growth 
and the gravity model. The allometric scaling relation between urban area and population is as below: 
a
p / D
D
b
aP
aP
A


,                               (7) 
where A refers to urban area (the corresponding dimension is Da), P to city population size within 
this area (the corresponding dimension is Dp), a denotes the proportionality coefficient, and b=Dp/Da 
is the allometric scaling exponent. The value of the scaling exponent is a well-known puzzle of 
traditional human geography. The dimension of urban area used to be regarded as Da=d=2. 
According to the principle of dimension consistency based on the idea of Euclidean geometry, if the 
dimension of city population is Dp=3, then b=2/3; if Dp=2 as given, then b=1 (Lee, 1989; Longley 
et al, 1991; Nordbeck, 1971). In short, the predicted value of b by the traditional theory is either 2/3 
or 1. However, the observed value of b always come between 2/3 and 1 rather than either 2/3 or 1 
(Chen, 2008a). The conflict between the theoretical value and the calculated value cannot be 

13 
 
interpreted by Euclidean geometry, but it can be readily construed by fractal geometry (Batty and 
Longley, 1994; Longley et al, 1991; Chen, 2010b; Chen and Xu, 1999). The gravity model is 
familiar to geographers and the basic and classical form is 

ij
j
i
ij
r
P
P
G
I 
,                                  (8) 
where Iij denotes the gravitation between places i and j, rij is the distance between the two places, Pi 
and Pj indicates the “masses” (size measurements) of places i and j, respectively, G refers to the 
gravity coefficient, and α to the distance-decay exponent. The exponent α cannot be interpreted with 
the dimensional concept based on Euclidean geometry so that the power-law impedance function 
was replaced by the exponential impedance function (Haggett et al, 1977; Haynes, 1975). However, 
the spatial interaction model based on the exponential function brought on a new problem, that is, 
the gravity model is based on the concept of action at a distance, but the exponential function 
suggests an effect of spatial localization instead of action at a distance (Chen, 2008b). This problem 
can also be solved by the concepts of fractal dimension (Chen, 2009; Chen, 2015). 
The ideas from fractals and scaling can be used to reinterpret and develop many classical theories 
and models. These years, a number of important geographical theories and models have been 
improved or reinterpreted by using the concepts from fractals, including central place theory 
(Arlinghaus, 1985; Arlinghaus and Arlinghaus, 1989; Batty and Longley, 1994; Chen, 2011; Chen, 
2014; Chen and Zhou, 2006), spatial interaction models (Chen, 2015), and spatial autocorrelation 
analysis (Chen, 2013a). Twenty year ago, Batty (1992) once observed: “Many of our theories in 
physical and human geography are being reinterpreted using ideas from fractals and tomorrow they 
will become as much a part of our education and experience as maps and statistics are today.” (page 
36) The above prediction has been being supported or even confirmed over and over again. 
4.3 New tools for geographical research 
Scientific research is nothing more than two processes: description and understanding. First, 
describe the characteristics of a system, and then try to understand its work mechanism (Gordon, 
2005). In order to describe a phenomenon precisely, we need mathematics and measurements, and 
in order to understand the mechanism deeply, we need laboratory experiments and computer 
simulation (Henry, 2002; Waldrop, 1992). Effective geographic spatio-temporal description and 

14 
 
understanding can lead to effective interpretation and prediction. As Fotheringham and O’Kelly 
(1989, page 2) once pointed out: “all mathematical modelling can have two major, sometimes 
contradictory, aims: explanation and prediction.” In fact, the main functions of science lie in 
explanation and prediction (Kac, 1969).  
The traditional paradigms of scientific research include mathematical theory (originated in 
ancient Greece) and laboratory experiment (originated in the Renaissance). The third important 
paradigm is computer simulation originated after World War II (Bak, 1996). Before the turn of the 
century, “there were three ways now to proceed in science: mathematical theory, laboratory 
experiment, and computer modeling” (Waldrop, 1992, page 268). Today, scientists are telling us the 
fourth paradigm has been emerging, and the paradigm can be termed data-intensive computing (Bell 
et al, 2009; Hey et al, 2009). Today, scientific paradigms have developed, including mathematical 
theory, experience and (laboratory) experiment, computer simulation, and data-intensive computing. 
In each paradigm of scientific methodology, we can find new tools for geographical research and 
spatial analysis (Table 5). Of course, laboratory experiments are still an exception. However, we can 
carry out on-the-spot investigation in light of the new way of geographical thinking so as to make 
up for the lack of geographical research which cannot be made by laboratory experiment. 
 
Table 5 The four paradigms in science and their applications to geographical research 
Type 
Paradigm 
Function 
Geographical 
research 
New 
tools 
for 
geography 
The first 
paradigm 
Mathematical 
theory 
Data 
processing and 
theoretical 
modeling 
Processing 
observational data 
fractal 
geometry, 
nonlinear 
mathematical 
methods, etc. 
The 
second 
paradigm 
Laboratory 
experiment 
and 
experience 
Finding 
out 
causal 
relationships 
Geographical systems 
are uncontrollable and 
the 
laboratory 
experiment is replaced 
by experience 
field 
investigation 
based on new thinking 
from 
fractals 
and 
scaling 
The third 
paradigm 
Computer 
simulation 
Finding 
out 
causal 
relationships 
and computer 
aided modeling 
It can be used to make 
up for the deficiency of 
laboratory experiment 
cellular 
automata 
(CA), multiple agent 
system (MAS), etc. 
The four Data-
Big 
data Processing big data 
scaling 
analysis, 

15 
 
paradigm 
intensive 
computing 
processing 
including 
allometry 
and complex network 
 
5 Conclusions 
Due to the shortage of methodology, the theoretical development of geography lags behind for a 
long time. However, things will change in the near future. Now, the barriers of theorization of 
geography are being partially removed, and to my thinking, geography may be on the threshold of 
theoretical revolution. The main viewpoints of this work can be summarized as follows. First, 
fractal geometry and a number of mathematical methods based on the ideas of scaling provide 
new tools for geographical description. A great many geographical phenomena have no 
characteristic scales, and cannot be characterized by means of conventional mathematical methods. 
Replacing characteristic length with scaling exponents, we can make spatial analyses of scale-free 
geographical systems from a new perspective. In particular, fractal geometry is the most important 
and powerful mathematical tool for geographical analysis among various new mathematical 
methods. The main difficulties of geographical mathematical modeling include spatial dimension, 
time lag, and interaction. Fractal geometry is one of the significant and useful tools for solving these 
problems. Effective description leads to effective understanding, which in turn results effective 
explanation and prediction. Second, computer modeling and simulation can be used to explore 
the causalities in geographical spatio-temporal processes. The geographic systems are 
uncontrollable, so we can’t analyze its evolution mechanisms with the help of systematic controlled 
experiments. Using computer simulation, we can find out the causal relationships hidden behind 
geographical spatial patterns and evolution processes. Computer simulation must be closely 
combined with mathematical modeling. Without valid theoretical modeling, computer simulation is 
blind, while without computer simulation, mathematical models may become lame. The traditional 
thinking of geographical mathematical modeling is based on the characteristic scales, and the 
corresponding computer simulation is also based on characteristic scale notion. In future, the idea 
of scaling should be introduced into computer simulation of geographical evolution. Third, the 
ideas and theories based on scaling can be employed to improve old theory and develop new 
theory in geography. As indicated above, many theories in physical and human geography can be 
reinterpreted and improved using ideas from fractals and scaling. Especially, new models can be 

16 
 
built and new theories can be developed by integrating ideas based on characteristic scales and those 
based on scaling. Complex geographic systems have two sides of unity of opposites: some 
phenomena bear characteristic scales, and some phenomena have no characteristic scales. In order 
to understand different geographical processes, we must adopt proper mathematical methods to 
describe different geographical phenomena. Traditional mathematical tools are not suitable for 
scaleful geographical phenomena, just as scaling analysis cannot be used for scale-free geographical 
phenomena. Only when the methods are used properly can good results be achieved in geographical 
modeling and analyses.  
Acknowledgements 
This research was sponsored by the National Natural Science Foundations of China (Grant No. 
41671167). The support is gratefully acknowledged. 
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