Computers, Environment and Urban Systems 38 (2013) 1–10

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Computers, Environment and Urban Systems

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Multifractal analysis of axial maps applied to the study of urbanmorphology

0198-9715/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compenvurbsys.2012.11.001

⇑ Corresponding author. Tel.: +34 957 218400; fax: +34 957 218455.E-mail addresses: g82arvia@uco.es (A.B. Ariza-Villaverde), fjhornero@uco.es (F.J.

Jiménez-Hornero), eduardo@uco.es (E.G.D. Ravé).

Ana B. Ariza-Villaverde, Francisco J. Jiménez-Hornero ⇑, Eduardo Gutiérrez De RavéDepartment of Graphic Engineering and Geomatics, University of Cordoba, Gregor Mendel Building (3rd Floor), Campus Rabanales, 14071 Córdoba, Spain

a r t i c l e i n f o

Article history:Received 8 March 2012Received in revised form 13 November 2012Accepted 13 November 2012Available online 11 January 2013

Keywords:Urban morphologyAxial mapsMultifractal analysisSandbox methodLacunarity

a b s t r a c t

Street layout is an important element of urban morphology that informs on urban patterns influenced bycity growing through the years under different planning regulations and different socio-economic con-texts. It is assumed by several authors that urban morphology has a fractal or monofractal nature. How-ever, not all the urban patterns can be considered as monofractal because of the presence of differentmorphologies. Therefore, a single fractal dimension may not always be enough to describe urban mor-phology. In this sense, a multifractal approach serves to tackle this problem by describing urban areasin terms of a set of fractal dimensions. With this aim in mind, two different neighbourhoods of the cityof Cordoba, in Andalusia (Spain), are analysed by using the Sandbox multifractal method and lacunarity.We analyse the street patterns represented by axial maps and obtained from the Space Syntax algorithm.The results suggest that the Rényi dimension spectrum is superior to a single fractal dimension todescribe the urban morphology of Cordoba, given the presence of regular and irregular street layoutsestablished under different planning and socio-economic regimes.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Urban spaces are commonly defined by street layout, which isan important component of urban morphology, as determined bythe distribution of buildings on a certain site. Street patterns havebeen classified as irregular, radial-concentric, rectangular and grid(Dickinson, 1961); star, satellite, linear, rectangular grid, other grid,baroque network and lacework (Lynch, 1981); regular, concentricand irregular (DTLR & CABE, 2001); and by over 100 additionaldescriptors for streets outlines, including radial, grid, tree and lin-ear (Marshall, 2005). City morphology is classified according to theway streets are arranged; over time, several urban morphologies ina major community are influenced by various planning regulationsand socio-economic conditions. Urban morphology provides infor-mation about the structural characteristics of a city; it provides in-sight into the structural origins of and impacts of historical changeon the chronological processes concerning construction and recon-struction of a city.

An understanding of urban morphology facilitates the projec-tion of future growth in municipal site. Therefore, different scien-tific approaches have focused on the development of statisticalmodels, such as the Geographical Information System (GIS)-basedapproach (Allen & Lu, 2003; Xiao et al., 2006), that can be usedto estimate future urban growth and obtain information about city

sustainability (Czerkauer-Yamu & Frankhauser, 2010; Van Diepen& Henk, 2003). By utilising graphic tools to quantify and definecharacteristics of urban morphology, Hillier and Hanson (1984)reproduce and simplify the spatial composition of a city by deter-mining the minimum number of axial lines required to cover allareas of an urban lattice; this approach graphically and accuratelyrepresents street networks through axial maps. This procedure hasbeen successfully applied to city modelling (Jiang & Claramunt,2002), urban design (Jeong & Ban, 2011), spatial distribution of ur-ban pollution (Croxford, Penn, & Hillier, 1996), prediction of hu-man movement (Jiang, 2007) and road network analysis (Duan &Wang, 2009; Hu, Jiang, Wang, & Wu, 2009).

Urban morphology has also been described by several authors asexhibiting a fractal nature (Batty, 2008; Batty & Longley, 1987,1994; Benguigui, Czamanski, Marinov, & Portugali, 2000; De Keers-maecker, Frankhauser, & Thomas, 2003; Feng & Chen, 2010; Fran-khauser, 1998). Based on this notion, the morphology of a city canbe described by a single fractal dimension. Milne (1990) argues thatnot all landscapes are fractal due to their contemporary patterns be-cause several past processes have individually influenced their com-plexity. Therefore, a single fractal dimension may not alwayssufficiently describe an urban area due to the presence of heteroge-neous fractal properties, as Encarnação, Gaudiano, Santos, Tenedó-rio, and Pacheco (2012) suggests. The multifractal analysisproposed in this research solves this dilemma by obtaining general-ised fractal dimensions, or Rényi spectra, to describe data. The re-search is a meaningful contribution to the study of urban patternsbecause the multifractal approach transforms irregular data into a

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compact form and amplifies small differences between variables(Folorunso, Puente, Rolston, & Pinzon, 1994). In addition, a multi-fractal framework exhibits a significant advantage over traditionalmethods through the integration of variability and scaling analyses(Kravchenko, Boast, & Bullock, 1999; Lee, 2002; Stanley & Meakin,1988).

Multifractal analyses have been successfully applied to thestudy of river network morphology with significant results (DeBartolo, Gaudio, & Gabriele, 2004; De Bartolo, Grabiele, & Gaudio,2000; De Bartolo, Veltri, & Primavera, 2006; Rinaldo, Rodriguez-Iturbe, Rigon, Ijjasz-Vasquez, & Bras, 1993). De Bartolo et al.(2004) has applied the Sandbox method to overcome limitationsdetected in the most frequently used multifractal fixed-sized algo-rithm, the Box-Counting algorithm (Mach, Mas, & Sagues, 1995), todescribe river networks. River networks and streets have commonproperties such as hierarchy and critical self-organisation (Chen,2009). Based on this context, this research proposes to describe ur-ban patterns by using the multifractal Sandbox method. Appar-ently, it is the first study that characterises different urban areasto obtain a set of fractal dimensions. Two neighbourhoods in thecity of Cordoba (Andalusia, southern Spain) are analysed. Thestreet networks are defined by the Space Syntax algorithm (Hillier& Hanson, 1984). To complete the multifractal analysis, a lacuna-rity study is conducted. Lacunarity describes the texture appear-ance of an image, on different scales, based on the distribution ofgaps or holes. It is applied to real data sets that may have a fractalor multifractal nature (Plotnick, Gardner, Hargrove, Prestegaard, &Perlmutter, 1996). This analysis is frequently utilised to developurban spatial configurations (Alves Junior & Barros Filho, 2005;Amorim, Barros Filho, & Cruz, 2009; Myint & Lam, 2005) and seg-regation; Wu & Sui, 2001).

The paper is organised as follows: Section 2 introduces the SpaceSyntax algorithm and the multifractal theory that is required tomodel and describe urban patterns. Section 3 details the study areasand discusses the main results of a multifractal analysis. Section 4includes conclusions and recommendations for further research.

2. Combination of axial maps and multifractal analysis todescribe urban morphology

2.1. Axial maps

Space Syntax is a set of tools that describes spatial configurationthrough connectivity lines that cover all areas of a plane. This set oflines comprises an axial map (Peponis, Wineman, Bafna, Rashid, &Kim, 1998; Turner, Penn, & Hillier, 2005), which is one of the pri-mary tools of Space Syntax. According to Turner (2006), an axialmap is an abstraction of the space between buildings, which is de-picted by straight lines and drawn according to a formal algorithm.The lines represent edges and intersections of lines represent junc-tions or connections between edges. Based on the algorithm, theaxial map is the minimal set of axial lines that are linked in sucha way that they completely cover the space.

Several authors have developed different computer programsfor the construction of axial maps based on the initial proposal of

Fig. 1. Different ways to define an axial line: (a) convex–convex ve

Hillier and Hanson (1984). Some programs have been imple-mented in GIS, such as Axman, which was created by Nick Daltonat University College in London (Major, Penn, & Hillier, 1997) andAxwoman, which was developed by Jiang, Claramunt, and Batty(1999). Both are used to draw axial lines with a computer and ana-lyse axial maps of urban and interior spaces; the main differencebetween the programs is that Axman is a Mac-based applicationand Axwoman is a Windows-based application. Other softwarehave been designed to generate axial maps automatically. Turneret al. (2005) introduced a universal-platform software, calledDepthmap, to perform a set of spatial network analyses designedto explain social processes within a built environment. AxialGen,which was developed by Jiang and Liu (2009), is a research proto-type for automatically generating axial maps to demonstrate thataxial lines constitute a true skeleton of an urban space. Althoughit is a good approximation of the axial map proposed by Hillierand Hanson (1984), this prototype has not been sufficiently tested.This research, which follows the method proposed by Turner et al.(2005), extracts axial maps that successfully demonstrate andimplement the algorithm proposed by Hillier and Hanson (1984)by obtaining accurate results with Depthmap. To translate formallanguage into algorithmic implementation, the definition of axialmaps given by Hillier and Hanson (1984) was clarified and rewrit-ten as ‘‘An axial map is the minimal set of lines such that the settaken together fully surveils the system and every axial line thatmay connect two otherwise unconnected lines is included’’. To cre-ate the axial map, Turner et al. (2005) establish two conditions: (i)the reduction from the all-lines map to a unique minimal axialgraph of the system and (ii) the ability to surveil the entire systemthrough axial lines and the preservation of topological rings.

Any possible axial lines are calculated from a map in which allstreets and blocks (polygons) are displayed. Every axial line is de-fined by joining two intervisible vertices. There are three differentpossibilities (Fig. 1): (i) both of the vertices are convex, (ii) one ver-tex is convex and one is concave or (iii) both vertices are concave.The reduction to a single minimal axial-line map is based on therule that if any line connects to a line, its neighbours do not jointhe two lines; the single line is retained or removed. When twolines have the same connection, the longest line is chosen andthe other one is removed. A second condition needs to be appliedto obtain the preservation of topological rings and surveillance ofthe entire system. To surveil the system, the algorithm choosesthose lines from which every point of the system is visible. There-fore, this new condition is added to the last criterion. To completethe topological rings, the algorithm executes a triangulationaround a polygon edge so that it is visible from the three axial linesthat are around the geometry (Fig. 2).

This research describes urban morphology as studied throughaxial maps and obtained with Depthmap software for two areasof Cordoba in Andalusia, which is in southern Spain (Fig. 3). Thiscity is situated at 37�500440 0 latitude and 04�500230 0 longitude; itsaverage elevation is 123 m above sea level. Cordoba, which islocated in the Guadalquivir River Valley, has an area of 1245 km2

and a population of 329,723 inhabitants (data provided by theStatistic National Institute). Cordoba was founded by the Romans

rtex; (b) convex–concave vertex; (c) concave–concave vertex.

Fig. 2. Example of a topological ring.

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in the first half of the 2nd century BC because of its privilegedgeographical location. The city has been walled since its founding.Its fortification was expanded when the Arabs conquered Cordoba.With the arrival of Christians to the city, the wall was enlargedfurther. Due to population growth, Cordoba began to expand be-yond its walls in 1905. Today, the city centre is situated in the areathat was previously walled.

Two neighbourhoods of Cordoba are studied, through a combi-nation of axial maps, to understand their urban morphology (Fig. 4)and multifractal analysis. These areas were chosen because theyexhibit similar densities of axial lines (Table 1).

2.2. Multifractal analysis

2.2.1. Sandbox methodThe Sandbox method introduced by Tél, Fülöp, and Vicsek

(1989) and developed by Vicsek (1990) and Vicsek, Family, and

Fig. 3. Location of the

Meakin (1990) allows the complete reconstruction of the multi-fractal spectrum for both positive and negative moment orders,q. It overcomes limitations of the Box-Counting algorithm, whosemain disadvantage is the incorrect determination of the fractaldimensions for negative moment orders (De Bartolo et al., 2004).

A Sandbox algorithm involves randomly selecting N points thatbelong to the street network and then counting, for each point i,the number of pixels Mi(R) that belong to the street network insidea region of a given radius R (a circle), which is centred at a specifiedpoint. The advantage of this method is that the circles are centredon the structure such that there are no boxes with few elementsinside (pixels), thereby avoiding border effects (Fig. 5).

By choosing arbitrary points as centres, the average value of themass and their qth moments over randomly distributed centrescan be computed as h½MðRÞ�qi, with q as the probability momentorder. Thus,

Xi

Mi

M0

� �q�1 Mi

M0/ R

L

� �ðq�1ÞDq

ð1Þ

where M0 stands for the total mass of the cluster or lattice mass,and L is the lattice size equal to 1 after normalisation. Based onFalconer (1990), this normalisation does not modify the measurebecause it is a geometrically invariant transformation. Consideringthe ratio Mi=M0 as a probability distribution on an approximatingfractal, the following averaged expression can be derived as:

MðRÞM0

� �q�1* +

/ RL

� �ðq�1ÞDq

ð2Þ

According to Eq. (2), the selection of the centres must be uni-form over the approximating fractal. The generalised fractaldimension Dq of moment order q is defined as (Tél et al., 1989):

DqðR=LÞ ¼ 1q� 1

limR=L!0

lnh½MðRÞ=M0�q�1ilnðR=LÞ ; for q–1 ð3Þ

De Bartolo et al. (2004) determined the solution for Dq whenq = 1 in the Taylor’s expansion around

city of Cordoba.

Fig. 4. Areas selected for this work.

Table 1Features and statistical analysis of the different axial maps.

Axial lines Axial lines length

Number of lines Area (km2) Number of lines/km2 Average CV Skewness Kurtosis

Area 1 169 1.009 167.55 146.06 0.604 1.766 0.371Area 2 56 0.343 163.40 267.849 0.615 0.768 0.628

4 A.B. Ariza-Villaverde et al. / Computers, Environment and Urban Systems 38 (2013) 1–10

D1ðR=LÞ ¼ limR=L!0

hln½MðRÞ=M0�ilnðR=LÞ ð4Þ

The generalised dimensions can be obtained through leastsquares linear regression as the slope of the scaling curveslnh½MðRÞ=M0�q�1i versus lnðR=LÞ for q – 1, and hln½MðRÞ=M0�i versuslnðR=LÞ for q = 1, with lnðR=LÞupper and lnðR=LÞlower as the inner andouter cut-off lengths, respectively. D0 is the fractal dimension, alsoknown as the capacity dimension, of the set over which the mea-sure is performed; D1 is the information dimension and D2 is thecorrelation fractal dimension (Grassberger, 1983; Grassberger &Procaccia, 1983). Dq is a decreasing function with respect to q fora multifractally distributed measure (Caniego, Martin, & San Jose,2003).

The relationship between the spectrum of the generalised frac-tal dimensions Dq and the multi-fractal spectrum f ðaÞwith a as theLipschitz–Hölder exponent, is given through the sequence of massexponents sq (Hentschel & Procaccia, 1983), which is a functionconnecting the moments of probability to the radius length ofthe covering regions, as given by the expression

sq ¼ ðq� 1ÞDq ð5Þ

The mass exponent sequences are interpolated with fifth-orderpolynomials to obtain the multifractal spectrum f ðaÞ by means ofthe Legendre transform, which is defined by the relations (Halsey,Jensen, Kadanoff, Procaccia, & Shraiman, 1986):

aq ¼ �dsq=dq

f ðaqÞ ¼ qaq þ sqð6Þ

The Lipschitz–Hölder, or singularity exponent aq, quantifies thestrength of the measure singularities. f ðaqÞ is an inverted parabolafor multifractally distributed measures, with a wider range of avalues when the heterogeneity of the distribution increases. Formonofractally distributed measures, a is equivalent for all regionsof the same size and the multifractal spectrum consists of a single

point (Kravchenko et al., 1999). The highest value of the multifrac-tal spectrum f ða0Þ corresponds to the fractal dimension D0. Bothprovide information about the degree of filled space.

2.2.2. LacunarityThe lacunarity concept was introduced by Mandelbrot (1982) to

differentiate texture patterns that may have the same fractaldimension but are different in appearance. It can be used with bothbinary and quantitative data in one, two and three dimensions. It isa measure of the gaps or holes in a space distribution. Thus, lacu-narity measures the spatial heterogeneity of pixels in an image. Ifan image has a large amount of gaps, it has a high lacunarity; incontrast, if an image has a translational invariance, it has a lowlacunarity. Several algorithms have been proposed to measure thisproperty (Gefen, Meir, & Aharony, 1983; Lin & Yang, 1986;Mandelbrot, 1982).

Regarding this issue, the current research is based on the use ofthe ‘‘Gliding Box’’ algorithm proposed by Allain and Cloitre (1991)and popularised by Plotonick, Gardner, and ÓNeill (1993). Thismethod consists of a square box of size r2 that slides over a spaceof total size M; the mass S is counted inside of the box at each pointin the sliding process. Beginning with the first pixel located in thefirst column and row of images, the box slices along each pixel ofthe image. In this research, the images are binary, thus, the masscorresponds to the number of pixels that fill the space, whose valueis 1. The process is repeated for each new box size until the box sizeequals the image size. The frequency distribution of the box massesis n(S, r). This frequency distribution is converted into a probabilitydistribution Q(S, r) by dividing each frequency value by the totalnumber of boxes for each size N(r). Then, the first and secondmoment of the distribution are defined as:

Zð1Þ ¼X

SQðS; rÞ

Zð2Þ ¼X

S2QðS; rÞð7Þ

The lacunarity is defined by the relationship between thesemoments as:

Fig. 5. Random distribution circles of radius R on the street network.

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KðrÞ ¼ Zð2Þ=Z2ð1Þ ð8Þ

The first moment can be described by the mean E(S) and thesecond moment can be described by the variance Var(S) of themasses as follows:

Zð1Þ ¼ EðSÞZð2Þ ¼ VarðSÞ þ E2ðSÞ

ð9Þ

Therefore, the lacunarity for each box size can be defined by

KðrÞ ¼ 1þ ðVarðSÞ=E2ðSÞÞ ð10Þ

This approach uses a measure based on a multiscale analysisthat is dependent on scale. In this research, lacunarity is calculatedin an area of 256 � 256 pixels located in the image centre to avoidborder effects (Fig. 6). For each area, the lacunarity is calculated forbox sizes ranging from r = 1 to r = 256 in multiples of 2.

3. Case study: multifractal analysis application to twoneighbourhoods in the city of Cordoba (Spain)

3.1. Description of study areas

Two neighbourhoods in the city of Cordoba, Spain, are studied.Area 1 presents an irregular morphology as illustrated by Fig. 7,which shows the corresponding axial map. This irregularity is aconsequence of its unique history (i.e., Islamic heritage combinedwith centralised administrative influences). Today, this area exhib-its typical Mudejar urban morphology, which is characterised byrectilinear main avenues and tortuous secondary streets. Thisneighbourhood reflects Roman, Arab and Christian settlementsand is located in the city centre; it is the main business area forthe community. Area 2 is a residential neighbourhood; it consistsof houses that were built in the 1950s and are based on a regularpattern (Fig. 7) that corresponds to modern planning and socio-economic requirements.

Fig. 6. Areas where lacunarity is calculated.

Fig. 7. Axial maps of the study areas.

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Table 1 shows the number of axial lines, density of axial linesper area and statistical parameters of the axial line length. Thesedata are extracted from the axial maps that were generated forboth areas. Although Area 1 has a slightly higher density of axial

lines, it displays shorter axial lines than Area 2. Table 2 showsthe statistical analysis of the block size for each study area. Theaverage block size has been normalised over the surface of eachzone to obtain a suitable comparison between the study areas. This

Table 2Block statistical analysis.

Numbers of blocks Average (km2/km2) Maximum Minimum CV

Area 1 105 7.20E�03 2.70E�02 2.75E�04 0.78Area 2 69 7.80E�03 6.33E�03 9.55E�04 0.42

Table 3Multifractal parameters obtained from applying the Sandbox method.

D0 D1 D2 R/Llower R/Lupper R2q¼0

Area 1 1.7654 1.7371 1.7168 �4.8 �2.2 0.9995Area 2 1.7575 1.7562 1.7543 �4.5 �2.5 0.9990

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table indicates that Area 1 has the highest number of blocks andhighest coefficient of variation of block size. However, an oppositebehaviour is detected for the coefficient of variation of Area 2 dueto its regular morphology.

3.2. Multifractal analysis results

The Sandbox method of axial-map analysis is applied using thealgorithm proposed by Tél et al. (1989), Vicsek (1990) and Vicseket al. (1990). To calculate the mass M(R) or number of networkpoints in a circle of a given radius R, the algorithm begins with acircle of a maximum radius of 0.25 and a minimum radius of0.0025 for all axial maps. The minimum radius is chosen so thattwo pixels or two street network points must be inside the circle.

Fig. 8 shows the scaling curves for q 2 ½�5; 5� at 0.25 incre-ments. This study is limited to this range of q values to avoid insta-bility in the multifractal parameters, because higher and lowermoment orders may magnify the influence of outliers in the mea-surements, as stated by Zeleke and Si (2005). These curves are fit-ted linearly to obtain the fractal dimension Dq. To obtain the bestfit for R2, the linear regression is cut between a lower limit (R/L)lower

and an upper limit (R/L)upper for q = 0 (Table 3). The sequences ofmass exponent sq are detailed in Fig. 9 for each area. As shown,the sq curve for Area 1 exhibits multifractal behaviour, due tothe different slopes for negative and positive moment orders. How-ever, for Area 2, sq is almost a straight line, which confirms a quasi-fractal nature. In all cases, the generalised fractal dimension Dq is adecreasing function with D0 > D1 > D2, as verified in Table 3,which confirms the multifractal nature (Fig. 10). Dq shows a strongdependence on the values of q for Area 1. However, for Area 2, Dq

tends to be constant, which denotes a much closer monofractal orsimply fractal behaviour (D0 � D1 � D2) and a more uniform streetdistribution than in Area 1. This finding influences the shape andlength of the multifractal spectra shown in Fig. 11. Points that be-

Fig. 8. Sandbox metho

long to the spectrum in Area 2 tend to be grouped compared withthe dissemination of points exhibited in Area 1, which denotes theexistence of a quasi-monofractal nature (one fractal dimension issufficient to describe this zone). Table 3 illustrates that both areashave similar space-filling values D0. They almost have equivalentfractal dimensions although they display distinct morphologies.Area 2 (quasi-fractal case) has a lower number of axial lines thanArea 1 (multifractal case) because it fills the space in a more effi-cient way.

To address two city areas with different morphologies and sim-ilar D0, lacunarity was calculated to distinguish texture patterns forthe two fractal networks. Lacunarity curves are displayed in Fig. 12and indicate a minimum value when the variance is 0 and a max-imum value of one when the box size is 1. Lacunarity curves arestraight and decrease almost linearly. This result corresponds tovariability in the geometric distribution. Area 1, which exhibits agreater dispersion of its elements, presents the highest lacunarityvalue and, therefore, a heterogeneous street distribution in everybox size. At the beginning, the curve of Area 2 is straight for lowbox-size values, which indicates heterogeneous lacunarity values.However, for a box size equal to 32 � 32, the lacunarity is morehomogeneous. Lacunarity can be related to factors that influenceurban morphology (e.g., block size). Area 1 shows the highest coef-ficient of variation of block size (Table 2) and the highest lacunarityvalues. Opposite results were found for Area 2.

4. Discussion

Previous works have considered city morphology to be fractal(Batty, 2008; Batty & Longley, 1994; Benguigui et al., 2000; DeKeersmaecker et al., 2003; Feng & Chen, 2010; Frankhauser,1998). The capacity dimension D0 of urban form can be observedas a ratio of filling a city space (Chen & Lin, 2009), which provides

d scaling curves.

Fig. 9. The mass exponent functions of each area evaluated at 0.25 increments in qvalues.

Fig. 10. The generalised dimension spectra. Vertical bars representing the standarderror of the computed multifractal average parameters.

Fig. 11. Multifractal spectra.

Fig. 12. Lacunarity curves for each area.

8 A.B. Ariza-Villaverde et al. / Computers, Environment and Urban Systems 38 (2013) 1–10

a new approach for performing urban spatial analysis. DimensionD0 is connected to the frequent measurements of a city shapeand vice versa, within a certain range of spatial scales (Chen, 2011).

Based on the results of this research, the prior notion that themorphology of a city can only be considered as fractal in natureis valid for city zones with regular morphology, such as Area 2 inwhich D0 � D1 � D2. In this case, the fractal dimensions have corre-sponding values (as a monofractal case), as illustrated in Table 3.However, when a city structure exhibits more complexity (Area1), a multifractal nature emerges with different fractal dimensions,such as D0 > D1 > D2, that provide more information about thestreet network distribution.

Table 3 shows that dimension D0 has similar values for bothstudy areas, which implies similar space fillings. However, their

morphologies are different because they can be derived from frac-tal dimensions D1 and D2. According to Davis, Marshak, Wiscombe,and Cahalan (1994), D1 provides a measure of the degree of heter-ogeneity in the spatial distribution of a variable. In addition, theinformation dimension characterises the distribution and intensi-ties of singularities (high values of street network density) with re-spect to the mean. If D1 is lower, the distribution of singularities inthe street network density will be sparse. To the contrary, if D1 in-creases, these singularities will have lower values that exhibit amore uniform distribution.

This finding is in agreement with the D1 values listed in Table 3for Areas 1 and 2. Area 1 yielded a higher D1 because it has moreblocks with several sizes and irregular shapes (Fig. 7a) than Area2, which has a smaller number of blocks that are similar in sizeand resemble rectangles (Fig. 7b). This finding is consistent with

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the CV values shown in Table 2 (CVArea 1 > CVArea 2). When D0 = D1,a monofractal case exists. This description might be valid for Area2; however, it is not valid for Area 1 (D0 > D1), which exhibits amultifractal nature.

Correlation dimension D2 describes the uniformity of the streetnetwork density among several selected zones (circles of radius R).D2 is related to the probability of finding pixels that belong to thestreet network within a given distance when beginning on a pixelthat belongs to this object. D2 is confirmed to be higher for Area 2(Table 3) and more uniform than Area 1, as shown by Fig. 7. As inthe case of D1, this finding is a result of relevant differences in thenumber, size and shape of blocks that determine the analysed mor-phologies of the neighbourhood. Area 2 can be considered quasi-fractal because D0 � D2, whereas Area 1 has a multifractal nature(D0 > D2), as previously stated.

According to these results, the Rényi dimension spectrum canbe used to describe urban morphology instead of a single-valuedfractal dimension. This situation is determined by different urbanmorphology generative processes (city growth according to severalplanning/socio-economic regulations) over time, which results inregular and irregular areas.

5. Conclusions

With the goal of performing a multifractal analysis of urbanmorphology, street layouts are extracted using a Space Syntaxalgorithm for two urban neighbourhoods, thereby obtaining axialmaps of each area to accurately represent their spatial configura-tions. The Sandbox multifractal method and lacunarity measure-ments are applied to describe street networks. Multifractalanalysis is efficient for characterising the morphology of street net-works with the advantage of parameters that are independent overa range of scales. According to the generalised dimension spectraobtained in this research, it is convenient to consider the existenceof the multifractal nature of some urban zones, especially whenaddressing cities with frequent irregular morphology; these areascannot be described by a single fractal dimension as previous re-search states. Instead, infinite fractal dimensions for these zonesshould be theoretically considered. In practice, capacity, informa-tion and correlation fractal dimensions are sufficient parametersfor characterising irregular morphologies.

This research is based on a case study of two neighbourhoods inthe same city. Although the regular and irregular urban patternsstudied here are frequently observed in other cities, the replicationof this analysis would establish the generality of the conclusionsreached in the case study of Cordoba.

The differences found for the fractal dimensions obtained forthe neighbourhoods in this research are influenced by urban pat-terns derived from planning regulations and diverse socio-eco-nomic situations, such as the spatial and temporal evolution ofland values (i.e., Hu, Cheng, & Wang, 2012). Thus, with the objec-tive of aiding urban planning and management, further analysesshould be performed to relate specific fractal dimensions to differ-ent planning and socio-economic regimes. The study of relation-ships between fractal dimensions and market context issuggested because the city street network is determined by busi-ness principles in many cases.

Acknowledgements

The ‘‘sequence-determines-credit’’ approach has been appliedto the order of the authors. The authors gratefully acknowledgethe support of the Spanish Ministry of Economy, CompetitivenessERDF Projects AGL2009-12936-C03-02, the Department of Innova-tion, Science and Business (Andalusian Autonomous Government)ERDF and the ESF Project P08-RNM-3989.

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