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Fractal dimension of European Cities: A comparison of the patterns of built-up areas in the urban core and the peri-urban ring

Dimension fractale des villes européennes : Une comparaison des modèles de zones bâties dans le noyau urbain et la périphérie
La dimensión fractal de las ciudades europeas. Una comparación de los modelos de las áreas construidas en el núcleo urbano y en la periferia
Apostolos Lagarias et Poulicos Prastacos

Résumés

Cette étude compare la forme urbaine des grandes villes européennes en utilisant les dimensions fractales. Les dimensions fractales sont calculées pour les zones bâties en utilisant les données disponibles dans l'ensemble de données d'Imperméabilité à Haute Résolution (EAA, 2012), et sont estimées selon deux méthodologies différentes : l'approche binaire qui prend en compte deux classes, les terres aménagées et les terres non aménagées, et l'approche “grayscale” qui prend en compte l'intensité de l'urbanisation, par le degré d'imperméabilisation des sols. Les dimensions fractales sont estimées pour les 60 villes européennes de plus d'un million d'habitants. En utilisant les limites définies dans l'atlas urbain, la superficie de chaque zone urbaine fonctionnelle (FUA) est divisée en deux parties, la zone urbaine centrale et la zone périurbaine. Les dimensions fractales sont calculées pour chacune de ces trois entités spatiales. Les résultats sont liés aux différences de forme urbaine et aux questions de planification telles que l'étalement urbain et le développement polycentrique. Les villes sont comparées entre différentes régions (Royaume-Uni-Irlande, Scandinavie, Centre, Sud, Est) et différents pays. Les différences régionales sont plus évidentes dans les zones périurbaines, les villes d'Europe de l'Est et de Scandinavie formant un groupe distinct avec des valeurs de dimension fractale plus faibles. Des valeurs fractales significativement plus élevées au niveau du FUA sont signalées dans les villes situées sur le corridor européen fortement urbanisé qui s'étend du sud du Royaume-Uni aux Pays-Bas, à la Belgique, à la région de la Ruhr et au nord de l'Italie. Un groupe secondaire de dimensions fractales FUA élevées est formé par les villes côtières d'Europe du Sud comme Porto, Lisbonne, Naples et Barcelone. Dans les zones urbaines centrales, les valeurs les plus élevées de la dimension fractale sont observées dans les plus grandes villes (Londres, Paris, Berlin) qui sont construites de manière compacte.

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1. Introduction

1Fractals are used to analyze urban form as described by the patterns of built-up areas (Batty & Longley, 1994; Frankhauser, 1998; Benguigui et al., 2000; Shen, 2002; Tannier & Pumain, 2005; Terzi & Kaya, 2011; Thomas et al, 2012), the boundaries of the urban areas (Lagarias, 2007; Tannier & Thomas, 2013) and other structural elements of the city such as transportation networks (Lu et al., 2016; Pavón-Domínguez et al., 2017) etc. The fractal dimension (D) characterizes the spatial organization of built-up areas (Thomas et al., 2008), and its values are between 1 and 2. A relatively homogeneous distribution of built-up patterns leads to higher fractal dimension values, while linear/contrasted/fragmented development patterns result in lower values (Frankhauser, 1998; Encarnação et al., 2012). Fractal dimension is therefore an indicator that captures some characteristics of the complexity of urban development patterns.

2Until recently, comparative fractal analysis of urban areas was a tedious task. Results of various older studies (Batty & Longley, 1994; Shen, 2002) were not directly comparable, since the datasets used for computing fractals were different and often of low resolution, and the methodological approaches not similar (i.e. urban area boundaries, computation procedures etc.). Notable exceptions are recent works by Sémécurbe et al. (2019) computing fractal dimension built patterns throughout mainland France, by Thomas et al., (2012) comparing patterns in eighteen urban agglomerations located in Belgium, Finland France, Germany, Italy and Switzerland, and by Yuan et al. (2019) estimating spatial metrics (among which fractal indexes) for 115 Chinese cities.

3However, with recently released high-resolution datasets covering the whole of Europe, it is possible to compute fractal dimensions using similar datasets and therefore identify urban form differences using fractals. Two such databases are the Urban Atlas with land use data (UA, https://land.copernicus.eu/​local/​urban-atlas) and the Imperviousness Density High Resolution Layer with information on the soil sealing degree (IMD HRL, https://land.copernicus.eu/​pan-european/​high-resolution-layers). Both datasets were developed by the European Environment Agency (EEA) and are distributed through the Copernicus Land Monitoring Service. UA is a vector database of the land use/land cover of 785 urban areas (cities in Europe and Turkey with population higher than 50,000). Urban fabric areas (including usually a mix of residential structures, office buildings, city centers etc.) are differentiated by the built-up density, as identified by soil sealing degree. The IMD HRL database is a raster database of 20m resolution that covers all of Europe, both urban and rural areas, and has information on the imperviousness/soil sealing degree (s.d.). Both databases consider only ground level development and do not include information on the height of the buildings.

4Fractal dimensions are usually computed using the binary methodology (Frankhauser 1998; Thomas et al, 2008). With this approach there are two possible land uses for a given urban area, developed and non-developed α-areas are not uniform and neglecting this factor results in fractal dimensions that do not describe urban form accurately. Lagarias and Prastacos (2020) proposed a methodology for computing fractals that takes into account built-up densities and estimated the fractal dimension of 14 large South European cities. This approach uses “grayscale” raster maps, in which pixels represent the development intensity (the sealing degree). Another issue with fractal dimensions is that in most studies they are computed for the whole urban area. Nonetheless, the distribution of built-up areas is not homogeneous across the urban space. There is always a small high density urban center and a large low density peri-urban ring and it is expected that fractal dimensions will be lower in the periphery (Thomas at al., 2008). There is therefore a need to study the relationship between the fractal dimension of the whole urban area and that of its subareas and develop a meaningful understanding of the factors that affect urban fractals. Thomas et al. (2012) and Sémécurbe et al. (2019) compute fractal dimensions at the local level using urban neighborhoods boundaries and 2km grid respectively.

5This study extends previous research (Lagarias & Prastacos, 2020), using the IMD HRL database to estimate fractal dimensions for the 60 largest European urban areas (population over one million), and compare results across different cities and regions (UK/Ireland, Scandinavia, Central South, East). The spatial extent of each urban area, the Functional Urban Area (FUA), is defined as in UA 2012. This is the first time high-resolution data as those provided by EEA and Copernicus, are used for a large-scale comparison of the fractal dimension of European cities. A second and probably most crucial contribution from this study is that urban form differences between the urban center (CORE) and the peri-urban ring (RING) areas are analyzed using fractals. The two fractal dimension sets are compared to that of the whole urban area and it is shown that they are highly interrelated.

6Fractals dimensions are estimated with Fraclac, a public domain plugin of the Image J software that analyzes binary and/or grayscale images (Karperien, 2004-2005). The fractal dimension of a grayscale image (mentioned as DG) represents differences in pixel values intensities; in the IMD HRL dataset these correspond to the built-up densities. Fractals dimensions are also estimated using binary maps (DB) and results of the two approaches are compared.

7This paper is organized in 5 sections. In the second section the methodology, the data used and the study areas are introduced, while in the third section the fractal analysis results are outlined and compared. In the fourth section the results are discussed to identify urban form differences between the case study cities, whereas in the conclusions’ section there is an overview of the major findings.

2. Methodology and Data

2.1 Data

8The IMD HRL dataset is a high-resolution raster map (available at 20m and 100m resolution) that covers all European countries and Turkey. Pixel values indicate the soil sealing degree (s.d.), the percentage of land covered by artificial/man-made structures. Sealing degree values are between 0 and 100 (100 being fully covered/developed land, 0 being open space). The IMD HRL dataset was first released for year 2006 and has been updated for 2009, 2012 and 2015. In this study, fractals dimensions are computed using the 2012 IMD HRL dataset.

9Fractal dimensions are computed using two different maps: a) Binary raster maps of artificial surfaces, that have “built-up” areas (pixels with soil sealing > 0, black) and “non-built” areas (pixels with soil sealing = 0) and b) Grayscale raster maps of built-up areas with pixel values representing the sealing degree. The 100 different s.d. levels were reclassified into 11 classes (0, 1-10, 11-20, …, 91-100) and a gray color scale was introduced to represent the 11 different built-up densities (0, 5, 15, 25, …, 85, 95% on the white-black scale). To maintain the high resolution of the IMD HLR database the maps were converted to TIFF images at a resolution of 889 dpi at a scale of 1:700.000, each pixel therefore corresponding to 20 m. Figure 1 presents the two representations of the urban CORE of Copenhagen.

Figure 1. Grayscale (left) and binary (right) representation of the artificially sealed surfaces in the CORE area of Copenhagen.

Figure 1. Grayscale (left) and binary (right) representation of the artificially sealed surfaces in the CORE area of Copenhagen.

10Using the IMD HRL dataset the following three indicators are estimated for the FUA, the CORE and the RING areas of the 60 European cities.

  1. The %developed (DEVEL): impervious areas (s.d. > 0) divided by total land.

  2. The %impervious (IMPER): impervious areas (s.d. > 0), weighted by their sealing degree and divided by total land.

  3. The mean sealing degree (MEANSD): The mean s.d. of the impervious areas (cells with s.d. > 0).

11By definition, the values of all three indicators are between 0-100. The first two indicators are highly related. Both are reported in this analysis since the DEVEL indicator accounts for the binary representation of land (developed/non-developed), whereas the IMPER considers the land developed as well as the intensity of development. The third indicator (MEANSD) expresses the average density of the built-up areas at the pixel/cell level. It is independent of the size of the case study area, and therefore it is affected to a smaller degree by boundary definition issues.

2.2 Methodology

12The most widely applied method for computing the fractal dimension of urban areas is the box-counting. It is an iterative procedure; a grid (r) is positioned over the map and at each iteration the number (N) of “occupied” cells (built-up areas in cities) is counted. The approach is repeated for different grid sizes (r). Results are then used to estimate the relationship Ν(r) = r–D, where D is the fractal dimension. The fractal dimension is estimated using log-linear regression between N and r. Fractal dimensions can be also computed using alternative methodologies such as dilation, correlation, radius-mass and others (Frankhauser, 1998; 2004).

13In this study the Fraclac plugin of Image J software (Karperien, 2004-2005) is used for computing binary and grayscale fractal dimensions. Values of grayscale image pixels are between 0 (white, undeveloped areas) and 255 (black, fully built-up areas), it is therefore possible to measure the intensity of the parameter represented by the pixel value (Karperien, 2004-2005). To determine the grayscale fractal dimension at each iteration of the box counting method a grid is superimposed on the image/map and differences in pixel intensity (δIi,j,ε) are estimated for all pixels within a box. DG is estimated as follows:

Ii,j,ε= 1 + δIi,j,ε, Iε = ∑ [1 + δIi,j,ε]

DG = limε→0 ln(Iε)/ln(1/ε).

14DG and the corresponding r2 value are computed from the log-log regression line of the sum of all I (i, j, ε) vs ε. In this study grid sizes sequence was defined as follows: 2, 4, 8, 16, 32, 64, 128, 256 (in pixels). The metric equivalent of this sequence is from 40 m to about 5 km. To account for potential bias from the initial grid position used, the D and associated r2 were computed for 10 different initial grid positions. The fractal dimension of the urban area was then calculated as the mean value of the 10 different runs. When estimating the grayscale fractal dimension of RING areas, the central zone (CORE) was considered as “background” and was excluded from the box-counting estimation.

2.3 Case study areas

  • 1 To be consistent with the use of the 2012 IMD HRL dataset, population data refer to the 2010-2012 p (...)

15All European cities with population over one million were selected as case study areas1. That is a total of 60 different urban areas; 25 cities in Central Europe (France, Germany, Belgium, Netherlands, Austria and Switzerland), 13 cities in South Europe (Greece, Italy, Spain and Portugal), 10 cities in East Europe (Poland, Hungary, Czech Republic, Romania, and Bulgaria), 4 cities in Scandinavia (Sweden, Finland, Denmark and Norway) and 8 cities in UK/Ireland.

16A critical issue in this research was the definition of the boundaries outlining urban areas. Urban areas are large and there is not a consensus on how far they extend from the urban center. Including a vast peri-urban area affects fractal dimensions, since in these areas there is usually a sharp contrast from the development patterns in the more central parts of the urban agglomeration. A related research issue was the definition of the boundaries of the CORE area. Rather than developing a new methodology for outlining urban boundaries it was decided to use established and widely used boundaries, the Functional area boundaries (FUA) and the CORE boundaries defined in Urban Atlas 2012.

17FUA boundaries follow administrative city boundaries and were initially specified in Urban Audit as Larger Urban Zones (LUZ). Municipalities in the periphery were included in the LUZ after an analysis of the commuting patterns (home to work trips, percentage of a municipality‘s residents commuting to the center). Detailed information on the definition of LUZ zones is available in Bretagnolle et al (2011). In 2012 EU and OECD developed a harmonized definition of city boundaries and commuting zones and introduced the FUA concept (Dijkstra & Poelman, 2012). To define FUA, various criteria were used. The most recent revision of FUAs was prepared by Eurostat in 2017 and includes 785 urban areas. The 2012 Urban Atlas database was then updated to reflect the new boundaries. In the same update CORE areas were defined for each urban area. To define CORE areas, first the high density cells (cells with at least 1,500 inhabitants per km2) were identified. To outline the urban center these cells were huddled to form clusters with 50,000 or more inhabitants and adjacent municipalities with at least fifty percent of their population in the urban center were added in the CORE (Dijkstra & Poelman (2012).

18In this study, three different spatial entities are identified for every urban area: a) the FUA area b) the Core city (CORE) and c) the peri-urban ring (RING). The characteristics of these three spatial entities are analyzed and fractal dimensions are computed for each one. To further analyze the relationship between area boundaries and fractal dimensions, CORE boundaries were also defined following an approach based on Urban Morphological Zones and CORE fractal dimensions estimated again. The characteristics of the 60 urban areas are tabulated in Appendix Table A1. Information presented includes population and the IMPER, DEVEL and MEAN S.D. indicators for each of the three spatial entities. Box-plots of selected variables are presented in Figure 2.

19In most cities the CORE area is less than 20% of the FUA area, and often less than 10%. Cities with relatively large CORE areas include Naples (70% of the FUA area), Porto (50%), Liverpool (49%), Lille (42%), Manchester (41%), West Midlands (37%), Milan (35%), Katowice (31%), Koln (25%), London (23%), Bucharest (22%) and Rome (21%). The average IMPER_FUA indicator for all cities is about 8. The highest values of this indicator are in Liverpool (19), Naples (18), Porto, Dusseldorf and Antwerp (17), Ruhrgebeit and West Midlands (16) and Koln (15). The values of the IMPER_CORE indicator are higher. The average IMPER_CORE indicator for all cities is 32 and the highest values are for the CORE areas of Paris (54), Antwerp (44) and Torino (44). In Scandinavian cities (Copenhagen is an exception), most of East European cities and Rome the IMPER_CORE is lower (less than 20). The equivalent indicator for the RING areas (IMPER_RING) is almost always less than 10. The highest values of this indicator are in Dusseldorf (17), Antwerp (14), Koln, Liverpool and Porto (12).

20MEANSD in Central European cities is 62/67/58 (FUA/CORE/RING), 44/51/41 in East European cities, 49/54/44 in Scandinavian cities, 63/68/58 in South European cities, and 58/61/53 in UK/Irish cities. South European cities are more densely built-up with MEANSD_FUA taking values between 54-69 (average value: 63). High MEANSD_FUA values are also observed in Central European cities (average value: 62), the range of values, however, is extended (48-72). In East European and Scandinavian cities the MEANSD_FUA is considerably lower (average values 44 and 49 respectively). In UK this indicator is relatively the same for all cities (values 57-62), while in Dublin MEANSD_FUA is considerably lower (47). The MEANSD_COREand MEANSD_RINGare highly correlated (r=0.91), an indication that cities with higher density central areas are also characterized by higher density development patterns in the peri-urban areas.

Figure 2. Box-plots of the IMPER and DEVEL indicators

Figure 2. Box-plots of the IMPER and DEVEL indicators

21Density patterns of urban areas, were compared using soil sealing “profiles” (Figure 3). The profiles are histograms of the sealing degree of the built-up areas aggregated to ten different s.d. classes (1-10, 11-20, .…., 91-100) and show in a single graph the CORE and RING areas adding up to the FUA area. Profiles comparison shows that the profiles of South European cities are asymmetrical to the right with densely built-up areas (s.d. > 60) covering large zones, whereas in Central Europe and UK/Ireland the profiles are relatively symmetrical with medium density areas (s.d. 30-70) accounting for the larger part of the urban area. In East European and Scandinavian cities the soil sealing profiles are asymmetrical to the left with low density urban areas (s.d. < 40) often representing almost 50% of the developed area. The graphs indicate that densities in all three spatial levels are higher in Central and South European cities and lower in East European and Scandinavian cities.

Figure 3. Soil sealing profiles for selected European cities

Figure 3. Soil sealing profiles for selected European cities

3. Results

3.1 Fractal dimensions of the 60 European cities

22Fractal dimensions were computed for the FUA, CORE and RING areas of every city using both binary (DB) and grayscale (DG) methodologies. Fractal dimensions of every urban area are reported in Table A2 of the Appendix, whereas average city fractal dimensions for different European regions are presented in Table 3. Maps of CORE and RING grayscale fractals are provided in Figures 4, 5 and 6, classified in 5 groups, based on the natural breaks method. The Pearson correlation values in the box-counting calculation of DG are higher than 0.99 for all cities except for the FUA areas of Ruhrgebeit, Dublin, Munich, Antwerp, Lyon and Zurich where the r is between 0.988 and 0.990. When implementing the binary methodology, the r is higher than 0.990 in all cities (FUA, CORE and RING areas).

23The average grayscale fractal dimensions of FUA areas (DG_FUA)is 1.36 and individual values are between 1.18 (Newcastle) and 1.55 (Porto). DG_FUA values are higher than 1.50 in 3 cities (Porto, Naples, Ruhrgebiet), while in 9 cities DG_FUA values are in the 1.45-1.50 range (Dusseldorf, Antwerp, Milan, Manchester, Liverpool, Koln, Barcelona, London, Lisbon). DG_FUA is less than 1.25 in 5 cities (Bremen, Stockholm, Dublin, Riga, Oslo and Newcastle). The average fractal dimension of CORE areas is 1.51, values range between 1.37 (Valencia) to 1.62 (Paris) and the highest values are in Central European and UK/Irish cities. DG_CORE values are significantly higher than the corresponding fractal of the FUA and RING areas. DG_RING values are significantly lower, with an average equal to 1.28 and values between 1.05 (Newcastle) and 1.42 (Dusseldorf). Average DG_RING values are higher in Southern (1.33) and Central European (1.30) cities and lower in Scandinavian (1.21) and East European cities (1.22). Average fractal dimensions of the cities in France, UK, Germany, Spain and Italy are also reported in Table 3; in each of these countries there is a meaningful number of sample cities (5 to 13) that permits the computation of a national average. The results show that differences among the national averages in these countries are marginal; for FUA areas the average D is the highest in Italian and UK cities, while for CORE areas the highest D is in French cities, whereas for RING areas the highest D is in Italian cities. Standard deviation is also tabulated in Table 3, and results show that it is always less than 0.10 for DG and less than 0.15 for DB.

Table 3. Average fractal dimensions of the 60 European cities by region and country

 

DG_ FUA

St.d.

DB_ FUA

St.d.

DG_ CORE

St.d.

DB_ CORE

St.d.

DG_ RING

St.d.

DB_ RING

St.d.

Regions

 

 

 

 

 

 

 

 

 

 

 

 

UK/Ireland

1.37

0.12

1.56

0.13

1.52

0.07

1.74

0.07

1.24

0.10

1.42

0.14

Scandinavia

1.30

0.10

1.48

0.12

1.48

0.07

1.70

0.07

1.21

0.10

1.37

0.13

Central

1.37

0.07

1.56

0.08

1.53

0.04

1.74

0.05

1.30

0.07

1.47

0.08

South

1.31

0.06

1.53

0.08

1.48

0.03

1.73

0.06

1.22

0.04

1.42

0.06

East

1.41

0.08

1.58

0.09

1.51

0.06

1.72

0.06

1.33

0.07

1.49

0.08

All cities

1.36

0.09

1.55

0.09

1.51

0.05

1.73

0.06

1.28

0.08

1.45

0.10

Countries

Germany (13 cities)

1.37

0.08

1.56

0.08

1.52

0.04

1.75

0.04

1.30

0.07

1.48

0.08

France (6 cities)

1.35

0.06

1.52

0.07

1.54

0.06

1.74

0.07

1.25

0.06

1.40

0.07

UK (7 cities)

1.39

0.11

1.58

0.13

1.53

0.06

1.75

0.07

1.25

0.10

1.44

0.14

Italy (5 cities)

1.42

0.08

1.61

0.08

1.51

0.03

1.73

0.04

1.33

0.06

1.50

0.06

Spain (5 cities)

1.36

0.08

1.52

0.10

1.48

0.07

1.68

0.08

1.30

0.10

1.45

0.11

24The comparison of the results obtained using the two alternative fractal computation methodologies, shows that binary fractal dimensions are always higher than grayscale ones. This was an expected finding since the grayscale approach considers not only the binary developed/non developed dichotomy but also the heterogeneity of built-up densities. The difference between the two fractal sets is about 10-20% for all cities, at all three spatial levels. This indicates that considering density levels results in a relatively homogenous decrease of fractal dimensions. To further examine this concept, cities were ranked on the basis of their fractal dimension (from higher to lower values, Appendix Table A3). At the FUA level the ranking of cities is almost the same for both methodologies with significant ranking differences for only a few cities. With the binary approach, cities with low MEANSD (Budapest, Bucharest, Zurich, Katowice, Warsaw, Berlin, Ostrava) increase their ranking by at least 5 positions, while the ranking of cities with relatively high MEANSD (Manchester, Madrid, Marseille, Manheim-Ludwinghafen and Prague) drops by more than 5 positions. The correlation (r) between the change in rankings and MEANSD is statistically significant and equal to 0.53, 0.47 and 0.51 at the FUA/CORE/RING level respectively. These findings indicate that the grayscale approach has an impact in fractal dimensions since it considers the heterogeneity between low and high density cells.

25For RING areas the rankings are about the same irrespective of the methodology used with important ranking changes only for Manchester, Liverpool, Helsinki, Zurich, Ostrava, Bucharest and Warsaw (increase in ranking) and Naples, Seville, Madrid, Nurnberg, Marseille, Leeds (decrease in ranking). For CORE areas the ranking differences are more significant. The ranking of Amsterdam, Lisbon, Porto, Liverpool, Copenhagen, Bordeaux and Lille drops by at least 10 positions when implementing the binary approach, while Turin, Krakow, Antwerp, Budapest, Warsaw and Bucharest, increase their ranking by more than 10 positions. In conclusion, results show that cities with lower MEANSD tend to move higher in the ranking order when implementing the binary fractal approach and vice-versa.

Figure 4. Fractal dimension values of the FUA areas of the 60 cities

Figure 4. Fractal dimension values of the FUA areas of the 60 cities

Figure 5. Fractal dimension values of the CORE areas of the 60 cities

Figure 5. Fractal dimension values of the CORE areas of the 60 cities

Figure 6. Fractal dimension values of the RING areas of the 60 cities

Figure 6. Fractal dimension values of the RING areas of the 60 cities

26Α bivariate plot comparing DG_CORE and DG_RING is presented in Figure 7. A clear regional pattern is not evident. However, a Low-Low relation (both CORE and RING) is the case for most cities in East Europe and Scandinavia, while most cities in South Europe, Copenhagen, Brussels, Amsterdam and Rotterdam are classified in the High-High group. Notable outliers are Valencia (low DG_CORE values and high DG_RING values) and Newcastle (low DG_RING values but DG_CORE values close to the mean). Cities in the Low-High group include Berlin, Toulouse, Bordeaux, Vienna, Munich, Prague, Manchester and West Midlands. In addition to Valencia, the cities of Liverpool, Zurich, Antwerp are cities are in High-Low group (low DG_CORE values and high DG_RING values).

Figure 7. Plot of DG_ CORE versus DG_ RING values

Figure 7. Plot of DG_ CORE versus DG_ RING values

3.2 Correlations between variables

27The Pearson correlation coefficients of the different fractal sets and the two built-up patterns indicators (IMPER and DEVEL) are tabulated in Table 5. The two fractal sets (DG and DB) are highly correlated. This is true for all three spatial entities CORE, RING and FUA. The correlation of DG and DB is 0.88/0.96/0.95 for the CORE/RING/FUA areas respectively. As discussed earlier the high correlations imply that the major determinant of the fractal of an urban area is the developed/undeveloped dichotomy. FUA fractals are highly correlated to the fractals of RING areas (0.86 grayscale approach, 0.88 binary), while the correlation of the FUA and CORE fractals is statistically significant but lower (0.43 grayscale, 0.39 binary). The high correlation of DFUA and DRING implies that the fractal dimension of an urban area is highly affected by the built-up patterns in the peri-urban areas.

Table 5. Pearson Correlation of different variables used in the analysis.

 

POP FUA

POP CORE

DG_ FUA

DB_ FUA

DG_ CORE

DB_ CORE

DG_ RING

DB_ RING

IMPER

FUA

IMPER

CORE

IMPER

RING

DEVEL

FUA

DEVEL CORE

DEVEL RING

POPFUA

1

0,82**

0,28*

0,32*

0,53**

0,51**

0,25

0,23

0,20

0,22

0,10

0,20

0,19

0,03

POPCORE

 

1

0,22

0,29*

0,55**

0,56**

0,11

0,14

0,18

0,25*

0,07

0,21

0,30*

0,05

DG_ FUA

 

 

1

0,95**

0,43**

0,29*

0,86**

0,87**

0,93**

0,21

0,73**

0,92**

0,11

0,54**

DB_ FUA

 

 

 

1

0,45**

0,39**

0,83**

0,88**

0,90**

0,24

0,72**

0,93**

0,21

0,55**

DG_ CORE

 

 

 

 

1

0,88**

0,33**

0,31*

0,37**

0,33*

0,24

0,35**

0,34**

0,14

DB_ CORE

 

 

 

 

 

1

0,24

0,27*

0,26*

0,47**

0,16

0,30

0,60**

0,12

DG_ RING

 

 

 

 

 

 

1

0,96**

0,79**

0,47**

0,90**

0,73**

0,35**

0,70**

DB_ RING

 

 

 

 

 

 

 

1

0,78**

0,44**

0,87**

0,77**

0,40**

0,70**

IMPER_FUA

 

 

 

 

 

 

 

 

1

0,23

0,79**

0,97**

0,12

0,62**

IMPER_CORE

 

 

 

 

 

 

 

 

 

1

0,47* 

0,17

0,93**

0,31*

IMPER_RING

 

 

 

 

 

 

 

 

 

1

0,75**

0,38**

0,83**

DEVEL_FUA

 

 

 

 

 

 

 

 

 

 

 

1

0,14

0,63**

DEVEL _CORE

 

 

 

 

 

 

 

 

 

 

 

 

1

0,32*

DEVEL _RING

 

 

 

 

 

 

 

 

 

 

 

 

 

1

* Statistically significant at the 0.05 level
** Statistically significant at the 0.01 level

28The fractal dimension of FUA areas is highly correlated to the IMPER_FUA (r=0.93) and the DEVEL_FUA (0.92) indicators. These two indicators are also highly correlated (0.97). DG_FUA is also highly correlated to the IMPER_RING (0.75), whereas the correlation with the equivalent CORE indicator is not statistically significant. The high correlation between fractal dimension and the IMPER and DEVEL indicators is also evident in the RING areas (0.90/0.93). For CORE areas the correlation between DG_CORE and IMPER_CORE is statistically significant but lower (r=0.33). Population in the CORE area (usually about 70% of the FUA population) is correlated to the fractal value of the CORE (DG_CORE) (r=0.55).

3.3 Morphological boundary definitions for CORE areas

29To assess how the boundaries used for the analysis (obtained from UA) affect the fractal dimensions, additional computations were carried for CORE areas and boundaries were defined using the Urban Morphological Zones (UMZ) specified by EEA (2006). To specify the CORE-UMZ area, UMZ polygons were aggregated using a 500m distance-threshold, then the central area cluster was extracted and a buffer of 500m was imposed. The Convex Hull of the central area cluster was then created and this identified the CORE-UMZ area. With this approach the CORE area increases in most cities. CORE-UMZ represents on average 29.6% of the FUA area, whereas when using the Urban Atlas boundaries (UA-CORE) it accounts for 11.5%. A comparison of the FUA, UA-CORE and CORE-UMZ boundaries for the city of Ostrava appears in Table 8.

Figure 8. FUA and CORE boundaries for the city of Ostrava; CORE area definition using the Urban Atlas and the Urban Morphological Zones approach.

Figure 8. FUA and CORE boundaries for the city of Ostrava; CORE area definition using the Urban Atlas and the Urban Morphological Zones approach.

30Grayscale fractal dimensions for CORE areas were then estimated using the CORE-UMZ (Convex Hull) boundaries. The resulting fractal dimensions are relatively similar to those estimated using the UA-CORE boundaries (Figure 9), for most cities the difference of the two values is less than 3%. The differences are higher in Paris, Brussels, Vienna and Lyon (DG_CORE-UMZ lower), and in Milano, Rome, Dublin, Zagreb, Marseille, Leeds, Stockholm and Valencia (DG_CORE-UMZ higher). The two fractal sets are correlated by r = 0.59.

Figure 9. Comparison of DG_CORE and DG_CORE-UMZ (CONVEX HULL) values, cities sorted by DG_CORE values.

Figure 9. Comparison of DG_CORE and DG_CORE-UMZ (CONVEX HULL) values, cities sorted by DG_CORE values.

4. Comparison of the urban form of European cities using fractal dimensions

31The results show that computing fractals separately for the CORE and RING areas brings forward important insights. The fractal dimension of the urban CORE is always significantly higher than the corresponding value of the RING area and that the FUA fractal dimension is highly correlated to that of the RΙΝG area. Peri-urban areas are dynamic parts of a city, usually under rapid transformation in terms of population increase, land use restructuring, infrastructure networks development etc. They account for 70-95% of the FUA area (Milan 65%, West Midlands 63%, Manchester 60%, Lille 57%, Liverpool 52%, Porto 50% and Naples 30% are notable exceptions), and built-up surfaces in these areas are between 1% in Newcastle to 17% in Dusseldorf. Central European cities (particularly those located in the Ruhr region) and cities in South Europe (Porto, Milan, Barcelona, Naples, and Lisbon) have the highest levels of peri-urban development (RING) as measured by the DEVEL and IMPER indicators. This is reflected in the corresponding fractal dimension values that take values between 1.33 and 1.42, higher than the mean DG_RING (1.26) estimated for RING areas in the sample of cities.

32Higher FUA fractal dimensions are reported in cities located on the highly urbanized European corridor spanning from South UK to the Netherlands, Belgium, the Ruhr region and Northern Italy. Similar high values are also observed in South UK (London, West Midlands Urban Area, Manchester and Liverpool) covering the north “blue banana” corridor of urbanization. East European and Scandinavian cities form a distinct group with lower fractal dimension values. This can be explained by the more contrasted urban development patterns between centre-periphery and the lower development percentages in these cities. In South European cities, the high fractal dimensions imply a homogeneous development pattern that is the result of urban expansion; the city of Porto is a typical example. The traditionally sharp divide between compactly built-up central city areas and the hinterland is changing because of peri-urbanization and coastalization processes that often take the form of low density development (Catalan et al, 2008; Membrado, 2015; Salvati & Morelli, 2014). Porto, Lisbon, Madrid, Barcelona, Naples and Milan are cities with urbanized peripheral RING areas characterized by sprawling development and this is reflected in higher fractal dimensions (DG_RING) (Figure 6).

33For CΟRΕ areas, the highest fractal dimensions are observed in the largest cities and particularly in the 3 mega-regions of London, Paris, Berlin, and the West Midlands Urban area (including the core of Birmingham, historically one of the main industrial cities in UK and part of a densely urbanized area). The highest CORE area binary fractals (DB_CORE) are between 1.78 – 1.87 (12 cities, Paris, London, West Midlands, Berlin, Budapest, Munich, Brussels, Bucharest, Athens, Warsaw, Hamburg, Vienna). Although comparing fractal values estimated from different types of data is not fully appropriate, it must be pointed out that the CORE fractal dimensions of these 12 cities are among the highest values reported in the literature (Batty & Longley 1994) and imply a very organized and compact structure. Τhe CORE binary average D for the 60 cities is 1.73 which compares to the average fractal value of 1.70 that Encarnação et al (2012) report for various cities worldwide. The FUA binary average fractal dimension of the 60 European cities is 1.55, which is substantially lower than the average for CORE areas (1.73). These comparisons bring again forwards the issue of the definition of the urban areas boundaries and how they affect fractal dimensions.

34Another important finding of this research is that particularly for FUA areas, the fractal dimension is highly correlated to the IMPER and DEVEL indicators. This seems surprising since these two indicators describe respectively the percentage of the urban area developed and the average development intensity and do not take into account the spatial distribution of the built-up patterns. High correlation between fractal dimension and developed area was also reported by Shen (2002) when comparing fractal values of a large sample of U.S. cities. These results might indicate that fractal dimensions are not very sensitive to the spatial distribution of the built-up patterns and that different urban forms can have similar fractal dimensions. An explanation for this might be that cities have a relatively similar structure with historical urban growth patterns affected by market mechanisms and planning constraints. Urban land expands from the high density core towards the low density peri-urban areas forming built-up patterns that have common morphological characteristics, particularly when they reach a similar overall level of development/density.

35Nonetheless, the CORE area results indicate that in cities with similar mean density the fractal dimensions are different. To better comprehend the contribution of fractal dimension as urban form indicators, cities of similar average development intensity (IMPER) are compared. For Valencia, Budapest and Porto although the IMPER_CORE indicator is the same (23-24), DG_CORE values are considerably different (values from 1.37 to 1.57). Similar differences are observed between Seville, Barcelona, Berlin, Frankfurt (IMPER_CORE 31-32, and fractal dimensions from 1.48 to 1.60) and other sets of cities. As shown in Figure 10, a relatively homogeneous distribution of low density development in Porto leads to higher fractal dimension (1.57 at the CORE level). In Palermo there is mono-centric development with large empty areas between urban zones, and this pattern results in a lower fractal (1.48), while in Budapest compact and continuously built-up patterns characterized by significant differences in densities lead to higher fractal dimension (1.53).

Figure 10. Comparison of 3 cities with similar mean density of development (IMPER = 24 in all cases) and different fractal dimensions on the CORE level (Porto/Budapest/Palermo).

Figure 10. Comparison of 3 cities with similar mean density of development (IMPER = 24 in all cases) and different fractal dimensions on the CORE level (Porto/Budapest/Palermo).

5. Conclusions

36In this study the urban form of a large set of European cities is compared using fractal analysis. With data from the 2012 IMD HRL database fractal dimensions are computed for the FUA, CORE and RING areas of the 60 European cities with population over one million. Geographic boundaries of the various spatial entities were defined following the 2012 edition of the Urban Atlas. Two different methodologies for computing fractal dimensions are implemented, binary and grayscale. The results of the grayscale box-counting approach are comparable to those of the binary one; however, the former approach also addresses the issue of different densities across an urban area.

37Results confirm that fractal dimension is higher in urban areas characterized by a relatively homogeneous distribution of developed areas and built-up densities. This is true for both the dense/compact central areas and also the less dense homogeneous peri-urban areas. Fractal dimensions are lower in cities with less urbanization (as expressed by the percent of land developed) and/or discontinuous or relatively linear/elongated built-up patterns in peri-urban zones.

38Methodological conclusions of this research are as follow:

  1. Accounting for built-up densities when estimating fractal dimensions (grayscale approach) results in fractal dimensions that are highly correlated with those obtained when only the developed/non developed dichotomy is considered (binary approach).

  2. Although the fractal dimensions obtained from the two different approaches are highly correlated, the grayscale approach provides additional information on the urban form particularly in areas that are characterized by relatively dense development. This is evident in the fractal dimension rankings of the cities; the ranking of cities with lower densities in the CORE city is substantially higher with the binary approach, whereas their ranking drops when the grayscale approach is used. The conclusion is that some urban patterns characteristics are not fully accounted for in the binary approach.

  3. The urban form differences of the CORE and RING areas are reflected in the fractal dimensions with the CORE having a significantly higher fractal dimension than the RING. Therefore, urban form differences should be identified not only for the whole urban area (FUA) but also for the CORE and RING areas.

  4. The fractal dimension of the FUA area is highly correlated to the fractal of the RING area, an indication that the fractal of the whole area is affected to a large degree by development patterns in the periphery.

  5. There is a very significant correlation between the fractal dimension of an urban area and area wide average indicators of development (impervious/developed land). This is particularly true for FUA and RING areas. An explanation for this might be that in large urban areas, such as those analyzed in this research, there are some common urban form characteristics that are attributed to the characteristic structure of cities consisting of a high density center(s) and lower density peripheral areas.

39Results show the similarities and the differences in the urban areas of the various European regions. Cities in South Europe are built more compactly with higher MEANSD values and soil sealing degree profiles asymmetrical to the right. The MEANSD indicator is also high in Central European and UK cities, but the soil sealing degree profiles are more symmetrical, while in cities in East Europe and Scandinavia the MEANSD is considerably lower and the soil sealing degree profiles asymmetrical to the left. These regional patterns are not clearly revealed in the fractal dimension values. The highest DG values are reported in cities located on the highly urbanized European corridor spanning from South UK to the Netherlands, the Ruhr region and Northern Italy. This implies that European cities in this urbanized corridor share some similar urban form characteristics as a result of a dense polycentric peri-urban structure. Another group of urban areas with high FUA fractal dimension includes Porto, Lisbon, Naples and Barcelona, coastal cities in South Europe. East European and Scandinavian cities are characterized by lower fractal dimensions. For CORE areas, the cities with the highest fractal dimensions are the mega-regions of London, Paris, Berlin and West Midlands followed by Munich, Copenhagen, and Brussels, cities with strong planning policies. Cities with homogeneous sprawl patterns (Porto, Lisbon) and compact mono-centric cities (Athens, Lyon) are following in the CORE city ranking.

40The results underline that in order to quantify and understand the complexity of urban patterns, there is a need for a more detailed comparative analysis of urban form. Urban form is affected by various factors; geomorphology, growth patterns in the past, socioeconomic processes, changing life styles, environmental and energy concerns and national and city level urban planning policies are the most significant. Comparative fractal analysis reveals urban form characteristics with respect to the organization of urban space and the differentiation between CORE and RING areas discussed in this study provides a valuable insight. Additional research is needed, however, on how urban fractals are related to the factors affecting urban form and translate the fractal analysis results into practical guidelines useful for the planning of existing cities.

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Annexe

Table A1. Characteristics of the 60 urban areas

 

POP_FUA 1

% POP CORE 2

% CORE Area

IMPER FUA

IMPER

IMPER RING

DEVEL FUA

DEVEL CORE

DEVEL RING

MEANSD FUA

CORE

UK/IRELAND

 

 

 

 

 

 

Dublin (IR)

1,793,902

69.4

13.2

3.0

14.4

1.2

6.2

26.5

3.1

47.6

London (UK)

12,100,598

67.5

22.6

12.7

35.4

6.1

22.2

57.9

11.8

57.1

Leeds (UK)

3,682,971

20.4

10.8

5.7

16.0

4.5

9.8

27.3

7.7

58.3

West Midlands (UK)

2,860,749

84.6

37.3

15.8

33.0

5.6

27.1

55.1

10.4

58.5

Manchester (UK)

2,773,420

99.3

40.9

11.1

22.2

3.5

18.2

35

6.6

61.2

Glasgow (UK)

1,787,515

33.1

5.2

7.4

39.8

5.1

12.4

63.2

9.6

59.7

Liverpool (UK)

1,506,935

70.4

48.5

19.3

27.6

11.4

31.1

43.2

19.7

61.9

Newcastle (UK)

1,145,347

72.4

7.4

2.9

26.9

1.0

4.9

44.2

1.8

58.9

SCANDINAVIA

 

 

 

 

 

 

Copenhagen (DK)

1,928,612

61.5

18.8

10.3

28.8

6.5

18.3

45.3

12

56.4

Helsinki (FL)

1,366,241

76.8

15.8

4.6

18.1

2.2

10.8

36.9

6.0

43.1

Oslo (NW)

1,256,554

47.1

6.1

1.8

10.0

1.8

3.9

19.2

4.1

45.9

Stockholm (SW)

2,091,473

75.8

18.9

2.7

8.8

1.5

5.5

16.8

2.8

49.8

CENTRAL EUROPE

 

 

 

 

 

 

Vienna (AT)

2,076,838

82.4

4.5

4.1

27.8

3.0

8.5

48.8

6.5

48.4

Brussels (BE)

2,565,164

87.9

5.0

7.7

41.3

6.0

13.8

61.6

11.3

56.0

Antwerp (BE)

1,079,819

91.8

17.1

16.6

43.8

11.0

23.3

53.6

17.1

71.0

Paris (FR)

11,755,918

56.7

6.3

8.4

54.2

5.3

12.4

69.2

8.6

68.1

Lyon (FR)

1,913,462

67.6

14.2

9.2

33.6

5.1

14.4

48.1

8.8

63.8

Marseille (FR)

1,725,142

59.1

14.3

6.0

19.5

3.7

9.2

28.6

6.0

64.8

Lille (FR)

1,349,579

79.0

42.4

10.5

19.5

3.8

18.1

32.2

7.7

57.7

Toulouse (FR)

1,235,068

57.0

8.8

4.1

26.2

2.0

7.3

41

4.0

56.4

Bordeaux (FR)

1,135,044

62.9

9.9

4.3

24.2

2.2

7.0

34.6

3.9

62.3

Berlin (DE)

5,097,712

64.4

5.1

4.3

32.7

2.8

7.3

51.9

4.9

59.3

Ruhrgebiet (DE)

5,135,136

N/A

 

15.9

26.2

60.7

Hamburg (DE)

3,228,054

52.8

10.1

6.4

27.0

4.1

11.2

44.3

7.5

57.5

Munich (DE)

2,727,106

37.4

5.7

6.5

36.7

4.7

11.0

58.2

8.2

59.2

Stuttgart (DE)

2,691,666

21.7

5.7

9.9

27.3

8.8

15.2

41.2

13.6

65.1

Frankfurt (DE)

2,574,812

26.0

5.8

8.8

32.1

7.4

13.1

44.7

11.2

67.1

Koln (DE)

1,920,101

52.4

25.0

15.0

28.2

10.6

24.1

42.8

17.9

62.2

Dusseldorf (DE)

1,530,700

38.4

18.1

16.8

29.6

14.0

27.0

45.3

22.9

62.3

Dresden (DE)

1,350,587

38.1

5.6

5.1

21.6

4.1

9.6

37.8

7.9

52.7

Nuremberg (DE)

1,310,225

37.1

6.3

7.0

34.2

5.2

10.9

47.8

8.4

64.4

Hannover (DE)

1,295,700

39.2

6.9

8.0

34.8

6.9

13.0

52.5

10.1

61.0

Bremen (DE)

1,243,924

43.3

5.5

4.1

27.5

2.7

6.9

41.1

4.9

59.6

Mannheim_Lud. (DE)

1,138,165

N/A

 

9.8

 

14.0

69.9

Amsterdam (NL)

2,485,103

5.4

9.7

10.5

27.8

8.6

15.7

38.6

13.3

66.6

Rotterdam (NL)

1,424,159

10.8

18.3

13.1

35.2

8.2

18.2

45.7

12.1

71.9

Zurich (SW)

1,203,998

49.0

17.5

11.0

26.3

7.7

20.4

45.2

15.2

53.6

EAST EUROPE

 

 

 

 

 

 

Sofia (BU)

1,535,819

78.2

7.9

2.4

16.7

1.2

5.8

33.6

3.4

41.8

Zagreb (CRO)

1,219,889

64.7

12.7

2.8

11.2

1.5

5.7

20.1

1.7

48.0

Prague (CZ)

2,156,097

58.9

8.7

6.3

28.2

4.2

9.9

39.8

7.0

64.2

Ostrava (HU)

1,145,851

25.9

15.0

5.4

16.1

3.6

14

34.3

10.4

38.9

Budapest (HU)

3,284,110

52.6

8.2

4.7

24.5

3.0

11.9

55.2

8.0

39.9

Warsaw (PL)

3,057,631

55.6

6.0

3.2

20.8

2.1

8.9

49.3

6.4

35.9

Krakow (PL)

2,585,884

74.8

30.9

5.8

12.8

1.7

14.4

30.3

7.3

33.3

Gdansk (PL)

1,384,015

54.8

8.7

2.8

15.1

3.4

8.5

38.7

5.7

56.5

Katowice (PL)

1,130,209

40.7

9.9

5.1

21.0

2.7

9.1

33.8

6.4

40.4

Bucharest (RO)

2,183,091

85.9

22.2

13.6

36.8

6.9

26.1

66.5

14.5

52.1

SOUTH EUROPE

 

 

 

 

 

 

Athens (GR)

4,164,475

66.2

12.2

9.4

38.8

5.3

14.8

55.2

9.2

63.4

Milan (IT)

4,135,172

74.2

35.1

14.5

21.6

10.6

24.2

34.9

18.4

59.9

Rome (IT)

4,086,779

64.1

20.8

6.1

15.8

3.6

10.8

25.8

6.9

56.6

Naples (IT)

3,545,562

79.1

70.2

18.4

23.9

8.9

29.7

37.7

16.0

61.9

Turin (IT)

1,761,392

49.5

7.6

10.4

44.0

7.6

17

61.8

13.3

61.4

Palermo (IT)

1,002,723

65.7

10.7

5.0

24.6

2.6

9.2

38.8

12.0

54.3

Lisbon (PT)

2,817,901

66.1

14.6

9.3

30.9

6.8

14.0

42.3

9.2

66.4

Porto (PT)

1,294,719

75.5

50.3

17.2

24.0

10.3

27.0

37.1

16.8

63.7

Madrid (ES)

6,569,530

9.4

7.7

6.5

23.2

5.1

10.4

35.2

8.3

62.6

Barcelona (ES)

4,917,162

11.8

17.1

12.5

31.6

8.5

18.7

43.8

13.6

66.7

Valencia (ES)

1,631,202

9.1

7.7

8.0

23.1

6.8

12.1

32.0

10.5

66.3

Seville (ES)

1,413,882

8.4

3.0

3.5

31.2

2.6

5.0

41.8

3.9

68.9

Bilbao (ES)

1,031,440

16.0

10.9

4.1

20.8

2.1

6.5

29.3

3.7

63.7

1 Population estimates from https://ec.europa.eu/​eurostat/​web/​cities/​data/​database (2012, reference date)

2CORE area population estimated using data available in UA 2012 that provide population estimate for each polygon

3 For the urban regions of Rugrgebeit (DE) and Mannheim-Ludwighafen CORE/RING areas are not defined in Urban Atlas, since these regions are a conurbation of many cities

Table A2. Fractal dimension of the 60 European cities

 

DG_FUA

DB_FUA

DG_CORE

DB_CORE

DG_UMZ-CORE

DG_RING

DB_RING

UK/IRELAND

 

Dublin (IR)

1.23

1.41

1.45

1.66

1.52

1.12

1.29

London (UK)

1.46

1.67

1.62

1.85

1.58

1.33

1.52

Leeds (UK)

1.33

1.49

1.44

1.67

1.52

1.27

1.42

West Midlands (UK)

1.48

1.69

1.59

1.82

1.58

1.26

1.45

Manchester (UK)

1.47

1.62

1.56

1.75

1.53

1.21

1.44

Glasgow (UK)

1.36

1.55

1.54

1.77

1.55

1.31

1.49

Liverpool (UK)

1.47

1.68

1.48

1.65

1.47

1.35

1.57

Newcastle (UK)

1.18

1.34

1.50

1.75

1.53

1.05

1.15

SCANDINAVIA

 

Copenhagen (DK)

1.44

1.64

1.57

1.77

1.58

1.36

1.55

Helsinki (FL)

1.31

1.52

1.48

1.73

1.50

1.19

1.40

Oslo (NW)

1.22

1.37

1.48

1.69

1.44

1.17

1.31

Stockholm (SW)

1.24

1.39

1.41

1.6

1.48

1.11

1.24

CENTRAL EUROPE

 

Vienna (AT)

1.28

1.47

1.54

1.78

1.44

1.23

1.42

Brussels (BE)

1.41

1.59

1.57

1.8

1.46

1.38

1.55

Antwerp (BE)

1.47

1.68

1.49

1.75

1.50

1.41

1.60

Paris (FR)

1.37

1.57

1.62

1.87

1.54

1.31

1.48

Lyon (FR)

1.39

1.57

1.56

1.77

1.51

1.29

1.46

Marseille (FR)

1.34

1.51

1.44

1.64

1.50

1.29

1.43

Lille (FR)

1.42

1.61

1.51

1.71

1.56

1.24

1.41

Toulouse (FR)

1.27

1.44

1.53

1.75

1.55

1.17

1.33

Bordeaux (FR)

1.29

1.44

1.54

1.72

1.57

1.19

1.32

Berlin (DE)

1.27

1.57

1.60

1.81

1.56

1.22

1.38

Ruhrgebiet (DE)

1.51

1.71

-

-

-

-

-

Hamburg (DE)

1.36

1.53

1.56

1.78

1.54

1.29

1.45

Munich (DE)

1.31

1.51

1.58

1.80

1.55

1.27

1.44

Stuttgart (DE)

1.39

1.58

1.49

1.72

1.49

1.36

1.54

Frankfurt (DE)

1.37

1.55

1.50

1.73

1.50

1.33

1.51

Koln (DE)

1.46

1.67

1.53

1.76

1.49

1.39

1.58

Dusseldorf (DE)

1.48

1.69

1.5

1.73

1.52

1.42

1.62

Dresden (DE)

1.30

1.50

1.48

1.71

1.48

1.26

1.45

Nuremberg (DE)

1.32

1.49

1.49

1.72

1.48

1.27

1.42

Hannover (DE)

1.36

1.55

1.53

1.77

1.52

1.31

1.49

Bremen (DE)

1.25

1.42

1.48

1.71

1.48

1.20

1.35

Mannheim_Lud. (DE)

1.38

1.55

-

-

-

-

-

Amsterdam (NL)

1.44

1.62

1.52

1.70

1.56

1.38

1.56

Rotterdam (NL)

1.44

1.62

1.53

1.74

1.54

1.34

1.51

Zurich (SW)

1.40

1.63

1.47

1.68

1.46

1.33

1.54

EAST EUROPE

 

Sofia (BU)

1.26

1.43

1.5

1.73

1.53

1.16

1.32

Zagreb (CRO)

1.26

1.42

1.45

1.62

1.51

1.18

1.33

Prague (CZ)

1.31

1.48

1.52

1.74

1.53

1.24

1.40

Ostrava (HU)

1.34

1.57

1.45

1.70

1.44

1.26

1.48

Budapest (HU)

1.34

1.59

1.53

1.81

1.50

1.28

1.48

Warsaw (PL)

1.25

1.55

1.51

1.79

1.48

1.19

1.42

Krakow (PL)

1.36

1.58

1.45

1.70

1.49

1.22

1.41

Gdansk (PL)

1.28

1.49

1.47

1.73

1.45

1.21

1.40

Katowice (PL)

1.30

1.48

1.47

1.68

1.46

1.22

1.39

Bucharest (RO)

1.43

1.67

1.51

1.79

1.49

1.29

1.51

SOUTH EUROPE

 

Athens (GR)

1.42

1.61

1.56

1.79

1.56

1.34

1.51

Milan (IT)

1.47

1.68

1.53

1.73

1.57

1.38

1.57

Rome (IT)

1.37

1.55

1.48

1.67

1.55

1.27

1.45

Naples (IT)

1.52

1.71

1.56

1.76

1.57

1.39

1.52

Turin (IT)

1.40

1.60

1.52

1.77

1.51

1.34

1.54

Palermo (IT)

1.34

1.51

1.48

1.70

1.46

1.26

1.42

Lisbon (PT)

1.45

1.62

1.56

1.74

1.57

1.36

1.53

Porto (PT)

1.55

1.72

1.57

1.75

1.61

1.41

1.57

Madrid (ES)

1.38

1.55

1.56

1.75

1.56

1.35

1.49

Barcelona (ES)

1.46

1.64

1.54

1.74

1.53

1.39

1.55

Valencia (ES)

1.39

1.57

1.37

1.57

1.51

1.36

1.53

Seville (ES)

1.27

1.42

1.48

1.70

1.51

1.24

1.36

Bilbao (ES)

1.28

1.43

1.44

1.63

1.45

1.16

1.30

Table A3. Ranking of the 60 European cities with respect their fractal dimension (highest fractal dimension rank 1)

 

Rank DG_FUA

Rank

Rank DG_CORE

Rank DB_CORE

Rank DG_RING

Rank DB_RING

DB_FUA

UK/IRELAND

 

 

 

Dublin (IR)

58

57

52

52

56

56

London (UK)

12

11

2

2

21

17

Leeds (UK)

39

45

56

51

30

36

West Midlands (UK)

4

4

4

3

35

31

Manchester (UK)

8

16

13

21

45

33

Glasgow (UK)

31

32

19

13

24

23

Liverpool (UK)

9

6

42

53

14

4

Newcastle (UK)

60

60

34

25

58

58

SCANDINAVIA

 

 

 

 

Copenhagen (DK)

14

13

6

17

13

10

Helsinki (FL)

43

39

44

34

50

45

Oslo (NW)

59

59

45

47

52

54

Stockholm (SW)

57

58

57

57

57

57

CENTRAL EUROPE

 

 

 

Vienna (AT)

47

49

16

12

41

38

Brussels (BE)

20

23

7

7

9

9

Antwerp (BE)

6

7

38

20

2

2

Paris (FR)

29

29

1

1

22

27

Lyon (FR)

24

26

10

14

26

28

Marseille (FR)

36

42

54

54

28

35

Lille (FR)

19

19

29

39

40

42

Toulouse (FR)

52

51

22

24

53

51

Bordeaux (FR)

46

50

17

36

48

52

Berlin (DE)

50

27

3

4

42

47

Ruhrgebiet (DE)

3

3

 

 

Hamburg (DE)

33

38

12

11

25

29

Munich (DE)

41

40

5

6

31

34

Stuttgart (DE)

25

25

37

38

12

11

Frankfurt (DE)

30

33

32

31

20

19

Koln (DE)

10

9

21

19

6

3

Dusseldorf (DE)

5

5

33

35

1

1

Dresden (DE)

44

43

46

40

34

30

Nuremberg (DE)

40

44

36

37

32

40

Hannover (DE)

32

35

23

16

23

24

Bremen (DE)

56

56

39

41

47

49

Mannheim_Lud. (DE)

27

34

 

 

Amsterdam (NL)

15

17

26

46

7

7

Rotterdam (NL)

16

15

20

28

17

18

Zurich (SW)

21

14

48

48

19

13

EAST EUROPE

 

 

 

Sofia (BU)

54

52

35

30

55

53

Zagreb (CRO)

53

54

53

56

51

50

Prague (CZ)

42

47

28

29

39

44

Ostrava (HU)

37

30

50

45

37

25

Budapest (HU)

35

22

24

5

29

26

Warsaw (PL)

55

36

31

10

49

39

Krakow (PL)

48

46

49

32

46

43

Gdansk (PL)

45

48

47

49

43

46

Katowice (PL)

34

24

51

42

44

41

Bucharest (RO)

17

10

30

8

27

20

SOUTH EUROPE)

 

 

 

Athens (GR)

18

20

11

9

18

21

Milan (IT)

7

8

25

33

8

5

Rome (IT)

28

31

43

50

33

32

Naples (IT)

2

2

15

18

4

16

Turin (IT)

22

21

27

15

16

12

Palermo (IT)

38

41

41

43

36

37

Lisbon (PT)

13

18

9

26

11

15

Porto (PT)

1

1

8

22

3

6

Madrid (ES)

26

37

14

23

15

22

Barcelona (ES)

11

12

18

27

5

8

Valencia (ES)

23

28

58

58

10

14

Seville (ES)

51

55

40

44

38

48

Bilbao (ES)

49

53

55

55

54

55

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Notes

1 To be consistent with the use of the 2012 IMD HRL dataset, population data refer to the 2010-2012 period (https://ec.europa.eu/eurostat/web/cities/data/database).

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Table des illustrations

Titre Figure 1. Grayscale (left) and binary (right) representation of the artificially sealed surfaces in the CORE area of Copenhagen.
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-1.png
Fichier image/png, 379k
Titre Figure 2. Box-plots of the IMPER and DEVEL indicators
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-2.png
Fichier image/png, 60k
Titre Figure 3. Soil sealing profiles for selected European cities
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-3.png
Fichier image/png, 119k
Titre Figure 4. Fractal dimension values of the FUA areas of the 60 cities
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-4.png
Fichier image/png, 273k
Titre Figure 5. Fractal dimension values of the CORE areas of the 60 cities
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-5.png
Fichier image/png, 269k
Titre Figure 6. Fractal dimension values of the RING areas of the 60 cities
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-6.png
Fichier image/png, 270k
Titre Figure 7. Plot of DG_ CORE versus DG_ RING values
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-7.png
Fichier image/png, 41k
Titre Figure 8. FUA and CORE boundaries for the city of Ostrava; CORE area definition using the Urban Atlas and the Urban Morphological Zones approach.
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-8.png
Fichier image/png, 296k
Titre Figure 9. Comparison of DG_CORE and DG_CORE-UMZ (CONVEX HULL) values, cities sorted by DG_CORE values.
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-9.png
Fichier image/png, 61k
Titre Figure 10. Comparison of 3 cities with similar mean density of development (IMPER = 24 in all cases) and different fractal dimensions on the CORE level (Porto/Budapest/Palermo).
URL http://journals.openedition.org/cybergeo/docannexe/image/37243/img-10.png
Fichier image/png, 180k
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Référence électronique

Apostolos Lagarias et Poulicos Prastacos, « Fractal dimension of European Cities: A comparison of the patterns of built-up areas in the urban core and the peri-urban ring », Cybergeo : European Journal of Geography [En ligne], Systèmes, Modélisation, Géostatistiques, document 987, mis en ligne le 23 juillet 2021, consulté le 01 août 2021. URL : http://journals.openedition.org/cybergeo/37243 ; DOI : https://doi.org/10.4000/cybergeo.37243

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Auteurs

Apostolos Lagarias

Postdoc Researcher, Regional Analysis Group, Institute of Applied and Computational Mathematics, Foundation for Research and Technology-Hellas (FORTH), Greece
Corresponding author: lagarias@iacm.forth.gr

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Poulicos Prastacos

Director of research, Regional Analysis Group, Institute of Applied and Computational Mathematics, Foundation for Research and Technology-Hellas (FORTH), Greece

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