# The Fractal Geometry of Urban Land Use: The Case of Ulaanbaatar City, Mongolia

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Data Source and Spatial Analysis Method of GIS

#### 2.3. Estimation of Fractal Geometry of Urban Land Use

## 3. Results

#### 3.1. Urban Land-Use Changes and Expansion

#### 3.2. Compactness Ratio and Fractal Dimension

#### 3.3. Fractal Geometry of Urban Land Use

## 4. Discussion

_{∆t}= 0.09) and slightly increased. This suggests that in 2000 the city’s built-up area boundary already had been in irregular shape, historically city expansion performed along the narrow river valleys as a result of specific geographical locations: mountainous terrain barrier, steep slope, and lack of plain areas. However, a slight increase of fractal dimension reveals that the city is in expansion and a more complex and asymmetrical urban foot print appeared over 10 years. In other words, the fractal dimension analysis was not particularly impressive for historically asymmetrical-linearly developed Ulaanbaatar city, where expansion along the river valleys has to increase the area-perimeter relationship in any case.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chen, Y. A New Model of Urban Population Density Indicating Latent Fractal Structure. Int. J. Urban Sustain. Dev.
**2010**, 1, 89–110. [Google Scholar] [CrossRef] - Myagmartseren, P.; Myagmarjav, I.; Otgongerel, B. Urban Compactness Measurement of the Ulaanbaatar City. Geogr. Issues
**2015**, 14, 51–59. [Google Scholar] - Zhan, Q.; Wang, J.; Gong, P.; Shi, P. Urban built-up land change detection with roads density and spectral information from multi-temporal Landsat TM data. Int. J. Remote Sens.
**2002**, 23, 3057–3078. [Google Scholar] [CrossRef] - Triantakonstantis, D.P. Urban Growth Prediction Modelling Using Fractals and Theory of Chaos. Open J. Civ. Eng.
**2012**, 2, 81–86. [Google Scholar] [CrossRef] - Batty, M.; Longley, P.A. The Fractal Simulation of Urban Structure. Environ. Plan. A Econ. Space
**1986**, 18, 1143–1179. [Google Scholar] [CrossRef] - Batty, M.; Xie, X. Self-Organized Criticality and Urban Development. Discret. Dyn. Nat. Soc.
**1999**, 3, 109–124. [Google Scholar] [CrossRef] - Batty, M. Exploring isovist fields: Space and shape in architectural and urban morphology. Environ. Plan. B Plan. Des.
**2001**, 28, 1231–1250. [Google Scholar] [CrossRef] - Benguigui, L.; Czamanski, D.; Marinov, M.; Portugali, Y. When and Where Is a City Fractal. Environ. Plan. B Plan. Des.
**2000**, 27, 507–519. [Google Scholar] [CrossRef] - Chen, Y.G.; Zhou, Y.X. Scaling laws and indications of self-organized criticality in urban systems. Chaos Solitons Fractals
**2008**, 35, 85–98. [Google Scholar] [CrossRef] - Fotheringham, A.S.; Batty, M.; Longley, P.A. Diffusion-Limited Aggregation and the Fractal Nature of Urban Growth. Pap. Reg. Sci. Assoc.
**1989**, 67, 55–69. [Google Scholar] [CrossRef] - Shen, G. Fractal Dimension and Fractal Growth of Urbanized Areas. Int. J. Geogr. Inf. Sci.
**2002**, 16, 437–519. [Google Scholar] [CrossRef] - Tannier, C.; Thomas, I.; Vuidel, G.; Frankhauser, P. Fractal Approach to Identifying Urban Boundaries. Geogr. Anal.
**2011**, 43, 211–227. [Google Scholar] [CrossRef] - Thomas, I.; Frankhauser, P.; Biernacki, C. The Morphology of Built-Up Landscapes in Wallonia (Belgium): A Classification Using Fractal Indices. Landsc. Urban Plan.
**2008**, 84, 99–115. [Google Scholar] [CrossRef] - Hastings, H.M.; Sugihara, G. Fractals: A User’s Guide for the Natural Sciences; Oxford University Press: Oxford, UK, 1994. [Google Scholar]
- Terzi, F.; Kaya, H.S. Analyzing Urban Sprawl Patterns through Fractal Geometry: The Case of Istanbul Metropolitan Area; Paper 144; Centre for Advanced Spatial Analysis: London, UK, 2008. [Google Scholar]
- Pastzo, V.; Marek, L.; Tucek, P.; Janoska, Z. Perspectives of fractal geometry in GIS analysis. GIS Ostrav.
**2011**, 1, 232–236. [Google Scholar] - De Keersmaecker, M.-L.; Frankhauser, P.; Thomas, I. Using fractal dimensions for characterizing intra-urban diversity: The example of Brussels. Geogr. Anal.
**2003**, 35, 3103–3128. [Google Scholar] [CrossRef] - Frankhauser, P. Fractal Aspects of Urban Systems. Sonderforschungsbereich
**1988**, 230, 67–76. [Google Scholar] - Theiler, J. Estimating fractal dimension. J. Opt. Soc. Am. A
**1990**, 7, 1055–1073. [Google Scholar] [CrossRef] - Mandelbrot, B. The Fractal Geometry of Nature; Freeman: San Francisco, CA, USA, 1983. [Google Scholar]
- Addison, P.S. Fractals and Chaos: An Illustrated Course; Institute of Physics Publishing: Bristol, UK, 1997. [Google Scholar]
- Thomas, I.; Frankhauser, P.; Keersmaecker, M.L.D. Fractal dimension versus density of built-up surfaces in the periphery of Brussels. Pap. Reg. Sci.
**2007**, 86, 287–308. [Google Scholar] [CrossRef] - Morency, C.; Chapleau, R. Fractal geometry for the characterisation of urban-related states: Greater Montreal Case. Harmon. Fractal Image Anal.
**2003**, 30–34. Available online: www.fch.vut.cz/lectures/imagesci/download_ejournal/09_C.Morency.pdf (accessed on 17 April 2018). - Myagmartseren, P.; Myagmarjav, I.; Erdenejargal, B. Land Use Changes in Urban Areas of Mongolia. In Proceedings of the IX International Conference Environment and Sustainable Development in Mongolian Plateau and Surrounding Regions, Ulan-Ude, Russia, 20–22 August 2013; Buryat State University: Ulan Ude, Russia, 2013; Volume 2, pp. 18–21. [Google Scholar]
- Myagmartseren, P.; Myagmarsuren Buyandelger, M.; Brandt, S.A. Implications of a Spatial Multicriteria Decision Analysis for Urban Development in Ulaanbaatar, Mongolia. Math. Probl. Eng.
**2017**, 2017, 2819795. [Google Scholar] [CrossRef] - Victorian Department of Planning and Community Development; Open Space Planners Network in Victoria. Open Space Planning and Design Guide; Park and Leisure Australia: Norwood, Australia, 2013. [Google Scholar]
- Myagmartseren, P.; Mayagmarjav, I.; Otgongerel, B. Multi-Criteria Decision Analysis for Cropland Suitability Survey. J. Geogr. Rev. Mong.
**2014**, 10, 1301–1336. [Google Scholar] - Ge, M.; Lin, Q. Realizing the Box-counting Method for Calculating Fractal Dimension of Urban Form Based on Remote Sensing Image. Geo-Spat. Inf. Sci.
**2009**, 12, 2652–2670. [Google Scholar] [CrossRef] - Torrens, P.M.; Alberti, M. Measuring Sprawl, Paper 27; Centre for Advanced Spatial Analysis: London, UK, 2000. [Google Scholar]
- Kaye, B.H. A Random Walk through Fractal Dimensions; VCH Publishers: New York, NY, USA, 1989. [Google Scholar]
- Frolov, Y.S. Measuring shape of geographical phenomena—History of issues. Sov. Geogr. Rev. Transl.
**1975**, 16, 676–687. [Google Scholar] - Herold, M.; Scepan, J.; Clarke, K. The use of remote sensing and landscape metrics to describe structures and changes in urban land uses. Environ. Plan. A
**2002**, 34, 1443–1458. [Google Scholar] [CrossRef] - Chu, J. Study on Quantitative Analysis of Urban Spatial Form; Southeast University Press: Nanjing, China, 2007. [Google Scholar]
- Purevtseren, G.; Myagmartseren, P.; Jigjidsuren, S. Master Land Use Plan of Ulaanbaatar City; Governor Office of Capital City: Ulaanbaatar, Mongolia, 2001. [Google Scholar]
- Fan, W.; Shi, Y.; Liu, Y. Application of GIS in quantifying the urban form: A case study of Shanghai. J. Digit. Content Technol. Appl.
**2012**, 6, 3443–3453. [Google Scholar] - Falconer, K.J. Fractal Geometry: Mathematical Foundations and Applications; John Wiley & Sons: New York, NY, USA, 1999. [Google Scholar]
- Akashi, T. Urban Land Use Planning System in Japan, 2nd ed.; JAICA: Tokyo, Japan, 2007; pp. 4–5. [Google Scholar]

**Figure 1.**Map over the study area. (

**a**) Location of Ulaanbaatar; (

**b**) map with forest, river and urban areas (grey = built-up area; green = forest; blue = river).

**Figure 4.**Change of fractal dimensions of urban form and sprawl index for Ulaanbaatar in 2000 and 2010.

**Table 1.**Data sources used to create the various information layers for the land-use classes of built-up area.

Features Considered | Source | Method of Creation |
---|---|---|

Built-up areas | Landsat 7 satellite image | Supervised classification |

Forest areas (urban internal part reclassed as open space) | Landsat 7 satellite image | Supervised classification |

Residential, industrial, mixed use (built-up area reclassed) | Land use classification map Land use zoning map | Created by Land Administration Department of Capital City with participation of authors |

Rivers | Aster GDEM (Global Digital Elevation Model) of 30 m | Geo-processing/stream delineation |

Vegetation, barren land (reclassed as open space) | Landsat 7 satellite image | Normalized Difference Vegetation Index (NDVI) |

Residential-sprawl (ger area), residential-middle/high rise | Cadastral map | Created by Agency of Land Affairs, Geodesy and Cartography |

Road network | Road map | Digitizing/georeferencing |

Land Use | 2000 | 2010 | ||
---|---|---|---|---|

1 | Residential | total | 33.2% | 58.7% |

sprawl | 28.2% | 51.8% * | ||

2 | Infrastructure | 5.2% | 9.4% | |

3 | Industry | 10.0% | 8.3% | |

4 | Service | 13.7% | 6.3% | |

5 | Special purpose and others (defense, public land, water buffer etc.) | 6.9% | 15.1% | |

6 | Unused and Reservation | 28.2% | 10.0% |

Year | Area (ha) | Perimeter | Feret’s Diameter | Compactness Index | Fractal Dimension |
---|---|---|---|---|---|

$\mathbf{L}\left(\mathbf{r}\right)={\displaystyle {\displaystyle \sum}_{\mathit{i}=1}^{\mathit{n}-1}}{\mathit{d}}_{\mathit{i},\mathit{i}+1}$ | $\mathbf{F}=\mathit{m}\mathit{a}{\mathit{x}}_{\mathit{i},\mathit{i}+1\left({\mathit{d}}_{\mathit{i},\mathit{i}+1}\right)}$ | $\mathbf{C}=2\sqrt{\mathsf{\pi}\mathbf{A}}/\mathbf{P}$ | $\mathbf{D}=\frac{2\mathbf{log}\mathbf{P}/4}{\mathbf{log}\mathbf{A}}$ | ||

2000 | 15,141 | 155,927 | 28,955 | 0.28 | 1.12 |

2010 | 50,459 | 762,116 | 68,750 | 0.10 | 1.21 |

ΔT | 35,318 | 606,189 | 39,795 | 018 | 0.09 |

Land Use | Area (ha) | Number of Parcels | Number of Points | Number of Common Points | ||||
---|---|---|---|---|---|---|---|---|

2000 | 2010 | 2000 | 2010 | 2000 | 2010 | 2000 | 2010 | |

Sprawl | 6070 | 26,108 | 8 | 31 | 1253 | 5509 | 691 | 2077 |

Residential | 1309 | 2941 | 8 | 10 | 302 | 774 | 302 | 701 |

Industrial | 1821 | 1184 | 5 | 3 | 226 | 205 | 210 | 205 |

Mixed land use | 1322 | 3839 | 13 | 10 | 408 | 941 | 408 | 926 |

Open space | 4619 | 16,387 | 14 | 22 | 775 | 2744 | 542 | 1823 |

Urban foot print | 15,141 | 50,459 | 48 | 76 | 2964 | 10,173 | 2153 | 5732 |

Land Use | Area (ha) A | Length of Perimeter L(r) = P | Feret’s Diameter F | Fractal Dimension D | Compactness Index C | |||||
---|---|---|---|---|---|---|---|---|---|---|

2000 | 2010 | 2000 | 2010 | 2000 | 2010 | 2000 | 2010 | 2000 | 2010 | |

Sprawl | 6070 | 26,108 | 210,835 | 773,559 | 28,955 | 58,200 | 1.21 | 1.26 | 0.13 | 0.07 |

Residential | 1309 | 2941 | 50,680 | 106,311 | 28,460 | 32,220 | 1.15 | 1.18 | 0.25 | 0.18 |

Industrial | 1821 | 1184 | 51,517 | 31,554 | 28,810 | 29,970 | 1.30 | 1.10 | 0.29 | 0.39 |

Mixed land use | 1322 | 3839 | 63,627 | 128,175 | 24,865 | 35,220 | 1.18 | 1.19 | 0.20 | 0.17 |

Open space | 4619 | 16,387 | 156,231 | 437,905 | 27,650 | 68,750 | 1.20 | 1.23 | 0.15 | 0.10 |

Urban foot print | 15,141 | 50,459 | 155,927 | 762,116 | 28,955 | 68,750 | 1.12 | 1.21 | 0.28 | 0.10 |

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**MDPI and ACS Style**

Purevtseren, M.; Tsegmid, B.; Indra, M.; Sugar, M.
The Fractal Geometry of Urban Land Use: The Case of Ulaanbaatar City, Mongolia. *Land* **2018**, *7*, 67.
https://doi.org/10.3390/land7020067

**AMA Style**

Purevtseren M, Tsegmid B, Indra M, Sugar M.
The Fractal Geometry of Urban Land Use: The Case of Ulaanbaatar City, Mongolia. *Land*. 2018; 7(2):67.
https://doi.org/10.3390/land7020067

**Chicago/Turabian Style**

Purevtseren, Myagmartseren, Bazarkhand Tsegmid, Myagmarjav Indra, and Munkhnaran Sugar.
2018. "The Fractal Geometry of Urban Land Use: The Case of Ulaanbaatar City, Mongolia" *Land* 7, no. 2: 67.
https://doi.org/10.3390/land7020067