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The detection of natural cities in the Netherlands—Nocturnal satellite imagery and Zipf’s law

Die Abgrenzung natürlicher Städte in den Niederlanden: Nachtsatellitenbilder und das Zipf-Gesetz


How to detect the true extent of cities in highly urbanized countries? This paper addresses the delineation of natural urban and non-urban space and its change based on a wider understanding of spatial heterogeneity. The Netherlands is selected as a case study. “Natural” means the extent of urban space irrespective of administrative boundaries. The database, used for this study, is radiance-calibrated nocturnal satellite imagery from the Defence Meteorological Satellite Program (DMSP). Extraction of cities is done by K-means segmentation. Based on the variance of luminosity it is possible to detect natural urban space. After removal of outliers in the skewed pixel distributions and after correction of “blooming” (over-glow of light emission) Zipf’s law is then applied as a test for segmentation adequacy. The comparative analysis for the years 1996 and 2011 shows that the rank-size distribution of natural cities is well confirmed by Zipf’s law, in contrast to that of administrative cities.


Wie lässt sich die wahre Ausdehnung von Städten in hochgradig urbanisierten Staaten erkennen? Die Studie behandelt die Differenzierung von natürlichem städtischen und nicht-städtischen Raum und seinem Wandel in einem erweiterten Sinne von räumlicher Heterogenität. Die Niederlande werden als Fallstudie betrachtet. „Natürlich“ meint hierbei die Unabhängigkeit städtischer Ausdehnung von administrativ gezogenen Grenzen. Die verwendete Datenbasis sind radianzkalibrierte Nachtsatellitenbilder des Defence Meteorological Satellite Program (DMSP). Die Extrahierung der Städte erfolgt unter Anwendung der K-means-Segmentierung. Auf Basis der Varianz der Lichtemission lässt sich so der natürliche städtische Raum sichtbar machen. Nach Beseitigung von statistischen Ausreißern in den schiefen Pixel-Verteilungen und nach Korrektur der Verzerrungen durch Überstrahlungseffekte wird die gewonnene Segmentierung mit Hilfe des Zipf-Gesetzes auf Angemessenheit getestet. Die vergleichende Analyse für die Jahre 1996 und 2011 zeigt, dass die Rangverteilung der natürlichen Städte durch das Zipf-Gesetz bestätigt wird, ganz im Gegensatz zur Rangverteilung der Städte in ihren Verwaltungsgrenzen.

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  1. 1.

    While spatial heterogeneity usually represents the first-order moments of space, spatial dependence is conceived as its second-order property. However, there are arguments that the relationship is contrary, namely that spatial heterogeneity is of higher order. Jiang (2015, p. 6) argues that “… Spatial heterogeneity is a kind of hidden order, which appears disordered on the surface, but possesses a deep order beneath. This kind of hidden order can be characterized by a power law or a heavy-tailed distribution in general. …”. So if just viewing the visible heterogeneity at the surface only part of its complex underpinnings is regarded.

  2. 2.

    In comparing levels of urbanization among EU countries with data from the European Commission and Eurostat, the Netherlands and the UK can be depicted at the top of the ranking (Eurostat 2017).

  3. 3.

    Care is definitively needed to use the term in an interdisciplinary dialogue as it had been also used in philosophical debates, such as the issue of reintegrating the natural with the urban (e.g. Stefanovic and Scharper 2012).

  4. 4. (accessed 23 February 2017).

  5. 5.

    Jiang et al. (2015) found exponents close to −1 in the range of any size, but notably those in the range above ten square kilometers, thus well-fitting Zipf’s law at the world, continental and national scales.

  6. 6.

    Analyses cover estimates for cities proper and city agglomerations. A problem with both approaches has been definition and international comparison.

  7. 7.

    This is also confirmed by Jiang et al. (2015) at world and European scale.

  8. 8.

    Further to that, the study of Jiang et al. (2015) finds substantially stronger evidence of Zipf’s law for industrialized countries vs. developing and emerging countries. Estimates by Brakman et al. (1999) and Soo (2005)—both for population based estimates—do not indicate a systematic difference between levels of economic development.

  9. 9.

    A plausible explanation for that is the influence of asymmetric industrial shocks in addition to symmetric policy and regional shocks (Gabaix 1999b), violating the assumption of scale invariance at least temporarily for the rank distribution of the bigger cities. By applying a maximum likelihood ratio test, Malevergne et al. (2011) show for the US cities, that the upper tail of the distribution is in fact Pareto and not lognormally distributed.

  10. 10.

    Cf: The radiance-calibrated composites are based on a limited set of observations where the gain of the detector was set much lower than its typical operational setting. The combination of those sparsely acquirable data at low gain settings with the operational data retrieved at high gain settings made it possible to produce a set of global nighttime lights products with no sensor saturation. However, the pixel values of the radiance-calibrated images are not digital numbers (DN). They are deemed unitless because of the lacking on-board calibration. The images thus show the true variation of luminosity but not the absolute radiance. The range in the images explored in this study is between 0 and 1,237 ( Since 2012, the 14-bit onboard-calibrated VIIRS images (Visible Infrared Imaging Radiometer Suite) are available. Those images are much more precise, but a long-run analysis of urbanization and evolution of Zipf’s law is not yet possible.

  11. 11.

    Inter-calibration is based on the methodology of Hsu et al. (2015). This study addressed the issue of missing inter-calibration (and thus comparability of images) by relating all radiance-calibrated composites to the annual one for 2006 by linear regression. 2006 was selected as it provides a better cloud-free coverage over the world. Hence the matrix of pixel values in the images is multiplied by different annual coefficients and constants.

  12. 12.

    Principally, with both, K-means as well as Isodata clustering it could be also possible to detect the typical peri-urban range or further differentiation from rural to urban, because it is not restricted to just two clusters. Isodata segmentation can be a superior choice when looking for inherent spectral clusters (Pandiya et al. 2013).

  13. 13.

    In case of selecting K> 2, the threshold separating the upper segment from the remaining distribution is preferred. Any other threshold would be rather arbitrary since the final classification is a binary one: urban vs. non-urban.

  14. 14.

    In accordance to Hsu et al. (2015) as described earlier.

  15. 15.

    The medcouple is a robust measure of distribution skewness of a distribution function F:\(MC(F)=\underset{x_{1}<m_{F}<x_{2}}{\text{med}}\cdot h(x_{1},x_{2})\), where mF is the median of F and the kernel function h is \(h(x_i,x_j)=\frac{(x_j-m_F)-(m_F-x_i)}{x_j-x_i}\).

  16. 16.

    Dissimilarity measure: continuous; Distance: Euclidian; the random seed is unimportant in univariate clustering as results do not differ.

  17. 17.

    The Gabaix-Ibragimov approach aims to remove the bias of the simple OLS estimates when samples are small. This coefficient is obtained by reducing the rank by ½ and then using the standard OLS approach, i.e.: ln (R1/2)=cα ln(S). The standard error of the Pareto exponent differs from the OLS standard error. It is asymptotically (2/n)1/2α (cf. Gabaix and Ibragimov 2011, p. 30).

  18. 18.

    The lower tails of the logarithmic regressions are concavely bowed. For the 2011 data (K = 2) a Shapiro-Francia normality test on the 50 lower tail observations of the log-transformed observations could not reject the Null hypothesis (= normal distribution) at the 95% level, while for 21 upper tail observations (see above) p < 0.0001, thus strongly rejecting the Null hypothesis.

  19. 19.

    The purpose of the Pseudo-F statistic is not to define a minimum threshold but rather to find the optimum choice of K in terms of intra- and inter-cluster variance. Hence, a two-cluster segmentation of the images is not per se inappropriate. Only the proportion of the variances is lower than for K = 3 or K = 4.

  20. 20.

    In the extraction of segmented space, the natural city area of Leeuwarden has a size of 85 pixels and belongs thus to the truncated tail of patches <100 pixels for the 2011 analysis. According to the Centraal Bureau voor de Statistiek, in 2011 the number of inhabitants of Leeuwarden was 94,838. In 1996, the city area of Dordrecht has a size of 109 pixels, thus selected within the tail with patches >100 pixels. The number of inhabitants of that city was 118,810.

  21. 21.

    The coefficient and the constant were originally estimated in a restricted study on twenty lighted settlements in California (Elvidge et al. 2004). Small et al. (2005, p. 287–288) confirmed those prior results by comparing it with an estimate on lateral blooming of small and large urbanized islands worldwide. The idea behind this exercise was the fact that urbanized island coasts represent an unambiguous boundary between potential sources of urban light emission and unlighted water. The results of both were consistent and were deemed by the authors to provide a basis for correcting the blooming problem. It is to be stressed, that this statistical correction still does not necessarily represent the true extent of natural Dutch cities. Blooming effects cannot be removed from the maps.

  22. 22.

    The distribution of lit areas below that threshold is flatter than for the non-adjusted data. The reason for that is a strongly stretched lower tail converting sizes <1 pixel into negative logarithmic values.

  23. 23.

    The map, showing the extent of administrative cities, is based on an available shapefile from 2015. It is not exactly congruent with the 2011 boundaries, but the boundaries of all highlighted municipalities remained constant during that period. Between 2011 and 2015 there have been few minor municipal reforms so-called “Gemeentelijke Herindelingen” described in: Wikipedia (2017).

  24. 24.

    Cf. Jiang et al. (2015, p. 509), saying that “… for all cities of a country to constitute a whole, the country must usually be of a certain size. A country might not be a legitimate unit for Zipf’s law …”.

  25. 25.

    The functional division of urban and rural space must have an economic foundation because of the uneven spatial distribution of production factors and accessibility (thus the non-homogeneity of space) and the resulting incentive for people to move (cf. Starrett 1978, p. 36; Brakman et al. 2009, pp. 51 ff.).


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The author thanks Rüdiger Budde and two anonymous referees for help and useful comments. An earlier draft version of this paper titled “Segmentation of urban and non-urban space in the Netherlands—Testing Zipf’s law with nocturnal satellite imagery” had been prepared for the ERSA 2017 conference in Groningen. It is accessible on:

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Correspondence to Rolf Bergs.

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Bergs, R. The detection of natural cities in the Netherlands—Nocturnal satellite imagery and Zipf’s law. Rev Reg Res 38, 111–140 (2018).

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  • Natural cities
  • Segmentation of space
  • Satellite imagery
  • Zipf’s law


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  • R 12