Fractal Description of the Temporal Fluctuation of PM2.5 and PM10 Concentrations and their Cross-correlation at Cotonou Autonomous Port and the “Boulevard de la Marina” area (Benin Republic, West Africa)

Document Type : Original Research Paper


1 Laboratoire de Physique du Rayonnement (LPR), Université d’Abomey-Calavi, Abomey-Calavi BP : 526 UAC, Bénin

2 International Chair in Mathematical Physics and Applications (UNESCO-Chaire UAC-CIPMA)

3 International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), Université d’Abomey-Calavi, Abomey-Calavi BP: 526 UAC, Bénin



The present study aims to provide baseline information on the temporal characteristics of PM2.5 and PM10 concentration time series variations, mainly on the cross-correlation between PM2.5 and PM10, using the improved mathematical and nonlinear methods. Firstly, the fractal theory such as fractal dimension is used to detect the pollution level in PM2.5 and PM10 time series. Secondly, the Multifractal Detrending Moving-Average Analysis (MFDMA) is used to analyze the multifractal characteristics of PM2.5 and PM10 concentrations. Thirdly, Multifractal Detrending Moving-Average cross-correlation Analysis (MFXDMA) is used to study the cross-correlation between PM2.5 and PM10 concentrations measured from January 1 to December 31, 2020, along the Boulevard de la Marina, one of the major roads in Cotonou. The results have indicated that: (1) PM10 and PM2.5 concentration time series are characterized by a fractal dimension, which can permit to detect the pollution levels and to analyze the differences in emissions sources; (2) there is a significant multifractal structure in the PM2.5 and PM10 concentration data and their fluctuations are long-range correlated, however, the multifractal properties and self-memory characteristics change with the months; (3) generally, the multifractal degree and the complexity of PM10 are much stronger than those of PM2.5. However, they present a similar multifractality degree in some months of the year; (4) except, in February, the cross-correlation between PM2.5 and PM10 time series in the months of the year presents multifractal characteristics with positive persistence; (5) the cross-correlation multifractal features show monthly variation. This paper provides the inter-relationship between air PM2.5 and PM10 time series which may help taking steps in controlling the air quality and management of the Cotonou port area environment.


Agbazo, M.N., Koto N’Gobi, G., Alamou, E., Kounouhewa, B. and Afouda, A. (2021). Assessing Nonlinear Dynamics and Trends in Precipitation by Ensemble Empirical Mode Decomposition (EEMD) and Fractal Approach in Benin Republic (West Africa). Hindawi, Complexity, vol. 2021, Article ID 3689397, 14 pages.
Agbazo, M.N., Koto N’Gobi, G., Alamou, E., Kounouhewa, B. and Afouda, A. (2019). Fractal analysis of the long-term memory in precipitation over Benin (West Africa). Advances in Meteorology, vol. 2019, no. 2, pp. 1-12.
Arianos, S. and Carbone, A. (2007). Detrending moving average algorithm: a closed-form approximation of the scaling law. Physica A 382, 9-15.
Awokola, B. I., Okello, G., Mortimer, K.J., Jewell, C.P., Erhart, A. and Semple, S. (2020)
Measuring air quality for advocacy in Africa (MA3): Feasibility and practicality of longitudinal ambient PM2.5 measurement using low-cost sensors. International journal of environmental research and public health, 17(19), 7243.
Biaou, A. (2004). De la méso-échelle à la micro-échelle: désagrégation spatio-temporelle
multifractale des précipitations. Thèse de l’Ecole des Mines de Paris, Paris, France.
Dossou, K.M.and Glehouenou-Dossou, B. (2007). The vulnerability to climate change of Cotonou (Benin) rising in sea level. Environ. Urban, 19, 65-79.
Djossou, J., Léon, J.F., Akpo, A.B., Liousse, C., Yoboué, V., Bedou, M. and Awanou, C.N. (2018). Mass concentration, optical depth and carbon composition of particulate matter in the major southern west African cities of Cotonou (Benin) and Abidjan (Côte d’Ivoire). Atmospheric Chemistry and Physics, 18(9), 6275-6291
Esposito, E., De Vito, S., Salvato, M., Bright, V., Jones, R.L. and Popoola, O. (2016). Dynamic neural network architectures for on-field stochastic calibration of indicative Low-cost air quality sensing systems. Sensors and Actuators B. Chemical, 231, 701-713.
Evans, M.J., Knippertz, P., Akpo, A., Allan, R.P., Amekudzi, L., Brooks, B., Chiu, J.C., Coe, H., Fink, A.H., Flamant, C., Jegede, O.O., Leal-Liousse, C., Lohou, F., Kalthoff, N., Mari, C., Marsham, J.H., Yoboué, V. and Reimann-Zumsprekel, C. (2018). Policy-relevant findings of the Dacciwa project, doi: 10.5281/zenodo.1476843.
Feder, J. (1988). Fractals. Plenum Press, New York.
Gao, X., Wang, X. and Shi, H. (2019). Multifractal cascade analysis on the nature of air pollutants concentration time series over China. Aerosol and Air Quality. Research, 19(9), 2100-2114.
Gu, G.F. and Zhou, W.X. (2010). Detrending moving average algorithm for multifractals. Phys. Rev. E 82.
Hubert, P. and Carbonnel, J.P. (1989). Dimensions fractales de l’occurrence de pluie en climat soudano-sahélien. Hydrologie Continentale. 4(1) : 3-10.
Ho, D.S., Juang, L.C., Liao, Y.Y., Wang, C.C., Lee, C.K., Hsu, T.C. and Yu, C.C. (2004). The temporal variations of PM10 concentration in Taipei: a fractal approach. Aerosol and Air Quality Research 4(1), 38-55.
Jiang, Z.Q. and Zhou, W.X. (2011). Multifractal detrending moving average cross-correlation analysis. Phys. Rev. E 84, 016106.
 Kounouhewa, B., Koto N’Gobi, G., Houngue, H., Müller, L., Wirtz, M., Yurtsever-Kneer, S. and Barbe, S. (2020). Cotonou’s next breath: Particulate matter monitoring and Capturing. Scientific African, 8, e00367.
Kinney, P.L. (2008). Climate change, air quality, and human health. American journal of preventive medicine. 35(5), 459-467.
Knippertz, P., Coe, H., Chiu, J.C., Evans, M.J., Fink, A. H., Kalthoff, N. and Marsham, J.H. (2015). The DACCIWA project: Dynamics-aerosol-chemistry-cloud interactions in West Africa. Bulletin of the American Meteorological Society, 96(9), 1451-1460.
 Liu, Z., Wang, L. and Zhu, H. (2015). A time–scaling property of air pollution indices: a case study of Shanghai. China. Atmospheric Pollution Research, 6(5), 886-892.
Lee, C. K. (2002). Multifractal characteristics in air pollutant concentration time series. Water Air Soil Pollut. 135, 389–409.
Lee, C.K., Ho, D.S., Yu, C.C. and Wang, C.C. (2003a). Fractal analysis of temporal variation of air pollutant concentration by box counting. Environ. Modell. Software 18, 243–251.
Lee, C.K., Ho, D.S., Yu, C.C., Wang, C.C. and Hsiao, Y. H. (2003b). Simple multifractal cascade model for air pollutant concentration (APC) time series. Environmetrics 14, 255-269.
Liu, L., Huang, G.H, Liu, Y., Fuller, G.A. and Zeng, G.M. (2003). A fuzzy-stochastic robust programming model for regional air quality management under uncertainty. Engineering Optimization, 35(2), 177-199.
Lovejoy, S., Schertzer, D. and Tsonis, A. A. (1987). Functional Box-counting and Multiple Elliptical Dimensions of Rain. Science, 235: 1036-1038.
Mandelbrot, B. (1982). The fractal geometry of nature. San Francisco: W. H. Freeman.
Makowiec, D. and Fulinski, A. (2010). Multifractal detrended fluctuation analysis as the estimator of long-range dependence. Acta Physica Polonica, vol. 41, pp. 1025-1050.
Mama, D.D., Biaou, A., Adounkpe, M., Ahomadegbe, J., Youssao, M., Kouazounde, A. and Kouanda, J. (2013). Transport urbain au (Benin) et pollution atmosphérique : Evaluation quantitative de certains polluants chimiques de (Cotonou). International Journal of Biological and Chemical Sciences, 7(1), 377-386.
Mayer, H. (1999). Air pollution in cities. Atmospheric Environment, 33(24-25), 4029-4037.
Movahed, M.S., Jafari, G.R., Ghasemi, F., Rahvar, S., Tabar, M.R.R. (2006). Multifractal detrended fluctuation analysis of sunspot time series. J. Stat. Mech. Theory Exp., P02003-P02003.
Nikolopoulos, D. Moustris, K. Petraki, E., Koulougliotis, D. and Cantzos, D. (2019). Fractal and long-memory traces in PM10 time series in Athens, Greece Environments, 6(3), 29.
Nikolopoulos, D. Moustris, K., Petraki, E. and Cantzos, D. (2021). Long-memory traces in PM10 time series in Athens, Greece: investigation through DFA and R/S analysis. Meteorology and Atmospheric Physics, 133(2), 261-279.
Nikolopoulos, D., Alam, A., Petraki, E., Papoutsidakis, M., Yannakopoulos, P. and Moustris, K. (2021). Stochastic and Self-Organisation Patterns in a 17-Year PM10 Time Series in Athens, Greece. Entropy 2021, 23, 307.
Peitgen, H. O., Jurgens, H.and Saupe, D. (2004). Chaos and Fractals. Springer, Berlin.
Shi, K., Liu, C. and Huang, Y. (2015). Multifractal processes and self-organized Criticality of PM2. 5 during a typical haze period in Chengdu, China. Aerosol and Air Quality Research, 15(3), 926-934.
Shi, K., Liu, C. Q., Ai, N.  S. and Zhang, X. H. (2008). Using three methods to investigate time-scaling properties in air pollution indexes time series. Nonlinear Anal. Real World Appl., 9, 693–707.
Shi, K. Liu, C. Q. and Ai, N. S. (2009). Monofractal and multifractal approaches in investigating temporal variation of air pollution indexes. Fractals, 17(04), 513-521.
Wang, L., Zhang, H., Mao, L., Li, S. and Wu, H. (2020). Assessing Spatiotemporal Characteristics of Urban PM2.5, Using Fractal Dimensions and Wavelet Analysis. Mathematical Problems in Engineering, 15 pages.
World Health Organization (2016). Ambient air pollution: A global assessment of exposure and burden of disease. Working Papers, World Health Organization.
World Health Organization (2018). How Air Pollution is Destroying Our Health.  Available online:
World Health Organization (2020). Ambient Air Pollution, Global Health Observatory (GHO) data. Available online.
Xepapadeas, A. (1992). Optimal taxes for pollution regulation: Dynamic, spatial and stochastic characteristics. Natural Resource Modeling, 6(2), 139-170.
Xie, S. and Bao, Z. (2004). Fractal and multifractal properties of geochemical fields. Mathematical Geology, vol. 36, no. 7, pp. 847–864.
Xie, H. and He, H. (2019). Multifractal Property Between PM2.5 and PM10 in Hongkong Port. Journal of Atmospheric and Environmental Optics.
Xue, Y., Pan, W., Lu, W. Z. and He, H. D. (2015). Multifractal nature of particulate matters (PMs) in Hong Kong urban air. Science of the Total Environment,532, 744-751.
Xu, L. Ivanov, P.C., Hu, K., Chen, Z., Carbone A. and Stanley, H.E. (2005). Quantifying Signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques. Phys. Rev. E 71- 051101.
Xu, H.C., Gu, G.F. and Zhou, W. X. (2017). Direct determination approach for the multifractal detrending moving average analysis. Phys. Rev. E 96-052201.
Zhang, C., Ni, Z., Ni, L., Li, J. and Zhou, L. (2016). Asymmetrical multifractal detrending moving average analysis in time series of PM2.5 concentration. Physica A 457, 322-330.
Zhang, C. Wang, X., Chen, S., Zou, L., Zhang, X. and Tang, C. (2019). A study of daily PM2.5 concentrations in Hong Kong using the EMD-based MFDFA method.  Physica A 530, 121182.
Zhao, D., Chen, H., Yu, E. and Luo T. (2019). PM2.5/PM10 Ratios in Eight Economic Regions and Their Relationship with Meteorology in China. Hindawi, Advances in Meteorology, vol, 2019,15 pages,
Zhou, W. X. (2008). Multifractal detrended cross-correlation analysis for two non-stationary signals. Phys. Rev. E 77, 066211.