State estimation of urban air pollution with statistical, physical, and super-learning graph models

Matthieu Dolbeault; Olga Mula; Agustin Somacal

doi:10.3934/acse.2024008

Article Contents

2024, Volume 2, Issue 2: 130-151. Doi: 10.3934/acse.2024008

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State estimation of urban air pollution with statistical, physical, and super-learning graph models

1.
RWTH Aachen, Germany
2.
TU Eindhoven, Netherlands
3.
Sorbonne Université, France

^*Corresponding author: Agustin Somacal
^*Corresponding author: Agustin Somacal

Received: February 2024

Revised: May 2024

Early access: June 2024

Published: June 2024

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Abstract

We consider the problem of real-time reconstruction of urban air pollution maps. The task is challenging due to the heterogeneous sources of available data, the scarcity of direct measurements, the presence of noise, and the large surfaces that need to be considered. In this work, we introduce different reconstruction methods based on posing the problem on city graphs. Our strategies can be classified as fully data-driven, physics-driven, or hybrid, and we combine them with super-learning models. The performance of the methods is tested in the case of the inner city of Paris, France.

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Mathematics Subject Classification: Primary: 62P12; Secondary: 68T07.

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Figure 1. Cropped Google Map screenshot of Paris and the $ 13 $ available stations in the study: red dots represent the projection of the station locations to the nearest vertex in the graph of streets, while blue crosses correspond to the exact positions of the stations

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Figure 2. Raw data from Google Maps: the image contains the city with its main landmarks, and some streets are highlighted with one of the four colors corresponding to traffic

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Figure 3. The metric graph downloaded from Open Street Maps, with the edges that never had Google Traffic activation in red, and the edges remaining after filtration in yellow

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Figure 4. Correlation between stations as a function of the distance. The vertical slashed red line marks the maximal separation between vertex and station (165m) which still lays in the zone of high correlation

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Figure 5. Root mean square error on tested stations for the different proposed methods

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Figure 6. Pollution map for the ensemble model at $ 8 $am on March 1st, 2023. Based on the predictions on the node, one can linearly extrapolate pollution values even outside of the graph edges. Note that the fine variations in pollutant concentration (between 45 and 55 $ \mu \text{g/m}^3 $) seem to trace the main circulation axes

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References

[1]	M. Arioli and M. Benzi, A finite element method for quantum graphs, IMA Journal of Numerical Analysis, 38 (2018), 1119-1163. doi: 10.1093/imanum/drx029.
[2]	G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186, American Mathematical Soc., 2013. doi: 10.1090/surv/186.
[3]	P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Data assimilation in reduced modeling, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 1-29. doi: 10.1137/15M1025384.
[4]	L. Breiman, Stacked regressions, Machine Learning, 24 (1996), 49-64. doi: 10.1007/BF00117832.
[5]	J. Cai, J. Chen, H. Cheng, S. Zi, J. Xiao, F. Xia and J. Zhao, The effects of thermal stratification on airborne transport within the urban roughness sublayer, International Journal of Heat and Mass Transfer, 184 (2022), 122289. doi: 10.1016/j.ijheatmasstransfer.2021.122289.
[6]	A. Cohen, W. Dahmen, R. DeVore, J. Fadili, O. Mula and J. Nichols, Optimal reduced model algorithms for data-based state estimation, SIAM J. Numer. Anal., 58 (2020), 3355-3381. doi: 10.1137/19M1255185.
[7]	A. Cohen, W. Dahmen, O. Mula and J. Nichols, Nonlinear reduced models for state and parameter estimation, SIAM/ASA J. Uncertain. Quantif., 10 (2022), 227-267. doi: 10.1137/20M1380818.
[8]	A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs, Acta Numer., 24 (2015), 1-159. doi: 10.1017/S0962492915000033.
[9]	A. Cohen, R. DeVore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE's, Anal. Appl., 9 (2011), 11-47. doi: 10.1142/S0219530511001728.
[10]	A. Cohen, M. Dolbeault, O. Mula and A. Somacal, Nonlinear approximation spaces for inverse problems, Anal. Appl., 21 (2023), 217-253. doi: 10.1142/S0219530522400140.
[11]	N. Cressie, Statistics for Spatial Data, John Wiley & Sons, 2015.
[12]	A. Criado, J. M. Armengol, H. Petetin, D. Rodriguez-Rey, J. Benavides, M. Guevara, C. Pérez García-Pando, A. Soret and O. Jorba, Data fusion uncertainty-enabled methods to map street-scale hourly NO₂ in Barcelona: A case study with CALIOPE-Urban v1.0, Geoscientific Model Development, 16 (2023), 2193-2213. doi: 10.5194/gmd-16-2193-2023.
[13]	W. Dahmen, M. Wang and Z. Wang, Nonlinear reduced DNN models for state estimation, Commun. Comput. Phys., 32 (2022), 1–40, arXiv: 2110.08951. doi: 10.4208/cicp.OA-2021-0217.
[14]	J. L. Eftang, A. T. Patera and E. M. Rönquist, An "hp" certified reduced basis method for parametrized elliptic partial differential equations, SIAM J. Sci. Comput., 32 (2010), 3170-3200. doi: 10.1137/090780122.
[15]	J. K. Hammond, R. Chakir, F. Bourquin and Y. Maday, PBDW: A non-intrusive reduced basis data assimilation method and its application to an urban dispersion modeling framework, Appl. Math. Model., 76 (2019), 1-25. doi: 10.1016/j.apm.2019.05.012.
[16]	J. S. Hesthaven, G. Rozza and B. Stamm, Certified reduced basis methods for parametrized partial differential equations, SpringerBriefs in Mathematics, 2015. doi: 10.1007/978-3-319-22470-1.
[17]	R. Kurtenbach, J. Kleffmann, A. Niedojadlo and P. Wiesen, Primary NO₂ emissions and their impact on air quality in traffic environments in Germany, Environmental Sciences Europe, 24 (2012), 1-8. doi: 10.1186/2190-4715-24-21.
[18]	J. Li, X. Zhang, J. Orlando, G. Tyndall and G. Michalski, Quantifying the nitrogen isotope effects during photochemical equilibrium between NO and NO₂: Implications for $\delta^15$N in tropospheric reactive nitrogen, Atmospheric Chemistry and Physics, 20 (2020), 9805-9819. doi: 10.5194/acp-20-9805-2020.
[19]	Y. Maday, A. T. Patera, J. D. Penn and M. Yano, A parameterized-background data-weak approach to variational data assimilation: Formulation, analysis, and application to acoustics, Internat. J. Numer. Methods Engrg., 102 (2015), 933-965. doi: 10.1002/nme.4747.
[20]	V. Mallet, G. Stoltz and B. Mauricette, Ozone ensemble forecast with machine learning algorithms, Journal of Geophysical Research: Atmospheres, 114 (2009). doi: 10.1029/2008JD009978.
[21]	R. Molinaro, Y. Yang, B. Engquist and S. Mishra, Neural inverse operators for solving PDE inverse problems, preprint, 2023, arXiv: 2301.11167.
[22]	O. Mula, Inverse problems: A deterministic approach using physics-based reduced models, In Model Order Reduction and Applications, Cetraro, Italy 2021, Springer Nature Switzerland, Cham, 2023, 73-124. doi: 10.1007/978-3-031-29563-8_2.
[23]	M. Niu, Y. Wang, S. Sun and Y. Li, A novel hybrid decomposition-and-ensemble model based on CEEMD and GWO for short-term PM2.5 concentration forecasting, Atmospheric Environment, 134 (2016), 168-180. doi: 10.1016/j.atmosenv.2016.03.056.
[24]	E. C. Polley and M. J. Van der Laan, Super Learner in Prediction, bepress, 2010.
[25]	G. Rozza, D. B. P. Huynh and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Eng., 15 (2008), 229-257. doi: 10.1007/s11831-008-9019-9.
[26]	T. J. Santner, B. J. Williams and W. I. Notz, The Design and Analysis of Computer Experiments, volume 1, Springer, 2003. doi: 10.1007/978-1-4757-3799-8.
[27]	A. Tilloy, V. Mallet, D. Poulet, C. Pesin and F. Brocheton, BLUE-based NO₂ data assimilation at urban scale, Journal of Geophysical Research, 118 (2013), 2,031-2,040. doi: 10.1002/jgrd.50233.
[28]	M. J. Van der Laan, E. C. Polley and A. E. Hubbard, Super learner, Statistical Applications in Genetics and Molecular Biology, 6 (2007), Art. 25, 23 pp. doi: 10.2202/1544-6115.1309.
[29]	Z. Zou, D. Kouri and W. Aquino, An adaptive local reduced basis method for solving PDEs with uncertain inputs and evaluating risk, Comput. Methods Appl. Mech. Engrg., 345 (2019), 302-322. doi: 10.1016/j.cma.2018.10.028.