On numerical computation of sensitivity of response functions to system inputs in variational data assimilation problems
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Victor Shutyaev
,
Tran Thu Ha
Abstract
Prediction of pollution in water flow is a very important task. To this end, it is imperative to be able to define the uncertainty in a model prediction. This is the purpose of sensitivity analysis whose role is to identify what uncertainty in the model outputs is attributable to the model inputs (parameters in this case). Traditionally, this is achieved by running the model perturbed by many random samples in the parameter space to determine their impact on the model outputs. It provides information on how much of the output variance is controlled by each parameter of the inputs.
The theoretical results related to the procedure based adjoint approach for computing a sensitivity of the response function (RF) to changes in the input source are presented in the paper. It is shown that this approach allows to compute, by one single integration of the adjoint equation over a given time interval, a sensitivity of the RF to any source located in the domain of interest. The proposed approach is applied to the 2D Saint-Venant flow equations for modelling the water pollution problem. A numerical experiment is formulated and implemented for the Thanh Nhan Lake in Hanoi for studying a sensitivity of some RF to observations. The numerical model is constructed by applying the well-known finite-volume method. Two appropriate optimization problems are introduced and solved on the basis of the BFGS algorithm. The numerical results show the efficiency of the proposed method and confirm the theoretical findings.
Dedicated to the memory of Francois-Xavier Le Dimet
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Funding: The work was supported by the Russian Science Foundation (project No. {20-11-20057}, in the part of research of Sections 1–2) and the Moscow Center of Fundamental and Applied Mathematics (agreement with the Ministry of Education and Science of the Russian Federation No. 075-15-2019-1624). The authors gratefully acknowledge the financial support from the Fund Nos. VAST01.09/21-22.
References
[1] H. Akimoto, Global air quality and pollution. Science 302 (2003), No. 5651, 1716–1719.10.1126/science.1092666Search in Google Scholar
[2] D. Allen, K. Pickering, and M. Fox-Rabinovitz, Evaluation of pollutant outflow and CO sources during TRACE-P using model-calculated, aircraft-based, and Measurements of Pollution in the Troposphere (MOPITT)-derived CO concentrations. J. Geophys. Res. 109 (2004), No. D15, D15S03.10.1029/2003JD004250Search in Google Scholar
[3] P. E. Biscaye, F. E. Grousset, A. M. Svensson, and A. Bory, Eurasian air pollution reaches eastern North America. Science 290 (2000), No. 5500, 2258–2259.10.1126/science.290.5500.2258Search in Google Scholar
[4] J. F. Bonnans, J. Ch. Gilbert, C. Lemaréchal, and C. A. Sagastizábal, Numerical Optimization, Theoretical and Numerical Aspects. 2nd edition. Springer, 2006.Search in Google Scholar
[5] B. Daryoush and D. N. Encyeh, Introduction of Frechet and Gateaux derivative. Appl. Math. Sci. 2 (2008), No. 20, 975–980.10.1111/j.1365-2648.1994.tb02455.xSearch in Google Scholar
[6] R. G. Derwent, M. E. Jenkin, S. M. Saunders, M. J. Pilling, P. G. Simms, N. R. Passant, G. R. Dollard, P. Dumitrean, and A. Kent, Photochemical ozone formation in north west Europe and its control. Atmospheric Environment 37 (2003), No. 14, 1983–1991.10.1016/S1352-2310(03)00031-1Search in Google Scholar
[7] R. G. Derwent, D. S. Stevenson, R. M. Doherty, W. J. Collins, and M. G. Sanderson, How is surface ozone in Europe linked to Asian and North American NOx emissions? Atmospheric Environment 42 (2008), No. 32, 7412–7422.10.1016/j.atmosenv.2008.06.037Search in Google Scholar
[8] B. N. Duncan, J. J. West, Y. Yoshida, A. M. Fiore, and J. R. Ziemke, The influence of European pollution on ozone in the Near East and northern Africa. Atmospheric Chemistry and Physics Discussion 8 (2008), No. 1, 1913–1950.10.5194/acp-8-2267-2008Search in Google Scholar
[9] A. M. Fiore, F. J. Dentener, O. Wild, et al., Multimodel estimates of intercontinental source-receptor relationships for ozone pollution. J. Geophys. Res. 114 (2009), No. D4, D04301.10.1029/2008JD010816Search in Google Scholar
[10] A. Fiore, D. J. Jacob, H. Liu, R. M. Yantosca, T. D. Fairlie, and Q. Li, Variability in surface ozone background over the United States: Implications for air quality policy. J. Geophys. Res. 108 (2003).10.1029/2003JD003855Search in Google Scholar
[11] J. C. Gilbert and C. Lemarechal, Some numerical experiments with variable-storage quasi-Newton algorithm. Math Program. 45 (1989), No. 3, 407–435.10.1007/BF01589113Search in Google Scholar
[12] A. Hakami, J. H. Seinfeld, C. Tianfeng, T. Youhua, G. R. Carmichael, and S. Adrian, Adjoint sensitivity analysis of ozone nonattainment over the continental United States. J. Environmental Sci. Technology 40 (2006), No. 12, 3855–3864.10.1021/es052135gSearch in Google Scholar PubMed
[13] E. Hendrik and S. Hauke, Ozone episode analysis by four-dimensional variational chemistry data assimilation. J. Geophys. Res. 106 (2001), No. D4, 3569–3590.10.1029/2000JD900448Search in Google Scholar
[14] D. K. Henze, J. H. Seinfeld, and D. T. Shindell, Inverse modeling and mapping US air quality influences of inorganic PM(2.5) precursor emissions using the adjoint of GEOS-Chem. J. Atmos. Chemistry Phys. 9 (2009), No. 16, 5877–5903.10.5194/acp-9-5877-2009Search in Google Scholar
[15] P. G. Hess and T. Vukicevic, Intercontinental transport, chemical transformations, and baroclinic systems. J. Geophys. Res. 108 (2003), No. D12, 4354.10.1029/2002JD002798Search in Google Scholar
[16] D. J. Jacob, J. L. Logan, and P. P. Murti, Effect of rising Asian emissions on surface ozone in the United States. Geophys. Res. Lett. 26 (2009), No. 14, 2175–2178.10.1029/1999GL900450Search in Google Scholar
[17] D. Jaffe, T. Anderson, D. Covert, R. Kotchenruther, B. Trost, J. Danielson, W. Simpson, T. Berntsen, S. Karlsdottir, D. Blake, J. Harris, G. Carmichael, and I. Uno, Transport of Asian air pollution to North America. Geophys. Res. Lett. 26 (2003), No. 6, 711–714.10.1029/1999GL900100Search in Google Scholar
[18] D. Jaffe, I. McKendry, T. Anderson, and H. Price, Six ‘new’ episodes of trans-Pacific transport of air pollutants. J. Atmospheric Environment 37 (2003), No. 6, 391–404.10.1016/S1352-2310(02)00862-2Search in Google Scholar
[19] J. E. Jonson, D. Simpson, H. Fagerli, and S. Solberg, Can we explain the trends in European ozone levels? Atmos. Chem. Phys. Discuss. 5 (2005), No. 4, 5957–5985.10.5194/acp-6-51-2006Search in Google Scholar
[20] F.-X. Le Dimet, V. P. Shutyaev, and T. H. Tran, General sensitivity analysis in data assimilation. Russ. J. Numer. Anal. Math. Modelling 29 (2014), No. 2, 107–127.10.1515/rnam-2014-0009Search in Google Scholar
[21] C. Licht, T. H. Tran, and Q. P. Vu, On some linears problems on shallow water flows. J. Differential and Integral Equations 22 (2009), No. 3-4, 275–283.Search in Google Scholar
[22] J.-L. Lions, Contrôle Optimal des Systès Gouvernés par des Équations aux Dérivées Partielles. Paris, Dunod, 1968.Search in Google Scholar
[23] P. T. Martien and R. A. Harley, Adjoint sensitivity analysis for a three-dimensional photochemical model: Application to southern California. J. Environmental Science & Technology 40 (2006), No. 13, 4200–4210.10.1021/es051026zSearch in Google Scholar PubMed
[24] P. T. Martien, R..A. Harley, and D. G. Cacuci, Adjoint sensitivity analysis for a three-dimensional photochemical model: Implementation and method comparison. J. Environmental Science & Technology 40 (2006), No. 8, 2663–2670.10.1021/es0510257Search in Google Scholar PubMed
[25] M. D. Morris, Factorial sampling plans for preliminary computational experiments. Technometrics 33 (1991), No. 2, 161–174.10.1080/00401706.1991.10484804Search in Google Scholar
[26] K. Nester and H.-J. Panitz, Sensitivity analysis by the adjoint chemistry transport model DRAISfor an episode in the Berlin Ozone (BERLIOZ) experiment. J. Atmos. Chem. Phys. 6 (2006), No. 8, 2091–2106.10.5194/acp-6-2091-2006Search in Google Scholar
[27] W. Rauch, M. Henze, L. Koncsos, P. Shanahan, L. Somlyody, and P. Vanrolleghem, River water quality modelling, I. State of the art. Wat. Sci. Tech. 38 (1998), No. 11, 237–244.10.2166/wst.1998.0473Search in Google Scholar
[28] J. H. Seinfeld, G. R. Carmichael, R. Arimoto, W. C. Conant, F. J. Brechtel, T. S. Bates, T. A. Cahill, A. D. Clarke, S. J. Doherty, P. J. Flatau, B. J. Huebert, J. Kim, K. M. Markowicz, P. K. Quinn, L. M. Russell, P. B. Russell, A. Shimizu, Y. Shinozuka, C. H. Song, Y. Tang, I. Uno, A. M. Vogelmann, R. J. Weber, J.-H. Woo, and X. Y. Zhang, ACE-ASIA: Regional climatic and atmospheric chemical effects of Asian dust and pollution. Bull. Amer. Meteorol. Soc. 85 (2004), No. 3, 367–380.10.1175/BAMS-85-3-367Search in Google Scholar
[29] P. A. Sleigh, P. H. Gaskell, M. Berzins, and N. G. Wright, An unstructured finite volume algorithm for predicting flow in rivers and estuaries. Computers & Fluids 27 (1998), 479–508.10.1016/S0045-7930(97)00071-6Search in Google Scholar
[30] I. M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Modelling Comput. Experiment 1 (1993), No. 4, 407–414.Search in Google Scholar
[31] H. Tracey, A. Fiore, and M. G. Hastings, Intercontinental transport of air pollution: Will emerging science lead to a new hemispheric treaty? Environmental Science and Technology 37 (2003), No. 20, 4535–4542.10.1021/es034031gSearch in Google Scholar PubMed
[32] T. H. Tran, D. T. Pham, V. L. Hoang, and H. P. Nguen, Pollution estimation based on the 2D water transport model and the singular evolutive interpolated filter. C. R. Mechanique 342 (2014), 106–124.10.1016/j.crme.2013.10.007Search in Google Scholar
[33] T. H. Tran, H. P. Nguen, F.-X. Le Dimet, and H. S. Hoang, Data assimilation and pollution forecasting in Burgers’ equation with model error function. Comptes Rendus Mécanique 347 (2019), No. 5, 423–444.10.1016/j.crme.2019.02.002Search in Google Scholar
[34] T. Vukićević and P. Hess, Analysis of tropospheric transport in the Pacific Basin using the adjoint technique. J. Geophys. Res. 105 (2000), No. D6, 7213–7230.10.1029/1999JD901110Search in Google Scholar
[35] K. Wang, Y. Zhang, C. Jang, S. Phillips, and B. Wang, Modeling intercontinental air pollution transport over the trans-Pacific region in 2001 using the Community Multiscale Air Quality modeling system, Geophys. Res. 114 (2009), No. D4, 1–23.10.1029/2008JD010807Search in Google Scholar
[36] O. Wild and A. Hajime, Intercontinental transport of ozone and its precursors in a three-dimensional global CTM. J. Geophys. Res. 106 (2001), No. D21, 27729–27744.10.1029/2000JD000123Search in Google Scholar
[37] K. E. Wilkening, A. B. Leonard, and M. Engle, Trans-Pacific air pollution. Science 290 (2000), No. 5489, 65–67.10.1126/science.290.5489.65Search in Google Scholar PubMed
[38] C. Xu and G. Gertner, Understanding and comparisons of different sampling approaches for the Fourier amplitudes sensitivity test (FAST). Computational Statistics and Data Analysis 55 (2011), 184–198.10.1016/j.csda.2010.06.028Search in Google Scholar PubMed PubMed Central
[39] J. Yienger, M. Galanter, T. A. Holloway, M. J. Phadnis, S. K. Guttikunda, G. R. Carmichael, W. J. Moxim, and H. Levy, The episodic nature of air pollution transport from Asia to North America. J. Geophys. Res. 105 (2000), No. D22, 26931–26945.10.1029/2000JD900309Search in Google Scholar
[40] L. Zhang, D. J. Jacob, M. Kopacz, D. K. Henze, K. Singh, and D. A. Jaffe, Intercontinental source attribution of ozone pollution at western U.S. sites using an adjoint method. J. Geophys. Res. Lett. 36 (2009), No. 11, L11810.10.1029/2009GL037950Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Transfer matrices and solution of the problem of stochastic dynamics of aerosol clusters by Monte Carlo method
- Constant upper bounds on the matrix exponential norm
- On the approximation of the diffusion operator in the ionosphere model with conserving the direction of geomagnetic field
- On numerical computation of sensitivity of response functions to system inputs in variational data assimilation problems
- Model reduction in Smoluchowski-type equations
Articles in the same Issue
- Frontmatter
- Transfer matrices and solution of the problem of stochastic dynamics of aerosol clusters by Monte Carlo method
- Constant upper bounds on the matrix exponential norm
- On the approximation of the diffusion operator in the ionosphere model with conserving the direction of geomagnetic field
- On numerical computation of sensitivity of response functions to system inputs in variational data assimilation problems
- Model reduction in Smoluchowski-type equations
