Far Eastern Mathematical Journal

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Comparative analysis of the error of the single scattering approximation when solving one inverse problem in two-dimensional and three- dimensional cases


Vornovskikh P.A., Prokhorov I.V.

2021, issue 2, P. 151-165
DOI: https://doi.org/10.47910/FEMJ202113


Abstract
The inverse problem for the nonstationary radiative transfer equation is considered, which consists in finding the scattering coefficient for a given time-angular distribution of the solution to the equation at a certain point. To solve this problem, the single scattering approximation in the pulsed sounding mode is used. A comparative analysis of the error in solving the inverse problem in the single scattering approximation for two-dimensional and three-dimensional models describing the process of high-frequency acoustic sounding in a fluctuating ocean is carried out. It is shown that in the two-dimensional case the error of the approximate solution significantly exceeds the error in the three-dimensional model.

Keywords:
radiative transfer equation, pulsed ocean sounding, scattering coefficient, inverse problem, Monte Carlo methods

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References

[1] P.A. Vornovskikh, A. Kim, I.V. Prokhorov, “Primenimost' priblizheniia odnokratnogo rasseianiia pri impul'snom zondirovanii neodnorodnoi sredy”, Komp'iuternye issledovaniia i modelirovanie, 12:5 (2020), 1063–1079.
[2] A. Isimaru, Rasprostranenie i rasseianie voln v sluchaino-neodnorodnykh sredakh, Mir, M., 1981.
[3] V.I. Mendus, G.A. Postnov, “Ob uglovom raspredelenii vysokochastotnykh dinamicheskikh shumov okeana”, Akusticheskii zhurnal, 39:6 (1993), 1107–1116.
[4] I.B. Andreeva, A.V. Belousov, “O dopustimosti ispol'zovaniia priblizheniia odnokratnogo rasseianiia akusticheskikh voln v zadachakh o skopleniiakh gidrobiontov”, Akusticheskii zhurnal, 42:4 (1996), 560–562.
[5] G. Bal, “Kinetics of scalar wave fields in random media”, Wave Motion, 43 (2005), 132–157.
[6] G. Bal, “Inverse transport theory and applications”, Inverse Problems, 25:5 (2009), 025019.
[7] I.V. Prokhorov, V.V. Zolotarev, I.B. Agafonov, “Zadacha akusticheskogo zondirovaniia vo fluktuiruiushchem okeane”, Dal'nevostochnyi matematicheskii zhurnal, 11:1 (2011), 76–87.
[8] I.V. Prokhorov, A.A. Sushchenko, “Issledovanie zadachi akusticheskogo zondirovaniia morskogo dna metodami teorii perenosa izlucheniia”, Akusticheskii zhurnal, 61:3 (2015), 400–408.
[9] D.S. Anikonov, A.E. Kovtaniuk, I.V. Prokhorov, Ispol'zovanie uravneniia perenosa v tomografii, Logos, M., 2000.
[10] D.S. Anikonov, V.G. Nazarov, I.V. Prokhorov, Poorly visible media in X-ray tomography, Inverse and Ill-Posed Problems Series, 38, VSP, Boston-Utrecht, 2002.
[11] A.B. Prilepko, A.L. Ivankov, “Obratnye zadachi opredeleniia koeffitsienta, indikatrisy rasseianiia i pravoi chasti nestatsionarnogo mnogoskorostnogo uravneniia perenosa”, Differentsial'nye uravneniia, 21:5 (1985), 870–885.
[12] V.G. Romanov, “Otsenka ustoichivosti v zadache ob opredelenii koeffitsienta oslableniia i indikatrisy rasseianiia dlia uravneniia perenosa”, Sib. matem. zhurn., 37:2 (1996), 361–377.
[13] S. Acosta, “Time reversal for radiative transport with applications to inverse and control problems”, Inverse Problems, 29 (2013), 085014.
[14] C. Wang, T. Zhou, “A hybrid reconstruction approach for absorption coefficient by fluo- rescence photoacoustic tomography”, Inverse Problems, 35 (2018), 025005.
[15] M. Bellassoued, Y. Boughanja, “An inverse problem for the linear Boltzmann equation with a time-dependent coefficient”, Inverse Problems, 35 (2019), 085003.
[16] W. Dahmen, F. Gruber, O. Mula, “An adaptive nested source term iteration for radiative transfer equations”, Math. Comp, 89 (2020), 1605–1646.
[17] Q. Li, W. Sun, “Applications of kinetic tools to inverse transport problems”, Inverse Problems, 36 (2020), 035011.
[18] L. Florescu, V.A. Markel, J.C. Schotland, “Single-scattering optical tomography: simulta- neous reconstruction of scattering and absorption”, Phys. Rev. E., 81 (2010), 016602.
[19] A. Kleinboehl, J.T. Schofield, W.A. Abdou, P.G.J. Irwin, de R.J. Kok, “A single-scattering approximation for infrared radiative transfer in limb geometry in the Martian atmosphere”, Journal of Quantitative Spectroscopy and Radiative Transfer, 112:10 (2011), 1568–1580.
[20] S. Moon, Y. Hristova, B. Kwon, “Single scattering tomography with curved detectors”, Journal Biomedical Physics and Engineering Express, 4 (2018), 045040.
[21] I.V. Prokhorov, A.A. Sushchenko, “Zadacha Koshi dlia uravneniia perenosa izlucheniia v neogranichennoi srede”, Dal'nevost. matem. zhurn., 18:1 (2018), 101–111.
[22] G.I. Marchuk, G.A. Mikhailov, M.A. Nazaraliev i dr., Metod Monte-Karlo v atmosfernoi optike, Nauka, Novosibirsk, 1976.
[23] G.A. Mikhailov, I.N. Medvedev, Optimizatsiia vesovykh algoritmov statisticheskogo modelirovaniia, Omega Print, Novosibirsk, 2011.
[24] S.M. Prigarin, “Statisticheskoe modelirovanie effektov, sviazannykh s mnogokratnym rasseianiem impul'sov nazemnykh i kosmicheskikh lidarov v oblachnoi atmosfere”, Optika atmosfery i okeana, 29:9 (2016), 747–751.
[25] A. Kim, I.V. Prokhorov, “Teoreticheskii i chislennyi analiz nachal'no-kraevoi zadachi dlia uravneniia perenosa izlucheniia s frenelevskimi usloviiami sopriazheniia”, Zhurnal vychislitel'noi matematiki i matematicheskoi fiziki, 58:5 (2018), 762–777.

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