# Some Inverse Problem for the Polarized-Radiation Transfer Equation

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## Abstract

An inverse problem for the steady vector transfer equation for polarized radiation is studied. For this problem, an attenuation factor is found from a given solution of the equation at a medium boundary. An approach is propounded to solve the inverse problem by using special external radiative sources. A formula is proposed which relates the Radon transform of an attenuation factor to a solution of the equation at the medium boundary. Numerical experiments show that the proposed reconstruction algorithm for the polarized-radiation transfer equation has an advantage over the similar method for the scalar case.

## Keywords

Inverse Problem Vector Function Transfer Equation Direct Problem Attenuation Factor
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## References

- 1.Marchuk G.I., Mikhailov G.A., Nazarliev M.A. et al. Monte–Carlo Method in Atmospheric Optics, Novosibirsk: Nauka, 1976. (in Russian)Google Scholar
- 2.Germogenova T.A., Konovalov N.V., and Kuz’mina M.G. Foundations of Mathematical Theory for Polarized Radiation Transfer (Rigorous Results), Proc. All-Union Symp. on Invariance Principle and Its Applications, Byurakan, 1981, Yerevan: Akad. Nauk Armyan. SSR, 1989, pp. 271–284. (in Russian)Google Scholar
- 3.Mikhailov G.A., Ukhinov S.A., and Chimaeva A.S. Variance of Standard Vector Estimate of Monte–Carlo Method in Polarized Radiation Transfer Theory, J. Comp. Math. and Math. Phys., 2006, vol. 46, no. 11., pp. 2099–2113.MathSciNetGoogle Scholar
- 4.Sushkevich T.A., Strelkov S.A., and Maksakova S.V. Mathematical Model of Polarized Radiation Transfer, J. Math. Model., 1998, vol. 10, no. 7. pp. 61–75. (in Russian)Google Scholar
- 5.Siewert C.E. Determination of the Single Scattering Albedo from Polarization Measurements of the Rayleigh Atmosphere, Astrophys. Space Sci., 1979, vol. 60, pp. 237–239.CrossRefGoogle Scholar
- 6.Siewert C.E. Solution of an Inverse Problem in Radiative Transfer with Polarization, J. Quant. Spectrosc. Radiat. Transfer, 1983, vol. 30, no. 6. pp. 523–526.MathSciNetCrossRefGoogle Scholar
- 7.Ukhinov S.A. and Yurkov D.I. Computation of the Parametric Derivatives of Polarized Radiation and the Solution of Inverse Atmospheric Optics Problems, Russ. J. Numer. Anal. Math. Model., 2002, vol. 17, no. 3, pp. 283–303.MathSciNetzbMATHGoogle Scholar
- 8.Anikonov D.S. and Prokhorov I.V. Determination of Transfer Equation Coefficient with Energy and Angular Singularities of External Radiation, Dokl. Math., 1992, vol. 327, no. 2, pp. 205–207.MathSciNetGoogle Scholar
- 9.Anikonov D.S., Prokhorov I.V., and Kovtanyuk A.E. Investigation of Scattering and Absorbing Media by the Methods of X-Ray Tomography, J. Inv. Ill-Posed Probl., 1993, vol. 1, no. 4, pp. 259–281.MathSciNetzbMATHCrossRefGoogle Scholar
- 10.Anikonov D.S., Kovtanyuk A.E., and Prokhorov I.V. Transport Equation and Tomography, Utrecht: VSP, 2002.zbMATHGoogle Scholar
- 11.Kovtanyuk A.E. and Prokhorov I.V. Tomography Problem for the Polarized-Radiation Transfer Equation, J. Inv. Ill-Posed Probl., 2006, vol. 14, no. 6, pp. 1–12.MathSciNetGoogle Scholar
- 12.Germogenova T.A. Local Properties of Solutions of the Transfer Equation, Moscow: Nauka, 1986. (in Russian)Google Scholar
- 13.Khachaturov A.A. Determination of the Measure of a Domain in an n-Dimensional Euclidian Space from Its Values for All Half-Spaces, Russ. Math. Surv., 1954, vol. 9, no. 3(61), pp. 205–212. (in Russian)Google Scholar
- 14.Natterer F. The Mathematics of Computerized Tomography, Stuttgart: B. G. Teubner and John Wiley and Sons, 1986.zbMATHGoogle Scholar

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