hydrophysicsФундаментальная и прикладная гидрофизикаFundamental and Applied Hydrophysics2073-66732782-5221St. Petersburg Research Center of the Russian Academy of Sciences10.59887/2073-6673.2026.19(2)-7qwjposhydrophysics-1555Research ArticleГИДРОАКУСТИКАHYDROACOUSTICSАналитическая модель диффузного отражения света от водной среды с резкоанизотропным рассеяниемAnalytical model for diffuse light reflection from an aqueous highly-forward-scattering mediumhttps://orcid.org/0000-0003-0707-8604МаринюкВ. В.MarinyukV. V.

Маринюк Виталий Владиславович, кандидат физико-математических наук, доцент; старший научный сотрудник

115409, Москва, Каширское шоссе, д. 3

1117997, Москва, Нахимовский проспект, д. 36

vmarinyuk@mail.ruhttps://orcid.org/0009-0007-5117-0877ПавловаМ. А.PavlovaM. A.

Павлова Мария Александровна, младший научный сотрудник

117997, Москва, Нахимовский проспект, д. 36

141701, Московская область, Долгопрудный, Институтский переулок, д. 9

pavlova.ma@phystech.eduРогозкинД. Б.RogozkinD. B.

Рогозкин Дмитрий Борисович, доктор физико-математических наук, главный научный сотрудник

127055, Москва, Сущевская улица, д. 22

dmrogozkin@gmail.comhttps://orcid.org/0000-0001-8489-2859ШеберстовС. В.SheberstovS. V.

Шеберстов Сергей Валентинович, старший научный сотрудник

117997, Москва, Нахимовский проспект, д. 36

sheberst@yandex.ruНациональный исследовательский ядерный университет «МИФИ»; Институт океанологии им. П.П. Ширшова РАНРоссияMoscow Engineering Physics Institute; Shirshov Institute of Oceanology, Russian Academy of SciencesRussian FederationИнститут океанологии им. П.П. Ширшова РАН; Московский физико-технический институтРоссияShirshov Institute of Oceanology, Russian Academy of Sciences; Moscow Institute of Physics and TechnologyRussian FederationВсероссийский научно-исследовательский институт автоматики им. Н.Л. Духова (ВНИИА)РоссияDukhov Research Institute of Automatics (VNIIA)Russian FederationИнститут океанологии им. П.П. Ширшова РАНРоссияShirshov Institute of Oceanology, Russian Academy of SciencesRussian Federation2026100820261927789Copyright © Маринюк В.В., Павлова М.А., Рогозкин Д.Б., Шеберстов С.В., 20262026Маринюк В.В., Павлова М.А., Рогозкин Д.Б., Шеберстов С.В.Marinyuk V.V., Pavlova M.A., Rogozkin D.B., Sheberstov S.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://hydrophysics.spbrc.ru/jour/article/view/1555

Цель настоящего исследования — разработка метода расчета коэффициента диффузного отражения света от сильнопоглощающей среды с резкоанизотропным рассеянием типа морской воды. Актуальность этого вопроса обусловлена широким использованием коэффициента диффузного отражения в обработке данных спектральных измерений яркости океана. Предлагаемый метод основан на разложении коэффициента отражения в ряд по кратности рассеяния на большой угол (по сравнению с характерным углом однократного рассеяния), что эквивалентно разложению в ряд по обратным степеням показателя поглощения. Для первых двух членов разложения получены явные выражения. Первый воспроизводит известный результат приближения квазиоднократного рассеяния, а второй соответствует вкладу квазидвукратного рассеяния (многократного малоуглового рассеяния до, между и после двух актов рассеяния на большой угол). Проведено сравнение найденного аналитического решения с результатами численного интегрирования уравнения переноса для индикатрисы Хеньи-Гринстейна, а также для двухчленной индикатрисы Хеньи-Гринстейна, моделирующей рассеяние света в морской воде. Показано, что сумма вкладов квазиоднократного и квазидвукратного рассеяний хорошо описывает дифференциальный коэффициент отражения в широком диапазоне углов падения и наблюдения.

The aim of the present study is to develop a method for calculating the diffuse component of the bidirectional reflectance of a strongly absorbing and highly-forward scattering medium like seawater. The relevance of this issue is due to the widespread use of the bidirectional reflectance in the processing of ocean color measurements. Explicit expressions for the first two terms of the expansion are derived. The first term reproduces the well-known result of quasi-single-scattering approximation, and the second corresponds to the contribution of quasi-double scattering (the small-angle multiple scattering before, between, and after two events of large-angle scattering). To validate our analytical results, we carry out numerical integration of the radiative transfer equation for the Henyey-Greenstein phase function, as well as for the two-term Henyey-Greenstein function that models light scattering in seawater. It is shown that for optical parameters typical to seawater, the first two terms of the expansion (i. e. the sum of quasi-single- and quasi-double-scattering contributions) prove to be sufficient to describe the bidirectional reflectance with high accuracy over a wide range of illumination/viewing angles.

яркость океанакоэффициент отраженияуравнение переноса излученияквазидвукратное рассеяниеWater-leaving radianceBRDFradiative transfer equationquasi-double scatteringВ.В. Маринюк, М.А. Павлова и С.В. Шеберстов выражают благодарность за финансовую поддержку со стороны ИО РАН (теоретическое исследование выполнено в рамках государственного задания Минобрнауки России для ИО РАН (тема № FMWE-2024-0015); численные расчеты проведены при поддержке гранта Российского научного фонда № 25-77-10065).V.V. Marinyuk, M.A. Pavlova, and S.V. Sheberstov acknowledge financial support from IO RAS (the theoretical study has been carried out as part of the state assignment of SIO RAS, theme № FMWE-2024-0015, and the numerical computations have been performed with the support of the Russian Science Foundation, grant № 25-77-10065).ReferencesMobley C.D. Light and water: radiative transfer in natural waters. San Diego, Calif: Academic; 1994. 592 p.Mobley CD. Light and Water: Radiative Transfer in Natural Waters. San Diego, Calif: Academic; 1994. 592 p.Werdell P.J., McKinna L.I.W., Boss E., et al. An overview of approaches and challenges for retrieving marine inherent optical properties from ocean color remote sensing // Progress in Oceanography. 2018. Vol. 160. P. 186–212. EDN YEYMUX. https://doi.org/10.1016/j.pocean.2018.01.001Werdell PJ, McKinna LIW, Boss E, et al. An overview of approaches and challenges for retrieving marine inherent optical properties from ocean color remote sensing. Progress in Oceanography. 2018;160:186–212. https://doi.org/10.1016/j.pocean.2018.01.001Groom S., Sathyendranath S., Ban Y., et al. Satellite Ocean Colour: Current Status and Future Perspective // Frontiers in Marine Science. 2019; Vol. 6. P. 485. https://doi.org/10.3389/fmars.2019.00485Groom S, Sathyendranath S, Ban Y, et al. Satellite ocean colour: current status and future perspective. Frontiers in Marine Science. 2019;6:485. https://doi.org/10.3389/fmars.2019.00485Morel A., Gentili B. Diffuse reflectance of oceanic waters. II Bidirectional aspects // Applied Optics. 1993. Vol. 32. P. 6864–6879. https://doi.org/10.1364/AO.32.006864Morel A, Gentili B. Diffuse reflectance of oceanic waters. II. Bidirectional aspects. Applied Optics. 1993;32:6864–6879. https://doi.org/10.1364/AO.32.006864Morel A., Gentili B. Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem // Applied Optics. 1996. Vol. 35. P. 4850–4862. https://doi.org/10.1364/AO.35.004850Morel A, Gentili B. Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem. Applied Optics. 1996;35:4850–4862.Garver S.A., Siegel D.A. Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation: 1. Time series from the Sargasso Sea // Journal of Geophysical Research. 1997. Vol. 102(C8). P. 18607–18625. https://doi.org/10.1029/96JC03243Garver SA, Siegel DA. Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation: 1. Time series from the Sargasso Sea. Journal of Geophysical Research. 1997;102(C8):18607–18625. https://doi.org/10.1029/96JC03243O’Reilly J.E., Maritorena S., Mitchell B.G. et al. Ocean color chlorophyll algorithms for SeaWiFS // Journal of Geophysical Research. 1998. Vol. 103. P. 24937–24953. https://doi.org/10.1029/98JC02160O’Reilly JE, Maritorena S, Mitchell BG, et al. Ocean color chlorophyll algorithms for SeaWiFS. Journal of Geophysical Research. 1998;103:24937–24953. https://doi.org/10.1029/98JC02160Maritorena S., Siegel D.A., Peterson A. Optimization of a semianalytical ocean color model for global-scale applications // Applied Optics. 2002. Vol. 41. P. 2705–2714. EDN LVCWIP. https://doi.org/10.1364/AO.41.002705Maritorena S, Siegel DA, Peterson A. Optimization of a semianalytical ocean color model for global-scale applications. Applied Optics. 2002;41:2705–2714. https://doi.org/10.1364/AO.41.002705Morel A., Antoine D., Gentili B. Bidirectional reflectance of oceanic waters: accounting for Raman emission and varying particle scattering phase function // Applied Optics. 2002. Vol. 41. P. 6289–6306. https://doi.org/10.1364/AO.41.006289Morel A, Antoine D, Gentili B. Bidirectional reflectance of oceanic waters: accounting for Raman emission and varying particle scattering phase function. Applied Optics. 2002;41:6289–6306. https://doi.org/10.1364/AO.41.006289Lee Z.P., Carder K.L., Arnone R.A. Deriving inherent optical properties from water color: a multi-band quasi-analytical algorithm for optically deep waters // Applied Optics. 2002. Vol. 41. P. 5755–5772. EDN LVCWJJ. https://doi.org/10.1364/AO.41.005755Lee ZP, Carder KL, Arnone RA. Deriving inherent optical properties from water color: a multi-band quasi-analytical algorithm for optically deep waters. Applied Optics. 2002;41:5755–5772. https://doi.org/10.1364/AO.41.005755Werdell P.J., Franz B.A., Sean W. et al. Generalized ocean color inversion model for retrieving marine inherent optical properties // Applied Optics. 2013. Vol. 52. P. 2019–2037. https://doi.org/10.1364/AO.52.002019Werdell PJ, Franz BA, Sean W, et al. Generalized ocean color inversion model for retrieving marine inherent optical properties. Applied Optics. 2013;52:2019–2037. https://doi.org/10.1364/AO.52.002019Lewis K.M., Arrigo K.R. Ocean color algorithms for estimating chlorophyll a, CDOM absorption, and particle backscattering in the Arctic Ocean // Journal of Geophysical Research: Oceans. 2020. Vol. 125. e2019JC015706. EDN OHWXJM. https://doi.org/10.1029/2019JC015706Lewis KM, Arrigo KR. Ocean color algorithms for estimating chlorophyll a, CDOM absorption, and particle backscattering in the Arctic Ocean. Journal of Geophysical Research: Oceans. 2020;125: e2019JC015706. https://doi.org/10.1029/2019JC015706Najah A., Al-Shehhi M.R. Performance of the Ocean Color Algorithms: QAA, GSM, and GIOP in Inland and Coastal Waters // Remote Sensing in Earth Systems Sciences. 2021. Vol. 4. P. 235–248. EDN MQHQVO. https://doi.org/10.1007/s41976-022-00068-3Najah A, Al-Shehhi MR. Performance of the ocean color algorithms: QAA, GSM, and GIOP in inland and coastal waters. Remote Sensing in Earth Systems Sciences. 2021;4:235–248. https://doi.org/10.1007/s41976-022-00068-3Moore T.S., Kolluru S., Tonizzo A., Twardowski M.S. Bio-optical inversion scheme based on the radiative transfer equation // Optics Express. 2024. Vol. 32. P. 41270–41296. https://doi.org/10.1364/OE.508090Moore TS, Kolluru S, Tonizzo A, Twardowski MS. Bio-optical inversion scheme based on the radiative transfer equation. Optics Express. 2024;32:41270–41296. https://doi.org/10.1364/OE.508090Gordon H.R., Brown O.B., Evans R.H. et al. A semianalytic radiance model of ocean color // Journal of Geophysical Research: Atmospheres. 1988. Vol. 93. P. 10909–10924. https://doi.org/10.1029/JD093iD09p10909Gordon HR, Brown OB, Evans RH, et al. A semianalytic radiance model of ocean color. Journal of Geophysical Research: Atmospheres. 1988;93:10909–10924. https://doi.org/10.1029/JD093iD09p10909Gordon H.R. Ocean color remote sensing: Influence of the particle phase function and the solar zenith angle // Eos, Transactions, American Geophysical Union. 1986. Vol. 67, No. 14 P. 1055.Gordon HR. Ocean color remote sensing: influence of the particle phase function and the solar zenith angle. EOS Trans actions, American Geophysical Union. 1986;67(14):1055.Gordon H.R. Simple calculation of the diffuse reflectance of the ocean // Applied Optics. 1973. Vol. 12. P. 2803–2804. https://doi.org/10.1364/AO.12.002803Gordon HR. Simple calculation of the diffuse reflectance of the ocean. Applied Optics. 1973;12:2803–2804. https://doi.org/10.1364/AO.12.002803Голубицкий Б.М., Левин И.М., Танташев М.В. Коэффициент яркости полубесконечного слоя морской воды // Известия Академии наук СССР. Физика атмосферы и океана. 1974. Т. 10. С. 1235–1238.Golubitsky BM, Levin IM, Tantashev MV. Luminosity coefficient of a semi-infinite layer of seawater. Izvestiya: Atmospheric and Oceanic Physics. 1974;10:1235–1238. (In Russ.)Зеге Э.П. Коэффициент яркости света, отраженного полубесконечной средой // Известия Академии наук СССР. Физика атмосферы и океана. 1983. Т. 19. С. 927–936.Zege EP. Luminosity coefficient of light reflected by a semi-infinite medium. Izvestiya: Atmospheric and Oceanic Physics. 1983;19:927–936. (In Russ.).Зеге Э.П., Иванов А.П., Кацев И.Л. Перенос изображения в рассеивающей среде. Минск: Наука и техника, 1985. 327 с.Zege EP, Ivanov AP, Katsev IL. Image Transfer Through a Scattering Medium. Berlin: Springer-Verlag; 1991. 349 p.Zaneveld J.R.V. A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties // Journal of Geophysical Research: Oceans. 1995. Vol. 100. P. 13135–13142. https://doi.org/10.1029/95JC00453Zaneveld JR.V. A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties. Journal of Geophysical Research: Oceans. 1995;100:13135–13142. https://doi.org/10.1029/95JC00453Jerome J.H., Bukata R.P., Miller J.R. Remote sensing reflectance and its relationship to optical properties of natural waters // International Journal of Remote Sensing. 1996. Vol. 17. P. 3135–3155. https://doi.org/10.1080/01431169608949135Jerome JH, Bukata RP, Miller JR. Remote sensing reflectance and its relationship to optical properties of natural waters. International Journal of Remote Sensing. 1996;17:3135–3155. https://doi.org/10.1080/01431169608949135Albert A., Mobley C.D. An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters // Optics Express. 2003. Vol. 11. P. 2873–2890. EDN MEKHSX. https://doi.org/10.1364/OE.11.002873Albert A, Mobley CD. An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters. Optics Express. 2003;11:2873–2890. https://doi.org/10.1364/OE.11.002873Park Y.-J., Ruddick K. Model of remote-sensing reflectance including bidirectional effects for case 1 and case 2 waters // Applied Optics. 2005. Vol. 44. P. 1236–1249. https://doi.org/10.1364/AO.44.001236Park Y-J, Ruddick K. Model of remote-sensing reflectance including bidirectional effects for case 1 and case 2 waters. Applied Optics. 2005;44:1236–1249. https://doi.org/10.1364/AO.44.001236Twardowski M., Tonizzo A. Ocean Color Analytical Model Explicitly Dependent on the Volume Scattering Function // Applied Sciences. 2018. Vol. 8(12). P. 2684. https://doi.org/10.3390/app8122684Twardowski M, Tonizzo A. Ocean color analytical model explicitly dependent on the volume scattering function. Applied Sciences. 2018;8(12):2684. https://doi.org/10.3390/app8122684Маринюк В.В., Рогозкин Д.Б., Шеберстов С.В. Распространение пучка света в поглощающей среде с крупномасштабными неоднородностями // Оптика и спектроскопия. 2014. Т. 117. С. 106–114. EDN SGVVHL. https://doi.org/10.7868/S0030403414070150Marinyuk VV, Rogozkin DB, Sheberstov SV. Propagation of a light beam in an absorbing medium with large-scale inhomogeneities. Optics and Spectroscopy. 2014;117:102–110. https://doi.org/10.1134/S0030400X14070157Marinyuk V.V., Rogozkin D.B., Sheberstov S.V. Optical beam spread in seawater // Optics Communications. 2025. Vol. 574. 131098. EDN SORGDZ. https://doi.org/10.1016/j.optcom.2024.131098Marinyuk VV, Rogozkin DB, Sheberstov SV. Optical beam spread in seawater. Optics Communications. 2025;574:131098. https://doi.org/10.1016/j.optcom.2024.131098Jin Z., Stamnes K. Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system // Applied Optics. 1994. Vol. 33. P. 431–442. https://doi.org/10.1364/AO.33.000431Jin Z, Stamnes K. Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system. Applied Op tics. 1994;33:431–442. https://doi.org/10.1364/AO.33.000431Kattawar G.W. A three-parameter analytic phase function for multiple scattering calculations // Journal of Quantitative Spectroscopy and Radiative Transfer. 1975. Vol. 15. P. 839–849. https://doi.org/10.1016/0022-4073(75)90095-3Kattawar GW. A three-parameter analytic phase function for multiple scattering calculations. Journal of Quantitative Spectroscopy and Radiative Transfer. 1975;15:839–849. https://doi.org/10.1016/0022-4073(75)90095-3Plass G.N., Kattawar G.W., Humphreys T.J. Influence of the oceanic scattering phase function on the radiance // Journal of Geophysical Research. 1985. Vol. 90(C2). P. 3347–3351. https://doi.org/10.1029/JC090iC02p03347Plass GN, Kattawar GW, Humphreys TJ. Influence of the oceanic scattering phase function on the radiance. Journal of Geophysical Research. 1985;90(C2):3347–3351. https://doi.org/10.1029/JC090iC02p03347Haltrin V.I. One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater // Applied Optics. 2002. Vol. 41. P. 1022–1028. EDN LWXKXZ. https://doi.org/10.1364/AO.41.001022Haltrin VI. One-parameter two-term Henyey–Greenstein phase function for light scattering in seawater. Applied Optics. 2002;41:1022–1028. https://doi.org/10.1364/AO.41.001022Case K.M., Zweifel P.F. Linear Transport Theory. Addison-Wesley, 1967. 342 p.Case KM, Zweifel PF. Linear Transport Theory. Addison-Wesley; 1967. 342 p.Ishimaru A. Wave Propagation and Scattering in Random Media. IEEE/OUP series on electromagnetic wave theory. New York: Academic Press, 1978. 267 p. EDN YFXUWL. https://doi.org/10.1016/B978-0-12-374701-3.X5001-7Ishimaru A. Wave Propagation and Scattering in Random Media. New York: Academic Press; 1978. 267 p. https://doi.org/10.1016/B978-0-12-374701-3.X5001-7Marinyuk V.V., Pavlova M.A., Rogozkin D.B., Sheberstov S.V. Characterization of the aerosol contribution to the top-of-atmosphere radiance for satellite ocean color retrievals // Applied Optics. 2024. Vol. 63. P. 7212–7224. EDN HTBUJP. https://doi.org/10.1364/AO.528718Marinyuk VV, Pavlova MA, Rogozkin DB, Sheberstov SV. Characterization of the aerosol contribution to the top-of-atmosphere radiance for satellite ocean color retrievals. Applied Optics. 2024;63:7212–7224. https://doi.org/10.1364/AO.528718Budak V.P., Klyuykov D.A., Korkin S.V. Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering. In: Kokhanovsky, A. (eds) Light Scattering Reviews 5. Springer Praxis Books. Springer, Berlin, Heidelberg. 2010. P. 147–203. https://doi.org/10.1007/978-3-642-10336-0_5Budak VP, Klyuykov DA, Korkin SV. Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering. In: Kokhanovsky A, editor. Light Scattering Reviews 5. Berlin, Heidelberg: Springer; 2010. p. 147–203. https://doi.org/10.1007/978-3-642-10336-0_5Mobley C.D., Sundman L.K., Boss E. Phase function effects on oceanic light fields // Applied Optics. 2002. Vol. 41. P. 1035–1050. https://doi.org/10.1364/AO.41.001035Mobley CD, Sundman LK, Boss E. Phase function effects on oceanic light fields. Applied Optics. 2002;41:1035–1050. https://doi.org/10.1364/AO.41.001035

The authors declare that there are no conflicts of interest present.