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In electromagnetic radiative transfer calculation, the accuracy and the computation time are usually determined by the representation of single-scattering phase function. Accurate calculation is time consuming even for spherical particle, thus, an analytic representation is commonly adopted to approximate the exact phase function and then accelerate the calculation. Most widely used single-scattering phase functions are the Henyey-Greenstein phase function and modified Henyey-Greenstein phase function (Henyey-Greenstein*). Although the Henyey-Greenstein phase function and the Henyey-Greenstein* phase function can represent the forward-scattering peak of Mie-scattering phase function well, they fail to reproduce the backscattering behavior, limiting the accuracy of the calculation. In order to better fit exact calculations and simulate the backward-scattering peak, we develop a new analytic expression based on the fundamental theory of electromagnetic scattering and radiation transmission. This phase function is an algebraic expression with one single free parameter (asymmetry factor), and can be expanded into Legendre polynomials. The new phase function converges to the Rayleigh phase function when the asymmetry factor approximates to 0, and it can approach to the Henyey-Greenstein phase function as the asymmetry factor is about 1. We compare the Henyey-Greenstein phase function, the Henyey-Greenstein* phase function, and the new phase function for different asymmetry factors, and find that the new phase function provides a more realistic description for the unpolarized light scattering from small particles. Furthermore, the calculated value for the ratio of the scattering intensity at 90 degree to that in the backward direction is more reasonable. We also investigate the effectiveness by approximating the scattering from polydispersed particles through comparing the new phase function, the Henyey-Greenstein* phase function, and the Mie-scattering phase function for three types of Derimendjian's polydispersions. Results show that the new phase function fits the Mie-scattering phase function much better than the Henyey-Greenstein* phase function. For the new phase function, the root-mean-square error is small for 73.3% data. By contrast, only 26.7% data fit the Mie-scattering phase function well for the Henyey-Greenstein* phase function. Similarly, the effectiveness of new function is most significant when calculating the ratio of the scattering intensity at 90 degree to that in the backward direction. In summary, the new Henyey-Greenstein* phase function provides a more accurate calculation for the scattering intensity in the backward direction, and is conducive to electromagnetic radiative transfer calculation. Furthermore, because the proposed phase function has the same basic form as the Heny-Greenstein phase function, reformatting radiative transfer model in terms of the new phase function should require relatively little effort.
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Keywords:
- scattering /
- phase function /
- Henyey-Greenstein phase function /
- new Henyey-Greenstein phase function
[1] Si F Q, Liu J G, Xie P H, Zhang Y J, Dou K, Liu W Q 2006 Acta Phys. Sin. 55 3165(in Chinese)[司福祺, 刘建国, 谢品华, 张玉钧, 窦科, 刘文清2006物理学报 55 3165]
[2] Xie Z L, Zhang R, Xiu X Q, Han P, Liu B, Chen L, Yu H Q, Jiang R L, Shi Y, Zhang Y D 2007 Acta Phys. Sin. 56 6717(in Chinese)[谢自力, 张荣, 修向前, 韩平, 刘斌, 陈琳, 俞慧强, 江若链, 施毅, 郑有炓2007物理学报 56 6717]
[3] Li Y, Song J, Zhang Z M 2003 Chin. Phys. Lett. 20 1189
[4] Li J N, Barker H, Yang P, Yi B Q 2015 J. Geophys. Res. 120 128
[5] Liu K, Zhang F, Wu K, Zhou X J, Zhang H, Liu R Q 2016 Acta Meteorol. Sin. 74 784(in Chinese)[刘琨, 张峰, 吴琨, 周秀骥, 张华, 刘仁强2016气象学报 74 784]
[6] Toublance D 1996 Appl. Opt. 35 3270
[7] Chen X H, Liu Q, Wei H L 2007 J. Light Scatt. 3 283(in Chinese)[陈秀红, 刘强, 魏合理2007光散射学报 3 283]
[8] Ding S, Xie Y, Yang P, Weng F, Liu Q, Baum B, Hu Y 2009 J. Quant. Spectrosc. Radiat. Transfer 110 1190
[9] Cai H, Huang N N 2003 Chin. Phys. Lett. 20 496
[10] Xu L Q, Li H, Xiao Z Y 2008 Acta Phys. Sin. 57 6030(in Chinese)[徐兰青, 李晖, 肖郑颖2008物理学报 57 6030]
[11] Liou K N (translated by Guo C L, Zhou S J) 2004 An Introduction to Atmospheric Radiation (2nd Ed.) (Beijing:China Meteorogical Press) pp108-109(in Chinese)[廖国男著(郭彩丽, 周诗健译) 2004大气辐射导论(第二版) (北京:气象出版社)第108–109页]
[12] Joseph J, Wiscombe W, Weinman J 1976 J. Atmos. Sci. 33 2452
[13] Wiscombe W 1977 J. Atmos. Sci. 34 1408
[14] Huang C J, Liu Y F, Wu Z S, Sun Y Q, Long S M 2009 Acta Phys Sin. 58 2397(in Chinese)[黄朝军, 刘亚峰, 吴振森, 孙彦清, 龙姝明2009物理学报 58 2397]
[15] Kattawar G W 1975 Quant. Spectrosc. Radiat. Transfer 15 839
[16] Cornette W M, Shanks J G 1992 Appl. Opt. 31 3152
[17] Bohre C F, Huffman D R 1983 Absorption and Scattering of Light by Small Particles (New York:John Wiley) p100
[18] McCartney E J 1988 Optics of the Atmosphere (Beijing:Science Press) p276(in Chinese)[麦克特尼E J著(潘乃先译) 1988大气光学(北京:科学出版社)第276页]
[19] Bai L, Tang S Q, Wu Z S, Xie P H, Wang S M 2010 Acta Phys. Sin. 59 1749(in Chinese)[白璐, 汤双庆, 吴振森, 谢品华, 汪世美2010物理学报 59 1749]
[20] Zhu M Z, Zhang H L, Jia H L, Yang J K, Cheng S L 2007 J. Light Scatt. 3 225(in Chinese)[朱孟真, 张海良, 贾红辉, 杨建坤, 常胜利2007光散射学报 3 225]
[21] Derimendjian D 1963 Scattering and Polarization Properties of Polydispersed Suspensions with Partial Absorption (New York:The MacMillan Company, Pursuant to a Special Arrangement with Pergamon Press) pp4-9
[22] Diermendjian D 1969 Electromagnetic Scattering on Spherical Polydispersions (New York:American Elsevier) pp31-78
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[1] Si F Q, Liu J G, Xie P H, Zhang Y J, Dou K, Liu W Q 2006 Acta Phys. Sin. 55 3165(in Chinese)[司福祺, 刘建国, 谢品华, 张玉钧, 窦科, 刘文清2006物理学报 55 3165]
[2] Xie Z L, Zhang R, Xiu X Q, Han P, Liu B, Chen L, Yu H Q, Jiang R L, Shi Y, Zhang Y D 2007 Acta Phys. Sin. 56 6717(in Chinese)[谢自力, 张荣, 修向前, 韩平, 刘斌, 陈琳, 俞慧强, 江若链, 施毅, 郑有炓2007物理学报 56 6717]
[3] Li Y, Song J, Zhang Z M 2003 Chin. Phys. Lett. 20 1189
[4] Li J N, Barker H, Yang P, Yi B Q 2015 J. Geophys. Res. 120 128
[5] Liu K, Zhang F, Wu K, Zhou X J, Zhang H, Liu R Q 2016 Acta Meteorol. Sin. 74 784(in Chinese)[刘琨, 张峰, 吴琨, 周秀骥, 张华, 刘仁强2016气象学报 74 784]
[6] Toublance D 1996 Appl. Opt. 35 3270
[7] Chen X H, Liu Q, Wei H L 2007 J. Light Scatt. 3 283(in Chinese)[陈秀红, 刘强, 魏合理2007光散射学报 3 283]
[8] Ding S, Xie Y, Yang P, Weng F, Liu Q, Baum B, Hu Y 2009 J. Quant. Spectrosc. Radiat. Transfer 110 1190
[9] Cai H, Huang N N 2003 Chin. Phys. Lett. 20 496
[10] Xu L Q, Li H, Xiao Z Y 2008 Acta Phys. Sin. 57 6030(in Chinese)[徐兰青, 李晖, 肖郑颖2008物理学报 57 6030]
[11] Liou K N (translated by Guo C L, Zhou S J) 2004 An Introduction to Atmospheric Radiation (2nd Ed.) (Beijing:China Meteorogical Press) pp108-109(in Chinese)[廖国男著(郭彩丽, 周诗健译) 2004大气辐射导论(第二版) (北京:气象出版社)第108–109页]
[12] Joseph J, Wiscombe W, Weinman J 1976 J. Atmos. Sci. 33 2452
[13] Wiscombe W 1977 J. Atmos. Sci. 34 1408
[14] Huang C J, Liu Y F, Wu Z S, Sun Y Q, Long S M 2009 Acta Phys Sin. 58 2397(in Chinese)[黄朝军, 刘亚峰, 吴振森, 孙彦清, 龙姝明2009物理学报 58 2397]
[15] Kattawar G W 1975 Quant. Spectrosc. Radiat. Transfer 15 839
[16] Cornette W M, Shanks J G 1992 Appl. Opt. 31 3152
[17] Bohre C F, Huffman D R 1983 Absorption and Scattering of Light by Small Particles (New York:John Wiley) p100
[18] McCartney E J 1988 Optics of the Atmosphere (Beijing:Science Press) p276(in Chinese)[麦克特尼E J著(潘乃先译) 1988大气光学(北京:科学出版社)第276页]
[19] Bai L, Tang S Q, Wu Z S, Xie P H, Wang S M 2010 Acta Phys. Sin. 59 1749(in Chinese)[白璐, 汤双庆, 吴振森, 谢品华, 汪世美2010物理学报 59 1749]
[20] Zhu M Z, Zhang H L, Jia H L, Yang J K, Cheng S L 2007 J. Light Scatt. 3 225(in Chinese)[朱孟真, 张海良, 贾红辉, 杨建坤, 常胜利2007光散射学报 3 225]
[21] Derimendjian D 1963 Scattering and Polarization Properties of Polydispersed Suspensions with Partial Absorption (New York:The MacMillan Company, Pursuant to a Special Arrangement with Pergamon Press) pp4-9
[22] Diermendjian D 1969 Electromagnetic Scattering on Spherical Polydispersions (New York:American Elsevier) pp31-78
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