Abstract

We report the results of an extensive study of the scattering of light by size and size–shape distributions of randomly oriented prolate and oblate spheroids with the index of refraction 1.5 + 0.02i typical of some mineral terrestrial aerosols. The scattering calculations have been carried out with Waterman’s T-matrix approach, as developed recently by Mishchenko [J. Opt. Soc. Am. A 8, 871 (1991); Appl. Opt. 32, 4562 (1993)]. Our main interest is in light scattering by polydisperse models of nonspherical particles because averaging over sizes provides more realistic modeling of natural ensembles of scattering particles and washes out the interference structure and ripple typical of monodisperse scattering patterns, thus enabling us to derive meaningful conclusions about the effects of particle nonsphericity on light scattering. Following Hansen and Travis [Space Sci. Rev. 16, 527 (1974)], we show that scattering properties of most physically plausible size distributions of randomly oriented nonspherical particles depend primarily on the effective equivalent-sphere radius and effective variance of the distribution, the actual shape of the distribution having a minor influence. To minimize the computational burden, we have adopted a computationally convenient power law distribution of particle equivalent-sphere radii n(r) ∝ r−3, r1rr2. The effective variance of the size distribution is fixed at 0.1, and the effective size parameter continuously varies from 0 to 15. We present results of computer calculations for 24 prolate and oblate spheroidal shapes with aspect ratios from 1.1 to 2.2. The elements of the scattering matrix for the whole range of size parameters and scattering angles are displayed in the form of contour plots. Computational results are compared with analogous calculations for surface-equivalent spheres, and the effects of particle shape on light scattering are discussed in detail.

© 1994 Optical Society of America

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  36. D. W. Cooper, J. W. Davis, R. L. Byers, “Measurements of depolarization by dry and humidified salt aerosols using a lidar analogue,” Aerosol Sci. 5, 117–123 (1974).
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  37. T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE 71, 1029–1078 (1983).
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  40. P. H. Herzegh, A. R. Jameson, “Observing precipitation through dual-polarization radar measurements,” Bull. Am. Meteorol. Soc. 73, 1365–1374 (1992).
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    [CrossRef]
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    [CrossRef]
  44. D. L. Coffeen, J. E. Hansen, “Polarization studies of planetary atmospheres,” in Planets, Stars and Nebulae Studied with Photopolarimetry, T. Gehrels, ed. (U. Arizona Press, Tucson, Ariz., 1974), pp. 518–592.
  45. T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki: Part II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  50. J. I. Hage, J. M. Greenberg, R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30, 1141–1152 (1991).
    [CrossRef] [PubMed]
  51. B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3)*,” Phys. Rev. D 8, 3661–3678 (1973).
    [CrossRef]
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    [CrossRef]
  57. M. I. Mishchenko, “Enhanced backscattering of polarized light from discrete random media: calculations in exactly the backscattering direction,” J. Opt. Soc. Am. A 9, 978–982 (1992).
    [CrossRef]
  58. W. M. F. Wauben, J. F. de Haan, J. W. Hovenier, “Influence of particle shape on the polarized radiation in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 50, 237–246 (1993).
    [CrossRef]

1994

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

F. Kuik, J. F. de Haan, J. W. Hovenier, “Single scattering of light by circular cylinders,” Appl. Opt. 33, 4906–4918 (1994).
[CrossRef] [PubMed]

1993

R. H. Zerull, B. Å. S. Gustafson, K. Schulz, E. Thiele-Corbach, “Scattering by aggregates with and without an absorbing mantle: microwave and analog experiments,” Appl. Opt. 32, 4088–4100 (1993).
[PubMed]

M. I. Mishchenko, “Light scattering by size/shape distributions of randomly oriented axially symmetric particles of size comparable to a wavelength,” Appl. Opt. 32, 4652–4666 (1993).
[CrossRef] [PubMed]

Y. J. Kaufman, “Aerosol optical thickness and atmospheric path radiance,” J. Geophys. Res. 98, 2677–2692 (1993).
[CrossRef]

W. M. F. Wauben, J. F. de Haan, J. W. Hovenier, “Influence of particle shape on the polarized radiation in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 50, 237–246 (1993).
[CrossRef]

1992

Note, however, that multiple scattering can also produce significant depolarization of backscattered light, even for spherical particles [e.g., M. I. Mishchenko, “Polarization characteristics of the coherent backscatter opposition effect,” Earth Moon Planets 58, 127–144 (1992)]. Therefore, in analyzing observational or laboratory data, one should be careful in assigning the origin of nonzero backscattering depolarization ratios solely to particle nonsphericity.
[CrossRef]

M. I. Mishchenko, “Enhanced backscattering of polarized light from discrete random media: calculations in exactly the backscattering direction,” J. Opt. Soc. Am. A 9, 978–982 (1992).
[CrossRef]

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

A. Lacis, J. Hansen, M. Sato, “Climate forcing by stratospheric aerosols,” Geophys. Res. Lett. 19, 1607–1610 (1992).
[CrossRef]

P. H. Herzegh, A. R. Jameson, “Observing precipitation through dual-polarization radar measurements,” Bull. Am. Meteorol. Soc. 73, 1365–1374 (1992).
[CrossRef]

W. L. Eberhard, “Ice-cloud depolarization of backscatter for CO2 and other infrared lidars,” Appl. Opt. 31, 6485–6490 (1992).
[CrossRef] [PubMed]

1991

J. I. Hage, J. M. Greenberg, R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30, 1141–1152 (1991).
[CrossRef] [PubMed]

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866(1991).
[CrossRef]

S. Y. Matrosov, “Theoretical study of radar polarization parameters obtained from cirrus clouds,” J. Atmos. Sci. 48, 1062–1070 (1991).
[CrossRef]

J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
[CrossRef] [PubMed]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991); J. Opt. Soc. Am. A 9, 497(E) (1992).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

1989

1988

1987

1986

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 1: Cross sections, single-scattering albedo, asymmetry factor, and backscattered fraction,” Appl. Opt. 25, 1235–1244 (1986).
[CrossRef] [PubMed]

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

1984

1983

W. Wiscombe, “Atmospheric radiation: 1975–1983,” Rev. Geophys. Space Phys. 21, 997–1021 (1983).
[CrossRef]

T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE 71, 1029–1078 (1983).
[CrossRef]

1981

1980

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semiempirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
[CrossRef]

S. J. Ostro, G. H. Pettengill, D. B. Campbell, “Radar observations of Saturn’s rings at intermediate tilt angles,” Icarus 41, 381–388 (1980); S. J. Ostro, G. H. Pettengill, D. B. Campbell, R. M. Goldstein, “Delay-Doppler radar observations of Saturn’s rings,” Icarus 49, 367–381 (1982).
[CrossRef]

1978

1976

R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).

1974

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974); J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983); M. I. Mishchenko, “Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 46, 171–181 (1991).
[CrossRef]

J. E. Hansen, J. W. Hovenier, “Interpretation of the polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974); M. I. Mishchenko, “Physical properties of the upper tropospheric aerosols in the equatorial region of Jupiter,” Icarus 84, 296–304 (1990).
[CrossRef]

K.-N. Liou, H. Lahore, “Laser sensing of cloud composition: a backscatter depolarization technique,” J. Appl. Meteorol. 13, 257–263 (1974).
[CrossRef]

D. W. Cooper, J. W. Davis, R. L. Byers, “Measurements of depolarization by dry and humidified salt aerosols using a lidar analogue,” Aerosol Sci. 5, 117–123 (1974).
[CrossRef]

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1973

B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3)*,” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

1971

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971); S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975); P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

1948

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Arao, K.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki: Part II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Asano, S.

Barber, P. W.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Byers, R. L.

D. W. Cooper, J. W. Davis, R. L. Byers, “Measurements of depolarization by dry and humidified salt aerosols using a lidar analogue,” Aerosol Sci. 5, 117–123 (1974).
[CrossRef]

Campbell, D. B.

S. J. Ostro, G. H. Pettengill, D. B. Campbell, “Radar observations of Saturn’s rings at intermediate tilt angles,” Icarus 41, 381–388 (1980); S. J. Ostro, G. H. Pettengill, D. B. Campbell, R. M. Goldstein, “Delay-Doppler radar observations of Saturn’s rings,” Icarus 49, 367–381 (1982).
[CrossRef]

Carlson, B. E.

M. I. Mishchenko, B. E. Carlson, A. A. Lacis, “Radiative properties of nonspherical aerosols,” in Proceedings of the Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 307–309.

Coffeen, D. L.

D. L. Coffeen, J. E. Hansen, “Polarization studies of planetary atmospheres,” in Planets, Stars and Nebulae Studied with Photopolarimetry, T. Gehrels, ed. (U. Arizona Press, Tucson, Ariz., 1974), pp. 518–592.

Cooper, D. W.

D. W. Cooper, J. W. Davis, R. L. Byers, “Measurements of depolarization by dry and humidified salt aerosols using a lidar analogue,” Aerosol Sci. 5, 117–123 (1974).
[CrossRef]

Cuzzi, J. N.

J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semiempirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
[CrossRef]

d’Almeida, G. A.

G. A. d’Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols (Deepak, Hamptom, Va., 1991).

Davis, J. W.

D. W. Cooper, J. W. Davis, R. L. Byers, “Measurements of depolarization by dry and humidified salt aerosols using a lidar analogue,” Aerosol Sci. 5, 117–123 (1974).
[CrossRef]

de Haan, J. F.

F. Kuik, J. F. de Haan, J. W. Hovenier, “Single scattering of light by circular cylinders,” Appl. Opt. 33, 4906–4918 (1994).
[CrossRef] [PubMed]

W. M. F. Wauben, J. F. de Haan, J. W. Hovenier, “Influence of particle shape on the polarized radiation in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 50, 237–246 (1993).
[CrossRef]

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

J. F. de Haan, “Effects of aerosols on the brightness and polarization of cloudless planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1987).

Domke, H.

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974); J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983); M. I. Mishchenko, “Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 46, 171–181 (1991).
[CrossRef]

Draine, B. T.

Eberhard, W. L.

Flatau, P. J.

Fry, E. S.

Fuller, K. A.

Glatter, O.

Goodman, J. J.

Greenberg, J. M.

Gustafson, B. Å. S.

Hage, J. I.

Hansen, J.

A. Lacis, J. Hansen, M. Sato, “Climate forcing by stratospheric aerosols,” Geophys. Res. Lett. 19, 1607–1610 (1992).
[CrossRef]

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

J. E. Hansen, J. W. Hovenier, “Interpretation of the polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974); M. I. Mishchenko, “Physical properties of the upper tropospheric aerosols in the equatorial region of Jupiter,” Icarus 84, 296–304 (1990).
[CrossRef]

D. L. Coffeen, J. E. Hansen, “Polarization studies of planetary atmospheres,” in Planets, Stars and Nebulae Studied with Photopolarimetry, T. Gehrels, ed. (U. Arizona Press, Tucson, Ariz., 1974), pp. 518–592.

Herb, P.

Herzegh, P. H.

P. H. Herzegh, A. R. Jameson, “Observing precipitation through dual-polarization radar measurements,” Bull. Am. Meteorol. Soc. 73, 1365–1374 (1992).
[CrossRef]

Hess, M.

Hill, A. C.

Hill, S. C.

Hofer, M.

Hovenier, J. W.

F. Kuik, J. F. de Haan, J. W. Hovenier, “Single scattering of light by circular cylinders,” Appl. Opt. 33, 4906–4918 (1994).
[CrossRef] [PubMed]

W. M. F. Wauben, J. F. de Haan, J. W. Hovenier, “Influence of particle shape on the polarized radiation in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 50, 237–246 (1993).
[CrossRef]

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

J. E. Hansen, J. W. Hovenier, “Interpretation of the polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974); M. I. Mishchenko, “Physical properties of the upper tropospheric aerosols in the equatorial region of Jupiter,” Icarus 84, 296–304 (1990).
[CrossRef]

Hu, C.-R.

Huffman, D. R.

Hunt, A. J.

Jameson, A. R.

P. H. Herzegh, A. R. Jameson, “Observing precipitation through dual-polarization radar measurements,” Bull. Am. Meteorol. Soc. 73, 1365–1374 (1992).
[CrossRef]

Kattawar, G. W.

Kaufman, Y. J.

Y. J. Kaufman, “Aerosol optical thickness and atmospheric path radiance,” J. Geophys. Res. 98, 2677–2692 (1993).
[CrossRef]

Koepke, P.

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Kuik, F.

F. Kuik, J. F. de Haan, J. W. Hovenier, “Single scattering of light by circular cylinders,” Appl. Opt. 33, 4906–4918 (1994).
[CrossRef] [PubMed]

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

F. Kuik, “Single scattering of light by ensembles of particles with various shapes,” Ph.D. dissertation (Free University, Amsterdam, 1992).

Lacis, A.

A. Lacis, J. Hansen, M. Sato, “Climate forcing by stratospheric aerosols,” Geophys. Res. Lett. 19, 1607–1610 (1992).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, B. E. Carlson, A. A. Lacis, “Radiative properties of nonspherical aerosols,” in Proceedings of the Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 307–309.

Lahore, H.

K.-N. Liou, H. Lahore, “Laser sensing of cloud composition: a backscatter depolarization technique,” J. Appl. Meteorol. 13, 257–263 (1974).
[CrossRef]

Liou, K.-N.

K.-N. Liou, H. Lahore, “Laser sensing of cloud composition: a backscatter depolarization technique,” J. Appl. Meteorol. 13, 257–263 (1974).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Matrosov, S. Y.

S. Y. Matrosov, “Theoretical study of radar polarization parameters obtained from cirrus clouds,” J. Atmos. Sci. 48, 1062–1070 (1991).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

M. I. Mishchenko, “Light scattering by size/shape distributions of randomly oriented axially symmetric particles of size comparable to a wavelength,” Appl. Opt. 32, 4652–4666 (1993).
[CrossRef] [PubMed]

M. I. Mishchenko, “Enhanced backscattering of polarized light from discrete random media: calculations in exactly the backscattering direction,” J. Opt. Soc. Am. A 9, 978–982 (1992).
[CrossRef]

Note, however, that multiple scattering can also produce significant depolarization of backscattered light, even for spherical particles [e.g., M. I. Mishchenko, “Polarization characteristics of the coherent backscatter opposition effect,” Earth Moon Planets 58, 127–144 (1992)]. Therefore, in analyzing observational or laboratory data, one should be careful in assigning the origin of nonzero backscattering depolarization ratios solely to particle nonsphericity.
[CrossRef]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991); J. Opt. Soc. Am. A 9, 497(E) (1992).
[CrossRef]

M. I. Mishchenko, B. E. Carlson, A. A. Lacis, “Radiative properties of nonspherical aerosols,” in Proceedings of the Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 307–309.

M. I. Mishchenko, L. D. Travis, “Light scattering by size/shape distributions of nonspherical particles of size comparable to a wavelength,” in Atmospheric Propagation and Remote Sensing II, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1968, 118–129 (1993).

Mugnai, A.

Nakajima, T.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki: Part II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Nakanishi, Y.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki: Part II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Oguchi, T.

T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE 71, 1029–1078 (1983).
[CrossRef]

Ostro, S. J.

S. J. Ostro, G. H. Pettengill, D. B. Campbell, “Radar observations of Saturn’s rings at intermediate tilt angles,” Icarus 41, 381–388 (1980); S. J. Ostro, G. H. Pettengill, D. B. Campbell, R. M. Goldstein, “Delay-Doppler radar observations of Saturn’s rings,” Icarus 49, 367–381 (1982).
[CrossRef]

Parkin, M. E.

Perry, R. J.

Peterson, B.

B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3)*,” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Pettengill, G. H.

S. J. Ostro, G. H. Pettengill, D. B. Campbell, “Radar observations of Saturn’s rings at intermediate tilt angles,” Icarus 41, 381–388 (1980); S. J. Ostro, G. H. Pettengill, D. B. Campbell, R. M. Goldstein, “Delay-Doppler radar observations of Saturn’s rings,” Icarus 49, 367–381 (1982).
[CrossRef]

Pollack, J. B.

J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semiempirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
[CrossRef]

Sassen, K.

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866(1991).
[CrossRef]

Sato, M.

A. Lacis, J. Hansen, M. Sato, “Climate forcing by stratospheric aerosols,” Geophys. Res. Lett. 19, 1607–1610 (1992).
[CrossRef]

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

Schaefer, R. W.

Schuerman, D. W.

Schulz, K.

Shettle, E. P.

G. A. d’Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols (Deepak, Hamptom, Va., 1991).

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Shiobara, M.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki: Part II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Stammes, P.

P. Stammes, “Light scattering properties of aerosols and the radiation inside a planetary atmosphere,” Ph.D. dissertation (Free University, Amsterdam, 1989).

Ström, S.

B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3)*,” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Tanaka, M.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki: Part II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Thiele-Corbach, E.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by size/shape distributions of nonspherical particles of size comparable to a wavelength,” in Atmospheric Propagation and Remote Sensing II, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1968, 118–129 (1993).

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

van de Hulst, H. C.

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

van der Mee, C. V. M.

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Vouk, V.

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Wang, R. T.

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971); S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975); P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

Wauben, W. M. F.

W. M. F. Wauben, J. F. de Haan, J. W. Hovenier, “Influence of particle shape on the polarized radiation in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 50, 237–246 (1993).
[CrossRef]

Wiscombe, W.

W. Wiscombe, “Atmospheric radiation: 1975–1983,” Rev. Geophys. Space Phys. 21, 997–1021 (1983).
[CrossRef]

Wiscombe, W. J.

Yamano, M.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki: Part II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Zerull, R. H.

Aerosol Sci.

D. W. Cooper, J. W. Davis, R. L. Byers, “Measurements of depolarization by dry and humidified salt aerosols using a lidar analogue,” Aerosol Sci. 5, 117–123 (1974).
[CrossRef]

Appl. Opt.

R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determinations of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2700–2710 (1978).
[CrossRef] [PubMed]

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

E. S. Fry, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
[CrossRef] [PubMed]

D. W. Schuerman, R. T. Wang, B. Å. S. Gustafson, R. W. Schaefer, “Systematic studies of light scattering. 1: Particle shape,” Appl. Opt. 20, 4039–4050 (1981).
[CrossRef] [PubMed]

S. C. Hill, A. C. Hill, P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,” Appl. Opt. 23, 1025–1031 (1984).
[CrossRef] [PubMed]

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 1: Cross sections, single-scattering albedo, asymmetry factor, and backscattered fraction,” Appl. Opt. 25, 1235–1244 (1986).
[CrossRef] [PubMed]

C.-R. Hu, G. W. Kattawar, M. E. Parkin, P. Herb, “Symmetry theorems on the forward and backward scattering Mueller matrices for light scattering from a nonspherical dielectric scatterer,” Appl. Opt. 26, 4159–4173 (1987).
[CrossRef] [PubMed]

W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2: Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).
[CrossRef] [PubMed]

P. Koepke, M. Hess, “Scattering functions of tropospheric aerosols: the effects of nonspherical particles,” Appl. Opt. 27, 2422–2430 (1988).
[CrossRef] [PubMed]

M. Hofer, O. Glatter, “Mueller matrix calculations for randomly oriented rotationally symmetric objects with low contrast,” Appl. Opt. 28, 2389–2400 (1989).
[CrossRef] [PubMed]

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 3: Variability in angular scattering patterns,” Appl. Opt. 28, 3061–3073 (1989).
[CrossRef] [PubMed]

J. I. Hage, J. M. Greenberg, R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30, 1141–1152 (1991).
[CrossRef] [PubMed]

M. I. Mishchenko, “Light scattering by size/shape distributions of randomly oriented axially symmetric particles of size comparable to a wavelength,” Appl. Opt. 32, 4652–4666 (1993).
[CrossRef] [PubMed]

F. Kuik, J. F. de Haan, J. W. Hovenier, “Single scattering of light by circular cylinders,” Appl. Opt. 33, 4906–4918 (1994).
[CrossRef] [PubMed]

R. H. Zerull, B. Å. S. Gustafson, K. Schulz, E. Thiele-Corbach, “Scattering by aggregates with and without an absorbing mantle: microwave and analog experiments,” Appl. Opt. 32, 4088–4100 (1993).
[PubMed]

W. L. Eberhard, “Ice-cloud depolarization of backscatter for CO2 and other infrared lidars,” Appl. Opt. 31, 6485–6490 (1992).
[CrossRef] [PubMed]

Astron. Astrophys.

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Astrophys. Space Sci.

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974); J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983); M. I. Mishchenko, “Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 46, 171–181 (1991).
[CrossRef]

Beitr. Phys. Atmos.

R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).

Bull. Am. Meteorol. Soc.

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866(1991).
[CrossRef]

P. H. Herzegh, A. R. Jameson, “Observing precipitation through dual-polarization radar measurements,” Bull. Am. Meteorol. Soc. 73, 1365–1374 (1992).
[CrossRef]

Earth Moon Planets

Note, however, that multiple scattering can also produce significant depolarization of backscattered light, even for spherical particles [e.g., M. I. Mishchenko, “Polarization characteristics of the coherent backscatter opposition effect,” Earth Moon Planets 58, 127–144 (1992)]. Therefore, in analyzing observational or laboratory data, one should be careful in assigning the origin of nonzero backscattering depolarization ratios solely to particle nonsphericity.
[CrossRef]

Geophys. Res. Lett.

A. Lacis, J. Hansen, M. Sato, “Climate forcing by stratospheric aerosols,” Geophys. Res. Lett. 19, 1607–1610 (1992).
[CrossRef]

Icarus

S. J. Ostro, G. H. Pettengill, D. B. Campbell, “Radar observations of Saturn’s rings at intermediate tilt angles,” Icarus 41, 381–388 (1980); S. J. Ostro, G. H. Pettengill, D. B. Campbell, R. M. Goldstein, “Delay-Doppler radar observations of Saturn’s rings,” Icarus 49, 367–381 (1982).
[CrossRef]

J. Appl. Meteorol.

K.-N. Liou, H. Lahore, “Laser sensing of cloud composition: a backscatter depolarization technique,” J. Appl. Meteorol. 13, 257–263 (1974).
[CrossRef]

J. Atmos. Sci.

J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semiempirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
[CrossRef]

S. Y. Matrosov, “Theoretical study of radar polarization parameters obtained from cirrus clouds,” J. Atmos. Sci. 48, 1062–1070 (1991).
[CrossRef]

J. E. Hansen, J. W. Hovenier, “Interpretation of the polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974); M. I. Mishchenko, “Physical properties of the upper tropospheric aerosols in the equatorial region of Jupiter,” Icarus 84, 296–304 (1990).
[CrossRef]

J. Geophys. Res.

Y. J. Kaufman, “Aerosol optical thickness and atmospheric path radiance,” J. Geophys. Res. 98, 2677–2692 (1993).
[CrossRef]

J. Meteorol. Soc. Jpn.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki: Part II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

W. M. F. Wauben, J. F. de Haan, J. W. Hovenier, “Influence of particle shape on the polarized radiation in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 50, 237–246 (1993).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

Nature (London)

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Opt. Commun.

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Opt. Lett.

Phys. Rev. D

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971); S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975); P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3)*,” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Proc. IEEE

T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE 71, 1029–1078 (1983).
[CrossRef]

Proc. R. Soc. London Ser. A

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Rev. Geophys. Space Phys.

W. Wiscombe, “Atmospheric radiation: 1975–1983,” Rev. Geophys. Space Phys. 21, 997–1021 (1983).
[CrossRef]

Space Sci. Rev.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Other

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

M. I. Mishchenko, L. D. Travis, “Light scattering by size/shape distributions of nonspherical particles of size comparable to a wavelength,” in Atmospheric Propagation and Remote Sensing II, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1968, 118–129 (1993).

G. A. d’Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols (Deepak, Hamptom, Va., 1991).

W. J. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (Goddard Space Flight Center, NASA, Greenbelt, Md., 1986).

V. K. Varadan, V. V. Varadan, eds., Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach (Pergamon, New York, 1980).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

J. F. de Haan, “Effects of aerosols on the brightness and polarization of cloudless planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1987).

P. Stammes, “Light scattering properties of aerosols and the radiation inside a planetary atmosphere,” Ph.D. dissertation (Free University, Amsterdam, 1989).

F. Kuik, “Single scattering of light by ensembles of particles with various shapes,” Ph.D. dissertation (Free University, Amsterdam, 1992).

M. I. Mishchenko, B. E. Carlson, A. A. Lacis, “Radiative properties of nonspherical aerosols,” in Proceedings of the Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 307–309.

D. L. Coffeen, J. E. Hansen, “Polarization studies of planetary atmospheres,” in Planets, Stars and Nebulae Studied with Photopolarimetry, T. Gehrels, ed. (U. Arizona Press, Tucson, Ariz., 1974), pp. 518–592.

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Figures (13)

Fig. 1
Fig. 1

Power law (———), gamma (— — —), and log-normal (……) size distributions having the same effective radius reff = 1.5 μm and effective variance νeff = 0.1.

Fig. 2
Fig. 2

Elements of the scattering matrix for power law (———) and log-normal (— — —) distributions of equivalent-sphere radii for randomly oriented oblate spheroids with aspect ratio 1.7. For both distributions, the effective radius is reff = 1.5 μm and the effective variance is νeff = 0.1. The refractive index of the spheroids is 1.5 + 0.02i, and the wavelength is 0.6283 μm.

Fig. 3
Fig. 3

Upper graph, efficiency factor for extinction Qext versus effective size parameter xeff for surface-equivalent polydispersions of spheres (———), randomly oriented prolate spheroids with aspect ratios ∊ = 1.5 (……) and 2.2 (— · — · —), randomly oriented oblate spheroids with aspect ratios ∊ = 1.5 (— — —) and ∊ = 2.2 (— · · · —), and an equiprobable mixture of 24 spheroidal (———). Lower graph, corresponding percent spherical–nonspherical differences ∊ext defined by Eq. (21).

Fig. 4
Fig. 4

Upper graph, efficiency factor for scattering Qsca versus effective size parameter xeff for surface-equivalent polydispersions of spheres (———), randomly oriented prolate spheroids with aspect ratios ∊ = 1.5 (……) and 2.2 (— · — · —), randomly oriented oblate spheroids with aspect ratios ∊ = 1.5 (— — —) and ∊ = 2.2 (— · · · —), and an equiprobable mixture of 24 spheroidal shapes (———). Lower graph, corresponding percent spherical–nonspherical differences.

Fig. 5
Fig. 5

Upper graph, efficiency factor for absorption Qabs versus effective size parameter xeff for surface-equivalent polydispersions of spheres (———), randomly oriented prolate spheroids with aspect ratios ∊ = 1.5 (……) and 2.2 (— · — · —), randomly oriented oblate spheroids with aspect ratios ∊ = 1.5 (— — —) and ∊ = 2.2 (— · · · —), and an equiprobable mixture of 24 spheroidal shapes (———). Lower graph, corresponding percent spherical–nonspherical differences.

Fig. 6
Fig. 6

Upper graph, single-scattering albedo ϖ versus effective size parameter xeff for surface-equivalent polydispersions of spheres (———), randomly oriented prolate spheroids with aspect ratios ∊ = 1.5 (……) and 2.2 (— · — · —), randomly oriented oblate spheroids with aspect ratios ∊ = 1.5 (— — —) and ∊ = 2.2 (— · · · —), and an equiprobable mixture of 24 spheroidal shapes (———). Lower graph, corresponding percent spherical–nonspherical differences.

Fig. 7
Fig. 7

Upper graph, asymmetry parameter of the phase function 〈cos Θ〉 versus effective size parameter xeff for surface-equivalent polydispersions of spheres (———), randomly oriented prolate spheroids with aspect ratios ∊ = 1.5 (……) and 2.2 (— · — · —), randomly oriented oblate spheroids with aspect ratios ∊ = 1.5 (— — —) and ∊ = 2.2 (— · · · —), and an equiprobable mixture of 24 spheroidal shapes (———). Lower graph, corresponding percent spherical–nonspherical differences.

Fig. 8
Fig. 8

Contour plots of ∊I [see Eq. (24)] as a function of scattering angle Θ and effective size parameter xeff for surface-equivalent polydispersions of (a) prolate and (b) oblate spheroids with the aspect ratio ∊ = 1.5, (c) prolate and (d) oblate spheroids with the aspect ratio ∊ = 2.2, and (e) an equiprobable mixture of 24 spheroidal shapes. (f) contour plot of the normalized phase function for polydisperse spherical particles.

Fig. 9
Fig. 9

Contour plots of the ratio F22/F11 × 100% as a function of scattering angle Θ and effective size parameter xeff for surface-equivalent polydispersions of (a) prolate and (b) oblate spheroids with the aspect ratio ∊ = 1.5, (c) prolate and (d) oblate spheroids with the aspect ratio ∊ = 2.2, and (e) an equiprobable mixture of 24 spheroidal shapes.

Fig. 10
Fig. 10

Contour plots of the ratio F33/F11 × 100% as a function of scattering angle Θ and effective size parameter xeff for surface-equivalent polydispersions of (a) prolate and (b) oblate spheroids with the aspect ratio ∊ = 1.5, (c) prolate and (d) oblate spheroids with the aspect ratio ∊ = 2.2, (e) an equiprobable mixture of 24 spheroidal shapes, and (f) spheres.

Fig. 11
Fig. 11

Contour plots of the ratio F44/F11 × 100% as a function of scattering angle Θ and effective size parameter xeff for surface-equivalent polydispersions of (a) prolate and (b) oblate spheroids with the aspect ratio ∊ = 1.5, (c) prolate and (d) oblate spheroids with the aspect ratio ∊ = 2.2, (e) an equiprobable mixture of 24 spheroidal shapes, and (f) spheres.

Fig. 12
Fig. 12

Contour plots of linear polarization −F12/F11 × 100% as a function of scattering angle Θ and effective size parameter xeff for surface-equivalent polydispersions of (a) prolate and (b) oblate spheroids with the aspect ratio ∊ = 1.5, (c) prolate and (d) oblate spheroids with the aspect ratio ∊ = 2.2, (e) an equiprobable mixture of 24 spheroidal shapes, and (f) spheres.

Fig. 13
Fig. 13

Contour plots of the ratio F34/F11 × 100% as a function of scattering angle Θ and effective size parameter xeff for surface-equivalent polydispersions of (a) prolate and (b) oblate spheroids with the aspect ratio ∊ = 1.5, (c) prolate and (d) oblate spheroids with the aspect ratio ∊ = 2.2, (e) an equiprobable mixture of 24 spheroidal shapes, and (f) spheres.

Tables (2)

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Table 1 Optical Cross Sections, Single-Scattering Albedo, and Asymmetry Parameter of the Phase Function for Power Law, Gamma, and Log-Normal Distributions of Equivalent-Sphere Radii for Randomly Oriented Oblate Spheroids with Aspect Ratio 1.7a

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Table 2 Maximum Values of the Ratio Δ and Backscattering Depolarization Ratios δL and δc Along with Corresponding Values of the Effective Size Parameter and Scattering Angle for Prolate and Oblate Spheroids with Varying Aspect Ratios and the Equiprobable Shape Mixture

Equations (32)

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r ( ϑ , ϕ ) = a [ sin 2 ϑ + a 2 b 2 cos 2 ϑ ] - 1 / 2 ,
r = 1 2 [ 2 a 2 + 2 a b arcsin e e ] 1 / 2
r = 1 2 [ 2 a 2 + b 2 e ln ( 1 + e 1 - e ) ] 1 / 2
e = ( 2 - 1 ) 1 / 2 .
F ( Θ ) = [ F 11 ( Θ ) F 12 ( Θ ) 0 0 F 12 ( Θ ) F 22 ( Θ ) 0 0 0 0 F 33 ( Θ ) F 34 ( Θ ) 0 0 - F 34 ( Θ ) F 44 ( Θ ) ] ,
1 4 π 4 π d Ω F 11 ( Θ ) = 1.
r eff = 1 G 0 d r r π r 2 n ( r ) ,
G = 0 d r π r 2 n ( r )
0 d r n ( r ) = 1.
ν eff = 1 G r eff 2 0 d r ( r - r eff ) 2 π r 2 n ( r ) .
n ( r ) = { 2 r 1 2 r 2 2 r 2 2 - r 1 2 r - 3 for r 1 r r 2 , 0 otherwise ,
n ( r ) = constant r ( 1 - 3 ν eff ) / ν eff exp ( - r r eff ν eff ) ,
n ( r ) = 1 ( 2 π ) 1 / 2 σ g 1 r exp [ - ( ln r - ln r g ) 2 2 σ g 2 ] ,
r eff = r 2 - r 1 ln ( r 2 / r 1 ) ,
ν eff = r 2 + r 1 2 ( r 2 - r 1 ) ln ( r 2 / r 1 ) - 1 ,
r eff = r g exp ( 5 σ g 2 / 2 ) ,
ν eff = exp ( σ g 2 ) - 1.
( q 2 - q 1 ) ln ( q 2 / q 1 ) = 1 ,
q 2 + q 1 = 2 ( ν eff + 1 ) .
Q ext = C ext π r eff 2 ,             Q sca = C sca π r eff 2 ,             Q abs = C abs π r eff 2 .
ext = C ext ( spherical ) - C ext ( nonspherical ) C ext ( spherical ) × 100 %
F 22 ( Θ ) / F 11 ( Θ ) 1 ,
F 33 ( Θ ) / F 11 ( Θ ) F 44 ( Θ ) / F 11 ( Θ ) .
I ( Θ ) = [ F 11 ( Θ ) ] spherical - [ F 11 ( Θ ) ] nonspher [ F 11 ( Θ ) ] spherical × 100 % .
F 33 - F 44 F 11 - F 22 ,
Δ = F 11 - F 22 + F 33 - F 44 F 11 × 100 %
Δ ( 180 ° ) = 0 ,
F 44 ( 180 ° ) - F 33 ( 180 ° ) = F 11 ( 180 ° ) - F 22 ( 180 ° ) .
F 44 ( 180 ° ) F 33 ( 180 ° ) .
F 11 ( 0 ) - F 22 ( 0 ) - F 33 ( 0 ) + F 44 ( 0 ) = 0.
δ L = F 11 ( 180 ° ) - F 22 ( 180 ° ) F 11 ( 180 ° ) + F 22 ( 180 ° ) ,
δ C = F 11 ( 180 ° ) + F 44 ( 180 ° ) F 11 ( 180 ° ) - F 44 ( 180 ° ) .

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