Abstract

We investigate the phase matrices of droxtals at wavelengths of 0.66 and 11 μm by using an improved geometrical-optics method. An efficient method is developed to specify the incident rays and the corresponding impinging points on the particle surface necessary to initialize the ray-tracing computations. At the 0.66-μm wavelength, the optical properties of droxtals are different from those of hexagonal ice crystals. At the 11-μm wavelength, the phase functions for droxtals are essentially featureless because of strong absorption within the particles, except for ripple structures that are caused by the phase interference of the diffracted wave.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. P. Yang, K. N. Liou, “Geometric-optics integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  42. Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
    [CrossRef] [PubMed]
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2003

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

S. Havemann, A. J. Baran, J. M. Edwards, “Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 707–720 (2003).
[CrossRef]

Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[CrossRef] [PubMed]

S. Reichardt, J. Reichardt, “Effect of multiple scattering on depolarization measurements with spaceborne lidars,” Appl. Opt. 42, 3620–3633 (2003).
[CrossRef] [PubMed]

2002

J. Reichardt, S. Reichardt, P. Yang, T. J. McGee, “Retrieval of polar stratospheric cloud microphysical properties from lidar measurements: dependence on particle shape assumptions,” J. Geophys. Res. 107D, 10.1029/2001JD001021 (2002).

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

2001

P. R. Lawson, B. A. Baker, C. G. Schmitt, T. J. Jensen, “An overview of microphysical properties of Arctic clouds observed in May and July 1998 during FIRE ACE,” J. Geophys. Res. 106, 14989–15014 (2001).
[CrossRef]

L. Liu, M. I. Mishchenko, “Constraints on PSC particle microphysics derived from lidar observations,” J. Quant. Spectrosc. Radiat. Transfer 70, 817–831 (2001).
[CrossRef]

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532–1547 (2001).
[CrossRef]

2000

A. J. Heymsfield, J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 57, 916–938 (2000).
[CrossRef]

D. Rosenfeld, W. L. Woodley, “Deep convective clouds with sustained supercooled liquid water down to -37.5 degrees,” Nature (London) 405, 440–442 (2000).
[CrossRef]

W. Sun, Q. Fu, “Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices,” Appl. Opt. 39, 5569–5578 (2000).
[CrossRef]

1999

1998

M. Hess, R. B. A. Koelemeijer, P. Stammes, “Scattering matrices of imperfect hexagonal crystals,” J. Quant. Spectrosc. Radiat. Transfer 60, 301–308 (1998).
[CrossRef]

P. Yang, K. N. Liou, “Single-scattering properties of complex ice crystals in terrestrial atmosphere,” Contrib. Atmos. Phys. 71, 223–248 (1998).

1996

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance, and optical thickness retrieval to ice-particle shape,” J. Geophys. Res. 101, 16973–16985 (1996).
[CrossRef]

E. J. Jensen, O. B. Toon, H. B. Selkirk, J. D. Spinhirne, M. R. Schoeberl, “On the formation and persistence of subvisible cirrus clouds near the tropical tropopause,” J. Geophys. Res. 101, 21361–21375 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

1995

J. Iaquinta, H. Isaka, P. Personne, “Scattering phase function of bullet rosette ice crystals,” J. Atmos. Sci. 52, 1401–1413 (1995).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics methods,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

1993

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
[CrossRef] [PubMed]

A. J. Heymsfield, L. M. Miloshevich, “Homogeneous ice nucleation and supercooled liquid water in orographic wave clouds,” J. Atmos. Sci. 50, 2335–2353 (1993).
[CrossRef]

M. I. Mishchenko, “Light scattering by size-shape distributions of randomly oriented axially symmetric particles of a size comparable to wavelength,” Appl. Opt. 32, 623–625 (1993).
[CrossRef]

1990

A. J. Heymsfield, K. M. Miller, J. D. Spinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

1989

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044–3050 (1989).
[CrossRef] [PubMed]

1986

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

1985

1984

1982

1979

1975

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirriform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

1970

T. Ohtake, “Unusual crystal in ice fog,” J. Atmos. Sci. 27, 509–511 (1970).
[CrossRef]

1954

W. C. Thuman, E. Robinson, “Studies of Alaskan ice-fog particles,” J. Meteorol. 11, 151–156 (1954).
[CrossRef]

Ackerman, S. A.

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

Baker, B. A.

P. R. Lawson, B. A. Baker, C. G. Schmitt, T. J. Jensen, “An overview of microphysical properties of Arctic clouds observed in May and July 1998 during FIRE ACE,” J. Geophys. Res. 106, 14989–15014 (2001).
[CrossRef]

Baran, A. J.

Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[CrossRef] [PubMed]

S. Havemann, A. J. Baran, J. M. Edwards, “Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 707–720 (2003).
[CrossRef]

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

Baum, B. A.

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[CrossRef] [PubMed]

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532–1547 (2001).
[CrossRef]

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

Cai, Q.

Edwards, J. M.

S. Havemann, A. J. Baran, J. M. Edwards, “Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 707–720 (2003).
[CrossRef]

Fu, Q.

Gao, B.-C.

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532–1547 (2001).
[CrossRef]

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

Greenler, R.

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, UK, 1980).

Havemann, S.

S. Havemann, A. J. Baran, J. M. Edwards, “Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 707–720 (2003).
[CrossRef]

Hess, M.

M. Hess, R. B. A. Koelemeijer, P. Stammes, “Scattering matrices of imperfect hexagonal crystals,” J. Quant. Spectrosc. Radiat. Transfer 60, 301–308 (1998).
[CrossRef]

Heymsfield, A.

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

Heymsfield, A. J.

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

A. J. Heymsfield, J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 57, 916–938 (2000).
[CrossRef]

A. J. Heymsfield, L. M. Miloshevich, “Homogeneous ice nucleation and supercooled liquid water in orographic wave clouds,” J. Atmos. Sci. 50, 2335–2353 (1993).
[CrossRef]

A. J. Heymsfield, K. M. Miller, J. D. Spinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirriform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Hu, Y.

Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[CrossRef] [PubMed]

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

Hu, Y. X.

Hu, Y.-X.

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

Huang, H.-L.

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[CrossRef] [PubMed]

Iaquinta, J.

A. J. Heymsfield, J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 57, 916–938 (2000).
[CrossRef]

J. Iaquinta, H. Isaka, P. Personne, “Scattering phase function of bullet rosette ice crystals,” J. Atmos. Sci. 52, 1401–1413 (1995).
[CrossRef]

Isaka, H.

J. Iaquinta, H. Isaka, P. Personne, “Scattering phase function of bullet rosette ice crystals,” J. Atmos. Sci. 52, 1401–1413 (1995).
[CrossRef]

Jayaweera, K.

Jensen, E. J.

E. J. Jensen, O. B. Toon, H. B. Selkirk, J. D. Spinhirne, M. R. Schoeberl, “On the formation and persistence of subvisible cirrus clouds near the tropical tropopause,” J. Geophys. Res. 101, 21361–21375 (1996).
[CrossRef]

Jensen, T. J.

P. R. Lawson, B. A. Baker, C. G. Schmitt, T. J. Jensen, “An overview of microphysical properties of Arctic clouds observed in May and July 1998 during FIRE ACE,” J. Geophys. Res. 106, 14989–15014 (2001).
[CrossRef]

Kahnert, F. M.

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

Koelemeijer, R. B. A.

M. Hess, R. B. A. Koelemeijer, P. Stammes, “Scattering matrices of imperfect hexagonal crystals,” J. Quant. Spectrosc. Radiat. Transfer 60, 301–308 (1998).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance, and optical thickness retrieval to ice-particle shape,” J. Geophys. Res. 101, 16973–16985 (1996).
[CrossRef]

Lawson, P. R.

P. R. Lawson, B. A. Baker, C. G. Schmitt, T. J. Jensen, “An overview of microphysical properties of Arctic clouds observed in May and July 1998 during FIRE ACE,” J. Geophys. Res. 106, 14989–15014 (2001).
[CrossRef]

Lee, Y. K.

Liou, K. N.

P. Yang, K. N. Liou, “Single-scattering properties of complex ice crystals in terrestrial atmosphere,” Contrib. Atmos. Phys. 71, 223–248 (1998).

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics methods,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

Q. Cai, K. N. Liou, “Theory of polarized light scattering by hexagonal ice crystals,” Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

K. N. Liou, Introduction to Atmospheric Radiation (Academic, New York, 1980).

K. N. Liou, Y. Takano, P. Yang, “2000: light scattering and radiative transfer by ice crystal clouds: applications to climate research,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 15, pp. 417–449.
[CrossRef]

Liu, L.

L. Liu, M. I. Mishchenko, “Constraints on PSC particle microphysics derived from lidar observations,” J. Quant. Spectrosc. Radiat. Transfer 70, 817–831 (2001).
[CrossRef]

Macke, A.

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[CrossRef]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance, and optical thickness retrieval to ice-particle shape,” J. Geophys. Res. 101, 16973–16985 (1996).
[CrossRef]

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
[CrossRef] [PubMed]

McFarquhar, G.

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

McFarquhar, G. M.

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

McGee, T. J.

J. Reichardt, S. Reichardt, P. Yang, T. J. McGee, “Retrieval of polar stratospheric cloud microphysical properties from lidar measurements: dependence on particle shape assumptions,” J. Geophys. Res. 107D, 10.1029/2001JD001021 (2002).

Miller, K. M.

A. J. Heymsfield, K. M. Miller, J. D. Spinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

Miloshevich, L.

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

Miloshevich, L. M.

A. J. Heymsfield, L. M. Miloshevich, “Homogeneous ice nucleation and supercooled liquid water in orographic wave clouds,” J. Atmos. Sci. 50, 2335–2353 (1993).
[CrossRef]

Mishchenko, M. I.

Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[CrossRef] [PubMed]

L. Liu, M. I. Mishchenko, “Constraints on PSC particle microphysics derived from lidar observations,” J. Quant. Spectrosc. Radiat. Transfer 70, 817–831 (2001).
[CrossRef]

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532–1547 (2001).
[CrossRef]

M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[CrossRef]

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance, and optical thickness retrieval to ice-particle shape,” J. Geophys. Res. 101, 16973–16985 (1996).
[CrossRef]

M. I. Mishchenko, “Light scattering by size-shape distributions of randomly oriented axially symmetric particles of a size comparable to wavelength,” Appl. Opt. 32, 623–625 (1993).
[CrossRef]

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Mueller, J.

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Muinonen, K.

Nasiri, S. L.

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532–1547 (2001).
[CrossRef]

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

Ohtake, T.

T. Ohtake, “Unusual crystal in ice fog,” J. Atmos. Sci. 27, 509–511 (1970).
[CrossRef]

Personne, P.

J. Iaquinta, H. Isaka, P. Personne, “Scattering phase function of bullet rosette ice crystals,” J. Atmos. Sci. 52, 1401–1413 (1995).
[CrossRef]

Raschke, E.

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Reichardt, J.

S. Reichardt, J. Reichardt, “Effect of multiple scattering on depolarization measurements with spaceborne lidars,” Appl. Opt. 42, 3620–3633 (2003).
[CrossRef] [PubMed]

J. Reichardt, S. Reichardt, P. Yang, T. J. McGee, “Retrieval of polar stratospheric cloud microphysical properties from lidar measurements: dependence on particle shape assumptions,” J. Geophys. Res. 107D, 10.1029/2001JD001021 (2002).

Reichardt, S.

S. Reichardt, J. Reichardt, “Effect of multiple scattering on depolarization measurements with spaceborne lidars,” Appl. Opt. 42, 3620–3633 (2003).
[CrossRef] [PubMed]

J. Reichardt, S. Reichardt, P. Yang, T. J. McGee, “Retrieval of polar stratospheric cloud microphysical properties from lidar measurements: dependence on particle shape assumptions,” J. Geophys. Res. 107D, 10.1029/2001JD001021 (2002).

Robinson, E.

W. C. Thuman, E. Robinson, “Studies of Alaskan ice-fog particles,” J. Meteorol. 11, 151–156 (1954).
[CrossRef]

Rosenfeld, D.

D. Rosenfeld, W. L. Woodley, “Deep convective clouds with sustained supercooled liquid water down to -37.5 degrees,” Nature (London) 405, 440–442 (2000).
[CrossRef]

Rossow, W. B.

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance, and optical thickness retrieval to ice-particle shape,” J. Geophys. Res. 101, 16973–16985 (1996).
[CrossRef]

Schmitt, C. G.

P. R. Lawson, B. A. Baker, C. G. Schmitt, T. J. Jensen, “An overview of microphysical properties of Arctic clouds observed in May and July 1998 during FIRE ACE,” J. Geophys. Res. 106, 14989–15014 (2001).
[CrossRef]

Schoeberl, M. R.

E. J. Jensen, O. B. Toon, H. B. Selkirk, J. D. Spinhirne, M. R. Schoeberl, “On the formation and persistence of subvisible cirrus clouds near the tropical tropopause,” J. Geophys. Res. 101, 21361–21375 (1996).
[CrossRef]

Selkirk, H. B.

E. J. Jensen, O. B. Toon, H. B. Selkirk, J. D. Spinhirne, M. R. Schoeberl, “On the formation and persistence of subvisible cirrus clouds near the tropical tropopause,” J. Geophys. Res. 101, 21361–21375 (1996).
[CrossRef]

Spinhirne, J. D.

E. J. Jensen, O. B. Toon, H. B. Selkirk, J. D. Spinhirne, M. R. Schoeberl, “On the formation and persistence of subvisible cirrus clouds near the tropical tropopause,” J. Geophys. Res. 101, 21361–21375 (1996).
[CrossRef]

A. J. Heymsfield, K. M. Miller, J. D. Spinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

Stammes, P.

M. Hess, R. B. A. Koelemeijer, P. Stammes, “Scattering matrices of imperfect hexagonal crystals,” J. Quant. Spectrosc. Radiat. Transfer 60, 301–308 (1998).
[CrossRef]

Sun, W.

Takano, Y.

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Y. Takano, K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed from ray tracing,” Appl. Opt. 24, 3254–3263 (1985).
[CrossRef] [PubMed]

K. N. Liou, Y. Takano, P. Yang, “2000: light scattering and radiative transfer by ice crystal clouds: applications to climate research,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 15, pp. 417–449.
[CrossRef]

Thuman, W. C.

W. C. Thuman, E. Robinson, “Studies of Alaskan ice-fog particles,” J. Meteorol. 11, 151–156 (1954).
[CrossRef]

Toon, O. B.

E. J. Jensen, O. B. Toon, H. B. Selkirk, J. D. Spinhirne, M. R. Schoeberl, “On the formation and persistence of subvisible cirrus clouds near the tropical tropopause,” J. Geophys. Res. 101, 21361–21375 (1996).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Tsay, S.-C.

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

Warren, S. G.

Weikman, H. K.

Wendling, P.

Wendling, R.

Winker, D. M.

Wiscombe, W.

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

Wiscombe, W. J.

Woodley, W. L.

D. Rosenfeld, W. L. Woodley, “Deep convective clouds with sustained supercooled liquid water down to -37.5 degrees,” Nature (London) 405, 440–442 (2000).
[CrossRef]

Yang, P.

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[CrossRef] [PubMed]

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

J. Reichardt, S. Reichardt, P. Yang, T. J. McGee, “Retrieval of polar stratospheric cloud microphysical properties from lidar measurements: dependence on particle shape assumptions,” J. Geophys. Res. 107D, 10.1029/2001JD001021 (2002).

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532–1547 (2001).
[CrossRef]

P. Yang, K. N. Liou, “Single-scattering properties of complex ice crystals in terrestrial atmosphere,” Contrib. Atmos. Phys. 71, 223–248 (1998).

P. Yang, K. N. Liou, “Geometric-optics integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics methods,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

K. N. Liou, Y. Takano, P. Yang, “2000: light scattering and radiative transfer by ice crystal clouds: applications to climate research,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 15, pp. 417–449.
[CrossRef]

Appl. Opt.

M. I. Mishchenko, “Light scattering by size-shape distributions of randomly oriented axially symmetric particles of a size comparable to wavelength,” Appl. Opt. 32, 623–625 (1993).
[CrossRef]

P. Wendling, R. Wendling, H. K. Weikman, “Scattering of solar radiation by hexagonal ice crystals,” Appl. Opt. 18, 2663–2671 (1979).
[CrossRef] [PubMed]

Q. Cai, K. N. Liou, “Theory of polarized light scattering by hexagonal ice crystals,” Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

S. G. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
[CrossRef] [PubMed]

Y. Takano, K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed from ray tracing,” Appl. Opt. 24, 3254–3263 (1985).
[CrossRef] [PubMed]

K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044–3050 (1989).
[CrossRef] [PubMed]

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
[CrossRef] [PubMed]

M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

W. Sun, Q. Fu, “Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices,” Appl. Opt. 39, 5569–5578 (2000).
[CrossRef]

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532–1547 (2001).
[CrossRef]

Y. K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. Hu, H.-L. Huang, W. J. Wiscombe, A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[CrossRef] [PubMed]

S. Reichardt, J. Reichardt, “Effect of multiple scattering on depolarization measurements with spaceborne lidars,” Appl. Opt. 42, 3620–3633 (2003).
[CrossRef] [PubMed]

Contrib. Atmos. Phys.

P. Yang, K. N. Liou, “Single-scattering properties of complex ice crystals in terrestrial atmosphere,” Contrib. Atmos. Phys. 71, 223–248 (1998).

J. Atmos. Sci.

A. J. Heymsfield, J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 57, 916–938 (2000).
[CrossRef]

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirriform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

T. Ohtake, “Unusual crystal in ice fog,” J. Atmos. Sci. 27, 509–511 (1970).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

A. J. Heymsfield, L. M. Miloshevich, “Homogeneous ice nucleation and supercooled liquid water in orographic wave clouds,” J. Atmos. Sci. 50, 2335–2353 (1993).
[CrossRef]

J. Iaquinta, H. Isaka, P. Personne, “Scattering phase function of bullet rosette ice crystals,” J. Atmos. Sci. 52, 1401–1413 (1995).
[CrossRef]

J. Geophys. Res.

J. Reichardt, S. Reichardt, P. Yang, T. J. McGee, “Retrieval of polar stratospheric cloud microphysical properties from lidar measurements: dependence on particle shape assumptions,” J. Geophys. Res. 107D, 10.1029/2001JD001021 (2002).

P. Yang, B.-C. Gao, B. A. Baum, W. Wiscombe, Y. Hu, S. L. Nasiri, A. Heymsfield, G. McFarquhar, L. Miloshevich, “Sensitivity of cirrus bidirectional reflectance in MODIS bands to vertical inhomogeneity of ice crystal habits and size distributions,” J. Geophys. Res. 106, 17267–17291 (2001).
[CrossRef]

P. R. Lawson, B. A. Baker, C. G. Schmitt, T. J. Jensen, “An overview of microphysical properties of Arctic clouds observed in May and July 1998 during FIRE ACE,” J. Geophys. Res. 106, 14989–15014 (2001).
[CrossRef]

E. J. Jensen, O. B. Toon, H. B. Selkirk, J. D. Spinhirne, M. R. Schoeberl, “On the formation and persistence of subvisible cirrus clouds near the tropical tropopause,” J. Geophys. Res. 101, 21361–21375 (1996).
[CrossRef]

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance, and optical thickness retrieval to ice-particle shape,” J. Geophys. Res. 101, 16973–16985 (1996).
[CrossRef]

J. Meteorol.

W. C. Thuman, E. Robinson, “Studies of Alaskan ice-fog particles,” J. Meteorol. 11, 151–156 (1954).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

S. Havemann, A. J. Baran, J. M. Edwards, “Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 707–720 (2003).
[CrossRef]

L. Liu, M. I. Mishchenko, “Constraints on PSC particle microphysics derived from lidar observations,” J. Quant. Spectrosc. Radiat. Transfer 70, 817–831 (2001).
[CrossRef]

P. Yang, B. A. Baum, A. J. Heymsfield, Y.-X. Hu, H.-L. Huang, S.-C. Tsay, S. A. Ackerman, “Single scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1169 (2003).
[CrossRef]

M. Hess, R. B. A. Koelemeijer, P. Stammes, “Scattering matrices of imperfect hexagonal crystals,” J. Quant. Spectrosc. Radiat. Transfer 60, 301–308 (1998).
[CrossRef]

Mon. Weather Rev.

A. J. Heymsfield, K. M. Miller, J. D. Spinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

Nature (London)

D. Rosenfeld, W. L. Woodley, “Deep convective clouds with sustained supercooled liquid water down to -37.5 degrees,” Nature (London) 405, 440–442 (2000).
[CrossRef]

Other

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, UK, 1980).

K. N. Liou, Introduction to Atmospheric Radiation (Academic, New York, 1980).

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

D. K. Lynch, K. Sassen, D. O. Starr, G. Stephens, eds., Cirrus (Oxford U. Press, New York, 2002).

K. N. Liou, Y. Takano, P. Yang, “2000: light scattering and radiative transfer by ice crystal clouds: applications to climate research,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 15, pp. 417–449.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Replicator images of small ice particles (dark objects in left image) observed for a wave cloud. The distance between the two adjacent lines shown in the right panel is 10 μm. The particle sizes are approximately 30 μm.

Fig. 2
Fig. 2

(a) Geometric representation of a droxtal, (b) cross section of a droxtal (side view).

Fig. 3
Fig. 3

(a) Conceptual diagram to illustrate the method used to determine whether an incident ray can intersect with a convex scattering particle in the ray-tracing calculation (left panel: the ray intersects with the particle; right panel: the ray passes the particle). (b) The results in which the sampled incident rays impinge on the surface of a droxtal and those that pass the particle without intersecting it.

Fig. 4
Fig. 4

All possible projections of a droxtal with various orientations relative to the incident direction.

Fig. 5
Fig. 5

Phase matrix computed by two models for randomly orientated hexagonal ice columns and plates.

Fig. 6
Fig. 6

Nonzero elements of the phase matrix for a droxtal at a wavelength of 0.66 μm with the geometric configuration of R = 50 μm, θ1 = 32.35°, and θ2 = 71.81°.

Fig. 7
Fig. 7

Nonzero elements of the phase matrix for a droxtal at a wavelength of 0.66 μm with the geometric configuration of R = 50 μm, θ1 = 1.5°, and θ2 = 85°.

Fig. 8
Fig. 8

Nonzero elements of the phase matrix for a droxtal at a wavelength 11 μm with the geometric configuration of R = 50 μm, θ1 = 32.35°, and θ2 = 71.81°.

Fig. 9
Fig. 9

Nonzero elements of the phase matrix for droxtals at 11 μm with the geometric configuration of R = 50 μm, θ1 = 1.5°, and θ2 = 85°.

Fig. 10
Fig. 10

Linear depolarization ratio for droxtal and hexagonal ice crystals.

Fig. 11
Fig. 11

Phase function of droxtals derived from the averaging of various combinations of θ1 and θ2.

Tables (1)

Tables Icon

Table 1 Minimum-Deviation Angles Associated with Various Combinations of the Particle Faces

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

a1=R sin θ1, a2=R sin θ2,
L1=R cos θ1, L2=R cos θ2,
h=a1L1-L2/a2-a1.
ni,x=sin α cosπ/2+i-1π/3,
ni,y=sin α sinπ/2+i-1π/3,
ni,z=cos α,
ϕi,j=π-cos-1nˆi · nˆj,
x=Rζ3 cos2πζ4,
y=Rζ3 sin2πζ4,
z=-L,
x0y0z0=Txyz,
T=cos θp cos β-sin βsin θp cos βcos θp sin βcos βsin θp sin β-sin θp0cos θp.
Sd=k2D4πcos θ+cos2 θ001+cos θ,
D=P expikrˆs · rd2r,
r=ε1-ηri+ηεri+1, i=1-12,
Di=trianglei expikrˆs·rd2r=|ri×ri+1|×0101expikrˆs·ε1-ηri+ηεri+1εdηdε =|ri×ri+1|ikrˆs·ri+1-riexpikrˆs·ri+1/2×sinkrˆs·ri+1/2krˆs·ri+1/2-expikrˆs·ri/2×sinkrˆs·ri/2krˆs·ri/2.
ri=-ri+6, i=1-6.
D=i=16Di+Di+6=i=15 2|ri×ri+1|×cos qi-1/qi-cos qi+1-1/qi+1qi+1-qi -2|r6×r1| cos q6-1/q6+cos q1-1/q1q6+q1,
θm=2 sin-1m sinΔ2-Δ,
δH=P11-P22P11+2P12+P22,
δV=P11-P22P11-2P12+P22.
P¯11θs=θ1θ2 P11θs; θ1, θ2W1θ1W2θ2σsθ1, θ2dθ1dθ2θ1θ2 W1θ1W2θ2σsθ1, θ2dθ1dθ2,

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