Abstract

In regions of deep tropical convection, ice particles often undergo aggregation and form complex chains. To investigate the effect of the representation of aggregates on electromagnetic scattering calculations, we developed an algorithm to efficiently specify the geometries of aggregates and to compute some of their geometric parameters, such as the projected area. Based on in situ observations, ice aggregates are defined as clusters of hexagonal plates with a chainlike overall shape, which may have smooth or roughened surfaces. An aggregate representation is developed with 10 ensemble members, each consisting of between 4–12 hexagonal plates. The scattering properties of an individual aggregate ice particle are computed using either the discrete dipole approximation or an improved geometric optics method, depending upon the size parameters. Subsequently, the aggregate properties are averaged over all geometries. The scattering properties of the aggregate representation closely agree with those computed from 1000 different aggregate geometries. As a result, the aggregate representation provides an accurate and computationally efficient way to represent all aggregates occurring within ice clouds. Furthermore, the aggregate representation can be used to study the influence of these complex ice particles on the satellite-based remote sensing of ice clouds. The computed cloud reflectances for aggregates are different from those associated with randomly oriented individual hexagonal plates. When aggregates are neglected, simulated cloud reflectances are generally lower at visible and shortwave-infrared wavelengths, resulting in smaller effective particle sizes but larger optical thicknesses.

© 2011 Optical Society of America

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2010

C. G. Schmitt and A. J. Heymsfield, “The dimensional characteristics of ice crystal aggregates from fractal geometry,” J. Atmos. Sci. 67, 1605–1616 (2010).
[CrossRef]

2009

A. J. Baran, “A review of the light scattering properties of cirrus,” J. Quant. Spectrosc. Radiat. Transfer 110, 1239–1260 (2009).
[CrossRef]

J. Um and G. M. McFarquhar, “Single-scattering properties of aggregates of plates,” Q. J. R. Meteorol. Soc. 135, 291–304 (2009).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, and R. Kahn, “Single-scattering properties of triaxial ellipsoidal particles for a size parameter range from the Rayleigh to geometric-optics regimes,” Appl. Opt. 48, 114–126 (2009).
[CrossRef]

G. Hong, P. Yang, B. A. Baum, A. J. Heymsfield, F. Z. Weng, Q. H. Liu, G. Heygster, and S. A. Buehler, “Scattering database in the millimeter and submillimeter wave range of 100–1000 GHz for nonspherical ice particles,” J. Geophys. Res. 114, D06201, doi:06210.01029/02008JD010451 (2009).
[CrossRef]

T. Nousiainen, E. Zubko, J. V. Niemi, K. Kupiainen, M. Lehtinen, K. Muinonen, and G. Videen, “Single-scattering modeling of thin, birefringent mineral-dust flakes using the discrete-dipole approximation,” J. Geophys. Res. 114, D07207, doi:07210.01029/02008JD011564 (2009).
[CrossRef]

2008

P. Yang, G. Hong, G. W. Kattawar, P. Minnis, and Y. X. Hu, “Uncertainties associated with the surface texture of ice particles in satellite-based retrieval of cirrus clouds. part II: effect of particle surface roughness on retrieved cloud optical thickness and effective particle size,” IEEE Trans. Geosci. Remote Sensing 46, 1948–1957 (2008).
[CrossRef]

P. Yang, G. W. Kattawar, G. Hong, P. Minnis, and Y. X. Hu, “Uncertainties associated with the surface texture of ice particles in satellite-based retrieval of cirrus clouds. part I: single-scattering properties of ice crystals with surface roughness,” IEEE Trans. Geosci. Remote Sensing 46, 1940–1947(2008).
[CrossRef]

P. Yang, Z. B. Zhang, G. W. Kattawar, S. G. Warren, B. A. Baum, H. L. Huang, Y. X. Hu, D. Winker, and J. Iaquinta, “Effect of cavities on the optical properties of bullet rosettes: implications for active and passive remote sensing of ice cloud properties,” J. Appl. Meteorol. Clim. 47, 2311–2330 (2008).
[CrossRef]

2007

J. Um and G. M. McFarquhar, “Single-scattering properties of aggregates of bullet rosettes in cirrus,” J. Appl. Meteorol. Clim. 46, 757–775 (2007).
[CrossRef]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transfer 106, 546–557 (2007).
[CrossRef]

T. Nousiainen and K. Muinonen, “Surface-roughness effect on single-scattering properties of wavelength-scale particles,” J. Quant. Spectrosc. Radiat. Transfer 106, 389–397 (2007).
[CrossRef]

2006

C. G. Schmitt, J. Iaquinta, and A. J. Heymsfield, “The asymmetry parameter of cirrus clouds composed of hollow bullet rosette-shaped ice crystals from ray-tracing calculations,” J. Appl. Meteorol. Clim. 45, 973–981 (2006).
[CrossRef]

A. J. Baran and L. C. Labonnote, “On the reflection and polarisation properties of ice cloud,” J. Quant. Spectrosc. Radiat. Transfer 100, 41–54 (2006).
[CrossRef]

V. Shcherbakov, J. F. Gayet, B. Baker, and P. Lawson, “Light scattering by single natural ice crystals,” J. Atmos. Sci. 63, 1513–1525 (2006).
[CrossRef]

2005

M. W. Gallagher, P. J. Connolly, J. Whiteway, D. Figueras-Nieto, M. Flynn, T. W. Choularton, K. N. Bower, C. Cook, R. Busen, and J. Hacker, “An overview of the microphysical structure of cirrus clouds observed during EMERALD-1,” Q. J. R. Meteorol. Soc. 131, 1143–1169 (2005).
[CrossRef]

P. J. Connolly, C. P. R. Saunders, M. W. Gallagher, K. N. Bower, M. J. Flynn, T. W. Choularton, J. Whiteway, and R. P. Lawson, “Aircraft observations of the influence of electric fields on the aggregation of ice crystals,” Q. J. R. Meteorol. Soc. 131, 1695–1712 (2005).
[CrossRef]

B. A. Baum, A. J. Heymsfield, P. Yang, and S. T. Bedka, “Bulk scattering properties for the remote sensing of ice clouds. part I: microphysical data and models,” J. Appl. Meteorol. 44, 1885–1895 (2005).
[CrossRef]

B. A. Baum, P. Yang, A. J. Heymsfield, S. Platnick, M. D. King, Y. X. Hu, and S. T. Bedka, “Bulk scattering properties for the remote sensing of ice clouds. part II: narrowband models,” J. Appl. Meteorol. 44, 1896–1911 (2005).
[CrossRef]

K. F. Evans, J. R. Wang, P. E. Racette, G. Heymsfield, and L. H. Li, “Ice cloud retrievals and analysis with the compact scanning submillimeter imaging radiometer and the cloud radar system during CRYSTAL FACE,” J. Appl. Meteorol. 44, 839–859 (2005).
[CrossRef]

A. J. Baran, V. N. Shcherbakov, B. A. Baker, J. F. Gayet, and R. P. Lawson, “On the scattering phase-function of non-symmetric ice-crystals,” Q. J. R. Meteorol. Soc. 131, 2609–2616 (2005).
[CrossRef]

M. Wendisch, P. Pilewskie, J. Pommier, S. Howard, P. Yang, A. J. Heymsfield, C. G. Schmitt, D. Baumgardner, and B. Mayer, “Impact of cirrus crystal shape on solar spectral irradiance: a case study for subtropical cirrus,” J. Geophys. Res. 110, D03202, doi:03210.01029/02004JD005294 (2005).
[CrossRef]

P. Yang, H. L. Wei, H. L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44, 5512–5523 (2005).
[CrossRef] [PubMed]

2004

Z. B. Zhang, P. Yang, G. W. Kattawar, S. C. Tsay, B. A. Baum, Y. X. Hu, A. J. Heymsfield, and J. Reichardt, “Geometrical-optics solution to light scattering by droxtal ice crystals,” Appl. Opt. 43, 2490–2499 (2004).
[CrossRef] [PubMed]

O. V. Kalashnikova and I. N. Sokolik, “Modeling the radiative properties of nonspherical soli-derived mineral aerosols,” J. Quant. Spectrosc. Radiat. Transfer 87, 137–166 (2004).
[CrossRef]

J. L. Stith, J. A. Haggerty, A. Heymsfield, and C. A. Grainger, “Microphysical characteristics of tropical updrafts in clean conditions,” J. Appl. Meteorol. 43, 779–794 (2004).
[CrossRef]

2003

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

2002

J. L. Stith, J. E. Dye, A. Bansemer, A. J. Heymsfield, C. A. Grainger, W. A. Petersen, and R. Cifelli, “Microphysical observations of tropical clouds,” J. Appl. Meteorol. 41, 97–117 (2002).
[CrossRef]

A. J. Heymsfield, A. Bansemer, P. R. Field, S. L. Durden, J. L. Stith, J. E. Dye, W. Hall, and C. A. Grainger, “Observations and parameterizations of particle size distributions in deep tropical cirrus and stratiform precipitating clouds: results from in situ observations in TRMM field campaigns,” J. Atmos. Sci. 59, 3457–3491 (2002).
[CrossRef]

2001

D. L. Mitchell, W. P. Arnott, C. Schmitt, A. J. Baran, S. Havemann, and Q. Fu, “Photon tunneling contributions to extinction for laboratory grown hexagonal columns,” J. Quant. Spectrosc. Radiat. Transfer 70, 761–776(2001).
[CrossRef]

R. P. Lawson, B. A. Baker, C. G. Schmitt, and T. L. 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]

1998

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

1996

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

G. M. McFarquhar and A. J. Heymsfield, “Microphysical characteristics of three anvils sampled during the Central Equatorial Experiment,” J. Atmos. Sci. 53, 2401–2423 (1996).
[CrossRef]

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

1995

K. F. Evans and G. L. Stephens, “Microwave radiative transfer through clouds composed of realistically shaped ice crystals. part I: single scattering properties,” J. Atmos. Sci. 52, 2041–2057 (1995).
[CrossRef]

1994

1993

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697(1993).
[CrossRef]

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

1991

A. L. Kosarev and I. P. Mazin, “An empirical model of the physical structure of upper-layer clouds,” Atmos. Res. 26, 213–228 (1991).
[CrossRef]

D. L. Mitchell, “Evolution of snow-size spectra in cyclonic storms. part II: deviations from the exponential form,” J. Atmos. Sci. 48, 1885–1899 (1991).
[CrossRef]

H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A. 43, 2093–2112 (1991).
[CrossRef] [PubMed]

1989

M. Kajikawa and A. J. Heymsfield, “Aggregation of ice crystals in cirrus,” J. Atmos. Sci. 46, 3108–3121 (1989).
[CrossRef]

1988

1987

R. A. Houze and D. D. Churchill, “Mesoscale organization and cloud microphysics in a Bay of Bengal depression,” J. Atmos. Sci. 44, 1845–1867 (1987).
[CrossRef]

1986

A. J. Heymsfield, “Ice particle evolution in the anvil of a severe thunderstorm during CCOPE,” J. Atmos. Sci. 43, 2463–2478(1986).
[CrossRef]

1982

1980

H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

1978

P. Dinh-Van and L. Phan-Cong, “Aggregation of small ice crystals in an electric field,” Atmos.-Ocean 16, 248–259 (1978).
[CrossRef]

1974

P. Hobbs, S. Chang, and J. Locatelli, “The dimension and aggregation of ice crystals in natural clouds,” J. Geophys. Res. 79, 2199–2206 (1974).
[CrossRef]

1973

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1963

H. R. Pruppacher, “The effects of electric fields on cloud physical processes,” J. Appl. Math. Phys. 14, 590–599(1963).
[CrossRef]

1954

Arnott, W. P.

D. L. Mitchell, W. P. Arnott, C. Schmitt, A. J. Baran, S. Havemann, and Q. Fu, “Photon tunneling contributions to extinction for laboratory grown hexagonal columns,” J. Quant. Spectrosc. Radiat. Transfer 70, 761–776(2001).
[CrossRef]

Baker, B.

V. Shcherbakov, J. F. Gayet, B. Baker, and P. Lawson, “Light scattering by single natural ice crystals,” J. Atmos. Sci. 63, 1513–1525 (2006).
[CrossRef]

Baker, B. A.

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P. J. Connolly, C. P. R. Saunders, M. W. Gallagher, K. N. Bower, M. J. Flynn, T. W. Choularton, J. Whiteway, and R. P. Lawson, “Aircraft observations of the influence of electric fields on the aggregation of ice crystals,” Q. J. R. Meteorol. Soc. 131, 1695–1712 (2005).
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J. L. Stith, J. E. Dye, A. Bansemer, A. J. Heymsfield, C. A. Grainger, W. A. Petersen, and R. Cifelli, “Microphysical observations of tropical clouds,” J. Appl. Meteorol. 41, 97–117 (2002).
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M. W. Gallagher, P. J. Connolly, J. Whiteway, D. Figueras-Nieto, M. Flynn, T. W. Choularton, K. N. Bower, C. Cook, R. Busen, and J. Hacker, “An overview of the microphysical structure of cirrus clouds observed during EMERALD-1,” Q. J. R. Meteorol. Soc. 131, 1143–1169 (2005).
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J. L. Stith, J. A. Haggerty, A. Heymsfield, and C. A. Grainger, “Microphysical characteristics of tropical updrafts in clean conditions,” J. Appl. Meteorol. 43, 779–794 (2004).
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A. J. Heymsfield, A. Bansemer, P. R. Field, S. L. Durden, J. L. Stith, J. E. Dye, W. Hall, and C. A. Grainger, “Observations and parameterizations of particle size distributions in deep tropical cirrus and stratiform precipitating clouds: results from in situ observations in TRMM field campaigns,” J. Atmos. Sci. 59, 3457–3491 (2002).
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J. L. Stith, J. E. Dye, A. Bansemer, A. J. Heymsfield, C. A. Grainger, W. A. Petersen, and R. Cifelli, “Microphysical observations of tropical clouds,” J. Appl. Meteorol. 41, 97–117 (2002).
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G. Hong, P. Yang, B. A. Baum, A. J. Heymsfield, F. Z. Weng, Q. H. Liu, G. Heygster, and S. A. Buehler, “Scattering database in the millimeter and submillimeter wave range of 100–1000 GHz for nonspherical ice particles,” J. Geophys. Res. 114, D06201, doi:06210.01029/02008JD010451 (2009).
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P. Yang, Z. B. Zhang, G. W. Kattawar, S. G. Warren, B. A. Baum, H. L. Huang, Y. X. Hu, D. Winker, and J. Iaquinta, “Effect of cavities on the optical properties of bullet rosettes: implications for active and passive remote sensing of ice cloud properties,” J. Appl. Meteorol. Clim. 47, 2311–2330 (2008).
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P. Yang, G. Hong, G. W. Kattawar, P. Minnis, and Y. X. Hu, “Uncertainties associated with the surface texture of ice particles in satellite-based retrieval of cirrus clouds. part II: effect of particle surface roughness on retrieved cloud optical thickness and effective particle size,” IEEE Trans. Geosci. Remote Sensing 46, 1948–1957 (2008).
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P. Yang, H. L. Wei, H. L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44, 5512–5523 (2005).
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Figures (14)

Fig. 1
Fig. 1

(a) Transformation from the o P x P y P z P to o x y z coordinate system. (b) Polar and azimuthal angles in the o x y z coordinate system.

Fig. 2
Fig. 2

Geometries of aggregates: (a) 1, (b) 2, (c) 3, (d) 4, and (e) 5.

Fig. 3
Fig. 3

Geometries of aggregates: (a) 6, (b) 7, (c) 8, (d) 9, and (e) 10.

Fig. 4
Fig. 4

(a), (b) Variation of ice crystal projected area and volume versus maximum dimension for aggregates 1–5. (c), (d) Variation of ice crystal projected area and volume versus maximum dimension for aggregates 6–10.

Fig. 5
Fig. 5

Extinction efficiency, absorption efficiency, single-scattering albedo, and asymmetry factor as functions of the size param eter for aggregate 1 at λ = 2.13 μm . The refractive index of ice at λ = 2.13 μm is 1.2673 + i 5.57 × 10 4 .

Fig. 6
Fig. 6

Same as Fig. 5, except that λ = 12.0 μm . The refractive index of ice at λ = 12.0 μm is 1.2799 + i 4.13 × 10 1 .

Fig. 7
Fig. 7

Scattering phase matrices for aggregate 1 at λ = 0.65 μm . The refractive index of ice at λ = 0.65 μm is 1.3080 + i 1.43 × 10 8 .

Fig. 8
Fig. 8

Scattering phase matrices for aggregate 10 at λ = 0.65 μm . The refractive index of ice at λ = 0.65 μm is 1.3080 + i 1.43 × 10 8 .

Fig. 9
Fig. 9

Scattering phase matrices for aggregates 1 and 10 at λ = 12.0 μm . The refractive index of ice at λ = 12.0 μm is 1.2799 + i 4.13 × 10 1 .

Fig. 10
Fig. 10

(a) Comparison of the scattering phase functions for the averaged values over 1000 aggregates (solid curve), the approximation using aggregates 6–10 (dashed curve), and aggregate 9 (dotted curve). (b) Comparison of the scattering phase functions for ice crystal surface under smooth, moderately rough, and very rough conditions. (c) Comparison of the scattering phase functions for the averaged values over 1000 aggregates (solid curve), the approximation using aggregates 1–5 (dashed curve), and aggregate 5 (dotted curve).

Fig. 11
Fig. 11

Lookup tables using 0.65 (x axis) and 2.13 μm (y axis) reflectances for (a) independent plates and the same ice crystals except that 30% plates form aggregates and (b) independent plates and the same ice crystals except that 90% plates form aggregates. The solar zenith and viewing zenith angles are 30 ° , respectively, and the relative azimuth angle is 90 ° . τ represents the cloud optical thickness.

Fig. 12
Fig. 12

Geometries of hexagonal particles.

Fig. 13
Fig. 13

Two types of faces for a hexagonal ice crystal.

Fig. 14
Fig. 14

Schematic illustrating the computation of the projected area of an aggregate ice crystal.

Tables (2)

Tables Icon

Table 1 Parameters Associated with the Five Aggregates with Small Particle Sizes a

Tables Icon

Table 2 Parameters Associated with the Five Aggregates with Large Particle Sizes a

Equations (48)

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L = 2.4883 a 0.474 ,
a = 20 + 20 ξ 1 ,
a = 40 + 40 ξ 2 ,
[ x P y P z P ] = [ x P y P z P ] + [ x 0 y 0 z 0 ] ,
[ x y z ] = R [ x P y P z P ] ,
R = [ cos γ sin γ 0 sin γ cos γ 0 0 0 1 ] · [ cos β 0 sin β 0 1 0 sin β 0 cos β ] · [ cos α sin α 0 sin α cos α 0 0 0 1 ] = [ cos α cos β cos γ sin α sin γ cos β cos γ sin α cos α sin γ cos γ sin β cos γ sin α + cos α cos β sin γ cos α cos γ cos β sin α sin γ sin β sin γ cos α sin β sin α sin β cos β ] ,
α = π ( 2 ξ 3 1 ) ,
β = cos 1 ( 2 ξ 4 1 ) ,
γ = π ( 2 ξ 5 1 ) ,
x 0 = d ξ 6 sin θ cos φ ,
y 0 = d ξ 6 sin θ sin φ ,
z 0 = d ξ 6 cos θ ,
θ = cos 1 ( 2 ξ 7 1 ) ,
φ = 2 π ξ 8 ,
β N = { cos 1 ( 2 ξ 9 1 ) for     N = 1 β N 1 + cos 1 [ 2.0 × ( 0.9 90 ξ 10 0.5 ) ] for     N > 1 ,
d = { D m 20 , λ 20 | m | for     X 1 λ 20 | m | for     1 < X 5 λ 10 | m | for     5 < X 15 λ 5 | m | for     X > 15 ,
X = π D s λ ,
Q e , edge ( λ ) = 2 c 1 ( λ π D m ) 2 / 3 ,
Q a , edge ( λ ) = 2 c 2 ( λ π D m ) 2 / 3 .
c 1 = 0.5 [ Q e , ADDA ( λ t ) Q e , IGOM ( λ t ) ] ( π D m λ t ) 2 / 3 ,
c 2 = 0.5 [ Q a , ADDA ( λ t ) Q a , IGOM ( λ t ) ] ( π D m λ t ) 2 / 3 ,
P 11 ( Θ , D m , λ ) ¯ = n = 1 M P 11 ( Θ , D m , λ , n ) C s ( D m , λ , n ) n = 1 M C s ( D m , λ , n ) ,
| t | = 0.1405 < t 0.05 = 1.96 ,
| t | = 0.5096 < t 0.05 = 1.96 ,
n ( D m ) = N 0 D m μ exp ( b + μ + 0.67 D m _ median D m ) ,
D e = 3 2 ( 1 f ) [ i = 1 24 D min D 1 V p i n ( D m ) d D m + j = 1 50 D 1 D max V p j n ( D m ) d D m ] + N a f D min D max V a n ( D m ) d D m ( 1 f ) [ i = 1 24 D min D 1 A p i n ( D m ) d D m + j = 1 50 D 1 D max A p j n ( D m ) d D m ] + N a f D min D max A a n ( D m ) d D m ,
P 11 = ( 1 f ) [ i = 1 24 D min D 1 P 11 , p i C s , p i n ( D m ) d D m + i = 25 74 D 1 D max P 11 , p i C s , p i n ( D m ) d D m ] + N a f D min D max P 11 , a C s , a n ( D m ) d D m ( 1 f ) [ i = 1 24 D min D 1 C s , p i n ( D m ) d D m + i = 25 74 D 1 D max C s , p i n ( D m ) d D m ] + N a f D min D max C s , a n ( D m ) d D m ,
τ = ( 1 f ) Δ z [ i = 1 24 D min D 1 C e , p i n ( D m ) d D m + i = 25 74 D 1 D max C e , p i n ( D m ) d D m ] + N a f Δ z D min D max C e , a n ( D m ) d D m ,
D = D ( P k A , F i B , k A = 1 , 2 , , 12 , i B = 1 , 2 , , 8 ) D ( P k B , F i A , k B = 1 , 2 , , 12 , i A = 1 , 2 , , 8 ) D ( L j A , L j B , i A = 1 , 2 , , 18 , j B = 1 , 2 , , 18 ) ,
D ( P k A , F i B , k A = 1 , 2 , , 12 , i B = 1 , 2 , , 8 ) = { | p k A p u | ( k A = 1 , 2 , , 12 , i B = 1 , 2 , , 8 ) for     P u F i B D ( P k A , L i B _ m 1 , m 1 = 1 , 2 , 4 ( or     6 ) ) ( k A = 1 , 2 , , 12 , i B = 1 , 2 , , 8 ) for     P u F i B ,
p u = p k A + f i B f i B · ( c i B p k A ) | f i B | 2 .
D ( P k A , L i B _ m 1 , m 1 = 1 , 2 , 4 ( or     6 ) ) = { | p k A p v | for     P v L i B _ m 1 | p k A p i B _ m 1 _ m 2 | ( m 2 = 1     and     2 ) ( m 1 = 1 , 2 , 4 ( or     6 ) ) for     P v L i B _ m 1 ,
p v = ( p k A p i B _ m 1 _ m 2 ) · ( p i B _ m 1 _ 1 p i B _ m 1 _ 2 ) | p i B _ m 1 _ 1 p i B _ m 1 _ 2 | 2 ( p i B _ m 1 _ 1 p i B _ m 1 _ 2 ) + p i B _ m 1 _ 1 ( m 1 = 1 , 2 , 4 ( or     6 ) ) .
D ( L j A , L j B , j A = 1 , 2 , , 18 , j B = 1 , 2 , , 18 ) = | p j A _ m 3 p j B _ m 4 | ( m 3 , m 4 = 1     and     2 ) D ( P j A _ m 3 , L j B , m 3 = 1     and     2 ) D ( P j B _ m 4 , L j A , m 4 = 1     and     2 ) D ( P w , L j A )     for     P w L j B ,
P w = p j A _ 1 + ( p j A _ 1 p j A _ 2 ) [ ( p j A _ 1 p j B _ 1 ) × ( p j B _ 2 p j B _ 1 ) ( p j A _ 2 p j A _ 1 ) × ( p j B _ 2 p j B _ 1 ) ] z ,
p j A _ 1 = p j A _ 1 + ( l j A × l j B ) ( l j A × l j B ) · ( p j B _ 1 p j A _ 1 ) | l j A × l j B | 2 ,
p j A _ 2 = p j A _ 2 + ( l j A × l j B ) ( l j A × l j B ) · ( p j B _ 1 p j A _ 2 ) | l j A × l j B | 2 .
{ i B = 1 8 D ( P k A , F i B , k A = 1 , 2 , , 12 ) 3 3 a B + L B i A = 1 8 D ( P k B , F i A , k B = 1 , 2 , , 12 ) 3 3 a A + L A j A = 1 18 L j A i B = 1 8 F i B = ,
{ L j A F i B for     m 5 = 1 4 D ( L j A , L i B _ m 5 ) a B + L B L j A F i B = for     m 5 = 1 4 D ( L j A , L i B _ m 5 ) = a B + L B ,
{ L j A F i B for     m 6 = 1 6 D ( L j A , L i B _ m 6 ) 3 3 a B L j A F i B = for     m 6 = 1 6 D ( L j A , L i B _ m 5 ) = 3 3 a B .
| p i p i 0 | = D m ξ A ,
p i · p i 0 = | p i 0 | 2 ,
( p i p i 0 ) · ( p B p i 0 ) = D m ξ A | p B p i 0 | cos ( 2 π ξ B ) ,
p B = ( 0 , 0 , | p i 0 | 2 ( p i 0 ) z ) .
{ P i L i l i = p i 0 ,
M i = { 1 for     L i j = 1 8 N F j 0 for     L i j = 1 8 N F j = ,
S = π D m 2 i = 1 N M i N L ,
S = 3 4 a ( 3 a + 2 L ) ,

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