Abstract

Both passive and active remote sensing of atmospheric precipitation are studied with the vector radiative transfer equations by making use of the Mie scattering phase functions and incorporating the raindrop-size distributions. For passive remote sensing we employ the Gaussian quadrature method to solve for the brightness temperatures. For active remote sensing an iterative approach carrying out to the second order in albedo is used to calculate for the bistatic scattering coefficients, the backscattering cross sections/unit volume, and the interchannel cross talks. The calculated results are plotted as a function of rainfall rates and compared to various available experimental data. The theoretical model is easily applied to the remote sensing of aerosol particles, smoke, fog, and haze at infrared and visible frequencies.

© 1983 Optical Society of America

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References

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  1. L. Tsang, J. A. Kong, J. Appl. Phys. 8, 3593 (1977).
    [CrossRef]
  2. L. Tsang, J. A. Kong, E. Njoku, D. H. Staelin, J. W. Waters, IEEE Trans. Antennas Propag. AP-5, 650 (1977).
    [CrossRef]
  3. T. T. Wilheit, A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, J. S. Theon, Appl. Meteorol. 16, 551 (1977).
    [CrossRef]
  4. H. H. K. Burke, K. R. Hardy, N. K. Tripp, Remote Sensing Environ. 12, 169 (1982).
    [CrossRef]
  5. R. K. Crane, H. H. K. Burke, ERT Document 3606 (Feb.1978).
  6. A. Deepak, U. O. Farrukh, A. Zardecki, Appl. Opt. 21, 439 (1982).
    [CrossRef] [PubMed]
  7. L. J. Battan, Radar Observation of the Atmosphere (U. Chicago Press, Chicago, 1973).
  8. T. Li, W. C. Jakes, J. A. Morrison, IEEE Trans. Antennas Propag. AP-5, 646 (1977).
  9. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).
  10. J. Hasted, “The Dielectric Properties of Water,” in Progress in Dielectrics, Vol. 3 (Wiley, New York, 1969).
  11. N. Abel, in Proceedings, AGARD NATO Conference 107 on Telecommunications Aspects on Frequencies Between 10 and 100 GHz, Gausdal, Norway, 18–21 Sept., 1972, A. W. Biggs, Ed.,

1982 (2)

H. H. K. Burke, K. R. Hardy, N. K. Tripp, Remote Sensing Environ. 12, 169 (1982).
[CrossRef]

A. Deepak, U. O. Farrukh, A. Zardecki, Appl. Opt. 21, 439 (1982).
[CrossRef] [PubMed]

1978 (1)

R. K. Crane, H. H. K. Burke, ERT Document 3606 (Feb.1978).

1977 (4)

L. Tsang, J. A. Kong, J. Appl. Phys. 8, 3593 (1977).
[CrossRef]

L. Tsang, J. A. Kong, E. Njoku, D. H. Staelin, J. W. Waters, IEEE Trans. Antennas Propag. AP-5, 650 (1977).
[CrossRef]

T. T. Wilheit, A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, J. S. Theon, Appl. Meteorol. 16, 551 (1977).
[CrossRef]

T. Li, W. C. Jakes, J. A. Morrison, IEEE Trans. Antennas Propag. AP-5, 646 (1977).

Abel, N.

N. Abel, in Proceedings, AGARD NATO Conference 107 on Telecommunications Aspects on Frequencies Between 10 and 100 GHz, Gausdal, Norway, 18–21 Sept., 1972, A. W. Biggs, Ed.,

Battan, L. J.

L. J. Battan, Radar Observation of the Atmosphere (U. Chicago Press, Chicago, 1973).

Burke, H. H. K.

H. H. K. Burke, K. R. Hardy, N. K. Tripp, Remote Sensing Environ. 12, 169 (1982).
[CrossRef]

R. K. Crane, H. H. K. Burke, ERT Document 3606 (Feb.1978).

Chang, A. T. C.

T. T. Wilheit, A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, J. S. Theon, Appl. Meteorol. 16, 551 (1977).
[CrossRef]

Crane, R. K.

R. K. Crane, H. H. K. Burke, ERT Document 3606 (Feb.1978).

Deepak, A.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

Farrukh, U. O.

Hardy, K. R.

H. H. K. Burke, K. R. Hardy, N. K. Tripp, Remote Sensing Environ. 12, 169 (1982).
[CrossRef]

Hasted, J.

J. Hasted, “The Dielectric Properties of Water,” in Progress in Dielectrics, Vol. 3 (Wiley, New York, 1969).

Jakes, W. C.

T. Li, W. C. Jakes, J. A. Morrison, IEEE Trans. Antennas Propag. AP-5, 646 (1977).

Kong, J. A.

L. Tsang, J. A. Kong, J. Appl. Phys. 8, 3593 (1977).
[CrossRef]

L. Tsang, J. A. Kong, E. Njoku, D. H. Staelin, J. W. Waters, IEEE Trans. Antennas Propag. AP-5, 650 (1977).
[CrossRef]

Li, T.

T. Li, W. C. Jakes, J. A. Morrison, IEEE Trans. Antennas Propag. AP-5, 646 (1977).

Morrison, J. A.

T. Li, W. C. Jakes, J. A. Morrison, IEEE Trans. Antennas Propag. AP-5, 646 (1977).

Njoku, E.

L. Tsang, J. A. Kong, E. Njoku, D. H. Staelin, J. W. Waters, IEEE Trans. Antennas Propag. AP-5, 650 (1977).
[CrossRef]

Rao, M. S. V.

T. T. Wilheit, A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, J. S. Theon, Appl. Meteorol. 16, 551 (1977).
[CrossRef]

Rodgers, E. B.

T. T. Wilheit, A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, J. S. Theon, Appl. Meteorol. 16, 551 (1977).
[CrossRef]

Staelin, D. H.

L. Tsang, J. A. Kong, E. Njoku, D. H. Staelin, J. W. Waters, IEEE Trans. Antennas Propag. AP-5, 650 (1977).
[CrossRef]

Theon, J. S.

T. T. Wilheit, A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, J. S. Theon, Appl. Meteorol. 16, 551 (1977).
[CrossRef]

Tripp, N. K.

H. H. K. Burke, K. R. Hardy, N. K. Tripp, Remote Sensing Environ. 12, 169 (1982).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, E. Njoku, D. H. Staelin, J. W. Waters, IEEE Trans. Antennas Propag. AP-5, 650 (1977).
[CrossRef]

L. Tsang, J. A. Kong, J. Appl. Phys. 8, 3593 (1977).
[CrossRef]

Waters, J. W.

L. Tsang, J. A. Kong, E. Njoku, D. H. Staelin, J. W. Waters, IEEE Trans. Antennas Propag. AP-5, 650 (1977).
[CrossRef]

Wilheit, T. T.

T. T. Wilheit, A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, J. S. Theon, Appl. Meteorol. 16, 551 (1977).
[CrossRef]

Zardecki, A.

Appl. Meteorol. (1)

T. T. Wilheit, A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, J. S. Theon, Appl. Meteorol. 16, 551 (1977).
[CrossRef]

Appl. Opt. (1)

ERT Document 3606 (1)

R. K. Crane, H. H. K. Burke, ERT Document 3606 (Feb.1978).

IEEE Trans. Antennas Propag. (2)

L. Tsang, J. A. Kong, E. Njoku, D. H. Staelin, J. W. Waters, IEEE Trans. Antennas Propag. AP-5, 650 (1977).
[CrossRef]

T. Li, W. C. Jakes, J. A. Morrison, IEEE Trans. Antennas Propag. AP-5, 646 (1977).

J. Appl. Phys. (1)

L. Tsang, J. A. Kong, J. Appl. Phys. 8, 3593 (1977).
[CrossRef]

Remote Sensing Environ. (1)

H. H. K. Burke, K. R. Hardy, N. K. Tripp, Remote Sensing Environ. 12, 169 (1982).
[CrossRef]

Other (4)

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

J. Hasted, “The Dielectric Properties of Water,” in Progress in Dielectrics, Vol. 3 (Wiley, New York, 1969).

N. Abel, in Proceedings, AGARD NATO Conference 107 on Telecommunications Aspects on Frequencies Between 10 and 100 GHz, Gausdal, Norway, 18–21 Sept., 1972, A. W. Biggs, Ed.,

L. J. Battan, Radar Observation of the Atmosphere (U. Chicago Press, Chicago, 1973).

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Figures (5)

Fig. 1
Fig. 1

Geometrical configuration of the problem.

Fig. 2
Fig. 2

Brightness temperature vs rain rate at 19.35 GHz. Data given in Ref. 3.

Fig. 3
Fig. 3

Brightness temperature vs rain rate at 37 GHz.

Fig. 4
Fig. 4

Backscattering cross section/volume vs rain rate at 70 GHz. Data given in Ref. 5.

Fig. 5
Fig. 5

Cross talk vs rain rate at 11 GHz. Data given in Ref. 8.

Equations (111)

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cos θ d dz [ I υ ( θ , z ) I h ( θ , z ) ] = ( κ ad + κ ag ) CT 1 ( κ ed + κ ag ) [ I υ ( θ , z ) I h ( θ , z ) ] + 0 π d θ sin θ [ ( υ , υ ) ( υ , h ) ( h , υ ) ( h , h ) ] [ I υ ( θ , z ) I h ( θ , z ) ] ,
[ I υ ( π θ , 0 ) I h ( π θ , 0 ) ] = 0
[ I υ ( θ , d ) I h ( θ , d ) ] = [ r υ ( θ ) I υ ( π θ , d ) r h ( θ ) I h ( π θ , d ) ] + CT 2 [ 1 r υ ( θ ) 1 r h ( θ ) ] ,
[ T B υ T Bh ] = 1 C [ I υ ( θ , 0 ) I h ( θ , 0 ) ]
[ T B υ T Bh ] = 1 C [ I υ ( π θ , d ) I h ( π θ , d ) ]
μ i d dz [ I υ ( θ i , z ) I h ( θ i , z ) ] = ( κ ad + κ ag ) CT 1 ( κ ed + κ ag ) [ I υ ( θ i , z ) I h ( θ i , z ) ] + j = N N a j [ ( υ i , υ j ) ( υ i , h j ) ( h i , υ j ) ( h i , h j ) ] [ I υ ( θ j , z ) I h ( θ j , z ) ] ,
cos θ d dz ( θ , ϕ , z ) = κ a ( θ , ϕ , z ) κ s ( θ , ϕ , z ) + 0 π d θ sin θ 0 2 π d ϕ P ̿ ( θ , ϕ ; θ , ϕ ) ( θ , ϕ , z ) ,
( θ , ϕ , z ) = [ I υ ( θ , ϕ , z ) I h ( θ , ϕ , z ) U ( θ , ϕ , z ) V ( θ , ϕ , z ) ] ,
( π θ , ϕ , z = 0 ) = oi δ ( cos θ cos θ oi ) δ ( ϕ ϕ oi ) ,
( θ , ϕ , z = d ) = 0 .
γ β α ( θ s , ϕ s ; θ oi , ϕ oi ) = 4 π I s β ( θ s , ϕ s ) cos θ s I oi α cos θ oi ,
σ β α 0 ( θ oi ) = cos θ oi γ β α ( θ oi , π + ϕ oi ; θ oi , ϕ oi ) = 4 π cos θ oi I oi β ( θ oi , π + ϕ oi ) I oi α .
P t = I o exp ( κ e d sec θ i ) + Ω r d Ω I s ( π θ , ϕ ) I o exp ( κ e d sec θ i ) + I s ( π θ i , ϕ i , d ) Ω r .
P s = Ω r d Ω I s ( π θ s , ϕ s ) I s ( π θ s , ϕ s , d ) Ω r ,
CX = P s P t = I s ( π θ s , ϕ s , d ) Ω r I o exp ( κ e d sec θ i ) + I s ( π θ i , ϕ i , d ) Ω r .
( θ , ϕ , z ) = sec θ exp ( κ e z sec θ ) d z dz exp ( κ e z sec θ ) × 0 π / 2 d θ sin θ 0 2 π d ϕ { P ̿ ( θ , ϕ ; θ , ϕ ) ( θ , ϕ , z ) + P ̿ ( θ , ϕ ; π θ , ϕ ) ( π θ , ϕ , z ) } ,
( π θ , ϕ , z ) = oi δ ( cos θ o cos θ oi ) δ ( ϕ o ϕ oi ) exp ( κ e z sec θ ) + sec θ exp ( κ e z sec θ ) z 0 dz exp ( κ e z sec θ ) 0 π / 2 d θ sin θ × 0 2 π d ϕ { P ̿ ( π θ , ϕ ; θ , ϕ ) ( θ , ϕ , z ) + P ̿ ( π θ , ϕ ; π θ , ϕ ) × ( π θ , ϕ , z ) } ,
( 0 ) ( θ , ϕ , z ) = 0 ,
( 0 ) ( π θ , ϕ , z ) = oi δ ( cos θ o cos θ oi ) × δ ( ϕ o ϕ oi ) exp ( κ e z sec θ ) .
( 1 ) ( θ , ϕ , z ) = exp ( κ e z sec θ ) sec θ P ̿ ( θ , ϕ ; π θ i , ϕ i ) oi × exp [ κ e z ( sec θ + sec θ i ) ] exp [ κ e d ( sec θ + sec θ i ) ] κ e ( sec θ + sec θ i ) ,
( 1 ) ( π θ , ϕ , z ) = exp ( κ e z sec θ ) sec θ P ̿ ( π θ , ϕ ; π θ i , ϕ i ) oi × 1 exp [ κ e z ( sec θ sec θ i ) ] κ e ( sec θ sec θ i ) .
o ( 1 ) ( θ os , ϕ os ) = ( 1 ) ( θ s , ϕ s , z = 0 ) = sec θ s P ̿ ( θ s , ϕ s ; π θ i , ϕ i ) × oi 1 exp [ κ e d ( sec θ s + sec θ i ) ] κ e ( sec θ s + sec θ i )
2 ( 1 ) ( π θ 2 s , ϕ 2 s ) = ( 1 ) ( π θ s , ϕ s , z = d ) = exp ( κ e d sec θ s ) sec θ s P ̿ ( π θ s , ϕ s , π θ i , ϕ i ) oi 1 exp [ κ e d ( sec θ s sec θ i ) ] κ e ( sec θ s sec θ i ) .
o ( 2 ) ( θ os , π + ϕ os ) = sec θ s 0 π / 2 d θ sin θ × 0 2 π d ϕ { P ̿ ( θ s , π + ϕ s ; θ , ϕ ) × 1 ( θ , ϕ ) + P ̿ ( θ s , π + ϕ s ; π θ , ϕ ) 2 ( θ , ϕ ) } ,
1 ( θ , ϕ ) = d 0 dz exp ( κ e z sec θ i ) ( 1 ) ( θ , ϕ , z ) = sec θ P ̿ ( θ , ϕ ; π θ i , ϕ i ) oi 1 κ e ( sec θ + sec θ i ) × { D 2 ( θ s , θ i ) D 1 ( θ s , θ ) exp [ κ e d ( sec θ + sec θ i ) ] } ,
2 ( θ , ϕ ) = d 0 dz exp ( κ e z sec θ i ) exp ( κ e z sec θ ) × sec θ P ̿ ( π θ , ϕ ; π θ i , ϕ i ) × oi 1 κ e ( sec θ sec θ i ) { 1 exp [ κ e z ( sec θ sec θ i ) ] } = sec θ P ̿ ( π θ , ϕ ; π θ i , ϕ i ) oi 1 κ e ( sec θ sec θ i ) × [ D 2 ( θ s , θ ) D 2 ( θ s , θ i ) ] ,
D 1 ( θ i , θ s ) = 1 exp [ κ e d ( sec θ i sec θ s ) ] κ e ( sec θ i sec θ s ) ,
D 2 ( θ s , θ i ) = 1 exp [ κ e d ( sec θ s + sec θ i ) ] κ e ( sec θ s + sec θ i ) .
os υ ( 2 ) ( θ os , π + ϕ os ) = I oi υ sec θ s 0 π / 2 d θ sin θ 0 2 π d ϕ sec θ × { [ P 11 ( θ s , π + ϕ s ; θ , ϕ ) K 1 ( θ , ϕ ) + P 11 ( θ s , π + ϕ s ; π θ , ϕ ) × L 1 ( θ , ϕ ) ] + [ P 12 ( θ s , π + ϕ s ; θ , ϕ ) K 2 ( θ , ϕ ) + P 12 ( θ s , π + ϕ s ; π θ , ϕ ) L 2 ( θ , ϕ ) ] + [ P 13 ( θ s , π + ϕ s ; θ , ϕ ) K 3 ( θ , ϕ ) + P 13 ( θ s , π + ϕ s ; π θ , ϕ ) × L 3 ( θ , ϕ ) ] + [ P 14 ( θ s , π + ϕ s ; θ , ϕ ) K 4 ( θ , ϕ ) + P 14 ( θ s , π + ϕ s ; π θ , ϕ ) L 4 ( θ , ϕ ) ] } ,
K 1 ( θ , ϕ ) = P 11 ( θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ + sec θ i ) × { D 2 ( θ s , θ i ) D 1 ( θ s , θ ) exp [ κ e d ( sec θ + sec θ i ) ] } ,
K 2 ( θ , ϕ ) = P 21 ( θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ + sec θ i ) × { D 2 ( θ s , θ i ) D 1 ( θ s , θ ) exp [ κ e d ( sec θ + sec θ i ) ] } ,
K 3 ( θ , ϕ ) = P 31 ( θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ + sec θ i ) × { D 2 ( θ s , θ i ) D 1 ( θ s , θ ) exp [ κ e d ( sec θ + sec θ i ) ] } ,
K 4 ( θ , ϕ ) = P 41 ( θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ + sec θ i ) × { D 2 ( θ s , θ i ) D 1 ( θ s , θ ) exp [ κ e d ( sec θ + sec θ i ) ] } ,
L 1 ( θ , ϕ ) = P 11 ( π θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ sec θ i ) × [ D 2 ( θ s , θ ) D 2 ( θ s , θ i ) ] ,
L 2 ( θ , ϕ ) = P 21 ( π θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ sec θ i ) × [ D 2 ( θ s , θ ) D 2 ( θ s , θ i ) ] ,
L 3 ( θ , ϕ ) = P 31 ( π θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ sec θ i ) × [ D 2 ( θ s , θ ) D 2 ( θ s , θ i ) ] ,
L 4 ( θ , ϕ ) = P 41 ( π θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ sec θ i ) × [ D 2 ( θ s , θ ) D 2 ( θ s , θ i ) ] ,
lim θ θ i 1 κ e ( sec θ sec θ i ) [ D 2 ( θ s , θ ) D 2 ( θ s , θ i ) ] = 1 κ e ( sec θ s + sec θ i ) × { D 2 ( θ s , θ i ) D 1 ( θ s , θ i ) exp [ κ e d ( sec θ s + sec θ i ) ] } .
σ υ υ = 4 π cos θ oi I os υ ( 1 ) ( θ i , π + ϕ i ) + I os υ ( 2 ) ( θ i , π + ϕ i ) I oi υ ,
osh ( 2 ) ( θ os , π + ϕ os ) = I oi υ sec θ s 0 π / 2 d θ sin θ 0 2 π d ϕ sec θ × { [ P 21 ( θ s , π + ϕ s ; θ , ϕ ) K 1 ( θ , ϕ ) + P 21 ( θ s , π + ϕ s ; π θ , ϕ ) × L 1 ( θ , ϕ ) ] + [ P 22 ( θ s , π + ϕ s ; θ , ϕ ) K 2 ( θ , ϕ ) + P 22 ( θ s , π + ϕ s ; π θ , ϕ ) L 2 ( θ , ϕ ) ] + [ P 23 ( θ s , π + ϕ s ; θ , ϕ ) K 3 ( θ , ϕ ) + P 23 ( θ s , π + ϕ s ; π θ , ϕ ) × L 3 ( θ , ϕ ) ] + [ P 24 ( θ s , π + ϕ s ; θ , ϕ ) K 4 ( θ , ϕ ) + P 24 ( θ s , π + ϕ s ; π θ , ϕ ) L 4 ( θ , ϕ ) ] } .
γ oh υ ( θ os ϕ os ; θ oi , ϕ oi ) = 4 π cos θ s I osh ( θ s , ϕ s ) cos θ oi I oi υ = 4 π sec θ s P 21 ( θ s , π + ϕ s ; π θ i , ϕ i ) D 2 ( θ s , θ i ) + 4 π sec θ s 0 π / 2 d θ sin θ sec θ [ F 1 ( θ s , θ ) + F 2 ( θ s , θ ) + F 3 ( θ s , θ ) + F 4 ( θ s , θ ) ] ,
F 1 ( θ s , θ ) = 0 2 π d θ [ P 21 ( θ s , π + ϕ s ; θ , ϕ ) K 1 ( θ , ϕ ) + P 21 ( θ s , π + ϕ s ; π θ , ϕ ) L 1 ( θ , ϕ ) ] ,
F 2 ( θ s , θ ) = 0 2 π d θ [ P 22 ( θ s , π + ϕ s ; θ , ϕ ) K 1 ( θ , ϕ ) + P 22 ( θ s , π + ϕ s ; π θ , ϕ ) L 1 ( θ , ϕ ) ] ,
F 3 ( θ s , θ ) = 0 2 π d θ [ P 23 ( θ s , π + ϕ s ; θ , ϕ ) K 1 ( θ , ϕ ) + P 23 ( θ s , π + ϕ s ; π θ , ϕ ) L 1 ( θ , ϕ ) ] ,
F 4 ( θ s , θ ) = 0 2 π d θ [ P 24 ( θ s , π + ϕ s ; θ , ϕ ) K 1 ( θ , ϕ ) + P 24 ( θ s , π + ϕ s ; π θ , ϕ ) L 1 ( θ , ϕ ) ] .
2 ( 2 ) ( π θ 2 s , ϕ 2 s ) = ( 2 ) ( π θ s , ϕ s , z = d ) = sec θ s exp ( κ e d sec θ s ) × d 0 d z exp ( κ e z sec θ s ) 0 π / 2 d θ sin θ 0 2 π d ϕ × { P ̿ ( π θ s , ϕ s ; θ , ϕ ) ( 1 ) ( θ , ϕ , z ) + P ̿ ( π θ s , ϕ s ; π θ , ϕ ) × ( 1 ) ( π θ , ϕ , z ) } = sec θ s exp ( κ e d sec θ s ) × 0 π / 2 d θ sin θ 0 2 π d ϕ × { P ̿ ( π θ s , ϕ s ; θ , ϕ ) B ̅ 1 ( θ , ϕ ) + P ̿ ( π θ s , ϕ s ; π θ , ϕ ) × B ̅ 2 ( θ , ϕ ) } ,
B ̅ 1 ( θ , ϕ ) = d 0 dz exp ( κ e z sec θ s ) ( 1 ) ( θ , ϕ , z ) = sec θ P ̿ ( θ , ϕ ; π θ i , ϕ i ) oi 1 κ e ( sec θ + sec θ i ) × { D 1 ( θ i , θ s ) exp [ κ e d ( sec θ i sec θ s ) ] exp [ κ e d ( sec θ + sec θ i ) ] κ e ( sec θ s + sec θ ) } ,
B ̅ 2 ( θ , ϕ ) = d 0 dz exp ( κ e d sec θ s ) ( 1 ) ( π θ , ϕ , z ) = sec θ P ̿ ( π θ , ϕ ; π θ i , ϕ i ) oi 1 κ e ( sec θ sec θ i ) × [ D 1 ( θ i , θ s ) D 1 ( θ , θ s ) ] .
I 2 s υ ( π θ 2 s , ϕ 2 s ) = I 2 s υ ( 1 ) ( π θ 2 s , ϕ 2 s ) + I 2 s υ ( 2 ) ( π θ 2 s , ϕ 2 s ) = sec θ s exp ( κ e d sec θ s ) P 11 ( π θ s , ϕ s ; π θ i , ϕ i ) I oi υ × D 1 ( θ i , θ s ) + I oi υ sec θ s exp ( κ e d sec θ s ) × 0 π / 2 d θ sin θ 0 2 π d ϕ = sec θ { [ P 11 ( π θ s , ϕ s ; θ , ϕ ) R 1 ( θ , ϕ ) + P 11 ( π θ s , ϕ s ; π θ , ϕ ) × S 1 ( θ , ϕ ) ] + [ P 12 ( π θ s , ϕ s ; θ , ϕ ) R 2 ( θ , ϕ ) + P 12 ( π θ s , ϕ s ; π θ , ϕ ) S 2 ( θ , ϕ ) ] + [ P 13 ( π θ s , ϕ s ; θ , ϕ ) R 3 ( θ , ϕ ) + P 13 ( π θ s , ϕ s ; π θ , ϕ ) × S 3 ( θ , ϕ ) ] + [ P 14 ( π θ s , ϕ s ; θ , ϕ ) R 4 ( θ , ϕ ) + P 14 ( π θ s , ϕ s ; π θ , ϕ ) S 4 ( θ , ϕ ) ] }
I 2 sh ( π θ 2 s , ϕ 2 s ) = I 2 sh ( 1 ) ( π θ 2 s , ϕ 2 s ) + I 2 sh ( 2 ) ( π θ 2 s , ϕ 2 s ) = sec θ s exp ( κ e d sec θ s ) P 21 ( π θ s , ϕ s ; π θ i , ϕ i ) I oi υ D 1 ( θ i , θ s ) + I oi υ sec θ s exp ( κ e d sec θ s ) 0 π / 2 d θ sin θ 0 2 π d ϕ sec θ × { [ P 21 ( π θ s , ϕ s ; θ , ϕ ) R 1 ( θ , ϕ ) + P 21 ( π θ s , ϕ s ; π θ , ϕ ) × S 1 ( θ , ϕ ) ] + [ P 22 ( π θ s , ϕ s ; θ , ϕ ) R 2 ( θ , ϕ ) + P 22 ( π θ s , ϕ s ; π θ , ϕ ) S 2 ( θ , ϕ ) ] + [ P 23 ( π θ s , ϕ s ; θ , ϕ ) R 3 ( θ , ϕ ) + P 23 ( π θ s , ϕ s ; π θ , ϕ ) S 3 ( θ , ϕ ) ] + [ P 24 ( π θ s , ϕ s ; θ , ϕ ) × R 4 ( θ , ϕ ) + P 24 ( π θ s , ϕ s ; π θ , ϕ ) S 4 ( θ , ϕ ) ] } ,
R 1 ( θ , ϕ ) = P 11 ( θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ + sec θ i ) × { D 1 ( θ i , θ s ) exp [ κ e d ( sec θ i sec θ s ) ] exp [ κ e d ( sec θ + sec θ i ) ] κ e ( sec θ s + sec θ ) } ,
R 2 ( θ , ϕ ) = P 21 ( θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ + sec θ i ) × { D 1 ( θ i , θ s ) exp [ κ e d ( sec θ i sec θ s ) ] exp [ κ e d ( sec θ + sec θ i ) ] κ e ( sec θ s + sec θ ) } ,
R 3 ( θ , ϕ ) = P 31 ( θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ + sec θ i ) × { D 1 ( θ i , θ s ) exp [ κ e d ( sec θ i sec θ s ) ] exp [ κ e d ( sec θ + sec θ i ) ] κ e ( sec θ s + sec θ ) } ,
R 4 ( θ , ϕ ) = P 41 ( θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ + sec θ i ) × { D 1 ( θ i , θ s ) exp [ κ e d ( sec θ i sec θ s ) ] exp [ κ e d ( sec θ + sec θ i ) ] κ e ( sec θ s + sec θ ) } ,
S 1 ( θ , ϕ ) = P 11 ( π θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ sec θ i ) × [ D 1 ( θ i , θ s ) D 1 ( θ , θ s ) ] ,
S 2 ( θ , ϕ ) = P 21 ( π θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ sec θ i ) × [ D 1 ( θ i , θ s ) D 1 ( θ , θ s ) ] ,
S 3 ( θ , ϕ ) = P 31 ( π θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ sec θ i ) × [ D 1 ( θ i , θ s ) D 1 ( θ , θ s ) ] ,
S 4 ( θ , ϕ ) = P 41 ( π θ , ϕ ; π θ i , ϕ i ) 1 κ e ( sec θ sec θ i ) × [ D 1 ( θ i , θ s ) D 1 ( θ , θ s ) ] .
lim θ θ i 1 κ e ( sec θ sec θ i ) [ D 1 ( θ i , θ s ) D 1 ( θ , θ i ) ] = 1 2 d 2 .
CX = I 2 s υ ( π θ s , ϕ s , d ) Ω r I o exp ( κ e d sec θ i ) + I 2 s υ ( π θ i , ϕ i d ) Ω r = cos θ i cos θ s exp [ κ e d ( sec θ s sec θ i ) ] [ D 1 ( θ i , θ s ) P 11 s + VV s ] Ω r cos θ i + [ d P 11 i + VV i ] Ω r ,
P 11 l = P 11 ( π θ l , ϕ l ; π θ i , ϕ i ) ,
VV l = 0 π / 2 d θ sin θ sec θ { V 1 ( θ l , θ ) + V 2 ( θ l , θ ) + V 3 ( θ l , θ ) + V 4 ( θ l , θ ) } ,
V 1 ( θ l , θ ) = 0 2 π d θ [ P 11 ( π θ l , ϕ l ; θ , ϕ ) R 1 ( θ , ϕ ) + P 11 ( π θ l , ϕ l ; π θ , ϕ ) S 1 ( θ , ϕ ) ] ,
V 2 ( θ l , θ ) = 0 2 π d ϕ [ P 12 ( π θ l , ϕ l ; θ , ϕ ) R 2 ( θ , ϕ ) + P 12 ( π θ l , ϕ l ; π θ , ϕ ) S 2 ( θ , ϕ ) ] ,
V 3 ( θ l , θ ) = 0 2 π d θ [ P 13 ( π θ l , ϕ l ; θ , ϕ ) R 3 ( θ , ϕ ) + P 13 ( π θ l , ϕ l ; π θ , ϕ ) S 3 ( θ , ϕ ) ] ,
V 4 ( θ l , θ ) = 2 π d θ [ P 14 ( π θ l , ϕ l ; θ , ϕ ) R 4 ( θ , ϕ ) + P 14 ( π θ l , ϕ l ; π θ , ϕ ) S 4 ( θ , ϕ ) ] .
σ ̿ = [ | C υ s υ i | 2 | C υ s h i | 2 Re [ C υ s h i * C υ s υ i ] Im [ C υ s h i * C υ s υ i ] | C h s υ i | 2 | C h s h i | 2 Re [ C h s h i * C h s υ i ] Im [ C h s h i * C h s υ i ] 2 Re [ C υ s υ i C h s υ i * ] 2 Re [ C υ s h i C h s h i * ] Re [ C υ s h i C h s υ i * + C υ s υ i C h s h i * ] Im [ C υ s υ i C h s h i * C υ s h i C h s υ i * ] 2 Im [ ] 2 Im [ ] Im [ ] Re [ ] ] ,
[ C υ s υ i C υ s h i C h s υ i C h s h i ] = i exp ( ik 1 r ) k 1 r n = 1 m = 0 n ( 2 δ m ) ( 2 n + 1 ) ( n m ) ! n ( n + 1 ) ( n + m ) ! , { ( A n m 2 sin θ sin θ i P n m ( cos θ ) P n m ( cos θ i ) + B n dP n m ( cos θ ) d θ dP n m ( cos θ i ) d θ i ) cos m ( ϕ ϕ i ) ( A n m sin θ P n m ( cos θ ) dP n m ( cos θ i ) d θ i + B n m sin θ i P n m ( cos θ i ) dP n m ( cos θ ) d θ ) sin m ( ϕ ϕ i ) ( A n m sin θ i P n m ( cos θ i ) dP n m ( cos θ ) d θ B n m sin θ P n m ( cos θ ) dP n m ( cos θ i ) d θ i ) sin m ( ϕ ϕ i ) ( A n dP n m ( cos θ ) d θ dP n m ( cos θ i ) d θ i + B n m 2 sin θ sin θ i P n m ( cos θ ) P n m ( cos θ i ) ) cos m ( ϕ ϕ i ) } ,
P ̿ ( θ s , ϕ s ; θ i , ϕ i ) = a min a max daN ( a ) σ ̿ ( θ s , ϕ s ; θ i , ϕ i ) = a min a max daN ( a ) [ P 11 P 12 P 13 P 11 P 21 P 22 P 23 P 24 P 31 P 32 P 33 P 34 P 41 P 42 P 43 P 44 ] ,
K ( θ , θ i ) = n = 1 ( 2 n + 1 ) n ( n + 1 ) B n dP n ( θ ) d θ dP n ( θ i ) d θ i ,
K h ( θ , θ i ) = n = 1 ( 2 n + 1 ) n ( n + 1 ) A n dP n ( θ ) d θ dP n ( θ i ) d θ i ,
Q m ( θ , θ i ) = n = m ( 2 n + 1 ) ( n m ) ! n ( n + 1 ) ( n + m ) ! ( A n m 2 sin θ sin θ i P n m ( θ ) P n m ( θ i ) + B n dP n m ( θ ) d θ dP n m ( θ i ) d θ i ) ,
Q m h ( θ , θ i ) = n = m 2 ( 2 n + 1 ) ( n m ) ! n ( n + 1 ) ( n + m ) ! ( A n dP n m ( θ ) d θ dP n m ( θ i ) d θ i + B n m 2 sin θ sin θ i P n m ( θ ) P n m ( θ i ) ) ,
Q m a ( θ , θ i ) = n = m 2 ( 2 n + 1 ) ( n m ) ! n ( n + 1 ) ( n + m ) ! m ( A n sin θ P n m ( θ ) dP n m ( θ i ) d θ i + B n sin θ i dP n m ( θ ) d θ P n m ( θ i ) ) ,
Q m b ( θ , θ i ) = n = m 2 ( 2 n + 1 ) ( n m ) ! n ( n + 1 ) ( n + m ) ! m ( A n sin θ i dP n m ( θ ) d θ P n m ( θ i ) + B n sin θ P n m ( θ ) dP n m ( θ i ) d θ i ) .
C υ s υ i = i exp ( ik 1 r ) k 1 r [ K + m = 1 Q m cos m ( ϕ ϕ i ) ] ,
C υ s h i = i exp ( ik 1 r ) k 1 r [ m = 1 Q m a sin m ( ϕ ϕ i ) ] ,
C h s υ i = i exp ( ik 1 r ) k 1 r [ m = 1 Q m b sin m ( ϕ ϕ i ) ] ,
C h s h i = i exp ( ik 1 r ) k 1 r [ K h + m = 1 Q m h cos m ( ϕ ϕ i ) ] .
P 11 ( θ , ϕ ; θ i , ϕ i ) = | C υ s υ i | 2 = 1 k 1 2 r 2 [ | K | 2 + m = 1 1 2 | Q m | 2 ] + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) [ 2 Re ( Q m * K ) n = 1 Re ( Q n Q n + m * ) ] + 1 k 1 2 r 2 m = 2 cos m ( ϕ ϕ i ) { n = 1 ( m 1 ) / 2 Re ( Q n Q m n * ) m = odd n = 1 ( m 2 ) / 2 Re ( Q n Q m n * ) + 1 2 | Q m / 2 | 2 m = even ,
P 12 ( θ , ϕ ; θ i , ϕ i ) = | C υ s h i | 2 = 1 k 1 2 r 2 m = 1 1 2 | Q m a | 2 + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) [ n = 1 Re ( Q n a Q n + m a * ) ] + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) { n = 1 ( m 1 ) / 2 Re ( Q n a Q m n * ) m = odd n = 1 ( m 2 ) / 2 Re ( Q n a Q m n a * ) + 1 2 | Q m / 2 a | 2 m = even ,
P 13 ( θ , ϕ ; θ i , ϕ i ) = Re ( C υ s h i * C υ s υ i ) = 1 k 1 2 r 2 m = 1 sin m ( ϕ ϕ i ) × Re [ Q m a * K + n = 1 1 2 ( Q n Q n + m a * Q n + m Q n a * ) + 1 2 n = 1 m 1 Q n Q m n a * ] ,
P 14 ( θ , ϕ ; θ i , ϕ i ) = Im ( C υ s h i * C υ s υ i ) = 1 k 1 2 r 2 m = 1 sin m ( ϕ ϕ i ) Im [ Q m a * K n = 1 1 2 ( Q n Q n + m a * Q n + m Q n a * ) 1 2 n = 1 m 1 Q n Q m n a * ] ,
P 21 ( θ , ϕ ; θ i , ϕ i ) = | C h s υ i | 2 = 1 k 1 2 r 2 m = 1 1 2 | Q m b | 2 + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) [ n = 1 Re ( Q n b Q n m b * ) ] 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) { n = 1 ( m 1 ) / 2 Re ( Q n b Q m n b * ) m = odd n = 1 ( m 1 ) / 2 Re ( Q n b Q m n b * ) + 1 2 | Q m / 2 b | 2 m = even ,
P 22 ( θ , ϕ ; θ i , ϕ i ) = | C h s h i | 2 = 1 k 1 2 r 2 [ | K h | 2 + m = 1 1 2 | Q m h | 2 + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) [ 2 Re ( Q m h * K h ) + n = 1 Re ( Q n h Q n + m h * ) ] + 1 k 1 2 r 2 m = 2 cos m ( ϕ ϕ i ) { n = 1 ( m 1 ) / 2 Re ( Q n h Q m n h * ) m = odd n = 1 ( m 2 ) / 2 Re ( Q n h Q m n h * ) + 1 2 | Q m / 2 h | 2 m = even ,
P 23 ( θ , ϕ ; θ i , ϕ i ) = Re ( C h s h i * C h s υ i ) = 1 k 1 2 r 2 m = 1 sin m ( ϕ ϕ i ) × Re [ Q m b K h * + n = 1 1 2 ( Q n b Q n + m h * Q n + m b Q n h * ) n = 1 m 1 Q h b Q m n h * ] ,
P 24 ( θ , ϕ ; θ i , ϕ i ) = Im ( C h s h i * C h s υ i ) = 1 k 1 2 r 2 m = 1 sin m ( ϕ ϕ i ) × Im [ Q m b K h * n = 1 1 2 ( Q n b Q n + m h * Q n + m b Q n h * ) + n = 1 m 1 Q n b Q m n h * ] ,
P 31 ( θ , ϕ ; θ i , ϕ i ) = 2 Re ( C υ s υ i C h s υ i * ) = 1 k 1 2 r 2 m = 1 sin m ( ϕ ϕ i ) × Re [ 2 Q m b * K + n = 1 ( Q m + n Q n b * Q n Q n + m b * ) n = 1 m 1 Q n Q m n b * ] ,
P 32 ( θ , ϕ ; θ i , ϕ i ) = 2 Re ( C υ s h i C h s h i * ) = 1 k 1 2 r 2 m = 1 sin m ( ϕ ϕ i ) × Re [ 2 Q m a K h * + n = 1 ( Q m + n a Q n h * Q n a Q m + n h * ) + n = 1 m 1 Q n a Q m n h * ] ,
P 33 ( θ , ϕ ; θ i , ϕ i ) = Re ( C υ s h i C h s υ i * + C υ s υ i C h s h i * ) = 1 k 1 2 r 2 Re [ KK h * + 1 2 m = 1 ( Q m Q m h * Q m a * Q m b ) ] + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) × Re [ KQ m h * + Q m K h * + n = 1 1 2 ( Q n Q n + m b * + Q n + m Q n h * Q n a * Q n + m b Q n + m a * Q n b ) + n = 1 m 1 1 2 ( Q n Q m n h * + Q n a * Q m n b ) ] ,
P 34 ( θ , ϕ ; θ i , ϕ i ) = Im ( C υ s h i C h s υ i * + C υ s υ i C h s h i * ) = 1 k 1 2 r 2 Im [ KK h * 1 2 m = 1 ( Q m Q m h * Q m a Q m b * ) ] + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) × Im [ KQ m h * Q m K h * + n = 1 1 2 ( Q n Q n + m h * Q n + m Q n h * + Q n a Q n + m b * + Q n + m a Q n b * ) + n = 1 m 1 1 2 ( Q n Q m n h * Q n a Q m n b * ) ] ,
P 41 ( θ , ϕ ; θ i , ϕ i ) = 2 Im ( C υ s υ i C h s υ i * ) = 1 k 1 2 r 2 m = 1 sin m ( ϕ ϕ i ) × Im [ 2 Q m b * K + n = 1 ( Q m + n Q n b * Q n Q n + m b * ) n = 1 m 1 Q n Q m n b * ] ,
P 42 ( θ , ϕ ; θ i , ϕ i ) = 2 Im ( C υ s h i C h s h i * ) = 1 k 1 2 r 2 m = 1 sin m ( ϕ ϕ i ) × Im [ 2 Q m a K h * + n = 1 ( Q m + n a Q n h * Q n a Q m + n h * ) + n = 1 m 1 Q n a Q m n h * ] ,
P 43 ( θ , ϕ ; θ i , ϕ i ) = Im ( C υ s h i C h s υ i * + C υ s υ i C h s h i * ) = 1 k 1 2 r 2 Im [ KK h * + 1 2 m = 1 ( Q m Q m h * + Q m a Q m b * ) ] + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) × Im [ KQ m h * + Q m K h * + n = 1 1 2 ( Q n Q n + m h * + Q n + m Q n h * + Q n a Q n + m b * + Q n + m a Q n b * ) + n = 1 m 1 1 2 ( Q n Q m n h * Q n a Q m n b * ) ] ,
P 44 ( θ , ϕ ; θ i , ϕ i ) = Re ( C υ s h i C h s υ i * + C υ s υ i C h s h i * ) = 1 k 1 2 r 2 Re [ KK h * + 1 2 m = 1 ( Q m Q m h * + Q m a Q m b * ) ] + 1 k 1 2 r 2 m = 1 cos m ( ϕ ϕ i ) × Re [ KQ m h * + Q m K h * + n = 1 1 2 ( Q n Q n + m h * + Q n + m Q n h * + Q n a Q n + m b + Q n + m a Q n b * ) + n = 1 m 1 1 2 ( Q n Q m n h * Q n a Q m n b * ) ] .
t n m ( cos θ ) = P n m ( cos θ ) sin θ ( ( 2 n + 1 ) ( n m ) ! 2 ( n + m ) ! ) 1 / 2 ,
s n m ( cos θ ) = ( ( 2 n + 1 ) ( n m ) ! 2 ( n + m ) ! ) d d θ P n m ( cos θ ) ,
K ( θ , θ i ) = n = 1 2 n ( n + 1 ) B n s n o ( cos θ ) s n o ( cos θ i ) ,
K h ( θ , θ i ) = n = 1 2 n ( n + 1 ) A n s n o ( cos θ ) s n o ( cos θ i ) ,
Q m ( θ , θ i ) = n = m 4 n ( n + 1 ) [ m 2 A n t n m ( cos θ ) t n m ( cos θ i ) + B n s n m ( cos θ ) s n m ( cos θ i ) ] ,
Q m h ( θ , θ i ) = n = m 4 n ( n + 1 ) [ A n s n m ( cos θ ) s n m ( cos θ i ) + m 2 B n t n m ( cos θ ) t n m ( cos θ i ) ] ,
Q m a ( θ , θ i ) = n = m 4 n ( n + 1 ) [ A n t n m ( cos θ ) s n m ( cos θ i ) + B n s n m ( cos θ ) t n m ( cos θ i ) ] ,
Q m b ( θ , θ i ) = n = m 4 m n ( n + 1 ) [ A n s n m ( cos θ ) t n m ( cos θ i ) + B n t n m ( cos θ ) s n m ( cos θ i ) ] ,
K ( θ , θ i ) = n = 1 M 2 n ( n + 1 ) B n s n o ( cos θ ) s n o ( cos θ i ) ,
K h ( θ , θ i ) = n = 1 M 2 n ( n + 1 ) A n s n o ( cos θ ) s n o ( cos θ i ) ,
Q m ( θ , θ i ) = n = m M 4 n ( n + 1 ) [ m 2 A n t n m ( cos θ ) t n m ( cos θ i ) + B n s n m ( cos θ ) s n m ( cos θ i ) ] ,
Q m h ( θ , θ i ) = n = m M 4 n ( n + 1 ) [ A n s n m ( cos θ ) s n m ( cos θ i ) + m 2 B n t n m ( cos θ ) t n m ( cos θ i ) ] ,
Q m a ( θ , θ i ) = n = m M 4 m n ( n + 1 ) [ A n t n m ( cos θ ) s n m ( cos θ i ) + B n s n m ( cos θ ) t n m ( cos θ i ) ] ,
Q m b ( θ , θ i ) = n = m M 4 m n ( n + 1 ) [ A n s n m ( cos θ ) t n m ( cos θ i ) + B n t n m ( cos θ ) s n m ( cos θ i ) ] .
P 11 ( θ , ϕ ; θ i , ϕ i ) = 1 k 1 2 r 2 [ | K | 2 + m = 1 M 1 2 | Q m | 2 ] + 1 k 1 2 r 2 m = 1 M cos m ( ϕ ϕ i ) × 2 Re ( Q m * K ) + m = 1 M 1 cos m ( ϕ ϕ i ) n = 1 M m Real ( Q n Q n + m * ) + 1 k 1 r n = 1 M 1 m = 2 n + 1 M + n cos m ( ϕ ϕ i ) Re ( Q n Q m n * ) + 1 k 1 2 r 2 m = 1 M [ cos 2 m ( ϕ ϕ i ) 1 2 | Q m | 2 ] ,
P 13 ( θ , ϕ ; θ i , ϕ i ) = 1 k 1 2 r 2 m = 1 M sin m ( ϕ ϕ i ) Re ( Q m a * K ) + 1 k 1 2 r 2 m = 1 M 1 sin m ( ϕ ϕ i ) Re [ n = 1 M m 1 2 ( Q n Q n + m a * Q n + m Q n a * ) ] + 1 k 1 2 r 2 n = 1 M m = n + 1 M + n sin m ( ϕ ϕ i ) Re ( 1 2 Q n Q m n a * ) .

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