Abstract

Several expressions for the mutual coherence function are obtained for waves scattered by a random layer with rough boundaries (including those waves transmitted through) by modifying the previous Bethe-Salpeter (B-S) equation for a random medium with one rough boundary. Considering that the medium and the boundaries are involved in the equation on exactly the same footing, two sets of addition formulas are prepared for the incoherent and coherent terms, respectively, to obtain various expressions of the mutual coherence function by exchanging the roles of the medium and the boundaries. The conventional method of solving the transport equation is shown to be available only when the separation of the boundaries is sufficiently large compared with the coherence distance of the wave, whereas the expressions obtained are available beyond this limit. The previous expression by Fung and Eom [ IEEE Trans. Antennas Propag. AP-29, 899 ( 1981)] is also shown to be reproduced with the present exact formulation, as long as one of the boundaries is perfectly random. The corresponding expressions for fixed scatterers embedded in a (homogeneous or semi-infinite) random medium are also obtained, which show an explicit shadowing effect. Summarized in the appendixes are basic equations including addition formulas, the Green function and the scattering matrix of the B-S equation, and a general theory of fixed scatterers embedded in a random medium.

© 1985 Optical Society of America

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References

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  1. A. K. Fung, M. F. Chen, “Scattering from a Rayleigh layer with an irregular interface,” Radio Sci. 16, 1337–1347 (1981).
    [Crossref]
  2. K. Furutsu, “Transport theory and boundary-value solutions. I. The Bethe-Salpeter equation and scattering matrices,” J. Opt. Soc. Am. A 2, 913–931 (1985).
    [Crossref]
  3. A. K. Fung, H. J. Eom, “A theory of wave scattering from an inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
    [Crossref]
  4. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  5. Equation (2.27) is formally reduced to Eq. (24) of Ref. 3 with the replacement of σ(23) → Rq, S0(12)→Γ, and I(0q)→ Tt or I(0q) → Tt* (depending on the direction of propagation). Therefore σ(23/q)→ RB and S0(12/q)→RT or S0(12/q)→QT [compare Eqs. (2.18) and Eqs. (4.26b) of Part I of this paper with Eqs. (16)–(20) of Ref. 3], with US(0q)U→S∼ or US(0q)U→S∼*. Here S0(12) is given by the series of Eq. (3.29b) of Part I with U(C)→ U, which means the multiple scattering by a perfectly random boundary [V(12)) = 0], as realized when the boundary is very rough. This case was investigated in detail by K. Furutsu, [“Statistical theory of scattering and propagation over a random surface,” IEE (London) Proc. Part F 130, 601–622 (1983)] for one side boundary, based on a new tangent plane method (as distinguished from the conventional). The scattering cross section was shown to be given by a series of multiple scattering consistent with shadowing effect [Eq. (113)].
  6. S. Twomey, H. Jacobowitz, H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
    [Crossref]
  7. H. B. Howell, H. Jacobowitz, “Matrix method applied to the multiple scattering of polarized light,” J. Atmos. Sci. 27, 1195–1206 (1970).
    [Crossref]
  8. K. Furutsu, “Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation,” Radio Sci. 10, 29–44 (1975).
    [Crossref]
  9. K. Furutsu, “Operator methods for time-dependent waves in random media with applications to the case of random particles,” J. Math. Phys. 21, 2764–2779 (1980).
    [Crossref]

1985 (1)

1981 (2)

A. K. Fung, M. F. Chen, “Scattering from a Rayleigh layer with an irregular interface,” Radio Sci. 16, 1337–1347 (1981).
[Crossref]

A. K. Fung, H. J. Eom, “A theory of wave scattering from an inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
[Crossref]

1980 (1)

K. Furutsu, “Operator methods for time-dependent waves in random media with applications to the case of random particles,” J. Math. Phys. 21, 2764–2779 (1980).
[Crossref]

1975 (1)

K. Furutsu, “Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation,” Radio Sci. 10, 29–44 (1975).
[Crossref]

1970 (1)

H. B. Howell, H. Jacobowitz, “Matrix method applied to the multiple scattering of polarized light,” J. Atmos. Sci. 27, 1195–1206 (1970).
[Crossref]

1966 (1)

S. Twomey, H. Jacobowitz, H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chen, M. F.

A. K. Fung, M. F. Chen, “Scattering from a Rayleigh layer with an irregular interface,” Radio Sci. 16, 1337–1347 (1981).
[Crossref]

Eom, H. J.

A. K. Fung, H. J. Eom, “A theory of wave scattering from an inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
[Crossref]

Fung, A. K.

A. K. Fung, H. J. Eom, “A theory of wave scattering from an inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
[Crossref]

A. K. Fung, M. F. Chen, “Scattering from a Rayleigh layer with an irregular interface,” Radio Sci. 16, 1337–1347 (1981).
[Crossref]

Furutsu, K.

K. Furutsu, “Transport theory and boundary-value solutions. I. The Bethe-Salpeter equation and scattering matrices,” J. Opt. Soc. Am. A 2, 913–931 (1985).
[Crossref]

K. Furutsu, “Operator methods for time-dependent waves in random media with applications to the case of random particles,” J. Math. Phys. 21, 2764–2779 (1980).
[Crossref]

K. Furutsu, “Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation,” Radio Sci. 10, 29–44 (1975).
[Crossref]

Howell, H. B.

H. B. Howell, H. Jacobowitz, “Matrix method applied to the multiple scattering of polarized light,” J. Atmos. Sci. 27, 1195–1206 (1970).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

Jacobowitz, H.

H. B. Howell, H. Jacobowitz, “Matrix method applied to the multiple scattering of polarized light,” J. Atmos. Sci. 27, 1195–1206 (1970).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

Twomey, S.

S. Twomey, H. Jacobowitz, H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

IEEE Trans. Antennas Propag. (1)

A. K. Fung, H. J. Eom, “A theory of wave scattering from an inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
[Crossref]

J. Atmos. Sci. (2)

S. Twomey, H. Jacobowitz, H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

H. B. Howell, H. Jacobowitz, “Matrix method applied to the multiple scattering of polarized light,” J. Atmos. Sci. 27, 1195–1206 (1970).
[Crossref]

J. Math. Phys. (1)

K. Furutsu, “Operator methods for time-dependent waves in random media with applications to the case of random particles,” J. Math. Phys. 21, 2764–2779 (1980).
[Crossref]

J. Opt. Soc. Am. A (1)

Radio Sci. (2)

A. K. Fung, M. F. Chen, “Scattering from a Rayleigh layer with an irregular interface,” Radio Sci. 16, 1337–1347 (1981).
[Crossref]

K. Furutsu, “Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation,” Radio Sci. 10, 29–44 (1975).
[Crossref]

Other (2)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Equation (2.27) is formally reduced to Eq. (24) of Ref. 3 with the replacement of σ(23) → Rq, S0(12)→Γ, and I(0q)→ Tt or I(0q) → Tt* (depending on the direction of propagation). Therefore σ(23/q)→ RB and S0(12/q)→RT or S0(12/q)→QT [compare Eqs. (2.18) and Eqs. (4.26b) of Part I of this paper with Eqs. (16)–(20) of Ref. 3], with US(0q)U→S∼ or US(0q)U→S∼*. Here S0(12) is given by the series of Eq. (3.29b) of Part I with U(C)→ U, which means the multiple scattering by a perfectly random boundary [V(12)) = 0], as realized when the boundary is very rough. This case was investigated in detail by K. Furutsu, [“Statistical theory of scattering and propagation over a random surface,” IEE (London) Proc. Part F 130, 601–622 (1983)] for one side boundary, based on a new tangent plane method (as distinguished from the conventional). The scattering cross section was shown to be given by a series of multiple scattering consistent with shadowing effect [Eq. (113)].

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Figures (3)

Fig. 1
Fig. 1

Geometry and notation of random layer for Eq. (2.1). The boundary space Rs (Fig. 1 of Part I) is omitted because we regard it to be infinitesimally thin.

Fig. 2
Fig. 2

Fixed scatterer embedded in a semi-infinite random medium for Eq. (3.1).

Fig. 3
Fig. 3

Layer composed of two independent layers of K(A) and K(B) for Eq. (A1).

Equations (204)

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F ( C ) = G * G U 1 = 1 + U V ( C ) , F ¯ ( C ) = U 1 * G * G = 1 + V ( C ) U , I ( A ) = U ( C ) + U ( C ) S ( A ) U ( C ) = U + U σ ( A ) U , F C ( A ) = [ 1 K ( A ) U ( C ) ] 1 K ( A ) , σ C ( A ) = V ( C ) + F ¯ ( C ) S ( A ) F ( C ) , U ( C ) = G * G = 1 + U V ( C ) U = F ( C ) U = U F ¯ ( C ) , V ( C ) = V ( 12 ) + V ( 23 ) ;
I 0 ( A ) = I ( A ) | 0 , S 0 ( A ) = S ( A ) | 0 ,
I ( 0 q ) = I ( q ) | 0 , S ( 0 q ) = S ( q ) | 0 , U = U ( C ) | 0 .
I ( q + 12 + 23 ) = U ( C ) [ 1 + ( K ( q ) + K ( 12 ) + K ( 23 ) ) I ( q + 12 + 23 ) ]
U a b ( C ) ( z | z ) = U a ( z z ) δ a b + U a ( z z a ) V a b ( C ) U b ( z b z ) .
U a c ; b d ( C ) ( z 1 ; z 2 | z 1 ; z 2 ) = U a c ( z 1 ; z 2 | z 1 ; z 2 ) δ a b δ c d + U a c ( z 1 ; z 2 | z a ; z c ) V a c ; b d ( C ) U b d ( z b ; z d | z 1 ; z 2 ) .
V ( C ) = V ( 12 ) + V ( 23 ) ,
I ( q + 12 + 23 ) = I ( q + 12 ) [ 1 + K ( 23 ) I ( q + 12 + 23 ) ]
I ( q + 12 + 23 ) = I ( q + 12 ) + I ( q + 12 ) S ( 23 / q + 12 ) I ( q + 12 ) .
I ( q + 12 ) = I ( q ) + I ( q ) S ( 12 / q ) I ( q ) ,
I ( q ) = I ( 0 q ) + I ( 0 q ) V ( C / q ) I ( 0 q ) ,
S ( 12 / q ) = [ 1 S ( 12 ) U ( C ) S ( q ) U ( C ) ] 1 S ( 12 ) ,
S ( 23 / q + 12 ) = [ 1 S ( 23 / q ) I ( q ) S ( 12 / q ) I ( q ) ] 1 S ( 23 / q ) .
I a b ( q + 12 + 23 ) = I a b ( q ) + i , j = 1 3 I a i ( q ) S i j ( 12 + 23 / q ) I j b ( q ) .
S ( 12 + 23 / q ) = S ( 12 / q ) + [ 1 + S ( 12 / q ) I ( q ) ] S ( 23 / q + 12 ) × [ I ( q ) S ( 12 / q ) + 1 ] .
I ( q + 12 + 23 ) = [ 1 + I ( q ) S ( 23 / q ) I ( q ) ] × [ 1 S ( 12 / q ) I ( q ) S ( 23 / q ) I ( q ) ] 1 [ S ( 12 / q ) I ( q ) + 1 ] ,
I a b ( q + 12 + 23 ) = I a ( 0 q ) δ a b + I a ( 0 q ) S a b ( 12 + 23 / q ) I b ( 0 q ) .
S 11 ( 12 + 23 / q ) = S 11 ( 12 / q ) + S 12 ( 12 / q ) I 2 ( 0 q ) S 22 ( 23 / q + 12 ) I 2 ( 0 q ) S 21 ( 12 / q ) ,
S 22 ( 23 / q + 12 ) = [ 1 S 22 ( 23 / q ) I 2 ( 0 q ) S 22 ( 12 / q ) I 2 ( 0 q ) ] 1 S 22 ( 23 / q )
S 33 ( 12 + 23 / q ) = S 33 ( 23 / q + 12 ) ,
I 31 ( q + 12 + 23 ) = I 3 ( 0 q ) S 32 ( 23 / q ) I 2 ( 0 q ) × [ 1 S 22 ( 12 / q ) I 2 ( 0 q ) S 22 ( 23 / q ) I 2 ( 0 q ) ] 1 S 21 ( 12 / q ) I 1 ( 0 q ) ,
I 31 ( q + 12 + 23 ) = I 3 ( 0 q ) S 32 ( 23 / q + 12 ) I 2 ( 0 q ) S 21 ( 12 / q ) I 1 ( 0 q )
= I 3 ( 0 q ) S 32 ( 23 / q ) I 2 ( 0 q ) S 21 ( 12 / q + 23 ) I 1 ( 0 q ) ,
S ( 12 / q ) = [ 1 S 0 ( 12 ) U S ( 0 q ) U ] 1 S 0 ( 12 ) [ V ( C ) = 0 ] ,
S 22 ( 12 / q ) = [ 1 S 0 , 22 ( 12 ) U 2 S 2 ( 0 q ) U 2 ] 1 S 0 , 22 ( 12 ) ,
S 21 ( 12 / q ) = [ 1 S 0 , 22 ( 12 ) U 2 S 2 ( 0 q ) U 2 ] 1 S 0 , 21 ( 12 ) ,
S 11 ( 12 / q ) = S 0 , 11 ( 12 ) + S 0 , 12 ( 12 ) U 2 S 22 ( q / 12 ) U 2 S 0 , 21 ( 12 ) ,
U 2 S 22 ( q / 12 ) U 2 = [ 1 U 2 S 2 ( 0 q ) U 2 S 0 , 22 ( 12 ) ] 1 U 2 S 2 ( 0 q ) U 2 .
I ( q + 12 ) = I ( 12 ) + I ( 12 ) S ( q / 12 ) I ( 12 )
= I ( 0 q ) + I ( 0 q ) σ C ( 12 / q ) I ( 0 q )
I ( q + 12 + 23 ) = I ( 0 q ) + I ( 0 q ) σ ( 12 + 23 / q ) I ( 0 q ) .
σ ( 12 + 23 / q ) = σ C ( 12 / q ) + [ 1 + σ C ( 12 / q ) I ( 0 q ) ] × S ( 23 / q + 12 ) [ I ( 0 q ) σ C ( 12 / q ) + 1 ] ,
S ( 23 / q + 12 ) = [ 1 S 0 ( 23 / q ) I ( 0 q ) σ C ( 12 / q ) I ( 0 q ) ] 1 S 0 ( 23 / q ) ,
I ( q + 12 + 23 ) = [ 1 + I ( 0 q ) S 0 ( 23 / q ) ] I ( 0 q ) × [ 1 σ C ( 12 / q ) I ( 0 q ) S 0 ( 23 / q ) I ( 0 q ) ] 1 [ σ C ( 12 / q ) I ( 0 q ) + 1 ] .
σ C ( 12 / q ) = [ 1 σ C ( 12 ) U S ( 0 q ) U ] 1 σ C ( 12 ) ,
σ C ( 12 ) = V ( C ) + [ 1 + V ( C ) U ] S c ( 12 ) [ U V ( C ) + 1 ] ,
S C ( 12 ) = [ 1 S 0 ( 12 ) U V ( C ) U ] 1 S 0 ( 12 ) ,
σ ( 12 + 23 / q ) = S 0 ( 12 / q ) + [ 1 + S 0 ( 12 / q ) I ( 0 q ) × [ 1 σ ( 23 / q ) I ( 0 q ) S 0 ( 12 / q ) I ( 0 q ) ] 1 σ ( 23 / q ) × [ I ( 0 q ) S 0 ( 12 / q ) + 1 ] [ V ( 12 ) = 0 ] .
I ( q + 12 + 23 ) = I ( 12 + 23 ) [ 1 + K ( q ) I ( q + 12 + 23 ) ] .
I ( q + 12 + 23 ) = I ( 12 + 23 ) + I ( 12 + 23 ) S ( q / 12 + 23 ) I ( 12 + 23 ) .
I ( 12 + 23 ) = U + U σ ( 12 + 23 ) U .
σ ( 12 + 23 ) = σ C ( 12 ) + [ 1 + σ C ( 12 ) U ] S ( 23 / 12 ) [ U σ C ( 12 ) + 1 ] ,
S ( 23 / 12 ) = [ 1 S 0 ( 23 ) U σ C ( 12 ) U ] 1 S 0 ( 23 ) ,
σ C ( 12 ) = S 0 ( 12 ) + [ 1 + S 0 ( 12 ) U ] V ( C / 12 ) [ U S 0 ( 12 ) + 1 ]
I a b ( q + 12 + 23 ) = I a ( 0 q ) δ a b + I a ( 0 q ) σ a b ( 12 + 23 / q ) I b ( 0 q ) .
σ ( 12 + 23 / q ) = [ 1 σ ( 12 + 23 ) U S ( 0 q ) U ] 1 σ ( 12 + 23 ) ,
σ ( 12 + 23 / q ) = σ ( 12 + 23 ) + σ ( 12 + 23 ) U S ( q / 12 + 23 ) U σ ( 12 + 23 ) ,
U S ( q / 12 + 23 ) U = [ 1 U S ( 0 q ) U σ ( 12 + 23 ) ] 1 U S ( 0 q ) U ,
U 2 S 2 ( q / 12 + 23 ) U 2 [ 1 U 2 S 2 ( 0 q ) U 2 σ 22 ( 12 + 23 ) ] 1 U 2 S 2 ( 0 q ) U 2 .
I ( q + 12 + 23 ) = I ( 0 q ) [ 1 σ ( 12 + 23 ) U S ( 0 q ) U ] 1 [ σ ( 12 + 23 ) U + 1 ] .
σ ( 12 + 23 ) = σ ( 12 ) + σ ( 23 ) .
σ ( 12 + 23 / q ) = σ ( 12 / q ) + [ 1 + σ ( 12 / q ) U S ( 0 q ) U ] × [ 1 σ ( 23 / q ) U S ( 0 q ) U σ ( 12 / q ) U S ( 0 q ) U ] 1 σ ( 23 / q ) × [ 1 + U S ( 0 q ) U σ ( 12 / q ) ] ,
σ ( 23 / q ) = [ 1 σ ( 23 ) U S ( 0 q ) U ] 1 σ ( 23 ) .
I ( q + 12 + 23 ) = I ( 0 q ) [ 1 + σ ( 23 / q ) U S ( 0 q ) U ] × [ 1 σ ( 12 / q ) U S ( 0 q ) U σ ( 23 / q ) U S ( 0 q ) U ] 1 × [ 1 σ ( 12 / q ) U S ( 0 q ) U ] [ 1 + σ ( 12 + 23 ) U ] ,
1 + σ ( 12 / q ) I ( 0 q ) + [ 1 + σ ( 12 / q ) U S ( 0 q ) U ] σ ( 23 ) U
σ 11 ( q + 12 + 23 ) = σ 11 ( 12 / q ) + σ 12 ( 12 / q ) U 2 S 2 ( 0 q ) U 2 × [ 1 σ 22 ( 23 / q ) U 2 S 2 ( 0 q ) U 2 σ 22 ( 12 / q ) U 2 S 2 ( 0 q ) U 2 ] 1 σ 22 ( 23 / q ) × U 2 S 2 ( 0 q ) U 2 σ 21 ( 12 / q )
I 31 ( q + 12 + 23 ) = I 3 ( 0 q ) + σ 32 ( 23 / q ) U 2 S 2 ( 0 q ) U 2 × [ 1 σ 22 ( 12 / q ) U 2 S 2 ( 0 q ) U 2 σ 22 ( 23 / q ) U 2 S 2 ( 0 q ) U 2 ] 1 × σ 21 ( 12 / q ) I 1 ( 0 q ) ,
σ c ( 12 ) = σ ( 12 ) + σ ( 23 ) ( 12 ) ,
σ ( 23 ) ( 12 ) = ( 1 + S 0 ( 12 ) U ) [ 1 + V ( 12 / 12 ) U S 0 ( 12 ) U ] × [ 1 V ( 23 / 12 ) U S 0 ( 12 ) U V ( 12 / 12 ) U S 0 ( 12 ) U ] 1 V ( 23 / 12 ) × [ 1 + U S 0 ( 12 ) U V ( 12 / 12 ) ] [ 1 + U S 0 ( 12 ) ] ,
V ( 12 / 12 ) = [ 1 V ( 12 ) U S 0 ( 12 ) U ] 1 V ( 12 ) ,
V ( 23 / 12 ) = [ 1 V ( 23 ) U S 0 ( 12 ) U ] 1 V ( 23 ) .
σ ( 12 + 23 ) σ ( 12 ) + σ ( 23 ) , γ 2 L 1 ,
I α ( q ) = I ( 0 q ) + I ( 0 q ) V ( α / q ) I ( 0 q ) .
V ( α / q ) = [ 1 V ( α ) U S ( 0 q ) U ] 1 V ( α ) ,
d Ω ̂ V ( α ) ( Ω ̂ | Ω ̂ ) = 0 ,
d Ω ̂ V ( α / q ) ( Ω ̂ | Ω ̂ ) = 0 ,
V ( α 1 + α 2 ) = V ( α 1 ) + V ( α 2 ) .
V ( α 1 + α 2 / q ) = V ( α 1 / q ) + [ 1 + V ( α 1 / q ) U S ( 0 q ) U ] × [ 1 V ( α 2 / q ) U S ( 0 q ) U V ( α 1 / q ) U S ( 0 q ) U ] 1 V ( α 2 / q ) × [ 1 + U S ( 0 q ) U V ( α 1 / q ) ] ,
V ( α 1 + α 2 + q ) V ( α 1 / q ) + V ( α 2 / q )
| V ( α 1 / q ) U S ( 0 q ) U V ( α 2 / q ) | | V ( α 2 / q ) | .
I α ( q + 12 ) = I ( 0 q ) + I ( 0 q ) σ ( 12 + α ) ( 12 / q ) I ( 0 q ) .
σ ( 12 + α ) ( 12 ) = σ ( 12 ) + σ ( α ) ,
σ ( α ) = [ 1 + S 0 ( 12 ) U ] [ 1 + V ( 12 / 12 ) U S 0 ( 12 ) U ] × [ 1 V ( α / 12 ) U S 0 ( 12 ) U V ( 12 / 12 ) U S 0 ( 12 ) U ] 1 V ( α / 12 ) × [ 1 + U S 0 ( 12 ) U V ( 12 / 12 ) ] [ 1 + U S 0 ( 12 ) ]
σ ( α / q ) = [ 1 σ ( α ) U S ( 0 q ) U ] 1 σ ( α ) .
σ ( 12 + α ) ( 12 / q ) = σ ( 12 / q ) + [ 1 + σ ( 12 / q ) U S ( 0 q ) U ] × [ 1 σ ( α / q ) U S ( 0 q ) U σ ( 12 / q ) U S ( 0 q ) U ] 1 σ ( α / q ) × [ 1 + U S ( 0 q ) U σ ( 12 / q ) ] .
σ ( α ) = [ 1 + S 0 ( 12 ) U ] V ( α / 12 ) [ U S 0 ( 12 ) + 1 ] ;
I ( A + B ) = U ( C ) [ 1 + ( K ( A ) + K ( B ) ) I ( A + B ) ] .
I ( A ) = U ( C ) [ 1 + K ( A ) I ( A ) ] ,
I ( B ) = U ( C ) [ 1 + K ( B ) I ( B ) ]
I ( A ) = U ( C ) + U ( C ) S C ( A ) U ( C ) , I ( B ) = U ( C ) + U ( C ) S C ( B ) U ( C )
S C ( A ) U ( C ) = K ( A ) I ( A ) , S C ( B ) U ( C ) = K ( B ) I ( B )
S C ( A ) = K ( A ) [ 1 + U ( C ) S C ( A ) ] = [ 1 K ( A ) U ( C ) ] 1 K ( A ) ,
I ( A + B ) = U ( C ) + U ( C ) S C ( A + B ) U ( C ) ,
I ( A + B ) = I ( A ) [ 1 + K ( B ) I ( A + B ) ] .
I ( A + B ) = I ( A ) + I ( A ) S ( B / A ) I ( A ) ,
S ( B / A ) I ( A ) = K ( B ) I ( A + B )
S ( B / A ) = K ( B ) [ 1 + I ( A ) S ( B / A ) ] .
[ 1 K ( B ) U ( C ) ] S ( B / A ) = K ( B ) [ 1 + U ( C ) S C ( A ) U ( C ) S ( B / A ) ] .
S ( B / A ) = S C ( B ) [ 1 + U ( C ) S C ( A ) U ( C ) S ( B / A ) ] ,
S ( B / A ) = [ 1 S C ( B ) U ( C ) S C ( A ) U ( C ) ] 1 S C ( B ) ,
S C ( A + B ) = S C ( A ) + [ 1 + S C ( A ) U ( C ) ] S ( B / A ) [ U ( C ) S C ( A ) + 1 ] ,
S ( B / A ) = S C ( B ) + S C ( B ) U ( C ) S ( A / B ) U ( C ) S C ( B ) ,
S ( A / B ) U ( C ) S C ( B ) = S C ( A ) U ( C ) S ( B / A ) = K ( A ) I ( A + B ) K ( B ) ,
I ( A + B ) = [ 1 + U ( C ) S C ( A ) ] × [ 1 U ( C ) S C ( B ) U ( C ) S C ( A ) ] 1 U ( C ) [ 1 + S C ( B ) U ( C ) ] ,
S C ( A + B ) = S C ( A ) + F ¯ C ( A ) S ( B / A ) F C ( A ) ,
I ( A + B ) = F C ( A + B ) U ( C ) = U ( C ) F ¯ C ( A + B ) .
F C ( A + B ) = F C ( A ) [ 1 U ( C ) S C ( B ) U ( C ) S C ( A ) ] 1 F C ( B ) ,
F ¯ C ( A + B ) = F ¯ C ( B ) [ 1 S C ( A ) U ( C ) S C ( B ) U ( C ) ] 1 F ¯ C ( A ) ,
I ( A ) = F C ( A ) U ( C ) = U ( C ) F ¯ C ( A )
F C ( A ) = 1 + U ( C ) S C ( A ) , F ¯ C ( A ) = 1 + S C ( A ) U ( C ) ,
S ( A + B / C ) = S ( A / C ) + [ 1 + S ( A / C ) I ( C ) ] S ( B / A + C ) × [ I ( C ) S ( A / C ) + 1 ] ,
S ( B / A + C ) = [ 1 S ( B / C ) I ( C ) S ( A / C ) I ( C ) ] 1 S ( B / C ) ,
S ( B / A + C ) = [ S ( B / C ) + S ( B / C ) I ( C ) S ( A / B + C ) I ( C ) S ( B / C ) ,
S ( A / B + C ) I ( C ) S ( B / C ) = S ( A / C ) I ( C ) S ( B / A + C ) = K ( A ) I ( A + B + C ) K ( B ) .
I ( A + B + C ) = [ 1 + I ( C ) S ( A / C ) ] I ( C ) [ 1 S ( B / C ) I ( C ) S ( A / C ) I ( C ) ] 1 × [ 1 + S ( B / C ) I ( C ) ] .
U ( C ) = U + U V ( C ) U = U F ( C ) = F ¯ ( C ) U ,
S C ( A ) = S 0 ( A ) [ 1 + U V ( C ) U S C ( A ) ] ,
S 0 ( A ) = K ( A ) [ 1 + U S 0 ( A ) ] .
S C ( A ) = S 0 ( A ) + S 0 ( A ) U V ( C / A ) U S 0 ( A )
V ( C / A ) U S 0 ( A ) = V ( C ) U S C ( A )
S 0 ( A ) U V ( C / A ) = S C ( A ) U V ( C )
V ( C / A ) = [ 1 V ( C ) U S 0 ( A ) U ] 1 V ( C ) .
V ( C ) = V ( C 1 ) + V ( C 2 ) ,
S C 1 + C 2 ( A ) = S 0 ( A ) { 1 + U [ V ( C 1 ) + V ( C 2 ) ] U S C 1 + C 2 ( A ) } ;
S C 1 + C 2 ( A ) = S C 1 ( A ) + S C 1 ( A ) U V ( C 2 / A / C 1 ) U S C 1 ( A ) .
V ( C 2 / A / C 1 ) U S C 1 ( A ) = V ( C 2 ) U S C 1 + C 2 ( A )
V ( C 2 / A / C 1 ) = [ 1 V ( C 2 / A ) U S 0 ( A ) U V ( C 1 / A ) U S 0 ( A ) U ] 1 V ( C 2 / A ) ,
S C 1 + C 2 ( A ) = S 0 ( A ) + S 0 ( A ) U V ( C 1 + C 2 / A ) U S 0 ( A ) .
V ( C 1 + C 2 / A ) = V ( C 1 / A ) + Y ¯ ( C 1 / A ) V ( C 2 / A / C 1 ) Y ( C 1 / A ) ,
Y ¯ ( C / A ) = 1 + V ( C / A ) U S 0 ( A ) U = [ 1 V ( C ) U S 0 ( A ) U ] 1 ,
Y ( C / A ) = 1 + U S 0 ( A ) U V ( C / A ) = [ 1 U S 0 ( A ) U V ( C ) ] 1 ,
S C ( A ) U = S 0 ( A ) U Y ¯ ( C / A ) ,
U S C ( A ) = Y ( C / A ) U S 0 ( A ) .
V ( C / A ) = V ( C ) Y ( C / A ) = Y ¯ ( C / A ) V ( C ) ,
U S C 1 + C 2 ( A ) = Y ( C 1 + C 2 / A ) U S 0 ( A ) ,
Y ( C 1 + C 2 / A ) = Y ( C 2 / A ) × [ 1 U S 0 ( A ) U V ( C 1 / A ) U S 0 ( A ) U V ( C 2 / A ) ] 1 Y ( C 1 / A ) ,
Y ¯ ( C 1 + C 2 / A ) = Y ¯ ( C 1 / A ) × [ 1 V ( C 2 / A ) U S 0 ( A ) U V ( C 1 / A ) U S 0 ( A ) U ] 1 Y ¯ ( C 2 / A ) ,
I C ( A ) = I 0 ( A ) + I 0 ( A ) V ( C / A ) I 0 ( A ) ,
S C ( A ) K ( A ) + K ( A ) I C ( A ) K ( A ) ,
S C ( A ) U ( C ) = K ( A ) I C ( A ) , U ( C ) S C ( A ) = I C ( A ) K ( A )
I C 1 + C 2 ( A ) = I C 1 ( A ) + I 0 ( A ) Y ¯ ( C 1 / A ) V ( C 2 / A / C 1 ) Y ( C 1 / A ) I 0 ( A )
= I C 1 ( A ) = F C 1 ( A ) U V ( C 2 / A / C 1 ) U F ¯ C 1 ( A )
K ( A ) I C ( A ) = S C ( A ) U ( C ) = S C ( A ) U F ¯ ( C ) = S 0 ( A ) U Y ¯ ( C / A ) F ¯ ( C ) ,
I C ( A ) = I 0 ( A ) Y ¯ ( C / A ) F ¯ ( C ) = F ( C ) Y ( C / A ) I 0 ( A ) ,
I 0 ( A ) Y ¯ ( C / A ) = F C ( A ) U ,
Y ¯ ( C / A ) F ¯ ( C ) = 1 + V ( C / A ) I 0 ( A ) .
U S 0 ( A + B ) U = U S 0 ( A ) U + I 0 ( A ) S 0 ( B / A ) I 0 ( A )
V ( C / A + B ) = V ( C / A ) [ 1 + I 0 ( A ) S 0 ( B / A ) I 0 ( A ) V ( C / A + B ) ] = [ 1 V ( C / A ) I 0 ( A ) S 0 ( B / A ) I 0 ( A ) ] 1 V ( C / A ) .
Y ¯ ( C / A + B ) = [ 1 V ( C / A ) I 0 ( A ) S 0 ( B / A ) I 0 ( A ) ] 1 Y ¯ ( C / A ) ;
U S 0 ( q + 12 ) U = U S ( 0 q ) U + I ( 0 q ) S 0 ( 12 / q ) I ( 0 q ) .
S 0 ( 12 / q ) = [ 1 S 0 ( 12 ) U S ( 0 q ) U ] S 0 ( 12 ) .
U S 0 ( q + 12 + 23 ) U = U S 0 ( q + 12 ) U + I 0 ( q + 12 ) S 0 ( 23 / q + 12 ) I 0 ( q + 12 )
= U S 0 ( 12 + 23 ) U + I 0 ( 12 + 23 ) S 0 ( q / 12 + 23 ) I 0 ( 12 + 23 ) ,
I 0 ( q + A ) U = U S 0 ( q + A ) U = U S ( 0 q ) U + I ( 0 q ) S 0 ( A / q ) I ( 0 q )
= U S 0 ( A ) U + I 0 ( A ) S 0 ( q / A ) I 0 ( A ) .
I ( q ) = I ( 0 q ) + I ( 0 q ) V ( C / q ) I ( 0 q )
I ( A ) = U + U σ ( A ) U
I ( q + A ) = I ( 0 q ) + I ( 0 q ) σ ( A / q ) I ( 0 q ) ,
σ ( A / q ) = [ 1 σ ( A ) U S ( 0 q ) U ] 1 σ ( A ) ,
σ C ( A ) = V ( C ) + [ 1 + V ( C ) U ] S C ( A ) [ U V ( C ) + 1 ]
= S 0 ( A ) + [ 1 + S 0 ( A ) U ] V ( C / A ) [ U S 0 ( A ) + 1 ] ,
S C ( A ) = [ 1 S 0 ( A ) U V ( C ) U ] 1 S 0 ( A )
σ ( A + B / q ) = σ ( A / q ) + [ 1 + σ ( A / q ) I ( 0 q ) ] S ( B / q + A ) [ I ( 0 q ) σ ( A / q ) + 1 ] ,
S ( B / q + A ) = [ 1 S 0 ( B / q ) I ( 0 q ) σ ( A / q ) I ( 0 q ) ] 1 S 0 ( B / q ) .
σ ( A + B ) = σ ( A ) + [ 1 + σ ( A ) U ] S ( B / A ) [ U σ ( A ) + 1 ]
= S 0 ( B ) + [ 1 + S 0 ( B ) U ] σ ( A / B ) [ U S 0 ( B ) + 1 ] ,
S ( B / A ) = [ 1 S 0 ( B ) U σ ( A ) U ] 1 S 0 ( B )
I C 1 + C 2 ( q + A ) = I C 1 ( q + A ) + I 0 ( q + A ) Y ¯ ( C 1 / q + A ) V ( C 2 / q + A / C 1 ) Y ( C 1 / q + A ) I 0 ( q + A ) .
σ C 1 + C 2 ( A / q ) = σ C 1 ( A / q ) + [ 1 + S 0 ( A / q ) I ( 0 q ) ] Y ¯ ( C 1 / q + A ) × V ( C 2 / q + A / C 1 ) Y ( C 1 / q + A ) [ I ( 0 q ) S 0 ( A / q ) + 1 ] .
Y ¯ ( C 1 / q + A ) = 1 + V ( C 1 / q + A ) U S 0 ( q + A ) U ,
Y ( C 1 / q + A ) = 1 + U S 0 ( q + A ) U V ( C 1 / q + A ) ,
V ( C / q + A ) = [ 1 V ( C / q ) I ( 0 q ) S 0 ( A / q ) I ( 0 q ) ] 1 V ( C / q )
= [ 1 V ( C / q ) I 0 ( A ) S 0 ( q / A ) I 0 ( A ) ] 1 V ( C / A ) ,
σ C 1 + C 2 ( A ) = σ C 1 ( A ) + [ 1 + S 0 ( A ) U ] Y ¯ ( C 1 / A ) × V ( C 2 / A / C 1 ) Y ( C 1 / A ) [ U S 0 ( A ) + 1 ] .
σ C 1 + C 2 ( A ) = σ C 1 ( A ) + Δ σ C 1 ( A )
V ( C / q ) = V ( C 1 / q ) + Y ¯ ( C 1 / q ) V ( C 2 / q / C 1 ) Y ( C 1 / q ) ,
I ( A + B ) = I ( A ) [ 1 + K ( B ) I ( A + B ) ] ,
S ( B / A ) = K ( B ) [ 1 + I ( A ) S ( B / A ) ] .
I ( A + B ) = I ( A ) + I ( A ) S ( B / A ) I ( A ) ,
S ( B / A ) = K ( B ) + K ( B ) I ( A + B ) K ( B ) ,
K ( B ) I ( A + B ) = S ( B / A ) I ( A ) ,
I ( A + B ) K ( B ) = I ( A ) S ( B / A ) .
K ( B ) [ 1 + I ( A ) S ( B / A ) ] = S ( B / A ) .
S ( B / A ) = [ 1 K ( B ) I ( A ) ] 1 K ( B ) ,
I ( A + B ) = I ( A ) [ 1 K ( B ) I ( A ) ] 1 = [ 1 I ( A ) K ( B ) ] 1 I ( A ) ,
[ L q ( x ̂ ) q α ( x ̂ ) ] g α ( x ̂ | x ̂ ) = δ ( x ̂ x ̂ ) ,
[ L q q α ] g α = 1 .
q g α = ( M + Δ q α ) G α ,
[ L M q α ] G α = 1 , q α = q α + Δ q α ;
T α = ( 1 q α G ) 1 q α , ( L M ) G = 1 ,
G α = G + G T α G ,
g α = G α [ ( q M Δ q α ) g α + 1 ] ,
( q M Δ q α ) q α = 0 .
q g α q G α q ,
Δ q α = q G T α G q ,
M = q G q .
Δ q α = n d β ̂ T β G T α G T β ,
M = n d β ̂ T β .
G α 1 + α 2 = G [ 1 + ( q α 1 + q α 2 ) G α ̂ 1 + α 2 ] = G α 1 [ 1 + q α 2 G α 1 + α 2 ]
= G α 1 [ 1 + T α 2 / α 1 G α 1 ] .
T α 2 / α 1 = q α 2 [ 1 + G α 1 T α 2 / α 1 ] ,
T α 2 / α 1 = T α 2 [ 1 + G T α 1 G T α 2 / α 1 ] .
T α 2 / α 1 = [ 1 T α 2 G T α 1 G ] 1 T α 2 .
T α 1 / α 2 = T α 1 + ( 1 + T α 1 G ) T α 2 / α 1 ( G T α 1 + 1 ) .
U ( C ) ( 1 ; 2 ) = G α * ( 1 ) G α ( 2 )
U ( C ) ( 1 ; 2 ) = U ( 1 ; 2 ) + U ( 1 ; 2 ) V ( α ) ( 1 ; 2 ) U ( 1 ; 2 ) .
V ( α ) ( 1 ; 2 ) = T α * ( 1 ) T α ( 2 ) + T α * ( 1 ) G ( 2 ) 1 + T α ( 2 ) G * ( 1 ) 1 ,
V ( α ) ( Ω ̂ | Ω ̂ ) = σ ( α ) ( Ω ̂ | Ω ̂ ) γ ( α ) ( Ω ̂ ) δ 2 ( Ω ̂ Ω ̂ ) ,
d Ω ̂ V ( α ) ( Ω ̂ | Ω ̂ ) = 0 .
σ ( α ) ( Ω ̂ | Ω ̂ ) = ( 4 π ) 2 | T α ( k Ω ̂ | Ω ̂ k ) | 2 ,
γ ( α ) ( Ω ̂ ) = ( 2 i k ) 1 [ T α * ( k Ω ̂ | k Ω ̂ ) T α ( k Ω ̂ | k Ω ̂ ) ] .
V ( α / q ) j = 1 N V ( α j / q ) .
I α ( q ) = U ( C ) + U ( C ) S α ( q ) U ( C ) ,
S α ( q ) = S ( 0 q ) + S ( 0 q ) U V ( α / q ) U S ( 0 q ) ,

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