Abstract

The most recent theoretical studies have shown that three-dimensional (3-D) radiation effects play an important role in the optical remote sensing of atmospheric aerosol and land surface reflectance. These effects may contribute notably to the error budget of retrievals in a broad range of sensor resolutions, introducing systematic biases in the land surface albedo data sets that emerge from the existing global observation systems. At the same time, 3-D effects are either inadequately addressed or completely ignored in data processing algorithms. Thus there is a need for further development of the radiative transfer theory that can rigorously treat both 3-D and surface anisotropy effects and yet be flexible enough to permit the development of fast forward and inversion algorithms. We describe a new theoretical solution to the 3-D radiative transfer problem with an arbitrary nonhomogeneous non-Lambertian surface. This solution is based on an exact semianalytical solution derived in operator form by the Green’s function method. The numerical implementation is based on several parameterizations that accelerate the solution dramatically while keeping its accuracy within several percent under most general conditions.

© 2002 Optical Society of America

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References

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  1. A. I. Lyapustin, “3-D effects in the remote sensing of surface albedo,” IEEE Trans. Geosci. Remote Sens. 39, 254–263 (2001).
    [CrossRef]
  2. A. I. Lyapustin, Y. J. Kaufman, “The role of adjacency effect in the remote sensing of aerosol over land,” J. Geophys. Res. 106, 11,909–11,916 (2001).
    [CrossRef]
  3. Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
    [CrossRef]
  4. D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97–125 (1984).
    [CrossRef]
  5. D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational considerations and results,” J. Quant. Spectrosc. Radiat. Transfer 32, 279–304 (1984).
    [CrossRef]
  6. W. A. Pearce, “A study of the effect of atmosphere on the Thematic Mapper observations,” Rep. 004-77 (EG&G/Washington Analytical Services Center, Inc., Riverdale, Md., 1977).
  7. G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
    [CrossRef]
  8. K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
    [CrossRef]
  9. T. A. Germogenova, The Local Properties of the Solution of the Transport Equation (Nauka, Moscow, 1986; in Russian).
  10. T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, 1990; in Russian).
  11. A. I. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt. 40, 3495–3501 (2001).
    [CrossRef]
  12. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  13. S. Chandrasekhar, “On the diffuse reflection of a pencil of radiation by a plane-parallel atmosphere,” Proc. Natl. Acad. Sci. USA 44, 933–940 (1958).
    [CrossRef] [PubMed]
  14. A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
    [CrossRef]
  15. A. I. Lyapustin, “Radiative transfer code SHARM-3D for radiance simulations over non-Lambertian nonhomogeneous surface: intercomparison study,” Appl. Opt. 41, 5607–5615 (2002).
    [CrossRef] [PubMed]
  16. T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
    [CrossRef]

2002

2001

A. I. Lyapustin, “3-D effects in the remote sensing of surface albedo,” IEEE Trans. Geosci. Remote Sens. 39, 254–263 (2001).
[CrossRef]

A. I. Lyapustin, Y. J. Kaufman, “The role of adjacency effect in the remote sensing of aerosol over land,” J. Geophys. Res. 106, 11,909–11,916 (2001).
[CrossRef]

A. I. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt. 40, 3495–3501 (2001).
[CrossRef]

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

1999

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[CrossRef]

1998

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

1997

Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
[CrossRef]

1984

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97–125 (1984).
[CrossRef]

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational considerations and results,” J. Quant. Spectrosc. Radiat. Transfer 32, 279–304 (1984).
[CrossRef]

1958

S. Chandrasekhar, “On the diffuse reflection of a pencil of radiation by a plane-parallel atmosphere,” Proc. Natl. Acad. Sci. USA 44, 933–940 (1958).
[CrossRef] [PubMed]

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chandrasekhar, S.

S. Chandrasekhar, “On the diffuse reflection of a pencil of radiation by a plane-parallel atmosphere,” Proc. Natl. Acad. Sci. USA 44, 933–940 (1958).
[CrossRef] [PubMed]

Chu, A.

Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
[CrossRef]

Darbinjan, R.

G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
[CrossRef]

Diner, D. J.

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97–125 (1984).
[CrossRef]

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational considerations and results,” J. Quant. Spectrosc. Radiat. Transfer 32, 279–304 (1984).
[CrossRef]

Elepov, B.

G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
[CrossRef]

Evans, K. F.

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

Germogenova, T. A.

T. A. Germogenova, The Local Properties of the Solution of the Transport Equation (Nauka, Moscow, 1986; in Russian).

Holben, B. N.

Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
[CrossRef]

Ioltuhovskii, A. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, 1990; in Russian).

Kargin, B.

G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
[CrossRef]

Kaufman, Y. J.

A. I. Lyapustin, Y. J. Kaufman, “The role of adjacency effect in the remote sensing of aerosol over land,” J. Geophys. Res. 106, 11,909–11,916 (2001).
[CrossRef]

Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
[CrossRef]

Knyazikhin, Yu.

Lyapustin, A. I.

A. I. Lyapustin, “Radiative transfer code SHARM-3D for radiance simulations over non-Lambertian nonhomogeneous surface: intercomparison study,” Appl. Opt. 41, 5607–5615 (2002).
[CrossRef] [PubMed]

A. I. Lyapustin, Y. J. Kaufman, “The role of adjacency effect in the remote sensing of aerosol over land,” J. Geophys. Res. 106, 11,909–11,916 (2001).
[CrossRef]

A. I. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt. 40, 3495–3501 (2001).
[CrossRef]

A. I. Lyapustin, “3-D effects in the remote sensing of surface albedo,” IEEE Trans. Geosci. Remote Sens. 39, 254–263 (2001).
[CrossRef]

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[CrossRef]

Marchuk, G.

G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
[CrossRef]

Martonchik, J. V.

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational considerations and results,” J. Quant. Spectrosc. Radiat. Transfer 32, 279–304 (1984).
[CrossRef]

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97–125 (1984).
[CrossRef]

Mikhailov, G.

G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
[CrossRef]

Muldashev, T. Z.

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[CrossRef]

Nazaraliev, N.

G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
[CrossRef]

Pearce, W. A.

W. A. Pearce, “A study of the effect of atmosphere on the Thematic Mapper observations,” Rep. 004-77 (EG&G/Washington Analytical Services Center, Inc., Riverdale, Md., 1977).

Remer, L. A.

Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
[CrossRef]

Strelkov, S. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, 1990; in Russian).

Sultangazin, U. M.

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[CrossRef]

Sushkevich, T. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, 1990; in Russian).

Tanre, D.

Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
[CrossRef]

Vermote, E. F.

Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

A. I. Lyapustin, “3-D effects in the remote sensing of surface albedo,” IEEE Trans. Geosci. Remote Sens. 39, 254–263 (2001).
[CrossRef]

J. Atmos. Sci.

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

J. Geophys. Res.

A. I. Lyapustin, Y. J. Kaufman, “The role of adjacency effect in the remote sensing of aerosol over land,” J. Geophys. Res. 106, 11,909–11,916 (2001).
[CrossRef]

Y. J. Kaufman, D. Tanre, L. A. Remer, E. F. Vermote, A. Chu, B. N. Holben, “Operational remote sensing of tropospheric aerosol over land from EOS moderate resolution imaging spectroradiometer,” J. Geophys. Res. 102, 17,051–17,068 (1997).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97–125 (1984).
[CrossRef]

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational considerations and results,” J. Quant. Spectrosc. Radiat. Transfer 32, 279–304 (1984).
[CrossRef]

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[CrossRef]

Proc. Natl. Acad. Sci. USA

S. Chandrasekhar, “On the diffuse reflection of a pencil of radiation by a plane-parallel atmosphere,” Proc. Natl. Acad. Sci. USA 44, 933–940 (1958).
[CrossRef] [PubMed]

Other

W. A. Pearce, “A study of the effect of atmosphere on the Thematic Mapper observations,” Rep. 004-77 (EG&G/Washington Analytical Services Center, Inc., Riverdale, Md., 1977).

G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
[CrossRef]

T. A. Germogenova, The Local Properties of the Solution of the Transport Equation (Nauka, Moscow, 1986; in Russian).

T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, 1990; in Russian).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

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Equations (50)

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ν+α zLz; r; s=σz4πΩχz, γLz; r; sds+Sλ4σzχz; γ0exp-τz/μ0,
L0; r; s=0, μ>0,
LH; r; s=Sλμ0 exp-τ0/μ0ρr; s0, s+1πΩ+LH; r; sρr; s, s×μds, μ<0.
Lz; r; s0, s=Dz; s0, s+Jz; r; s.
Lˆ3Jz; r; s=SˆJz; r; s,
J+0; r=0,J-H; r=RrI+0H+RrJ+H; r.
I+0H=πSλ exp-τ0/μ0δs-s0+DH; s0, s, μ>0.
Jz; r; s=k1 Jkz; r; s.
Lˆ3Jkz; r; s=SˆJkz; r; s,
J+k0; r=0;J-kH; r=RrJ+k-1H; r,
Jkz; r; s=-+dr Ω- G3z; r-r; s1, sJk×H; r; s1ds1.
Lˆ3G3=SˆG3,
G+30; r-r=0, G-3H; r-r=δr-rδs-s1.
Γˆz;r3s=-+dr Ω-ds1G3z; r-r; s1, s.
J+kH; r=ΓˆH;r3+J-kH; r.
J-k+1H; r=RrJ+kH; r=RrΓˆH;r3+J-kH; r.
J-H; r=k1 J-kH; r=k0RrΓˆH;r3+kRrI+0H=Iˆ-RrΓˆH;r3+-1RrI+0H,
Jk+1H; r; s=1πΩ+dsμρr; s, s×-+dr Ω- G3H; r-r; s1, sJkH; r; s1ds1.
J-z; r-rs=Γˆz;r-rs3-Iˆ-RrΓˆH,r3+-1RrI+0H,
J˜NLz; r; s/J˜z; r; s qmaxc01-q¯c0,
J-H; r=k0R¯+R˜rΓˆH;r3+kR¯+R˜rI+0HJ¯-H+k0R¯ΓˆH;r3+kR˜rI+0H+k0l=0kR¯ΓˆH;r3+k-lR˜rΓˆH;r3+R¯ΓˆH;r3+lR¯I+0H.
k0l=0kR¯ΓˆH;r3+k-lR˜rΓˆH1+R¯lΓˆH1+R¯I+0H.
G1z; s1, s=-+ G3z; r-r; s1, sdr,
J˜-H; r Iˆ-R¯ΓˆH;r3+-1R˜rIˆ-ΓˆH1+R¯-1I+0H.
J˜-z; p=F J˜-z; r=Γˆz;p3-Iˆ-R¯ΓˆH;p3+-1R˜p×Iˆ-ΓˆH1+R¯-1I+0H.
Γˆz;p3-=Ω-ds1G3z; p; s1, s, μ<0.
Lˆ3pG3p=SˆG3p,
G+3p0=0;G-3pH=δs-s1,
G3z; p; s1, s=expiprsG3dz; p; s1, sμ>0exp-τ0-τz/|μ1|δs-s1+G3dz; p; s1, sμ<0.
Lz; r; s=Dz; s0, s+J¯z; s0, s+F-1J˜z; p; s.
R˜pIˆ-ΓˆH1+R¯-1I+0H=R˜pk0ΓˆH1+R¯kI+0H=q˜pk0c0q¯kE0μ0=q˜pαE0μ0,
R¯ΓˆH;p3+R˜pIˆ-ΓˆH1+R¯-1I+0H=q¯πΩ+Ω- μG3dH; p; s1, sds1ds×q˜pαE0μ0=q¯cpq˜pαE0μ0,
J˜H; r; s=k0R¯ΓˆH;r3+kR˜pIˆ-ΓˆH1+R¯-1I+0H=k0q¯cpk×q˜pαE0μ0=q˜mpαE0μ0,
Γˆz;p3-I-R¯ΓˆH;p3+-1R˜pIˆ-ΓˆH1+R¯-1I+0H=q˜mp×αE0μ0Ω- G3z; p; s1, sds1=q˜mpαE0μ0Ψz; p; s.
Ψz; p; s=Ω- G3z; p; s1, sds1=expiprsΩ- G3dz; p; s1, sds1μ>0exp-τ0-τz/|μ|+Ω- G3dz; p; s1, sds1μ<0
Lˆ3pΨp=SˆΨp,
Ψp+0=0, Ψp-H=1.
J˜z; r; s=αE0μ0q˜mr-rsexp-τ0-τz/|μ|+12π2-+ q˜mpAz; p; sexp-ipr-rs-Φz; p; sdp,
R˜pIˆ-ΓˆH1+R¯-1I+0H=R˜pI+0H+R˜pk1ΓˆH1+R¯kI+0H.
R˜pk0ΓˆH1+R¯kΓˆH1+R¯I+0HR˜pΓˆH1+R¯Iδ++D+1-η¯,
R˜pΓˆH1+R¯Iδ+Sλμ0 exp-τ0/μ0c0ρ˜1p; sρ¯2s0,R˜pΓˆH1+R¯D+η¯R˜pD+,
ρ1s=12πΩ+ ρs, sds,ρ2s0=12πΩ- ρs0, sds.
R˜pIˆ-ΓˆH1+R¯-1I+0HSλμ0 exp-τ0/μ0ρ˜p; s0, s+αc0ρ˜1p; sρ¯2s0+απΩ+ DH; s0, s×ρ˜ p; s, sμds,
α=1-q¯θ0c0-1.
J˜0; r; sF-1expiprsJ˜H; p; sexp-τ0/|μ|+Ω- G3pd0; p; s1, sJ˜H; p; s1ds1.
qμ0; r=FH; r/F¯H; μ0,
F¯H; μ0=πE0μ0=πSλμ0 exp-τ0/μ0+Ω+ DH; s0, sμds,
FH; r=Ω- μds Sλμ0 exp-τ0/μ0ρr; s0, s+1πΩ+ μρr; s, sDH; s0, sds.
Lr-rs; s0, sD0; s0, s+exp-τ0/|μ|JH; r; s+Ω- G1d0; s1, sJ¯H, s1ds1+αE0μ02π2-+q˜θ0, pA0; p; s1-q¯θ0cp×exp-i[pr-rs-Φ0; p; sdp,
JH; r; sSλμ0 exp-τ0/μ0ρr; s0, s+αc0ρ1r; μρ¯2μ0+απΩ+ DH; s0, sρr; s, sμds.

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