Abstract

We present a retrieval scheme that can be used to derive the aerosol phase function and single-scattering albedo from the sky radiance over land. The retrieval algorithm iteratively corrects the aerosol volume scattering function, the product of the single-scattering albedo and the phase function, based on the difference between the measured sky radiance and the radiance calculated by solving the radiative transfer equation. It is tested first under ideal conditions, i.e., the approximations made in the retrieval algorithm totally agree with actual conditions assumed in creating the pseudodata for sky radiance. It is then tested under more realistic conditions to assess its susceptibility to measurement errors and effects of conditions not recognized in the retrieval algorithm, e.g., surface horizontal inhomogeneity, departures of the surface from Lambertian, and aerosol horizontal inhomogeneity. These simulations show that, in most cases, this scheme can retrieve the aerosol single-scattering albedo with high accuracy (within 1%) and can therefore be used to identify strongly absorbing aerosols. It can also produce meaningful retrievals of most aerosol phase functions: less than 5% error at 865 nm and less than 10% at 443 nm in most cases. Typically, the error in the volume scattering function is small for scattering angles ≲90°, then increases for larger angles. Disappointing results in both the single-scattering albedo and the scattering phase function occur at 443 nm, either when there are large calibration errors in the radiometer used to measure the sky radiance or when the land reflection properties are significantly inhomogeneous.

© 1998 Optical Society of America

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1997 (3)

1996 (3)

1995 (2)

1994 (7)

H. R. Gordon, M. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33, 443–452 (1994).
[CrossRef] [PubMed]

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sensing 32, 598–615 (1994).
[CrossRef]

M. Wang, H. R. Gordon, “Estimating aerosol optical properties over the oceans with the multiangle imaging spectroradiometer: some preliminary studies,” Appl. Opt. 33, 4042–4057 (1994).
[CrossRef] [PubMed]

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Eraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase functions of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99D, 10341–10356 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheriods with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

H. R. Gordon, M. Wang, “Influence of oceanic whitecaps on atmospheric correction of SeaWiFS,” Appl. Opt. 33, 7754–7763 (1994).
[CrossRef] [PubMed]

S. F. Biggar, P. N. Slater, D. I. Gellman, “Uncertainties in the in-flight calibration of sensors with reference to measured ground sites in the 0.4 to 1.1 μm range,” Remote Sensing Environ. 48, 245–252 (1994).
[CrossRef]

1993 (1)

1992 (2)

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

N. J. McCormick, “Inverse radiative transfer problems: a review,” Nucl. Sci. Eng. 112, 185–198 (1992).

1991 (1)

M. Wendisch, W. von Hoyningen-Huene, “High speed version of the method of ’successive order of scattering‘ and its application to remote sensing,” Beitr. Phys. Atmos. 64, 83–91 (1991).

1990 (1)

W. A. Hoppel, J. W. Fitzgerald, G. M. Frick, R. E. Larson, E. J. Mack, “Aerosol size distributions and optical properties found in the marine boundary layer over the Atlantic Ocean,” J. Geophys. Res. 95D, 3659–3686 (1990).
[CrossRef]

1989 (5)

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sensing 27, 145–152 (1989).
[CrossRef]

D. J. Diner, C. J. Bruegge, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, P. J. Sellers, J. Clark, J. A. Daniels, E. D. Danielson, V. G. Duval, K. P. Klaasen, G. W. L. A. D. I. Nakamoto, R. Pagano, T. H. Reilly, “MISR: a multi-angle imaging spectroradiometer for geophysical and climatological research from EOS,” IEEE Trans. Geosci. Remote Sensing 27, 200–214 (1989).
[CrossRef]

H. R. Gordon, D. J. Castaño, “Aerosol analysis with the coastal zone color scanner: a simple method for including multiple scattering effects,” Appl. Opt. 28, 1320–1326 (1989).
[CrossRef] [PubMed]

Y. Sasano, E. V. Browell, “Light scattering characteristics of various aerosol types derived from multiple wavelength lidar observations,” Appl. Opt. 28, 1670–1679 (1989).
[CrossRef] [PubMed]

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘Fisheye’ camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

1987 (1)

R. J. Charlson, J. E. Lovelock, M. O. Andreae, S. G. Warren, “Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate,” Nature (London) 326, 655–661 (1987).
[CrossRef]

1986 (1)

D. S. Kimes, W. W. Newcomb, R. F. Nelson, J. B. Schutt, “Directional reflectance factor distributions of a hardwood and pine forest canopy,” IEEE Trans. Geosci. Remote Sensing 24, 281–293 (1986).
[CrossRef]

1985 (2)

D. S. Kimes, W. W. Newcomb, C. J. Tucker, “Directional reflectance factor distributions for cover types of Northern Africa,” Remote Sensing Environ. 18, 1–19 (1985).
[CrossRef]

K. P. Bowman, A. J. Krueger, “A global climatology of total ozone from the nimbus 7 total ozone mapping spectrometer,” J. Geophys. Res. 90D, 7967–7976 (1985).
[CrossRef]

1984 (1)

R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

1983 (4)

1982 (1)

1979 (1)

M. D. King, B. M. Herman, “Determination of the ground albedo and the index of absorption of atmospheric particulates by remote sensing. Part I: Theory,” J. Atmos. Sci. 36, 163–173 (1979).
[CrossRef]

1976 (1)

M. D. King, D. M. Byrne, “A method for inferring total ozone content from the spectral variation of optical depth obtained with a solar radiometer,” J. Atmos. Sci. 35, 2242–2251 (1976).
[CrossRef]

1974 (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1954 (1)

Abreu, L. W.

F. X. Kenizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric Transmittance/Radiance: The LOWTRAN 6 Model,” AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscomb Air Force Base, Mass., 1983).

Ackerman, T. P.

D. J. Diner, C. J. Bruegge, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, P. J. Sellers, J. Clark, J. A. Daniels, E. D. Danielson, V. G. Duval, K. P. Klaasen, G. W. L. A. D. I. Nakamoto, R. Pagano, T. H. Reilly, “MISR: a multi-angle imaging spectroradiometer for geophysical and climatological research from EOS,” IEEE Trans. Geosci. Remote Sensing 27, 200–214 (1989).
[CrossRef]

Andreae, M. O.

R. J. Charlson, J. E. Lovelock, M. O. Andreae, S. G. Warren, “Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate,” Nature (London) 326, 655–661 (1987).
[CrossRef]

Barnes, W. L.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sensing 27, 145–152 (1989).
[CrossRef]

Biggar, S. F.

P. N. Slater, S. F. Biggar, K. J. Thome, D. I. Gellman, P. R. Spyak, “Vicarious radiometric calibration of EOS sensors,” J. Atmos. Oceanic Technol. 13, 349–359 (1996).
[CrossRef]

S. F. Biggar, P. N. Slater, D. I. Gellman, “Uncertainties in the in-flight calibration of sensors with reference to measured ground sites in the 0.4 to 1.1 μm range,” Remote Sensing Environ. 48, 245–252 (1994).
[CrossRef]

Boi, P.

Bowman, K. P.

K. P. Bowman, A. J. Krueger, “A global climatology of total ozone from the nimbus 7 total ozone mapping spectrometer,” J. Geophys. Res. 90D, 7967–7976 (1985).
[CrossRef]

Bréon, F. M.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sensing 32, 598–615 (1994).
[CrossRef]

Bricaud, A.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sensing 32, 598–615 (1994).
[CrossRef]

Browell, E. V.

Bruegge, C. J.

D. J. Diner, C. J. Bruegge, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, P. J. Sellers, J. Clark, J. A. Daniels, E. D. Danielson, V. G. Duval, K. P. Klaasen, G. W. L. A. D. I. Nakamoto, R. Pagano, T. H. Reilly, “MISR: a multi-angle imaging spectroradiometer for geophysical and climatological research from EOS,” IEEE Trans. Geosci. Remote Sensing 27, 200–214 (1989).
[CrossRef]

Buis, J. P.

B. N. Holben, T. F. Eck, I. Slutsker, D. Tanre, J. P. Buis, A. Setzer, E. Vermote, J. A. Reagan, Y. J. Kaufman, T. Nakajima, F. Lavenu, “Multi-band automatic Sun and sky scanning radiometer system for measurement of aerosols,” presented at the Sixth International Symposium on Physical Measurements and Signatures in Remote Sensing, Val-d’Isère, France, 17–21 January 1994.

Buriez, J. C.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sensing 32, 598–615 (1994).
[CrossRef]

Byrne, D. M.

M. D. King, D. M. Byrne, “A method for inferring total ozone content from the spectral variation of optical depth obtained with a solar radiometer,” J. Atmos. Sci. 35, 2242–2251 (1976).
[CrossRef]

Castaño, D. J.

Cess, R. D.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Charlson, R. J.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

R. J. Charlson, J. E. Lovelock, M. O. Andreae, S. G. Warren, “Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate,” Nature (London) 326, 655–661 (1987).
[CrossRef]

Chetwynd, J. H.

F. X. Kenizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric Transmittance/Radiance: The LOWTRAN 6 Model,” AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscomb Air Force Base, Mass., 1983).

Clark, J.

D. J. Diner, C. J. Bruegge, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, P. J. Sellers, J. Clark, J. A. Daniels, E. D. Danielson, V. G. Duval, K. P. Klaasen, G. W. L. A. D. I. Nakamoto, R. Pagano, T. H. Reilly, “MISR: a multi-angle imaging spectroradiometer for geophysical and climatological research from EOS,” IEEE Trans. Geosci. Remote Sensing 27, 200–214 (1989).
[CrossRef]

Clough, S. A.

F. X. Kenizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric Transmittance/Radiance: The LOWTRAN 6 Model,” AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscomb Air Force Base, Mass., 1983).

Coakley, J. A.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Cox, C.

d’Almeida, G. A.

G. A. d’Almeida, P. Koepke, E. P. Shettle , Atmospheric Aerosols—Global Climatology and Radiative Characteristics (A. Deepak, Hampton, Va., 1991).

Daniels, J. A.

D. J. Diner, C. J. Bruegge, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, P. J. Sellers, J. Clark, J. A. Daniels, E. D. Danielson, V. G. Duval, K. P. Klaasen, G. W. L. A. D. I. Nakamoto, R. Pagano, T. H. Reilly, “MISR: a multi-angle imaging spectroradiometer for geophysical and climatological research from EOS,” IEEE Trans. Geosci. Remote Sensing 27, 200–214 (1989).
[CrossRef]

Danielson, E. D.

D. J. Diner, C. J. Bruegge, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, P. J. Sellers, J. Clark, J. A. Daniels, E. D. Danielson, V. G. Duval, K. P. Klaasen, G. W. L. A. D. I. Nakamoto, R. Pagano, T. H. Reilly, “MISR: a multi-angle imaging spectroradiometer for geophysical and climatological research from EOS,” IEEE Trans. Geosci. Remote Sensing 27, 200–214 (1989).
[CrossRef]

Davies, R.

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[CrossRef]

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[CrossRef]

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P. N. Slater, S. F. Biggar, K. J. Thome, D. I. Gellman, P. R. Spyak, “Vicarious radiometric calibration of EOS sensors,” J. Atmos. Oceanic Technol. 13, 349–359 (1996).
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P. N. Slater, S. F. Biggar, K. J. Thome, D. I. Gellman, P. R. Spyak, “Vicarious radiometric calibration of EOS sensors,” J. Atmos. Oceanic Technol. 13, 349–359 (1996).
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Travis, L. D.

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Figures (11)

Fig. 1
Fig. 1

Phase functions and single-scattering albedos for the aerosol models used in this study: (a) λ = 865 nm and (b) λ = 443 nm.

Fig. 2
Fig. 2

Comparison between the true (curves) and retrieved (open circles) V a (Θ) for the C80 aerosol model: (a) λ = 865 nm and (b) λ = 443 nm.

Fig. 3
Fig. 3

Retrieval errors as a function of τ a under ideal conditions: (a) and (b) error in the aerosol volume scattering function at λ = 865 nm and λ = 443 nm, respectively; (c) and (d) error in the aerosol single-scattering albedo at λ = 865 nm and λ = 443 nm, respectively.

Fig. 4
Fig. 4

BRDF for two surfaces: (a) and (b) stepped grass surface in bands 580–680 and 730–1100 nm, respectively, at θ0 = 63°; (c) and (d) irrigated wheat surface in bands 580–680 and 730–1100 nm, respectively, at θ0 = 59°.

Fig. 5
Fig. 5

Retrieval errors for non-Lambertian surfaces: (a) and (b) stepped grass surface at 865 and 443 nm, respectively; (c) and (d) irrigated wheat surface at 865 and 443 nm, respectively.

Fig. 6
Fig. 6

Fig. 6. | Δ V a / V a | ¯ and δω0 for the checkerboard model.

Fig. 7
Fig. 7

Errors in V a and ω0: (a) and (b) error in V a at 865 and 443 nm, respectively; (c) and (d) error in ω0 at 865 and 443 nm, respectively.

Fig. 8
Fig. 8

Effect of vertical structure: (a) 865 nm for U00 with pseudodata created with the one-layer model; (b) 443 nm for U00 with pseudodata created with the one-layer model; (c) 865 nm for U00 with pseudodata created with the modified two-layer model; (d) 443 nm for U00 with pseudodata created with the modified two-layer model. In each case the original two-layer model is used for retrieval.

Fig. 9
Fig. 9

Retrieved errors when there is a +5% systematic error in the sky radiance: (a) λ = 865 nm and (b) λ = 443 nm.

Fig. 10
Fig. 10

Retrieval errors when there is a 1% random error in the sky radiance for (a) λ = 865 nm and (b) λ = 443 nm.

Fig. 11
Fig. 11

Retrieval errors when there is a -10% error in the land albedo for (a) λ = 865 nm and (b) λ = 443 nm.

Tables (2)

Tables Icon

Table 1 Albedos of Different types of Land Surface

Tables Icon

Table 2 Summary of the Error in ω0 at 865 nm

Equations (45)

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

V ( ξ ˆ 0 ξ ˆ ) = ω 0 4 π   P ( ξ ˆ 0 ξ ˆ ) ,
L t = L r exp - τ a / μ + L a exp ( - τ r / μ 0 ) + L ra ,
Δ L t ( ξ ˆ i ) L t ( c ) ( ξ ˆ i ) - L t ( m ) ( ξ ˆ i ) ,
Δ L t ( ξ ˆ i ) = Δ L a ( ξ ˆ i ) exp ( - τ r / μ 0 ) + Δ L ra ( ξ ˆ i ) .
L t c [ ξ ˆ i ;   V Θ + Δ V Θ ] - L t ( m ) ( ξ ˆ i ) = L t c [ ξ ˆ i ;   V Θ ] - L t m ( ξ ˆ i ) + j δ L t c [ ξ ˆ i ;   V Θ ] δ V ( Θ j )   Δ V ( Θ j ) .
Δ L t ( ξ ˆ i ) = L a ( ξ ˆ i ) exp ( - τ r / μ 0 ) V ( Θ i )   Δ V ( Θ i ) ,
V a Θ i new c = V a Θ i old c + C Δ V a Θ i c ,
ω 0 = 4 π   V a Θ d Ω ,
P a Θ = 4 π   V a Θ ω 0 .
Δ ω 0 = ω 0 c - ω 0 t ω 0 t ,
Δ V a / V a = 1 N i = 1 N V a Θ i c - V a Θ i t V a Θ i t ,
L t = L 10 + L 20 ,
L 10 d ( ξ ˆ ) = F 0 τ | ξ ˆ · n ˆ |   V a ( ξ ˆ 0 ξ ˆ ) .
L 10 u ( ξ ˆ ) = F 0 τ a - τ | ξ ˆ · n ˆ |   V a ( ξ ˆ 0 ξ ˆ ) ,
L 20 ( ξ ˆ ) = F 0 τ a 2 2 4 π V a ( ξ ˆ 0     ξ ˆ ) V a ( ξ ˆ     ξ ˆ ) | ξ ˆ · n ˆ | | ξ ˆ · n ˆ | d Ω ( ξ ˆ ) .
V a ( ξ ˆ     ξ ˆ ) = V a ( Θ ) - k δ ( ξ ˆ - ξ ˆ ) ,
Δ L t ( ξ ˆ ) = kF 0 τ a 2 V a ( ξ ˆ 0 ξ ˆ ) | ξ ˆ 0 · n ˆ | | ξ ˆ · n ˆ | .
Δ L t ( ξ ˆ ) = k   τ a L 10 ( ξ ˆ ) | ξ ˆ 0 · n ˆ | .
Δ L t ( ξ ˆ ) = k τ a | ξ ˆ 0 · n ˆ | [ L a 10 ( ξ ˆ ) + L r 10 ( ξ ˆ ) ] ,
Δ V a ( Θ ) V a ( Θ ) = Δ L t L a exp ( - τ r / ξ ˆ 0 · n ˆ ) ,
Δ V a Θ V a Θ = k   τ a μ 0 L a 10 + L r 10 L a exp - τ r / μ 0 ,
Δ V a Θ V a Θ     1 a + b τ a + ,
R ( ξ ˆ 0 ξ ˆ ) = π L ref ( ξ ˆ ) E 0 ( ξ ˆ 0 ) | ξ ˆ 0 · n ˆ | ,
R ( ξ ˆ 0 ξ ˆ ) = ω l ,
R L ( ξ ˆ ,   ξ ˆ 0 ) π L ref E total ( ξ ˆ 0 ) ,
ω l ( ξ ˆ 0 ) = 1 π ξ ˆ · n ˆ > 0   R ( ξ ˆ 0 ξ ˆ ) | ξ ˆ · n ˆ | d Ω ( ξ ˆ ) ,
L 1 s ( ξ ˆ ) = F 0 | ξ ˆ 0 · n ˆ | π z 0 z 1 - + - +   c a ω l ( x ,   y ) × V a ( ξ ˆ ξ ˆ ) | ξ ˆ · n ˆ | 2 | ξ ˆ · n ˆ | r 2 ( ξ ˆ ,   z ;   x ,   y ) d x d y d z ,
L 10 2 - lyr ( ξ ˆ ) = F 0 μ 0 exp ( - τ a / μ ) × exp ( - τ r / μ 0 ) - exp ( - τ r / μ ) μ - μ 0 V r ( Θ ) ,
L 10 1 - lyr ( ξ ˆ ) = F 0 μ 0 × exp [ - τ r + τ a / μ 0 ] - exp - τ r + τ a / μ μ - μ 0 × τ r V r Θ τ a + τ r .
L 10 2 - lyr ( ξ ˆ ) = L 10 1 - lyr ( ξ ˆ ) = F 0 τ r exp [ - τ a + τ r / μ 0 ] μ 0 V r Θ .
L 2 - lyr ( ξ ˆ ) L 1 - lyr ( ξ ˆ ) = τ r + τ a τ r × [ exp ( - τ r / μ 0 ) - exp - τ r / μ ] exp - τ a / μ exp [ - τ r + τ a / μ 0 ] - exp - τ r + τ a / μ .
L 2 - lyr ( ξ ˆ ) L 1 - lyr ( ξ ˆ ) = τ r + τ a τ r exp ( - τ a / μ ) exp ( - τ a / μ 0 ) .
L ( m ) ( ξ ˆ i ) = L ( t ) ( ξ ˆ i ) + ρ 0 L ( t ) ( ξ ˆ i ) ,
L ( m ) ( ξ ˆ i ) = L ( t ) ( ξ ˆ i ) + ρ i L ( t ) ( ξ ˆ i ) ,
P ( ρ i ) = 1 2 π σ 2 exp ( - ρ i 2 / 2 σ 2 ) ,
L ( t ) ( ξ ˆ ) = τ a | ξ ˆ · n ˆ |   V a ( ξ ˆ 0 ξ ˆ ) + τ r | ξ ˆ · n ˆ |   V r ( ξ ˆ 0 ξ ˆ ) ,
V a ( ξ ˆ 0 ξ ˆ ) = L ( t ) ( ξ ˆ ) | ξ ˆ · n ˆ | - τ r V r ( ξ ˆ 0 ξ ˆ ) τ a .
Δ V a ( ξ ˆ 0 ξ ˆ ) V a ( ξ ˆ 0 ξ ˆ ) = ρ 0 τ a V a ( ξ ˆ 0 ξ ˆ ) + τ r V r ( ξ ˆ 0 ξ ˆ ) τ a V a ( ξ ˆ 0 ξ ˆ ) .
L t ( ξ ˆ ) = L 10 ( ξ ˆ ) + L 1 s ( ξ ˆ ) + L 20 ( ξ ˆ ) ,
L 10 ( ξ ˆ ) = F 0 τ a | ξ ˆ · n ˆ |   V a ( ξ ˆ 0 ξ ˆ ) ,
L 1 s ( ξ ˆ ) = F 0 τ a ω l | ξ ˆ 0 · n ˆ | | ξ ˆ · n ˆ | ξ ˆ · n ˆ > 0   V a ( ξ ˆ ξ ˆ ) d Ω ( ξ ˆ ) ,
L 20 ( ξ ˆ ) = F 0 τ a 2 2 4 π V a ( ξ ˆ 0 ξ ˆ ) V a ( ξ ˆ ξ ˆ ) | ξ ˆ · n ˆ | | ξ ˆ · n ˆ | d Ω ( ξ ˆ ) .
Δ L 10 ( ξ ˆ i ) = δ i L 10 ( ξ ˆ i ) .
Δ L t ( ξ ˆ i ) = δ i L s 0 .
δ i = L t ( ξ ˆ i ) L 10 ( ξ ˆ i ) Δ L t ( ξ ˆ i ) L t ( ξ ˆ i ) ,

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