Abstract

A physical algorithm is developed to solve the radiative transfer problem in the solar reflective spectral domain. This new code, Advanced Modeling of the Atmospheric Radiative Transfer for Inhomogeneous Surfaces (AMARTIS), takes into account the relief, the spatial heterogeneity, and the bidirectional reflectances of ground surfaces. The resolution method consists of first identifying the irradiance and radiance components at ground and sensor levels and then modeling these components separately, the rationale being to find the optimal trade off between accuracy and computation times. The validity of the various assumptions introduced in the AMARTIS model are checked through comparisons with a reference Monte Carlo radiative transfer code for various ground scenes: flat ground with two surface types, a linear sand dune landscape, and an extreme mountainous configuration. The results show a divergence of less than 2% between the AMARTIS code and the Monte Carlo reference code for the total signals received at satellite level. In particular, it is demonstrated that the environmental and topographic effects are properly assessed by the AMARTIS model even for situations in which the effects become dominant.

© 2000 Optical Society of America

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References

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  1. E. Vermote, D. Tanré, J. L. Deuzé, M. Herman, J. J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
    [CrossRef]
  2. A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran 7,” (Spectral Sciences, Burlington, Mass., 1989).
  3. B. N. Holben, C. O. Justice, “The topographic effect on spectral response from nadir-pointing sensors,” Photogramm. Eng. Remote Sens. 46, 1191–1200 (1980).
  4. D. S. Kimes, J. A. Kirchner, “Modeling the effects of various radiant transfers in mountainous terrain on sensor response,” IEEE Trans. Geosci. Remote Sens. GE-19, 100–108 (1981).
    [CrossRef]
  5. C. Proy, D. Tanré, P. Y. Deschamps, “Evaluation of topographic effects on remotely sensed data,” Remote Sens. Environ. 30, 21–32 (1989).
    [CrossRef]
  6. T. Kusaka, Y. Kawata, “Atmospheric and topographic correction of satellite data over mountainous terrain,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New York N.J., 1994), Vol. 1, pp. 58–60.
  7. R. Richter, “Correction of atmospheric and topographic effects for high spatial resolution satellite imagery,” Int. J. Remote Sens. 18, 1099–1111 (1997).
    [CrossRef]
  8. C. Miesch, X. Briottet, Y. H. Kerr, F. Cabot, “Monte Carlo approach for solving the radiative transfer over rugged and heterogeneous areas,” Appl. Opt. 36, 7419–7430 (1999).
    [CrossRef]
  9. B. M. Herman, S. R. Browning, “A numerical solution to the equation of radiative transfer,” J. Atmos. Sci. 22, 559–566 (1965).
    [CrossRef]
  10. D. Tanré, M. Herman, P. Y. Deschamps, “Influence of the background contribution upon space measurements of ground reflectance,” Appl. Opt. 20, 733–741 (1981).
  11. P. N. Reinersman, K. L. Carder, “Monte Carlo simulations of the atmospheric point-spread function with an application to correction for adjacency effect,” Appl. Opt. 34, 4453–4471 (1995).
    [CrossRef] [PubMed]
  12. J. Dozier, J. Frew, “Atmospheric corrections to satellite radiometric data over rugged terrain,” Remote Sens. Environ. 11, 191–205 (1981).
    [CrossRef]
  13. ASTER (Advanced Spaceborne Thermal Emission and Reflection Radiometer) spectral library, NASA, http://speclib.jpl.nasa.gov/ ; cognizant scientist, simon.j.hook@jpl.nasa.gov.
  14. R. Mc Clatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical properties of the atmosphere,” (U.S. AFCRL, L. G. Hanscom Field, Bedford, Mass., 1971).
  15. H. Cosnefroy, M. Leroy, X. Briottet, “Selection and characterization of Saharan and Arabian desert sites for the calibration of optical satellite sensors,” Remote Sens. Environ. 58, 101–114 (1996).
    [CrossRef]

1999 (1)

C. Miesch, X. Briottet, Y. H. Kerr, F. Cabot, “Monte Carlo approach for solving the radiative transfer over rugged and heterogeneous areas,” Appl. Opt. 36, 7419–7430 (1999).
[CrossRef]

1997 (2)

R. Richter, “Correction of atmospheric and topographic effects for high spatial resolution satellite imagery,” Int. J. Remote Sens. 18, 1099–1111 (1997).
[CrossRef]

E. Vermote, D. Tanré, J. L. Deuzé, M. Herman, J. J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

1996 (1)

H. Cosnefroy, M. Leroy, X. Briottet, “Selection and characterization of Saharan and Arabian desert sites for the calibration of optical satellite sensors,” Remote Sens. Environ. 58, 101–114 (1996).
[CrossRef]

1995 (1)

1989 (1)

C. Proy, D. Tanré, P. Y. Deschamps, “Evaluation of topographic effects on remotely sensed data,” Remote Sens. Environ. 30, 21–32 (1989).
[CrossRef]

1981 (3)

D. S. Kimes, J. A. Kirchner, “Modeling the effects of various radiant transfers in mountainous terrain on sensor response,” IEEE Trans. Geosci. Remote Sens. GE-19, 100–108 (1981).
[CrossRef]

D. Tanré, M. Herman, P. Y. Deschamps, “Influence of the background contribution upon space measurements of ground reflectance,” Appl. Opt. 20, 733–741 (1981).

J. Dozier, J. Frew, “Atmospheric corrections to satellite radiometric data over rugged terrain,” Remote Sens. Environ. 11, 191–205 (1981).
[CrossRef]

1980 (1)

B. N. Holben, C. O. Justice, “The topographic effect on spectral response from nadir-pointing sensors,” Photogramm. Eng. Remote Sens. 46, 1191–1200 (1980).

1965 (1)

B. M. Herman, S. R. Browning, “A numerical solution to the equation of radiative transfer,” J. Atmos. Sci. 22, 559–566 (1965).
[CrossRef]

Berk, A.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran 7,” (Spectral Sciences, Burlington, Mass., 1989).

Bernstein, L. S.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran 7,” (Spectral Sciences, Burlington, Mass., 1989).

Briottet, X.

C. Miesch, X. Briottet, Y. H. Kerr, F. Cabot, “Monte Carlo approach for solving the radiative transfer over rugged and heterogeneous areas,” Appl. Opt. 36, 7419–7430 (1999).
[CrossRef]

H. Cosnefroy, M. Leroy, X. Briottet, “Selection and characterization of Saharan and Arabian desert sites for the calibration of optical satellite sensors,” Remote Sens. Environ. 58, 101–114 (1996).
[CrossRef]

Browning, S. R.

B. M. Herman, S. R. Browning, “A numerical solution to the equation of radiative transfer,” J. Atmos. Sci. 22, 559–566 (1965).
[CrossRef]

Cabot, F.

C. Miesch, X. Briottet, Y. H. Kerr, F. Cabot, “Monte Carlo approach for solving the radiative transfer over rugged and heterogeneous areas,” Appl. Opt. 36, 7419–7430 (1999).
[CrossRef]

Carder, K. L.

Cosnefroy, H.

H. Cosnefroy, M. Leroy, X. Briottet, “Selection and characterization of Saharan and Arabian desert sites for the calibration of optical satellite sensors,” Remote Sens. Environ. 58, 101–114 (1996).
[CrossRef]

Deschamps, P. Y.

C. Proy, D. Tanré, P. Y. Deschamps, “Evaluation of topographic effects on remotely sensed data,” Remote Sens. Environ. 30, 21–32 (1989).
[CrossRef]

D. Tanré, M. Herman, P. Y. Deschamps, “Influence of the background contribution upon space measurements of ground reflectance,” Appl. Opt. 20, 733–741 (1981).

Deuzé, J. L.

E. Vermote, D. Tanré, J. L. Deuzé, M. Herman, J. J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

Dozier, J.

J. Dozier, J. Frew, “Atmospheric corrections to satellite radiometric data over rugged terrain,” Remote Sens. Environ. 11, 191–205 (1981).
[CrossRef]

Fenn, R. W.

R. Mc Clatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical properties of the atmosphere,” (U.S. AFCRL, L. G. Hanscom Field, Bedford, Mass., 1971).

Frew, J.

J. Dozier, J. Frew, “Atmospheric corrections to satellite radiometric data over rugged terrain,” Remote Sens. Environ. 11, 191–205 (1981).
[CrossRef]

Garing, J. S.

R. Mc Clatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical properties of the atmosphere,” (U.S. AFCRL, L. G. Hanscom Field, Bedford, Mass., 1971).

Herman, B. M.

B. M. Herman, S. R. Browning, “A numerical solution to the equation of radiative transfer,” J. Atmos. Sci. 22, 559–566 (1965).
[CrossRef]

Herman, M.

E. Vermote, D. Tanré, J. L. Deuzé, M. Herman, J. J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

D. Tanré, M. Herman, P. Y. Deschamps, “Influence of the background contribution upon space measurements of ground reflectance,” Appl. Opt. 20, 733–741 (1981).

Holben, B. N.

B. N. Holben, C. O. Justice, “The topographic effect on spectral response from nadir-pointing sensors,” Photogramm. Eng. Remote Sens. 46, 1191–1200 (1980).

Justice, C. O.

B. N. Holben, C. O. Justice, “The topographic effect on spectral response from nadir-pointing sensors,” Photogramm. Eng. Remote Sens. 46, 1191–1200 (1980).

Kawata, Y.

T. Kusaka, Y. Kawata, “Atmospheric and topographic correction of satellite data over mountainous terrain,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New York N.J., 1994), Vol. 1, pp. 58–60.

Kerr, Y. H.

C. Miesch, X. Briottet, Y. H. Kerr, F. Cabot, “Monte Carlo approach for solving the radiative transfer over rugged and heterogeneous areas,” Appl. Opt. 36, 7419–7430 (1999).
[CrossRef]

Kimes, D. S.

D. S. Kimes, J. A. Kirchner, “Modeling the effects of various radiant transfers in mountainous terrain on sensor response,” IEEE Trans. Geosci. Remote Sens. GE-19, 100–108 (1981).
[CrossRef]

Kirchner, J. A.

D. S. Kimes, J. A. Kirchner, “Modeling the effects of various radiant transfers in mountainous terrain on sensor response,” IEEE Trans. Geosci. Remote Sens. GE-19, 100–108 (1981).
[CrossRef]

Kusaka, T.

T. Kusaka, Y. Kawata, “Atmospheric and topographic correction of satellite data over mountainous terrain,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New York N.J., 1994), Vol. 1, pp. 58–60.

Leroy, M.

H. Cosnefroy, M. Leroy, X. Briottet, “Selection and characterization of Saharan and Arabian desert sites for the calibration of optical satellite sensors,” Remote Sens. Environ. 58, 101–114 (1996).
[CrossRef]

Mc Clatchey, R.

R. Mc Clatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical properties of the atmosphere,” (U.S. AFCRL, L. G. Hanscom Field, Bedford, Mass., 1971).

Miesch, C.

C. Miesch, X. Briottet, Y. H. Kerr, F. Cabot, “Monte Carlo approach for solving the radiative transfer over rugged and heterogeneous areas,” Appl. Opt. 36, 7419–7430 (1999).
[CrossRef]

Morcrette, J. J.

E. Vermote, D. Tanré, J. L. Deuzé, M. Herman, J. J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

Proy, C.

C. Proy, D. Tanré, P. Y. Deschamps, “Evaluation of topographic effects on remotely sensed data,” Remote Sens. Environ. 30, 21–32 (1989).
[CrossRef]

Reinersman, P. N.

Richter, R.

R. Richter, “Correction of atmospheric and topographic effects for high spatial resolution satellite imagery,” Int. J. Remote Sens. 18, 1099–1111 (1997).
[CrossRef]

Robertson, D. C.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran 7,” (Spectral Sciences, Burlington, Mass., 1989).

Selby, J. E. A.

R. Mc Clatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical properties of the atmosphere,” (U.S. AFCRL, L. G. Hanscom Field, Bedford, Mass., 1971).

Tanré, D.

E. Vermote, D. Tanré, J. L. Deuzé, M. Herman, J. J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

C. Proy, D. Tanré, P. Y. Deschamps, “Evaluation of topographic effects on remotely sensed data,” Remote Sens. Environ. 30, 21–32 (1989).
[CrossRef]

D. Tanré, M. Herman, P. Y. Deschamps, “Influence of the background contribution upon space measurements of ground reflectance,” Appl. Opt. 20, 733–741 (1981).

Vermote, E.

E. Vermote, D. Tanré, J. L. Deuzé, M. Herman, J. J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

Volz, F. E.

R. Mc Clatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical properties of the atmosphere,” (U.S. AFCRL, L. G. Hanscom Field, Bedford, Mass., 1971).

Appl. Opt. (3)

IEEE Trans. Geosci. Remote Sens. (2)

E. Vermote, D. Tanré, J. L. Deuzé, M. Herman, J. J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

D. S. Kimes, J. A. Kirchner, “Modeling the effects of various radiant transfers in mountainous terrain on sensor response,” IEEE Trans. Geosci. Remote Sens. GE-19, 100–108 (1981).
[CrossRef]

Int. J. Remote Sens. (1)

R. Richter, “Correction of atmospheric and topographic effects for high spatial resolution satellite imagery,” Int. J. Remote Sens. 18, 1099–1111 (1997).
[CrossRef]

J. Atmos. Sci. (1)

B. M. Herman, S. R. Browning, “A numerical solution to the equation of radiative transfer,” J. Atmos. Sci. 22, 559–566 (1965).
[CrossRef]

Photogramm. Eng. Remote Sens. (1)

B. N. Holben, C. O. Justice, “The topographic effect on spectral response from nadir-pointing sensors,” Photogramm. Eng. Remote Sens. 46, 1191–1200 (1980).

Remote Sens. Environ. (3)

C. Proy, D. Tanré, P. Y. Deschamps, “Evaluation of topographic effects on remotely sensed data,” Remote Sens. Environ. 30, 21–32 (1989).
[CrossRef]

J. Dozier, J. Frew, “Atmospheric corrections to satellite radiometric data over rugged terrain,” Remote Sens. Environ. 11, 191–205 (1981).
[CrossRef]

H. Cosnefroy, M. Leroy, X. Briottet, “Selection and characterization of Saharan and Arabian desert sites for the calibration of optical satellite sensors,” Remote Sens. Environ. 58, 101–114 (1996).
[CrossRef]

Other (4)

ASTER (Advanced Spaceborne Thermal Emission and Reflection Radiometer) spectral library, NASA, http://speclib.jpl.nasa.gov/ ; cognizant scientist, simon.j.hook@jpl.nasa.gov.

R. Mc Clatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical properties of the atmosphere,” (U.S. AFCRL, L. G. Hanscom Field, Bedford, Mass., 1971).

T. Kusaka, Y. Kawata, “Atmospheric and topographic correction of satellite data over mountainous terrain,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New York N.J., 1994), Vol. 1, pp. 58–60.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran 7,” (Spectral Sciences, Burlington, Mass., 1989).

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Figures (19)

Fig. 1
Fig. 1

Irradiance components at ground level.

Fig. 2
Fig. 2

Diffuse sky radiance incident upon a rugged terrain.

Fig. 3
Fig. 3

Radiance components at sensor level.

Fig. 4
Fig. 4

Ground surface intercepted by the IFOV.

Fig. 5
Fig. 5

Diffuse upward path.

Fig. 6
Fig. 6

Seashore landscape.

Fig. 7
Fig. 7

Irradiance components at ground level on the seashore. In this and the figures below, the solid, lighter curves represent the Monte Carlo model and the dashed, darker curves represent the physical model. RMSE, root-mean-square error.

Fig. 8
Fig. 8

Radiance components received by the sensor over the seashore.

Fig. 9
Fig. 9

Linear sand dunes.

Fig. 10
Fig. 10

Irradiance terms over linear sand dunes.

Fig. 11
Fig. 11

Radiance terms observed over linear sand dunes.

Fig. 12
Fig. 12

Deep valley between snow-covered hilltops.

Fig. 13
Fig. 13

Reflectance models of snow and grass. Negative zenith values are used for backward scattering.

Fig. 14
Fig. 14

Irradiance terms over the deep valley.

Fig. 15
Fig. 15

Environment and adjacency irradiance over the deep valley.

Fig. 16
Fig. 16

Radiance components over the deep valley.

Fig. 17
Fig. 17

Topographic effect on the sky diffuse radiances.

Fig. 18
Fig. 18

Downward radiance.

Fig. 19
Fig. 19

Upward radiance.

Equations (34)

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Edirect=Edirect+Ediffuse+Ecoupling+Edirect-reflected+Ediffuse-reflected.
EdirectP=ETOAoP, θs, ϕsNˆP · nˆs×exp-τP/μsTgP, θs,
Ediffuse flat groundM=TgM, θs×Ω=2π LdzM, θ, ϕcos θdω,
EdiffuseP=TgP, θsΩskyP LdzP, θ, ϕcos θdω+Ω¯skyP LdzP, θ, ϕKP, θ, ϕcos θdω=TgP, θsΩ=2π LdzP, θ, ϕKP, θ, ϕ×cos θdω,
EcouplingP=GEdirectM+EdiffuseMN×dncouplingM, PdSPdSM.
Edirect-reflectedP=VP EdirectMρMNˆM, nˆs, nˆMPπ×exp-τM, PTgM, PVMP×NˆM · nˆMPNˆP · nˆPMrMP2dSM,
Ediffuse-reflectedP=VP EdiffuseMaMNˆM, nˆMPπ×exp-τM, PTgM, PVMP×NˆM · nˆMPNˆP · nˆPMrMP2dSM.
Ltotal=Ldirect+Ldiffuse+Latm.
Ldirecti=1ΩiΩi LgroundPdω, nˆνPdω×exp-τPdω/μνPdωTgPdω, θνPdωdω,
Ldirect=Ldirect-direct+Ldiffuse-direct+Lcoupling-direct+Ldirect-reflected-direct+Ldiffuse-reflected-direct.
Ldirect-directi=1ΩiΩi EdirectPdωρNˆPdω, nˆs, nˆνPdωπ×exp-τPdω/μνPdω×TgPdω, θνPdωdω,
ρ¯diffuse-directP, nˆνP=Ω=2π LdzP, θ, ϕKP, θ, ϕρPNˆP, nˆθ, ϕ, nˆνPcos θdωΩ=2π LdzP, θ, ϕKP, θ, ϕcos θdω.
Ldiffuse-directi=1ΩiΩi EdiffusePdω×ρ¯diffuse-directPdω, nˆνPdωπ×exp-τPdω/μνPdω×TgPdω, θνPdωdω.
Lcoupling-directi=1ΩiΩi EcouplingPdω×aPdωNˆPdω, nˆνPdωπ×exp-τPdω/μνPdω×TgPdω, θνPdωdω.
Ldirect-reflected-directi=1ΩiΩiVPdω EdirectMρMNˆM, nˆs, nˆMPdωπ TgM, Pdωexp-τM, Pdω×NˆM · nˆMPdωNˆPdω · nˆMPdωrMPdω2 VMPdωρPdωNˆPdω, nˆPdω, nˆνPdωπdSM×exp-τPdω/μνPdωTgPdω, θνPdωdω.
Ldiffuse-reflected-directP=1ΩiΩiVPdω EdiffuseMaMNˆM, nˆMPdωπ TgM, Pdωexp-τM, Pdω×NˆM · nˆMPdωNˆPdω · nˆPdωMrMPdω2 VMPdωρPdωNˆPdω, nˆPdωM, nˆνPdωπdSM×exp-τPdω/μνPdωTgPdω, θνPdωdω.
CdP, i=1NdndP, idSP.
Ldiffusei=G LgroundPCdP, iTgP, θνPdSP,
Ldiffuse=Ldirect-diffuse+Ldiffuse-diffuse+Lcoupling-diffuse+Ldirect-reflected-diffuse+Ldiffuse-reflected-diffuse.
Ldirect-diffusei=G EdirectPaPNˆP, nˆsπ×CdP, iTgP, θνPdSP.
Ldiffuse-diffusei=G EdiffusePSPπ CdP, iTg×P, θνPdSP.
Lcoupling-diffusei=G EcouplingPSPπ CdPTg×P, θνPdSP.
Ldirect-reflected-diffusei=GVP EdirectMρMNˆM, nˆs, nˆMPπ TgM, Pexp-τM, P×NˆM · nˆMPNˆP · nˆPMrMP2 VMPaPNˆP, nˆPMπdSMCdP, iTgP, nˆνPdSP,
Ldiffuse-reflected-diffusei=GVP EdiffuseMaMNˆM, nˆMPπ TgM, Pexp-τM, P×NˆM · nˆMPNˆP · nˆPMrMP2 VMPaPNˆP, nˆPMπdSMCdP, iTgP, nˆvPdSP.
LP relief=LP-LQ exp-|τP-τQ|/μ,
dLMθ, ϕ=LTOAθs, ϕsexp-τ/μspθs, ϕs, θ, ϕ4π×exp-τM-τ/μdτμs,
LMθ, ϕ=LTOAθs, ϕsexp-τM/μpθs, ϕs, θ, ϕ4πμsμsμμ-μs×1-exp-τM1/μs-1/μ.
LP reliefθ, ϕ=LP-LQ exp-τP-τQ/μ=LP1-1-exp-τP1/μs-1/μ1-exp-τQ1/μs-1/μ.
dLMθ, ϕ=LTOAθs, ϕsexp-τ/μspθs, ϕs, θ, ϕ4π×exp-τ-τM/μdτμs,
LMθ, ϕ=LTOAθs, ϕsexp-τM/μpθs, ϕs, θ, ϕ4πμs×μsμμs+μ1-exp-τatm-τM1/μs+1/μ.
LP reliefθ, ϕ=LP-LQ exp-τQ-τP/μ=LP1-1-exp-τatm-τQ1/μs+1/μ1-exp-τatm-τP1/μs+1/μ×exp-τQ-τP/μ-τQ-τP/μs.
LP=Lπ/2, ϕ0 exp-σerdr,
LP relief=1π/2, ϕ0r0 exp-σerdr=LP0r0 exp-σerdr0 exp-σerdr=LP1-exp-σer0.
KL=1-1-exp-τP1/μs-1/μ1-exp-τQ1/μs-1/μ,  KL=1-1-exp-τatm-τQ1/μs+1/μ1-exp-τatm-τP1/μs+1/μ×exp-τQ-τP/μ-τQ-τP/μs,  KL=1-exp-σer0.

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