Abstract

This work analyses and solves for issues encountered when retrieving surface emissivity in LWIR (750 to 1250 cm-1) and MWIR (2000 to 3500 cm-1) bands under outdoor conditions. The Spectral Smoothness method, which takes advantage of high spectral resolution measurements to solve for temperature emissivity separation, and which has already proven its efficiency in the LWIR domain, was applied in an experimental campaign to assess its ability to operate both in the LWIR and MWIR domains. In the MWIR band, directional behaviour of surface emissivity is shown to be a source of systematic errors in the retrieved emissivity and a new method, called SmaC (SMoothness And Continuity), corrects for this error by providing quantitative estimates on the deviation of the surface from Lambertian behavior.

©2007 Optical Society of America

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References

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  1. K. Kanani, Utilisation de la très haute résolution spectrale pour la mesure en environnement extérieur de l’émissivité de surface dans la bande infrarouge 3–13µm. Méthodes et validation expérimentale, Thèse de Doctorat, (Université Louis Pasteur, Strasbourg N°4946, 2005).
  2. P. Dash, F.-M. Göttsche, F.-S. Olesen, and H. Fischer, “Land surface temperature and emissivity estimation from passive sensor data: theory and practice current trends,” Int. J. Remote Sens. 23, 2563–2594 (2002).
    [Crossref]
  3. C. C. Borel, “Surface emissivity and temperature retrieval for a hyperspectral sensor,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing, (IEEE, 1998) pp. 504–509.
  4. C. Salvaggio and C. J. Miller, “Comparison of field and laboratory collected midwave and longwave infrared emissivity spectra/data reduction techniques,” Proc. SPIE,  4381, 549–558 (2001).
    [Crossref]
  5. N. Bower, Measurement of Land Surface Emissivity and Temperature in the Thermal Infrared using a Ground-Based Interferometer, PhD Thesis, (Curtin University of Technology, Perth, Australia, 2001).
    [PubMed]
  6. P. M. Ingram and A. H. Muse, “Sensitivity of Iterative Spectrally Smooth Temperature/Emissivity Separation to Algorithmic Assumptions and Measurement Noise,” IEEE Transaction on Geoscience and Remote Sensing 39, 2158–2167 (2001).
    [Crossref]
  7. Z.-L. Li, F. Becker, M. Stoll, and Z. Wan, “Evaluation of six methods for extracting relative emissivity spectra from thermal infrared images,” Remote Sens. Environ. 69, 197–214 (1999).
    [Crossref]
  8. J. Cuenca and J. A. Sobrino, “Experimental measurements for studying angular and spectral variation of thermal infrared emissivity,” Appl. Opt. 43, 4568–4602 (2004).
    [Crossref]
  9. W. C. Snyder, Z. Wan, Y. Zhang, and Y.-Z. Feng, “Thermal Infrared (3–14µm) Bidirectional ReflectanceMeasurements of Sands and Soils,” Remote Sens. Environ. 60, 101–109 (1997).
    [Crossref]
  10. G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
    [Crossref]

2004 (1)

J. Cuenca and J. A. Sobrino, “Experimental measurements for studying angular and spectral variation of thermal infrared emissivity,” Appl. Opt. 43, 4568–4602 (2004).
[Crossref]

2002 (1)

P. Dash, F.-M. Göttsche, F.-S. Olesen, and H. Fischer, “Land surface temperature and emissivity estimation from passive sensor data: theory and practice current trends,” Int. J. Remote Sens. 23, 2563–2594 (2002).
[Crossref]

2001 (2)

C. Salvaggio and C. J. Miller, “Comparison of field and laboratory collected midwave and longwave infrared emissivity spectra/data reduction techniques,” Proc. SPIE,  4381, 549–558 (2001).
[Crossref]

P. M. Ingram and A. H. Muse, “Sensitivity of Iterative Spectrally Smooth Temperature/Emissivity Separation to Algorithmic Assumptions and Measurement Noise,” IEEE Transaction on Geoscience and Remote Sensing 39, 2158–2167 (2001).
[Crossref]

1999 (1)

Z.-L. Li, F. Becker, M. Stoll, and Z. Wan, “Evaluation of six methods for extracting relative emissivity spectra from thermal infrared images,” Remote Sens. Environ. 69, 197–214 (1999).
[Crossref]

1997 (1)

W. C. Snyder, Z. Wan, Y. Zhang, and Y.-Z. Feng, “Thermal Infrared (3–14µm) Bidirectional ReflectanceMeasurements of Sands and Soils,” Remote Sens. Environ. 60, 101–109 (1997).
[Crossref]

Acharya, P.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Adler-Golden, S.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Anderson, G.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Becker, Z.-L. Li, F.

Z.-L. Li, F. Becker, M. Stoll, and Z. Wan, “Evaluation of six methods for extracting relative emissivity spectra from thermal infrared images,” Remote Sens. Environ. 69, 197–214 (1999).
[Crossref]

Berk, A.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Bernstein, L.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Borel, C. C.

C. C. Borel, “Surface emissivity and temperature retrieval for a hyperspectral sensor,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing, (IEEE, 1998) pp. 504–509.

Bower, N.

N. Bower, Measurement of Land Surface Emissivity and Temperature in the Thermal Infrared using a Ground-Based Interferometer, PhD Thesis, (Curtin University of Technology, Perth, Australia, 2001).
[PubMed]

Chetwynd, J.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Cuenca, J.

J. Cuenca and J. A. Sobrino, “Experimental measurements for studying angular and spectral variation of thermal infrared emissivity,” Appl. Opt. 43, 4568–4602 (2004).
[Crossref]

Dash, P.

P. Dash, F.-M. Göttsche, F.-S. Olesen, and H. Fischer, “Land surface temperature and emissivity estimation from passive sensor data: theory and practice current trends,” Int. J. Remote Sens. 23, 2563–2594 (2002).
[Crossref]

Dothe, H.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Felde, G.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Feng, Y.-Z.

W. C. Snyder, Z. Wan, Y. Zhang, and Y.-Z. Feng, “Thermal Infrared (3–14µm) Bidirectional ReflectanceMeasurements of Sands and Soils,” Remote Sens. Environ. 60, 101–109 (1997).
[Crossref]

Fischer, H.

P. Dash, F.-M. Göttsche, F.-S. Olesen, and H. Fischer, “Land surface temperature and emissivity estimation from passive sensor data: theory and practice current trends,” Int. J. Remote Sens. 23, 2563–2594 (2002).
[Crossref]

Gardner, J.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Göttsche, F.-M.

P. Dash, F.-M. Göttsche, F.-S. Olesen, and H. Fischer, “Land surface temperature and emissivity estimation from passive sensor data: theory and practice current trends,” Int. J. Remote Sens. 23, 2563–2594 (2002).
[Crossref]

Hoke, M.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Ingram, P. M.

P. M. Ingram and A. H. Muse, “Sensitivity of Iterative Spectrally Smooth Temperature/Emissivity Separation to Algorithmic Assumptions and Measurement Noise,” IEEE Transaction on Geoscience and Remote Sensing 39, 2158–2167 (2001).
[Crossref]

Kanani, K.

K. Kanani, Utilisation de la très haute résolution spectrale pour la mesure en environnement extérieur de l’émissivité de surface dans la bande infrarouge 3–13µm. Méthodes et validation expérimentale, Thèse de Doctorat, (Université Louis Pasteur, Strasbourg N°4946, 2005).

Leong, L.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Matthew, M.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Mello, J.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Miller, C. J.

C. Salvaggio and C. J. Miller, “Comparison of field and laboratory collected midwave and longwave infrared emissivity spectra/data reduction techniques,” Proc. SPIE,  4381, 549–558 (2001).
[Crossref]

Muse, A. H.

P. M. Ingram and A. H. Muse, “Sensitivity of Iterative Spectrally Smooth Temperature/Emissivity Separation to Algorithmic Assumptions and Measurement Noise,” IEEE Transaction on Geoscience and Remote Sensing 39, 2158–2167 (2001).
[Crossref]

Olesen, F.-S.

P. Dash, F.-M. Göttsche, F.-S. Olesen, and H. Fischer, “Land surface temperature and emissivity estimation from passive sensor data: theory and practice current trends,” Int. J. Remote Sens. 23, 2563–2594 (2002).
[Crossref]

Pukall, B.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Ratkowski, A.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Richtsmeier, S.

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

Salvaggio, C.

C. Salvaggio and C. J. Miller, “Comparison of field and laboratory collected midwave and longwave infrared emissivity spectra/data reduction techniques,” Proc. SPIE,  4381, 549–558 (2001).
[Crossref]

Snyder, W. C.

W. C. Snyder, Z. Wan, Y. Zhang, and Y.-Z. Feng, “Thermal Infrared (3–14µm) Bidirectional ReflectanceMeasurements of Sands and Soils,” Remote Sens. Environ. 60, 101–109 (1997).
[Crossref]

Sobrino, J. A.

J. Cuenca and J. A. Sobrino, “Experimental measurements for studying angular and spectral variation of thermal infrared emissivity,” Appl. Opt. 43, 4568–4602 (2004).
[Crossref]

Stoll, M.

Z.-L. Li, F. Becker, M. Stoll, and Z. Wan, “Evaluation of six methods for extracting relative emissivity spectra from thermal infrared images,” Remote Sens. Environ. 69, 197–214 (1999).
[Crossref]

Wan, Z.

Z.-L. Li, F. Becker, M. Stoll, and Z. Wan, “Evaluation of six methods for extracting relative emissivity spectra from thermal infrared images,” Remote Sens. Environ. 69, 197–214 (1999).
[Crossref]

W. C. Snyder, Z. Wan, Y. Zhang, and Y.-Z. Feng, “Thermal Infrared (3–14µm) Bidirectional ReflectanceMeasurements of Sands and Soils,” Remote Sens. Environ. 60, 101–109 (1997).
[Crossref]

Zhang, Y.

W. C. Snyder, Z. Wan, Y. Zhang, and Y.-Z. Feng, “Thermal Infrared (3–14µm) Bidirectional ReflectanceMeasurements of Sands and Soils,” Remote Sens. Environ. 60, 101–109 (1997).
[Crossref]

Appl. Opt. (1)

J. Cuenca and J. A. Sobrino, “Experimental measurements for studying angular and spectral variation of thermal infrared emissivity,” Appl. Opt. 43, 4568–4602 (2004).
[Crossref]

IEEE Transaction on Geoscience and Remote Sensing (1)

P. M. Ingram and A. H. Muse, “Sensitivity of Iterative Spectrally Smooth Temperature/Emissivity Separation to Algorithmic Assumptions and Measurement Noise,” IEEE Transaction on Geoscience and Remote Sensing 39, 2158–2167 (2001).
[Crossref]

Int. J. Remote Sens. (1)

P. Dash, F.-M. Göttsche, F.-S. Olesen, and H. Fischer, “Land surface temperature and emissivity estimation from passive sensor data: theory and practice current trends,” Int. J. Remote Sens. 23, 2563–2594 (2002).
[Crossref]

Proc. SPIE (1)

C. Salvaggio and C. J. Miller, “Comparison of field and laboratory collected midwave and longwave infrared emissivity spectra/data reduction techniques,” Proc. SPIE,  4381, 549–558 (2001).
[Crossref]

Remote Sens. Environ. (2)

Z.-L. Li, F. Becker, M. Stoll, and Z. Wan, “Evaluation of six methods for extracting relative emissivity spectra from thermal infrared images,” Remote Sens. Environ. 69, 197–214 (1999).
[Crossref]

W. C. Snyder, Z. Wan, Y. Zhang, and Y.-Z. Feng, “Thermal Infrared (3–14µm) Bidirectional ReflectanceMeasurements of Sands and Soils,” Remote Sens. Environ. 60, 101–109 (1997).
[Crossref]

Other (4)

G. Anderson, A. Berk, P. Acharya, M. Matthew, L. Bernstein, J. Chetwynd, H. Dothe, S. Adler-Golden, A. Ratkowski, G. Felde, J. Gardner, M. Hoke, S. Richtsmeier, B. Pukall, J. Mello, and L. Leong, “MODTRAN4: Radiative transfer modeling for remote sensing,” Proc. SPIE4049 (2000).
[Crossref]

N. Bower, Measurement of Land Surface Emissivity and Temperature in the Thermal Infrared using a Ground-Based Interferometer, PhD Thesis, (Curtin University of Technology, Perth, Australia, 2001).
[PubMed]

C. C. Borel, “Surface emissivity and temperature retrieval for a hyperspectral sensor,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing, (IEEE, 1998) pp. 504–509.

K. Kanani, Utilisation de la très haute résolution spectrale pour la mesure en environnement extérieur de l’émissivité de surface dans la bande infrarouge 3–13µm. Méthodes et validation expérimentale, Thèse de Doctorat, (Université Louis Pasteur, Strasbourg N°4946, 2005).

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Figures (22)

Fig. 1.
Fig. 1. Field campaign: the spectroradiometer is located in an insulated chamber. The samples to measure slide on rails into the radiometer’s field of view

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Fig. 2.
Fig. 2. Laboratory measurement of the Infragold brdf at 3.39µm, for θv =0°

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Fig. 3.
Fig. 3. Relative difference, in LWIR band, between the total irradiance I ↓ assessed from the radiance measurement on the reflective panel and the irradiance I ↓[SIMU] simulated with MODTRAN.

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Fig. 4.
Fig. 4. Spectral differences, in the MWIR band, between the total irradiance I ↓ assessed from the radiance measurement on the reflective panel (assumed either Lambertian or directionnal) and I ↓[SIMU] simulated with MODTRAN. The “Rfl lamber” curve correponds to I ↓ assessed by assuming the reflective panel Lambertian, whereas the “Rfl dir” curve corresponds to I ↓ assessed by assuming the reflective panel directionnal.

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Fig. 5.
Fig. 5. Results of emissivity retrieved by SpSm in comparison to laboratory measurements, for each sample and for three spectral domains: the deviation bars are centred on Δ M/L (Eq. 14) and have a half-length equal to σm (Eq. 15).

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Fig. 6.
Fig. 6. SpSm retrieved emissivity example of stone#1 sample. Comparison with laboratory measurement

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Fig. 7.
Fig. 7. SpSm retrieved emissivity example of negev sample. Comparison with laboratory measurement

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Fig. 8.
Fig. 8. SmaC discontinuity measurement

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Fig. 9.
Fig. 9. SmaC retrieved emissivity example of stone#1 and negev samples in MWIR band. Comparison with laboratory measurement

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Fig. 10.
Fig. 10. Results of emissivity retrieved by SmaC and SpSm in DIIa , in comparison to laboratory measurements (same legend as for figure 5)

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Fig. 11.
Fig. 11. Results of emissivity retrieved by SmaC and SpSm in DIIb , in comparison to laboratory measurements (same legend as for figure 5)

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Fig. 12.
Fig. 12. Emissivities and directional factors of slate retrieved by SmaC.

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Fig. 13.
Fig. 13. Emissivities and directional factors of wood retrieved by SmaC.

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Fig. 14.
Fig. 14. Emissivities and directional factors of BB retrieved by SmaC.

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Fig. 15.
Fig. 15. Emissivities and directional factors of sand#1 retrieved by SmaC.

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Fig. 16.
Fig. 16. Emissivities and directional factors of negev retrieved by SmaC.

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Fig. 17.
Fig. 17. Emissivities and directional factors of stone#1 retrieved by SmaC.

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Fig. 18.
Fig. 18. Emissivities and directional factors of stone#2 retrieved by SmaC.

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Fig. 19.
Fig. 19. Emissivities and directional factors of stone#3 retrieved by SmaC.

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Fig. 20.
Fig. 20. Emissivities and directional factors of pstyr retrieved by SmaC.

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Fig. 21.
Fig. 21. Emissivities and directional factors of sand#2 retrieved by SmaC.

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Fig. 22.
Fig. 22. Emissivities and directional factors of SiC retrieved by SmaC.

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Tables (2)

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Table 1. Main characteristics and composition information of the surfaces mesured during the field campaign.

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Table 2. Spectral domains considered for the performance assessment

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Equations (29)

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

R ( ν ) = ε ( ν ) B ( ν , T ) + ( 1 − ε ( ν ) ) I ↓ ( ν ) π
R ( θ v , φ v , ν ) = ε ( θ v , φ v , ν ) B ( ν , T )
+ ∫ 2 π brd f ( θ i , φ i , θ v , φ v , ν ) R env ↓ ( θ i , φ i , ν ) cos ⁡ θ i d Ω i
+ brd f ( θ s , φ s , θ v , φ v , ν ) I dir ↓ ( θ s , φ s , ν )
R gld ( θ v , φ v , ν ) = ( 1 − ρ gld ( θ v , ϕ v , ν ) ) B ( ν , T gld )
+ ∫ 2 π brd f gld ( θ i , φ i , θ v , φ v , ν ) R env ↓ ( θ i , φ i , ν ) cos ⁡ θ i d Ω i
+ brd f gld ( θ s , φ s , θ v , φ v , ν ) I dir ↓ ( θ s , φ s , ν )
ρ gld ( θ v , φ v ,ν ) = ∫ 2 π brd f gld ( θ i , φ i , θ v , φ v , ν ) cos ⁡ θ i d Ω i
R gld = ( 1 − ρ gld ) B ( ν , T gld ) + ∫ 2 π brd f gld ( θ i , φ i ) R env ↓ ( θ i , φ i ) cos ⁡ θ i d Ω i
R gld = ( 1 − ρ gld ) B ( T gld ) + ρ gld I env ↓ ⁄ π
with I env ↓ = ∫ 2 π R env ↓ cos ⁡ θ i d Ω i
I ↓ = π R gld − ( 1 − ρ gld ) B ( T gld ) ρ gld
R gld = ( 1 − ρ gld ) B ( T gld ) + ρ gld I env ↓ ⁄ π + brd f gld ( θ s , φ s ) I dir ↓
b r ˜ d f gld ( θ s , φ s ) =
〈 R gld − ( 1 − ρ gld ) B ( T gld ) − ρ gld I env ↓ [ SIMU ] ⁄ π I dir ↓ [ SIMU ] 〉 2735 − 2880
I dir ↓ = R gld − ( 1 − ρ gld ) B ( T gld ) + ρ gld I env ↓ ⁄ π b r ˜ d f gld ( θ s , φ s )
ε ( T ) = L − I ↓ ⁄ π B ( T ) − I ↓ ⁄ π
Sm ( T ) = ∫ ∣ ∂ ε ( T ) ∂ ν ∣ d ν
Δ M ⁄ L = 1 N D ∑ i = 1 N D [ M [ i ] − ε L [ i ] ]
with M [ i ] = 1 N m ∑ i = 1 N m ε j [ i ]
σ m = 1 N m N D − 1 ∑ i = 1 N D ∑ j = 1 N m ( ε j [ i ] − M [ i ] ) 2
R ( θ v , φ v , ν ) = ε ( θ v , φ v , ν ) B ( ν , T ) + ( 1 − ε ( θ v , φ v , ν ) ) I env ↓ ( ν ) ⁄ π
+ brd f ( θ s , φ s , θ v , φ v , ν ) I dir ↓ ( θ s , φ s , ν )
brd f ( θ v , φ v , θ i , φ i , ν ) = 1 π × f ˜ ( θ i , φ i , θ v , φ v ) × ρ ( θ v , φ v , ν )
R ( θ v , φ v , ν ) = ε ( θ v , φ v , ν ) B ( ν , T )
+ 1 π ( 1 − ε ( θ v , φ v , ν ) ) [ I env ↓ ( ν ) + f ˜ ( θ s , φ s , θ v , φ v ) I dir ↓ ( θ s , φ s , ν ) ]
ε ( θ v , φ v , ν ) = π R ( θ v , φ v , ν ) − I ˜ ↓ ( θ s , φ s ,ν ) π B ( ν , T ) − I ˜ ↓ ( θ s , φ s ,ν )
with I ˜ ↓ ( θ s , φ s , ν ) = I env ↓ ( ν ) + f ˜ ( θ s , φ s , θ v , φ v ) I dir ↓ ( θ s , φ s , ν )
Δ = ∣ lim ν → ν c − ε ( ν ) − lim ν → ν c + ε ( ν ) ∣

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