Abstract

The radiant-intensity function of a ground-based light source, otherwise called here the city emission function (CEF), is a pivotal modulator of night-sky radiance and is one of the key factors affecting light pollution propagation in a nighttime environment. It is difficult or rather impossible to compute a CEF from databases that are usually incomplete in their description of artificial sources other than public lighting. However, we have developed an indirect remote-sensing method to retrieve the CEF from sky-brightness measurements made at a local meridian that intersects a horizontal circle at the azimuthal position of a city or town. The inversion algorithm is validated for sensitivity and specificity in the reproduction of the initial emission functions, and demonstrates that the solution model succeeds in reconstructing different types of CEFs. The numerical inversion of the integral equation was successful even under extreme conditions, including elevated ground reflectance or zero uplight cases. The good numerical convergence and applicability of the model to experimental data is demonstrated by processing the radiance patterns obtained under natural conditions. The model provides a great opportunity for systematic retrievals of CEF needed by astronomers, physicists, environmental designers, city and suburban planners, as well as biologists and illumination engineers who deal with public lighting systems and must recognize the potentially adverse effects of night-lighting characteristics on the environment.

© 2017 Optical Society of America

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References

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2016 (2)

M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016).
[Crossref]

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

2015 (1)

M. Aubé, “Physical behavior of anthropogenic light propagation into the nocturnal environment,” Philos. Trans. R. Soc. London B 370, 20140117 (2015).
[Crossref]

2014 (4)

M. Kocifaj, “Modeling the night-sky radiances and inversion of multi-angle and multi-spectral radiance data,” J. Quant. Spectrosc. Radiat. Transfer 139, 35–42 (2014).
[Crossref]

M. Kocifaj, “Night sky luminance under clear sky conditions: theory vs. experiment,” J. Quant. Spectrosc. Radiat. Transfer 139, 43–51 (2014).
[Crossref]

D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
[Crossref]

M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014).
[Crossref]

2013 (1)

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

2012 (1)

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

2010 (2)

N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010).
[Crossref]

C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
[Crossref]

2009 (2)

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

2007 (3)

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

M. Kocifaj, “Light-pollution model for cloudy and cloudless night skies with ground-based light sources,” Appl. Opt. 46, 3013 (2007).
[Crossref]

2006 (1)

M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
[Crossref]

2004 (1)

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

2002 (1)

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

2001 (1)

P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
[Crossref]

1997 (1)

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

1993 (2)

C. Dellago and H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I, basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[Crossref]

H. Horvath and C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements–II, case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[Crossref]

1992 (1)

G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[Crossref]

1990 (1)

1986 (1)

R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364–375 (1986).
[Crossref]

Adam, M.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Arboledas, L. A.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Aubé, M.

M. Aubé, “Physical behavior of anthropogenic light propagation into the nocturnal environment,” Philos. Trans. R. Soc. London B 370, 20140117 (2015).
[Crossref]

Auriol, F.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

Balois, J. Y.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

Baugh, K.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Baugh, K. E.

C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
[Crossref]

Box, G. P.

G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[Crossref]

Box, M. A.

G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[Crossref]

Cakoni, F.

F. Cakoni and D. Colton, Quantitative Methods in Inverse Scattering Theory: An Introduction (Springer, 2006).

Cinzano, P.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Colton, D.

F. Cakoni and D. Colton, Quantitative Methods in Inverse Scattering Theory: An Introduction (Springer, 2006).

Davis, D. R.

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Davis, R.

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Dellago, C.

C. Dellago and H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I, basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[Crossref]

H. Horvath and C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements–II, case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[Crossref]

Deuzé, J. L.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

Dubovik, O.

R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

Duriscoe, D.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Duriscoe, D. M.

D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
[Crossref]

Elvidge, C. D.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
[Crossref]

C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
[Crossref]

Fadeev, V. Y.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Falchi, F.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Fischer, J.

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Francçois, P.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

Furgoni, R.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Gangl, M.

M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
[Crossref]

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Garstang, R. H.

R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364–375 (1986).
[Crossref]

Goloub, P.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

Gushchin, G. P.

G. P. Gushchin, The Methods, Instrumentation and Results of Atmospheric Spectral Measurements (Gidrometeoizdat, 1988).

Hölker, F.

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Horvath, H.

M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
[Crossref]

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
[Crossref]

C. Dellago and H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I, basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[Crossref]

H. Horvath and C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements–II, case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Jovanovic, O.

M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
[Crossref]

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Kabanov, M. V.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Kahn, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

Kaller, W.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Kasahara, M.

P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
[Crossref]

Keith, D. M.

C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
[Crossref]

Kocifaj, M.

M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016).
[Crossref]

M. Kocifaj, “Night sky luminance under clear sky conditions: theory vs. experiment,” J. Quant. Spectrosc. Radiat. Transfer 139, 43–51 (2014).
[Crossref]

M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014).
[Crossref]

M. Kocifaj, “Modeling the night-sky radiances and inversion of multi-angle and multi-spectral radiance data,” J. Quant. Spectrosc. Radiat. Transfer 139, 35–42 (2014).
[Crossref]

M. Kocifaj, “Light-pollution model for cloudy and cloudless night skies with ground-based light sources,” Appl. Opt. 46, 3013 (2007).
[Crossref]

M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
[Crossref]

Kokhanovsky, A. A.

A. A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions, 3rd ed. (Praxis, 2004).

Kovalev, V. A.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Krekov, G. M.

V. E. Zuev and G. M. Krekov, Optical Models of the Atmosphere (Gidrometeoizdat, 1986).

Kuechly, H. U.

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Kyba, C. C. M.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Léon, J. F.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

Levy, R. C.

R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

Lindemann, C.

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Lockwood, G. W.

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Lockwood, W.

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Loeb, N. G.

N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010).
[Crossref]

Luginbuhl, C.

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Luginbuhl, C. B.

D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
[Crossref]

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

Moore, C.

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Nair, N.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Olmo, F. J.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Ondov, J. M.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Pahlow, M.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Panchenko, M. V.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Parlange, M. B.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Pesava, P.

P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
[Crossref]

Pick, K.

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Pkhalagov, Y. A.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Portnov, B. A.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Remer, L. A.

R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

Richman, A.

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Ruhtz, T.

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Rybnikova, N. A.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Sánchez, C.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Sauerzopf, H.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Sealey, K. M.

G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[Crossref]

Seidl, S.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Selders, J.

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Shifrin, K. S.

K. S. Shifrin, “Study of the properties of matter from single scattering,” in Theoretical and Applied Problems in the Scattering of Light (Nauka i Tekhnika, 1971) (in Russian).

Solano-Lamphar, H. A.

M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016).
[Crossref]

M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014).
[Crossref]

Stamnes, K.

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 2006).

Su, W.

N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010).
[Crossref]

Teillet, P. M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Thomas, G. E.

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 2006).

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

Tuttle, B. T.

C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
[Crossref]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, 2002).

Uzhegov, V. N.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Veretennikov, V. V.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Verwaerde, C.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Waquet, F.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

West, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

Wolter, C.

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Zuev, V. E.

V. E. Zuev and G. M. Krekov, Optical Models of the Atmosphere (Gidrometeoizdat, 1986).

Žwak, Z.

Z. Žwak, data provided (personal communication, 2012).

AIP Conf. Proc. (1)

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

Appl. Opt. (2)

Atmos. Environ. (1)

M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
[Crossref]

J. Aerosol Sci. (3)

C. Dellago and H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I, basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[Crossref]

H. Horvath and C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements–II, case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[Crossref]

P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
[Crossref]

J. Atmos. Sci. (1)

G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[Crossref]

J. Climate (1)

N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010).
[Crossref]

J. Geophys. Res. (5)

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (2)

M. Kocifaj, “Modeling the night-sky radiances and inversion of multi-angle and multi-spectral radiance data,” J. Quant. Spectrosc. Radiat. Transfer 139, 35–42 (2014).
[Crossref]

M. Kocifaj, “Night sky luminance under clear sky conditions: theory vs. experiment,” J. Quant. Spectrosc. Radiat. Transfer 139, 43–51 (2014).
[Crossref]

Light. Res. Technol. (1)

D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
[Crossref]

Mon. Not. R. Astron. Soc. (1)

M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014).
[Crossref]

Philos. Trans. R. Soc. London B (1)

M. Aubé, “Physical behavior of anthropogenic light propagation into the nocturnal environment,” Philos. Trans. R. Soc. London B 370, 20140117 (2015).
[Crossref]

Publ. Astron. Soc. Pac. (4)

M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016).
[Crossref]

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364–375 (1986).
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Figures (6)

Fig. 1.
Fig. 1. Decimal logarithms of the kernel functions K(zE;z)/S computed after Eq. (12) for τm=0.15, τa=0.2, ga=0.85, and Ωa=0.9. (a) D=1  km, (b) D=10  km, (c) D=50  km. The remaining parameters are the same as in the text (Section 2).
Fig. 2.
Fig. 2. Same as Fig. 1, but for K(zE;z)/S as a function of τa, ga, and Ωa. (a) τa=0.5 (solid line), τa=0.1 (short dashes), τa=0.03 (long dashes); (b) ga=0.5 (solid line), ga=0.7 (short dashes), ga=0.95 (long dashes); (c) Ωa=0.7 (solid line), Ωa=0.5 (short dashes), Ωa=0.3 (long dashes).
Fig. 3.
Fig. 3. (a) Synthetic radiance data (black triangles) and radiance function generated using CEF found in the process of inversion. (b) CEF used as an input to the regularization algorithm (solid line) and CEF obtained in the process of minimization.
Fig. 4.
Fig. 4. Synthetic CEFs (solid lines) of different types computed in accordance with Eq. (20) and the corresponding regularized solutions (dashed lines).
Fig. 5.
Fig. 5. CEF (in arbitrary units) obtained from a field experiment with local lights turned off (blue short dashed line) and turned on (black long dashed line). Triangles are used for CEF determined from background-compensated radiance data, while Garstang’s fit is demonstrated as the red solid line.
Fig. 6.
Fig. 6. (a) Sky radiance (in arbitrary units) taken in the city of Žilina and the reconstructed one. (b) Solution to the inverse problem.

Equations (20)

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zE=0π/2K(zE;z,a)I(zE)dzE=L(z,a).
Kλ(zE;z,a)=M(z)St[h(zE),zE]t[h(zE),z]h2(zE)×dh(zE)dzEΓ[h(zE),zE,z,a]cos3zEds,
tan2(zE)=tan2(z)+Di2h22Dihtan(z)cos(acityaipixel),
aipixel=acityarccos(D2+Di2Ri22DiD),
xi=Dcos(acity)+Ricos(acityφi),yi=Dsin(acity)+Risin(acityφi),
Kλ(zE;z,a)=M(z)Stλ[h(zE),zE]tλ[h(zE),z]h2(zE)×dh(zE)dzEΓ[h(zE),zE,z,a]cos3zE,
d[tan2(zE)]dh=2tanzEcos2zEdzEdh=2D2h3+2Dh2tanz,
dhdzE=D1+tan2zE(tanztanzE)2,
h(z)=Dtanz+tanzE.
Γλ(h,θ)=τmHmexp{h(z)Hm}3(1+cos2θ)16π+ΩaτaHaexp{h(z)Ha}1ga24π(1+ga22gacosθ)3/2,
t[h(zE),zE]t[h(zE),z]=exp{[M(zE)+M(z)]×[τm+τaτmexp(h(z)Hm)τaexp(h(z)Ha)]},
K(zE;z)=SDcos3zEcosz(1+tan2zE)4π×exp{coszE+coszcoszcoszE×[τm+τaτmexp(h(z)Hm)τaexp(h(z)Ha)]}×{τmHmexp[h(z)Hm]3[1+cos2(z+zE)]4+τaHaexp[h(z)Ha]Ωa(1ga2)[1+ga2+2gacos(z+zE)]3/2},
KI=L,
I=K1L.
Kmn=Km(zn)=Gm1,m(zn)+Gm,m(zn)+Gm+1,m(zn),m1,mMK1n=K1(zn)=G1,1(zn)+G2,1(zn),KMn=KM(zn)=GM1,M(zn)+GM,M(zn),
Gm,p(zn)=(zE,m1+zE,m)/2(zE,m+zE,m+1)/2K(zE;zn)ωm(zE)(zEzE,p)ωm(zE,p)dzE,
ωm(zE)=(zEzE,m1)(zEzE,m)(zEzE,m+1),
Bmn=(zE,mzE,m1)1,n=m1,m=2M,Bmn=1zE,mzE,m1+1zE,m+1zE,m,n=m,m=1M,Bmn=(zE,nzE,n1)1,n=m+1,m=1M1,Bmn=0,|nm|>1,m,n=1M.
Kcos(zE;z)=K(zE;z)coszECEF(zE)=I(zE)coszE,
CEF(zE)=GEF(zE)[2G(1F)coszE+0.554FzE4],

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