MOCRA: a Monte Carlo code for the simulation of radiative transfer in the atmosphere

Margherita Premuda; Elisa Palazzi; Fabrizio Ravegnani; Daniele Bortoli; Samuele Masieri; Giorgio Giovanelli

doi:10.1364/OE.20.007973

MOCRA: a Monte Carlo code for the simulation of radiative transfer in the atmosphere

Margherita Premuda, Elisa Palazzi, Fabrizio Ravegnani, Daniele Bortoli, Samuele Masieri, and Giorgio Giovanelli

Optics Express
Vol. 20,
Issue 7,
pp. 7973-7993
(2012)
•https://doi.org/10.1364/OE.20.007973

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Margherita Premuda, Elisa Palazzi, Fabrizio Ravegnani, Daniele Bortoli, Samuele Masieri, and Giorgio Giovanelli, "MOCRA: a Monte Carlo code for the simulation of radiative transfer in the atmosphere," Opt. Express 20, 7973-7993 (2012)

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Related Topics
Table of Contents Category
- Atmospheric and Oceanic Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Radiative transfer (010.5620)
- Probability theory, stochastic processes, and statistics (000.5490)

About this Article
History
- Original Manuscript: August 1, 2011
- Revised Manuscript: January 2, 2012
- Manuscript Accepted: January 2, 2012
- Published: March 22, 2012

Accessible

Open Access

Abstract

This paper describes the radiative transfer model (RTM) MOCRA (MOnte Carlo Radiance Analysis), developed in the frame of DOAS (Differential Optical Absorption Spectroscopy) to correctly interpret remote sensing measurements of trace gas amounts in the atmosphere through the calculation of the Air Mass Factor. Besides the DOAS-related quantities, the MOCRA code yields: 1- the atmospheric transmittance in the vertical and sun directions, 2- the direct and global irradiance, 3- the single- and multiple- scattered radiance for a detector with assigned position, line of sight and field of view. Sample calculations of the main radiometric quantities calculated with MOCRA are presented and compared with the output of another RTM (MODTRAN4). A further comparison is presented between the NO2 slant column densities (SCDs) measured with DOAS at Evora (Portugal) and the ones simulated with MOCRA. Both comparisons (MOCRA-MODTRAN4 and MOCRA-observations) gave more than satisfactory results, and overall make MOCRA a versatile tool for atmospheric radiative transfer simulations and interpretation of remote sensing measurements.

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References

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[Crossref]
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[Crossref]
J. Pukıte, S. Kühl, T. Deutschmann, W. Wilms-Grabe, C. Friedeburg, U. Platt, and T. Wagner, “Retrieval of stratospheric trace gases from SCIAMACHY limb measurements,” Proceedings of the First Atmospheric Science Conference, 8–12 May, ESA/ESRIN, Frascati, Italy, ESA SP-628, (2006). http://earth.esa.int/workshops/atmos2006/participants/1148/paper proc Frasc 2.pdf
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[Crossref]
J. Puķīte, S. Kühl, T. Deutschmann, U. Platt, and T. Wagner, “Accounting for the effect of horizontal gradients in limb measurements of scattered sunlight,” Atmos. Chem. Phys. 8(12), 3045–3060 (2008).
[Crossref]
G. Hönninger, C. von Friedeburg, and U. Platt, “Multi axis differential optical absorption spectroscopy (MAX-DOAS),” Atmos. Chem. Phys. 4(1), 231–254 (2004), http://www.atmos-chem-phys.net/4/231/2004/acp-4-231-2004.pdf .
[Crossref]
A. Petritoli, F. Ravegnani, G. Giovanelli, D. Bortoli, U. Bonafè, I. Kostadinov, and A. Oulanovsky, “Off-axis measurements of atmospheric trace gases by use of an airborne ultraviolet-visible spectrometer,” Appl. Opt. 41(27), 5593–5599 (2002).
[Crossref] [PubMed]
G. Giovanelli, E. Palazzi, A. Petritoli, D. Bortoli, I. Kostadinov, F. Margelli, S. Pagnutti, M. Premuda, F. Ravegnani, and G. Trivellone, “Perspectives of 2D and 3D mapping of Atmospheric Pollutants over Urban Areas by means of airborne DOAS Spectrometer,” Ann. Geophys. 49(1), 133–142 (2006).
M. Premuda, S. Masieri, D. Bortoli, I. Kostadinov, A. Petritoli, and G. Giovanelli, “Evaluation of vessel emissions in a lagoon area with ground based Multi axis DOAS measurements,” Atmos. Environ. 45(29), 5212–5219 (2011), doi:, http://dx.doi.org/10.1016/j.atmosenv.2011.05.067 .
[Crossref]
E. Palazzi, A. Petritoli, G. Giovanelli, I. Kostadinov, D. Bortoli, F. Ravegnani, and S. S. Sackey, “PROMSAR: A backward Monte Carlo spherical RTM for the analysis of DOAS remote sensing measurements,” Adv. Space Res. 36(5), 1007–1014 (2005).
[Crossref]
T. Wagner, J. P. Burrows, T. Deutschmann, B. Dix, C. von Friedeburg, U. Frieß, F. Hendrick, K.-P. Heue, H. Irie, H. Iwabuchi, Y. Kanaya, J. Keller, C. A. McLinden, H. Oetjen, E. Palazzi, A. Petritoli, U. Platt, O. Postylyakov, J. Pukite, A. Richter, M. van Roozendael, A. Rozanov, V. Rozanov, R. Sinreich, S. Sanghavi, and F. Wittrock, “Comparison of box-air-mass-factors and radiances for Multiple-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) geometries calculated from different UV/visible radiative transfer models,” Atmos. Chem. Phys. 7(7), 1809–1833 (2007).
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2011 (1)

M. Premuda, S. Masieri, D. Bortoli, I. Kostadinov, A. Petritoli, and G. Giovanelli, “Evaluation of vessel emissions in a lagoon area with ground based Multi axis DOAS measurements,” Atmos. Environ. 45(29), 5212–5219 (2011), doi:, http://dx.doi.org/10.1016/j.atmosenv.2011.05.067 .
[Crossref]

2010 (1)

B. Mayer, S. W. Hoch, and C. D. Whiteman, “Validating the MYSTIC three-dimensional radiative transfer model with observations from the complex topography of Arizona's Meteor Crater,” Atmos. Chem. Phys. Discuss. 10(5), 13373–13405 (2010), doi:.
[Crossref]

2009 (1)

M. Premuda, S. Masieri, D. Bortoli, F. Margelli, F. Ravegnani, A. Petritoli, I. Kostadinov, G. Giovanelli, and E. Cupini, “A Monte Carlo simulation of radiative transfer in the atmosphere applied to ToTaL-DOAS,” Proc. SPIE 7475, 74751A, 74751A-12 (2009).
[Crossref]

2008 (3)

D. Nowak, L. Vuilleumier, C. N. Long, and A. Ohmura, “Solar irradiance computations compared with observations at the Baseline Surface Radiation Network Payerne site,” J. Geophys. Res. 113(D14), D14206 (2008), doi:.
[Crossref]

S. Kühl, J. Puķīte, T. Deutschmann, U. Platt, and T. Wagner, “SCIAMACHY Limb Measurements of NO2, BrO and OClO, Retrieval of vertical profiles: Algorithm, first results, sensitivity and comparison studies,” Adv. Space Res. 42(10), 1747–1764 (2008).
[Crossref]

J. Puķīte, S. Kühl, T. Deutschmann, U. Platt, and T. Wagner, “Accounting for the effect of horizontal gradients in limb measurements of scattered sunlight,” Atmos. Chem. Phys. 8(12), 3045–3060 (2008).
[Crossref]

2007 (1)

T. Wagner, J. P. Burrows, T. Deutschmann, B. Dix, C. von Friedeburg, U. Frieß, F. Hendrick, K.-P. Heue, H. Irie, H. Iwabuchi, Y. Kanaya, J. Keller, C. A. McLinden, H. Oetjen, E. Palazzi, A. Petritoli, U. Platt, O. Postylyakov, J. Pukite, A. Richter, M. van Roozendael, A. Rozanov, V. Rozanov, R. Sinreich, S. Sanghavi, and F. Wittrock, “Comparison of box-air-mass-factors and radiances for Multiple-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) geometries calculated from different UV/visible radiative transfer models,” Atmos. Chem. Phys. 7(7), 1809–1833 (2007).
[Crossref]

2006 (1)

G. Giovanelli, E. Palazzi, A. Petritoli, D. Bortoli, I. Kostadinov, F. Margelli, S. Pagnutti, M. Premuda, F. Ravegnani, and G. Trivellone, “Perspectives of 2D and 3D mapping of Atmospheric Pollutants over Urban Areas by means of airborne DOAS Spectrometer,” Ann. Geophys. 49(1), 133–142 (2006).

2005 (1)

E. Palazzi, A. Petritoli, G. Giovanelli, I. Kostadinov, D. Bortoli, F. Ravegnani, and S. S. Sackey, “PROMSAR: A backward Monte Carlo spherical RTM for the analysis of DOAS remote sensing measurements,” Adv. Space Res. 36(5), 1007–1014 (2005).
[Crossref]

2004 (1)

G. Hönninger, C. von Friedeburg, and U. Platt, “Multi axis differential optical absorption spectroscopy (MAX-DOAS),” Atmos. Chem. Phys. 4(1), 231–254 (2004), http://www.atmos-chem-phys.net/4/231/2004/acp-4-231-2004.pdf .
[Crossref]

2002 (2)

A. Petritoli, F. Ravegnani, G. Giovanelli, D. Bortoli, U. Bonafè, I. Kostadinov, and A. Oulanovsky, “Off-axis measurements of atmospheric trace gases by use of an airborne ultraviolet-visible spectrometer,” Appl. Opt. 41(27), 5593–5599 (2002).
[Crossref] [PubMed]

A. Petritoli, G. Giovanelli, I. Kostadinov, F. Ravegnani, D. Bortoli, P. Bonasoni, F. Evangelisti, U. Bonafe, and F. Calzolari, “Tropospheric and stratospheric NO2 amount deduced by slant column measurements at Mt. Cimone Station,” Adv. Space Res. 29(11), 1691–1695 (2002).
[Crossref]

1999 (2)

H. K. Roscoe, P. V. Johnston, M. Van Roozendael, A. Richter, A. Sarkissian, J. Roscoe, K. E. Preston, J.-C. Lambert, C. Hermans, W. DeCuyper, S. Dzienus, T. Winterrath, J. Burrows, F. Goutail, J.-P. Pommereau, E. D’Almeida, J. Hottier, C. Coureul, R. Didier, I. Pundt, L. M. Bartlett, C. T. McElroy, J. E. Kerr, A. Elokhov, G. Giovanelli, F. Ravegnani, M. Premuda, I. Kostadinov, F. Erle, T. Wagner, K. Pfeilsticker, M. Kenntner, L. C. Marquard, M. Gil, O. Puentedura, M. Yela, D. W. Arlander, B. A. Kastad Hoiskar, C. W. Tellefsen, K. Karlsen Tornkvist, B. Heese, R. L. Jones, S. R. Aliwell, and R. A. Freshwater, “Slant column measurements of O3 and NO2 during NDSC intercomparison of zenith-sky UV-visible spectrometer in June 1996,” J. Atmos. Chem. 32(2), 281–314 (1999).

S. Chatigny, M. Morin, D. Asselin, Y. Painchaud, and P. Beaudry, “Hybrid Monte Carlo for photon transport through optically thick scattering media,” Appl. Opt. 38(28), 6075–6086 (1999).
[Crossref] [PubMed]

1996 (1)

E. Tinet, S. Avrillier, and J. M. Tualle, “Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13(9), 1903–1915 (1996).
[Crossref]

1995 (2)

A. De Matteis and S. Pagnutti, “Controlling correlations in parallel Monte Carlo,” Parallel Comput. 21(1), 73–84 (1995).
[Crossref]

A. Sarkissian, H. K. Roscoe, and D. J. Fish, “Ozone Measurement by Zenith-Sky Spectrometers: an Evaluation of Errors in Air-Mass Factors calculated by Radiative Transfer Models,” J. Quant. Spectrosc. Radiat. Transf. 54(3), 471–480 (1995).
[Crossref]

1993 (2)

A. De Matteis and S. Pagnutti, “Long-range Correlation-analysis of the Wichmann-Hill random number generator,” Stat. Comput. 3(2), 67–70 (1993).
[Crossref]

L. M. Perliski and S. Solomon, “On the Evaluation of Air Mass Factors for Atmospheric Near-Ultraviolet and Visible Absorptiion Spectroscopy,” J. Geophys. Res. 98(D6), 10363–10374 (1993).
[Crossref]

1988 (1)

K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27(12), 2502–2509 (1988).
[Crossref] [PubMed]

1987 (1)

S. Solomon, A. Schmeltekopf, and R. W. Sanders, “On the interpretation of zenith sky absorption measurements,” J. Geophys. Res. 92(D7), 8311–8319 (1987).
[Crossref]

1980 (1)

U. Platt and D. Perner, “Direct measurement of atmospheric CH2O, HNO2, O3, NO2, and SO2 by differential optical absorption in the near UV,” J. Geophys. Res. 85(C12), 7453–7458 (1980).
[Crossref]

1979 (1)

U. Platt, D. Perner, and H. W. Patz, “Simultaneous measurement of atmospheric CH2O, O3 e NO2 by differential optical absorption,” J. Geophys. Res. 84(C10), 6329–6335 (1979).
[Crossref]

1978 (2)

A. De Matteis and R. Simonini, “A New Monte Carlo Approach to the Adjoint Boltzmann Equation,” Nucl. Sci. Eng. 65, 93–105 (1978).

A. De Matteis and R. Simonini, “A Monte Carlo Biasing Scheme for the Adjoint Photon Transport,” Nucl. Sci. Eng. 67, 309–316 (1978).

1975 (1)

J. F. Noxon, “Nitrogen dioxide in the stratosphere and troposphere measured by ground-based absorption spectroscopy,” Science 189(4202), 547–549 (1975).
[Crossref] [PubMed]

Aliwell, S. R.

Arlander, D. W.

Asselin, D.

Avrillier, S.

Bartlett, L. M.

Beaudry, P.

Blättner, W. G.

W. G. Blättner, H. G. Horak, D. G. Collins, and M. B. Wells, “Monte carlo studies of the sky radiation at twilight,” Appl. Opt. 13(3), 534–547 (1974).
[Crossref] [PubMed]

Bonafe, U.

Bonafè, U.

Bonasoni, P.

Bortoli, D.

Burrows, J.

Burrows, J. P.

Calzolari, F.

Chatigny, S.

Collins, D. G.

W. G. Blättner, H. G. Horak, D. G. Collins, and M. B. Wells, “Monte carlo studies of the sky radiation at twilight,” Appl. Opt. 13(3), 534–547 (1974).
[Crossref] [PubMed]

Coureul, C.

Cupini, E.

D’Almeida, E.

De Matteis, A.

A. De Matteis and S. Pagnutti, “Controlling correlations in parallel Monte Carlo,” Parallel Comput. 21(1), 73–84 (1995).
[Crossref]

A. De Matteis and S. Pagnutti, “Long-range Correlation-analysis of the Wichmann-Hill random number generator,” Stat. Comput. 3(2), 67–70 (1993).
[Crossref]

A. De Matteis and R. Simonini, “A Monte Carlo Biasing Scheme for the Adjoint Photon Transport,” Nucl. Sci. Eng. 67, 309–316 (1978).

A. De Matteis and R. Simonini, “A New Monte Carlo Approach to the Adjoint Boltzmann Equation,” Nucl. Sci. Eng. 65, 93–105 (1978).

DeCuyper, W.

Deutschmann, T.

Didier, R.

Dix, B.

Dzienus, S.

Elokhov, A.

Erle, F.

Evangelisti, F.

Fish, D. J.

Freshwater, R. A.

Frieß, U.

Gil, M.

Giovanelli, G.

Goutail, F.

Heese, B.

Hendrick, F.

Hermans, C.

Heue, K.-P.

Hoch, S. W.

Hönninger, G.

Horak, H. G.

W. G. Blättner, H. G. Horak, D. G. Collins, and M. B. Wells, “Monte carlo studies of the sky radiation at twilight,” Appl. Opt. 13(3), 534–547 (1974).
[Crossref] [PubMed]

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Fig. 1

Multi-region 3D spherical geometry: a) a set of three cones with the vertex at the Erth centre; b) vertical half planes having the z axis as intersection line with azimuth angle ψi and ψj defined in (x,y) plane; c) a horizontal section of a possible geometry with three cones and 5, 7 half planes for the second and third cone, respectively; d) a vertical section with three cones of aperture θ1, θ2, θ3 and ground heights h1, h2 and h3.

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Fig. 2

AMFs for O3 (left) and for NO2 (right) calculated as in Eq. (13) (AMFSA, red line) and Eq. (14) (AMFWA, blue line). The red line for NO2 has been shifted upward of 0.5 to make it distinguishable from the blue line.

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Fig. 3

AMFs computed for ozone and nitrogen dioxide at three different wavelengths: 310 nm (blue), 435 nm (green), 550 nm (red). The atmospheric model used was that of MODTRAN4 for mid latitude winter, with urban aerosol extinction, visibility = 35 Km.

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Fig. 4

Absolute absorption cross sections for O3 and NO2 in the 250-650 nm spectral range at a 273 °K temperature.

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Fig. 5

Comparison between simulated (blue) and measured (cyan) NO₂ slant column densities. The measurements were carried out on 22 august 2009 at Evora (38°33′36”N, 7°54′ W), Portugal.

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Tables (3)

Table 1 Point Detector: MOCRA - MODTRAN-4 Comparison for vis = 5 Km (for wavelength and atmospheric conditions see text)*

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Table 2 Point Detector: MOCRA - MODTRAN-4 Comparison for vis = 23 Km ^a

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Table 3 Horizontal Looking Point Detector Radiance, MOCRA-MODTRAN 4 Comparison, Mid Latitude Winter Model, Urban Extinction. Units: watt/ (cm²-ster-cm⁻¹)

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Equations (16)

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\begin{matrix} I (r, Ω) = Γ (r_{s}, Ω) \exp [- \int_{0}^{| r - r_{s} |} d s^{″} k (r - s^{″} Ω)] + \\ \int_{0}^{| r - r_{s} |} d s^{'} Q (r - s^{'} Ω, Ω) \exp [- \int_{0}^{s^{'}} d s^{″} k (r - s^{″} Ω)] \end{matrix}

Q (r, Ω) = S (r) + \int_{4 π} d Ω^{'} k_{s} (r, Ω \cdot Ω^{'}) I (r, Ω^{'}) l .

{k^{'}}_{a} = k_{a} [1 - \exp (- h ν / K T)]

τ (r, r_{s}) = \int_{0}^{| r - r_{s} |} d s^{″} k (r - s^{″} Ω)

I_{T} = \int I (z) d z

I_{T} = \sum_{i} Δ z_{i} \int_{Δ z_{i}} I (z) (d z / Δ z_{i})

\begin{matrix} I_{T} = \int_{Δ Ω} I (Ω) d Ω \\ = Δ Ω \int_{Δ Ω} I (Ω) d Ω / Δ Ω \end{matrix}

k_{M s} (λ) = \frac{24 π^{3}}{N λ^{4}} {[\frac{n^{2} - 1}{n^{2} + 2}]}^{2} \frac{6 + 3 δ}{6 - 7 δ}

p (θ) = \frac{3}{16 π} \frac{2}{(2 + δ)} [(1 + δ) + (1 - δ) \cos^{2} θ]

p (θ) = \frac{1}{4 π} \frac{1 - g^{2}}{{(1 + g^{2} + 2 g \cos θ)}^{\frac{3}{2}}}

g = \iint_{4 π} p (θ) \cos θ d Ω

F O M = \frac{1}{t σ_{μ}^{2}}

A M F = - \frac{\ln (\frac{I}{I^{*}})}{δ_{a}}

A M F_{W A} = \frac{δ_{O P T}}{δ_{a}}

δ_{O P T} = \frac{\sum_{i} D_{i} I_{i}^{*}}{\sum_{i} I_{i}^{*}}

A M F_{b} = \frac{d S C D_{g}}{d V C D_{b}} = - \frac{1}{h_{b} σ_{b}} \frac{d \ln I_{g}}{d n_{b}} = \frac{1}{h_{b}} \frac{d \ln I_{g}}{d β_{b}} .

		Single scattering	Total scattering	Ground reflected	Total scattering plus ground
		radiance	radiance	radiance	reflected radiance
α^a = 180°	MODTRAN	2.12E-08	2.54E-07	1.78E-07	4.32E-07
ψ^b = 0°	MOCRA	2.12E-08	2.61E-07	1.78E-07	4.39E-07
θ^c = 30°	Diff. (%)	0.0	2.7	0.0	1.6
α^a = 180°	MODTRAN	9.09E-09	1.36E-07	7.75E-08	2.14E-07
ψ^b = 0°	MOCRA	9.07E-09	1.37E-07	7.74E-08	2.15E-07
θ^c = 60°	Diff. (%)	−0.2	0.7	−0.1	0.4
α^a = 150°	MODTRAN	2.03E-08	2.81E-07	1.58E-07	4.39E-07
ψ^b = 90°	MOCRA	2.03E-08	2.88E-07	1.57E-07	4.45E-07
θ^c = 30°	Diff. (%)	0.0	2.5	−0.6	−1.4
α^a = 150°	MODTRAN	1.04E-08	1.57E-07	6.87E-08	2.26E-07
ψ^b = 90°	MOCRA	1.04E-08	1.59E-07	6.85E-08	2.28E-07
θ^c = 60°	Diff. (%)	0.0	1.3	−0.3	0.9

		Single scattering	Total scattering	Ground reflected	Total scattering plus ground
		radiance	radiance	radiance	reflected radiance
α = 180°	MODTRAN	1.16E-08	6.85E-08	3.95E-07	4.64E-07
ψ = 0°	MOCRA	1.17E-08	6.77E-08	3.90E-07	4.57E-07
θ = 30°	Diff. (%)	0.9	−1.2	−1.3	−1.5
α = 180°	MODTRAN	6.75E-09	3.93E-08	1.97E-07	2.37E-07
ψ = 0°	MOCRA	6.78E-09	3.84E-08	1.95E-07	2.33E-07
θ = 60°	Diff. (%)	0.4	−2.7	−2.3	−1.7
α = 150°	MODTRAN	1.16E-08	7.68E-08	3.87E-07	4.64E-07
ψ = 90°	MOCRA	1.17E-08	7.53E-08	3.82E-07	4.57E-07
θ = 30°	Diff. (%)	0.9	−2.0	−1.3	−1.5
α = 150°	MODTRAN	1.16E-08	7.68E-08	3.87E-07	4.64E-07
ψ = 90°	MOCRA	1.17E-08	7.53E-08	3.82E-07	4.57E-07
θ = 60°	Diff. (%)	0.9	−2.0	−1.3	−1.5

		MOCRA		MODTRAN 4		Differences
	Solar source	Mean		Mean	Mean	Standard
λ (nm)	(watt/cm²·cm⁻¹)	Radiance		Radiance	Difference (%)	deviation (%)
310	5.1080E-07		6.272E-10	6.179E-10	2.4		4.7
435	3.2927E-06		6.131E-08	6.171E-08	1.3		2.6
550	6.1134E-06		1.648E-07	1.668E-07	−0.2		1.3
700	7.2381E-06		2.446E-07	2.502E-07	−1.8		0.5

		Single scattering	Total scattering	Ground reflected	Total scattering plus ground
		radiance	radiance	radiance	reflected radiance
α^a = 180°	MODTRAN	2.12E-08	2.54E-07	1.78E-07	4.32E-07
ψ^b = 0°	MOCRA	2.12E-08	2.61E-07	1.78E-07	4.39E-07
θ^c = 30°	Diff. (%)	0.0	2.7	0.0	1.6
α^a = 180°	MODTRAN	9.09E-09	1.36E-07	7.75E-08	2.14E-07
ψ^b = 0°	MOCRA	9.07E-09	1.37E-07	7.74E-08	2.15E-07
θ^c = 60°	Diff. (%)	−0.2	0.7	−0.1	0.4
α^a = 150°	MODTRAN	2.03E-08	2.81E-07	1.58E-07	4.39E-07
ψ^b = 90°	MOCRA	2.03E-08	2.88E-07	1.57E-07	4.45E-07
θ^c = 30°	Diff. (%)	0.0	2.5	−0.6	−1.4
α^a = 150°	MODTRAN	1.04E-08	1.57E-07	6.87E-08	2.26E-07
ψ^b = 90°	MOCRA	1.04E-08	1.59E-07	6.85E-08	2.28E-07
θ^c = 60°	Diff. (%)	0.0	1.3	−0.3	0.9

		Single scattering	Total scattering	Ground reflected	Total scattering plus ground
		radiance	radiance	radiance	reflected radiance
α = 180°	MODTRAN	1.16E-08	6.85E-08	3.95E-07	4.64E-07
ψ = 0°	MOCRA	1.17E-08	6.77E-08	3.90E-07	4.57E-07
θ = 30°	Diff. (%)	0.9	−1.2	−1.3	−1.5
α = 180°	MODTRAN	6.75E-09	3.93E-08	1.97E-07	2.37E-07
ψ = 0°	MOCRA	6.78E-09	3.84E-08	1.95E-07	2.33E-07
θ = 60°	Diff. (%)	0.4	−2.7	−2.3	−1.7
α = 150°	MODTRAN	1.16E-08	7.68E-08	3.87E-07	4.64E-07
ψ = 90°	MOCRA	1.17E-08	7.53E-08	3.82E-07	4.57E-07
θ = 30°	Diff. (%)	0.9	−2.0	−1.3	−1.5
α = 150°	MODTRAN	1.16E-08	7.68E-08	3.87E-07	4.64E-07
ψ = 90°	MOCRA	1.17E-08	7.53E-08	3.82E-07	4.57E-07
θ = 60°	Diff. (%)	0.9	−2.0	−1.3	−1.5

Abstract

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