Abstract

This paper describes the implementation of an efficient and accurate multiple scattering parameterization within the lowtran and fascode transmittance/radiance models. The parameterization is based on a stream approximation in which the local radiance field needed to evaluate the multiple scattering source function is estimated from the local radiation fluxes. The latter are calculated based on a parameterized two-flux for individual layers and an adding method for combining layers. Because of the line-by-line nature of fascode, it is straightforward to implement the multiple scattering treatment. For lowtran, an interface scheme was developedusing the k-distribution method to match the multiple scattering approach to the band model calculation of gas absorption. The interface scheme represents the lowtran band model by a sum of pseudomonochromatic calculations. The approach is valid for any band model for which k-distribution parameters can be evaluated. The accuracy of the multiple scattering parameterization has been demonstrated by comparing it with more detailed calculations for a variety of atmospheric conditions. RMS errors in radiance considering all possible viewing angles are <20%. In addition, to insure consistency between models, overlapping lowtran and fascode spectral regions are compared. Finally, it is demonstrated that the implemented multiple scattering parameterization corrects lowtran’s previous underestimation of path radiance for long horizon paths where multiple scattering is significant.

© 1987 Optical Society of America

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References

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  1. F. X. Kneizys et al., “Atmospheric Transmittance/Radiance: Computer Code lowtran6,” Report AFGL-TR-83-0287 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731, 1983).
  2. S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric Radiance and Transmittance: fascod2,” in Proceedings, Sixth Conference on Atmospheric Radiation (1986).
  3. A. Ben-Shalom, B. Barzilai, D. Cabib, A. D. Devir, S. G. Lipson, U. P. Oppenheim, “Sky Radiance at Wavelengths Between 7 and 14 μm: Measurement, Calculation, and Comparison with lowtran-4 Predictions,” Appl. Opt. 19, 838 (1980).
    [CrossRef] [PubMed]
  4. W. L. Ridgway, R. A. Moose, A. C. Cogley, “Single and Multiple scattered Solar Radiation,” Report AFGL-TR-82-0299 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731), ADA 126 323 (1982).
  5. R. G. Isaacs, “The Role of Radiative Transfer Theory in Visibility Modeling: Efficient Approximate Techniques,” Atmos. Environ. 15, 1827 (1981).
    [CrossRef]
  6. J. V. Dave, “Transfer of Visible Radiation in the Atmosphere,” Atmos. Environ. 15, 1805 (1981).
    [CrossRef]
  7. G. Stephens, “The Parameterization of Radiation for Numerical Weather Prediction and Climate Models,” Mon. Weather Rev. 112, 826 (1984).
    [CrossRef]
  8. A. Arking, K. Grossman, “The Influence of Line Shape and Band Structure on Temperatures in Planetary Atmospheres,” J. Atmos. Sci. 29, 937 (1972).
    [CrossRef]
  9. J. E. Hansen, “Radiative Transfer by Doubling Very Thin Layers,” Astrophys. J. 155, 565 (1969).
    [CrossRef]
  10. W. J. Wiscombe, G. W. Grams, “The Backscattered Fraction in Two-Stream Approximations,” J. Atmos. Sci. 33, 2440 (1976).
    [CrossRef]
  11. W. E. Meador, W. R. Weaver, “Two-Stream Approximations to Radiative Transfer in Planetary Atmospheres: A Unified Description of Existing Methods and a New Improvement,” J. Atmos. Sci. 37, 630 (1980).
    [CrossRef]
  12. W. C. Wang, P. B. Ryan, “Overlapping Effect of Atmospheric H2O, CO2 and O3 on the CO2 Radiative Effect,” Tellus 35B, 81 (1983).
    [CrossRef]
  13. W. J. Wiscombe, J. W. Evans, “Exponential-Sum Fitting of Radiative Transmission Functions,” J. Comput. Phys. 24, 416 (1977).
    [CrossRef]
  14. S. Bakan, P. Koepke, H. Quenzel, “Radiation Calculations in Absorption Bands: Comparison of Exponential Series and Path Length Distribution Method,” Beitr. Phys. Atmos. 51, 28 (1978).
  15. A. A. Lacis, W. C. Wang, J. E. Hansen, “Correlated K-Distribution Method for Radiative Transfer in Climate Models: Application to Effect of Cirrus Cloud on Climate,” NASA Publication 2076, E. R. Kreins, Ed., 416 pp. (1970).
  16. G. Yamamoto, M. Tanaka, S. Asano, “Radiative Transfer in Water Clouds in the Infrared Region,” J. Atmos. Sci. 27, 282 (1970).
    [CrossRef]
  17. G. Yamamoto, M. Tanaka, S. Asano, “Radiative Heat Transfer in Water Clouds in the Infrared Radiative,” J. Quant. Spectrosc. Radiat. Transfer 11, 697 (1971).
    [CrossRef]
  18. J. E. Hansen, “Exact and Approximate Solutions for Multiple Scattering by Cloudy and Hazy Planetary Atmospheres,” J. Atmos. Sci. 36, 478 (1969b).
    [CrossRef]
  19. R. G. Isaacs, W.-C. Wang, R. D. Worsham, S. Goldenberg, “Multiple Scattering Treatment for Use in the lowtran and fascode Models,” Report AFGL-TR-86-0073 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731 (1986).
  20. K. N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmospheres,” J. Atmos. Sci. 30, 1303 (1973).
    [CrossRef]
  21. J. V. Dave, “Development of Programs for Computing Characteristics of Ultraviolet Radiation. Programs I–IV,” Final Report, Contract NAS5-21680 (NASA Goddard Space Flight Center, Greenbelt, MD 20771, 1972).

1984

G. Stephens, “The Parameterization of Radiation for Numerical Weather Prediction and Climate Models,” Mon. Weather Rev. 112, 826 (1984).
[CrossRef]

1983

W. C. Wang, P. B. Ryan, “Overlapping Effect of Atmospheric H2O, CO2 and O3 on the CO2 Radiative Effect,” Tellus 35B, 81 (1983).
[CrossRef]

1981

R. G. Isaacs, “The Role of Radiative Transfer Theory in Visibility Modeling: Efficient Approximate Techniques,” Atmos. Environ. 15, 1827 (1981).
[CrossRef]

J. V. Dave, “Transfer of Visible Radiation in the Atmosphere,” Atmos. Environ. 15, 1805 (1981).
[CrossRef]

1980

A. Ben-Shalom, B. Barzilai, D. Cabib, A. D. Devir, S. G. Lipson, U. P. Oppenheim, “Sky Radiance at Wavelengths Between 7 and 14 μm: Measurement, Calculation, and Comparison with lowtran-4 Predictions,” Appl. Opt. 19, 838 (1980).
[CrossRef] [PubMed]

W. E. Meador, W. R. Weaver, “Two-Stream Approximations to Radiative Transfer in Planetary Atmospheres: A Unified Description of Existing Methods and a New Improvement,” J. Atmos. Sci. 37, 630 (1980).
[CrossRef]

1978

S. Bakan, P. Koepke, H. Quenzel, “Radiation Calculations in Absorption Bands: Comparison of Exponential Series and Path Length Distribution Method,” Beitr. Phys. Atmos. 51, 28 (1978).

1977

W. J. Wiscombe, J. W. Evans, “Exponential-Sum Fitting of Radiative Transmission Functions,” J. Comput. Phys. 24, 416 (1977).
[CrossRef]

1976

W. J. Wiscombe, G. W. Grams, “The Backscattered Fraction in Two-Stream Approximations,” J. Atmos. Sci. 33, 2440 (1976).
[CrossRef]

1973

K. N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmospheres,” J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

1972

A. Arking, K. Grossman, “The Influence of Line Shape and Band Structure on Temperatures in Planetary Atmospheres,” J. Atmos. Sci. 29, 937 (1972).
[CrossRef]

1971

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Heat Transfer in Water Clouds in the Infrared Radiative,” J. Quant. Spectrosc. Radiat. Transfer 11, 697 (1971).
[CrossRef]

1970

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Transfer in Water Clouds in the Infrared Region,” J. Atmos. Sci. 27, 282 (1970).
[CrossRef]

1969

J. E. Hansen, “Exact and Approximate Solutions for Multiple Scattering by Cloudy and Hazy Planetary Atmospheres,” J. Atmos. Sci. 36, 478 (1969b).
[CrossRef]

J. E. Hansen, “Radiative Transfer by Doubling Very Thin Layers,” Astrophys. J. 155, 565 (1969).
[CrossRef]

Anderson, G. P.

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric Radiance and Transmittance: fascod2,” in Proceedings, Sixth Conference on Atmospheric Radiation (1986).

Arking, A.

A. Arking, K. Grossman, “The Influence of Line Shape and Band Structure on Temperatures in Planetary Atmospheres,” J. Atmos. Sci. 29, 937 (1972).
[CrossRef]

Asano, S.

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Heat Transfer in Water Clouds in the Infrared Radiative,” J. Quant. Spectrosc. Radiat. Transfer 11, 697 (1971).
[CrossRef]

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Transfer in Water Clouds in the Infrared Region,” J. Atmos. Sci. 27, 282 (1970).
[CrossRef]

Bakan, S.

S. Bakan, P. Koepke, H. Quenzel, “Radiation Calculations in Absorption Bands: Comparison of Exponential Series and Path Length Distribution Method,” Beitr. Phys. Atmos. 51, 28 (1978).

Barzilai, B.

Ben-Shalom, A.

Cabib, D.

Clough, S. A.

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric Radiance and Transmittance: fascod2,” in Proceedings, Sixth Conference on Atmospheric Radiation (1986).

Cogley, A. C.

W. L. Ridgway, R. A. Moose, A. C. Cogley, “Single and Multiple scattered Solar Radiation,” Report AFGL-TR-82-0299 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731), ADA 126 323 (1982).

Dave, J. V.

J. V. Dave, “Transfer of Visible Radiation in the Atmosphere,” Atmos. Environ. 15, 1805 (1981).
[CrossRef]

J. V. Dave, “Development of Programs for Computing Characteristics of Ultraviolet Radiation. Programs I–IV,” Final Report, Contract NAS5-21680 (NASA Goddard Space Flight Center, Greenbelt, MD 20771, 1972).

Devir, A. D.

Evans, J. W.

W. J. Wiscombe, J. W. Evans, “Exponential-Sum Fitting of Radiative Transmission Functions,” J. Comput. Phys. 24, 416 (1977).
[CrossRef]

Goldenberg, S.

R. G. Isaacs, W.-C. Wang, R. D. Worsham, S. Goldenberg, “Multiple Scattering Treatment for Use in the lowtran and fascode Models,” Report AFGL-TR-86-0073 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731 (1986).

Grams, G. W.

W. J. Wiscombe, G. W. Grams, “The Backscattered Fraction in Two-Stream Approximations,” J. Atmos. Sci. 33, 2440 (1976).
[CrossRef]

Grossman, K.

A. Arking, K. Grossman, “The Influence of Line Shape and Band Structure on Temperatures in Planetary Atmospheres,” J. Atmos. Sci. 29, 937 (1972).
[CrossRef]

Hansen, J. E.

J. E. Hansen, “Radiative Transfer by Doubling Very Thin Layers,” Astrophys. J. 155, 565 (1969).
[CrossRef]

J. E. Hansen, “Exact and Approximate Solutions for Multiple Scattering by Cloudy and Hazy Planetary Atmospheres,” J. Atmos. Sci. 36, 478 (1969b).
[CrossRef]

A. A. Lacis, W. C. Wang, J. E. Hansen, “Correlated K-Distribution Method for Radiative Transfer in Climate Models: Application to Effect of Cirrus Cloud on Climate,” NASA Publication 2076, E. R. Kreins, Ed., 416 pp. (1970).

Isaacs, R. G.

R. G. Isaacs, “The Role of Radiative Transfer Theory in Visibility Modeling: Efficient Approximate Techniques,” Atmos. Environ. 15, 1827 (1981).
[CrossRef]

R. G. Isaacs, W.-C. Wang, R. D. Worsham, S. Goldenberg, “Multiple Scattering Treatment for Use in the lowtran and fascode Models,” Report AFGL-TR-86-0073 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731 (1986).

Kneizys, F. X.

F. X. Kneizys et al., “Atmospheric Transmittance/Radiance: Computer Code lowtran6,” Report AFGL-TR-83-0287 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731, 1983).

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric Radiance and Transmittance: fascod2,” in Proceedings, Sixth Conference on Atmospheric Radiation (1986).

Koepke, P.

S. Bakan, P. Koepke, H. Quenzel, “Radiation Calculations in Absorption Bands: Comparison of Exponential Series and Path Length Distribution Method,” Beitr. Phys. Atmos. 51, 28 (1978).

Lacis, A. A.

A. A. Lacis, W. C. Wang, J. E. Hansen, “Correlated K-Distribution Method for Radiative Transfer in Climate Models: Application to Effect of Cirrus Cloud on Climate,” NASA Publication 2076, E. R. Kreins, Ed., 416 pp. (1970).

Liou, K. N.

K. N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmospheres,” J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

Lipson, S. G.

Meador, W. E.

W. E. Meador, W. R. Weaver, “Two-Stream Approximations to Radiative Transfer in Planetary Atmospheres: A Unified Description of Existing Methods and a New Improvement,” J. Atmos. Sci. 37, 630 (1980).
[CrossRef]

Moose, R. A.

W. L. Ridgway, R. A. Moose, A. C. Cogley, “Single and Multiple scattered Solar Radiation,” Report AFGL-TR-82-0299 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731), ADA 126 323 (1982).

Oppenheim, U. P.

Quenzel, H.

S. Bakan, P. Koepke, H. Quenzel, “Radiation Calculations in Absorption Bands: Comparison of Exponential Series and Path Length Distribution Method,” Beitr. Phys. Atmos. 51, 28 (1978).

Ridgway, W. L.

W. L. Ridgway, R. A. Moose, A. C. Cogley, “Single and Multiple scattered Solar Radiation,” Report AFGL-TR-82-0299 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731), ADA 126 323 (1982).

Ryan, P. B.

W. C. Wang, P. B. Ryan, “Overlapping Effect of Atmospheric H2O, CO2 and O3 on the CO2 Radiative Effect,” Tellus 35B, 81 (1983).
[CrossRef]

Shettle, E. P.

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric Radiance and Transmittance: fascod2,” in Proceedings, Sixth Conference on Atmospheric Radiation (1986).

Stephens, G.

G. Stephens, “The Parameterization of Radiation for Numerical Weather Prediction and Climate Models,” Mon. Weather Rev. 112, 826 (1984).
[CrossRef]

Tanaka, M.

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Heat Transfer in Water Clouds in the Infrared Radiative,” J. Quant. Spectrosc. Radiat. Transfer 11, 697 (1971).
[CrossRef]

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Transfer in Water Clouds in the Infrared Region,” J. Atmos. Sci. 27, 282 (1970).
[CrossRef]

Wang, W. C.

W. C. Wang, P. B. Ryan, “Overlapping Effect of Atmospheric H2O, CO2 and O3 on the CO2 Radiative Effect,” Tellus 35B, 81 (1983).
[CrossRef]

A. A. Lacis, W. C. Wang, J. E. Hansen, “Correlated K-Distribution Method for Radiative Transfer in Climate Models: Application to Effect of Cirrus Cloud on Climate,” NASA Publication 2076, E. R. Kreins, Ed., 416 pp. (1970).

Wang, W.-C.

R. G. Isaacs, W.-C. Wang, R. D. Worsham, S. Goldenberg, “Multiple Scattering Treatment for Use in the lowtran and fascode Models,” Report AFGL-TR-86-0073 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731 (1986).

Weaver, W. R.

W. E. Meador, W. R. Weaver, “Two-Stream Approximations to Radiative Transfer in Planetary Atmospheres: A Unified Description of Existing Methods and a New Improvement,” J. Atmos. Sci. 37, 630 (1980).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, J. W. Evans, “Exponential-Sum Fitting of Radiative Transmission Functions,” J. Comput. Phys. 24, 416 (1977).
[CrossRef]

W. J. Wiscombe, G. W. Grams, “The Backscattered Fraction in Two-Stream Approximations,” J. Atmos. Sci. 33, 2440 (1976).
[CrossRef]

Worsham, R. D.

R. G. Isaacs, W.-C. Wang, R. D. Worsham, S. Goldenberg, “Multiple Scattering Treatment for Use in the lowtran and fascode Models,” Report AFGL-TR-86-0073 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731 (1986).

Yamamoto, G.

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Heat Transfer in Water Clouds in the Infrared Radiative,” J. Quant. Spectrosc. Radiat. Transfer 11, 697 (1971).
[CrossRef]

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Transfer in Water Clouds in the Infrared Region,” J. Atmos. Sci. 27, 282 (1970).
[CrossRef]

Appl. Opt.

Astrophys. J.

J. E. Hansen, “Radiative Transfer by Doubling Very Thin Layers,” Astrophys. J. 155, 565 (1969).
[CrossRef]

Atmos. Environ.

R. G. Isaacs, “The Role of Radiative Transfer Theory in Visibility Modeling: Efficient Approximate Techniques,” Atmos. Environ. 15, 1827 (1981).
[CrossRef]

J. V. Dave, “Transfer of Visible Radiation in the Atmosphere,” Atmos. Environ. 15, 1805 (1981).
[CrossRef]

Beitr. Phys. Atmos.

S. Bakan, P. Koepke, H. Quenzel, “Radiation Calculations in Absorption Bands: Comparison of Exponential Series and Path Length Distribution Method,” Beitr. Phys. Atmos. 51, 28 (1978).

J. Atmos. Sci.

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Transfer in Water Clouds in the Infrared Region,” J. Atmos. Sci. 27, 282 (1970).
[CrossRef]

A. Arking, K. Grossman, “The Influence of Line Shape and Band Structure on Temperatures in Planetary Atmospheres,” J. Atmos. Sci. 29, 937 (1972).
[CrossRef]

W. J. Wiscombe, G. W. Grams, “The Backscattered Fraction in Two-Stream Approximations,” J. Atmos. Sci. 33, 2440 (1976).
[CrossRef]

W. E. Meador, W. R. Weaver, “Two-Stream Approximations to Radiative Transfer in Planetary Atmospheres: A Unified Description of Existing Methods and a New Improvement,” J. Atmos. Sci. 37, 630 (1980).
[CrossRef]

J. E. Hansen, “Exact and Approximate Solutions for Multiple Scattering by Cloudy and Hazy Planetary Atmospheres,” J. Atmos. Sci. 36, 478 (1969b).
[CrossRef]

K. N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmospheres,” J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

J. Comput. Phys.

W. J. Wiscombe, J. W. Evans, “Exponential-Sum Fitting of Radiative Transmission Functions,” J. Comput. Phys. 24, 416 (1977).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

G. Yamamoto, M. Tanaka, S. Asano, “Radiative Heat Transfer in Water Clouds in the Infrared Radiative,” J. Quant. Spectrosc. Radiat. Transfer 11, 697 (1971).
[CrossRef]

Mon. Weather Rev.

G. Stephens, “The Parameterization of Radiation for Numerical Weather Prediction and Climate Models,” Mon. Weather Rev. 112, 826 (1984).
[CrossRef]

Tellus

W. C. Wang, P. B. Ryan, “Overlapping Effect of Atmospheric H2O, CO2 and O3 on the CO2 Radiative Effect,” Tellus 35B, 81 (1983).
[CrossRef]

Other

W. L. Ridgway, R. A. Moose, A. C. Cogley, “Single and Multiple scattered Solar Radiation,” Report AFGL-TR-82-0299 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731), ADA 126 323 (1982).

F. X. Kneizys et al., “Atmospheric Transmittance/Radiance: Computer Code lowtran6,” Report AFGL-TR-83-0287 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731, 1983).

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric Radiance and Transmittance: fascod2,” in Proceedings, Sixth Conference on Atmospheric Radiation (1986).

A. A. Lacis, W. C. Wang, J. E. Hansen, “Correlated K-Distribution Method for Radiative Transfer in Climate Models: Application to Effect of Cirrus Cloud on Climate,” NASA Publication 2076, E. R. Kreins, Ed., 416 pp. (1970).

J. V. Dave, “Development of Programs for Computing Characteristics of Ultraviolet Radiation. Programs I–IV,” Final Report, Contract NAS5-21680 (NASA Goddard Space Flight Center, Greenbelt, MD 20771, 1972).

R. G. Isaacs, W.-C. Wang, R. D. Worsham, S. Goldenberg, “Multiple Scattering Treatment for Use in the lowtran and fascode Models,” Report AFGL-TR-86-0073 (Air Force Geophysics Laboratory, Hanscom AFB, MA 01731 (1986).

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Figures (10)

Fig. 1
Fig. 1

AFGL transmittance/radiance code structure: (a) present models (solid line), (b) multiple scattering models (dashed line).

Fig. 2
Fig. 2

Comparison of lowtran transmission (line) to that calculated from the k-distribution model assuming no overlap (+) and including overlap (▲).

Fig. 3
Fig. 3

Comparison of fascode, fascode MS, and DOM for upward radiance [units: ergs s−1 cm−2 st−1 (cm−1) −1].

Fig. 4
Fig. 4

Same as in Fig. 3 but for downward radiance.

Fig. 5
Fig. 5

lowtran 6 upward radiance (solar, 0.55 μm) with and without multiple scattering compared to exact results. Solar zenith angle is 60°.

Fig. 6
Fig. 6

Same as in Fig. 5 but for downward radiance.

Fig. 7
Fig. 7

Percent error of lowtran 6 multiple scatter calculations relative to exact (Dave) results for optical depths between 0.25 and 1.00 with a surface reflectance of 0.0.

Fig. 8
Fig. 8

Comparisons of lowtran and fascode upward thermal radiance (with multiple scattering) between 850 and 1150 cm−1.

Fig. 9
Fig. 9

Comparison of lowtran and fascode downward thermal radiance (with multiple scattering) between 850 and 1150 cm−1.

Fig. 10
Fig. 10

Downward thermal radiance comparison between 8 and 13.5 μm: Blackbody radiance of lowest atmospheric layer, standard lowtran (without multiple scattering), Ben-Shalom modification, and lowtran (with multiple scattering). Viewing path is from the surface to the horizon.

Tables (4)

Tables Icon

Table I Set of ki, and Δgi Values Used to Approximate the lowtran 6 Transmission Data

Tables Icon

Table II Optical Properties and Layer and Surface Temperatures for fascode Comparison Cases

Tables Icon

Table III Case 2: Radiance Budgets [Unattenuated Surface Emission Corresponds to 120 Radiance Units; Units = erg s−1 cm−2 sr−1 (cm−1)−1]

Tables Icon

Table IV Comparison of Solar Multiply Scattered Emergent Fluxes (Normalized to π Units of Incident Irradiance) from lowtran Subroutine (L) and Exact Calculation (E)

Equations (34)

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

μ d d τ I ν ( τ , μ , ϕ ) = I ν ( τ , μ , ϕ ) - J ν ( τ , μ , ϕ ) .
τ ν ( z ) = z [ k a ( z ) + k s ( z ) + σ a ( z ) + σ s ( z ) ] d z .
J ( τ , μ , ϕ ) = J 0 ( τ , μ , ϕ ) + J MS ( τ , μ , ϕ ) ,
J 0 ( τ , μ , ϕ ) = ω 0 ( τ ) 4 π π F exp ( - τ / μ 0 ) P ( Ω ; - Ω 0 ) + [ 1 - ω 0 ( τ ) ] B [ θ ( τ ) ] ,
J MS ( τ , μ , ϕ ) = ω 0 ( τ ) 4 π Ω P ( Ω ; Ω ) I ( τ , Ω ) d Ω .
ω 0 ( τ ) = Δ τ s / Δ τ ,
I b ( 0 , - μ , ϕ ) = 0.0.
F ± ( τ ) = 0 2 π 0 1 I ( τ , ± μ , ϕ ) μ d μ d ϕ .
I b ( τ * , μ , ϕ ) = r 4 [ π F μ 0 exp ( - τ * / μ 0 ) + 0 2 π 0 1 I ( τ * , - μ , ϕ ) μ d μ d ϕ ] + ( 1 - r ) B [ θ ( τ * ) ] .
I ( τ , + μ , ϕ ) = I b ( τ * , μ , ϕ ) exp [ - ( τ * - τ ) / μ ] + τ τ * J ( t , μ , ϕ ) exp [ - ( t - τ ) / μ ] d t μ ,
I ( τ , - μ , ϕ ) = I b ( 0 , - μ , ϕ ) exp ( - τ / μ ) + 0 τ J ( t , μ , ϕ ) exp [ - ( τ - t ) / μ ] d t μ ,
I ( τ , μ , ϕ ) = { r π [ π μ 0 F exp ( - τ * / μ 0 ) + 0 1 0 2 π I ( τ * , - μ , ϕ ) μ d μ d ϕ ] + ( 1 - r ) B [ θ ( τ * ) ] } exp - ( τ * - τ ) / μ + 0 τ * J ( t , μ , ϕ ) exp [ - ( τ - t ) / μ ] d t μ ,
I ( τ , - μ , ϕ ) = 0 τ J ( t , μ , ϕ ) exp [ - ( τ - t ) / μ ] d t μ .
J MS ( τ , μ , ϕ ) ω 0 ( τ ) 4 π [ I + ( τ ) Ω + P ( Ω , Ω + ) d Ω + + I - ( τ ) Ω - P ( Ω , Ω - ) d Ω - ] .
I ± ( τ ) = F ± ( τ ) / π
J MS ( τ , ± μ , ϕ ) ω 0 ( τ ) π { F ± ( τ ) [ 1 - β ( μ ) ] + F ( τ ) β ( μ ) } .
J SA N ( ± μ ) = J 0 + ω 0 N π { F N ± [ 1 - β N ( μ , g N ) ] + F N β N ( μ , g N ) } .
R = F + / μ 0 π F ,
T = F - / μ 0 π F + exp ( - τ / μ ) .
F 1 N + = F N + + T N ( F 1 N - 1 + + F N - R N - 1 + ) ( 1 - R N R N - 1 + ) - 1 ,
R N + = R N + R N - 1 + T N 2 ( 1 - R N R N - 1 + ) - 1 .
F N + = ( F 1 N + + F 1 N + 1 - R N + ) ( 1 - R N + R N + 1 - ) - 1 ,
F N + 1 - = ( F 1 N + 1 - + F 1 N + R N + 1 + ) ( 1 - R N + R N + 1 - ) - 1 .
T Δ ν ( u ) 1 Δ ν Δ ν exp ( - k u ) d ν 0 f ( k ) exp ( - k u ) d k 0 1 exp ( - k u ) d g i = 1 N exp ( - k i u ) Δ g i ,
T Δ ν ( u 1 , u 2 ) = T 1 T 2 T m c T m s T a e ,
T 1 = T ( H 2 O + ) = i = j M exp ( - k 1 i u 1 ) Δ g 1 i ,
T 2 = T ( O 3 ) = i = 1 N exp ( - k 2 j u 2 ) Δ g 2 j .
T 1 = T ( H 2 O + ) = j = 1 6 T 1 j Δ g 2 j .
T 11 = [ T 11 Δ g 11 + T 12 Δ g 12 + T 13 Δ g 13 + T 13 Δ g 13 + T 14 ( g 21 - g 13 ) ] / Δ g 21 ,
T 1 · T 2 = [ j = 1 6 exp ( - k 1 j u 1 ) Δ g 2 j ] · [ j = 1 6 exp ( - k 2 j u 2 ) Δ g 2 j ] .
T 1 · T 2 j = 1 6 exp [ - k 1 j u 1 + k 2 j u 2 ] Δ g 2 j .
T j = T 1 j · T 2 j ,             i = 1 , 2 , 6.
τ j = - ln ( T j ) .
k j ( P , θ ) = k j ( P 0 , θ 0 ) P P 0 θ 0 / θ ,

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