Abstract

The k distribution for the exponential band model has been known for some time but requires intensive computation. Here a new expression is given that can be evaluated rapidly, and example calculations for water vapor are presented.

© 2001 Optical Society of America

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References

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  1. W. M. Irvine, “The formation of absorption bands and the distribution of photon optical paths in a scattering atmosphere,” Bull. Astron. Inst. Neth. 17, 266–279 (1964).
  2. S. Bakan, H. Quenzel, “Path length distributions of photons scattered in turbid atmospheres,” Beit. Phys. Atmos. 49, 272–284 (1976).
  3. S. Bakan, P. Koepke, H. Quenzel, “Radiation calculations in absorption bands: comparison of exponential series and path length distribution—method,” Beit. Phys. Atmos. 51, 28–30 (1976).
  4. W. Wiscombe, J. Evans, “Exponential sum fitting of radiative transmission functions,” J. Comput. Phys. 24, 416–444 (1977).
    [CrossRef]
  5. R. Goody, R. West, L. Chen, D. Crisp, “The correlated-k method for radiation calculations in nonhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 42, 539–550 (1989).
    [CrossRef]
  6. A. A. Lacis, V. Oinas, “A description of the correlated k-distribution method for modelling nongrey gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres,” J. Geophys. Res. 96, 9027–9063 (1991).
    [CrossRef]
  7. Q. Fu, K. N. Liou, “On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres,” J. Atmos. Sci. 49, 2139–2166 (1992).
    [CrossRef]
  8. D. P. Kratz, F. G. Rose, “Accounting for absorption in the spectral range of the CERES window channel,” J. Quant. Spectrosc. Radiat. Transfer 61, 83–95 (1999).
    [CrossRef]
  9. R. M. Goody, Y. L. Yung, Atmospheric Radiation (Oxford U. Press, Oxford, 1989).
  10. R. M. Goody, “A statistical model for water vapour absorption,” Q. J. R. Meteorol. Soc. 78, 165–169 (1952).
    [CrossRef]
  11. G. A. Domoto, “Frequency integration for radiative transfer problems involving homogeneous non-gray gases: the inverse transmission function,” J. Quant. Spectrosc. Radiat. Transfer 14, 935–942 (1974).
    [CrossRef]
  12. P. van Dooren, L. de Ridder, “An adaptive algorithm for numerical integration over an n-dimensional cube,” J. Comput. Appl. Math. 2, 207–217 (1976).
    [CrossRef]
  13. O. Marin, R. O. Buckius, “Wideband correlated-k method applied to absorbing, emitting, and scattering media,” J. Thermophys. Heat Transfer 10, 364–371 (1996).
    [CrossRef]
  14. I. S. Gradshteyn, I. M. Ryzhik, eds., Tables of Integrals, Series, and Products, corrected and enlarged edition (Academic, San Diego, Calif., 1980).
  15. E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, M. J. Morcrette, “The second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
    [CrossRef]

1999 (1)

D. P. Kratz, F. G. Rose, “Accounting for absorption in the spectral range of the CERES window channel,” J. Quant. Spectrosc. Radiat. Transfer 61, 83–95 (1999).
[CrossRef]

1997 (1)

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

1996 (1)

O. Marin, R. O. Buckius, “Wideband correlated-k method applied to absorbing, emitting, and scattering media,” J. Thermophys. Heat Transfer 10, 364–371 (1996).
[CrossRef]

1992 (1)

Q. Fu, K. N. Liou, “On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres,” J. Atmos. Sci. 49, 2139–2166 (1992).
[CrossRef]

1991 (1)

A. A. Lacis, V. Oinas, “A description of the correlated k-distribution method for modelling nongrey gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres,” J. Geophys. Res. 96, 9027–9063 (1991).
[CrossRef]

1989 (1)

R. Goody, R. West, L. Chen, D. Crisp, “The correlated-k method for radiation calculations in nonhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 42, 539–550 (1989).
[CrossRef]

1977 (1)

W. Wiscombe, J. Evans, “Exponential sum fitting of radiative transmission functions,” J. Comput. Phys. 24, 416–444 (1977).
[CrossRef]

1976 (3)

S. Bakan, H. Quenzel, “Path length distributions of photons scattered in turbid atmospheres,” Beit. Phys. Atmos. 49, 272–284 (1976).

S. Bakan, P. Koepke, H. Quenzel, “Radiation calculations in absorption bands: comparison of exponential series and path length distribution—method,” Beit. Phys. Atmos. 51, 28–30 (1976).

P. van Dooren, L. de Ridder, “An adaptive algorithm for numerical integration over an n-dimensional cube,” J. Comput. Appl. Math. 2, 207–217 (1976).
[CrossRef]

1974 (1)

G. A. Domoto, “Frequency integration for radiative transfer problems involving homogeneous non-gray gases: the inverse transmission function,” J. Quant. Spectrosc. Radiat. Transfer 14, 935–942 (1974).
[CrossRef]

1964 (1)

W. M. Irvine, “The formation of absorption bands and the distribution of photon optical paths in a scattering atmosphere,” Bull. Astron. Inst. Neth. 17, 266–279 (1964).

1952 (1)

R. M. Goody, “A statistical model for water vapour absorption,” Q. J. R. Meteorol. Soc. 78, 165–169 (1952).
[CrossRef]

Bakan, S.

S. Bakan, P. Koepke, H. Quenzel, “Radiation calculations in absorption bands: comparison of exponential series and path length distribution—method,” Beit. Phys. Atmos. 51, 28–30 (1976).

S. Bakan, H. Quenzel, “Path length distributions of photons scattered in turbid atmospheres,” Beit. Phys. Atmos. 49, 272–284 (1976).

Buckius, R. O.

O. Marin, R. O. Buckius, “Wideband correlated-k method applied to absorbing, emitting, and scattering media,” J. Thermophys. Heat Transfer 10, 364–371 (1996).
[CrossRef]

Chen, L.

R. Goody, R. West, L. Chen, D. Crisp, “The correlated-k method for radiation calculations in nonhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 42, 539–550 (1989).
[CrossRef]

Crisp, D.

R. Goody, R. West, L. Chen, D. Crisp, “The correlated-k method for radiation calculations in nonhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 42, 539–550 (1989).
[CrossRef]

de Ridder, L.

P. van Dooren, L. de Ridder, “An adaptive algorithm for numerical integration over an n-dimensional cube,” J. Comput. Appl. Math. 2, 207–217 (1976).
[CrossRef]

Deuze, J. L.

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

Domoto, G. A.

G. A. Domoto, “Frequency integration for radiative transfer problems involving homogeneous non-gray gases: the inverse transmission function,” J. Quant. Spectrosc. Radiat. Transfer 14, 935–942 (1974).
[CrossRef]

Evans, J.

W. Wiscombe, J. Evans, “Exponential sum fitting of radiative transmission functions,” J. Comput. Phys. 24, 416–444 (1977).
[CrossRef]

Fu, Q.

Q. Fu, K. N. Liou, “On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres,” J. Atmos. Sci. 49, 2139–2166 (1992).
[CrossRef]

Goody, R.

R. Goody, R. West, L. Chen, D. Crisp, “The correlated-k method for radiation calculations in nonhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 42, 539–550 (1989).
[CrossRef]

Goody, R. M.

R. M. Goody, “A statistical model for water vapour absorption,” Q. J. R. Meteorol. Soc. 78, 165–169 (1952).
[CrossRef]

R. M. Goody, Y. L. Yung, Atmospheric Radiation (Oxford U. Press, Oxford, 1989).

Herman, M.

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

Irvine, W. M.

W. M. Irvine, “The formation of absorption bands and the distribution of photon optical paths in a scattering atmosphere,” Bull. Astron. Inst. Neth. 17, 266–279 (1964).

Koepke, P.

S. Bakan, P. Koepke, H. Quenzel, “Radiation calculations in absorption bands: comparison of exponential series and path length distribution—method,” Beit. Phys. Atmos. 51, 28–30 (1976).

Kratz, D. P.

D. P. Kratz, F. G. Rose, “Accounting for absorption in the spectral range of the CERES window channel,” J. Quant. Spectrosc. Radiat. Transfer 61, 83–95 (1999).
[CrossRef]

Lacis, A. A.

A. A. Lacis, V. Oinas, “A description of the correlated k-distribution method for modelling nongrey gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres,” J. Geophys. Res. 96, 9027–9063 (1991).
[CrossRef]

Liou, K. N.

Q. Fu, K. N. Liou, “On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres,” J. Atmos. Sci. 49, 2139–2166 (1992).
[CrossRef]

Marin, O.

O. Marin, R. O. Buckius, “Wideband correlated-k method applied to absorbing, emitting, and scattering media,” J. Thermophys. Heat Transfer 10, 364–371 (1996).
[CrossRef]

Morcrette, M. J.

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

Oinas, V.

A. A. Lacis, V. Oinas, “A description of the correlated k-distribution method for modelling nongrey gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres,” J. Geophys. Res. 96, 9027–9063 (1991).
[CrossRef]

Quenzel, H.

S. Bakan, P. Koepke, H. Quenzel, “Radiation calculations in absorption bands: comparison of exponential series and path length distribution—method,” Beit. Phys. Atmos. 51, 28–30 (1976).

S. Bakan, H. Quenzel, “Path length distributions of photons scattered in turbid atmospheres,” Beit. Phys. Atmos. 49, 272–284 (1976).

Rose, F. G.

D. P. Kratz, F. G. Rose, “Accounting for absorption in the spectral range of the CERES window channel,” J. Quant. Spectrosc. Radiat. Transfer 61, 83–95 (1999).
[CrossRef]

Tanre, D.

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

van Dooren, P.

P. van Dooren, L. de Ridder, “An adaptive algorithm for numerical integration over an n-dimensional cube,” J. Comput. Appl. Math. 2, 207–217 (1976).
[CrossRef]

Vermote, E. F.

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

West, R.

R. Goody, R. West, L. Chen, D. Crisp, “The correlated-k method for radiation calculations in nonhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 42, 539–550 (1989).
[CrossRef]

Wiscombe, W.

W. Wiscombe, J. Evans, “Exponential sum fitting of radiative transmission functions,” J. Comput. Phys. 24, 416–444 (1977).
[CrossRef]

Yung, Y. L.

R. M. Goody, Y. L. Yung, Atmospheric Radiation (Oxford U. Press, Oxford, 1989).

Beit. Phys. Atmos. (2)

S. Bakan, H. Quenzel, “Path length distributions of photons scattered in turbid atmospheres,” Beit. Phys. Atmos. 49, 272–284 (1976).

S. Bakan, P. Koepke, H. Quenzel, “Radiation calculations in absorption bands: comparison of exponential series and path length distribution—method,” Beit. Phys. Atmos. 51, 28–30 (1976).

Bull. Astron. Inst. Neth. (1)

W. M. Irvine, “The formation of absorption bands and the distribution of photon optical paths in a scattering atmosphere,” Bull. Astron. Inst. Neth. 17, 266–279 (1964).

IEEE Trans. Geosci. Remote Sens. (1)

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

J. Atmos. Sci. (1)

Q. Fu, K. N. Liou, “On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres,” J. Atmos. Sci. 49, 2139–2166 (1992).
[CrossRef]

J. Comput. Appl. Math. (1)

P. van Dooren, L. de Ridder, “An adaptive algorithm for numerical integration over an n-dimensional cube,” J. Comput. Appl. Math. 2, 207–217 (1976).
[CrossRef]

J. Comput. Phys. (1)

W. Wiscombe, J. Evans, “Exponential sum fitting of radiative transmission functions,” J. Comput. Phys. 24, 416–444 (1977).
[CrossRef]

J. Geophys. Res. (1)

A. A. Lacis, V. Oinas, “A description of the correlated k-distribution method for modelling nongrey gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres,” J. Geophys. Res. 96, 9027–9063 (1991).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (3)

D. P. Kratz, F. G. Rose, “Accounting for absorption in the spectral range of the CERES window channel,” J. Quant. Spectrosc. Radiat. Transfer 61, 83–95 (1999).
[CrossRef]

R. Goody, R. West, L. Chen, D. Crisp, “The correlated-k method for radiation calculations in nonhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 42, 539–550 (1989).
[CrossRef]

G. A. Domoto, “Frequency integration for radiative transfer problems involving homogeneous non-gray gases: the inverse transmission function,” J. Quant. Spectrosc. Radiat. Transfer 14, 935–942 (1974).
[CrossRef]

J. Thermophys. Heat Transfer (1)

O. Marin, R. O. Buckius, “Wideband correlated-k method applied to absorbing, emitting, and scattering media,” J. Thermophys. Heat Transfer 10, 364–371 (1996).
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

R. M. Goody, “A statistical model for water vapour absorption,” Q. J. R. Meteorol. Soc. 78, 165–169 (1952).
[CrossRef]

Other (2)

I. S. Gradshteyn, I. M. Ryzhik, eds., Tables of Integrals, Series, and Products, corrected and enlarged edition (Academic, San Diego, Calif., 1980).

R. M. Goody, Y. L. Yung, Atmospheric Radiation (Oxford U. Press, Oxford, 1989).

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Figures (2)

Fig. 1
Fig. 1

Water-vapor transmission spectrum calculated from the k distributions of the the 6S exponential band model at several temperatures and pressures. Dotted curve, transmittance at the lower pressure and temperature as discussed in the text.

Fig. 2
Fig. 2

Absolute error for the evaluation of the transmittance by use of Eq. (20). The source of the error is that a low-order Gauss quadrature has been used for a function with large high-order derivatives.

Equations (20)

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fk=18π2πs/bexp-kπb/s0k0 I04πbxd1/2×exp-sx2/4πk-kdxH3sπb4kd21/2×1k21k-kexp-πsb/4kd2dk,
Tm=exp-N0SmΔν1+SmΠα0-1/2,
Πα0=4N0j=1j=N0Sjαj2j=1j=N0 Sj,
Tm=expαm1+βm.
fk=-1exp-γμexp+γ/μ.
ft=-1exp+γ/μ.
ft=δt+γ220F2, 32, 2, α2t4+γπt0F2, 12, 32, α2t4,
fk=0kα exp-α2/4βk-k-k-k/β2πβk-k31/2×exp-k/βα22β30F2, 32, 2, α2k4β3+αβ3πk0F2, 12, 32, α2k4β3dk+α2πβk3exp-α24βk-kβ.
fk=α exp-k/β2πβ×α22β3n=0n=Γ3/2Γ2Γ3/2+nΓ2+nα24β2n1n!×0kk-xnx-3/2 exp-α2/4βxdx+απβ3n=0n=Γ1/2Γ2Γ1/2+nΓ2+nα24β2n1n!×0kk-xn-1/2x-3/2×exp-α2/4βxdx+exp-α2/4βkk3/2.
0u xν-1u-xμ-1exp-b/xdx=bν-1/2u2μ+ν-1/2×exp-b/2uΓμW1-2μ-ν/2,ν/2-b/u,
fk=α exp-k/β2πβ×α24β-3/4exp-α2/4βkn=0n= Ānk2n+1/2×W-1/4-n,-3/4α24βk+α24β-3/4exp-α2/4βkn=0n= B¯nΓn+1/2n!×k2n-1/2W+1/4-n,-3/4α24βk+exp-α2/4βkk3/2,
Ān=α22β3Γ3/2Γ2Γ3/2+nΓ2+nα24β2n, B¯n=απβ3Γ1/2Γ2Γ1/2+nΓ2+nα24β2n.
1/2-λ-μWμ-1,λz=Wμ,λz-z Wμ-1/2,λ+1/2z, 1/2+λ-μWμ-1,λz=Wμ,λz-z Wμ-1/2,λ-1/2z.
Wλ,μz=Γ-2μΓ1/2-μ-λ Mλ,μz+Γ+2μΓ1/2+μ-λ Mλ,-μz.
Mλ,μz=exp-z/2z1/2+μM1/2+μ-λ, 1+2μ; z,
Ma, b; z=1+ab z+aa+1bb+1z22!+aa+1a+2bb+1b+2z33!+,
M1/4,-1/4z=M-1/4,-3/4z=z1/4 exp-z/2, M1/4,-3/4z=M-1/4,-1/4z=z-1/4 exp+z/2.
gk=0k fkdk.
dg/dk=f
Tm=0 fkexp-kmdk=01exp-kgmdg.

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