Abstract

The recently developed finite-element method for solution of the radiative transfer equation has been extended to compute the full azimuthal dependence of the radiance in a vertically inhomogeneous plane-parallel medium. The physical processes that are included in the algorithm are multiple scattering and bottom boundary bidirectional reflectivity. The incident radiation is a parallel flux on the top boundary that is characteristic for illumination of the atmosphere by the Sun in the UV, visible, and near-infrared regions of the electromagnetic spectrum. The theoretical basis is presented together with a number of applications to realistic atmospheres. The method is shown to be accurate even with a low number of grid points for most of the considered situations. The fortran code for this algorithm is developed and is available for applications.

© 1995 Optical Society of America

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References

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  1. V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
    [CrossRef]
  2. R. F. Harrington, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.
  3. O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989), Chap. 3, pp. 49–51.
  4. K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 30, 387–399 (1981).
    [CrossRef]
  5. K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
    [CrossRef]
  6. K. Stamnes, S. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
    [CrossRef] [PubMed]
  7. K. N. Liou, “Applications of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3440 (1975).
    [CrossRef]
  8. J. Lenoble, “Computational methods,” in Radiative Transfer in Scattering and Absorbing Atmospheres, J. Lenoble, ed. (Deepak, Hampton, Va., 1985), Chap. 3, pp. 36–39.
  9. F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).
  10. F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).
  11. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980), Chap. 6, pp. 176–196.
  12. C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de Diverses Methodes de Resolution de l’Equation de transfert du Rayonnement dans un Milieu Diffusant,” J. Quant. Spectrosc. Radiat. Transfer 13, 1421–1431 (1973).
    [CrossRef]
  13. V. V. Sobolev, Scattering of Light in Planetary Atmospheres (Nauka, Moscow, 1972), in Russian.
  14. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  15. T. C. Grenfell, “A radiative transfer model for sea ice with vertical structure variations,” J. Geophys. Res. 86, 16,991–17,001 (1991).

1994 (1)

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

1991 (1)

T. C. Grenfell, “A radiative transfer model for sea ice with vertical structure variations,” J. Geophys. Res. 86, 16,991–17,001 (1991).

1988 (1)

1981 (2)

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 30, 387–399 (1981).
[CrossRef]

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[CrossRef]

1975 (1)

K. N. Liou, “Applications of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3440 (1975).
[CrossRef]

1974 (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1973 (1)

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de Diverses Methodes de Resolution de l’Equation de transfert du Rayonnement dans un Milieu Diffusant,” J. Quant. Spectrosc. Radiat. Transfer 13, 1421–1431 (1973).
[CrossRef]

Abreu, L. W.

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Anderson, G. P.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Chetwynd, J. H.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

Clough, S. A.

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Dale, H.

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[CrossRef]

Devaux, C.

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de Diverses Methodes de Resolution de l’Equation de transfert du Rayonnement dans un Milieu Diffusant,” J. Quant. Spectrosc. Radiat. Transfer 13, 1421–1431 (1973).
[CrossRef]

Fenn, R. W.

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

Fouquart, Y.

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de Diverses Methodes de Resolution de l’Equation de transfert du Rayonnement dans un Milieu Diffusant,” J. Quant. Spectrosc. Radiat. Transfer 13, 1421–1431 (1973).
[CrossRef]

Gallery, W. O.

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Grenfell, T. C.

T. C. Grenfell, “A radiative transfer model for sea ice with vertical structure variations,” J. Geophys. Res. 86, 16,991–17,001 (1991).

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.

Herman, M.

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de Diverses Methodes de Resolution de l’Equation de transfert du Rayonnement dans un Milieu Diffusant,” J. Quant. Spectrosc. Radiat. Transfer 13, 1421–1431 (1973).
[CrossRef]

Jayaweera, K.

Kisselev, V. B.

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

Kneizys, F. X.

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Lenoble, J.

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de Diverses Methodes de Resolution de l’Equation de transfert du Rayonnement dans un Milieu Diffusant,” J. Quant. Spectrosc. Radiat. Transfer 13, 1421–1431 (1973).
[CrossRef]

J. Lenoble, “Computational methods,” in Radiative Transfer in Scattering and Absorbing Atmospheres, J. Lenoble, ed. (Deepak, Hampton, Va., 1985), Chap. 3, pp. 36–39.

Liou, K. N.

K. N. Liou, “Applications of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3440 (1975).
[CrossRef]

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980), Chap. 6, pp. 176–196.

Perona, G.

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

Roberti, L.

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

Selby, J. E. A.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

Shettle, E. P.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

Sobolev, V. V.

V. V. Sobolev, Scattering of Light in Planetary Atmospheres (Nauka, Moscow, 1972), in Russian.

Stamnes, K.

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

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[CrossRef]

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 30, 387–399 (1981).
[CrossRef]

Swanson, R. A.

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 30, 387–399 (1981).
[CrossRef]

Taylor, R. L.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989), Chap. 3, pp. 49–51.

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Tsay, S.

Wiscombe, W.

Zienkiewicz, O. C.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989), Chap. 3, pp. 49–51.

Appl. Opt. (1)

J. Atmos. Sci. (2)

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 30, 387–399 (1981).
[CrossRef]

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[CrossRef]

J. Geophys. Res. (2)

K. N. Liou, “Applications of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3440 (1975).
[CrossRef]

T. C. Grenfell, “A radiative transfer model for sea ice with vertical structure variations,” J. Geophys. Res. 86, 16,991–17,001 (1991).

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

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de Diverses Methodes de Resolution de l’Equation de transfert du Rayonnement dans un Milieu Diffusant,” J. Quant. Spectrosc. Radiat. Transfer 13, 1421–1431 (1973).
[CrossRef]

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Other (7)

V. V. Sobolev, Scattering of Light in Planetary Atmospheres (Nauka, Moscow, 1972), in Russian.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989), Chap. 3, pp. 49–51.

J. Lenoble, “Computational methods,” in Radiative Transfer in Scattering and Absorbing Atmospheres, J. Lenoble, ed. (Deepak, Hampton, Va., 1985), Chap. 3, pp. 36–39.

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough, R. W. Fenn, “Atmospheric transmittance/radiance: computer code lowtran 6,” Environmental Research Paper AFGL-TR-83-0187 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1983).

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users Guide to lowtran 7,” Environmental Research Paper AFGL-TR-88-0177 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980), Chap. 6, pp. 176–196.

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Figures (2)

Fig. 1
Fig. 1

Some of the phase functions from the lowtran database (i = 1, 70) used in the computations: ......., i = 4 rural aerosol; – – –, i = 18 urban aerosol; – · –, i = 51 advective fog; ——, i = 55 volcanic particles.

Fig. 2
Fig. 2

Downgoing intensity at z = 0 km corresponding to the case in Subsection 4.B (urban aerosol with visibility of 5 km, λ = 0.55 μm, Λ s = 0.5, and g s = 0.6) with A, η0 = 0.2; B, η0 = 0.64; C, η0 = 0.9.

Tables (7)

Tables Icon

Table 1 Mean Deviations of the FEM and DOM Intensity Values for Different Numbers of Grid Points for Rural Aerosol with 23-km Visibility, λ= 0.55 μm, Λ s = 0.5, and g s = 0.6a

Tables Icon

Table 2 Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Advective Fog with 0.2-km Visibility, λ= 0.55 μm, Λ s = 0.8, and g s = 0

Tables Icon

Table 3 Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Stratocumulus Clouds with λ = 0.55 μm, Λ s = 0.7, and g s = 0.7

Tables Icon

Table 4 Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Rural Aerosol and Extreme Volcanic Profile with λ = 0.4 μm, Λ s = 0.7, and g s = 0.7

Tables Icon

Table 5 Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Rural Aerosol and Cirrus Clouds with λ = 0.4 μm, Λ s = 0.8, and gs = 0.8

Tables Icon

Table 6 CPU Times on a VAX 7000-610 for Computation of One Harmonic for Different Values of N with Two Interpolation Angles and Levels

Tables Icon

Table 7 Maximum Value of Each Harmonic for Some Atmospheric Models that Were Measured over All the Interpolation Angles and Levels as a Function of Harmonic Number

Equations (60)

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I ( τ , η , ϕ ) = m = 0 M I m ( τ , η ) cos m ϕ .
{ I m + ( τ , η ) = I m ( τ , η ) if η > 0 , I m + ( τ , η ) = 0 if η < 0 ,
{ I m - ( τ , η ) = I m ( τ , η ) if η < 0 , I m - ( τ , η ) = 0 if η > 0.
η d I m ± ( τ , η ) / d τ = - I m ± ( τ , η ) + Λ ( l ) / 2 - 1 0 p l m ± ( η , η ) I m - ( τ , η ) d η + Λ ( l ) / 2 0 1 p l m ± ( η , η ) I m + ( τ , η ) d η + Λ ( l ) S / 4 p l m ± ( η , η 0 ) exp ( - τ / η 0 ) ,
p l m ( η , η ) = 1 / ( 2 π ) 0 2 π p l ( γ ) cos ( m ϕ ) d ϕ ,
cos γ = cos θ cos θ + sin θ sin θ cos ( ϕ - ϕ ) ,
I m - ( τ , η ) = i = 1 N I i m ( τ ) b i m ( η ) , I m + ( τ , η ) = i = N + 1 2 N I i m ( τ ) b i m ( η ) ,
b i m ( η ) = P m m ( η ) { 1 - η - η i / h if η - η i < h 0 otherwise ,
- 1 1 P k m ( η ) P k m ( η ) d η = 2 / ( 2 k + 1 ) ,
A m d I m ( τ ) / d τ = B m I m ( τ ) + S Λ / 4 R m exp ( - τ / η 0 ) ,
A m = ( A m ( 1 ) 0 0 - A m ( 1 ) ) ,             B m = ( B m ( 1 ) B m ( 2 ) B m ( 2 ) B m ( 1 ) ) .
p ( γ ) = m = 0 x m P m ( γ ) ,
p m ( η , η ) = k = m x k m P k m ( η ) P k m ( η ) ,
B i , j m = - B ^ i , j m + Λ / 2 k = m x k J N - j + 1 k , m J N - i + 1 k , m for i = 1 , , N ; j = 1 , , N , B i , j m = ( - 1 ) m Λ / 2 k = m ( - 1 ) k x k J 2 N - j + 1 k , m J N - i + 1 k , m for i = 1 , , N ; j = N + 1 , , 2 N ,
J i k , m = ( i - 2 ) h i h b 2 N - i + 1 m ( η ) P k m ( η ) d η ,             i = 1 , , N ,
B ^ i , j m = n j - 1 n j + 1 b i m ( η ) b j m ( η ) d η ,             i , j = 1 , , N .
R i k , m = ( i - 1 ) h i h P m m ( η ) P k m ( η ) d η , S i k , m = ( i - 1 ) h i h η P m m ( η ) P k m ( η ) d η , T i k , m = ( i - 1 ) h i h η 2 P m m ( η ) P k m ( η ) d η .
R i k + 1 , m ( t ) = { 1 / [ ( k + m + 1 ) ( k - m + 1 ) ] } 1 / 2 × { ( 2 k + 1 ) S i k , m - [ ( k + m ) ( k - m ) ] 1 / 2 R i k - 1 , m } ,
S i k + 1 , m ( t ) = { 1 / [ ( k + m + 1 ) ( k - m + 1 ) ] } 1 / 2 × { ( 2 k + 1 ) T i k , m - [ ( k + m ) ( k - m ) ] 1 / 2 S i k - 1 , m } ,
T i k , m = 1 / ( 2 m + 3 ) { 2 [ ( m + 1 ) / ( 2 m + 1 ) ] 1 / 2 Q i k , m + R i k , m } Q i k , m = 1 / ( k + m + 3 ) { t P m + 2 m ( t ) P k m ( t ) + [ ( k + m ) ( k - m ) ] 1 / 2 / ( m - k + 2 ) P m + 2 m ( t ) P k - 1 m ( t ) - ( 2 m + 1 ) / ( m - k + 2 ) P m + 1 m ( t ) P k m ( t ) } ( i - 1 ) h i h ,
I ( l ) ( τ ) = X ( l ) { Λ ( l ) S / 4 exp ( - τ / η 0 ) W ( l ) X ( l ) t R ( l ) + D ( l ) ( τ ) N 1 ( l ) } ,
D ( l ) ( τ ) = diag { exp [ - α j ( l ) τ ] } ,
W ( l ) = diag { 1 / [ α j ( l ) - 1 / η 0 ] }
- α ( l ) A X ( l ) = B ( l ) X ( l )
X ( l ) = ( X 1 ( l ) X 2 ( l ) X 2 ( l ) X 1 ( l ) ) .
N ( l ) = ( D 1 ( l ) [ τ ( l - 1 ) ] 0 0 D 1 ( l ) [ - τ ( l ) ] ) × [ N 1 ( l ) + Λ ( l ) S / 4 ( E 0 0 0 ) W ( l ) X ( l ) R ( l ) ] ,
I ( l ) ( τ ) = X ( l ) { ( D 1 ( l ) [ τ - τ ( l - 1 ) ] 0 0 D 1 ( l ) [ τ ( l ) - τ ] ) N ( l ) + Λ ( l ) S / 4 ( K ( l ) ( τ ) 0 0 E exp ( - τ / η 0 ) ) × ( E 0 0 W 2 ( l ) ) X ( l ) t R ( l ) } ,
K ( l ) ( τ ) = [ E exp ( - τ / η 0 ) - D 1 ( l ) ( τ ) ] W 1 ( l ) ,
I ( l ) [ τ ( l ) ] = I ( l + 1 ) [ τ ( l ) ] .
I ( 1 ) + ( 0 ) = 0.
I [ τ ( L ) , η , ϕ ] η < 0 = 1 / π 0 2 π d ϕ 0 1 r ( η , ϕ ; η , ϕ ) × I in [ τ ( L ) , η , ϕ ] η d η ,
I ( L ) m - [ τ ( L ) , η ] = 2 0 1 r m ( η , η ) I ( L ) m + [ τ ( L ) , η ] η d η + S η 0 r m ( η , η 0 ) exp [ - τ ( L ) / η 0 ] ,
r m ( η , η ) = 1 / ( 2 π ) 0 2 π r ( γ ) cos m ϕ d ϕ ,
r m ( η , η ) = k = m r k P k m ( η ) P k m ( η ) ,
B ^ I 1 ( L ) - = S I 2 ( L ) + + R S ,
F = F + - F - , 1 2 π F + = 0 1 I 0 ( τ , η ) η d η = h 2 { ( η N + 1 - 1 3 h ) I N + 1 0 + 2 j = N + 2 2 N - 1 η j I j 0 + ( η 2 N + 1 3 h ) I 2 N 0 } = F · I 0 + 1 2 π F - = 0 - 1 I 0 ( τ , η ) η d η = - h 2 { ( η 1 + 1 3 h ) I 1 0 + 2 j = 2 N η j I j 0 + ( η N - 1 3 h ) I N 0 } = F · I 0 - ,
F · G ( i ) < T i ,
[ B ^ X 1 ( L ) - S X 2 ( L ) ] D 1 ( L ) [ τ ( L ) - τ ( L - 1 ) ] N 1 ( L ) + [ B ^ X 2 ( L ) - S X 1 ( L ) ] N 2 ( L ) = - S Λ L / 4 { [ B ^ X 1 ( L ) - S X 2 ( L ) ] K ( L ) [ τ ( L ) ] × [ X 1 ( L ) t R 1 ( L ) + X 2 ( L ) t R 2 ( L ) ] + exp [ - τ ( L ) / η 0 ] × [ B ^ X 2 ( L ) - S X 1 ( L ) ] W 2 ( L ) [ X 2 ( L ) t R 1 ( L ) + X 1 ( L ) t R 2 ( L ) ] } + S η 0 exp [ - τ ( L ) / η 0 ] R S ,
UN = B ,
N = ( N ( 1 ) N ( 2 ) N ( L ) ) ,
( X 2 ( 1 ) Y 1 ( 1 ) 0 0 0 0 0 Y 1 ( 1 ) X 2 ( 1 ) - X 1 ( 2 ) - Y 2 ( 2 ) 0 0 0 Y 2 ( 1 ) X 1 ( 1 ) - X 2 ( 2 ) - Y 1 ( 2 ) 0 0 0 0 0 Y 1 ( 2 ) X 2 ( 2 ) 0 0 0 0 0 Y 2 ( 2 ) X 1 ( 2 ) 0 0 0 0 0 0 0 X 2 ( L - 1 ) - X 1 ( L ) - Y 2 ( L ) 0 0 0 0 X 1 ( L - 1 ) - X 2 ( L ) - Y 1 ( L ) 0 0 0 0 0 G 1 G 2 )
Y 1 ( 2 ) ( l ) = X 1 ( 2 ) ( l ) D 1 ( l ) [ τ ( l ) - τ ( l - 1 ) ] , G 1 = [ B ^ X 1 ( L ) - S X 2 ( L ) ] D 1 ( L ) [ τ ( L ) - τ ( L - 1 ) ] , G 2 = B ^ X 2 ( L ) - S X 1 ( L ) .
A ( 1 ) = ( E 0 0 B ^ ) ( X 2 X 1 D 1 ( τ 0 ) X 1 D 1 ( τ 0 ) X 2 ) .
V = - Λ S / 4 ( E 0 0 B ^ ) ( 0 X 1 X 1 K ( τ 0 ) exp ( - τ 0 / η 0 ) X 2 ) × ( E 0 0 W 2 ) X t R ,
I = X [ ( K ( τ ) 0 0 exp ( - τ 0 / η 0 ) ) Z 1 + ( D 1 ( τ ) 0 0 D 1 ( τ 0 - τ ) ) Z 2 ] ,
Z 1 = Λ S / 4 ( E 0 0 W 2 ) X t R , Z 2 = - Λ S / 4 ( X 2 X 1 D 1 ( τ 0 ) X 1 D 1 ( τ 0 ) X 2 ) - 1 × ( 0 X 1 X 1 K ( τ 0 ) exp ( - τ 0 / η 0 ) X 2 ) × ( E 0 0 W 2 ) X t R .
I m ( τ , η ) = exp ( - τ / η ) / η 0 τ exp ( τ / η ) Q m ( η , τ ) d τ ,
Q m ( η , τ ) = Λ / 2 - 1 1 p m ( η , η ) I ^ m ( τ , η ) d η + S Λ / 4 p m ( η , η 0 ) exp ( - τ / η 0 ) ,
Q ( l ) ( η , τ ) = F ( l ) · I ( l ) + S Λ ( l ) / 4 × exp ( - τ / η 0 ) k = m x k ( l ) P k m ( η ) P k m ( η 0 ) ,
F i ( l ) ( η ) = Λ ( l ) / 2 k = m ( - 1 ) k + m x k ( l ) P k m ( η ) J N - i + 1 k , m ,             i = 1 , , N , F i ( l ) ( η ) = Λ ( l ) / 2 k = m x k ( l ) P k m ( η ) J 2 N - i + 1 k , m ,             i = N + 1 , , 2 N .
I ( τ , η ) = exp ( - τ / η ) / η { P ( s ) ( τ , η ) + l = 1 s - 1 P ( l ) [ τ ( l ) , η ] } ,
P ( l ) ( τ , η ) = F ( l ) X ( l ) [ ( H 1 ( l ) ( τ ) 0 0 H 2 ( l ) ( τ ) ) N ( l ) + S Λ ( l ) / 4 ( H 3 ( l ) ( τ ) 0 0 E H 4 ( l ) ( τ ) ) × ( E 0 0 W 2 ( l ) ) X ( l ) t R ( l ) ] + S Λ ( l ) / 4 H 4 ( l ) k = m x k ( l ) P k m ( η ) P k m ( η 0 ) ,
H 1 ( l ) ( τ ) = exp ( τ / η ) diag ( exp { - [ τ - τ ( l - 1 ) ] / η } - exp { - α j ( l ) [ τ - τ ( l - 1 ) ] } α j ( l ) - 1 / η ) , H 2 ( l ) ( τ ) = exp ( τ / η ) diag ( exp { - α j ( l ) [ τ ( l ) - τ ] } - exp { - [ τ - τ ( l - 1 ) ] / η - α j ( l ) [ τ ( l ) - τ ( l - 1 ) ] } α j ( l ) + 1 / η ) H 3 ( l ) ( τ ) = [ E H 4 ( l ) ( τ ) - exp ( τ / η ) diag ( exp { - [ τ - τ ( l - 1 ) ] / η - α j ( l ) τ ( l - 1 ) } - exp ( - α j ( l ) τ ) α j ( l ) - 1 / η ) ] W 1 ( l ) , H 4 ( l ) ( τ ) = exp ( τ / η ) η 0 η η 0 - η ( exp ( - τ / η 0 ) - exp { - [ τ - τ ( l - 1 ) ] / η - τ ( l - 1 ) / η 0 } )
I ( τ , η ) = - exp ( - τ / η ) / η τ τ ( L ) exp ( τ / η ) Q ( l ) ( η , τ ) d τ + I [ τ ( L ) , η ] exp { [ τ ( L ) - τ ] / η } ,
H 1 ( l ) ( τ ) = exp ( τ / η ) diag ( exp { - α j ( l ) [ τ - τ ( l - 1 ) ] } - exp { [ τ ( l ) - τ ] / η - α j ( l ) [ τ ( l ) - τ ( l - 1 ) ] } α j ( l ) - 1 / η ) , H 2 ( l ) ( τ ) = exp ( τ / η ) diag ( exp { [ τ ( l ) - τ ] / η } - exp { - α j ( l ) [ τ ( l ) - τ ] } α j ( l ) + 1 / η ) , H 3 ( l ) ( τ ) = [ E H 4 ( l ) ( τ ) - exp ( τ / η ) diag ( exp [ - α j ( l ) τ ] - exp { [ τ ( l ) - τ ] / η - α j ( l ) τ ( l ) } α j ( l ) - 1 / η ) ] W 1 ( l ) , H 4 ( l ) ( τ ) = exp ( τ / η ) η 0 η η 0 - η [ exp { [ τ ( l ) - τ ] / η - τ ( l ) / η 0 } - exp ( - τ / η 0 ) ] .
tr = exp ( - K e h ) ;
tr s mo = exp ( - K s mo h ) ;
tr ae - hy = exp ( - K e ae - hy h ) ;
abs ae - hy = 1 - Tr a ae - hy = 1 - exp ( - K a ae - hy h ) ,
p ( γ ) = [ p ( γ ) ae - hy K s ae - hy + p ( γ ) mo K s mo ] / K s ,

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