Abstract

A technique is developed to model radiative transfer in three-dimensional natural clouds with a standard discrete ordinates finite-element method modified to evaluate cell-surface-averaged radiances. A log-least-squares-based scale transformation is used to improve the discrete phase-function model. We handle dense media by assuming constant diffuse radiances over input faces to cubic cells, allowing analytical forms for transmittance factors. Transmission equations are combined with diffuse volumetric single-scattering calculations to support evaluations of cell energy balance. Energy not accounted for volumetrically is treated with surface-based effects. Results produced show accurate flux computations at over 30 optical depths per modeled cell. Comparisons with nonuniform cloud Monte Carlo calculations show less than 1% rms error and correlations greater than 0.999 for cases in which cloud-density fluctuations are resolved.

© 1998 Optical Society of America

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  36. S. G. O’Brien, D. H. Tofsted, “Visualization of dense cloud radiation data in modeling and simulations,” in Modeling, Simulation, and Visualization of Sensory Response for Defense Applications, N. L. Faust, J. D. Illgen, eds., Proc. SPIE3085, 82–93 (1997).
    [CrossRef]
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    [CrossRef]

1998

K. F. Evans, “The spherical harmonics discrete ordinates method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

S. G. O’Brien, D. H. Tofsted, “Physics-based visualization of dense natural clouds. II. Cloud-rendering algorithm,” Appl. Opt. 37, 7680–7688 (1998).
[CrossRef]

1996

K.-S. Kuo, R. C. Weger, R. M. Welch, S. K. Cox, “The Picard iterative approximation to the solution of the integral equation of radiative transfer—Part II. Three-dimensional geometry,” J. Quant. Spectosc. Radiat. Transf. 55, 195–213 (1996).
[CrossRef]

R. N. Byrne, R. C. J. Somerville, B. Subasilar, “Broken-cloud enhancement of solar radiation absorption,” J. Atmos. Sci. 53, 878–886 (1996).
[CrossRef]

1994

J. Li, D. J. W. Geldart, P. Chylek, “Solar radiative transfer in clouds with vertical internal inhomogeneity,” J. Atmos. Sci. 51, 2542–2552 (1994).
[CrossRef]

J. Li, D. J. W. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

J. C. Chai, H. S. Lee, S. V. Patankar, “Improved treatment of scattering using the discrete ordinates method,” Trans. ASME 116, 260–263 (1994).
[CrossRef]

1993

1992

N. El Wakil, J. F. Sakadura, “Some improvements of the discrete ordinates method for the solution of the radiative transfer equation in multidimensional anisotropically scattering media,” Dev. Radiat. Heat Trans. (ASME) 203, 119–127 (1992).

1991

W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” Fundam. Radiat. Heat Trans. (ASME) 160, 89–96 (1991).

T. Kobayashi, “Reflected solar flux for horizontally inhomogeneous atmospheres,” J. Atmos. Sci. 48, 2436–2447 (1991).
[CrossRef]

1988

K. Stamnes, S.-C. 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]

G. L. Stephens, “Radiative transfer through arbitrarily shaped optical media. Part I: a general method of solution,” J. Atmos. Sci. 45, 1818–1836 (1988).
[CrossRef]

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transf. 40, 51–69 (1988).
[CrossRef]

1987

1986

G. L. Stephens, “Radiative transfer in spatially heterogeneous, two-dimensional, anisotropically scattering media,” J. Quant. Spectrosc. Radiat. Transf. 36, 51–67 (1986).
[CrossRef]

A. Zardecki, S. A. W. Gerstl, R. E. DeKinder, “Two- and three-dimensional radiative transfer in the diffusion approximation,” Appl. Opt. 25, 3508–3515 (1986).
[CrossRef] [PubMed]

1985

1984

R. M. Welch, B. A. Wielicki, “Stratocumulus cloud field reflected fluxes: the effect of cloud shape,” J. Atmos. Sci. 41, 3085–3103 (1984).
[CrossRef]

1983

1982

Harshvardhan, J. A. Weinman, R. Davies, “Transport of infrared radiation in cuboidal clouds,” J. Atmos. Sci. 38, 2500–2512 (1982).

1981

B. H. J. McKellar, M. A. Box, “The scaling group of the radiative transfer equations,” J. Atmos. Sci. 38, 1063–1068 (1981).
[CrossRef]

1980

M. Gube, J. Schmetz, E. Raschke, “Solar radiative transfer in a cloud field,” Contrib. Atmos. Phys. 53, 23–34 (1980).

1978

R. Davies, “The effect of finite geometry on the three-dimensional transfer of solar irradiance in clouds,” J. Atmos. Sci. 35, 1712–1725 (1978).
[CrossRef]

1977

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

1976

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative reflux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

1974

K.-C. Ng, “Hypernetted chain solutions for the classical one-component plasma up to Γ = 7000,” J. Chem. Phys. 61, 2680–2689 (1974).
[CrossRef]

T. B. McKee, S. K. Cox, “Scattering of visible radiation by finite clouds,” J. Atmos. Sci. 31, 1885–1892 (1974).
[CrossRef]

1970

J. F. Potter, “The delta function approximation in radiative transfer,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

Asrar, G.

R. B. Myneni, A. Marshak, Y. Knyazikhin, G. Asrar, “Discrete ordinates method for photon transport in leaf canopies,” in Photon-Vegetation Interactions, R. B. Myneni, J. Ross, eds. (Springer-Verlag, New York, 1991), pp. 45–110.
[CrossRef]

Box, M. A.

B. H. J. McKellar, M. A. Box, “The scaling group of the radiative transfer equations,” J. Atmos. Sci. 38, 1063–1068 (1981).
[CrossRef]

Byrne, R. N.

R. N. Byrne, R. C. J. Somerville, B. Subasilar, “Broken-cloud enhancement of solar radiation absorption,” J. Atmos. Sci. 53, 878–886 (1996).
[CrossRef]

Carlson, B. G.

B. G. Carlson, K. D. Lathrop, “Transport theory: the method of discrete ordinates,” in Computing Methods in Reactor Physics, H. Greenspan, C. H. Kelber, D. Okrent, eds. (Gordon & Breach, New York, 1968), pp. 171–266.

Chai, J. C.

J. C. Chai, H. S. Lee, S. V. Patankar, “Improved treatment of scattering using the discrete ordinates method,” Trans. ASME 116, 260–263 (1994).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chylek, P.

J. Li, D. J. W. Geldart, P. Chylek, “Solar radiative transfer in clouds with vertical internal inhomogeneity,” J. Atmos. Sci. 51, 2542–2552 (1994).
[CrossRef]

J. Li, D. J. W. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

Cox, S. K.

K.-S. Kuo, R. C. Weger, R. M. Welch, S. K. Cox, “The Picard iterative approximation to the solution of the integral equation of radiative transfer—Part II. Three-dimensional geometry,” J. Quant. Spectosc. Radiat. Transf. 55, 195–213 (1996).
[CrossRef]

T. B. McKee, S. K. Cox, “Scattering of visible radiation by finite clouds,” J. Atmos. Sci. 31, 1885–1892 (1974).
[CrossRef]

Davies, R.

Harshvardhan, J. A. Weinman, R. Davies, “Transport of infrared radiation in cuboidal clouds,” J. Atmos. Sci. 38, 2500–2512 (1982).

R. Davies, “The effect of finite geometry on the three-dimensional transfer of solar irradiance in clouds,” J. Atmos. Sci. 35, 1712–1725 (1978).
[CrossRef]

DeKinder, R. E.

El Wakil, N.

N. El Wakil, J. F. Sakadura, “Some improvements of the discrete ordinates method for the solution of the radiative transfer equation in multidimensional anisotropically scattering media,” Dev. Radiat. Heat Trans. (ASME) 203, 119–127 (1992).

Embury, J. F.

Evans, K. F.

K. F. Evans, “The spherical harmonics discrete ordinates method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

Fiveland, W. A.

W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” Fundam. Radiat. Heat Trans. (ASME) 160, 89–96 (1991).

Geldart, D. J. W.

J. Li, D. J. W. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

J. Li, D. J. W. Geldart, P. Chylek, “Solar radiative transfer in clouds with vertical internal inhomogeneity,” J. Atmos. Sci. 51, 2542–2552 (1994).
[CrossRef]

Gerstl, S. A. W.

Goldenberg, S.

Gube, M.

M. Gube, J. Schmetz, E. Raschke, “Solar radiative transfer in a cloud field,” Contrib. Atmos. Phys. 53, 23–34 (1980).

Haferman, J. L.

Harshvardhan,

Harshvardhan, J. A. Weinman, R. Davies, “Transport of infrared radiation in cuboidal clouds,” J. Atmos. Sci. 38, 2500–2512 (1982).

Isaacs, R. G.

Jayaweera, K.

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative reflux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Knyazikhin, Y.

R. B. Myneni, A. Marshak, Y. Knyazikhin, G. Asrar, “Discrete ordinates method for photon transport in leaf canopies,” in Photon-Vegetation Interactions, R. B. Myneni, J. Ross, eds. (Springer-Verlag, New York, 1991), pp. 45–110.
[CrossRef]

Kobayashi, T.

T. Kobayashi, “Reflected solar flux for horizontally inhomogeneous atmospheres,” J. Atmos. Sci. 48, 2436–2447 (1991).
[CrossRef]

Krajewski, W. F.

Kuo, K.-S.

K.-S. Kuo, R. C. Weger, R. M. Welch, S. K. Cox, “The Picard iterative approximation to the solution of the integral equation of radiative transfer—Part II. Three-dimensional geometry,” J. Quant. Spectosc. Radiat. Transf. 55, 195–213 (1996).
[CrossRef]

Lathrop, K. D.

B. G. Carlson, K. D. Lathrop, “Transport theory: the method of discrete ordinates,” in Computing Methods in Reactor Physics, H. Greenspan, C. H. Kelber, D. Okrent, eds. (Gordon & Breach, New York, 1968), pp. 171–266.

Lee, H. S.

J. C. Chai, H. S. Lee, S. V. Patankar, “Improved treatment of scattering using the discrete ordinates method,” Trans. ASME 116, 260–263 (1994).
[CrossRef]

Lewis, E. E.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, (Wiley, New York, 1984).

Li, J.

J. Li, D. J. W. Geldart, P. Chylek, “Solar radiative transfer in clouds with vertical internal inhomogeneity,” J. Atmos. Sci. 51, 2542–2552 (1994).
[CrossRef]

J. Li, D. J. W. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

Liou, K.-N.

K.-N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

Marshak, A.

R. B. Myneni, A. Marshak, Y. Knyazikhin, G. Asrar, “Discrete ordinates method for photon transport in leaf canopies,” in Photon-Vegetation Interactions, R. B. Myneni, J. Ross, eds. (Springer-Verlag, New York, 1991), pp. 45–110.
[CrossRef]

McKee, T. B.

T. B. McKee, S. K. Cox, “Scattering of visible radiation by finite clouds,” J. Atmos. Sci. 31, 1885–1892 (1974).
[CrossRef]

McKellar, B. H. J.

B. H. J. McKellar, M. A. Box, “The scaling group of the radiative transfer equations,” J. Atmos. Sci. 38, 1063–1068 (1981).
[CrossRef]

Miller, W. F.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, (Wiley, New York, 1984).

Myneni, R. B.

R. B. Myneni, A. Marshak, Y. Knyazikhin, G. Asrar, “Discrete ordinates method for photon transport in leaf canopies,” in Photon-Vegetation Interactions, R. B. Myneni, J. Ross, eds. (Springer-Verlag, New York, 1991), pp. 45–110.
[CrossRef]

Nakajima, T.

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transf. 40, 51–69 (1988).
[CrossRef]

Ng, K.-C.

K.-C. Ng, “Hypernetted chain solutions for the classical one-component plasma up to Γ = 7000,” J. Chem. Phys. 61, 2680–2689 (1974).
[CrossRef]

O’Brien, S. G.

S. G. O’Brien, D. H. Tofsted, “Physics-based visualization of dense natural clouds. II. Cloud-rendering algorithm,” Appl. Opt. 37, 7680–7688 (1998).
[CrossRef]

D. H. Tofsted, S. G. O’Brien, “Dense cloud radiative transfer scenarios and model validation,” in Cloud Impacts on DoD Operations and Systems, 1997 Proceedings, PL-TR-97-2112, (Phillips Lab, Hanscom Air Force Base, Mass., 1997), pp. 85–88.

S. G. O’Brien, D. H. Tofsted, “Visualization of dense cloud radiation data in modeling and simulations,” in Modeling, Simulation, and Visualization of Sensory Response for Defense Applications, N. L. Faust, J. D. Illgen, eds., Proc. SPIE3085, 82–93 (1997).
[CrossRef]

Patankar, S. V.

J. C. Chai, H. S. Lee, S. V. Patankar, “Improved treatment of scattering using the discrete ordinates method,” Trans. ASME 116, 260–263 (1994).
[CrossRef]

Potter, J. F.

J. F. Potter, “The delta function approximation in radiative transfer,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

Raschke, E.

M. Gube, J. Schmetz, E. Raschke, “Solar radiative transfer in a cloud field,” Contrib. Atmos. Phys. 53, 23–34 (1980).

Sakadura, J. F.

N. El Wakil, J. F. Sakadura, “Some improvements of the discrete ordinates method for the solution of the radiative transfer equation in multidimensional anisotropically scattering media,” Dev. Radiat. Heat Trans. (ASME) 203, 119–127 (1992).

Sánchez, A.

Schmetz, J.

M. Gube, J. Schmetz, E. Raschke, “Solar radiative transfer in a cloud field,” Contrib. Atmos. Phys. 53, 23–34 (1980).

Smith, T. F.

Somerville, R. C. J.

R. N. Byrne, R. C. J. Somerville, B. Subasilar, “Broken-cloud enhancement of solar radiation absorption,” J. Atmos. Sci. 53, 878–886 (1996).
[CrossRef]

Stamnes, K.

Stephens, G. L.

G. L. Stephens, “Radiative transfer through arbitrarily shaped optical media. Part I: a general method of solution,” J. Atmos. Sci. 45, 1818–1836 (1988).
[CrossRef]

G. L. Stephens, “Radiative transfer in spatially heterogeneous, two-dimensional, anisotropically scattering media,” J. Quant. Spectrosc. Radiat. Transf. 36, 51–67 (1986).
[CrossRef]

Subasilar, B.

R. N. Byrne, R. C. J. Somerville, B. Subasilar, “Broken-cloud enhancement of solar radiation absorption,” J. Atmos. Sci. 53, 878–886 (1996).
[CrossRef]

Tanaka, M.

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transf. 40, 51–69 (1988).
[CrossRef]

Tofsted, D. H.

S. G. O’Brien, D. H. Tofsted, “Physics-based visualization of dense natural clouds. II. Cloud-rendering algorithm,” Appl. Opt. 37, 7680–7688 (1998).
[CrossRef]

S. G. O’Brien, D. H. Tofsted, “Visualization of dense cloud radiation data in modeling and simulations,” in Modeling, Simulation, and Visualization of Sensory Response for Defense Applications, N. L. Faust, J. D. Illgen, eds., Proc. SPIE3085, 82–93 (1997).
[CrossRef]

D. H. Tofsted, S. G. O’Brien, “Dense cloud radiative transfer scenarios and model validation,” in Cloud Impacts on DoD Operations and Systems, 1997 Proceedings, PL-TR-97-2112, (Phillips Lab, Hanscom Air Force Base, Mass., 1997), pp. 85–88.

Tsay, S.-C.

Wang, W.-C.

Weger, R. C.

K.-S. Kuo, R. C. Weger, R. M. Welch, S. K. Cox, “The Picard iterative approximation to the solution of the integral equation of radiative transfer—Part II. Three-dimensional geometry,” J. Quant. Spectosc. Radiat. Transf. 55, 195–213 (1996).
[CrossRef]

Weinman, J. A.

Harshvardhan, J. A. Weinman, R. Davies, “Transport of infrared radiation in cuboidal clouds,” J. Atmos. Sci. 38, 2500–2512 (1982).

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative reflux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Welch, R. M.

K.-S. Kuo, R. C. Weger, R. M. Welch, S. K. Cox, “The Picard iterative approximation to the solution of the integral equation of radiative transfer—Part II. Three-dimensional geometry,” J. Quant. Spectosc. Radiat. Transf. 55, 195–213 (1996).
[CrossRef]

R. M. Welch, B. A. Wielicki, “Stratocumulus cloud field reflected fluxes: the effect of cloud shape,” J. Atmos. Sci. 41, 3085–3103 (1984).
[CrossRef]

Wielicki, B. A.

R. M. Welch, B. A. Wielicki, “Stratocumulus cloud field reflected fluxes: the effect of cloud shape,” J. Atmos. Sci. 41, 3085–3103 (1984).
[CrossRef]

Wiscombe, W.

Wiscombe, W. J.

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative reflux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Worsham, R. D.

Zardecki, A.

Appl. Opt.

Contrib. Atmos. Phys.

M. Gube, J. Schmetz, E. Raschke, “Solar radiative transfer in a cloud field,” Contrib. Atmos. Phys. 53, 23–34 (1980).

Dev. Radiat. Heat Trans. (ASME)

N. El Wakil, J. F. Sakadura, “Some improvements of the discrete ordinates method for the solution of the radiative transfer equation in multidimensional anisotropically scattering media,” Dev. Radiat. Heat Trans. (ASME) 203, 119–127 (1992).

Fundam. Radiat. Heat Trans. (ASME)

W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” Fundam. Radiat. Heat Trans. (ASME) 160, 89–96 (1991).

J. Atmos. Sci.

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

B. H. J. McKellar, M. A. Box, “The scaling group of the radiative transfer equations,” J. Atmos. Sci. 38, 1063–1068 (1981).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Maximum allowable g value ensuring positive–definite PFN condition on order L Legendre expansions of HGPF, with and without δ-M correction.

Fig. 2
Fig. 2

Division of a cell for transmission purposes.

Fig. 3
Fig. 3

Comparison of transmittance functions G i (τ) for i values ranging from 0 to 3.

Fig. 4
Fig. 4

Fractional efficiencies Λ for various streaming models.

Fig. 5
Fig. 5

RMSE’s for volume-averaged and linear-interpolated illuminations, and direct-radiation surface correction factor (1 - Λ0).

Fig. 6
Fig. 6

Comparison of Monte Carlo model results with DOM RT results with δ-M-based PFN form.

Fig. 7
Fig. 7

Comparison of Monte Carlo model results with DOM RT results with LLS-based PFN form.

Fig. 8
Fig. 8

Volume partitioning method with q and R defining geometric properties of the cloud regions.

Fig. 9
Fig. 9

Monte Carlo RT fractional flux exiting volume top (FTE T ) for varying σ̃Δ and R, with 0° zenith incident radiation.

Fig. 10
Fig. 10

Monte Carlo RT fractional flux exiting volume sides (FTE S ) for varying σ̃Δ and R, with 0° zenith incident radiation.

Fig. 11
Fig. 11

Underresolved cases of 43 and 83 resolution models.

Fig. 12
Fig. 12

Critically resolved and overresolved cases of all resolution models.

Tables (1)

Tables Icon

Table 1 RMS Errors for All Casesa

Equations (38)

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d / d s + σ ˜ I = σ ˜ B + J + J 0 ,
J r , Ω ˆ = ω ˜   4 π   I r ,   Ω ˆ P ˜ Ω ˆ ,   Ω ˆ d Ω ˆ ,
J 0 r ,   Ω ˆ = ω ˜ F 0 T 0 P ˜ Ω ˆ ,   Ω ˆ 0 .
P ˜ μ = l = 0   α l X l P l μ ,     X l = 2 π   - 1 1   P ˜ μ P l μ d μ , α l = 2 l + 1 4 π ,
P μ = f   N ρ 2 π ρ 2 exp - θ 2 / 2 ρ 2 + 1 - f l = 0 L   α l c l P l μ ,
N - 1 ρ = ρ - 2 0 π exp - θ 2 / 2 ρ 2 sin θ d θ ,
E 2 = 2 π   - 1 1 ln P ˜ μ - ln P μ 2 d μ ,
ln x = ln x , x > P min , = ln P min + 10 6 x - P min , x P min ,
c l = X l - fN ρ Q l ρ / 1 - f ,
Q l ρ = 0 π / ρ exp - u 2 / 2 P l cos ρ u sin ρ u ρ d u .
c l = e l X l - fN ρ Q l ρ / 1 - f ,
δ l = X l / f 1 - e l + NQ l e l ,
I s = I 0 - J exp - τ + J ,
I s = s - 1 I 0 + σ J / s - 1 + σ ,
I + = 2 I o - I - = s - 1 I - + σ 2 - I - s - 1 + σ .
I s = I 0 T s + B ¯ + J ¯ + J ¯ 0 1 - T s ,
I 0 T s = n   I i , n T i , m , n W i , m , n ,
G k τ = l = 0 k ! k + l ! - τ l ,
T - Z , + X = T - X , + Z = 2 δ G 1 τ M + β G 2 τ M / 2 δ + β .
T - Z , + Y = T - Y , + Z = 2 γ G 1 τ M + α G 2 τ M / 2 γ + α .
T ¯ X = δ G 1 τ M + β G 2 τ M ,
T ¯ Y = γ G 1 τ M + α G 2 τ M ,
T ¯ Z = γ δ G 0 τ M + α δ + β γ G 1 τ M + α β G 2 τ M .
J ¯ Ω ˆ i = ω   j 4 π w j 8   I Ω ˆ j v P Ω ˆ i ,   Ω ˆ j ,
Γ A = G 1 τ M ,     Γ B = Γ C = G 2 τ M ,     Γ D = G 3 τ M .
Γ X τ M = α δ   G 2 τ M 2 + β   G 3 τ M 3 ,
Γ Y τ M = β γ   G 2 τ M 2 + α   G 3 τ M 3 ,
Γ Z τ M = γ δ G 1 τ M + α δ + β γ G 2 τ M 2 + α β   G 3 τ M 3 ,
I Ω ˆ j v = n   Γ j , n I j , n .
F i , m = Δ 2 1 3 π 2   I i , m .
F T = Δ 2 1 3 π 2   G 2 τ M I j , n .
F S = ω Δ 2 1 3 π 2 1 - G 2 τ M G 3 τ M I j , n ,
Λ τ M = 1 - 1 - G 2 τ M 1 - G 3 τ M ,
F j = π w j 2   | χ j | Δ 2 I j ,
F S , j = ω ϑ j π w j 2   Δ 2 I j   i π w i 2   | χ i | P i , j = ω S j τ M F j ,
ϑ j = S j τ M / K j ,
K j = 1 | χ j | i π w i 2   | χ i | P i , j .
J i , m s = k   I k , m ω k ϑ k P i , k .

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