Abstract

Two Monte Carlo methods for atmospheric radiative transfer are presented. One is a backward three-dimensional code suitable for the visible region of the electromagnetic spectrum in which the source is considered to be a parallel flux incident on the upper boundary. The second is a forward plane-parallel code for the microwave region in which radiation is given by thermal emission. The biasing techniques used to reduce the computational time and keep the statistical oscillations relatively small are described. The results are tested by comparison with the results of equivalent codes, when available, and the speed of convergence is analyzed.

© 1997 Optical Society of America

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References

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  1. J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
    [CrossRef]
  2. T. C. Grenfell, “A radiative transfer model for sea ice with vertical structure variations,” J. Geophys. Res. 86, 16991–17001 (1991).
    [CrossRef]
  3. 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]
  4. J. Zhonghai, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431–442 (1994).
    [CrossRef]
  5. 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]
  6. V. B. Kisselev, L. Roberti, G. Perona, A finite element algorithm for radiative transfer in vertically inhomogeneous media: numerical scheme and applications, Appl. Opt. 34, 8460–8471 (1995).
    [CrossRef] [PubMed]
  7. J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (A. Deepak, Hampton, Va., 1985), Chap. 3, 31–34.
  8. L. L. House, L. W. Avery, “The Monte Carlo technique applied to radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 9, 1579–1591 (1969).
    [CrossRef]
  9. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [CrossRef] [PubMed]
  10. D. G. Collins, W. G. Blattner, M. B. Wells, H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of the radiation emerging form spherical-shell atmospheres,” Appl. Opt. 11, 2684–2696 (1972).
    [CrossRef] [PubMed]
  11. L. Roberti, J. Haferman, C. Kummerow, “Microwave radiative transfer through horizontally inhomogeneous precipitating clouds,” J. Geophys. Res., 9916,707–16,718 (1994).
    [CrossRef]
  12. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
    [CrossRef] [PubMed]
  13. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Orlando, 1980), 149–151.
  14. H. R. Gordon, “Bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 19, 4133–4148 (1987).
    [CrossRef]
  15. J. A. Weinman, R. Davis, “Thermal microwave radiances from horizontally finite clouds of hydrometeors,” J. Geophys. Res. 83, 3099–3107 (1978).
    [CrossRef]
  16. W.-K. Tao, J. Simpson, S.-T. Soong, “Statistical properties of a cloud ensemble: a numerical study,” J. Atmos. Sci. 44, 3175–3187 (1987).
    [CrossRef]
  17. C. Kummerow, “On the accuracy of the Eddington approximation for radiative transfer in the microwave frequencies,” J. Geophys. Res. 98, 2757–2765 (1993).
    [CrossRef]
  18. S. Chandrasekhar, Remote Sensing (Dover, New York, 1960), 24–42.
  19. G. M. Heymsfield, R. Fulton, “Passive microwave and infrared structure of mesoscale convective systems,” Meteorol. Atmos. Phys. 54, 123–139, (1994).
    [CrossRef]
  20. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, New York, 1985), 138–142.
  21. L. Tsang, K.-H. Ding, “Polarimetric signatures of a layer of random nonspherical discrete scatterers overlying a homogeneous half-space based on first and second order vector radiative transfer theory,” IEEE Trans. Geosci. Remote Sensing 29, 242–253 (1991).
    [CrossRef]
  22. H. T. Chuah, H. S. Tan, “A Monte Carlo method for backscatter from a half-space random medium,” IEEE Trans. Geosci. Remote Sensing 27, 86–93 (1989).
    [CrossRef]

1995 (1)

1994 (4)

J. Zhonghai, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431–442 (1994).
[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]

L. Roberti, J. Haferman, C. Kummerow, “Microwave radiative transfer through horizontally inhomogeneous precipitating clouds,” J. Geophys. Res., 9916,707–16,718 (1994).
[CrossRef]

G. M. Heymsfield, R. Fulton, “Passive microwave and infrared structure of mesoscale convective systems,” Meteorol. Atmos. Phys. 54, 123–139, (1994).
[CrossRef]

1993 (2)

1991 (3)

T. C. Grenfell, “A radiative transfer model for sea ice with vertical structure variations,” J. Geophys. Res. 86, 16991–17001 (1991).
[CrossRef]

L. Tsang, K.-H. Ding, “Polarimetric signatures of a layer of random nonspherical discrete scatterers overlying a homogeneous half-space based on first and second order vector radiative transfer theory,” IEEE Trans. Geosci. Remote Sensing 29, 242–253 (1991).
[CrossRef]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

1989 (1)

H. T. Chuah, H. S. Tan, “A Monte Carlo method for backscatter from a half-space random medium,” IEEE Trans. Geosci. Remote Sensing 27, 86–93 (1989).
[CrossRef]

1988 (1)

1987 (2)

W.-K. Tao, J. Simpson, S.-T. Soong, “Statistical properties of a cloud ensemble: a numerical study,” J. Atmos. Sci. 44, 3175–3187 (1987).
[CrossRef]

H. R. Gordon, “Bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 19, 4133–4148 (1987).
[CrossRef]

1978 (1)

J. A. Weinman, R. Davis, “Thermal microwave radiances from horizontally finite clouds of hydrometeors,” J. Geophys. Res. 83, 3099–3107 (1978).
[CrossRef]

1975 (1)

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

1972 (1)

1969 (1)

L. L. House, L. W. Avery, “The Monte Carlo technique applied to radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 9, 1579–1591 (1969).
[CrossRef]

Avery, L. W.

L. L. House, L. W. Avery, “The Monte Carlo technique applied to radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 9, 1579–1591 (1969).
[CrossRef]

Blattner, W. G.

Chandrasekhar, S.

S. Chandrasekhar, Remote Sensing (Dover, New York, 1960), 24–42.

Chuah, H. T.

H. T. Chuah, H. S. Tan, “A Monte Carlo method for backscatter from a half-space random medium,” IEEE Trans. Geosci. Remote Sensing 27, 86–93 (1989).
[CrossRef]

Collins, D. G.

Dave, J. V.

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

Davis, R.

J. A. Weinman, R. Davis, “Thermal microwave radiances from horizontally finite clouds of hydrometeors,” J. Geophys. Res. 83, 3099–3107 (1978).
[CrossRef]

Ding, K.-H.

L. Tsang, K.-H. Ding, “Polarimetric signatures of a layer of random nonspherical discrete scatterers overlying a homogeneous half-space based on first and second order vector radiative transfer theory,” IEEE Trans. Geosci. Remote Sensing 29, 242–253 (1991).
[CrossRef]

Fulton, R.

G. M. Heymsfield, R. Fulton, “Passive microwave and infrared structure of mesoscale convective systems,” Meteorol. Atmos. Phys. 54, 123–139, (1994).
[CrossRef]

Gentili, B.

Gordon, H. R.

Grenfell, T. C.

T. C. Grenfell, “A radiative transfer model for sea ice with vertical structure variations,” J. Geophys. Res. 86, 16991–17001 (1991).
[CrossRef]

Haferman, J.

L. Roberti, J. Haferman, C. Kummerow, “Microwave radiative transfer through horizontally inhomogeneous precipitating clouds,” J. Geophys. Res., 9916,707–16,718 (1994).
[CrossRef]

Heymsfield, G. M.

G. M. Heymsfield, R. Fulton, “Passive microwave and infrared structure of mesoscale convective systems,” Meteorol. Atmos. Phys. 54, 123–139, (1994).
[CrossRef]

Horak, H. G.

House, L. L.

L. L. House, L. W. Avery, “The Monte Carlo technique applied to radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 9, 1579–1591 (1969).
[CrossRef]

Jayaweera, K.

Jin, Z.

Kattawar, G. W.

Kisselev, V. B.

V. B. Kisselev, L. Roberti, G. Perona, A finite element algorithm for radiative transfer in vertically inhomogeneous media: numerical scheme and applications, Appl. Opt. 34, 8460–8471 (1995).
[CrossRef] [PubMed]

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]

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, New York, 1985), 138–142.

Kummerow, C.

L. Roberti, J. Haferman, C. Kummerow, “Microwave radiative transfer through horizontally inhomogeneous precipitating clouds,” J. Geophys. Res., 9916,707–16,718 (1994).
[CrossRef]

C. Kummerow, “On the accuracy of the Eddington approximation for radiative transfer in the microwave frequencies,” J. Geophys. Res. 98, 2757–2765 (1993).
[CrossRef]

Lenoble, J.

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (A. Deepak, Hampton, Va., 1985), Chap. 3, 31–34.

Liou, K. N.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Orlando, 1980), 149–151.

Mobley, C. D.

Morel, A.

Perona, G.

V. B. Kisselev, L. Roberti, G. Perona, A finite element algorithm for radiative transfer in vertically inhomogeneous media: numerical scheme and applications, Appl. Opt. 34, 8460–8471 (1995).
[CrossRef] [PubMed]

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]

Reinersman, P.

Roberti, L.

V. B. Kisselev, L. Roberti, G. Perona, A finite element algorithm for radiative transfer in vertically inhomogeneous media: numerical scheme and applications, Appl. Opt. 34, 8460–8471 (1995).
[CrossRef] [PubMed]

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]

L. Roberti, J. Haferman, C. Kummerow, “Microwave radiative transfer through horizontally inhomogeneous precipitating clouds,” J. Geophys. Res., 9916,707–16,718 (1994).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, New York, 1985), 138–142.

Simpson, J.

W.-K. Tao, J. Simpson, S.-T. Soong, “Statistical properties of a cloud ensemble: a numerical study,” J. Atmos. Sci. 44, 3175–3187 (1987).
[CrossRef]

Soong, S.-T.

W.-K. Tao, J. Simpson, S.-T. Soong, “Statistical properties of a cloud ensemble: a numerical study,” J. Atmos. Sci. 44, 3175–3187 (1987).
[CrossRef]

Stamnes, K.

Stavn, R.

Tan, H. S.

H. T. Chuah, H. S. Tan, “A Monte Carlo method for backscatter from a half-space random medium,” IEEE Trans. Geosci. Remote Sensing 27, 86–93 (1989).
[CrossRef]

Tao, W.-K.

W.-K. Tao, J. Simpson, S.-T. Soong, “Statistical properties of a cloud ensemble: a numerical study,” J. Atmos. Sci. 44, 3175–3187 (1987).
[CrossRef]

Tsang, L.

L. Tsang, K.-H. Ding, “Polarimetric signatures of a layer of random nonspherical discrete scatterers overlying a homogeneous half-space based on first and second order vector radiative transfer theory,” IEEE Trans. Geosci. Remote Sensing 29, 242–253 (1991).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, New York, 1985), 138–142.

Tsay, S.

Weinman, J. A.

J. A. Weinman, R. Davis, “Thermal microwave radiances from horizontally finite clouds of hydrometeors,” J. Geophys. Res. 83, 3099–3107 (1978).
[CrossRef]

Wells, M. B.

Wiscombe, W.

Zhonghai, J.

Appl. Opt. (7)

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]

J. Zhonghai, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef]

V. B. Kisselev, L. Roberti, G. Perona, A finite element algorithm for radiative transfer in vertically inhomogeneous media: numerical scheme and applications, Appl. Opt. 34, 8460–8471 (1995).
[CrossRef] [PubMed]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

D. G. Collins, W. G. Blattner, M. B. Wells, H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of the radiation emerging form spherical-shell atmospheres,” Appl. Opt. 11, 2684–2696 (1972).
[CrossRef] [PubMed]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

H. R. Gordon, “Bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 19, 4133–4148 (1987).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (2)

L. Tsang, K.-H. Ding, “Polarimetric signatures of a layer of random nonspherical discrete scatterers overlying a homogeneous half-space based on first and second order vector radiative transfer theory,” IEEE Trans. Geosci. Remote Sensing 29, 242–253 (1991).
[CrossRef]

H. T. Chuah, H. S. Tan, “A Monte Carlo method for backscatter from a half-space random medium,” IEEE Trans. Geosci. Remote Sensing 27, 86–93 (1989).
[CrossRef]

J. Atmos. Sci. (2)

W.-K. Tao, J. Simpson, S.-T. Soong, “Statistical properties of a cloud ensemble: a numerical study,” J. Atmos. Sci. 44, 3175–3187 (1987).
[CrossRef]

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

J. Geophys. Res. (4)

T. C. Grenfell, “A radiative transfer model for sea ice with vertical structure variations,” J. Geophys. Res. 86, 16991–17001 (1991).
[CrossRef]

L. Roberti, J. Haferman, C. Kummerow, “Microwave radiative transfer through horizontally inhomogeneous precipitating clouds,” J. Geophys. Res., 9916,707–16,718 (1994).
[CrossRef]

C. Kummerow, “On the accuracy of the Eddington approximation for radiative transfer in the microwave frequencies,” J. Geophys. Res. 98, 2757–2765 (1993).
[CrossRef]

J. A. Weinman, R. Davis, “Thermal microwave radiances from horizontally finite clouds of hydrometeors,” J. Geophys. Res. 83, 3099–3107 (1978).
[CrossRef]

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

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]

L. L. House, L. W. Avery, “The Monte Carlo technique applied to radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 9, 1579–1591 (1969).
[CrossRef]

Meteorol. Atmos. Phys. (1)

G. M. Heymsfield, R. Fulton, “Passive microwave and infrared structure of mesoscale convective systems,” Meteorol. Atmos. Phys. 54, 123–139, (1994).
[CrossRef]

Other (4)

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, New York, 1985), 138–142.

S. Chandrasekhar, Remote Sensing (Dover, New York, 1960), 24–42.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Orlando, 1980), 149–151.

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (A. Deepak, Hampton, Va., 1985), Chap. 3, 31–34.

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Figures (8)

Fig. 1
Fig. 1

Geometry used for the 3-D Monte Carlo method in the visible.

Fig. 2
Fig. 2

Intensity values as a function of the polar angle cosine at τ = τ1 + τ2 + τ3 = τmax with A, ϕ = 0 and B, ϕ = 180 and τ = 0 with C, ϕ = 0 and D, ϕ 180. Solid curve, FEM N = 64; circles, Monte Carlo NP = 105. Optical parameters are in Table 1; ΛS = 0, ηo = 0.64, ϕo = 0.

Fig. 3
Fig. 3

Intensity values as a function of the polar angle cosine at τ = 0 with A, ϕ = 0 and B, ϕ = 180. Solid curve, FEM N = 64; circles, Monte Carlo NP = 105. Optical parameters are in Table 1; Lambertian surface with ΛS = 1, ηo = 0.64, ϕo = 0.

Fig. 4
Fig. 4

Intensity values as a function of the polar angle cosine A, at τ = τmax and B, at τ = 0 with ϕ = 0. Solid curve, Monte Carlo NP = 105. Optical parameters are in Table 1; specular surface with ΛS = 1, ηo = 0.64, ϕo = 0.

Fig. 5
Fig. 5

Mean deviation (percent) between the Monte Carlo results obtained with the indicated number of photons and the results obtained with the FEM code and N = 64. Solid curve, τ = τmax η = 0.5; dashed curve, τ = τmax η < 0; dotted curve, τ = 0 η = -0.5. The surface is Lambertian with ΛS = 1, ηo = 0.5, ϕo = 0. The Optical parameters for A and B are listed in Tables 1 and 2, respectively.

Fig. 6
Fig. 6

Geometry used for the 3-D Monte Carlo method in the visible when there is a variation of refractive index in the vertical layering.

Fig. 7
Fig. 7

Intensity values as a function of the polar angle cosine for ϕ = 0, A and B, just above and, C and D, below the air–water inteface. Monte Carlo results with NP = 505 solid curve with and dashed curve without refractive index variation. The optical parameters are listed in Table 3; ηo = 0.65, ϕo = 0. The surface is perfectly absorbing.

Fig. 8
Fig. 8

Mean deviation (percent) between the (solid curve) backward and (dashed curve) the forward Monte Carlo results in the microwave, obtained with the indicated number of photons and the results obtained with the backward code and NP = 107. Results in A and B are for a subcloud in the heavily and the lightly rainy portion of the cloud, respectively (f = 85 GHz, θview = 53°).

Tables (4)

Tables Icon

Table 1 Optical parameters for the computations in Figs. 24, 5A, and Table 3

Tables Icon

Table 2 Optical parameters for the computations in Fig. 5B and Table 3

Tables Icon

Table 3 CPU times in seconds on a Digital Alpha 255 workstation for the intensity computation with the FEM and Monte Carlo (MC) code with the indicated number of gridpoints (N), photons (NP), and threshold values (Tr)

Tables Icon

Table 4 Optical parameters for the computations in Fig. 7

Equations (20)

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

τcoll=-lnr
τcoll=-ln1-r1-exp-τo,
Intni=SWni exp-τSunpΘSun.
Intni=SWniexp-τSunpΘSun+exp-τsurf-sunΛspΘSun,
nc=lnTr/lnΛ.
Int=1NPn=1NPi=1InIntni,
Intni=exp-τSunSWnipΘSun+exp-τR-SunSWnipΘR-SunRC.
Intni=exp-τT-SunSWnipΘT-SunTC.
Tr1=STSSTS+TB+TTOT, Tr2=STS+TBSTS+TB+TTOT,
TTOTK=0HTOT hThkexthdh,
INTn0=πSTSW0.
ϕ=2πR, η=cos θ=-R0.5.
INTn0=πTBW0,
ϕ=2πR, η=cos θ=R0.5.
INTn0=4πhkexthTh,
ϕ=2πR, η=cos θ=2R-1.
TBn0=expl-τviewINTn0peΘview/ηview,
TBni=exp-τviewINTn0WnipsΘin-view/ηview,
TB=1NPAn=1NPAi=0InTBnAiHTOT+1NPBn=1NPBi=0InTBnBi+1NPSn=1NPSn=0InTBnSi,
kext=kext-hIh+kext-hIh/Ih+Iv

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