Abstract

Satellite-sensor-based microwave brightness temperatures for a three-dimensional raining cloud over a reflecting surface are computed by using a radiative transfer model based on the discrete-ordinates solution procedure. The three-dimensional model applied to a plane layer is validated by comparison with results from a one-dimensional model that is available in the literature. Results examining the effects of cloud height, rainfall rate, surface reflectance, rainfall footprint area, and satellite viewing position on one- and three-dimensional brightness temperature calculations are reported. The numerical experiments indicate that, under certain conditions, three-dimensional effects are significant in the analysis of satellite-sensor-based rainfall retrieval algorithms. The results point to the need to consider carefully three-dimensional effects as well as surface reflectance effects when interpreting satellite-measured radiation data.

© 1993 Optical Society of America

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  1. J. Simpson, R. F. Adler, G. North, “A proposed tropical rainfall measuring mission (TRMM) satellite,” Bull. Am. Meteorol. Soc. 69, 278–295 (1988).
    [CrossRef]
  2. P. Arkin, P. Ardanuy, “Estimation of climatological scale precipitation from space: a review,” J. Climate 2, 1229–1238 (1989).
    [CrossRef]
  3. J. C. Alishouse, R. R. Ferraro, J. V. Fiore, “Inference of oceanic rainfall properties from the Nimbus 7 SMMR,” J. Appl. Meteorol. 29, 551–560 (1990).
    [CrossRef]
  4. C. D. Kummerow, J. A. Weinman, “Determining microwave brightness temperatures from precipitating horizontal finite and vertically structured clouds,” J. Geophys. Res. 93, 3720–3728 (1988).
    [CrossRef]
  5. T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
    [CrossRef]
  6. T. T. Wilheit, A. T. C. Chang, L. S. Chiu, “Retrieval of monthly rainfall indices from microwave radiometric measurements using probability distribution functions,” J. Atmos. Ocean. Technol. 8, 118–136 (1991).
    [CrossRef]
  7. A. Mugnai, H. J. Cooper, E. A. Smith, G. J. Tripoli, “Simulation of microwave brightness temperatures of an evolving hailstorm at SSM/I frequencies,” Bull. Am. Meteorol. Soc. 71, 2–13 (1990).
    [CrossRef]
  8. R. F. Adler, H-Y. M. Yeh, N. Prasad, W-K. Tao, J. Simpson, “Microwave simulations of a tropical rainfall system with a three-dimensional cloud model,” J. Appl. Meteorol. 30, 924–953 (1991).
    [CrossRef]
  9. T. T. Wilheit, A. T. C. Chang, M. S. B. Rao, J. S. Theon, “A satellite technique for quantitatively mapping rainfall rates over the oceans,” J. Appl. Meteorol. 16, 551–560 (1977).
    [CrossRef]
  10. J. A. Weinman, C. D. Kummerow, “A radiative transfer model of microwave radiances from horizontally finite clouds containing ice and liquid hydrometeor layers,” in Tropical Rainfall Measurements, J. S. Theon, N. Fugono, eds. (Deepack, Hampton, Va., 1988), pp. 325–336.
  11. B. G. Carlson, K. D. Lathrop, “Transport theory, the method of discrete-ordinates,” in Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, D. Okrent, eds. (Gordon & Breach, New York, 1968), pp. 171–266.
  12. S. A. W. Gerstl, A. Zardecki, “Discrete-ordinates finite-element method for atmospheric radiative transfer and remote sensing,” Appl. Opt. 24, 81–93 (1985).
    [CrossRef] [PubMed]
  13. 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]
  14. W. A. Fiveland, “Three-dimensional radiative heat transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer 2, 309–316 (1988).
    [CrossRef]
  15. W. A. Fiveland, A. S. Jamaluddin, “Three-dimensional spectral heat transfer solutions by the discrete-ordinates method,” in Heat Transfer Phenomena in Radiation, Combustion, and Fires, R. K. Shah, ed. (American Society of Mechanical Engineers, New York, 1989), Vol. 106, pp. 43–48.
  16. T. K. Kim, H. S. Lee, “Radiative transfer in two-dimensional anisotropic scattering media with collimated incidence,” J. Quant. Spectrosc. Radiat. Transfer 42, 225–238 (1989).
    [CrossRef]
  17. A. Sánchez, W. F. Krajewski, T. F. Smith, A General Purpose Radiative Transfer Model for Application to Remote Sensing in Multi-Dimensional Systems, IIHR Rep. 355 (Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, Iowa, 1992).
  18. W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” in Fundamentals of Radiation Heat Transfer, W. A. Fiveland, A. L. Crosbie, A. M. Smith, T. F. Smith, eds. (American Society of Mechanical Engineers, New York, 1991), Vol. 160, pp. 89–96.
  19. 1985 ASHRAE Handbook—Fundamentals (American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, Ga., 1985).
  20. F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing Fundamentals and Radiometry (Artech, Norwood, Mass., 1981), Vol. I.
  21. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  22. J. A. Lane, J. A. Saxton, “Dielectric dispersion in pure polar liquids at very high radio frequencies,” Proc. R. Soc. London Ser. A 213, 400–408 (1952).
    [CrossRef]
  23. A. Sánchez, T. F. Smith, W. F. Krajewski, “Dimensionality issues in modeling with the discrete-ordinates method,” submitted to J. Heat Transfer.
  24. H. W. Barker, J. A. Davies, “Solar radiative fluxes for broken cloud fields above reflecting surfaces,” J. Atmos. Sci. 49, 749–761 (1992).
    [CrossRef]
  25. A. Davis, P. Gabriel, S. Lovejoy, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer 3. Numerical results and meteorological applications,” J. Geophys. Res. 95, 11,729–11,742 (1990).
    [CrossRef]

1992 (1)

H. W. Barker, J. A. Davies, “Solar radiative fluxes for broken cloud fields above reflecting surfaces,” J. Atmos. Sci. 49, 749–761 (1992).
[CrossRef]

1991 (2)

T. T. Wilheit, A. T. C. Chang, L. S. Chiu, “Retrieval of monthly rainfall indices from microwave radiometric measurements using probability distribution functions,” J. Atmos. Ocean. Technol. 8, 118–136 (1991).
[CrossRef]

R. F. Adler, H-Y. M. Yeh, N. Prasad, W-K. Tao, J. Simpson, “Microwave simulations of a tropical rainfall system with a three-dimensional cloud model,” J. Appl. Meteorol. 30, 924–953 (1991).
[CrossRef]

1990 (3)

A. Mugnai, H. J. Cooper, E. A. Smith, G. J. Tripoli, “Simulation of microwave brightness temperatures of an evolving hailstorm at SSM/I frequencies,” Bull. Am. Meteorol. Soc. 71, 2–13 (1990).
[CrossRef]

J. C. Alishouse, R. R. Ferraro, J. V. Fiore, “Inference of oceanic rainfall properties from the Nimbus 7 SMMR,” J. Appl. Meteorol. 29, 551–560 (1990).
[CrossRef]

A. Davis, P. Gabriel, S. Lovejoy, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer 3. Numerical results and meteorological applications,” J. Geophys. Res. 95, 11,729–11,742 (1990).
[CrossRef]

1989 (2)

T. K. Kim, H. S. Lee, “Radiative transfer in two-dimensional anisotropic scattering media with collimated incidence,” J. Quant. Spectrosc. Radiat. Transfer 42, 225–238 (1989).
[CrossRef]

P. Arkin, P. Ardanuy, “Estimation of climatological scale precipitation from space: a review,” J. Climate 2, 1229–1238 (1989).
[CrossRef]

1988 (4)

J. Simpson, R. F. Adler, G. North, “A proposed tropical rainfall measuring mission (TRMM) satellite,” Bull. Am. Meteorol. Soc. 69, 278–295 (1988).
[CrossRef]

C. D. Kummerow, J. A. Weinman, “Determining microwave brightness temperatures from precipitating horizontal finite and vertically structured clouds,” J. Geophys. Res. 93, 3720–3728 (1988).
[CrossRef]

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]

W. A. Fiveland, “Three-dimensional radiative heat transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer 2, 309–316 (1988).
[CrossRef]

1985 (1)

1982 (1)

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

1977 (1)

T. T. Wilheit, A. T. C. Chang, M. S. B. Rao, J. S. Theon, “A satellite technique for quantitatively mapping rainfall rates over the oceans,” J. Appl. Meteorol. 16, 551–560 (1977).
[CrossRef]

1952 (1)

J. A. Lane, J. A. Saxton, “Dielectric dispersion in pure polar liquids at very high radio frequencies,” Proc. R. Soc. London Ser. A 213, 400–408 (1952).
[CrossRef]

Adler, R. F.

R. F. Adler, H-Y. M. Yeh, N. Prasad, W-K. Tao, J. Simpson, “Microwave simulations of a tropical rainfall system with a three-dimensional cloud model,” J. Appl. Meteorol. 30, 924–953 (1991).
[CrossRef]

J. Simpson, R. F. Adler, G. North, “A proposed tropical rainfall measuring mission (TRMM) satellite,” Bull. Am. Meteorol. Soc. 69, 278–295 (1988).
[CrossRef]

Alishouse, J. C.

J. C. Alishouse, R. R. Ferraro, J. V. Fiore, “Inference of oceanic rainfall properties from the Nimbus 7 SMMR,” J. Appl. Meteorol. 29, 551–560 (1990).
[CrossRef]

Ardanuy, P.

P. Arkin, P. Ardanuy, “Estimation of climatological scale precipitation from space: a review,” J. Climate 2, 1229–1238 (1989).
[CrossRef]

Arkin, P.

P. Arkin, P. Ardanuy, “Estimation of climatological scale precipitation from space: a review,” J. Climate 2, 1229–1238 (1989).
[CrossRef]

Austin, G. L.

A. Davis, P. Gabriel, S. Lovejoy, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer 3. Numerical results and meteorological applications,” J. Geophys. Res. 95, 11,729–11,742 (1990).
[CrossRef]

Barker, H. W.

H. W. Barker, J. A. Davies, “Solar radiative fluxes for broken cloud fields above reflecting surfaces,” J. Atmos. Sci. 49, 749–761 (1992).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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. N. Kelber, D. Okrent, eds. (Gordon & Breach, New York, 1968), pp. 171–266.

Chang, A. T. C.

T. T. Wilheit, A. T. C. Chang, L. S. Chiu, “Retrieval of monthly rainfall indices from microwave radiometric measurements using probability distribution functions,” J. Atmos. Ocean. Technol. 8, 118–136 (1991).
[CrossRef]

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

T. T. Wilheit, A. T. C. Chang, M. S. B. Rao, J. S. Theon, “A satellite technique for quantitatively mapping rainfall rates over the oceans,” J. Appl. Meteorol. 16, 551–560 (1977).
[CrossRef]

Chiu, L. S.

T. T. Wilheit, A. T. C. Chang, L. S. Chiu, “Retrieval of monthly rainfall indices from microwave radiometric measurements using probability distribution functions,” J. Atmos. Ocean. Technol. 8, 118–136 (1991).
[CrossRef]

Cooper, H. J.

A. Mugnai, H. J. Cooper, E. A. Smith, G. J. Tripoli, “Simulation of microwave brightness temperatures of an evolving hailstorm at SSM/I frequencies,” Bull. Am. Meteorol. Soc. 71, 2–13 (1990).
[CrossRef]

Davies, J. A.

H. W. Barker, J. A. Davies, “Solar radiative fluxes for broken cloud fields above reflecting surfaces,” J. Atmos. Sci. 49, 749–761 (1992).
[CrossRef]

Davis, A.

A. Davis, P. Gabriel, S. Lovejoy, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer 3. Numerical results and meteorological applications,” J. Geophys. Res. 95, 11,729–11,742 (1990).
[CrossRef]

Ferraro, R. R.

J. C. Alishouse, R. R. Ferraro, J. V. Fiore, “Inference of oceanic rainfall properties from the Nimbus 7 SMMR,” J. Appl. Meteorol. 29, 551–560 (1990).
[CrossRef]

Fiore, J. V.

J. C. Alishouse, R. R. Ferraro, J. V. Fiore, “Inference of oceanic rainfall properties from the Nimbus 7 SMMR,” J. Appl. Meteorol. 29, 551–560 (1990).
[CrossRef]

Fiveland, W. A.

W. A. Fiveland, “Three-dimensional radiative heat transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer 2, 309–316 (1988).
[CrossRef]

W. A. Fiveland, A. S. Jamaluddin, “Three-dimensional spectral heat transfer solutions by the discrete-ordinates method,” in Heat Transfer Phenomena in Radiation, Combustion, and Fires, R. K. Shah, ed. (American Society of Mechanical Engineers, New York, 1989), Vol. 106, pp. 43–48.

W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” in Fundamentals of Radiation Heat Transfer, W. A. Fiveland, A. L. Crosbie, A. M. Smith, T. F. Smith, eds. (American Society of Mechanical Engineers, New York, 1991), Vol. 160, pp. 89–96.

Fung, A. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing Fundamentals and Radiometry (Artech, Norwood, Mass., 1981), Vol. I.

Gabriel, P.

A. Davis, P. Gabriel, S. Lovejoy, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer 3. Numerical results and meteorological applications,” J. Geophys. Res. 95, 11,729–11,742 (1990).
[CrossRef]

Gerstl, S. A. W.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jamaluddin, A. S.

W. A. Fiveland, A. S. Jamaluddin, “Three-dimensional spectral heat transfer solutions by the discrete-ordinates method,” in Heat Transfer Phenomena in Radiation, Combustion, and Fires, R. K. Shah, ed. (American Society of Mechanical Engineers, New York, 1989), Vol. 106, pp. 43–48.

Jayaweera, K.

Kim, T. K.

T. K. Kim, H. S. Lee, “Radiative transfer in two-dimensional anisotropic scattering media with collimated incidence,” J. Quant. Spectrosc. Radiat. Transfer 42, 225–238 (1989).
[CrossRef]

King, J. L.

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

Krajewski, W. F.

A. Sánchez, W. F. Krajewski, T. F. Smith, A General Purpose Radiative Transfer Model for Application to Remote Sensing in Multi-Dimensional Systems, IIHR Rep. 355 (Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, Iowa, 1992).

A. Sánchez, T. F. Smith, W. F. Krajewski, “Dimensionality issues in modeling with the discrete-ordinates method,” submitted to J. Heat Transfer.

Krupp, B. M.

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

Kummerow, C. D.

C. D. Kummerow, J. A. Weinman, “Determining microwave brightness temperatures from precipitating horizontal finite and vertically structured clouds,” J. Geophys. Res. 93, 3720–3728 (1988).
[CrossRef]

J. A. Weinman, C. D. Kummerow, “A radiative transfer model of microwave radiances from horizontally finite clouds containing ice and liquid hydrometeor layers,” in Tropical Rainfall Measurements, J. S. Theon, N. Fugono, eds. (Deepack, Hampton, Va., 1988), pp. 325–336.

Lane, J. A.

J. A. Lane, J. A. Saxton, “Dielectric dispersion in pure polar liquids at very high radio frequencies,” Proc. R. Soc. London Ser. A 213, 400–408 (1952).
[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. N. Kelber, D. Okrent, eds. (Gordon & Breach, New York, 1968), pp. 171–266.

Lee, H. S.

T. K. Kim, H. S. Lee, “Radiative transfer in two-dimensional anisotropic scattering media with collimated incidence,” J. Quant. Spectrosc. Radiat. Transfer 42, 225–238 (1989).
[CrossRef]

Lovejoy, S.

A. Davis, P. Gabriel, S. Lovejoy, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer 3. Numerical results and meteorological applications,” J. Geophys. Res. 95, 11,729–11,742 (1990).
[CrossRef]

Milman, A. S.

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

Moore, R. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing Fundamentals and Radiometry (Artech, Norwood, Mass., 1981), Vol. I.

Mugnai, A.

A. Mugnai, H. J. Cooper, E. A. Smith, G. J. Tripoli, “Simulation of microwave brightness temperatures of an evolving hailstorm at SSM/I frequencies,” Bull. Am. Meteorol. Soc. 71, 2–13 (1990).
[CrossRef]

Nieman, R. A.

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

North, G.

J. Simpson, R. F. Adler, G. North, “A proposed tropical rainfall measuring mission (TRMM) satellite,” Bull. Am. Meteorol. Soc. 69, 278–295 (1988).
[CrossRef]

Prasad, N.

R. F. Adler, H-Y. M. Yeh, N. Prasad, W-K. Tao, J. Simpson, “Microwave simulations of a tropical rainfall system with a three-dimensional cloud model,” J. Appl. Meteorol. 30, 924–953 (1991).
[CrossRef]

Rao, M. S. B.

T. T. Wilheit, A. T. C. Chang, M. S. B. Rao, J. S. Theon, “A satellite technique for quantitatively mapping rainfall rates over the oceans,” J. Appl. Meteorol. 16, 551–560 (1977).
[CrossRef]

Rodgers, E. B.

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

Sánchez, A.

A. Sánchez, T. F. Smith, W. F. Krajewski, “Dimensionality issues in modeling with the discrete-ordinates method,” submitted to J. Heat Transfer.

A. Sánchez, W. F. Krajewski, T. F. Smith, A General Purpose Radiative Transfer Model for Application to Remote Sensing in Multi-Dimensional Systems, IIHR Rep. 355 (Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, Iowa, 1992).

Saxton, J. A.

J. A. Lane, J. A. Saxton, “Dielectric dispersion in pure polar liquids at very high radio frequencies,” Proc. R. Soc. London Ser. A 213, 400–408 (1952).
[CrossRef]

Schertzer, D.

A. Davis, P. Gabriel, S. Lovejoy, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer 3. Numerical results and meteorological applications,” J. Geophys. Res. 95, 11,729–11,742 (1990).
[CrossRef]

Siddalingaiah, H.

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

Simpson, J.

R. F. Adler, H-Y. M. Yeh, N. Prasad, W-K. Tao, J. Simpson, “Microwave simulations of a tropical rainfall system with a three-dimensional cloud model,” J. Appl. Meteorol. 30, 924–953 (1991).
[CrossRef]

J. Simpson, R. F. Adler, G. North, “A proposed tropical rainfall measuring mission (TRMM) satellite,” Bull. Am. Meteorol. Soc. 69, 278–295 (1988).
[CrossRef]

Smith, E. A.

A. Mugnai, H. J. Cooper, E. A. Smith, G. J. Tripoli, “Simulation of microwave brightness temperatures of an evolving hailstorm at SSM/I frequencies,” Bull. Am. Meteorol. Soc. 71, 2–13 (1990).
[CrossRef]

Smith, T. F.

A. Sánchez, T. F. Smith, W. F. Krajewski, “Dimensionality issues in modeling with the discrete-ordinates method,” submitted to J. Heat Transfer.

A. Sánchez, W. F. Krajewski, T. F. Smith, A General Purpose Radiative Transfer Model for Application to Remote Sensing in Multi-Dimensional Systems, IIHR Rep. 355 (Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, Iowa, 1992).

Stamnes, K.

Stratigos, J. S.

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

Tao, W-K.

R. F. Adler, H-Y. M. Yeh, N. Prasad, W-K. Tao, J. Simpson, “Microwave simulations of a tropical rainfall system with a three-dimensional cloud model,” J. Appl. Meteorol. 30, 924–953 (1991).
[CrossRef]

Theon, J. S.

T. T. Wilheit, A. T. C. Chang, M. S. B. Rao, J. S. Theon, “A satellite technique for quantitatively mapping rainfall rates over the oceans,” J. Appl. Meteorol. 16, 551–560 (1977).
[CrossRef]

Tripoli, G. J.

A. Mugnai, H. J. Cooper, E. A. Smith, G. J. Tripoli, “Simulation of microwave brightness temperatures of an evolving hailstorm at SSM/I frequencies,” Bull. Am. Meteorol. Soc. 71, 2–13 (1990).
[CrossRef]

Tsay, S-C.

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing Fundamentals and Radiometry (Artech, Norwood, Mass., 1981), Vol. I.

Weinman, J. A.

C. D. Kummerow, J. A. Weinman, “Determining microwave brightness temperatures from precipitating horizontal finite and vertically structured clouds,” J. Geophys. Res. 93, 3720–3728 (1988).
[CrossRef]

J. A. Weinman, C. D. Kummerow, “A radiative transfer model of microwave radiances from horizontally finite clouds containing ice and liquid hydrometeor layers,” in Tropical Rainfall Measurements, J. S. Theon, N. Fugono, eds. (Deepack, Hampton, Va., 1988), pp. 325–336.

Wilheit, T. T.

T. T. Wilheit, A. T. C. Chang, L. S. Chiu, “Retrieval of monthly rainfall indices from microwave radiometric measurements using probability distribution functions,” J. Atmos. Ocean. Technol. 8, 118–136 (1991).
[CrossRef]

T. T. Wilheit, A. T. C. Chang, J. L. King, E. B. Rodgers, R. A. Nieman, B. M. Krupp, A. S. Milman, J. S. Stratigos, H. Siddalingaiah, “Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm Cora,” J. Appl. Meteorol. 21, 1137–1145 (1982).
[CrossRef]

T. T. Wilheit, A. T. C. Chang, M. S. B. Rao, J. S. Theon, “A satellite technique for quantitatively mapping rainfall rates over the oceans,” J. Appl. Meteorol. 16, 551–560 (1977).
[CrossRef]

Wiscombe, W.

Yeh, H-Y. M.

R. F. Adler, H-Y. M. Yeh, N. Prasad, W-K. Tao, J. Simpson, “Microwave simulations of a tropical rainfall system with a three-dimensional cloud model,” J. Appl. Meteorol. 30, 924–953 (1991).
[CrossRef]

Zardecki, A.

Appl. Opt. (2)

Bull. Am. Meteorol. Soc. (2)

J. Simpson, R. F. Adler, G. North, “A proposed tropical rainfall measuring mission (TRMM) satellite,” Bull. Am. Meteorol. Soc. 69, 278–295 (1988).
[CrossRef]

A. Mugnai, H. J. Cooper, E. A. Smith, G. J. Tripoli, “Simulation of microwave brightness temperatures of an evolving hailstorm at SSM/I frequencies,” Bull. Am. Meteorol. Soc. 71, 2–13 (1990).
[CrossRef]

J. Appl. Meteorol. (4)

R. F. Adler, H-Y. M. Yeh, N. Prasad, W-K. Tao, J. Simpson, “Microwave simulations of a tropical rainfall system with a three-dimensional cloud model,” J. Appl. Meteorol. 30, 924–953 (1991).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry for a raining cloud.

Fig. 2
Fig. 2

Control volumes.

Fig. 3
Fig. 3

Three-dimensional model validation of plane layer (20 layers).

Fig. 4
Fig. 4

Dimensionality effects: three-dimensional versus one-dimensional.

Fig. 5
Fig. 5

Dimensionality effects: brightness temperature difference.

Fig. 6
Fig. 6

Surface reflectance effect (∊0 = 1.0).

Fig. 7
Fig. 7

Effect of rain footprint.

Fig. 8
Fig. 8

Viewing position effects.

Tables (1)

Tables Icon

Table 1 Model Parameters

Equations (22)

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d I d ζ = β I ( ζ ) + S ( ζ ) ,
S ( ζ ) = a I b + s 4 π 0 4 π I ( ζ , ω ¯ ) Φ ( ω , ω ¯ ) d ω .
1 4 π 0 4 π Φ ( ω ¯ , ω ¯ ) d ω = 1 .
I + = 0 I b + + ρ 0 π 0 2 π I ( ω ¯ ) η d ω .
I i P = μ i Δ x I i xr + δ i Δ y I i yr + γ i Δ z I i zr + α S μ i Δ x + δ i Δ y + γ i Δ z + α β ,
I i xe = [ I i P + ( α 1 ) I i xr ] / α .
S = a I b P + s 4 π j = 1 K w j I j P Φ i j ,
I + = 0 I b + + ρ 0 π Ω i < 0 I Ω i w i .
Φ i j = n = 0 M 1 ( 2 n + 1 ) b n P n ( cos θ i j ) ,
cos θ i j = μ i μ j + δ i δ j + γ i γ j .
Φ ¯ ij = Φ i j / 1 4 π i = 1 K w i Φ i j .
T b = I λ 4 C 2 2 C 1 ,
T ( y ) = α t ( y H t ) + T t ,
P υ = ϕ P s ( T ) ,
N ( r ) = N 0 exp ( Λ r ) ,
Λ = 81.56 R 0.21 .
Ñ ( r ) = [ D a ( 0 ) D a ( y ) ] 0.4 N ( r ) .
γ ( y ) = γ g ( y ) + γ c ( y ) + γ r ( y ) ,
a g ( y ) = a H 2 O ( y ) + a O 2 ( y ) ,
γ r ( y ) = r min r max Q r π r 2 Ñ ( r ) d r ,
a c = a 1 D c ,
a 1 = 6 π λ Im ( n 2 1 n 2 + 2 ) ,

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