Abstract

The phase function is an important parameter that affects the distribution of scattered radiation. In Rayleigh scattering, a scatterer is approximated by a dipole, and its phase function is analytically related to the scattering angle. For the Henyey–Greenstein (HG) approximation, the phase function preserves only the correct asymmetry factor (i.e., the first moment), which is essentially important for anisotropic scattering. When the HG function is applied to small particles, it produces a significant error in radiance. In addition, the HG function is applied only for an intensity radiative transfer. We develop a combined HG and Rayleigh (HG–Rayleigh) phase function. The HG phase function plays the role of modulator extending the application of the Rayleigh phase function for small asymmetry scattering. The HG–Rayleigh phase function guarantees the correct asymmetry factor and is valid for a polarization radiative transfer. It approaches the Rayleigh phase function for small particles. Thus the HG–Rayleigh phase function has wider applications for both intensity and polarimetric radiative transfers. For microwave radiative transfer modeling in this study, the largest errors in the brightness temperature calculations for weak asymmetry scattering are generally below 0.02 K by using the HG–Rayleigh phase function. The errors can be much larger, in the 1–3 K range, if the Rayleigh and HG functions are applied separately.

© 2006 Optical Society of America

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References

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2005

Q. Liu and F. Weng, "One-dimensional retrieval algorithm of temperature, water vapor, and cloud water profiles from advanced microwave sounding unit (AMSU)," IEEE Trans. Geosci. Remote Sens. 43, 1087-1095 (2005).
[CrossRef]

K. N. Liou, S. C. Ou, Y. Takano, and Q. Liu, "A polarized delta-four-stream approximation for infrared and microwave radiative transfer: Part I," J. Atmos. Sci. 62, 2542-2554 (2005).
[CrossRef]

2003

F. Weng and Q. Liu, "Satellite data assimilation in numerical weather prediction models. Part I: forward radiative transfer and Jocobian modeling in cloudy atmospheres," J. Atmos. Sci. 60, 2633-2646 (2003).
[CrossRef]

M. D. Goldberg, Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, and M. Divakarla, "AIRS near-real-time products and algorithms in support of operational numerical weather prediction," IEEE Trans. Geosci. Remote Sens. 41, 379-389 (2003).
[CrossRef]

1998

F. Rabier, J. N. Thepaut, and P. Courtier, "Extended assimilation and forecast experiments with a four dimensional variational assimilation scheme," Q. J. R. Meteorol. Soc. 124, 1861-1887 (1998).
[CrossRef]

1991

K. F. Evans and G. L. Stephens, "A new polarized atmospheric radiative transfer model," J. Quant. Spectrosc. Radiat. Transfer 46, 413-423 (1991).
[CrossRef]

1969

J. W. Hovenier, "Symmetry relationship of scattering of polarized light in a slab of randomly oriented particles," J. Atmos. Sci. 26, 488-499 (1969).
[CrossRef]

1941

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Courtier, P.

F. Rabier, J. N. Thepaut, and P. Courtier, "Extended assimilation and forecast experiments with a four dimensional variational assimilation scheme," Q. J. R. Meteorol. Soc. 124, 1861-1887 (1998).
[CrossRef]

Divakarla, M.

M. D. Goldberg, Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, and M. Divakarla, "AIRS near-real-time products and algorithms in support of operational numerical weather prediction," IEEE Trans. Geosci. Remote Sens. 41, 379-389 (2003).
[CrossRef]

Evans, K. F.

K. F. Evans and G. L. Stephens, "A new polarized atmospheric radiative transfer model," J. Quant. Spectrosc. Radiat. Transfer 46, 413-423 (1991).
[CrossRef]

Goldberg, M. D.

M. D. Goldberg, Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, and M. Divakarla, "AIRS near-real-time products and algorithms in support of operational numerical weather prediction," IEEE Trans. Geosci. Remote Sens. 41, 379-389 (2003).
[CrossRef]

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Hovenier, J. W.

J. W. Hovenier, "Symmetry relationship of scattering of polarized light in a slab of randomly oriented particles," J. Atmos. Sci. 26, 488-499 (1969).
[CrossRef]

Liang, S.

S. Liang, Quantitative Remote Sensing of Land Surface (Wiley, 2004).

Liou, K. N.

K. N. Liou, S. C. Ou, Y. Takano, and Q. Liu, "A polarized delta-four-stream approximation for infrared and microwave radiative transfer: Part I," J. Atmos. Sci. 62, 2542-2554 (2005).
[CrossRef]

K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, 1992).

P. Yang and K. N. Liou, "Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles," in Light Scattering by Nonspherical particles: Theory, Measurements, and Applications, M. Mishchenko, ed. (Academic, 2000), Chap. 7.
[CrossRef]

Liu, Q.

Q. Liu and F. Weng, "One-dimensional retrieval algorithm of temperature, water vapor, and cloud water profiles from advanced microwave sounding unit (AMSU)," IEEE Trans. Geosci. Remote Sens. 43, 1087-1095 (2005).
[CrossRef]

K. N. Liou, S. C. Ou, Y. Takano, and Q. Liu, "A polarized delta-four-stream approximation for infrared and microwave radiative transfer: Part I," J. Atmos. Sci. 62, 2542-2554 (2005).
[CrossRef]

F. Weng and Q. Liu, "Satellite data assimilation in numerical weather prediction models. Part I: forward radiative transfer and Jocobian modeling in cloudy atmospheres," J. Atmos. Sci. 60, 2633-2646 (2003).
[CrossRef]

Q. Liu, Q. Weng, and F. Weng, "Advanced doubling-adding method for radiative transfer in planetary atmosphere," J. Atmos. Sci. (to be published).

Mag., Philos.

Lord Rayleigh, Philos. Mag. 41, 107-120 (1871).

McMillin, L. M.

M. D. Goldberg, Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, and M. Divakarla, "AIRS near-real-time products and algorithms in support of operational numerical weather prediction," IEEE Trans. Geosci. Remote Sens. 41, 379-389 (2003).
[CrossRef]

Ou, S. C.

K. N. Liou, S. C. Ou, Y. Takano, and Q. Liu, "A polarized delta-four-stream approximation for infrared and microwave radiative transfer: Part I," J. Atmos. Sci. 62, 2542-2554 (2005).
[CrossRef]

Qu, Y.

M. D. Goldberg, Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, and M. Divakarla, "AIRS near-real-time products and algorithms in support of operational numerical weather prediction," IEEE Trans. Geosci. Remote Sens. 41, 379-389 (2003).
[CrossRef]

Rabier, F.

F. Rabier, J. N. Thepaut, and P. Courtier, "Extended assimilation and forecast experiments with a four dimensional variational assimilation scheme," Q. J. R. Meteorol. Soc. 124, 1861-1887 (1998).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, Philos. Mag. 41, 107-120 (1871).

Stephens, G. L.

K. F. Evans and G. L. Stephens, "A new polarized atmospheric radiative transfer model," J. Quant. Spectrosc. Radiat. Transfer 46, 413-423 (1991).
[CrossRef]

Takano, Y.

K. N. Liou, S. C. Ou, Y. Takano, and Q. Liu, "A polarized delta-four-stream approximation for infrared and microwave radiative transfer: Part I," J. Atmos. Sci. 62, 2542-2554 (2005).
[CrossRef]

Thepaut, J. N.

F. Rabier, J. N. Thepaut, and P. Courtier, "Extended assimilation and forecast experiments with a four dimensional variational assimilation scheme," Q. J. R. Meteorol. Soc. 124, 1861-1887 (1998).
[CrossRef]

Weng, F.

Q. Liu and F. Weng, "One-dimensional retrieval algorithm of temperature, water vapor, and cloud water profiles from advanced microwave sounding unit (AMSU)," IEEE Trans. Geosci. Remote Sens. 43, 1087-1095 (2005).
[CrossRef]

F. Weng and Q. Liu, "Satellite data assimilation in numerical weather prediction models. Part I: forward radiative transfer and Jocobian modeling in cloudy atmospheres," J. Atmos. Sci. 60, 2633-2646 (2003).
[CrossRef]

Q. Liu, Q. Weng, and F. Weng, "Advanced doubling-adding method for radiative transfer in planetary atmosphere," J. Atmos. Sci. (to be published).

Weng, Q.

Q. Liu, Q. Weng, and F. Weng, "Advanced doubling-adding method for radiative transfer in planetary atmosphere," J. Atmos. Sci. (to be published).

Wolf, W.

M. D. Goldberg, Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, and M. Divakarla, "AIRS near-real-time products and algorithms in support of operational numerical weather prediction," IEEE Trans. Geosci. Remote Sens. 41, 379-389 (2003).
[CrossRef]

Yang, P.

P. Yang and K. N. Liou, "Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles," in Light Scattering by Nonspherical particles: Theory, Measurements, and Applications, M. Mishchenko, ed. (Academic, 2000), Chap. 7.
[CrossRef]

Zhou, L.

M. D. Goldberg, Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, and M. Divakarla, "AIRS near-real-time products and algorithms in support of operational numerical weather prediction," IEEE Trans. Geosci. Remote Sens. 41, 379-389 (2003).
[CrossRef]

Astrophys. J.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

Q. Liu and F. Weng, "One-dimensional retrieval algorithm of temperature, water vapor, and cloud water profiles from advanced microwave sounding unit (AMSU)," IEEE Trans. Geosci. Remote Sens. 43, 1087-1095 (2005).
[CrossRef]

M. D. Goldberg, Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, and M. Divakarla, "AIRS near-real-time products and algorithms in support of operational numerical weather prediction," IEEE Trans. Geosci. Remote Sens. 41, 379-389 (2003).
[CrossRef]

J. Atmos. Sci.

F. Weng and Q. Liu, "Satellite data assimilation in numerical weather prediction models. Part I: forward radiative transfer and Jocobian modeling in cloudy atmospheres," J. Atmos. Sci. 60, 2633-2646 (2003).
[CrossRef]

J. W. Hovenier, "Symmetry relationship of scattering of polarized light in a slab of randomly oriented particles," J. Atmos. Sci. 26, 488-499 (1969).
[CrossRef]

K. N. Liou, S. C. Ou, Y. Takano, and Q. Liu, "A polarized delta-four-stream approximation for infrared and microwave radiative transfer: Part I," J. Atmos. Sci. 62, 2542-2554 (2005).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

K. F. Evans and G. L. Stephens, "A new polarized atmospheric radiative transfer model," J. Quant. Spectrosc. Radiat. Transfer 46, 413-423 (1991).
[CrossRef]

Q. J. R. Meteorol. Soc.

F. Rabier, J. N. Thepaut, and P. Courtier, "Extended assimilation and forecast experiments with a four dimensional variational assimilation scheme," Q. J. R. Meteorol. Soc. 124, 1861-1887 (1998).
[CrossRef]

Other

K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, 1992).

Lord Rayleigh, Philos. Mag. 41, 107-120 (1871).

S. Liang, Quantitative Remote Sensing of Land Surface (Wiley, 2004).

Q. Liu, Q. Weng, and F. Weng, "Advanced doubling-adding method for radiative transfer in planetary atmosphere," J. Atmos. Sci. (to be published).

P. Yang and K. N. Liou, "Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles," in Light Scattering by Nonspherical particles: Theory, Measurements, and Applications, M. Mishchenko, ed. (Academic, 2000), Chap. 7.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Difference between the original (g) and modified (G) asymmetry factor.

Fig. 2
Fig. 2

(a) Comparisons of the phase function, (b) linear polarization of the phase function, (c) difference in the brightness temperature for NOAA-18 AMSUA∕MHS channels and (d) differences for SSMIS using the original (black curve), the HG–Rayleigh (blue curve), the Rayleigh (yellow curve), and the HG phase functions (red curve).

Tables (2)

Tables Icon

Table 1 Comparisons of the Brightness Temperatures a

Tables Icon

Table 2 Comparisons of the Brightness Temperatures a

Equations (24)

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

2 × 2
P ( Θ , g ) = C   Rayleigh ( Θ ) HG ( Θ ,   G ( g ) )
= C 3 8 ( 1 + cos 2 Θ ) × 1 2 1 G 2 ( g ) [ 1 + G 2 ( g ) 2 G ( g ) cos Θ ] 3 / 2 .
0 π P ( Θ , g ) sin Θ d Θ = 1 ,
0 π cos Θ P ( Θ , g ) sin Θ d Θ = g ,
C = 4 2 + G 2 ,
G = 5 9 g + 1 2 ( 10 9 g + 250 729 g 3 ) + Δ 3 Δ 1 2 ( 10 9 g + 250 729 g 3 ) 3 ,
Δ = [ 1 2 ( 10 9 g 250 729 g 3 ) ] 2 + [ 1 3 ( 4 25 27 g 2 ) ] 3 .
( 2 n + 1 ) cos Θ P n = ( n + 1 ) P n + 1 + n P n 1 ,
P ( Θ , g ) = 4 2 + G 2 3 8 ( 1 + cos 2 Θ ) HG ( Θ ,   G ) = 3 4 + 2 G 2 n = 0 [ n ( n 1 ) 2 n 1 G n 2 + ( n + 2 ) ( n + 1 ) 2 n + 3 G n + 2 + ( n + 1 ) 2 2 n + 3 G n + 5 n 2 1 2 n 1 G n ] P n ( cos Θ ) .
g G
g G
P ( Θ , g ) = 4 2 + G 2 1 2 1 G 2 ( g ) ( 1 + G 2 ( g ) 2 G ( g ) cos Θ ) 3 / 2 3 8
× [ 1 + cos 2 Θ 1 + cos 2 Θ 0 0 1 + cos 2 Θ 1 + cos 2 Θ 0 0 0 0 cos Θ 0 0 0 0 cos Θ ]
[ I , Q , U , V ] t
183   GHz
300   μm
100   μm
90   μm
600   hPa
3 .2   K
183   GHz
276   μm
1 .5   K

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